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UBC Theses and Dissertations

Computer-aided formula optimization Vázquez Benítez, María Cecilia 1990

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COMPUTER-AIDED FORMULA OPTIMIZATION by Maria C e c i l i a Vazquez B e n i t e z B i o c h e m i c a l Engineer Manager i n Food P r o c e s s i n g , The I n s t i t u t e of Technology and Higher S t u d i e s of Monterrey, Mexico, 1984 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (The Department of Food Science)  We accept t h i s t h e s i s as conforming to the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA AUGUST 1990 Maria C e c i l i a Vazquez B e n i t e z  In  presenting  degree freely  at  of  department publication  in  partial  fulfilment  University of  British  Columbia,  for  and  the  available  copying  this  this or of  thesis  reference  thesis by  this  for  his  scholarly  or  thesis  for  her  of  FOOD SCIENCE  The University of British Columbia Vancouver, Canada  Date  DE-6  (2/88)  srPTFMRFR  t  purposes  representatives.  the  requirements  I agree  I further  financial gain  permission.  Department  study.  of  that  agree  may  be  It  is  shall not  that  the  permission  granted  an  advanced  Library shall  by  understood be  for  allowed  the  for  make  extensive  head  that without  it  of  copying my  my or  written  ABSTRACT  The  purpose of  optimization the  this  computer  meat p r o c e s s i n g  such a program that  meet  cost been  found  p r o g r a m had  would  be  as  well  as  product  constraints  part  of  the  the  modified  version  of  was  found  to  be  function  i t suitable second  statistically  the  problems for  part  formula of  this  significant model  fat,  programming,  quality  formulations  of  the  nonlinear  allowable  ingredients  has  equations,  the  e q u a t i o n s as  t o make i t an  objective  e f f e c t i v e formula  frankfurter  equations  the  IBM  the  were  involved  were  formula  nonlinear  constrained,  purposes. the  development  The  three  through  beef  mixture  observation  given  special cubic  an  model  extreme v e r t i c e s d e s i g n . for three  components was  a  meat.  S p e c i f i c q u a l i t y p a r a m e t e r s were e v a l u a t e d by  of  ingredients  p o u l t r y meat and generated  for  experimentation. points  on  FORPLEX  p r e d i c t i o n equations  formulation. deboned  The  of  linearly  optimization study  Box.  optimization  that  BASIC  FORPLEX i s b a s e d  Complex method o f  quality  mechanically  Ingredient-quality  canonical  in  linear  (FORPLEX). The  effective in  function  were: pork  best  study e s t a b l i s h e d  program  3-ingredient  to  s p e c i f i c a t i o n s within  by  formula  for quality control  to handle nonlinear  computer  The  a  the  better  optimization  making  used  to e s t a b l i s h a  method.  first  objective  for  q u a l i t y as  able  was  In c o n t r a s t  search  explained  t o be  optimization The  industry.  Since  to  project  p r o g r a m t o be  predetermined  ranges.  functions  research  at  Scheffe's fitted  to  the  experimental  statistical assessed of  data  validity  by  residuals.  multiple  e r r o r of the estimate  o f 17  regression  a d e q u a t e t o be u s e d  order  between  standard  analysis,  (b)  were u s e d :  correlation  coefficient  developed  were  f o r p r e d i c t i o n purposes. of  i n g r e d i e n t p r o p o r t i o n s and t h e q u a l i t y techniques  was  and a n a l y s i s o f  models  t o have a b e t t e r u n d e r s t a n d i n g  different  The  f o r p r e d i c t i o n purposes  of variance, adjusted  Fourteen  considered  multiple regression analysis.  of the equations  analysis  determination,  In  using  (a)  the  relationship  parameters,  response  three  surface  contour  a n a l y s i s and ( c ) s c a t t e r p l o t  matrices  analys i s . The  third  part  optimization  of  program. S e v e r a l performed.  In  of t h i s  frankfurter  each  quality  formulation  FORPLEX was a b l e constraints target  formulations  meet  should that  the  FORPLEX  optimization t r i a l s  quality.  either  best  combinations  were  measures of t h e f o r m u l a t i o n s ' s e l e c t e d based selected.  quality  on them.  values  using  quality  individually  to find  computational  of  in  when t h e t a r g e t v a l u e s  In both  formulations  a l l the  a  target  cases  the  t h a t met  the  i t was d i f f i c u l t  since  computed  were i n d i v i d u a l l y  a l l the t a r g e t q u a l i t y values.  be s e l e c t e d c a r e f u l l y  on  D i f f e r e n c e s between p r e d i c t e d and  existed  D i f f e r e n c e s e x i s t e d because to  different  were  were  imposed  quality  formulations  trial,  values  or  c o n s i s t e d of the  frankfurter formulation  p a r a m e t e r s were c o n s i d e r e d Target  study  failure  f o r the  Target  optimum selected.  formulations  quality  to obtain  values  formulations  meet t h e t a r g e t q u a l i t y a s c l o s e l y a s p o s s i b l e l a y n o t w i t h  the  performance  target  quality  Five seven  of  optimum f o r m u l a t i o n s  limit  The  optimum  quality.  The l e a s t - c o s t  adequacy  on  technique  in  the  with  by i n c r e a s i n g t h e  The p r e d i c t e d  quality  was c l o s e t o i t s r e s p e c t i v e  formulations  showed,  i n general,  values  t h e models f o r p r e d i c t i n g  the  s e t i n the  of  could  f o r the e f f e c t  n o t be e v a l u a t e d  since  f r o z e n f o r 6 months. of extended  frozen  quality the  The m o d e l s storage  on  of meat did the  of the f o r m u l a t i o n s .  Results  replace  were compared  from t h e t a r g e t q u a l i t y  formulations  account  based  by FORPLEX  w h i c h were f o u n d  formulation  i n g r e d i e n t s had been s t o r e d  quality  of  formulations.  frankfurter  not  found  formulations  considerable departure FORPLEX  the s e l e c t i o n  of the f a t binding c o n s t r a i n t .  o f e a c h FORPLEX target  but with  values.  least-cost  lower  t h e FORPLEX  of  this  study  indicated that  the  Complex method  for  food  linear  (FORPLEX)  formulation.  formula is  the  The FORPLEX  optimization more  may  be  suitable able  programming computer p r o g r a m s c u r r e n t l y b e i n g  the processed  meat  industry.  -iv-  to used  TABLE OF CONTENTS Page ABSTRACT  1i V  TABLE OF CONTENTS L I S T OF TABLES  Ix  L I S T OF FIGURES  x i i  ACKNOWLEDGMENTS  X V i i  INTRODUCTION  1  LITERATURE REVIEW A. F o r m u l a o p t i m i z a t i o n 1. E x p e r i m e n t a l f o r m u l a o p t i m i z a t i o n 2. C o m p u t a t i o n a l f o r m u l a o p t i m i z a t i o n 2.1. L i n e a r programming 2.1.1. A p p l i c a t i o n s i n i n d u s t r i e s r e l a t e d t o t h e food i n d u s t r y 2.1.2. A p p l i c a t i o n s i n f o o d f o r m u l a t i o n 2.1.3. A p p l i c a t i o n s i n t h e meat p r o c e s s i n g industry 2.1.3.1. L i m i t a t i o n s o f l i n e a r programming a s a meat f o r m u l a o p t i m i z a t i o n method B. Q u a l i t y p r e d i c t i o n m o d e l s 1. Q u a n t i t a t i v e s t r u c t u r e - a c t i v i t y r e l a t i o n s h i p s (QSAR) a p p r o a c h 2. I n g r e d i e n t - q u a l i t y r e l a t i o n s h i p s a p p r o a c h 2.1. M i x t u r e d e s i g n s C. N o n l i n e a r c o n s t r a i n e d o p t i m i z a t i o n 1. N o n l i n e a r c o n s t r a i n e d o p t i m i z a t i o n t e c h n i q u e s 2. The Complex method 2.1. The g e n e r a l o p t i m i z a t i o n p r o b l e m 2.2. The Complex method a l g o r i t h m . . 2.3. M o d i f i c a t i o n s t o t h e Complex method 2.4. A p p l i c a t i o n s o f t h e Complex method D. Comminuted meat p r o d u c t s 1. P r o d u c t d e s c r i p t i o n 2. P r o c e s s i n g s t e p s 3. Some f a c t o r s t h a t a f f e c t f i n a l p r o d u c t characteristics 3.1. C o m p o s i t i o n a l f a c t o r s 3.2. P r o c e s s i n g f a c t o r s  5 5 6 8 9  MATERIALS AND METHODS A. E x p e r i m e n t a l m e t h o d o l o g y B. I n g r e d i e n t s C. P r o x i m a t e a n a l y s i s  11 12 15 23 24 25 26 31 33 33 35 35 36 39 45 46 46 49 51 51 56 58 58 60 62  -v-  D. E. F.  G.  H.  1. Determination o£ moisture 2. Determination of crude f a t 3. Determination of p r o t e i n Experimental d e s i g n Frankfurter preparation Q u a l i t y parameters e v a l u a t e d 1. Determination of pH 2. Emulsion s t a b i l i t y a n a l y s i s 3. Per cent weight l o s s a f t e r p r o c e s s i n g and storage 4. Consumer cook t e s t 5. J u i c i n e s s e v a l u a t i o n 6. Texture e v a l u a t i o n 6.1. Texture p r o f i l e a n a l y s i s 6.2. Shear f o r c e S t a t i s t i c a l analysis 1. Regression a n a l y s i s 2. C o r r e l a t i o n a n a l y s i s 3. Response s u r f a c e contour a n a l y s i s O p t i m i z a t i o n methods 1. F e a s i b l e p o i n t computer program (FPOINT) 2. Formula o p t i m i z a t i o n computer program (FORPLEX) 2.1. Program d e s c r i p t i o n 2.1.1. General d e s c r i p t i o n 2.1.2. D e s c r i p t i o n of parameters 2.1.3. Summary of user requirements 2.1.4. New r o u t i n e s 2.1.4.1. Generation of random numbers 2.1.4.2. R e f l e c t i o n through best p o i n t 2.2. L i m i t a t i o n s of the formula o p t i m i z a t i o n algorithm 2.2.1. E q u a l i t y c o n s t r a i n t s 2.2.2. E n d l e s s c o n t r a c t i o n due to i m p l i c i t constraint violation 2.3. O p t i m i z a t i o n of c o n s t r a i n e d mathematical models 2.4. O p t i m i z a t i o n of f r a n k f u r t e r f o r m u l a t i o n s 3. Formula o p t i m i z a t i o n using l i n e a r programming  RESULTS AND DISCUSSION A. O p t i m i z a t i o n of c o n s t r a i n e d mathematical models B. Development of i n g r e d i e n t - q u a l i t y r e l a t i o n s h i p s f o r a 3 - i n g r e d i e n t model f r a n k f u r t e r f o r m u l a t i o n 1. Proximate a n a l y s i s 2. Q u a l i t y parameters evaluated 2.1. Product weight l o s s at d i f f e r e n t stages of the f r a n k f u r t e r p r e p a r a t i o n process 2.2. Emulsion s t a b i l i t y a n a l y s i s 2.3. J u i c i n e s s c h a r a c t e r i s t i c s of the cooked frankfurters 2.4. T e x t u r a l parameters of the cooked frankfurters  -vi-  Page 62 62 62 63 65 71 71 71 74 74 75 76 76 76 77 77 83 83 83 83 86 86 86 91 93 94 94 95 • . 95 95 96 96 97 98 99 99 130 130 133 134 137 141 144  Page 150 . 153 153 189 211 221  C.  2.5 D e t e r m i n a t i o n o f pH 3. Q u a l i t y p r e d i c t i o n m o d e l s 3.1. R e g r e s s i o n a n a l y s i s 3.2. R e s p o n s e s u r f a c e c o n t o u r a n a l y s i s 3.3. C o r r e l a t i o n a n a l y s i s 3.4. S c a t t e r p l o t m a t r i c e s a n a l y s i s C o m p u t a t i o n a l o p t i m i z a t i o n of f r a n k f u r t e r formulations 1. O p t i m i z a t i o n of f r a n k f u r t e r f o r m u l a t i o n s u s i n g t h e new f o r m u l a o p t i m i z a t i o n computer p r o g r a m (FORPLEX) 1.1. S i n g l e o b j e c t i v e o p t i m i z a t i o n 1.2. M u l t i - o b j e c t i v e o p t i m i z a t i o n 1.2.1. O p t i m i z a t i o n o f f r a n k f u r t e r f o r m u l a t i o n s where c o m b i n a t i o n s o f two t o f i v e q u a l i t y p a r a m e t e r s were c o n s i d e r e d m e a s u r e s of t h e formulations* q u a l i t y . Target q u a l i t y v a l u e s were c a l c u l a t e d f r o m t a r g e t points 1.2.2. O p t i m i z a t i o n o f f r a n k f u r t e r f o r m u l a t i o n s where c o m b i n a t i o n s o f f i v e q u a l i t y p a r a m e t e r s were c o n s i d e r e d m e a s u r e s o f t h e formulations' q u a l i t y . Target q u a l i t y v a l u e s were s e t i n d i v i d u a l l y 1.2.3. O p t i m i z a t i o n o f f r a n k f u r t e r f o r m u l a t i o n s when a q u a l i t y p a r a m e t e r was c o n s i d e r e d a constraint 2. O p t i m i z a t i o n o f f r a n k f u r t e r f o r m u l a t i o n s u s i n g l i n e a r programming 3. C o m p a r i s o n o f FORPLEX and l i n e a r programming computed optimum f o r m u l a t i o n s 3.1. C o m p a r i s o n i n t e r m s o f p r e d i c t e d q u a l i t y 3.2. C o m p a r i s o n i n t e r m s of c o s t 4. C o m p a r i s o n o f FORPLEX w i t h l i n e a r programming f o r meat f o r m u l a o p t i m i z a t i o n 5. E x p e r i m e n t a l v e r i f i c a t i o n o f t h e p r e d i c t e d q u a l i t y v a l u e s of two computed optimum f o r m u l a t i o n s  SUMMARY AND  CONCLUSIONS  APPENDIX B. APPENDIX C.  236 245 246  248  255 274 285 291 291 302 304 307 313  REFERENCES APPENDIX A.  236  319 Explanation proportions  o f how t o r e a d t h e i n g r e d i e n t i n t r i a n g u l a r graphs  L o t u s 1-2-3 analysis  template  Listing  of the  for texture  333  profile 337  FPOINT computer  -vii-  program  341  APPENDIX D. L i s t i n g  o f t h e FORPLEX computer  APPENDIX E . N o m e n c l a t u r e and d e f i n i t i o n parameters  -viii-  program  of the  Page 344  quality 350  L I S T OF  TABLES  Table 1. I n g r e d i e n t s and t h e i r l i m i t s used f o r the extreme v e r t i c e s d e s i g n Table 2. Extreme v e r t i c e s experimental d e s i g n Table 3. Weights of i n g r e d i e n t s used i n each formulation(g) Table 4. D e f i n i t i o n a l and working formulas f o r m u l t i p l e r e g r e s s i o n a n a l y s i s of v a r i a n c e f o r mixture models Table 5. O p t i m i z a t i o n r e s u l t s of t e s t problem 1 Table 6. O p t i m i z a t i o n r e s u l t s of t e s t problem 1 u s i n g d i f f e r e n t random number seeds Table 7. O p t i m i z a t i o n r e s u l t s of t e s t problem 2 Table 8. O p t i m i z a t i o n r e s u l t s of t e s t problem 3 Table 9. O p t i m i z a t i o n r e s u l t s of t e s t problem 4 Table 10. O p t i m i z a t i o n r e s u l t s of t e s t problem 5 with broad l i m i t s on the independent v a r i a b l e s Table 11. O p t i m i z a t i o n r e s u l t s of t e s t problem 5 with narrow l i m i t s on the independent v a r i a b l e s Table 12. O p t i m i z a t i o n r e s u l t s of t e s t problem 6 Table 13. O p t i m i z a t i o n r e s u l t s of t e s t problem 7 Table 14. O p t i m i z a t i o n r e s u l t s of t e s t problem 8 Table 15. O p t i m i z a t i o n r e s u l t s of t e s t problem 9 Table 16. O p t i m i z a t i o n r e s u l t s of t e s t problem 10 Table 17. O p t i m i z a t i o n r e s u l t s of t e s t problem 11 Table 18. Proximate composition and pH of raw ingredients Table 19. Experimental data f o r product weight l o s s a t d i f f e r e n t stages of the f r a n k f u r t e r p r e p a r a t i o n process Table 20. Experimental data f o r emulsion s t a b i l i t y analysis Table 21. Experimental data f o r j u i c i n e s s c h a r a c t e r i s t i c s of the cooked f r a n k f u r t e r s Table 22. Experimental data f o r t e x t u r a l parameters of the cooked f r a n k f u r t e r s Table 23. Regression s t a t i s t i c s corresponding t o the "best" model f i t t e d to per cent weight l o s s a f t e r p r o c e s s i n g (Shrink) data Table 24. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o per cent weight l o s s a f t e r 13 days under vacuum packaged storage (Vacuum s h r i n k ) data Table 25. Regression s t a t i s t i c s corresponding t o the "best" model f i t t e d t o per cent weight l o s s a f t e r the consumer cook t e s t (Cook s h r i n k ) data Table 26. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o emulsion s t a b i l i t y (Tmloss) data -ix-  Page 64 66 68 81 101 102 104 106 109 113 114 117 119 122 123 126 128 131 135 138 143 147 158  160  161 162  Page Table Table  Table  Table  Table  Table Table  Table  Table Table Table Table Table Table Table Table Table Table  Table  27. R e g r e s s i o n s t a t i s t i c s c o r r e s p o n d i n g t o t h e " b e s t " model f i t t e d t o e m u l s i o n s t a b i l i t y (Twloss) data 28. R e g r e s s i o n s t a t i s t i c s c o r r e s p o n d i n g t o t h e " b e s t " model f i t t e d t o e m u l s i o n s t a b i l i t y ( E S ) data 29. R e g r e s s i o n s t a t i s t i c s c o r r e s p o n d i n g t o t h e " b e s t " model f i t t e d t o p e r c e n t e x p r e s s i b l e f l u i d ( E x f l u i d ) data 30. R e g r e s s i o n s t a t i s t i c s c o r r e s p o n d i n g t o t h e " b e s t " model f i t t e d t o p e r c e n t e x p r e s s i b l e water (Exwater) d a t a . . . . . 31. R e g r e s s i o n s t a t i s t i c s c o r r e s p o n d i n g t o t h e " b e s t " model f i t t e d t o p e r c e n t e x p r e s s i b l e f a t (Exfat) data 32. R e g r e s s i o n s t a t i s t i c s c o r r e s p o n d i n g t o t h e " b e s t " model f i t t e d t o pH d a t a 33. R e g r e s s i o n s t a t i s t i c s c o r r e s p o n d i n g t o t h e " b e s t " model f i t t e d t o h a r d n e s s a t f i r s t compression (Hardl) data 34. R e g r e s s i o n s t a t i s t i c s c o r r e s p o n d i n g t o t h e " b e s t " model f i t t e d t o h a r d n e s s a t s e c o n d c o m p r e s s i o n (Hard2) d a t a 35. R e g r e s s i o n s t a t i s t i c s c o r r e s p o n d i n g t o t h e " b e s t " model f i t t e d t o maximum s h e a r f o r c e (Shear) data 36. R e g r e s s i o n s t a t i s t i c s c o r r e s p o n d i n g t o t h e " b e s t " model f i t t e d t o f i r m n e s s ( F i r m ) d a t a 37. R e g r e s s i o n s t a t i s t i c s c o r r e s p o n d i n g t o t h e " b e s t " model f i t t e d t o c o h e s i v e n e s s ( C o h e s ) data 38. R e g r e s s i o n s t a t i s t i c s c o r r e s p o n d i n g t o t h e " b e s t " model f i t t e d t o gumminess (Gummy) d a t a 39. R e g r e s s i o n s t a t i s t i c s c o r r e s p o n d i n g t o t h e " b e s t " model f i t t e d t o c h e w i n e s s (Chewy) d a t a 40. Q u a l i t y p r e d i c t i o n m o d e l s 41. C o r r e l a t i o n s between p r o x i m a t e c o m p o s i t i o n o f t h e meat b l o c k s and raw e m u l s i o n s and t h e q u a l i t y parameters evaluated 42. C o r r e l a t i o n s between t h e q u a l i t y p a r a m e t e r s evaluated 43. C o r r e l a t i o n s between t e x t u r a l p a r a m e t e r s 44. Optimum c o m b i n a t i o n s o f i n g r e d i e n t s p r o p o r t i o n s f o r maximum and minimum v a l u e s o f t h e q u a l i t y p a r a m e t e r s and t a r g e t d i f f e r e n c e v a l u e s 45. F r a n k f u r t e r f o r m u l a o p t i m i z a t i o n t r i a l s where p a i r s o f q u a l i t y p a r a m e t e r s were c o n s i d e r e d measures o f t h e f o r m u l a t i o n ' q u a l i t y . T a r g e t q u a l i t y v a l u e s were c a l c u l a t e d from t a r g e t p o i n t 1 (Xx=0.250, X = 0.200, X » = 0 . 5 5 0 ) 2  -x-  164 166 167  169  171 173  175  177  179 181 182 184 186 188 212 217 220 247  250  Table 46. F r a n k f u r t e r formula o p t i m i z a t i o n t r i a l s where three and four q u a l i t y parameters were c o n s i d e r e d measures of the f o r m u l a t i o n s ' q u a l i t y . Target q u a l i t y v a l u e s were c a l c u l a t e d from t a r g e t p o i n t 2 (Xr=0.150, Xa=0.100, X =0.750) Table 47. F r a n k f u r t e r formula o p t i m i z a t i o n t r i a l s where f i v e q u a l i t y parameters were c o n s i d e r e d measures of the f o r m u l a t i o n s ' q u a l i t y . Target q u a l i t y v a l u e s were c a l c u l a t e d from t a r g e t p o i n t 2 (X =0.150 X = 0.100, X s 0.750) Table 48. F r a n k f u r t e r formula o p t i m i z a t i o n t r i a l s where f i v e q u a l i t y parameters were c o n s i d e r e d measures of the f o r m u l a t i o n s ' q u a l i t y . Target q u a l i t y values were s e t i n d i v i d u a l l y Table 49. F r a n k f u r t e r formula o p t i m i z a t i o n t r i a l s where a q u a l i t y parameter was c o n s i d e r e d a c o n s t r a i n t Table 50. O b j e c t i v e f u n c t i o n and c o n s t r a i n t equations used i n the o p t i m i z a t i o n of f r a n k f u r t e r f o r m u l a t i o n s u s i n g l i n e a r programming Table 51. L e a s t - c o s t f r a n k f u r t e r f o r m u l a t i o n s Table 52. Comparison of FORPLEX and l i n e a r programming for the o p t i m i z a t i o n of meat f o r m u l a t i o n s Table 53. Experimental v e r i f i c a t i o n of the p r e d i c t e d q u a l i t y values of Formula.. Table 54. Experimental v e r i f i c a t i o n of the p r e d i c t e d q u a l i t y values of LP1 3  1  /  2  -xi-  Page  252  254  259 276 288 289 305 309 310  L I S T OF FIGURES Page Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure  1. A t w o - d i m e n s i o n a l c a s e o f t h e Complex method s e a r c h f o r t h e optimum 2. F l o w c h a r t o f t h e Complex method a l g o r i t h m 3. E x t r e m e v e r t i c e s e x p e r i m e n t a l d e s i g n 4. F l o w c h a r t o f t h e f r a n k f u r t e r p r e p a r a t i o n s t e p s and q u a l i t y p a r a m e t e r s e v a l u a t e d 5. F l o w c h a r t o f t h e F u n c t i o n s u b r o u t i n e o f FPOINT computer p r o g r a m 6. F l o w c h a r t o f t h e FORPLEX a l g o r i t h m 7. C o n t o u r p l o t o f t e s t p r o b l e m 5. I m p l i c i t c o n s t r a i n t s a r e r e p r e s e n t e d by d o t t e d l i n e s 8. C o n t o u r p l o t o f t e s t p r o b l e m s 6 and 7. I m p l i c i t c o n s t r a i n t s a r e r e p r e s e n t e d by s t r a i g h t lines 9. C o n t o u r p l o t o f t e s t p r o b l e m s 8 and 9. I m p l i c i t c o n s t r a i n t s a r e r e p r e s e n t e d by s t r a i g h t lines 10. C o n t o u r p l o t o f t e s t p r o b l e m s 10 and 11. I m p l i c i t c o n s t r a i n t s a r e r e p r e s e n t e d by s t r a i g h t lines 11. Mean pH v a l u e s o f t h e raw e m u l s i o n s . R e p l and Rep2 a r e r e p l i c a t i o n 1 and 2 r e s p e c t i v e l y 12. P l o t o f r e s i d u a l s f o r t h e S h r i n k model 13. P l o t o f r e s i d u a l s f o r t h e T m l o s s model 14. P l o t o f r e s i d u a l s f o r t h e T w l o s s model 15. P l o t o f r e s i d u a l s f o r t h e E x f l u i d model 16. P l o t o f r e s i d u a l s f o r t h e E x w a t e r model 17. P l o t o f r e s i d u a l s f o r t h e E x f a t model 18. P l o t o f r e s i d u a l s f o r t h e pH model 19. P l o t o f r e s i d u a l s f o r t h e H a r d l model 20. P l o t o f r e s i d u a l s f o r t h e Hard2 model 21. P l o t o f r e s i d u a l s f o r t h e S h e a r model 22. P l o t o f r e s i d u a l s f o r t h e Cohes model 23. P l o t o f r e s i d u a l s f o r t h e Gummy model 24. P l o t o f r e s i d u a l s f o r t h e Chewy model 25. R e s p o n s e s u r f a c e c o n t o u r p l o t f o r t h e S h r i n k model 26. R e s p o n s e s u r f a c e c o n t o u r p l o t f o r t h e T m l o s s model 27. R e s p o n s e s u r f a c e c o n t o u r p l o t f o r t h e T w l o s s model 28. R e s p o n s e s u r f a c e c o n t o u r p l o t f o r t h e ES model 29. R e s p o n s e s u r f a c e c o n t o u r p l o t f o r t h e E x f l u i d model 30. R e s p o n s e s u r f a c e c o n t o u r p l o t f o r t h e E x w a t e r model 31. R e s p o n s e s u r f a c e c o n t o u r p l o t f o r t h e E x f a t model -xii-  40 41 67 72 85 87 I l l 116 121 125 152 159 163 165 168 170 172 174 176 178 180 183 185 187 192 193 19 5 196 198 199 200  F i g u r e 32. Response s u r f a c e contour p l o t for the pH model F i g u r e 33. Response s u r f a c e contour p l o t f o r the Hardl model F i g u r e 34. Response s u r f a c e contour p l o t f o r the Hard2 model F i g u r e 35. Response s u r f a c e contour p l o t f o r the Gummy model F i g u r e 36. Response s u r f a c e contour p l o t f o r the Shear model F i g u r e 37. Response s u r f a c e contour p l o t f o r the Cohes model F i g u r e 38. Response s u r f a c e contour p l o t f o r the Chewy model F i g u r e 39. R e l a t i o n s h i p s between the ingredients p r o p o r t i o n s and the proximate composition of the meat blocks and raw emulsions, added i c e and pH of the raw emulsions. Pork f a t , XI; MDPM, X2; Beef meat, X3. Proximate composition of meat b l o c k : moisture, f a t , p r o t e i n and f a t to-protein ratio (FP); composition of raw emulsion: moisture (moistem) F i g u r e 40. R e l a t i o n s h i p s between the ingredients p r o p o r t i o n s and the quality parameters that describe product weight loss, emulsion s t a b i l i t y and j u i c i n e s s c h a r a c t e r i s t i c s . Pork f a t , XI; MDPM, X2; Beef meat, X3. Nomenclature and d e f i n i t i o n of the q u a l i t y parameters are g i v e n i n Appendix E F i g u r e 41. R e l a t i o n s h i p s between the proximate composition of the meat blocks and raw emulsions and the pH of the raw emulsions, and the q u a l i t y parameters t h a t d e s c r i b e product weight l o s s , emulsion s t a b i l i t y and j u i c i n e s s c h a r a c t e r i s t i c s . Proximate composition of meat block: moisture, f a t , p r o t e i n and f a t - t o protein ratio (FP); composition of raw emulsion: moisture (moistem). Nomenclature and d e f i n i t i o n of the q u a l i t y parameters are given i n Appendix E F i g u r e 42. R e l a t i o n s h i p s between the ingredients p r o p o r t i o n s and the t e x t u r a l parameters. Pork f a t , XI; MDPM, X2; Beef meat, X3. Nomenclature and d e f i n i t i o n of the q u a l i t y parameters a r e given i n Appendix E  -xiii-  Page 202 203 204 205 207 208 209  223  225  226  229  Page Figure  Figure Figure  Figure Figure Figure  Figure  Figure  43. R e l a t i o n s h i p s between the proximate composition o f t h e meat blocks a n d raw e m u l s i o n s a n d t h e pH o f t h e raw e m u l s i o n s , a n d the t e x t u r a l parameters. Proximate composition o f meat b l o c k : m o i s t u r e , f a t , p r o t e i n a n d f a t to-protein ratio ( F P ) ; composition o f raw e m u l s i o n : m o i s t u r e (moistem). Nomenclature and d e f i n i t i o n of the q u a l i t y parameters a r e given in Appendix E 44. R e l a t i o n s h i p s between some q u a l i t y p a r a m e t e r s . Nomenclature and d e f i n i t i o n o f t h e q u a l i t y parameters a r e g i v e n i n Appendix E 45. R e l a t i o n s h i p s between t h e q u a l i t y p a r a m e t e r s that describe product weight l o s s , emulsion stability and juiciness characteristics. Nomenclature and d e f i n i t i o n o f t h e q u a l i t y parameters a r e g i v e n i n Appendix E 46. R e l a t i o n s h i p s between t h e t e x t u r a l p a r a m e t e r s . Nomenclature and d e f i n i t i o n o f t h e q u a l i t y parameters a r e g i v e n i n Appendix E 47. F l o w c h a r t o f t h e F u n c t i o n s u b r o u t i n e o f t h e FORPLEX p r o g r a m 48. R e s p o n s e surface contour l i n e s corresponding to the target quality values of Twloss, Exwater, E x f a t , H a r d l , a n d Cohes s e t i n t r i a l 1. The b l a c k area represents the constrained region which i s given by t h e proximate c o m p o s i t i o n a n d c o s t c o n s t r a i n t s . The optimum formulation (Formula) i s r e p r e s e n t e d by a c l o s e d symbol 49. R e s p o n s e surface contour l i n e s corresponding to the target quality values of Shrink, Exfluid, S h e a r , Cohes a n d Gummy s e t i n t r i a l 2. The b l a c k a r e a represents the constrained region which i s given by t h e proximate c o m p o s i t i o n and c o s t c o n s t r a i n t s . The optimum formulation (Form2) i s represented by a c l o s e d symbol 50. R e s p o n s e surface contour l i n e s corresponding to the target quality values of Tmloss, Exwater, H a r d l , Gummy a n d Chewy s e t i n t r i a l 3. The b l a c k a r e a represents the constrained region which i s given by t h e proximate c o m p o s i t i o n a n d c o s t c o n s t r a i n t s . The optimum formulation (Form3) i s represented by a c l o s e d symbol  -xiv-  230 232  233  235 243  261  263  266  Page Figure  51.  Figure  52.  Figure  53.  Figure  54.  Figure  55.  Figure  56.  Figure  57.  Figure  58.  Figure  59.  Response surface contour l i n e s corresponding to the t a r g e t q u a l i t y v a l u e s of Twloss, ES, Exfat, Hardl and Cohes s e t i n t r i a l 4. The black area r e p r e s e n t s the c o n s t r a i n e d region which i s g i v e n by the proximate c o m p o s i t i o n and c o s t c o n s t r a i n t s . The optimum f o r m u l a t i o n (Form4) i s r e p r e s e n t e d by a c l o s e d s y m b o l Response surface contour l i n e s corresponding to the t a r g e t q u a l i t y v a l u e s of Twloss, ES, Exfat, Hardl and Cohes s e t i n t r i a l 5. The black area r e p r e s e n t s the c o n s t r a i n e d region which i s g i v e n by the proximate c o m p o s i t i o n and c o s t c o n s t r a i n t s . The optimum f o r m u l a t i o n (Form4*) i s r e p r e s e n t e d by a c l o s e d symbol Response surface contour l i n e s corresponding t o t h e t a r g e t q u a l i t y v a l u e s of S h r i n k , S h e a r , Cohes and Gummy s e t i n t r i a l 1. The b l a c k a r e a represents the constrained region which is given by t h e p r o x i m a t e c o m p o s i t i o n , c o s t and quality (Exfluid) c o n s t r a i n t s . The optimum f o r m u l a t i o n i s r e p r e s e n t e d by a c l o s e d symbol Response surface contour l i n e s corresponding to the target quality values of Exwater, Exfat, Hardl and Cohes s e t i n t r i a l 2. The black area r e p r e s e n t s the c o n s t r a i n e d region which i s g i v e n by t h e p r o x i m a t e c o m p o s i t i o n , c o s t and q u a l i t y ( T w l o s s ) constraints. The optimum f o r m u l a t i o n i s r e p r e s e n t e d by a c l o s e d symbol Response surface contour l i n e s corresponding to the t a r g e t q u a l i t y v a l u e s of S h r i n k , E x f a t , and Cohes set in t r i a l 3. The black area represents the constrained region which is g i v e n by the proximate c o m p o s i t i o n , c o s t and quality (Hardl) constraints. The optimum f o r m u l a t i o n i s r e p r e s e n t e d by a c l o s e d symbol Response surface contour p l o t f o r the TBV equation. The black area represents the constrained region which i s given by the proximate composition c o n s t r a i n t s D i f f e r e n c e s between s p e c i f i e d t a r g e t q u a l i t y values and the p r e d i c t e d q u a l i t y values of F o r m u l a and t h e l e a s t - c o s t f o r m u l a t i o n s D i f f e r e n c e s between s p e c i f i e d t a r g e t q u a l i t y values and the predicted q u a l i t y values of Form2 and t h e l e a s t - c o s t f o r m u l a t i o n s D i f f e r e n c e s between s p e c i f i e d t a r g e t q u a l i t y values and the predicted q u a l i t y values of Form3 and t h e l e a s t - c o s t f o r m u l a t i o n s  -xv-  269  272  277  280  282  287 292 293 294  Page F i g u r e 60. D i f f e r e n c e s between s p e c i f i e d t a r g e t q u a l i t y values and the p r e d i c t e d q u a l i t y v a l u e s of Form4 and the l e a s t - c o s t f o r m u l a t i o n s F i g u r e 61. D i f f e r e n c e s between s p e c i f i e d t a r g e t q u a l i t y values and the p r e d i c t e d q u a l i t y values of Form4* and the l e a s t - c o s t f o r m u l a t i o n s F i g u r e 62. Cost ($/kg of meat block) of the optimum formulations found by FORPLEX and the leastcost formulations F i g u r e A l . General mixture problem f o r three i n g r e d i e n t s F i g u r e A2. Experimental f e a s i b l e mixture space f o r three ingredients F i g u r e A3. O p t i m i z a t i o n f e a s i b l e mixture space f o r three ingredients F i g u r e B l . Lotus 1-2-3 template f o r t e x t u r e p r o f i l e analysis  -xvi-  295 296 303 334 335 336 338  ACKNOWLEDGEMENTS  I wish  t o express  my g r a t e f u l  e n c o u r a g e m e n t and g u i d a n c e  thanks  during the course  been a n i n t e r e s t i n g  and r e w a r d i n g  I would a l s o  t o express  my c o m m i t t e e , S.T.  sincere  quality  Many  are  also  importantly,  due  Durance and  Packers,  of the  V a n c o u v e r , B.C.,  project.  to  Eleanore  Wellwood,  who  I want t o t h a n k my h u s b a n d , G u i l l e r m o , f o r  support  and f a i t h  j o y s and sorrows,  belief  in  my a b i l i t y this  I dedicate this  T.  thesis.  my  that  J . Vanderstoep,  cooperation with t h i s  to finish  work. I t h a s  s u g g e s t i o n s and e n c o u r a g e m e n t .  shared  courage  for his  go t o Ray S e b a s t i a n and t h e s t a f f  p r o o f - r e a d my  unfailing  of t h i s  Nakai  experience.  l a b of Intercontinental  thanks  meticulously Most  thanks  control  for a l l t h e i r  helpful  S.  my a p p r e c i a t i o n t o t h e members o f  D r s . E. Li-Chan,  Chen, f o r t h e i r  My  his  like  t o Dr.  i n me. G u i l l e r m o , e s p e c i a l l y , has failures  ans s u c c e s s e s ,  t o overcome a n y p r o b l e m work.  has g i v e n  and h i s me  the  I t i s t o him and t o o u r s o n E m i l i o  thesis.  -xvii-  INTRODUCTION  The  food  industry  cannot  work with f i x e d  blending  because of c o n s t a n t l y v a r y i n g composition of the raw Implementation  of o p t i m i z a t i o n methods i n day-to-day  is  if  essential  Computational  the product  is  to  o p t i m i z a t i o n techniques  survive are  less  than experimental o p t i m i z a t i o n techniques, for  routine  programming used  quality is  f o r food  control  w i l l meet predetermined ingredients.  purposes.  the  market.  Currently,  linear  technique  being  industry  uses  l e a s t - c o s t meat f o r m u l a t i o n s t h a t  product s p e c i f i c a t i o n s with the a v a i l a b l e  The product s p e c i f i c a t i o n s ,  on the f o r m u l a t i o n s ,  formulation  time consuming  The meat p r o c e s s i n g  l i n e a r programming to determine  ingredients.  and thus are s u i t a b l e  the most popular computer-aided formulation.  in  schemes  or c o n s t r a i n t s  are s e t by government r e g u l a t i o n s ,  imposed product  l a b e l c l a i m s , market demands, company p o l i c i e s and standards, and they  must be s a t i s f i e d  i n order to market the product  (Rust  and  Olson, 1987). In sausage f o r m u l a t i o n s , the c o n s t r a i n t s are placed to  restrict:  (a)  i n g r e d i e n t contents (e.g.  (b)  proximate  composition  (e.g.  fat binding capacity).  (e.g.  beef meat c o n t e n t ) ,  f a t content)  and  (c)  quality  Advantages of u s i n g l i n e a r programming i n f o r m u l a t i n g low-cost meat  products have been seen  i n more a c c u r a t e c o n t r o l of  costs,  u n i f o r m i t y i n composition and maintenance of q u a l i t y  (Pearson  Tauber,  been  1984a).  Although  l i n e a r programming has  and  proven  u s e f u l as an o p t i m i z a t i o n technique f o r meat product f o r m u l a t i o n , -1-  i t presents (A) but the  several l i m i t a t i o n s :  L i n e a r programming p l a c e s f a r l e s s emphasis on  provide (B)  Because of t h i s ,  q u a l i t y parameters,  the  such as bind value  functions  (Nakai and  Bind value  constants  have  objective always  of  ingredients However,  the  (Parks i t has  by simple l i n e a r  been found to  1985;  been reported value  that  capacity  p r o t e i n or s a l t - s o l u b l e  constants  consider  emulsify  fat.  only  the  protein  which  quality as  raw  Tauber  meat 1984a).  commonly  emulsifying  content,  do  not  f i n i s h e d products (Parks et  Dempster 1981). the  the  e.g.,  Furthermore  bind  value  a b i l i t y of s a l t - s o l u b l e p r o t e i n s  Water-binding p r o p e r t i e s and  ingredients,  explain quality  parameters  constants,  a c c u r a t e l y p r e d i c t the q u a l i t y of the Comer and  of  Pearson and  into  1985;  functions;  fat binding  performance  incorporated and  Quality  have been used by meat processors  al.  bind  constants.  (a measure of the  functional et  ingredients  Arteaga, 1990).  meat i n g r e d i e n t s )  indicators  meat  as  l i n e a r programming does not  u s u a l l y not d e s c r i b e d  more a c c u r a t e l y  al.  by d e a l i n g with  L i n e a r r e l a t i o n s h i p s have been assumed between  instead nonlinear  of  quality,  reduction  the best q u a l i t y product.  parameters are  (C)  f i n a l product  q u a l i t y parameters as c o n s t r a i n t s r a t h e r than  functions.  and  heavy emphasis on c o s t  are  s t a b l e comminuted meat products,  -2-  important  gelation a b i l i t y  properties  have not been  in  considered.  to of  forming  since  q u a l i t y assurance should  with c o s t r e d u c t i o n , allowable accurate  a  be emphasized at l e a s t e q u a l l y  b e t t e r approach may  be to search  c o s t l i m i t s f o r the best q u a l i t y f o r m u l a t i o n . q u a l i t y p r e d i c t i o n equations are r e q u i r e d  objective functions. through  mixture  describing  to d e f i n e  the q u a l i t y of food  have been  proven  useful  products as a f u n c t i o n  l i n e a r programming cannot  functions order  another  to optimize  The  Complex  method that has to l i n e a r and formula  of  a meat product f o r m u l a t i o n method of Box  direct  been used to optimize  nonlinear  purposes.  equations that d e s c r i b e quality  nonlinear  Nonlinear  it  in  optimization  functions it  and  can  subject  s u i t a b l e for  quality  product s p e c i f i c a t i o n s and equations  objective  replace  search  objective functions  prediction  their  effectively.  c o n s t r a i n t s . T h i s makes  can be used as  necessary,  is a  in  nonlinear.  nonlinear  o p t i m i z a t i o n technique should  optimization  equations  manipulate  the  generated  composition. These r e l a t i o n s h i p s have been found to be Since  However,  Ingredient-quality relationships  experimentation  within  prediction the  cost, be  linear and,  used  if as  constraints.  The  main o b j e c t i v e of t h i s t h e s i s was  optimization  quality  formula  computer program to be used for q u a l i t y c o n t r o l  the meat p r o c e s s i n g best  to e s t a b l i s h a  i n d u s t r y . Such a program would search  formulations  that  s p e c i f i c a t i o n s within allowable  meet  predetermined  c o s t ranges.  -3-  for  in the  product  To  fulfill  this objective,  t h i s study was d i v i d e d  in  three  main p a r t s : (1) E s t a b l i s h m e n t of the formula o p t i m i z a t i o n computer program, (2) Development 3-ingredient  of  ingredient-quality  relationships  model f r a n k f u r t e r f o r m u l a t i o n  for  a  through mixture  experimentation, (3) O p t i m i z a t i o n  of s e v e r a l h y p o t h e t i c a l  frankfurter  formulations  using the formula o p t i m i z a t i o n computer program.  The program  second o b j e c t i v e was t o compare t h i s formula with  frankfurter  linear  programming  formulations.  -4-  for  the  optimization  optimization  of  LITERATURE REVIEW  A. FORMULA OPTIMIZATION In the food i n d u s t r y the b l e n d i n g of d i f f e r e n t raw i n g r e d i e n t s i s u s u a l l y r e q u i r e d to produce a food product. i n g r e d i e n t s and the l e v e l s t o be used are a f f e c t the f i n a l methods  of  food  a  time"  experimental  al.,  approaches.  often  The t r a d i t i o n a l  of i n g r e d i e n t s t o  the  methods  involve  appropriate  t o y i e l d an a c c e p t a b l e  labour  numerous  optimization traditional  (Smith  et  disadvantages.  i n t e n s i v e , time consuming,  expensive  and  have  mathematically  become  methods  available  for  formula  founded  procedures  for  and  replacing  the  are  optimization.  Formula  o p t i m i z a t i o n can be d e s c r i b e d as a s e t of a c t i v i t i e s t h a t the c h o i c e of the best f e a s i b l e product  and Norback,  1985).  formulations  must  To be  established c r i t e r i o n .  obtain  evaluated  the  best  formulation product,  leads (Gordon  different  and compared i n terms  of an  The choice of the c r i t e r i o n to use should  be based on the o p t i m i z a t i o n o b j e c t i v e s , maximum  of  the best f o r m u l a t i o n i s not determined.  Statistically  to  give  proportion  product  These two approaches present s e v e r a l  They are s u b j e c t i v e , and  These  until  i s found  1984).  f a c t o r s that  are the " t r i a l and e r r o r " and "one v a r i a b l e  trials  ingredients  important  characteristics.  f i n d i n g the best combination  the best food product at  product  The c h o i c e of the  consumer a c c e p t a b i l i t y ,  minimum c o s t . -5-  and these would be: (a)  (b) maximum q u a l i t y ,  and (c)  Formula o p t i m i z a t i o n methods can and  be d i v i d e d  computational o p t i m i z a t i o n methods.  those  that  require  is  The  computed  1. Experimental formula  used  of for  (RSM)  through  statistical  the  determine  of  characteristics.  Evans,  optimization  i s Response 1983).  RSM  Surface can be  method that uses q u a n t i t a t i v e data  linear regression analysis  (Giovanni,  models are u s e f u l f o r d e s c r i b i n g l e v e l s , the  techniques Methodology  defined from  1983).  the  independent  models can  of  ingredients  models  can  be used to  be  within  predict the  derivatized  effects  v a r i a b l e s on  appropiate  the  RSM  provides  developing  food  an  evaluated  combination  range of i n g r e d i e n t s s t u d i e d . response  y i e l d the most d e s i r e d response. RSM  c h a r a c t e r i s t i c s of the  multiple  characteristics).  responses f o r any  and/or  surfaces  efficient  have an  (Nakai and  and  can that  be will  i n f l u e n c e on  the  Arteaga, 1990).  systematic  products with s p e c i f i c -6-  The  i s u s e f u l i n cases when only  ingredients  food product  a  of changes i n the  generated to determine optimum i n g r e d i e n t combinations  a r e s t r i c t e d number of food  as  These mathematical  response, the dependent v a r i a b l e , ( i . e . product The  the  established  experimental designs to determine mathematical models by  ingredient  the  optimization  formula o p t i m i z a t i o n and  use  the product  the most popular experimental  (Norback  to  are  l a t t e r methods are those i n which  mathematical models t h a t d e s c r i b e  One  former methods  experimentation i n order  optimum f o r m u l a t i o n . optimization  The  i n t o experimental  procedure  characteristics,  for so i t  found  has  area.  The food  early  1970's  discussed (1967),  application  extensive  the  product development-  food  i n d u s t r y has been a major user  (Myers et a l . ,  the  in  1989).  of RSM s i n c e  Several  p u b l i c a t i o n s have  use of RSM i n food f o r m u l a t i o n s t u d i e s .  f o r example,  optimized  the  Kissell  the f o r m u l a t i o n of a white l a y e r  cake by v a r y i n g the l e v e l s of seven i n g r e d i e n t s .  Min and Thomas  (1980)  between  used  ingredients  RSM t o determine the (fat,  stabilizer  physical characteristics (1982)  described  the  of  relationship  and corn syrup s o l i d s )  a  dairy  a p p l i c a t i o n of  topping.  RSM  developing  in  based on sensory  developed  an i c e cream f o r m u l a t i o n c o n t a i n i n g blends  oil  milk  fat.  f o l l o w e d t o develop Mixture also  is  product  interest  cost-reduced  extensive  have  subset  he  of RSM,  has  area.  This  food  of  food  response  depends  upon  of the  Hence, the response i s  of the mixture  ( C o r n e l l , 1981). the  use  of  food f o r m u l a t i o n s . Huor e t a l . (1980)  p r o p o r t i o n s of watermelon,  j u i c e i n f r u i t punches.  the  been p u b l i s h e d r e g a r d i n g  designs t o optimize the  a  In t h i s case,  ingredients.  a f u n c t i o n of the composition  optimized  procedure  f o r m u l a t i o n s u s i n g RSM.  characteristics)  p r o p o r t i o n s of the d i f f e r e n t  mixture  the  (1984)  of s a f f l o w e r  u s e f u l i n the o p t i m i z a t i o n  characteristics.  studies  food  a l l the i n g r e d i e n t s have a s t r o n g i n f l u e n c e on  ( i . e . product  Several  outlined  the  Henika  Tong et a l .  a p p l i c a t i o n s i n the  particularly  f o r m u l a t i o n s when the  (1983)  response s u r f a c e methodology,  enjoyed  technique  e v a l u a t i o n data.  Flshken  and  whipped  products  and  three  Johnson and Zabik  -7-  pineapple and (1981) found  orange  ranges of  albumen food  p r o t e i n s t h a t optimized  cake  formulation.  foaming and  Rockower et a l .  volume  (1983)  of an optimized  p r o p o r t i o n s of t u r b o t , sodium a l g i n a t e , soy f l o u r and concentrate  i n minced f i s h  2. Computational  soy p r o t e i n  formula o p t i m i z a t i o n  manner, computational  i n the most r a p i d  o p t i m i z a t i o n methods must be used.  mathematical  o b j e c t i v e s and,  the  patties.  When optimum f o r m u l a t i o n s must be determined  case,  angel  models  that  i f necessary,  describe  the  In t h i s  optimization  the c o n s t r a i n t s on the f o r m u l a t i o n  must be known i n advance. Two  computational  optimize and  ingredient blending:  linear  computational  have  been  used to  simplex o p t i m i z a t i o n  programming.  Computational blending  o p t i m i z a t i o n techniques  simplex  optimization  was  used to  r a t i o of three strawberry essences  with  optimize  reconstituted  j u i c e concentrate to simulate the aroma of f r e s h strawberry (Aishima et a l . , optimize  the  1987).  blending  Datta  (1989)  ratios  of  juice  a p p l i e d t h i s technique  wines  to  the  standarize  to  aroma  quality. The based  development  of t h i s technique  optimization  on the aroma c h a r a c t e r i s t i c s of the food product  recent. Applications method rely  for blending  appear on  of  this  technique as  a  is  quality  quite control  to hold promise f o r food products which c u r r e n t l y  blending  procedures  based  on  sensory  example, wines, processed cheese, c o f f e e , and -8-  fruit  testing, juices  for (Nakai  and  Arteaga, 1990). The  second  and  most  popular  computational  technique used f o r formula o p t i m i z a t i o n detail discussion  2.1.  Linear  i s given  i n the  i s l i n e a r programming.  following  programming  technique.  section.  The  (LP)  term  is  a  mathematical  "programming"  comes from  economics, where a "program" r e f e r s to a plan a specified,  LP  Evans,  i t must be d e s c r i b e d  f u n c t i o n and  i n the  involved and  by  The  field  (Wolfe  For a problem to be solved  l i n e a r equations both i n the The  objective  term " l i n e a r e q u a t i o n "  linear  are three  optimization  components t h a t  Evans, 1983):  function  i s the sum  of constant m u l t i p l e s  linear  the d e s i r e d  This  function represents  function  is  q u a n t i t a t i v e l y and (2) D e c i s i o n  the  1969). Before d e s c r i b i n g the general  also called decision variables.  be maximized  has  i n a l l of  the v a r i a b l e s of i n t e r e s t ,  t h a t can  by  occur  (Norback and  objective  of  i n which no products of v a r i a b l e s  programming problem, there  (1)  the  of a c t i o n , that i s ,  an equation of f i r s t degree and  Debling,  must be d e f i n e d  1982).  constraints.  been used to d e s c r i b e  (Skinner  programming  or programmed, s o l u t i o n procedure to be used  K o e l l i n g , 1983;  variables  A  programming  Linear  and  optimization  or minimized used  to  s e l e c t the  o b j e c t i v e of the  (Wolfe and  compare  the  This  problem  Koelling, possible  of  1983).  solutions  optimum.  v a r i a b l e s are s p e c i f i e d parameters which a f f e c t the  performance of the system.  They can -9-  be  adjusted  to achieve  the  objective  (Evans,  1982)  and a r e a l s o known as r e s o u r c e s ,  input  v a r i a b l e s , v a r i a b l e s of i n t e r e s t or independent v a r i a b l e s . (3) C o n s t r a i n t s  are l i n e a r f u n c t i o n s  which place  the system and the d e c i s i o n v a r i a b l e s , the  objective  compared  function.  t o some c o n s t a n t .  r e s t r i c t i n g the  The c o n s t r a i n t  functions  The of  general  of  less  than or equal t o  LP problem may be s t a t e d as f o l l o w s :  m l i n e a r i n e q u a l i t i e s or equations i n n  c o n s t r a i n t s and maximize or  of the d e c i s i o n  always  (Norback and Evans, 1983).  f i n d non-negative values of the d e c i s i o n all  value of  are  The comparison i s i n terms  than or equal to (<), equal to (=), and g r e a t e r (>) e x p r e s s i o n s  r e s t r i c t i o n s on  variables"  "Given  decision  a set  variables,  v a r i a b l e s which  satisfy  minimize some l i n e a r combination  (Harper  and Wanninger J r . , 1970b).  M a t h e m a t i c a l l y , t h i s can be w r i t t e n  Maximize  subject  n (or minimize) = E c±X± i =l to  m E aijXi j=l  (1)  < b-, xi  >  (2) (3)  0  i = l , 2 , . . .,n j=1,2,...,m where Xi  = decision variable  Ci  = c o e f f i c i e n t s of the d e c i s i o n v a r i a b l e s i n the o b j e c t i v e function  a t j = c o e f f i c i e n t of the i t h d e c i s i o n v a r i a b l e i n the j t h - 1 0 -  constraint L i n e a r programming problems with two d e c i s i o n v a r i a b l e s can be s o l v e d by g r a p h i c a l s o l u t i o n . solve  LP problems,  Koelling an  (1983)  algorithm  Although LP computer  L i n e a r a l g e b r a can a l s o be used to  however i t has been d e s c r i b e d by  as an i m p r a c t i c a l method.  programs  can be s o l v e d by  based  are widely used s i n c e the  on  the simplex method  LP  problems.  t h i s method manually, time  s o l v e these problems i s s i g n i f i c a n t l y reduced. programs  and  The simplex method i s  t h a t i s more commonly used t o s o l v e problems  Wolfe  can  required  Standard be  LP to  computer  found  at  many  computer c e n t e r s , and s e v e r a l software packages are a v a i l a b l e for personal  computers  Services  of  software: Center,  (Nakai  the U n i v e r s i t y  LIP,  MINOS,  and Arteaga  of B r i t i s h Columbia  LINDO, XMP,  MPSX,  Restrepo  Computing  has s e v e r a l  LPXMP (UBC  LP  Computing  mentioned  objective functions. expressed  i n Nakai and Arteaga  (1990),  (1987), Bender and Kramer (1983), Bender et a l . (1976).  2.1.1. A p p l i c a t i o n s  and  i n i n d u s t r i e s r e l a t e d to the food i n d u s t r y  above,  LP  can  solve  problems  a l l the r e s t r i c t i o n s are expressed  Since  many  i n t h i s way,  in  which the  using  food i n d u s t r y r e l a t e d problems  LP has been a p p l i e d  t r a n s p o r t a t i o n schedules 1982;  The  1988).  D e t a i l s on LP t h e o r y can be found  As  1990).  linear can  be  i n determining optimum  (Nakai and Arteaga, 1990;  Bender et a l .  Skinner and D e b l i n g , 1969); i n the food s e r v i c e i n d u s t r y to  minimize  the manufacturing  c o s t of menu items while m a i n t a i n i n g -11-  q u a l i t y and  composition c o n s t r a i n t s  (Norback,  1982); i n the  mixing i n d u s t r y to s e l e c t the amount of i n g r e d i e n t s to to  meet n u t r i e n t  Miller, best  1988);  specifications i n the  combination of  (Varvarigos  and  at  minimum  cost  be  feed mixed  (Pesti  and  f i s h farming i n d u s t r y f o r determining  production  Home, 1986);  options  that  maximize  among d i e t i c i a n s and  the  profit  nutritionists  to f i n d the most economical combination of food  items  to ensure proper n u t r i t i o n a l balance  Arteaga, 1990).  2.1.2. A p p l i c a t i o n s  i n food  L i n e a r programming formulation cent  of  has  problems an  (Nakai and  in a  formulation been used e x t e n s i v e l y i n  solving  to determine the optimum q u a n t i t y  ingredient  meal  to  be  used,  subject  to  food  or pernecessary  c o n s t r a i n t s , to e i t h e r maximize or minimize a s p e c i f i e d o b j e c t i v e (Smith  et  imposed  on  al.,  1984).  the  formulation  specifications requirements produce  the  and  (Smith best  maximization and  The  c o n s t r a i n t s are  and  legal, et  of  restrictions  are g e n e r a l l y based compositional  a l . , 1984).  The  on  and  objective  p o s s i b l e product at a minimum  product  functional i s u s u a l l y to  cost.  Profit  maximization of consumer a c c e p t a b i l i t y have a l s o  been used as o b j e c t i v e f u n c t i o n s Linear  the  programming has  (Beausire  et a l . , 1988).  been used f o r f o r m u l a t i n g  various  types  foods:  (A) Cereal-based Linear  foods  programming  high-protein  has  i n f a n t foods.  been u t i l i z e d  to  V a l e n c i a et a l . - 1 2 -  develop (1988)  low-cost, used LP  to  develop three low-cost chickpea based high  nutritive  ingredient  value.  levels  infant  C o n s t r a i n t s were used  and to f u l f i l l  quality  was  (1972)  c e r e a l - b a s e d foods.  controlled  In  amino a c i d to c e r t a i n l i m i t s .  optimized  the  of  restrict  several  the  for lysine  used LP to develop this  by r e s t r i c t i n g the  essential  blending  to  minimum requirements  and s u l f u r amino a c i d s . Cavins et a l . several least-cost  food f o r m u l a t i o n s o£  study  protein  percentage of  Hsu et a l .  p l a n t and  each  (1977  animal  a;b)  protein  sources to o b t a i n low-cost f o r m u l a t i o n s f o r bread, pasta, c o o k i e s and an extruded corn-meal functional  snack. R e s t r i c t i o n s were placed t o meet  standards and n u t r i t i o n a l  (1988) commented on the a p p l i c a t i o n of low-cost,  high  c e r e a l s and  protein  quality.  et a l .  formulate n u t r i t i o n a l l y - i m p r o v e d corn-based  combinations (1990)  snacks.  objectives  formulation;  were  to  constraints  develop were  a  placed  least-cost, on  batch  of  used LP t o Smith et a l .  (1984) a p p l i e d LP to the r e f o r m u l a t i o n of E n g l i s h - s t y l e The  al.  LP f o r the f o r m u l a t i o n of  bakery products u s i n g  legumes. Alraeida-Dominguez  V a l e n c i a et  crumpets.  shelf-stable size,  water  a c t i v i t y and moisture, carbohydrate and p r o t e i n content. (B) Meat products The  formulation  industry Evans  of meat products i s a problem which  has commonly s o l v e d by l i n e a r programming  1983).  restrictions  (Norback  and  Many p u b l i c a t i o n s have r e p o r t e d the use of LP  f o r m u l a t i n g meat products, where the o b j e c t i v e formulation  the meat  i s to minimize  c o s t while meeting a s e t of composition and with  the a v a i l a b l e -13-  ingredients  (Norback  and  in the  quality Evans  1983). in  Some simple examples of sausage  Nakai  and  Arteaga (1990),  Skinner and D e b l i n g (1969). low-cost, f r e s h  f o r m u l a t i o n s can be found  Norback and Evans  Beausire e t a l . (1988)  t u r k e y bratwurst,  a coarse ground  u s i n g LP. An a c c e p t a b i l i t y c o n s t r a i n t  and  formulated a type  sausage,  f u n c t i o n was used to a t t a i n  maximum consumer a c c e p t a b i l i t y of the product. protein,  (1983),  Restrictions  f a t and moisture were a l s o used. A d e t a i l e d  on  description  of the a p p l i c a t i o n of LP i n the f o r m u l a t i o n of a l o w - c h o l e s t e r o l , low-fat  beef  stew was g i v e n by  Bender  et  al.  (1976).  The  o b j e c t i v e i n t h i s case was t o formulate a low-cost beef stew that met  the  nutritional  cholesterol Details  recommendations  of  Other  Ice  the  application  cream  of LP  i n the  i n the f o l l o w i n g  meat p r o c e s s i n g  section.  f o r m u l a t i o n s and beer blends have been s u c c e s f u l l y  and Evans (1983)  stocks and water must be meeting  low  food products  formulated using l i n e a r Norback  f a t and  diets.  i n d u s t r y w i l l be reviewed (C)  f o r low  product  programming  (Norback  d e s c r i b e an example blended to g i v e a  specifications  and Evans, where  1983).  four  low-cost beer  f o r a l c o h o l content,  beer while  specific  g r a v i t y , c o l o r and hop r e s i n c o n t e n t . Other  examples of the a p p l i c a t i o n of l i n e a r  food f o r m u l a t i o n o p t i m i z a t i o n can  be  programming f o r  found i n Bender and Kramer  (1983), Norback and Evans (1983), Bender e t a l . (1982), Bender e t al.  (1976), and Harper and Wanninger  -14-  (1970b).  2.1.3. A p p l i c a t i o n s Due  to  In the meat p r o c e s s i n g Industry  continuous v a r i a t i o n s  ingredients,  i n the composition of  the food i n d u s t r y cannot work with f i x e d  schemes. T h e r e f o r e , the implementation day-to-day  food  as l i n e a r programming optimization  techniques,  quality control In the meat and  and  i f the product  suitable  purposes. p r o c e s s i n g i n d u s t r y there i s  a  large  cuts of meats that can be used as raw  c h o i c e of  ingredients,  characteristics.  i n terms of f a t and moisture content processed of  Moreover,  composition,  to  standards.  formulate  Moreover,  the  meat  especially  (Pearson and Tauber, 1984a)  meat products are s t r i c t l y  their  techniques  regulated  processor  products  must  that  on  meet  replacing  ingredients  (Rust  formulation" d e f i n e the at  the  costly and  the p r o p o r t i o n of expensive meat  use  ingredients 1987).  i n the meat  The  with term  processing  T h i s has  less  costly  "least-cost industry  to  of l i n e a r programming t o formulate meat products  lowest  specifications  meat  Olson,  has been used  product exerting  i n g r e d i e n t s i n the f o r m u l a t i o n s (Rust and Olson, 1987). to  the  implement  the c o s t of meat has i n c r e a s e d ,  pressure on p r o c e s s o r s to reduce  led  such  for routine  composition of the raw i n g r e d i e n t s i s h i g h l y v a r i a b l e ,  basis  i s to  consuming than experimental  thus are  each with t h e i r own composition and  Since  blending  o p t i m i z a t i o n techniques  are l e s s time  raw  of o p t i m i z a t i o n methods i n  formulation i s e s s e n t i a l  s u r v i v e i n the market. Computational  species  the  possible  cost  with the a v a i l a b l e -15-  while  meeting  ingredients.  all  product  Since 1958  the  meat  processing  programming emphasis  Industry  to  being  has  determine placed  large production  on  been  using  least-cost  formulations  emulsion-type products  volume (Pearson and  Tauber,  of using l i n e a r programming i n f o r m u l a t i n g been  seen  in  composition  more accurate  and  computerized  c o n t r o l of  maintenance  of  with  more  to  their  due  1984a).  linear  Advantages  low-cost products have costs,  uniformity  q u a l i t y (Pearson  in  and  Tauber,  formulation  problems  1984a). L i n e a r programming has  been used i n meat  because the components of the problem can required  by l i n e a r programming  the o b j e c t i v e f u n c t i o n and linear  functions  the processor available  i n g r e d i e n t s and  (Pearson and  and  (3)  Tauber,  problem  Evans,  constraints  can  To  1984a).  As  (2)  Both as  problem,  (1)  list  of  composition of each  the  finished  in a l l optimization  must be mathematically d e f i n e d  (A) O b j e c t i v e  1983).  way  be expressed  following information:  their costs,  i n the  formulate the  s p e c i f i c a t i o n s of  o b j e c t i v e f u n c t i o n and  in  terms  product  techniques, of:  (A)  (B) c o n s t r a i n t s .  function  In l e a s t - c o s t f o r m u l a t i o n formulation  (Norback and  ingredients.  must have the  ingredient,  the  of the  the  be s p e c i f i e d  i s minimized.  problems, the t o t a l c o s t of the meat Therefore,  the o b j e c t i v e f u n c t i o n to  minimize i s a l i n e a r  f u n c t i o n t h a t comprises a l l the  (decision variables)  that may  c o s t of each i n g r e d i e n t .  be used i n the  formulation  T h i s f u n c t i o n takes the  assuming there are n d i f f e r e n t  ingredients  -16-  ingredients and  the  f o l l o w i n g form,  n cost = E C 1 X 1  minimize  (4)  i =l where c± = i s the c o s t of the i t h i n g r e d i e n t Xi  = i s the weight or per cent  of the i t h i n g r e d i e n t  to be computed Since  s e v e r a l r e s t r i c t i o n s are placed  least-cost  formula  will  probably  i n the f o r m u l a t i o n ,  not  contain  the  a l l possible  ingredients. (B)  Constraints  The  constraints  regulations, policies produced  are product s p e c i f i c a t i o n s s e t by government  product  label claims,  demands,  company  f o r each  product  and standards t h a t must be s a t i s f i e d i n order  to maintain q u a l i t y (Rust and Olson, 1987).  In sausage f o r m u l a t i o n s (a)  market  ingredient  quality.  content,  Mathematically  the c o n s t r a i n t s are placed  to  (b) proximate composition, the c o n s t r a i n t s are the  control and (c)  equations  or  i n e q u a l i t i e s that express the r e s t r i c t i o n s on the f o r m u l a t i o n , (a) Ingredient Ingredient ingredient That i s , be  constraints constraints  are  used to c o n t r o l the amount of an  or a combination of i n g r e d i e n t s  i n the  formulation.  the use of an i n g r e d i e n t or i n g r e d i e n t s combination can  l i m i t e d e i t h e r to a minimum or maximum, to a f i x e d l e v e l or to  a s p e c i f i e d range (Pearson and Tauber, imposed  by  the  meat  processor  1984a).  i n order  to  These l i m i t s are attain  a  desired  q u a l i t y or by the a v a i l a b i l i t y of the i n g r e d i e n t s . The  most  common  ingredient -17-  constraint  i s the  one  that  states  that the sum  u n i t s of weight  of the I n g r e d i e n t s must equal u n i t y  or some other s p e c i f i e d batch  C o n s t r a i n t s can be developed  quality  restrict  size.  s i n c e f r e e z i n g decreases  the  i n terms of b i n d i n g , f l a v o u r and c o l o u r  (Pearson and Tauber, 1984a).  Constraints sausages  100  by the meat processor to r e s t r i c t  the use of f r o z e n meat i n g r e d i e n t s , q u a l i t y of the raw meats  or  on  have  the  use  of  mechanically  a l s o be developed.  the use of deboned red meat  deboned  American f e d e r a l  meat  in  regulations  to a l e v e l of 20%  (Pearson  and Tauber, 1984a). The use of m e c h a n i c a l l y deboned meat has been found to have a negative e f f e c t on the q u a l i t y of f r a n k f u r t e r s a t l e v e l s higher than 20% Constraints ingredients sausages  are  such  are  (Pearson and Tauber, 1984a).  also as  used  to l i m i t  the  use  of  non-meat  f i l l e r s and b i n d e r s s i n c e t h e i r l e v e l s  s t r i c t l y r e g u l a t e d (Rakosky,  1989;  in  Long et a l . ,  1982). (b) Composition  constraints  C o n s t r a i n t s on  composition are used to s p e c i f y  the  required  f i n a l composition of the product  (Pearson and Tauber, 1984a).  products such as cooked  American and Canadian  regulations content of limited  specify the  to  sausages  the l i m i t s f o r f a t ,  finished  28%,  product.  plus  10%  In the USA,  four  of the f i n i s h e d weight  government  and  fat  minimum p r o t e i n content i s s e t  moisture content should not exceed protein  protein  moisture  content  is  11%  and  at  times the percentage (Pearson  In  and  of  Tauber,  1984a). In Canada, f a t content i s l i m i t e d to 25%, minimum p r o t e i n -18-  content i s s e t at 11% (Meat I n s p e c t i o n  and  moisture content should  Regulations,  not exceed  60%  C.R.C. 1978).  (c) Q u a l i t y c o n s t r a i n t s Using  only  processor  ingredient  can  composition  be c o n f i d e n t  but  not  used  constraints  of producing products  with  to meet c e r t a i n q u a l i t y a t t r i b u t e s ,  the  have  Tauber,  been  the  1984a).  The  color Color  Tauber,  maintain  the  1984a). fat  c o n s t r a i n t s have meat  products  c o n s t r a i n t s have been used  i n a bound s t a t e to ensure  ingredients  where the c o e f f i c i e n t s of these f u n c t i o n s are the and  bind  values  i n d i c a t o r s of the q u a l i t y and meat i n g r e d i e n t s Bind values  bind values proteins and  are  have  Pearson and  s i n c e S a f f l e introduced  sausage  and  expressed  (Comer, 1979).  of  the  developed the  so as raw  Tauber, 1984a).  have been used as a measure of f a t b i n d i n g q u a l i t y  formulations  Carpenter,  1987;  t h i s concept  (Porteus,  are based on the e m u l s i f y i n g  (Parks  been  f u n c t i o n a l performance of  (Parks et a l . 1985;  of meat Ingredients for least-cost  that  functions  meat  These  color  linear  stable  products.  called  are  binding  the d i f f e r e n t i n g r e d i e n t s  Binding  constraints  most w i d e l y  and  used to maintain the c o l o r i n t e n s i t y of the  (Pearson and  uniform and  constraints  (Pearson and  the  (Pearson  s i n c e the c o l o r i s d i l u t e d by blending  to  constraints  have developed q u a l i t y c o n s t r a i n t s .  quality  been  composition  n e c e s s a r i l y uniform q u a l i t y  Tauber, 1984a). In order processors  and  c a p a c i t y of s a l t Comer and  -19-  1964  1979). S a f f l e ' s soluble  Dempster,  i n grams of f a t e m u l s i f i e d per gram  S a f f l e ' s bind value  in  of  system i s based on the  1981) meat theory  of emulsion formation  i n f i n e l y comminuted meat products.  theory holds t h a t d u r i n g chopping, and  water form the continuous  fat  globules  are d i s p e r s e d .  phase  Carpenter, Bind  (1) by  coats the f a t  upon thermal p r o c e s s i n g  (Parks  1987).  values  have been  proteins  of the emulsion i n which the  The s o l u b l e p r o t e i n  g l o b u l e s which are then s t a b i l i z e d and  s a l t - s o l u b l e muscle  This  f o r meat i n g r e d i e n t s f o l l o w i n g  obtained  by the f o l l o w i n g procedure  Saffle's  system  (Lauck, 1975):  Fat b i n d i n g c a p a c i t y of s a l t - s o l u b l e p r o t e i n s i s determined titrating  (Carpenter (2)  liquid  f a t i n t o s a l i n e e x t r a c t of  and S a f f l e ,  meat  proteins  1964).  The f r a c t i o n of the t o t a l p r o t e i n which i s s a l t - s o l u b l e i s  determined (3)  (Carpenter  and S a f f l e ,  Multiplication  soluble  protein  results  in  a  of f a t b i n d i n g per u n i t  by the f r a c t i o n percent product  characterizes  a  1964). weight  of  salt-  of s a l t - s o l u b l e p r o t e i n  c a l l e d bind value constant.  u n i t weight of t o t a l p r o t e i n  T h i s , value  for  fat  binding  capacity. (4)  For  each meat i n g r e d i e n t ,  multiplied  by  the corresponding  the f r a c t i o n of  total  bind value constant  protein  results  in  amount  of  the bind value per u n i t weight of meat i n g r e d i e n t . The  bind  ingredient ingredients  value m u l t i p l i e d by in  the  results  the  corresponding  meat f o r m u l a t i o n and  summed  i n the t o t a l bind value  for  f o r the  a l l meat  formulation  (Lauck, 1975). In a d d i t i o n  to  S a f f l e ' s bind value system -20-  other  bind  value  systems  have  Kramlich et  been al.  developed  (1973)  have  experience  (Kramlich et a l .  on  developed  bind  (1979).  Clifton  1973).  The  e.g.  constant  emulsion  0  system  (1967),  In a d d i t i o n , on  bind value  t o .1.0  own  systems  of  are  while  i s based  on  c a p a c i t y and emulsion  meat  their  (1967) and Kramlich et a l . (1973)  value  p r o t e i n content,  and  ranking systems based  r e l a t i v e bind value s c a l e s ,  (1979)  Anderson  and Porteus  processors  Anderson and C l i f t o n  by  based  Porteous  salt-soluble  s t a b i l i t y of the  meat i n g r e d i e n t s . Although  bind  formulation,  values  S a f f l e are:  (1979)  e.g.  textural  and  (2)  were  specific protein,  water  determined and  s t a b i l i t y of for  meat  f a t composition.  questioned the accuracy of S a f f l e ' s bind v a l u e s .  Dempster  predicting  s t a t e d t h a t the bind values r e p o r t e d by  products  i n g r e d i e n t s with  tests  l i m i t a t i o n s to t h e i r  (1) not accurate i n p r e d i c t i n g emulsion  emulsion-type  Comer and  useful in least-cost  s e v e r a l authors have i n d i c a t e d  use. Parks et a l . (1985)  Porteous  have been proven  (1981)  emulsion functional  mentioned t h a t the f u n c t i o n a l p r o p e r t y  capacity test,  are  performance ( y i e l d ,  performance)  of meat  of  limited  cook  ingredients  use  stability  in and  i n comminuted meat  p r o d u c t s . The q u e s t i o n of whether or not a comminuted meat system can be  viewed  as  an  emulsion  has  increased  r e l i a b i l i t y of the bind value constant systems  doubt about the (Li-Chan et a l . ,  1987) . The quality  use of an u n r e l i a b l e bind value as an i n d i c a t o r of product can  result  in  either -21-  poor  product  quality  due  to  underestimation of the requirement of f u n c t i o n a l i n g r e d i e n t s , needlessly  high  product c o s t due  for expensive i n g r e d i e n t s  meat  p r o p e r t i e s and important  to  gelation a b i l i t y  consider  emulsify of  fat.  been overlooked,  and  there  is  functional  a  and  need  Comer and  for  contribution  especially  I t has  new  of  Dempster  which are products,  constants been widely  important  (1981)  concepts f o r  meat  and  water h o l d i n g c a p a c i t y and  Dempster  binding  role  products.  Parks et a l . (1985)  and  bind  that nonmeat i n g r e d i e n t s a l s o p l a y an  i n the q u a l i t y of meat  emphasized  estimating  nonmeat  the  ingredients,  g e l a t i o n phenomena.  (1981) mentioned t h a t using a bind value  Comer  scale  r e f l e c t s the g e l a t i o n p r o p e r t i e s of i n g r e d i e n t s would perhaps more  effective  ingredients. bind value  estimating  the  Based on t h i s need,  s c a l e based  quality factors the  in  ingredients  f u n c t i o n a l behavior of  they developed a  that r e f l e c t the o v e r a l l f u n c t i o n a l i n the meat products.  a  performance  bind of  value  system  both meat and  that  includes  the  effects  of  values  f i r s t attempt to the  nonmeat i n g r e d i e n t s  -22-  be  on a r b i t r a r y  Although the bind the  that  hypothetical  on t o t a l p r o t e i n content and  were s u b j e c t i v e l y determined, t h e i r work was develop  of  f u n c t i o n a l performance  for these products have not been e s t a b l i s h e d .  that  Water  meat i n g r e d i e n t s ,  Furthermore, the  i n g r e d i e n t s has  recognized  need  o n l y the a b i l i t y  i n forming s t a b l e comminuted meat  have not been c o n s i d e r e d . of nonraeat  constants  proteins  properties  of the  ( G i l l e t t et a l . , 1977).  In a d d i t i o n , bind value salt-soluble  to o v e r e s t i m a t i o n  or  in  functional comminuted  meat  products.  Comer and performance  Allan-Wojtas  (1988)  of i n g r e d i e n t s  s t a t e d that r e l a t i v e  functional  i s dependent upon the composition  the t o t a l system.  This  of bind values  or  any  system that uses s i n g u l a r numerical values  to  other  ranking  estimate  individual  should  questioned.  be  i n d i c a t e s t h a t the use  of  functional  performance  in  meat  products  2.1.3.1. L i m i t a t i o n s of l i n e a r programming as a meat o p t i m i z a t i o n method Linear but  programming  places  f a r l e s s emphasis on  s t r o n g emphasis on c o s t  f i n a l product  quality,  the q u a l i t y parameters o n l y as c o n s t r a i n t s . does not Arteaga constant  always 1990).  provide  the  best  Furthermore,  systems  c o n s i d e r a t i o n has  has  been  been given  with c o s t best  q u a l i t y assurance should  formulation.  and  of  current and  bind  no  contribution  of  two  problems  costs within in  attempting  value  objective ingredient  f i n i s h e d product.  be emphasized at l e a s t be to search  equally for  Q u a l i t y p r e d i c t i o n equations may  used as the o b j e c t i v e f u n c t i o n s to be maximized while the p r o c e s s i n g  with  q u a l i t y product. (Nakai  r e d u c t i o n , a b e t t e r approach may  quality  by d e a l i n g  LP  questioned to the  reduction  Because of t h i s ,  the use  f u n c t i o n a l i t y to the q u a l i t y of the Since  formula  allowable this.  p r e d i c t i o n equations are needed.  -23-  be  restricting  ranges. However, there Firstly,  the  accurate  Q u a l i t y parameters are  are  quality usually  not d e s c r i b e d functions  by  have  simple been  linear  functions;  found to e x p l a i n q u a l i t y  (Nakai and Arteaga, 1990).  For example,  as  are  rheological properties  composition  (Nakai,  nonlinearly  related  Secondly,  linear  to  should  in  it  to  product  (Fishken,  manipulate  t h e r e f o r e another  such  acceptability  levels  cannot  order  accurately  r e l a t e d to  product  ingredient  programming and  formulation  and  more  q u a l i t y parameters  nonlinearly  1987)  objective functions, replace  instead quadratic  is  1983).  nonlinear  o p t i m i z a t i o n method  optimize  a  meat  product  effectively.  B. QUALITY PREDICTION MODELS To  understand  the  behavior  of  the  system  under  study,  mathematical  models are u s u a l l y used. G e n e r a l l y , the o b j e c t i v e i s  to d e s c r i b e  the c a u s e - e f f e c t r e l a t i o n s h i p aiming at c o n t r o l l i n g ,  and p o s s i b l y manipulating the system Experimenters may of  a  system  as  attempt a  independent response.  variables A  to i d e n t i f y and  function  (independent v a r i a b l e s ) .  (Harper and Wanninger  of  the  describe variables  are  responsible  for  description  r e l a t i o n s h i p takes place i s c a l l e d a model  Harper  and Wanninger,  of  response interest  I t i s then assumed that changes  mathematical  d e r i v i n g the s p e c i f i c  the  model  is  1970a).  -24-  and  the  i n the in  how  the this  process f o r  c a l l e d m o d e l l i n g (Deming, I t i s important to bear  t h a t the model i s simply an approximation of 1989 ) .  changes  implying  1969).  the system  in  1989; mind  (Deming,  Mathematical models 1989;  can  be d i v i d e d  Harper and Wanninger, 1970a):  based on some known  mechanism  into  (1)  responsible  two  types  (Denting,  m e c h a n i s t i c models are f o r the r e l a t i o n s h i p  between f a c t o r s and responses. These models are based on chemical and the An  physical  theory, and, (2)  empirical  models are developed when  r e l a t i o n s h i p between the f a c t o r s and responses i s not known. assumption  i s made that  the true  function  response y with n number of f a c t o r s , X i  f,  relating  (i=l,...n)  y = f(Xi,...X„)  can  (5)  be approximated by a known and simple mathematical model. Two approaches of e m p i r i c a l  describing  modeling  The  need  knowledge  of  Generation  relationships chemistry,  (Stuper e t a l . , 1979;  models  relationships  of  has r e c e i v e d  in  pharmacology , b i o l o g y  and  use  and  structure-  i n areas such  as  biochemistry  Brown a t a l . , 1988). S t u d i e s on s t r u c t u r e -  r e l a t i o n s h i p s are based on the formation that  structure  quantitative  attention  linear  free-energy  hydrophobic, e l e c t r o n i c and s t e r i c ) as a c t i v i t y as the dependent v a r i a b l e  of  empirical  r e l a t e d parameters ( i . e . the independent  variables  (Stuper e t a l . 1979).  S t r u c t u r e - a c t i v i t y r e l a t i o n s h i p studies the  helpful  (QSAR) a p p r o a c h  the r e l a t i o n s h i p between molecular  activity.  analytical  activity  been  t o p r e d i c t b i o l o g i c a l a c t i v i t y of compounds r e q u i r e s  biological activity  have  food product c h a r a c t e r i s t i c s .  l. Quantitative structure-activity  and  a  have been performed i n  food s c i e n c e a r e a . The QSAR approach i n t h i s area i s based on -25-  the development of e m p i r i c a l models ( i . e . polynomials) variation  in  predicted  from  solubility,  activity well  (i.e.  defined  protein  where the  functionality)  physicochemical  d i s p e r s i b i l i t y , hydrophobicity)  can  properties (Nakai and  be  (e.g.  Li-Chan,  1988) . Li-Chan  et  functionality water  binding  al.  (1987)  of meat p r o t e i n s , properties  physicochemical properties and  developed  moisture,  pH,  and  e.g.  equations  to  gel strength,  fat binding  capacity  predict cookloss,  from  their  i n c l u d i n g the contents of f a t , p r o t e i n  protein  solubility,  hydrophobicity  and  s u l f h y d r y l group content. These m u l t i v a r i a t e p r e d i c t i o n equations included  squared  terms of the  independent v a r i a b l e s  product terms. N o n l i n e a r i t y of the that  an  optimum  necessary the  functionality  functionality;  is inferior  cross  independent v a r i a b l e s suggests  balance of the p h y s i c o c h e m i c a l  f o r the best  and  (Li-Chan  properties  below or above t h i s et a l . ,  is  value  1987;  Nakai,  characteristics  requires  1987 ) .  2.  I n q r e d i e n t - a u a l i t v r e l a t i o n s h i p s approach The  need  knowledge and/or  to p r e d i c t food  of  the  processing  example,  in  product  r e l a t i o n s h i p between c o n d i t i o n s and  food  formulation  ingredient  composition  product c h a r a c t e r i s t i c s .  s t u d i e s an experimenter might  interested  in studying  the e f f e c t of product composition on  yield  and  c e r t a i n q u a l i t y c h a r a c t e r i s t i c s of  These  relationships  need  to  be q u a n t i f i e d to  -26-  a  food have  For be the  product. a  better  understanding of  the magnitude of the r e l a t i o n s h i p  factors  t h a t a f f e c t s the system  (yield,  product q u a l i t y c h a r a c t e r i s t i c s )  between  ( i n g r e d i e n t s ) and the (Harper and  system of  accurately.  the  1989;  Thompson, 1982;  they  empirical  can d e s c r i b e the  I t i s assumed t h a t the response as a f u n c t i o n  ingredients  (Deming,  that  responses Wanninger,  1969). G e n e r a l l y , these r e l a t i o n s h i p s are not known and models are p o s t u l a t e d with the hope  the  can  be d e s c r i b e d  Myers et a l . Harper  1989;  a  polynomial  F l o r o s and Chinnan,  and Wanninger,  used are the f i r s t - o r d e r  by  model 1988;  1970a). The most commonly  polynomial: n  Y  =  |3o + E  |3iX±  (6)  i =l and  the second-order  polynomial  n n Y = 13= + E (3iX± + E p n X i * i=l i=l In  order to approximate  n-1 n + E E P^XiX-i i = l j=i+l  the r e l a t i o n s h i p between  and responses with a p o l y n o m i a l ,  given by an a p p r o p r i a t e  experimental d e s i g n , needs to be performed. collected,  the parameters  the  method  l e a s t squares  derived  of  from  information  this  about  ingredients  or with any form of model, some  p r e s e l e c t e d number of experimental runs,  are  (7)  Once the o b s e r v a t i o n s  i n the model can be estimated (Cornell,  general  1981).  procedure  the product.  can  by  The  equations  yield  valuable  They can be used f o r  different  purposes  such as p r e d i c t i o n , o p t i m i z a t i o n of the f o r m u l a t i o n , and  response  surface a n a l y s i s .  The experimental  strategy  and a n a l y s i s  -27-  of  response s u r f a c e  methodology r e v o l v e s around surface  experiments  response using  the procedure s t a t e d above.  attempt  to i d e n t i f y  and/or  Response  evaluate  of a system as a f u n c t i o n of the v a r i a b l e s of  different  mathematical models and  experimental designs (Myers et a l . ,  1989;  different Thompson,  interest  classes  the  of one  the most popular response s u r f a c e experimental design i s  central  generate  composite  r o t a t a b l e d e s i g n (CCRD).  ingredient-quality studies  al.,  M i t t a l and Usborne,  1988;  (Vazquez-Arteaga and Nakai,  d e s i g n s . For example, effects  of  four  In  order  r e l a t i o n s h i p s s e v e r a l meat  formulation  the  of  1982).  In the case when the f a c t o r s are completely independent another  the  1986)  Vazquez-Arteaga ingredients  product Bawa  et  have used t h i s c l a s s  of  and Nakai  (beef,  1989;  to  (1989)  studied  mechanically  deboned  p o u l t r y meat, soy p r o t e i n i s o l a t e and wheat f l o u r ) on cook y i e l d , emulsion  s t a b i l i t y and t e x t u r a l  c h a r a c t e r i s t i c s of  frankfurter  f o r m u l a t i o n s . G r a p h i c a l r e p r e s e n t a t i o n of the equations developed helped to g a i n  an understanding  l e v e l s of the i n g r e d i e n t s were  of  the  changed.  product q u a l i t y as the The equations were a l s o  used to optimize the q u a l i t y of the f o r m u l a t i o n . In  the  case  interdependent,  of mixture problems  where  the  variables  are  the response of the system depends o n l y on  the  p r o p o r t i o n s of the components of the mixture and not on the t o t a l amount  of  the  ingredients  (Agreda and Agreda,  1981).  These type of problems  call  1989;  Cornell,  f o r c a n o n i c a l polynomials and  f o r a s p e c i f i c c l a s s of response s u r f a c e designs known as mixture designs  (Agreda  and Agreda,1989; -28-  Cornell,  1981). In a mixture  problem,  the  amounts of  the mixture,  mixture, must sum v a r i a b l e s to be 1981). more  Xi,  factors,  and,  related  to  proportionate the  to u n i t y . I t i s t h i s c o n s t r a i n t that causes  the  interdependent  one  represent  i f expressed as f r a c t i o n s of  Food formulations than  (l=l,...k)  can  ingredient. the  (Agreda and  Agreda, 1989;  be considered Therefore  ingredient  Cornell,  mixtures or blends of  product  proportions  quality  can  using  be  canonical  polynomials. Johnson  and  cake volume and egg  Zabik  (1981)  reported  p r e d i c t i o n equations  for  cake tenderness as a f u n c t i o n of a mixture of s i x  albumen p r o t e i n s . Rockower et a l .  (1983) reported  prediction  equations f o r s e v e r a l t e x t u r a l a t t r i b u t e s of minced f i s h  patties  as a f u n c t i o n of f i v e  protein  concentrate, reported  soy  ingredients  f l o u r and  (two  f i s h species,  soy  sodium a l g i n a t e ) . Huor et a l .  (1980)  p r e d i c t i o n equations f o r a c c e p t a b i l i t y of a f r u i t  formulation  c o n t a i n i n g watermelon, pineapple  and  punch  orange j u i c e s .  Generation of i n g r e d i e n t - q u a l i t y r e l a t i o n s h i p s through mixture experimentation  could  be the most a p p r o p r i a t e  study of f u n c t i o n a l performance of i n comminuted meat products. Dempster the  (1981) suggested,  functional  Wojtas  nonmeat  new  emphasized  concepts are needed  to  Comer  to  and  estimate  of i n g r e d i e n t s .  Comer and  that  performance  functional  the  ingredients  As Parks et a l . (1985) and  contributions  (1988)  meat and  approach  Allanof  ingredients  i s dependent on the composition of the meat system as  a  and  whole,  suggested that  i n g r e d i e n t s should  be evaluated  the  functional  i n the a c t u a l meat  -29-  effect  of  products.  the  Researchers  at  used t h i s approach comminuted  the to  sausages  Food Research predict  Institute  in B r i s t o l  q u a l i t y c h a r a c t e r i s t i c s of f i n e l y  as a f u n c t i o n of three  ingredients:  meat,  f a t and added water.  color,  shear and cooking l o s s helped to understand  the  ingredient  (Anonymous, As  levels  on  each  quality  the e f f e c t of  parameter  before,  equations  derived  experiments  can be used to optimize  study.  Agreda  Agreda  computerized  evaluated  and  optimization  (1989)  the  commented  algorithms  through  with  (1989)  reviewed a  set  of  on  the  models  under use  of  developed Myers  et  the a p p l i c a t i o n s of o p t i m i z a t i o n methods to response  s i m u l t a n e o u s l y . Khuri and Conlon for  response  system  through mixture designs to optimize mixture responses.  method  for  1985).  mentioned  optimize  lean  The p r e d i c t i o n equations found  surface  al.  have  functions  (1981) developed an  the simultaneous o p t i m i z a t i o n  of  f u n c t i o n s represented by polynomial models. t h i s o p t i m i z a t i o n method could be a p p l i e d mixture models.  -30-  derived  through  RSM  optimization  several  response  They suggested  that  f o r the o p t i m i z a t i o n of  2.1  Mixture In  of  designs  mixture  the t o t a l  systems,  the components are s t a t e d as p r o p o r t i o n s  ( i . e . unity),  and  t h e r e f o r e the p r o p o r t i o n of each  component X± i n the mixture must l i e between 0 and  1.0  (Cornell,  1981) . k  Xi  = 1.0  (8)  0 < Xi  < 1.0  (9)  E  i =l  As mentioned b e f o r e , (e.g.  mixture  mixture  a  s p e c i a l c l a s s of experimental designs  designs)  systems  so  i s needed to c o l l e c t  o b s e r v a t i o n s of  that a maximum amount of i n f o r m a t i o n can  be  obtained from a minimum number of experimental runs. A v a r i e t y of mixture  designs  have  been  developed  for  specific  purposes.  S i m p l e x - l a t t i c e and s i m p l e x - c e n t r o i d designs can be used when the proportions from  zero  possible  of a l l the components i n the mixture can take values to  u n i t y and a l l blends among  (Cornell,  1981).  where some or a l l of either  a  1990;  frequently situations  1981).  where  is  i n meat  proportions  l i m i t a± and an upper l i m i t  0 < at < X i constraint  ingredient  regulatory considerations.  -31-  by  are  b±  < bt < 1  o f t e n caused  by  Arteaga,  Such s i t u a t i o n s are encountered  formulations,  are  exist  the component p r o p o r t i o n s are r e s t r i c t e d  c o n s t r a i n e d between a lower  This  ingredients  lower bound and/or an upper bound (Nakai and  Cornell,  product  However,  the  (10) economic,  technical  or  McLean design  and  which  problems.  Anderson (1966) was  Their  found to be u s e f u l procedure permits  i n t e r e s t by using the extreme of  the  developed the  hyperpolyhedron  drawback of t h i s design  for  extreme  vertices  constrained  mixture  e x p l o r a t i o n of the r e g i o n  vertices,  edge and  of  face c e n t r o i d s  d e f i n e d by the c o n s t r a i n t s .  A  major  i s that the number of experimental  points  i s q u i t e l a r g e when the number of components i s more than 5 (Snee and  Marquardt,  symmetric-simplex (1968) the  design  Saxena  and  approach  Nigam  and  p o i n t s u n i f o r m l y cover  Nigam,  1977).  However,  about the e f f i c i e n c y of using a  (1977)  reported by  f o r c o n s t r a i n e d mixture problems.  experimental  (Saxena  1974).  used  the  and  Das  Murty  By using these the  constrained  C o r n e l l (1981)  had  symmetric-simplex design  extreme v e r t i c e s design with a d d i t i o n a l boundary p o i n t s .  -32-  designs region doubts over an  C.  NONLINEAR CONSTRAINED OPTIMIZATION  1• Nonlinear  constrained  Constrained  optimization  optimization  techniques  c o n s i s t s of maximizing or minimizing  a known o b j e c t i v e f u n c t i o n while s a t i s f y i n g a s e t of c o n s t r a i n t s . When  the  functions  objective of the  f u n c t i o n and/or c o n s t r a i n t s  independent v a r i a b l e s then n o n l i n e a r  optimization  is required.  f o r handling  these problems. G i l l  on numerical methods optimization. handling  programming  into  linear  (e.g.  (e.g.  quadratic  are those that transform an unconstrained applying points and  a  (Saguy,  barrier  minimization  a  and  main  methods.  constrained  groups:  mathematical  optimization  function  (SUMT)  the  problem i n t o  at  proposed by Umeda (1969).  He  1983).  the  A  methods  analysis,  generalized  methods  include p a t t e r n search  Newton-Raphson method and -33-  different methods  methods, mathematical programming and  Search  penalty  unconstrained  optimization  classified  by  nonfeaslble  sequential  (Saguy,  constrained  methods.  nonlinear  Examples of these methods are the  technique  for  o b j e c t i v e f u n c t i o n s are modified  f u n c t i o n methods and  search  techniques  l i n e a r programming) and  c l a s s i f i c a t i o n scheme for  groups:  the  constrained  Mathematical programming  to the o b j e c t i v e  1983).  nonlinearly  programming). Transformation methods  problem. The  penalty  constrained  Murray (1974) e d i t e d a book  classifies two  transformation  i s c l a s s i f i e d as  and  linearly (1981)  constraints  nonlinear  S p e c i a l techniques have been developed  for  Schwefel  programming and  are  in  was three  variational  method,  gradient  ridge methods  (Umeda,  1969 ).  V a r i a t i o n a l methods c o n s i s t of  P o n t r y a g i n ' s maximum p r i n c i p l e (Umeda, are  those that s e l e c t the d i r e c t i o n  v a l u e s of respect the  partial derivatives to  the  majority  the  first  objective (Swann, most  the  and  quickly.  1981)  and  function  i s to employ an  The  derivatives of  c o n s t r a i n t s can  methods  have  derivatives  when the  methods evaluation  is  based and  on  1974).  One  direct-search  The  progress  comparison of the  sequence of t r i a l p o i n t s . unconstrained  or  the  of  of  solution  is  the that  i f so, lengthy  useful or  towards  the  in  when the are  direct-search optimum  f u n c t i o n values at  most  not  direct-  f u n c t i o n values  s t r a t e g y of  procedures -34-  in  alternative  proven  difficult,  objective the  the  procedure which does  is  (Swann,  An  Such methods are known as  are d i s c o n t i n u o u s  and  constraints  involve a  1974).  optimization  D i r e c t search  error  the  most important  problems f o r which d i f f e r e n t i a t i o n  to  Newton-type  d i f f e r e n t i a b l e , and  (Swann,  d e r i v a t i v e values.  "riding  are a number of drawbacks  and  calculation  and  l i k e l y to lead to the  should be continuous and  search methods.  subject  f u n c t i o n with  c o n s t r a i n t s make use  i n c e r t a i n cases  methods are  the  are some examples.  second  of these techniques.  of the  for  the  However there  complicated  approach call  and  1974). These  evaluation  objective  Murray, 1974)  sometimes  functions  functions  search by using  of the methods f o r handling and  implementation  the  method (Schwefel,  Quasi-Newton methods ( G i l l and  of  the  Gradient methods  independent v a r i a b l e s . H e m i s t i t c h i n g  constraints"  The  1969).  of  of  those r e l a t e d to  successful  i s the simplex  of  by a the  method  originally modified  proposed  this  by Spendley e t  method  to  handle  procedure the Complex method  al.  (1962).  constraints  (constrained  Box  and  simplex).  (1965)  termed  his  Box's Complex  method has had wide acceptance because of the simple way i n which the c o n s t r a i n t s are handled the g l o b a l optimum Ghani,  and the high p r o b a b i l i t y of l o c a t i n g  (Saguy et a l . , 1984;  Kuester and Mize,  1973;  1972).  2. The Complex method  2.1. The general The  optimization  minimizing)  some  independent constraints  on  chosen  the  problem  problem  variables  to m n o n l i n e a r the  optimization  nonlinear  (equation  independent  implicit  consists  maximizing  objective  11)  function  subject  variables  constraints  independent v a r i a b l e s )  of  to  (equation  (or of  n  explicit 12)  and  ( c o n s t r a i n t s on f u n c t i o n s of  (equation 13).  y = fo(Xi,X ,...X„) Li < X i < Ui  (11) (12)  2  i=1,2,...,n L* < g* (Xo.,X , . . .X„) < U*  (13)  2  k=l,2,..., m Where the  Lt  and U i represent  the lower and upper  independent v a r i a b l e s , and L *  upper  l i m i t s placed  variables  and  and U* represent  on the n o n l i n e a r  are  either  l i m i t s placed  on  the lower and  f u n c t i o n s of the independent  constants  or  functions  of  the  independent v a r i a b l e s (Box, 1965). M a t h e m a t i c a l l y speaking, i t i s important t o p o i n t out t h a t the -35-  term "nonlinear  model" r e f e r s to a model i n which the  appear n o n l i n e a r l y (Ratkowsky,  1983;  parameters  Draper and Smith, 1981), f o r  example y where 8 i this  and 0  study  2  ex (Si -  e ) 2  are the parameters t o be e s t i m a t e d .  the term "nonlinear  as f i r s t - and second-order  The Complex method  The  model"  w i l l be used  consists  of  search  more  independent  to  in  denote  polynomials.  algorithm i s taken from Box  Complex method handles the c o n s t r a i n t s by the (i.e.  However,  i n the independent v a r i a b l e s such  d e s c r i p t i o n that f o l l o w s  figure  (14)  8  models t h a t are not f i r s t - o r d e r  2.2.  te-St - e" it]  unit)  than  variables)  use  (1965). of a  c a l l e d the complex.  n + 1  v e r t i c e s (n i s  which  c o n t r a c t s when  flexible  This the  The  figure  number  constraints  of are  v i o l a t e d . The Complex method tends to f i n d  the g l o b a l optimum by  evaluating  f u n c t i o n value  and  comparing  the  objective  f e a s i b l e v e r t i c e s of the complex and by  a new  method,  feasible there  i s no  which each vertex  point. attempt  replacing  In c o n t r a s t  to Spendley's  t o preserve  is equidistant  the worst  the r e g u l a r  from a l l other  points  at a l l vertex Simplex  figure in (Friedman,  1971). The  first  step  i s t o generate an  feasible  region,  required  that a f e a s i b l e s t a r t i n g p o i n t  initial  complex  with a number of v e r t i c e s K > (n+1). (i.e.  in it  the is  does not v i o l a t e  any c o n s t r a i n t ) , X n , be known. The a d d i t i o n a l K - l p o i n t s , X u , -36-  are  obtained  one  at a time by the use  and  the ranges f o r each of the Xi*  = L  pseudo-random  numbers  independent v a r i a b l e s by means of  + ri(Ui  t  o£  - Li)  (15) i l , 2 , . . .,n =  j 2, • ••, K =  where  r  is  t  a  pseudo-random  d i s t r i b u t e d over the  interval  number  (0,1).  deviate  L± and  rectangularly  U± are the  lower  and  i s f e a s i b l e with respect  to  upper l i m i t , r e s p e c t i v e l y , of X i . Each the  p o i n t generated i n t h i s way  explicit  implicit trial  constraints,  constraints.  but may  I f an  point, Xi^(bad),  v i o l a t e one  implicit  constraint  i s moved halfway  or more  of  the  is violated,  the  towards the c e n t r o i d  of  the remaining v e r t i c e s X i ( n e w ) = ( X ( b a d ) + X ,=)/2 3  where Xio,  The  i d  the c o o r d i n a t e s are d e f i n e d  centroid  numerical  of the c e n t r o i d of the  Xi=  =  is  the  K E X i j-1  K-l  the  points,  - X (bad))  (17)  i S  p o i n t each of whose  step  constraints  assuming  coordinates  the  of the  i s repeated as necessary u n t i l satisfied.  feasible  generating  is  the  points  worst.  are  s a t i s f a c t o r y p o i n t w i l l be After  a  average of the corresponding c o o r d i n a t e s  This c o n t r a c t i o n  by  remaining  by  of the complex, except the  implicit  (16)  4  region  is  Box  (1965)  convex,  a l l the  states ultimately,  that a  found. initial -37-  complex the  following  procedure  i s repeated; of  the o b j e c t i v e  the complex.  Xid(worst),  The  vertex  i s replaced  in  having the  points  worst  a t each  vertex  response  value,  times as f a r from  the  as the r e f l e c t i o n of the worst  the new p o i n t , Xi->(new), being c o l i n e a r  the c e n t r o i d ,  Xi-j (worst),  with the r e j e c t e d p o i n t , the  i s evaluated  by a p o i n t a>l  c e n t r o i d of the remaining point  function  Xt = ,  and the c e n t r o i d ,  of  retained vertices Xi.-,(new) = o ( X i c - X i  If t h i s new p o i n t ,  not  variable, inside  the  expansion step  satisfy  the  function  variable  i s evaluated  function  value  function  value or i f an  c o n s t r a i n t s on  i s r e s e t to a value  violated limit of  the new p o i n t implicit  repeats i n constraint  is  points  to  give a new  trial  point.  repeated as necessary u n t i l a b e t t e r  The  collapses The  function  yield  (0.000001) If, the  the worst  violated,. the  the  remaining step  is  response value or a f e a s i b l e  continue.  equal values to  evaluations  within  a  of the  specified  u s u a l l y occurs a t the optimum when the  into i t s centroid  usual  independent  contraction  Convergence i s assumed when f i v e s u c c e s s i v e  This  an  does  Thus as long as the complex has not c o l l a p s e d  i n t o the c e n t r o i d , progress w i l l  criteria.  point,  the above  giving  moved halfway towards the c e n t r o i d of  objective  and  to y i e l d a f e a s i b l e point.  is  i s obtained.  worst  Delta  vertex  point  (18)  i e  i s repeated. I f , however, the p o i n t  one of the e x p l i c i t that  (worst) ) + X  which i s r e p l a c i n g the previous  is a f e a s i b l e point described  3  complex  (Box, 1965).  method f o r checking t h a t the g l o b a l -38-  rather  than  a  l o c a l maximum or minimum has from d i f f e r e n t s t a r t i n g converge  been found Is to r e s t a r t the  points  on the same r e s u l t ,  and a  With the Complex method, there starting will  use  of  the  contractions  progress  to  The  use  of  f a c t o r a>l  complex  and  towards the c e n t r o i d . initial  K>n+1 of  the  points  aids  complex,  complex tends to c o l l a p s e and  number  tends  initiators  It also  in  enables  maintaining  s i n c e , with K=n+1  f l a t t e n along  However i f n i s greater  K=2n.  1965). Box  the  of «=1.3  For  cause  a  rapid  p o i n t i s f a r away from  (Box,  be  to  1965).  thus to compensate f o r  which i s encountered and  found.  the  1965).  dimensionality  (1965)  the  full  points,  the  first  constraint  recommends a than 5,  value  less points  used. the  two  schematically  dimensional  i n Figure  procedure i s given  A  been  i s no d i f f i c u l t y i n using the same  reflection  be made when the  optimum (Box,  2.3.  i f they a l l  g l o b a l optimum has  p o i n t s i n c e d i f f e r e n t pseudo-random  c o n t i n u a l enlargement of the  may  that,  i n i t i a t e the problem with a d i f f e r e n t complex (Box,  The  the  to i n f e r  search  1.  case, A  in Figure  the  flow chart  procedure  is  shown  I l l u s t r a t i n g the above  2.  M o d i f i c a t i o n s to the Complex method number  of  m o d i f i c a t i o n s have been made  Complex method to overcome the In the  Complex method,  a  to  l i m i t a t i o n s of the  the  original  algorithm.  r e f l e c t i o n point must be  moved i n  halfway towards the c e n t r o i d of the remaining p o i n t s when the -39-  new  Figure  1.  A t w o - d i m e n s i o n a l c a s e o f t h e Complex method s e a r c h f o r t h e optimum (Umeda, T. 1 9 6 9 ) .  -40-  (^Start)  \ r — Feasible  Generate  starting point  initial  complex  /  (• Yes  Explicit constraints violated  >  v ;  Move p o i n t i n a d i s t a n c e DELTA inside the violated constraint  No No Initial complex generated  Implicit constraints violated  Yes  Yes Evaluate objective f u n c t i o n a t complex points  F i n d worst point i n complex  |t Yes  Check c o n v e r g e n c e criteria Move 1/2 way towards t h e centroid _.  No  Calculate a l l points  c e n t r o i d of except worst  R e f l e c t worst point through c e n t r o i d  Figure  2. F l o w  chart  o f t h e Complex  -41-  method  algorithm  No  ® Yes Explicit constraints violated 1•  Move 1 / 2 way towards the centroid  No  Move p o i n t i n a d i s t a n c e DELTA inside the v i o l a t e d constraint  Yes Implicit constraints violated  No  Evaluate  objective  function  Yes New p o i n t as worst  repeats point  No  - 4 2 -  p o i n t repeats  as being  the worst.  The  c o n t r a c t i o n i s c a r r i e d out  u n t i l a p o i n t b e t t e r than the r e j e c t e d one all  p o i n t s on the  are  worse  causes  than  the  line  from the c e n t r o i d to the p r o j e c t e d  the o r i g i n a l p o i n t ,  projected  point  c e n t r o i d , thus t e r m i n a t i n g Guin (1968) and  i s found. I f by chance  a p p l i c a t i o n of  e v e n t u a l l y to  point  this  coincide  rule  with  the  the s e a r c h . To overcome t h i s s i t u a t i o n  M i t c h e l l and  Kaplan (1968)  recommend t h a t i f the  r e f l e c t i o n f a c t o r a i s found to have been reduced below a c e r t a i n quantity trial  without  o b t a i n i n g a b e t t e r f u n c t i o n value,  p o i n t should  point)  and  (1971)  and  number  of  be returned  to i t s o r i g i n a l  Friedman and  Pinder  (1972)  through the best  point.  The falls  Friedman and  Complex into  a  fails  reflecting  Pinder,  would 1972;  otherwise  non-feasible  region;  this  searching  non-convex spaces (Guin,  checking  i f the c e n t r o i d i s found to be not  but the best p o i n t  new  complex c o n s t r u c t e d .  if  the  centroid  is  r e s t a r t e d by g e n e r a t i n g  a  be  fixed allowed  the c e n t r o i d  1968).  often  not  feasible  a new  initial  -43-  the  stopped  centroid  occurs  Guin (1968) feasible.  of the complex should M i t c h e l l and  have  to  Friedman, 1971).  i n f i n d i n g the optimum when  all  (worst  These m o d i f i c a t i o n s allow the search  i n s i t u a t i o n s where i t  1968;  should  this  Friedman  recommend t h a t  c o n t r a c t i o n s toward the c e n t r o i d p o i n t i s generated by  (Guin,  position  the second worst point r e j e c t e d i n s t e a d .  a f t e r which a new  continue  then  when  suggests If i t i s ,  be d i s c a r d e d  and  a  Kaplan (1968) suggest that  the search be stopped, complex.  and  Mitchell nature  o£  account  Kaplan (1968)  the  a  initial  starting  optimum. where  and  They  complex,  p o i n t that  since  the  i t does not  i s in close  Umeda  complex (1969)  seemed to have no major  is  random  take  proximity  developed a non-random i n i t i a l i z a t i o n  the i n i t i a l  However,  were concerned with  t o the  procedure  i n f l u e n c e d by the s t a r t i n g  s t a t e d that the effect  on  the  initial  into  point.  configuration  r e s u l t s and the r a t e of  convergence. One disadvantage of the Complex method i s the convergence  to  an  computations needed  optimum  point  and  the  relatively many  iterative  to a r r i v e a t the optimum (Saguy e t a l . , 1984;  Saguy, 1983; Umeda and Ichikawa, 1971). Umeda and Ichikawa proposed  that  determination function  algorithm  order t o improve  the  convergence  (1971)  rate  the  of the r e f l e c t i o n p o i n t must take i n t o account the  values  introduced  compared  in  slow  Ghani  (1972)  the expansion and c o n t r a c t i o n moves i n t o the  Complex  based  at  7  each v e r t e x of the complex.  on Nelder and  Mead's  h i s Complex method to that of  f u n c t i o n e v a l u a t i o n s were  needed  Simplex Box's  algorithm. Complex.  He Fewer  t o reach the optimum responses  in a number of examples. The and  s t o p p i n g c r i t e r i o n was m o d i f i e d by Umeda (1969) Ichikawa ABS  and  Umeda  (1971) to be (worst response - best response) worst response  -44-  < 0,001  (19)  2.4.  A p p l i c a t i o n s of the Complex method  The broad In  Complex  method has been found  the chemical f o r the  engineering area,  optimal  design  equipment o p t i m i z a t i o n .  Stevens (1972)  chemical  chemical  processes  an a b s o r b e r - s t r i p p e r  been  and f o r  system.  Friedman  (1971)  optimized  Adelman  investment a  of a  catalytic-  process.  In the area of food s c i e n c e and technology the been used to optimize s e v e r a l food processes  1984).  1984).  the Complex method has  maximized per cent r e t u r n i n  plant.  polymerization  of  many  Umeda (1969) used the Box Complex method  for the optimal design of  has  to  and d i f f e r e n t o p t i m i z a t i o n problems (Saguy e t a l . ,  used  and  t o be a p p l i c a b l e  Mishkin e t a l .  (1984)  Complex method (Saguy e t  al.,  used the Complex method t o d e r i v e  optimal temperature p r o f i l e s d u r i n g food d e h y d r a t i o n . S u l l i v a n e t al.  (1981) optimized a c a r r o t d e h y d r a t i o n process t o achieve  q u a l i t y low moisture and  color.  simulation objective  the  used the Complex method together with a  program  optimize a  that  to study  process.  optimization  (1987)  fermentation process.  was t o maximize  the  profit  The  of the  The Complex method has a l s o been used i n  of food f o r m u l a t i o n s .  Moskowitz  and  Jacobs  c i t e d the a p p l i c a t i o n s of the Complex to maximize t e x t u r e  preference  of  acceptable  level  a  pie crust formulation of storage s t a b i l i t y .  a l s o used to f i n d p i e c r u s t profiles  l o s s of v i t a m i n A  Saguy (1982)  of  fermentation  c a r r o t p i e c e s with l i t t l e  high  formulations  (Moskowitz and Jacobs,  1987).  -45-  while  maintaining  The Complex method with s p e c i f i c  an was  sensory  D.  COMMINUTED  1. Product  MEAT  PRODUCTS  description  Comminuted  meat products such as wieners,  bologna are f i n e l y comminuted, cooked, smoked  or  unsmoked  ingredients  s e m i s o l i d sausages,  (Pearson and Tauber,  are raw s k e l e t a l meats,  n i t r i t e s and p o s s i b l y f i l l e r s , meat products are produced  fat tissue,  and water,  and f u r t h e r s t a b i l i z e d  Comminuted  meat  products  emulsion p r o d u c t s "  physical  finite  of two  droplets  meat  having  phase)  containing  emulsion  diameters  as  et a l . , (1968) a  phase  of  the  is a  raw  - 4 6 -  "fine  batter  biphase  one being  between  0.1  system  dispersed and  as  10  )im  (Comer  and  1985). d e s c r i b e d the b a s i c  mixture  protein  as  (1968) r e p o r t e d that  of  finely  i s f a t and the continuous  solubilized  salt  p r o d u c t s " (Comer and A l l a n -  properties  An  to  d i s p e r s e d as a f a t - i n - w a t e r emulsion,  discontinuous  salt,  Comminuted  w i t h i n the continuous phase  emulsion  constituents  water,  referred  immiscible l i q u i d s ,  Hansen (1960) and S a f f l e a  and  emulsions.  A l l a n - W o j t a s , 1988; Asghar  of  basic  1982).  are o f t e n  or "emulsion-type  structures  true  (discontinuous  The  by thermal p r o c e s s i n g ( L a c r o i x  1988) s i n c e Hansen (1960) and S a f f l e  consisting  either  matrix obtained while chopping the meat with  and Castaigne, 1985; M o r r i s s e y e t a l . ,  resembled  and  by the d i s p e r s i o n of f i n e p a r t i c l e s of  a  the  1984b).  extenders and b i n d e r s .  fat within  Wojtas,  frankfurters  components.  divided  meat  i n which the  phase The  structure  i s water  salt-soluble  proteins  of  the  meat a c t as e m u l s i f y i n g agents  p r o t e i n membrane surrounding emulsion the  (Saffle, 1963;  1968;  capacity  Carpenter  Swift  et  emulsification  as  Several  (1982)  1988;  1982;  However,  an  This  model  systems  Swift and  Sulzbacher,  (1968)  considered  responsible  for  the  products.  1985;  products  (Comer  Acton et a l . ,  and 1983;  (1983), they do r e t a i n some of the  emulsion.  Photomicrographs of  show a proteinaceous membrane 1988;  Jones and Mandigo,  Borchert et a l . ,  not  a  1982). However, as p o i n t e d out by M o r r i s s e y et  f a t g l o b u l e s (Barbut, al.,  factor  Asghar et a l . ,  of  emulsions  Saffle  in  systems i n the c l a s s i c a l sense  and Acton et a l .  characteristics cooked  1961). primary  1968).  have commented t h a t comminued meat  not t r u e emulsion  M o r r i s s e y et a l . ,  proteins  S a f f l e , 1964;  in finished  authors  Allan-Wojtas,  al.  the  forming  by numerous s t u d i e s r e g a r d i n g  of meat  and  al.,  s t a b i l i t y observed  are  fat globules (Saffle,  theory has been supported  emulsifying  by  all  u n i f o r m l y surrounded  fat  1967;  raw  and  surrounding  the  1982;  Helmer and  d r o p l e t s are of  Swasdee et  Saffle,  uniform  1963).  size  or  by p r o t e i n membranes (Swasdee et a l . ,  Borchert et a l . , 1967). M o r r i s s e y et a l . (1982)  proposed  are 1982;  a model  f o r the i n t e r f a c i a l p r o t e i n membrane i n comminuted meat products. They  suggest  extend globule, around  into  that the  while the  constituents.  fat The  the hydrophobic liquid  of  myosin  f a t l a y e r at the s u r f a c e  actomyosin globule  heads  of  molecules the  concentrates i n a m u l t i l a y e r in  association  with  other  p r o t e i n m u l t i l a y e r entraps water and -47-  fat  region muscle  possesses  sufficient  viscosity,  coheslveness and e l a s t i c i t y to  the raw b a t t e r ( M o r r i s s e y et a l . ,  stabilize  1982).  Asghar et a l . (1985) r e d e f i n e d the s o - c a l l e d meat emulsions meat  suspensions,  multiphase is  and  system  described  them  as  consisting  i n which the continuous phase ( c a l l e d  proteins  proteins,  with  solid  compounds,  as  functions  both  immobilized w i t h i n the  c h e m i c a l l y and  I t g i v e s the product i t s  moistness,  appearance,  (Schmidt e t a l . ,  matrix.  mechanically  d i s p e r s e d phase i n the raw and cooked  to  1984;  matrix  products  characteristic  in  1981)  factor  mobility  t h a t emulsion for  to  texture, product  bite,  identity  the  Ziegler  that the  interfacial  and  protein protein  that prevents f a t c o a l e s c e n c e  once  Comer and Allan-Wojtas  out  responsible  the  have suggested  contrast  i s the p r i n c i p a l  completed.  the  ( M o r r i s s e y et a l . ,  o v e r a l l q u a l i t y and  Lee et a l . ,  diminishing  T h i s matrix  1981).  formation,  membrane,  pointed  insoluble  stabilize  S e v e r a l r e s e a r c h e r s (Comer and Allan-Wojtas, 1988; Acton,  and  f a t p a r t i c l e s and other i n s o l u b l e components of muscle  t i s s u e d i s p e r s e d and  1982).  such  a  matrix)  a complex h y d r o p h i l i c c o l l o i d a l aqueous s o l u t i o n of s a l t s  soluble  by  of  as  the  fat  distribution  (1988) and Acton et a l .  formation i s not the  is  (1983)  primary  factor  the s t a b i l i t y of the f i n i s h e d p r o d u c t s .  Water  b i n d i n g and g e l a t i o n a b i l i t y of i n s t a b i l i z i n g the cooked  meat  product.  - 4 8 -  p r o t e i n s p l a y a major r o l e  2 . Processing steps Excellent steps al.  d e s c r i p t i o n s of comminuted meat products p r o c e s s i n g  have been w r i t t e n by Pearson and Tauber  (1982) and S a f f l e  (1984b),  Long  et  (1968).  The o p e r a t i o n a l p r o c e s s i n g of comminuted meat products u s u a l l y begins with g r i n d i n g of the meat i n g r e d i e n t s through a grinder in  p l a t e and pork  grinder plate.  uniform  distribution  during  The lean ground meats a r e mixed of  the  meat  ingredients.  chopping  is essential 1968).  to  One t h i r d  s a l t - c u r e mix and seasoning are added  chopped.  and  temperature  reaches  comminution 1988).  depends  The  emulsifier  chopping i s continued  on  t o the meats  i f t h i s equipment  completion  a vacuum d u r i n g  i t i s recommended  emulsion of  the  (Whiting,  I f a vacuum  that  a p r e c h i l l e d vacuum mixer. emulsion  the  an  i s a v a i l a b l e . The temperature of the  been to  and  tissue  i s then passed through  has  transferred  keep  fatty  The  emulsion should not be over 18<>C. used,  to  the meat  finished not  emulsion  temperature r a t h e r than time  product from the chopper  mix i s  Temperature  until  14.4-15.6°C.  a  of the i c e or c o l d  The remainder of the i c e i s s l o w l y added  added  give  The  prevent  meat temperature near 4<>C. When a l l the i c e i s added is  to  (also c a l l e d s i l e n t c u t t e r ) .  breakdown (Townsend e t a l . , water,  in  trimmings and f a t t y t i s s u e through a 3/8  t r a n s f e r r e d t o a chopper control  1/8  formation has been  the  emulsion  Continuous found  chopper be  use of  t o increase  product s t a b i l i t y , decrease product shrinkage and promote uniform textural strength  (Whiting, 1988; T a n t i k a r n j a t h e p e t a l . , -49-  1983).  The  emulsion  is  t r a n s f e r r e d to  n a t u r a l or s y n t h e t i c c a s i n g s . casings under  and  linked  stuffers  for  F r a n k f u r t e r s are s t u f f e d  a c c o r d i n g to the d e s i r e d  vacuum produces a denser product with  texture  than does nonvacuum s t u f f i n g  (A) D r y i n g .  into  better  into  i n t o long  length.  (Whiting,  s t e p i s thermal p r o c e s s i n g of the product. p r o c e s s i n g can be d i v i d e d  extruding  Stuffing  b i n d i n g and  1988).  The  next  The stages of thermal  ( M i t t a l et a l . ,  1987):  Drying i s used to promote s k i n formation as w e l l as  to c o n d i t i o n the s u r f a c e of the product f o r smoke d e p o s i t i o n color (B)  development. Smoking.  The purposes  of smoking are development of f l a v o r  and c o l o r , p r o t e c t i o n from o x i d a t i o n and (C)  Resting.  penetration (D)  and  The  resting  period  preservation. is  used  allow smoke  i n t o the product.  Cooking.  The purposes  of cooking a r e :  (a)  microorganisms  and  improvement of storage l i f e ,  and  of  the meat p r o t e i n s ,  gelation  (c)  the  s t a b i l i z a t i o n of the  red  color  modification  (e)  (b) d e n a t u r a t i o n of  (d)  cured products and  d e s t r u c t i o n of  improvement  p e e l a b i l i t y of the f i n a l product, in  to  of  the  product  by s p r a y i n g the  product  texture. (E)  Chilling.  Chilling  i s performed  with  c o l d water f o l l o w e d by r e f r i g e r a t i o n .  wash  off  excess  smoke and to  make  the  The purposes are product  to  mechanically  stable. There are different  no  types  uniform  thermal  processing  of smokehouses are used -50-  (Long  schedules et  because  al.,  1982;  S a f f l e , 1968), S a f f l e  (1968) mentions a common thermal p r o c e s s i n g  schedule where the smokehouse i s r a i s e d 82°C. The r e l a t i v e humidity may range  5 <>C/15  min from 60°C t o  between 30  t o g r e a t e r than  80%. When the product reaches an i n t e r n a l temperature r e l a t i v e humidity should be h i g h . be  done e i t h e r  T h i s steam cook o p e r a t i o n can  i n the smokehouse or i n a  al.,  1982).  until  i t reaches an i n t e r n a l temperature  1982;  Saffle,  (NaCl)  (Sofos,  flavor  these c o n d i t i o n s  of 68°C (Long e t a l . ,  f i n a l product  characteristics  factors  i s e s s e n t i a l f o r the manufacture of meat products  1986). I t f u n c t i o n s as a p r e s e r v a t i v e ,  texture,  stability,  and cooking y i e l d  ( L a c r o i x and Castaigne,  and  enhances product  influences  1985; Sofos, 1983b).  and s o l u b i l i z e s m y o f i b r i l l a r p r o t e i n s ( M o r r i s s e y e t Comer  and  (Long et  1968).  Compositional  Salt  steam cooker  The product should be held under  3. Some f a c t o r s t h a t a f f e c t  3.1.  of 63°C the  Allan-Wojtas,  1988;  Acton e t  al.,  product  It extracts al.,  1983;  1982; Sofos,  1983a;b). During e x t r a c t i o n an i n c r e a s e i n water-binding c a p a c i t y of the m y o f i b r i l l a r p r o t e i n s occurs,  due t o the " s a l t i n g - i n "  sodium c h l o r i d e and t i s s u e d i s i n t e g r a t i o n al., several  1983).  (Schut, 1976;  S w e l l i n g of the m y o f i b r i l s  characteristics  t h i c k e n i n g and v i s c o s i t y  of  occurs,  the meat emulsion  (Morrissey et a l . , - 5 1 -  e f f e c t of Acton e t  which a f f e c t s such  as  body,  1982) which i n t u r n  s t a b i l i z e s the meat emulsion solubilized  proteins  ingredients  (Sofos,  (Smith,  can a l s o e m u l s i f y and 1983  a;b).  p r o t e i n s form the matrix,  thermal  processing,  will  is  needed  a v a i l a b l e to form the g e l (Whiting, formulations  fat  to  enough  t y p i c a l l y contain  used.  Sofos  frankfurter reducing  1983).  (1983a;b)  Flaked  2.5-3.0%  NaCl  levels  below  reduced s h e l f l i f e  2.0%  Hand et a l . , 1987;  and  an  1984;  e f f e c t on product  1988), and  1983). gives  results  proteins  salt  1954),  in increased  poor p e e l a b i l i t y ,  and  good  texture  ( L a c r o i x and  Castaigne,  1985;  in his  1988;  1988),  tenderness  (Uram et a l . ,  (Comer and  Allan-Wojtas,  Castaigne,  the meat  cooking  (Barbut,  Allan-Wojtas,  solubilization  ( L a c r o i x and  s t a b i l i t y to  be  unacceptable  and  e x t r a c t i o n of  1985;  Acton et a l . ,  I n c o r p o r a t i o n of water i n t o the meat b a t t e r at 15 a  can  Sofos 1983a;b).  (Comer and  j u i c i n e s s and  p a r t i c i p a t e s i n the  myofibrillar  or granulated  C l a r k e et a l . , 1987;  S w i f t et a l . ,  to ensure a good  s o f t e r products.  Water a d d i t i o n i n c r e a s e s y i e l d has  salt,  s t u d i e s have demonstrated that  l o s e s , reduced emulsion s t a b i l i t y , flavor,  protein  (or b r i n e )  found no d i f f e r e n c e between them  f o r m u l a t i o n s . Several  during  1987b).  c o n c e n t r a t i o n , a 4% b r i n e i s g e n e r a l l y necessary (Acton et a l . ,  salt  within i t .  based on t o t a l meat weight. In terms of e f f e c t i v e s a l t  batter  other  these  phase  have  These  and  which upon c o a g u l a t i o n  hold the d i s p e r s e d  solubilization  Frankfurter  bind  1968).  After extraction,  soluble  Sufficient  1988; saffle,  emulsion  during  to  25%  cooking  Acton et a l . , 1983). At lower water  l e v e l s , e x t r a c t i o n of meat p r o t e i n s i s poor and -52-  the meat emulsion  cannot reach i t s optimal s t a b i l i t y , product. causes  On the other hand, dilution  thus y i e l d i n g an unacceptable  a d d i t i o n of higher l e v e l s of water  of the p r o t e i n s and s a l t s which  decrease i n water h o l d i n g c a p a c i t y  (Schut, 1976)  i n product shrinkage (Uram e t a l . ,  1984).  results  in a  and an i n c r e a s e  Ice and/or  c o l d water  are used t o absorb the heat generated d u r i n g comminution i n order to  prevent  coagulation  of the p r o t e i n s  (Pearson  and  Tauber,  1984b). Fat  i n comminuted  mouthfeel,  meat  products  j u i c i n e s s and f l a v o r  r e d u c t i o n i n c r e a s e s the toughness and  Mittal,  (Anonymous, A  Hand e t a l . ,  (Schut, 1976).  solid-liquid  (Townsend e t a l . ,  1968).  of the f i n i s h e d product 1987)  General f a t c h a r a c t e r i s t i c s such  ratio, solid  made with pork suet and beef  Schut  (1976) tallow  while those made with pork back f a t and b e l l y f a t  i n a s t a b l e product.  i n c o r p o r a t i o n of s o f t  point  an  comminuted  cites  were u n s t a b l e ,  heats  ranges have  1983;  a study where sausages  Schut, 1976).  fats  as the  c o n s i s t e n c y and l i q u i d v i s c o s i t y ,  meat products (Acton e t a l . ,  non-uniform  (Barbut  i n behaviour between the animal  on the behaviour of f a t s d u r i n g p r o c e s s i n g of  resulted  Fat  and reduces meat f l a v o r  f u s i o n and t r a n s i t i o n and m e l t i n g temperature  effect  to texture,  1989).  substantial difference  exists  of  1989;  contributes  Lee e t a l .  (1981)  found that the  f a t d e s t a b i l i z e d the meat emulsion due to a  dispersion  of f a t .  chopping temperature  I t i s recommended that the end  f o r f o r m u l a t i o n s having pork and beef  f a t s be between 16 t o 18°C whereas a maximum of 10 to 12°C should - 5 3 -  be  used  f o r p o u l t r y meat p r o d u c t s  Comminuted kinds  meat  (Acton  e t a l . , 1983).  products may be prepared  from one  or  of raw s k e l e t a l muscle meats and/or raw or cooked  more  poultry  meat. Raw or cooked p o u l t r y meat s h a l l not comprise more than 15% of  the  total  products  ingredients.  These products may a l s o c o n t a i n  and m e c h a n i c a l l y separated meats.  by-  M e c h a n i c a l l y deboned  red meats are r e s t r i c t e d  to a maximum of 20% (Pearson and Tauber,  1984a).  be  Limits  must  placed  i n g r e d i e n t s such as by-products At high  l e v e l s of  by  color,  Thomsen and Zeuthen, 1988;  meat  but can s t i l l  r a t e s , low storage (Whiting, significant during review  frozen  functional  protein-protein  changes  of  the  major  freezing used  showed t h a t  of meats occur  Powrie (1973) made an e x t e n s i v e i n meat  decrease  water h o l d i n g c a p a c i t y (Powrie, One  (1980)  properties  during  as an i n c r e a s e  i n s o l u b i l i t y and  in  reduced  1973).  considerations  -54-  frozen storage.  p r o t e i n s take place d u r i n g  These changes are manifested interaction,  fresh  thawing r a t e s are  1976). M i l l e r e t a l .  changes of m y o f i b r i l l a r  storage.  1984a; Comer and  f u n c t i o n a l p r o p e r t i e s than  extended f r o z e n s t o r a g e . of the p r o t e i n  meats.  1975). Frozen meats are a l s o  temperature and slow  l o s s e s i n the  on  (Harding  have adequate f u n c t i o n a l i t y i f f a s t  1988; Schut,  Conformational  and s t a b i l i t y  Pearson and Tauber,  meat has poorer  processor  i n g r e d i e n t s can a l t e r the  flavor  Dempster, 1981; D h i l l o n and Maurer, Frozen  meat  and m e c h a n i c a l l y separated  i n c o r p o r a t i o n these  product y i e l d , t e x t u r e ,  used.  the  in selecting  the  meat  ingredients  f o r a comminuted meat product has  the meat to "bind" 1968). "high et  or e m u l s i f y  f a t and  r e t a i n moisture  I t i s common i n the meat p r o c e s s i n g binding",  al.,  "medium b i n d i n g " ,  1983;  Saffle,  1968).  and  been the a b i l i t y  "low  industry  bind values  well  the  m y o f i b r i l l a r p r o t e i n content present  skeletal  1983).  muscle  of  "low  meat from b u l l ,  binding  trimmings In  cheek  meats"  lips,  v e a l and tripe,  fillers,  added), i t i s the meat p r o t e i n s t h a t are suitable  t e x t u r e and  (Deng et a l . , myosin,  are  s t a b i l i t y to the  1981).  The  considered  processed meat products Whiting, batter during and  include  to  (Smith,  The  water  provide  water  binding  1988;  Kinsella  et a l . ,  Z i e g l e r and  (1983), M o r r i s s e y  extensively  Saffle,  1968).  extenders or  relied  comminuted  reviewed the  tongue  binders  meat products actomyosin  Li-Chan,  important capacity  1988; in  raw while  f i n i s h e d product, g e l a t i o n and  1983).  Acton  important  Nakai and  (1984),  et a l . , (1982),  fat  properties  Li-Chan  (1988),  Acton et a l . ,  (1983),  and  functional properties  -55-  and  f u n c t i o n a l i t y in  fat holding  more  and  upon to impart a  Nakai and  c a p a c i t y are the  Acton  (1988),  and  "medium  and  functional properties binding  include  hearts  the most  1988;  of  as  meats  pork trimmings  myofibrillar proteins,  the thermal process and  (Smith, Smith  1987b).  mutton,  Acton et a l . , 1983;  " a l l meat systems" ( i . e . no  i n the  b i n d i n g meats"  cows and  meat,  ox  (Rakosky, 1989;  meats  e s t a b l i s h e d by S a f f l e very c l o s e l y  Examples of "high  b i n d i n g meats" i n c l u d e  to r e f e r to  Such a c l a s s i f i c a t i o n of  the  (Acton et a l . ,  (Saffle,  b i n d i n g meats" (Acton  follows as  of  Schut  (1976) have  of meat p r o t e i n s  i n the p r o c e s s i n g of meat  3.2.  products.  Processing factors  The  most s t u d i e d p r o c e s s i n g v a r i a b l e s have been the end  chopping thermal  temperature  and  the temperature and  humidity  and  Saffle  (1963)  reported that  emulsion  breakdown  occurs at comminution temperatures above 16°C and  was  protein  noted  denaturation.  instability coincided  Townsend et a l .  (1968)  not due that  the  of emulsions comminuted at temperatures above 18.5<>C with the onset  of  temperature  of m e l t i n g of the high melting p o r t i o n s  protein-protein exceeded  15°C.  i n t e r f e r r e d with water and i n c r e a s e i n water and  C a r r o l l and depends  on  Lee  (1981)  the  interaction This  became  protein-protein  interaction  Lee et  a l . (1981)  s u s t a i n t h a t the s t a b i l i t y of the of the g e l and  the  and  t h a t optimal s t a b i l i t y was  Mandigo (1982) a l s o found  a f i n a l comminution temperature of thermal  increase in  of  temperature  results  in  occurs a  (Acton et a l . , 1983). -56-  Jones  achieved  Heat induced  due  a  solto  an  three-dimensional  p r o t e i n network where f a t and water are p h y s i c a l l y and stabilized  fat  16°C.  the meat p r o t e i n s which  of  formation.  p r o c e s s i n g of comminuted meat products  to-gel transformation  and  product  patterns  of the g e l matrix  During  as  f a t b i n d i n g by the p r o t e i n l e a d i n g to  fat separation.  rigidity  the  greater  d i s t r i b u t i o n at the beginning  at  to  f a t s . Deng et a l . (1981) e x p l a i n e d that d u r i n g comminution  extent  an  during  processing.  Helmer  of  point  chemically  gelation is a  two-  s t e p process which i n v o l v e s p a r t i a l u n f o l d i n g of p r o t e i n followed by  aggregation i n t o a continuous  the formation of a aggregation step  temperature  h i g h l y ordered g e l ,  step proceeds  (Whiting,  network  1987b;  at  (Kinsella,  Kinsella,  1983).  factors  that  temperature  affect  formation  of  of p r e c i p i t a t e s  and humidity  the s t a b i l i t y and  unfolding  Extreme c o n d i t i o n s  which lack the c h a r a c t e r i s t i c s of a g e l ( K i n s e l l a , processing,  For  i t i s necessary that the  a r a t e slower than the  and r a p i d h e a t i n g lead to  thermal  1983).  1983). During are  t e x t u r e of  processing  the  finished  product. There  is  humidity  have  t e x t u r e and Saffle  an  adverse  1967).  Cooking  T h i s procedure stages  on  the  and  emulsion  high  stability,  (Monagle et a l . ,  1974;  should be increased  allows r a p i d s u r f a c e d e h y d r a t i o n i n the  ( M i t t a l and  Monagle et a l .  temperature  temperatures  of cooking which,  allows s k i n formation 1967).  effect  c o l o r of the f i n i s h e d product  et a l . ,  stepwise. initial  general agreement t h a t high  (1974)  together with Blaisdel,  low  1983;  humidity,  S a f f l e et a l . ,  r e p o r t e d t h a t low humidity and a  steady r a t e of i n c r e a s e i n the smokehouse temperature a more a c c e p t a b l e product. M i t t a l and  Blaisdell  resulted in  (1983)  developed  a model to p r e d i c t the weight l o s s of f r a n k f u r t e r s d u r i n g  thermal  processing  product  under  v a r i o u s c o n d i t i o n s and  with  various  compositions. M i t t a l et a l . (1987) modeled the e f f e c t s of v a r i o u s r a t e s of i n c r e a s e of smokehouse temperature  and  on the  stability,  water h o l d i n g  capacity,  emulsion  t e x t u r a l parameters and sensory a t t r i b u t e s of -57-  relative  humidity  shrinkage,  frankfurters.  MATERIALS AND METHODS  A. EXPERIMENTAL METHODOLOGY The  main  optimization  o b j e c t i v e of t h i s study was t o e s t a b l i s h a  formula  program  i n the  meat p r o c e s s i n g would  search  predetermined ranges.  to  f o r the best product  quality  control  least-cost  formulations  specifications  This new method may r e p l a c e  To f u l f i l l  quality  i n d u s t r y . Such a new formula o p t i m i z a t i o n program  programs c u r r e n t l y being finding  be used f o r  within  that  meet  allowable  cost  l i n e a r programming  used i n the meat p r o c e s s i n g  computer  industry for  formulations.  the main o b j e c t i v e , t h i s study was d i v i d e d  i n three  main p a r t s : (1) Establishment A  of the Formula O p t i m i z a t i o n  computer program  computer program based on Box's Complex method was  i n IBM BASIC. For the purpose  written  of t h i s study the computer program  was named "FORPLEX". The Complex method has been used to optimize nonlinear making quality  functions subject  to  l i n e a r and n o n l i n e a r  i t s u i t a b l e f o r formula o p t i m i z a t i o n p r e d i c t i o n equations,  constraints  purposes.  Nonlinear  which have been found t o  explain  q u a l i t y b e t t e r , can be used as o b j e c t i v e f u n c t i o n s and the l i n e a r equations t h a t  describe  product s p e c i f i c a t i o n s  and i f necessary  q u a l i t y p r e d i c t i o n equations can be used as c o n s t r a i n t s . (2)  Development  of  ingredient-quality  3 - i n g r e d i e n t model f r a n k f u r t e r A three  relationships  for a  formulation  i n g r e d i e n t model f r a n k f u r t e r f o r m u l a t i o n -58-  was chosen t o  test  the  suitability  optimization. ingredients the a  Quality  of  the  Complex  method  for  formula  p r e d i c t i o n models as a f u n c t i o n  i n the f o r m u l a t i o n were developed.  of  As mentioned  l i t e r a t u r e review the process of m o d e l l i n g i s c a r r i e d set  fashion.  represent  the  An e m p i r i c a l model  should  be  of the food product.  An  adequate  in  out i n  postulated  r e l a t i o n s h i p between the i n g r e d i e n t s and  characteristics  the  to  quality  experimental  design should then be used which p r o v i d e s o b s e r v a t i o n p o i n t s from which the data can be c o l l e c t e d to which the model can be After  testing  equations In  can be used  t h i s study,  performed taken and  the s t a t i s t i c a l v a l i d i t y  of the equations  these  f o r p r e d i c t i o n and o p t i m i z a t i o n purposes.  generation  through mixture  since  fitted.  of  quality  prediction  experimentation.  models  was  T h i s approach  was  the r e l a t i o n s h i p between the i n g r e d i e n t p r o p o r t i o n s  the q u a l i t y of the f o r m u l a t i o n could be q u a n t i f i e d .  and Allan-Wojtas ingredients  (1988) suggested,  is  dependent  on  As Comer  f u n c t i o n a l performance  the  composition  of  of the  the  meat  f o r m u l a t i o n as a whole, making mixture experimentation a s u i t a b l e approach  to study the i n g r e d i e n t - q u a l i t y r e l a t i o n s h i p s of sausage  formulations. components  Scheffe's was  experimental  data  formulation)  of  regression several  the  formulation.  The  postulated  an  extreme  for  was  fitted  design  q u a l i t y parameters  at  each  point  using  "experimental d a t a "  evaluated  -59-  that  cubic  experimental  vertices  The term  parameters  special  model  c o l l e c t e d at each  analysis. quality  canonical  three to (i.e.  multiple refers  to  experimental  evaluated were: (a)  product  weight l o s s at d i f f e r e n t process,  (b)  processing,  stages  stability  of  of the the  frankfurter  raw  emulsions  (c) j u i c i n e s s of the cooked sausages,  t e x t u r a l c h a r a c t e r i s t i c s of the cooked sausages and s i g n i f i c a n c e and  adequacy of  each  assessed by a n a l y s i s of v a r i a n c e , of d e t e r m i n a t i o n  and  space (3)  thermal  (d)  specific  (e)  pH.  The  p r e d i c t i o n model  adjusted  multiple  a n a l y s i s of r e s i d u a l s .  a f u n c t i o n of the  to  quality  The  was  coefficient  models developed  were u s e f u l i n p r e d i c t i n g the q u a l i t y of f r a n k f u r t e r as  preparation  formulations  3 i n g r e d i e n t s used only w i t h i n the  mixture  studied. Optimization  of f r a n k f u r t e r formulations  using  the FORPLEX  program The  quality  optimize FORPLEX  prediction  several hypothetical computer program.  the m u l t i - o b j e c t i v e The  second  t h i s new The  equations  obtained  f r a n k f u r t e r formulations  optimization  to the  These equations were used as p a r t  of  f u n c t i o n and  o b j e c t i v e was  was  used using  were a l s o used as c o n s t r a i n t s .  to compare l i n e a r  o p t i m i z a t i o n method f o r formula  FORPLEX  were  compared  with  found  by  to the ones obtained  from l i n e a r  and  optimization.  l i n e a r programming f o r  of f r a n k f u r t e r f o r m u l a t i o n s .  optimum f o r m u l a t i o n s  programming  In  the  a d d i t i o n , computed  the FORPLEX program were compared programming.  B.INGREDIENTS Lean beef (MDPM), and  meat,  f r o z e n mechanically  deboned  s k i n l e s s pork f a t were obtained -60-  from  poultry  meat,  Intercontinental  Packers L i m i t e d , visible twice  Vancouver,B.C. The lean beef meat was trimmed of  f a t and connective  tissue,  c u t i n t o cubes  through a 6.3 mm p l a t e using a Hobart  (Hobart Manufacturing Company L i m i t e d , o b t a i n a uniform blend. cut  and  (Model 84142) g r i n d e r  Ont)  and mixed by hand t o  The pork f a t was trimmed of meat t r a c e s ,  The f r o z e n MDPM was p a r t i a l l y thawed a t 1 oc f o r 16  was used d i r e c t l y without f u r t h e r g r i n d i n g . These  were  analyzed  study the  referred  proportions meat block  ingredients: to  as  the "meat  block"  of  the  weighed  into  as d e s c r i b e d  the r e l a t i v e  Ont). A  i n g r e d i e n t bags  i n cardboard  250 g of  i n s e c t i o n s D and E. Each of these individually  R  three  f r e e z e r bags (DCP  f o u r t h f r e e z e r bag was used t o hold  of  boxes,  inZiploc  will  frankfurter  s p e c i f i e d i n the experimental plan to t o t a l  Canada Inc. P a r i s ,  placed  ingredients  beef meat, MDPM and pork f a t ,  The meats and f a t were  components was packaged  three  hrs  f o r proximate composition. For the purpose of t h i s  three  formulations.  the  ground  i n p i e c e s and ground once through a 9.4 mm p l a t e and mixed by  hand.  be  and  each f o r m u l a t i o n .  The bags  f r o z e n and s t o r e d a t  -30°C  were until  required. Table s a l t L t d . , Ont) was  (NaCl) ("Windsor" brand,  The Canadian S a l t Company  was purchased a t a l o c a l r e t a i l  from BDH Chemicals  (Toronto,  -61-  Ont).  store.  Sodium  nitrite  C.  PROXIMATE ANALYSIS  1. Determination of moisture Moisture content (24.002). placed  was determined using the AOAC (1980) method  Samples of beef meat,  i n aluminum  aluminum f o i l .  sample  MDPM and p o r k f a t  pans  Drying was performed  and  (ca 2 g)  were  covered  with  partially  i n a vacuum oven a t 90°C with  a vacuum of 27 inches u n t i l constant weight contents were c a l c u l a t e d as a percentage  (5 h r s ) . The moisture  (wet b a s i s ) .  2. D e t e r m i n a t i o n o f c r u d e f a t The  crude  f a t content was  determined  e x t r a c t i o n with a G o l d f i s h apparatus following Samples  by  (Laconco,  petroleum Kansas C i t y ,  the method d e s c r i b e d by Pomeranz and Meloan of  d r i e d beef meat,  ether MO)  (1978).  MDPM and pork f a t (ca 1 gr)  were  used. The crude f a t contents were c a l c u l a t e d as a percentage (wet basis).  3. Determination of p r o t e i n Protein MDPM using  content was determined on d r i e d - d e f a t t e d  and pork f a t samples. the r a p i d  Samples (5 t o 8 mg)  M i c r o - K j e l d a h l method of  Concon  beef were and  meat, digested Soltess  (1973). Digested samples were analyzed f o r t o t a l n i t r o g e n content using  an  Auto  Tarrytown, NY).  Analyzer II system  (Technicon  Instruments  Co.,  The crude p r o t e i n content was then c a l c u l a t e d by -62-  multiplying  the t o t a l n i t r o g e n content by a f a c t o r of 6.25.  p r o t e i n contents were c a l c u l a t e d as a percentage  D. EXPERIMENTAL A  v i s u a l i z a t i o n of the  model  frankfurter  was  f a c t o r space and contour  The three i n g r e d i e n t s used were:  for  which each i n g r e d i e n t was the  allowed  constraint  u n i t y . The model used was components f i t t e d  to  MDPM, ( X ) a  As  of the  to data c o l l e c t e d  vertices  design,  (1966).  S e l e c t i o n of t h i s design was sum  response s u r f a c e s .  at  ingredients  of the mixture  was  cubic for  the p o i n t s of  as d e s c r i b e d by McLean based  and  i n a l l mixture  Scheffes's canonical special  extreme  the  vary.  f o r the sum  easy  upper l i m i t s w i t h i n  3  maintaining  chosen  pork f a t , ( X i ) ;  beef meat, ( X ) . Table 1 shows the lower and  three  basis).  DESIGN  three-component  experiments  (wet  The  and  an  Anderson  on i t s c a p a b i l i t y of  components equal  while a l l o w i n g f o r v a r i a t i o n of the component  to  unity,  levels within  the  constra i n t s . The  design c o n s i s t e d of 10  vertices, program Arteaga,  experimental  four edge c e n t r o i d s and was  used  1990).  f i v e extreme  one c e n t r a l p o i n t . A  to compute the extreme The  points:  vertices  (Nakai  and  edge c e n t r o i d s were found by a v e r a g i n g  the  l e v e l s of extreme v e r t i c e s having a common i n g r e d i e n t central  p o i n t of the hyper-polyhedron  the extreme v e r t i c e s .  The  design  level.  represented the average  experimental d e s i g n was  order to estimate the experimental e r r o r adequately. the  computer  of  replicated in In a d d i t i o n ,  provided a measure of the lack of f i t of the -63-  The  model,  Table 1. I n g r e d i e n t s and t h e i r v e r t i c e s design.  l i m i t s used f o r the extreme  Limits Ingredient  Coded v a r i a b l e  Lower  Upper  Pork f a t  Xi  0.05  0.30  Mechanically deboned p o u l t r y meat  X  2  0.00  0.40  Beef meat  X  3  0.50  0.95  -64-  since  the  number of experimental  p o i n t s was  greater  than  the  number of parameters (Snee, 1971). The i n g r e d i e n t p r o p o r t i o n s f o r the 10  f o r m u l a t i o n s are r e p o r t e d  design and  is illustrated  A2  i n Table  i n F i g u r e 3. Refer  f o r an e x p l a n a t i o n on how  to  2 and the experimental  to Appendix A F i g u r e s A l read  the p r o p o r t i o n s  of  components i n t r i a n g u l a r graphs.  E. FRANKFURTER The  PREPARATION  experimental  design was r e p l i c a t e d ,  the 10 f o r m u l a t i o n s were prepared  i n a random o r d e r .  For the p r e p a r a t i o n of the f r a n k f u r t e r s , and  pork  fat  were  thawed  overnight  F r a n k f u r t e r emulsions were formulated to  protein ratio,  sodium  nitrite  2.5%  1°C  prior  to  t o c o n t a i n a 4:1  a food processor  Total  chopping  in  each f o r m u l a t i o n .  (Cuisinart,  time was 105  France)  sec  and  than  maintain  constant  chopping  chopping c o n d i t i o n s ,  temperature  c o n t r o l l e d the completion  was  the  to  temperatures,  protein  processing  making  This  caused  final  In order rather  factor High  that levels  the r e q u i r e d  lower  chopping temperature i m p r a c t i c a l  -65-  3  i n a 1°C the  of the chopping o p e r a t i o n .  ratio.  Table  chopping time  of i c e were needed i n some f o r m u l a t i o n s t o maintain moisture  0.01%  Chopping  temperature of the 20 f o r m u l a t i o n s d i d not exceed 16°C. to  use.  moisture  of the weight of the meat i n g r e d i e n t s .  was performed i n room.  at  the meat i n g r e d i e n t s  e f f e c t i v e s a l t c o n c e n t r a t i o n and  shows the i n g r e d i e n t l e v e l s used  cold  w i t h i n each r e p l i c a t e  emulsion as  the  Table 2. Extreme v e r t i c e s experimental  Formulation No.*-  design.  Ingredient proportions Xa.  1 2 3 4 5 6 7 8 9 10  X  0 .10 0.05 0.20 0.30 0.13 0.05 0 . 30 0 . 30 0 .18 0 .05  13  2  0 .40 0.40 0.30 0.20 0.22 0.22 0.09 0.00 0.00 0 . 00  *• Each f o r m u l a t i o n was r e p l i c a t e d f o r a t o t a l of 20 f o r m u l a t i o n s Xi=Pork f a t ; X = M e c h a n i c a l l y deboned p o u l t r y meat; B  2  X3=Beef meat  -66-  X  3  0 . 50 0.55 0 . 50 0.50 0.65 0 . 73 0.61 0.70 0.82 0.95  MDPM  FAT  B E E F  F i g u r e 3. E x t r e m e v e r t i c e s e x p e r i m e n t a l  -67-  design  Table  3. Weights  of i n g r e d i e n t s used  Meat block  (250 g)-  Formulation No. 1 2 3 4 5 6 7 8 9 10  i n each f o r m u l a t i o n ( g ) .  Ice  25.0 12.5 50.0 75.0 33 . 3 12.5 75.0 75.0 43 . 8 12.5  100.0 100. 0 75.0 50 . 0 54 . 3 54.3 23 . 0 0.0 0.0 0. 0  125.0 137.5 125.0 125.0 162.5 183 . 3 152.0 175. 0 206 . 3 237 . 5  18 .1 19.5 20.3 22.5 26 . 3 28.6 27.9 32 . 5 36.0 39 . 4  Salt  4. 5 4. 7 4. 2 4.0 4.7 5.0 4. 2 4.3 4.9 5.4  Xi=Pork f a t ; X = M e c h a n i c a l l y deboned p o u l t r y meat; X =Beef meat 2  3  -68-  Nitrite  0.023 0.024 0 .020 0 . 018 0 . 022 0.024 0.018 0.018 0 .021 0.024  controlling The  factor.  s a l t and sodium n i t r i t e were weighed and mixed  The meat I n g r e d i e n t s were t r a n s f e r r e d  to a c h i l l e d  together.  chopping  bowl.  The mixed d r y i n g r e d i e n t s and h a l f the crushed i c e were u n i f o r m l y distributed Chopping  over the meat i n g r e d i e n t s and chopped  was  performed  s i d e s of the bowl, at  each p e r i o d . 10  for  sec,  mix  The remainder  done.  Final  after  chopping  emulsion the  diameter, 10 cm long  Ont.).  an e l e c t r i c These  frankfurters  The  placed  in  5  the  f a t was added and chopped f o r  cellulose  10  casings  Immediately into  with a s t r i n g 7  cm  in  at  length  both were  o p e r a t i o n was performed were  1°C c o l d room  transferred  for less  1.5  cm  (Viskase, Lindsey,Ont),  gun ( P r o c t o r - S i l e x Canada Inc.,  tied  frankfurters  were r e c o r d e d .  emulsions were s t u f f e d  to  chopped  sec c o n s e c u t i v e chopping p e r i o d s were  raw  The s t u f f i n g  a  sec.  scrape  of the i c e was added and  temperatures  food  were of  formulation. room.  pork  a d d i t i o n a l 10  Four  interruptions to  45  the emulsion and measure the temperature  a f t e r which  sec.  using  with b r i e f  for  Picton,  ends.  Eight  produced  per  i n a lO^C c o l d  t o p l a s t i c bags and  than 2  h  until  thermal  p r o c e s s i n g . Emulsion samples from each f o r m u l a t i o n were c o l l e c t e d in p l a s t i c bags from the food gun and kept  in  ice  f o r emulsion  s t a b i l i t y a n a l y s i s and pH measurement. The  raw f r a n k f u r t e r s were hung on a cooking rack and placed i n  a preheated,  mechanical c o n v e c t i o n , h o r i z o n t a l a i r - f l o w  oven model O V - 4 9 0 A - 2 cooked  electric  (Blue M E l e c t r i c Co., Blue I s l a n d , IL)  and  a t 70<>C f o r 20 min. The f r a n k f u r t e r s were then t r a n s f e r r e d -69-  to  a steam  probe was  pot.  For each f o r m u l a t i o n ,  inserted  into  the  center  one of  T-type thermocouple  one  frankfurter.  thermocouple-thermometer model 450-ATT (Omega E n g i n e e r i n g , Stamford, CT) was  68-69°C was  reached.  were cold-water in  a IOC  thermal  A f t e r thermal  showered,  c o l d room  internal  processing,  The  temperature of the f r a n k f u r t e r s  placed i n p o l y e t h y l e n e bags and s t o r e d  overnight.  p r o c e s s i n g and a f t e r  p r o c e s s i n g weight  Inc.,  used to r e c o r d the f r a n k f u r t e r temperature.  f r a n k f u r t e r s were steam-cooked u n t i l an  A  l o s s as a  Weights  were recorded  before  overnight c o l d storage to determine percentage  (Shrink).  Shrink(%)  defined  as:  (weight  before p r o c e s s i n g - weight a f t e r c o l d storage) x weight before p r o c e s s i n g  was  100 (20)  The  next day  m a t e r i a l was  f r a n k f u r t e r s were hand peeled and  surface  c a r e f u l l y wiped o f f and weighed. Peeled f r a n k f u r t e r s  were packaged i n p o l y e s t e r / l a c q u e r lam./polyethylene Packaging,  Toronto,  Ont.)  and  vacuum s e a l e d with  vacuum s e a l e r (Sepp Haggenmuller KG,  Allgau,  W.  pouches a  consumer  cook t e s t was  performed.  Germany).  Percentage  a f t e r 13 days under vacuum packaged storage was d e f i n e d as: (weight  weight  (Vacuum s h r i n k ,  before - weight a f t e r vacuum storage) weight before vacuum storage  x  (DRG  Multivac  vacuum packaged f r a n k f u r t e r s were s t o r e d f o r 13 days at 1°C the  fatty  The until loss %)  100 (21)  -70-  F i g u r e 4 shows the flow c h a r t of the f r a n k f u r t e r p r e p a r a t i o n .  F. QUALITY PARAMETERS EVALUATED As a measure of the q u a l i t y of the f o r m u l a t i o n s the quality These  parameters  were e v a l u a t e d at each  parameters  are commonly used  following  experimental  point.  to e v a l u a t e the q u a l i t y  of  comminuted meat products.  1. Determination  of pH  For the d e t e r m i n a t i o n of pH of raw was  blended  ("blend"  emulsions,  g  sample  s e t t i n g ) i n an O s t e r i z e r G a l a x i e VII  blender  (Sunbeam C o r p o r a t i o n L i m i t e d , Toronto, Ont.)  f o r 10  mL d e i o n i z e d - d i s t i l l e d water. A F i s h e r model 420 S c i e n t i f i c Co.,  Pittsburgh,  electrode  used  was  to  PA)  a 10  sec with  pH-meter  100  (Fisher  with a combined g l a s s / r e f e r e n c e  t e s t the  pH  of  each  formulation  in  treatment  was  duplicate.  2. Emulsion The  stability  determined The was  stability  by two  emulsion  followed  of  after  bottles  were placed  probe. The  each  to  thermal  s t a b i l i t y t e s t r e p o r t e d by S a f f l e et a l .  were  in  the emulsions  methods.  samples  bottle  analysis  some m o d i f i c a t i o n s .  placed i n 20%  fat  equipped  were l e f t  with  g  emulsion  bottles.  i n a water bath held at 80°C.  batch was  bottles  d i v i s i o n Paley  Eighteen  (1967)  a T-type  One  control  thermocouple  i n the water bath f o r 30 min  -71-  The  after  Composition of the formulation g i v e n by t h e e x p e r i m e n t a l p l a n Water was added t o m a i n t a i n a 4:1 m o i s t u r e t o p r o t e i n r a t i o . S a l t was added a t a 2.5% effective concentration. N i t r i t e was added a t 0.01% o f meats. M i x i n g and c h o p p i n g i n a f o o d p r o c e s s o r f o r 105 s e c . Emuls i o n stability tests Stuffing in cellulose casings Dry and s t e a m c o o k i n g . I n t e r n a l t e m p e r a t u r e 68°C Showering Overnight  storage  @1°C Shrink  Peeling Vacuum  packaging  T Storage  @1<>C f o r 13 d a y s Vacuum  Consumer cook  Cook Texture  Figure  shrink  test  and j u i c i n e s s  shrink  tests  4. F l o w c h a r t o f t h e f r a n k f u r t e r p r e p a r a t i o n s t e p s and q u a l i t y parameters evaluated  -72-  the c o n t r o l reached an i n t e r n a l temperature schedule  resulted  approximately centrifuged Babcock  in  79°c.  an  The  internal  Indiana).  (Garver  Immediately  r e l e a s e d was  emulsion  b o t t l e s were f i l l e d  f o r 5 min a t "speed  centrifuge  o£ 60°c. T h i s cooking  No.  with  2" s e t t i n g  Manufacturing  after  temperatures 70<>C  of  water,  i n a heated (70<>C) Co.  Union  c e n t r i f u g a t i o n the per  City,  cent  fat  read d i r e c t l y from the stem of the P a l e y f a t b o t t l e .  Since the b o t t l e s used are designed  f o r 9 gr samples the per cent  fat  This test  reading  duplicate  was  divided  by two.  was  performed  in  f o r each f o r m u l a t i o n .  The emulsion developed  stability  was  a l s o determined  by Townsend et a l .  (1968)  with  using the  procedure  some m o d i f i c a t i o n s .  Twenty g emulsion  samples were p l a c e d i n 2.5x10.3cm polypropylene  c e n t r i f u g e tubes.  The tubes were c e n t r i f u g e d at 4°C f o r 5 min at  lOOOxG i n a S o r v a l l RC-2B automatic r e f r i g e r a t e d S o r v a l l , Inc., Norwalk,  CT). The  p a r a f i l m and aluminum f o i l at  80°C.  type for  One  thermocouple  probe.  temperature volume  of  preliminary  as  79°C.  in  an  %v/w  to be  measured  unbound f a t ,  experiments  i n order water  to  and  showed t h a t the f l u i d -73-  internal  T h i s method  water and p r o t e i n a c e o u s s o l i d s  thermal treatment  with a  of  emulsion  requires  released  the  during  express solids.  Tbath  i n t e r n a l temperature  resulted  approximately  of f a t ,  stability  schedule  equipped  with held  tubes were l e f t i n the water  30 min a f t e r the c o n t r o l reached an  60oc. T h i s cooking  a water bath  i n each batch was The  (Ivan  tubes were t i g h t l y covered  and were placed i n  c o n t r o l tube  centrifuge  emulsion However,  released  during  thermal treatment never developed a l i p i d the  solid  Therefore, as  material  difficult  to  measure  accurately.  the emulsion s t a b i l i t y r e s u l t s could not be expressed  suggested  by Townsend et a l .  weights of the  cooked  room temperature thermal  was  l a y e r and the volume of  emulsions  were r e c o r d e d .  treatment  (1968).  Instead,  the  a f t e r d r a i n i n g and Assuming  final  c o o l i n g to  the weight l o s s during  was due s o l e l y to water  loss,  results  were  expressed  as per cent water l o s s of o r i g i n a l moisture content of  the  block  meat  and as per cent weight  performed i n d u p l i c a t e f o r each  loss.  This  test  was  formulation.  A f t e r p r e l i m i n a r y experiments the above cooking schedules were found to produce the l e a s t v a r i a t i o n w i t h i n r e p l i c a t e  samples.  3. Per cent weight l o s s a f t e r p r o c e s s i n g and storage P r o c e s s i n g weight l o s s as a percentage (Shrink) weight l o s s a f t e r  13  days  under vacuum packaged  and per cent storage (Vacuum  s h r i n k ) were c a l c u l a t e d using equation 20 and 21, r e s p e c t i v e l y .  4. Consumer cook  test  F r a n k f u r t e r s were weighed  and placed i n 500  mL b o i l i n g  water  for 5 min. The f r a n k f u r t e r s were removed, d r a i n e d w e l l and cooled to  room  weighed room  temperature i n p l a s t i c boats. and placed  These samples were  then  i n beakers covered with saran wrap and l e f t at  temperature f o r l e s s than 3 h  until  j u i c i n e s s and  texture  e v a l u a t i o n t e s t s were performed. Per  cent  weight  loss  a f t e r the consumer -74-  cook  test  (Cook  s h r ink,  %) was de £ i ned as : (weight before - weight a f t e r cooking) weight before cooking  5. J u i c i n e s s  of  Lee and P a t e l  the f r a n k f u r t e r s was  (1984)  cook  s l i c e s of 1 cm  test  were  (Instron  cm  axis  to  Corp.  was  Canton,  The sample was  The amount  determined  (Whatman No.  by  analyzed  determined  by  oven. to  MA)  was  f r a n k f u r t e r s per Universal  used to  measure  A  weighed  and compressed  the compression of f l u i d  the  using a  deformation r a t e of  head  expressed d u r i n g of  50  along i t s  was  reversed  compression  filter  The  fluid  collected  in  f o r moisture and f a t c o n t e n t .  paper  weight  l o s s of the  a mechanical c o n v e c t i o n  sample weight. in  the  Fat  was  filter  the f i l t e r  Moisture was  f i l t e r papers a f t e r horizontal a i r  flow  R e s u l t s were expressed as % e x p r e s s i b l e water  remaining  Testing  41) and expressed i n terms of % e x p r e s s i b l e f l u i d i n  papers was  in  Two  weight g a i n of two sheets  r e l a t i o n to sample weight.  70°C  determination.  (Model 1122)  flat plate.  deformation,  automatically.  this  the  under s i n g l e compression of the samples  used. 70%  for  An I n s t r o n  diameter u n i a x i a l  mm/min was  used  of  Samples from  i n length per f r a n k f u r t e r and two  expressible f l u i d 5  measured by the method  with some m o d i f i c a t i o n s .  f o r m u l a t i o n were used. Machine  (22)  evaluation  Juiciness  consumer  x 100  in  5 hrs  at  electric relation  estimated by the weight of dry matter paper  after  drying.  Results  expressed as % e x p r e s s i b l e f a t i n r e l a t i o n to sample weight.  -75-  were  6. Texture e v a l u a t i o n  6.1.  Texture p r o f i l e  Texture was  profile  performed  analysis analysis  (TPA) as d e s c r i b e d by Bourne (1978)  u s i n g an I n s t r o n (Model  Machine equipped  with a 500  1122)  kg load c e l l .  Universal  Testing  Two s l i c e s of 1 cm i n  l e n g t h per f r a n k f u r t e r and two f r a n k f u r t e r s per f o r m u l a t i o n from the  consumer  cook t e s t were used.  mm/min and c h a r t speed  of 200  A  mm/min were used.  compressed twice using a 5 cm diameter of  the f r a n k f u r t e r  was  cross-head speed  i n i t i a l height.  uniaxial  of 100  Each sample was flat  p l a t e to 75%  The output from the I n s t r o n  t r a n s m i t t e d d i r e c t l y to an IBM-compatible PC through a  acquisition Labtech  Acquire  Corporation, were  system  and software software  Wilmington,  (PC-LabCard  package.  MA).  m a n i p u l a t i o n t o the spreadsheet Development hardness  Co.,  ( f i r s t and second  springiness, calculated template  6.2 Shear  Cambridge,  and  software  MA).  peaks),  cohesiveness, from  Laboratory  The raw data  i n i t i a l l y s t o r e d as ASCII f i l e s  Technologies  (voltage  transferred  readings) f o r data  1-2-3" (Lotus  The t e x t u r a l  parameters of  fracturability,  profiles  and using  firmness,  chewiness a  were  Lotus  1-2-3  (see Appendix B ) .  force  Shear f o r c e was measured using a s i n g l e shear compression mounted  and  Lotus  gumminess  the t e x t u r a l  Model PCL-718  data  on  an I n s t r o n (Model  1122)  -76-  Universal  Testing  cell  Machine  equipped  with a 500  kg load c e l l .  consumer  cook t e s t were used.  Three f r a n k f u r t e r s from the  The ends were removed  from  each  f r a n k f u r t e r and a 3 cm long center core was removed using a No. 7 (1.3 cm i n t e r n a l diameter) cork b o r e r . A mm/min  and  expressed  chart  speed  of 200  as the maximum f o r c e  cross-head speed of 100  mm/min were used.  i n Newtons  required  through the c r o s s - s e c t i o n of the f r a n k f u r t e r s . system,  software and the Lotus 1-2-3  a l s o used i n t h i s  G. STATISTICAL  Shear  The  to  was shear  acquisition  template used f o r TPA were  test.  ANALYSIS  S t a t i s t i c a l a n a l y s i s was performed using SYSTAT, a s t a t i s t i c a l program  package  (Wilkinson,  1988a).  A l l tests  of s t a t i s t i c a l  s i g n i f i c a n c e were made a t the p r o b a b i l i t y l e v e l of ct = 0.05.  1. R e g r e s s i o n a n a l y s i s The  average  of  n  determinations  evaluated per r e p l i c a t e was used  per q u a l i t y  parameter  i n the r e g r e s s i o n a n a l y s i s .  data obtained were t r e a t e d by m u l t i p l e r e g r e s s i o n a n a l y s i s l e a s t squares methodology. Scheffe's model  canonical  fitted  Y = 0iXi  The using  The MGLH module of SYSTAT was used.  s p e c i a l c u b i c f o r 3 components  was the  t o data c o l l e c t e d a t each experimental p o i n t .  + 0 Xa 2  + OaXa  +  B a . 2 X 3 . X 2 + 0a. XxXa 3  + BaaXsXa  +  GxaaXiXaXa  (23) The (3 parameters a r e the p a r t i a l r e g r e s s i o n c o e f f i c i e n t s when  estimated  i n d i c a t e the e f f e c t s of the i n g r e d i e n t s -77-  which  on the  dependent (e.g.  v a r i a b l e (response).  0a.)  represents  mixture  component  (e.g.  (3 )  components  (e.g.  0X23) the  three  0±d  0u*  are  synergistic or  effect  ternary  (Cornell, Not  (quality can  has  be  the  In  this  evaluated test of  on  H :  study  used  the  antagonistic effect  effect  caused  by  1 9 8 1 ) . When  mixture  on  two  has  the  the  a  binary response  on  3 components A number  response  required  to  e a c h of  the  on of  w h i c h of t h e  the  of t h e  i n the  and  parameters  0 coefficient  i s not  a l t e r n a t e hypothesis  full  model  statistical  components  do  therefore  can  f o r each  f o l l o w i n g manner. A i n e a c h model,  value  SYSTAT. T h i s r a t i o  parameter  were  model.  reduction  individual  using  of t h e  and  tests  the  null  is  -78-  response  Student's  using  i t s standard  significantly different  0i. = 0 the  (Cornell,  of  subscript  effect  If i t i s negative  to determine  performed  estimated  computed  an  single  effect  ternary  parameters e v a l u a t e d ) .  from the  was  the  the  reponse.  effect  have a s i g n i f i c a n t eliminated  0  the  b i n a r y or  a  triple  blending  ( i . e . XiXsXa)  the  of  double s u b s c r i p t s  the  partial regression coefficients  describe  procedures  be  on  mixture  7  adequately  not  nonlinear  subscripts  1981).  all  responses  positive  effect  blending  (3 t e r m w i t h  components  single  0 terms w i t h  nonlinear  The the  blending  The  the  XxXa).  13 t e r m s w i t h  linear  Xi) .  represents  blending or  (e.g.  represent  ia  (i.e.  the  The  the error  hypothesis from  zero  t-  ratio was  H«: the parameter i s s i g n i f i c a n t l y d i f f e r e n t 0i  *  from  0  If  the c a l c u l a t e d value of  was  equal to or g r e a t e r than the c r i t i c a l value  where N was  | t | f o r the estimated  concluded  regression response  0  coefficient  (to.o»,m,««-P>) /  the t o t a l number of o b s e r v a t i o n s and p the number  parameters i n the model, the n u l l hypothesis was i t was  zero  t h a t the  coefficient  and  component tested  rejected.  of  Thus,  a s s o c i a t e d with the p a r t i a l  had a s i g n i f i c a n t  effect  on  the  t h e r e f o r e should not be removed from the model (Zar,  1984). The  approach  study to a r r i v e significant component  t h a t was at  the "best"  higher-degree term  terms  (e.g.  0iXi).  A f t e r one  taken to reduce  equation was  terms f i r s t  ( i . e . 0123X1X2X3)  012X1X2)  and  term was  the f u l l model i n t h i s to remove the  starting  remaining  hypothesis  was  for  each  coefficient  in  tested  model and  repeated u n t i l a l l the estimated  The  terms.  partial  the r e d u c t i o n  0 coefficients  to be s i g n i f i c a n t l y d i f f e r e n t  The  null  regression  procedure  was  i n the model were  from zero (Zar, 1984).  s i g n i f i c a n c e and adequacy of the q u a l i t y p r e d i c t i o n models  found were assessed by a n a l y s i s coefficient  of  determination  (Draper arid Smith, The  (e.g.  dropped from the model a new r e g r e s s i o n the  concluded  three  component  l a s t l y the one component terms  computed u t i l i z i n g  the new  the  followed by the two  equation was  then  with  least  of (R«x)  variance,  adjusted multiple  and a n a l y s i s of  residuals  1981).  n u l l hypothesis to be t e s t e d -79-  for regression analysis  for  and Snee, 1974; C o r n e l l ,  mixture models i s (Marquadt  Ho.* the response does not depend on the mixture  and  (3d = B = >  j = 1,2,...,q  BJ  j = q+l,...,p (other terms)  = 0  (linear  1981):  components  terms)  the a l t e r n a t e h y p o t h e s i s i s  H » : the response does depend on the mixture Definitional  and working  components.  formulas f o r m u l t i p l e  regression  a n a l y s i s of v a r i a n c e f o r mixture models are shown i n Table 4. The  F  Fisher  fit/mean  square  ratio pure  f o r lack of f i t (mean error)  statistically significant Smith, 1 9 8 1 ) . Ho:  there S l»c=K 1  was used to t e s t  lack of f i t (Deming,  the  lack  o£  models f o r  1989;  Draper and  F r a t i o t e s t s the n u l l h y p o t h e s i s  This  i s no s i g n i f i c a n t o£  square  f i t ~  S pur« !  lack of f i t =  arcox  0  where s* stands f o r v a r i a n c e , and the a l t e r n a t e h y p o t h e s i s i s H«: If  there i s s i g n i f i c a n t  the c a l c u l a t e d  than the c r i t i c a l rejected  value of F lack of f i t was equal to or  model  P  I f the F r a t i o appeared  mean  t o be adequate  value  and the n u l l  was t e s t e d based  square r e g r e s s i o n t o mean square r e s i d u a l  regression). value  appeared  f o r lack of f i t was not  r e g r e s s i o n f o r the mixture model of  greater  value (Fo.o»,ii),«- -i.i), the n u l l h y p o t h e s i s was  and i t was concluded that the model  inadequate. the  lack of f i t  The n u l l  (Fa.o»,(u,p-i,ii-p)  significant  hypothesis f o r on the F - r a t i o (F-test for  h y p o t h e s i s was r e j e c t e d a t <* = 0.05  of the F - r a t i o was equal to or g r e a t e r than  i f the  the c r i t i c a l  and i t was concluded t h a t the response -80-  t o be  Table 4. D e f i n i t i o n a l and working formulas f o r m u l t i p l e r e g r e s s i o n a n a l y s i s of v a r i a n c e f o r mixture models.*-  Source of variation  Degrees of freedom  Sum of squares(SS)  Mean square(MS)  Regression  p-1  SSR= E(y - y )*  SSR/(p-l)  Residual  N-p  SSE= Z(y^ - & ) *  SSE/(N-p)  Lack of fit  N-p-1  SSLF=  SSLF/(N-p-l)  Pure e r r o r Total  *• p N y y y„ y. a  (  = = = = =  1 N-1  SSE-SSPE  SSPE= El/2(y . - y^)* j  F test  MSR/MSE  MSLF/MSPE  SSPE/1  SST= E(y - y ) « K  number of parameters i n the model t o t a l number of o b s e r v a t i o n s p r e d i c t e d response value at the uth o b s e r v a t i o n o v e r a l l response average of N o b s e r v a t i o n s observed response value at the uth o b s e r v a t i o n and y^ = observed reponse value at the j t h design p o i n t f o r r e p l i c a t e 1 and 2 r e s p e c t i v e l y  Source: Deming (1989), Draper and.Smith and Marquadt and Snee (1974)  -81-  (1981), C o r n e l l  (1981)  depended on the mixture The  adjusted  determined  components ( C o r n e l l , 1981).  multiple  coefficient  of  determination  was  as f o l l o w s : R**. = 1 - [ ( S S E / ( N - p ) ) / ( S S T / ( N - l ) ) ]  This  statistic  has been used  that  do not c o n t a i n a Bo  term)  variation  i n the response  statistic  takes i n t o account  fitted  for canonical  (24)  polynomials  as a measure of how  2  values i s e x p l a i n e d by the model. the number of parameters  adjusted multiple c o e f f i c i e n t  better  error  of e s t i m a t e ,  Y on X.  of  the model f i t s the observed  with which the f i t t e d of  (p)  This i n the  the standard e r r o r of the estimate were a l s o  i n t o c o n s i d e r a t i o n i n d e c i d i n g the "best" e q u a t i o n .  the  much of the  model.  The R A . and  model's  (models  The  S,  i s an o v e r a l l  The  taken  closer a  d e t e r m i n a t i o n i s to 1, responses.  The  standard  i n d i c a t i o n of the  accuracy  r e g r e s s i o n equation p r e d i c t s the dependence  s m a l l e r the value of S, the more a c c u r a t e w i l l  be  the p r e d i c t i o n s (Zar, 1984). The  models  residuals.  were  SYSTAT  further performs  checked the  i n d i c a t e s i f there are o u t l i e r s . corresponding  observations  using  analysis  the of  analysis  of  residuals  and  When o u t l i e r s were found,  the  from the 2 r e p l i c a t e s  were  removed  from the data, a f t e r which the data were reanalyzed without  these  o b s e r v a t i o n s . R e s i d u a l s were analyzed by examining  plots  residual  ( r e s i d u a l s vs. p r e d i c t e d v a l u e s ) with the GRAPH module of SYSTAT. The assumption independent  and  for residual analysis  i s that  f o l l o w a normal d i s t r i b u t i o n -82-  the  residuals  are  ( C o r n e l l , 1981). The  r e s i d u a l s should be seen  to be  randomly d i s t r i b u t e d about  However,  i t isdifficult  (Cornell,  1981). I f the p l o t s show a d e t e r m i n i s t i c  the  dispersion  then  (Joglekar  2. C o r r e l a t i o n Pearson's  such d i s t r i b u t i o n  with  pattern  of the r e s i d u a l s changes with the p r e d i c t e d  the assumption  violated  t o see  underlying  the a n a l y s i s  of  zero. N<30 or i f values  residuals  is  computed  to  and May, 1987).  analysis correlation  coefficients  (r)  were  compare the r e l a t i o n s h i p between the proximate composition of the formulations  and  the q u a l i t y parameters evaluated  and  each p a i r of q u a l i t y parameters e v a l u a t e d . CORR module was  between  of  S Y S T A T  used.  3. Response s u r f a c e Contour p l o t s using  contour  of estimated  the p r e d i c t i o n  obtained using  analysis  models  SYGRAPH  a  response found.  graphics  values were The  contour  program package  constructed plots  were  (Wilkinson,  1988b).  H. OPTIMIZATION METHODS  I. F e a s i b l e To  start  requires  point  computer program  (FPOINT)  the search f o r the optimum,  an i n i t i a l  point  the Complex  that does not v i o l a t e the  In problems with complicated and numerous c o n s t r a i n t s -83-  method  constraints. i t i s often  very  difficult  Vaessen  (1984)  determine This is  to e s t a b l i s h a f e a s i b l e p o i n t . described  a f e a s i b l e point  f u n c t i o n i s the sum violated  should  be  does the  not make use f u n c t i o n can  simplex based  Minimization  i s found. Any  be used  (Box,  (CSO)  38  and  39  implicit  l i m i t s of the  When  value  of  f i r s t derivatives  Arteaga,  of Morgan and  to allow  which s t a r t s  c o n s t r a i n t s are  1990)  of  which i s  Deming (1974)  was  than  f o r the simplex  i n l i n e 46 was  involved  (upper  to  found. modified.  i n the  s i n c e the Simplex a l g o r i t h m  f a c t o r s ) through the  (Nakai and  more and  a  o p t i m i z a t i o n technique which  not v i o l a t i n g the e x p l i c i t c o n s t r a i n t s  routine  function  when a response f u n c t i o n of zero was  f u n c t i o n computation, of  constraint  this  f u n c t i o n has  (Nakai and  were modified  f u n c t i o n subroutine  Only the  problems.  f o l l o w i n g changes:  stop the search 2) The  of  to  1965). In t h i s study computational  on the Simplex a l g o r i t h m  Lines  lower  point.  and  used  for constraint optimization  of the c o n t i n u i t y of the  optimization  used with the 1)  o b j e c t i v e f u n c t i o n to be  so t h a t when the  zero, a f e a s i b l e p o i n t  (1965)  of the amounts by which each  by a given performed  the  Box  objective takes and  care lower  prohibit-range-trespassing  Arteaga, 1990).  one  s t a r t i n g p o i n t was  upper l i m i t s of the  f a c t o r s or  required, implicit  either  the  constraints  were narrowed. A flow c h a r t of the The  f u n c t i o n subroutine  computer program i s l i s t e d  i n Appendix  -84-  i s shown i n F i g u r e C.  5.  Subroutine  starts i  1  Function  3  = 0  Compute c o n s t r a i n t * value at a c t u a l vertex  Is c o n s t r a i n t v a l u e h i g h e r t h a n upper l i m i t of c o n s t r a i n t i  No.  Yes Function  = Function  + c o n s t r a i n t value  - upper  Is c o n s t r a i n t v a l u e lower t h a n lower l i m i t of c o n s t r a i n t i ,  limit  No  Yes st_  Function  No  = Function  + lower  limit  - c o n s t r a i n t value  Last constraint  Yes  * Minimize Figure  Function  5. F l o w c h a r t o f t h e F u n c t i o n computer p r o g r a m -85-  subroutine  o f FPOINT  2. Formula o p t i m i z a t i o n computer  program  (FORPLEX)  A computer program of the Complex method of Box was w r i t t e n i n IBM  BASIC  (1973).  based  on the FORTRAN program  of  Kuester  and  M o d i f i c a t i o n s to t h i s program were performed to  i t s e f f i c i e n c y and i t s output and to overcome  Mize  improve  limitations  of the  program and of the o r i g i n a l Complex method. The l i t e r a t u r e review ( s e c t i o n C.2.)  g i v e s complete d e s c r i p t i o n of the Complex method.  The formula o p t i m i z a t i o n computer program  (FORPLEX)  f i n d s the  maximum of a n o n l i n e a r f u n c t i o n of one or more v a r i a b l e s , to e x p l i c i t  and  constraints.  If  implicit  (linear  minimization  of  or  nonlinear)  the  objective  r e q u i r e d , maximization of the negative of the  subject  inequality function  function  is  (max. of  - f ( x ) ) should be performed. F i g u r e 6 shows a d e t a i l e d  flow chart of the FORPLEX a l g o r i t h m .  The computer  program i s l i s t e d  2.1. Program  description  2.1.1. General The  description  program  subroutines subroutines  consists  (CONSX,  of  CHECK,  a  main  program,  CENTR)  and  two  three user  general supplied  (FUNCTION and CONSTRAINT).  (A) Main program (a)  i n Appendix D.  functions  E s t a b l i s h m e n t of the dimension of the f o l l o w i n g which can be modified a c c o r d i n g  parameters  to the requirements of each  p a r t i c u l a r problem: X(K,M), R(K,N), F ( K ) , G(M), H(M), XC(N). -86-  f S t a r t "\ 1 /Feasible  Generate  s t a r t i n g point 1 initial  /  complex Yes  Move p o i n t i n a d i s t a n c e DELTA inside the violated constraint  Explicit constraints violated  No  J  K—  No Init i a l complex generated  Implicit constraints violated  X  Yes  Yes  Evaluate objective f u n c t i o n a t complex points  F i n d w o r s t and b e s t p o i n t s i n complex Yes Check c o n v e r g e n c e criteria Move 1/2 way towards t h e centroid  No  Calculate a l l points  c e n t r o i d of except worst  I  R e f l e c t worst point through centroid  Figure  6. F l o w  chart  o f t h e FORPLEX  -87-  algorithm  No  1  Yes \.  Explicit constraints violated  Move p o i n t i n a d i s t a n c e DELTA inside the v i o l a t e d constraint  NO  Move 1/2 way towards the centroid  > *  Yes  Implicit constraints violated  No  No  r  Evaluate  objective  function  New p o i n t r e p e a t s as worst p o i n t  Has i t b e e n contracted 5 o r more times  No Yes  >  G e n e r a t e a new p o i n t by r e f l e c t i n g the centroid through the best point  -88-  (b)  interaction information  with  the user  to o b t a i n  the  following  t o s t a r t the o p t i m i z a t i o n :  (1) number of independent v a r i a b l e s  (factors) involved  i n the  o b j e c t i v e f u n c t i o n computation. (2) t o t a l number of c o n s t r a i n t s (3) complex s i z e (4)  maximum  number  iteration  of  iterations  i s defined  as  s e l e c t a f e a s i b l e point the  performed.  the c a l c u l a t i o n s  An  required  that does not repeat  worst f u n c t i o n value.  initial  t o be  to  in yielding  I t e r a t i o n s are counted  after  complex i s computed.  (5) p r i n t i n g procedure (6) convergence c r i t e r i a v a l u e s : ALPHA, BETA and GAMMA (7) lower and upper l i m i t s of f a c t o r s (8) lower and upper l i m i t s of i m p l i c i t  constraints  (9) f e a s i b l e s t a r t i n g p o i n t (c) Generation of random numbers (d) P r i n t o u t of i n f o r m a t i o n (e) A l l i n f o r m a t i o n  entered  by the user  gathered i s t r a n s f e r r e d t o the other  subroutines (f)  P r i n t o u t of the f i n a l value coordinates  of the f u n c t i o n ,  and the t o t a l number of  the f a c t o r s  iterations  s o l u t i o n has converged t o w i t h i n the a l l o w a b l e  when the  range  (BETA),  or when the maximum number of i t e r a t i o n s i s exceeded The IC,  parameters d e f i n e d  i n main program are:  N, M,  IPRINT, ALPHA, BETA, GAMMA, DELTA, I, G, H, X, R -89-  K,  ITMAX,  (B) Subroutine CONSX CONSX program  i s the primary s u b r o u t i n e . and  FUNCTION,  coordinates  It i s called  a l l other s u b r o u t i n e s  CONSTRAINTS).  The  from the  (CHECK,  computations performed  main  CENTR, in  this  s u b r o u t i n e are as f o l l o w s : (a) computes i n i t i a l  complex  (b) p r i n t o u t of i n i t i a l  complex  (c) f i n d s p o i n t s with worst and best (d) checks convergence  computes c o n t r a c t i o n  (g) f i n d s  value  criteria  (e) computes r e f l e c t i o n (f)  function  point point  i f r e f l e c t i o n or  contraction  p o i n t s repeat as  worst  points (h)  computes new p o i n t by r e f l e c t i n g the c e n t r o i d best  (i)  of  coordinates printout message  function  values,  a t each i t e r a t i o n of f u n c t i o n  when  BETA, LLY,  and  centroid  desires  factors  coordinates  and  no improvement i s observed by r e f l e c t i n g  a  the  point  parameters used i n CONSX are: GAMMA,  factors  i f user  value,  c e n t r o i d through the best The  the  point  printout  (j)  through  N, M, K, ITMAX, IT, ALPHA,  DELTA, X, I, R, F, I E V l , IEV2, G, H,  XC,  IPRINT,  CC, CNT, TT, KODE, K l , KOUNT  (C) Subroutine CHECK This  subroutine  checks  against  explicit  and  implicit  c o n s t r a i n t s and a p p l i e s c o r r e c t i o n i f v i o l a t i o n s are found. -90-  (a)  i£ e x p l i c i t c o n s t r a i n t s are v i o l a t e d , distance  (b)  moves p o i n t s  in a  DELTA i n s i d e the v i o l a t e d c o n s t r a i n t  i f i m p l i c i t constraints are v i o l a t e d ,  contracts  the p o i n t  halfway towards the c e n t r o i d . The parameter used i n CHECK a r e :  N,  M, K, X, G, H, I, IEV1,  KODE, XC, DELTA, KT, TT, CC (D) Subroutine CENTR This the worst  s u b r o u t i n e c a l c u l a t e s the c e n t r o i d of a l l p o i n t s  except  point.  The parameters used i n CENTR a r e : N, IEV1, XC, X, K l (E) Subroutine FUNCTION This function  is a  user  supplied  subroutine  where  the  objective  is specified.  The parameters used i n FUNCTION a r e : X, F, I . (F) Subroutine CONSTRAINT This  is a  user  supplied  subroutine  where  the  implicit  (optimization  factors)  c o n s t r a i n t s are s p e c i f i e d . The parameters used i n CONSTRAINT a r e : X, I.  2.1.2. D e s c r i p t i o n  of parameters  N  Number of independent v a r i a b l e s  M  T o t a l number of c o n s t r a i n t s  K  Number of p o i n t s  ITMAX  Maximum number of i t e r a t i o n s  IC  Number of i m p l i c i t  ALPHA  Reflection factor  ( v e r t i c e s ) i n the complex  constraints  -91-  BETA  Convergence parameter. Maximum allowed between best and  GAMMA  difference  worst response values  Convergence parameter. Consecutive i t e r a t i o n s with "same" response value  DELTA  E x p l i c i t constraint violation  IPRINT  Code used to c o n t r o l p r i n t i n g . IPRINT=1 causes a l l i t e r a t i o n s to be p r i n t e d , until  X  correction  IPRINT=0 suppresses p r i n t i n g  f i n a l s o l u t i o n i s obtained  Optimization  f a c t o r s and  c o n s t r a i n t s depended  variables R  Pseudo-random numbers between 0 and  F  Objective  IT  I t e r a t i o n number  IEVl  Point  number with worst f u n c t i o n  IEV2  Point  number with best  G  Lower l i m i t  of  constraints  H  Upper l i m i t of  constraints  1  function  function  value value  XC  Centroid  I  Point  KODE  Code used to determine i f i m p l i c i t c o n s t r a i n t s  number  p r o v i d e d . KODE=0 when no  implicit constraints  p r o v i d e d , KODE=l when i m p l i c i t  constraints  are are  are  provided Kl  Do  loop  limit  KT  Code used to d e s c r i b e  i f implicit  constraints  v i o l a t e d . KT=0 no v i o l a t i o n occurs, -92-  KT=1  are  violation  occurs TT  Code used t o determine  i f c e n t r o i d needs t o be  computed. TT=1 t o be computed, TT=0 not t o be computed CC  Number of c o n t r a c t i o n s performed  LLY  Number of c o n t r a c t i o n s performed when p o i n t repeats as worst  i n each  iteration  point  Code used t o determine i f r e f l e c t i o n through the best  CNT  p o i n t has been performed KOUNT  code used t o check convergence  criteria  2.1.3. Summary of user requirements (A)  Determine  values f o r N,  M,  K, ITMAX,  IC, ALPHA,  BETA,  GAMMA, DELTA, IPRINT Recommendations f o r s p e c i f y i n g the f o l l o w i n g  parameters  K = 2N ALPHA =1.3 BETA = small number, magnitude of f u n c t i o n value times 1 0  - 4  GAMMA = 5 DELTA = small number, magnitude of f a c t o r values times 10 ~* (B)  Specify  upper and lower l i m i t s f o r e x p l i c i t  constraints. If a particular  and i m p l i c i t  f a c t o r or c o n s t r a i n t has no upper or  lower l i m i t , the user should provide a reasonable estimate of the 1imit. (C)  Determine  initial  feasible  starting  point  u s i n g FPOINT  computer program. (D) A d j u s t DIM (dimension) statement as necessary ( l i n e 20) -93-  (E)  Specify  optimization and  objective  f u n c t i o n i n FUNCTION  f a c t o r s should  be d e s c r i b e d  the dependent v a r i a b l e being  (F)  Specify  implicit  optimized  constraint  subroutine.  Implicit  constraints  optimization  factors.  The  described  X (I,factor  as  as  X(I factor  The  number)  f  as F ( l ) .  functions are  in  CONSTRAINT  functions  optimization  number)  subroutine.  and the  factors  of  the  should  be  implicit  constraint  dependent v a r i a b l e s as X ( I f a c t o r numbers + 1) f  2.1.4. New  routines  2.1.4.1. Generation of random numbers Every  pseudo-random  random draws Seeding  number  generator produces a  that e v e n t u a l l y repeats  In IBM BASIC seeding  can be done using two  (1) RND(A) where A i s a negative A i s a number between -32768 used s i n c e seeding numbers  order each and  1984).  functions:  number and (2) RANDOMIZE A where  and 32767.  RANDOMIZE f u n c t i o n  was  with RND(A) gave i d e n t i c a l sequences no matter In t h i s study RANDOMIZE 3 was used  in  t o generate the same sequence of pseudo-random numbers  in  run. thus  were used.  This allowed f o r mistakes t o be followed  correction  future a p p l i c a t i o n s , be  al.,  of  must always be done to generate a p a r t i c u l a r sequence of  numbers.  what  (Modianos et  series  used  to  reseed  program i s run.  more  easily  to the program to be made i f r e q u i r e d .  For  RANDOMIZE and RANDOMIZE TIMER f u n c t i o n s can the random number generator  each  In t h i s study the term "random number" -94-  time  the  was used  to denote pseudo-random numbers.  2.1.4.2. R e f l e c t i o n through best The Friedman  modification (1971)  Kuester and This through  the  reflection  best  XIH  Limitations  2.2.1. E q u a l i t y  algorithm;  variable  (1972)  was  Xu(best),  added to  the  the  after a fixed  centroid  are  reflection point,Xij,  i s defined  + oi(Xi.d (best)  Xi.d(best)  X  number  repeat  as  ,  the worst  by:  - X )  formula o p t i m i z a t i o n  i c  of  performed when  points  of the  by  centroid,  subsequent c o n t r a c t i o n  =  done  algorithm.  point,  (25)  l o  algorithm  constraints  constraints instead,  constraints (Vaessen,  incorporated  cannot  be  handled  they have to be can  T h i s can  linear +  X  2  as  removed.  be removed by using  1984).  Xi  be  Pinder  method  c o n s i s t s of r e f l e c t i n g the  f o l l o w i n g example. The  can  o r i g i n a l Complex  Friedman and  (5) towards and  Equality  the  Mize Complex  p o i n t s . T h i s new  equality  and  modification  contractions  2.2.  to  point  be b e t t e r  such  by  Simple  them to explained  this linear  remove with  a the  constraint +  X  3  = 1  by r e p l a c i n g i n the  o c c u r r e n c e s of the v a r i a b l e Xa with the 1 - Xa. - Xa  -95-  (26) objective  function  all  expression (27)  2.2.2. E n d l e s s c o n t r a c t i o n Box to  (1965)  commented  violation  of  necessary u n t i l all  the  point  on  the  the  line  step  should  constraints, eventually  algorithm  d o e s not  a s o l u t i o n to t h i s  be  done.  In  "breaking"  2.3.  endless this  The  objective first  program) function  tested  problems author  using  reported  were u s e d .  were v e r i f i e d  with  (LINDO s o f t w a r e  Ctrl  of  suitability  computer  case the  (pressing  Optimization  computation  C or  search Ctrl  constrained  of  the  to  locate  the  i n the The  Break  literature  and  (LINDO S y s t e m s ,  objective  f u n c t i o n - l i n e a r l y constrained  surface  analysis  constrained  p r o b l e m s were  nonlinear used.  -96-  with  the  If  with  this  point  will by  program.  (i.e.,  linear  and  FORPLEX nonlinear  constrained  models.  the by  i n the  case  problems,  objective  by  FORPLEX  linear  or  was  Mathematical  some f o r m u l a t e d  given  Inc.))  rule  models  linearly  mathematical  this  terminated  keys) the  of  chance  optimization  contraction be  as  reflection  problem.  Complex method  were  optimum r e s u l t s  the  formula  should  r e s u l t s obtained  when  the  optimum  that  constrained  The  mathematical  modified  problems  of  i f by  coincide  the  occurs  search.  to  thus t e r m i n a t i n g  problem  to  due  repeated  a p p l i c a t i o n of  centroid,  provide  be  centroid  violation  performed  i s f o u n d . However,  from t h e  point  constraints  contraction  constraints  implicit  projected  implicit  the  f e a s i b l e point  v i o l a t e the  causes  to  that  implicit  a  points  due  the  program  programming of by  linear response  function-linearly  2.4.  Optimization The  of f r a n k f u r t e r  formula o p t i m i z a t i o n  to o b t a i n product  "best  quality"  specifications  i n g r e d i e n t l e v e l s and The  computer program  formulations for:  (a)  that  proximate  (FORPLEX) met  the q u a l i t y of f r a n k f u r t e r formulations  formulations  constraints a  predetermined  composition,  by f i t t i n g  and  that met  target  optimization  predict  at each p o i n t computed were d e f i n e d  the product s p e c i f i c a t i o n s and  whose p r e d i c t e d  predetermined  formulation  q u a l i t y was quality.  as c l o s e as Several  t r i a l s were performed using  f o r each o p t i m i z a t i o n  absolute  The value  f u n c t i o n subroutine  overall  response  was  predicted  The  l i n e a r equations that d e s c r i b e d  and  the c o n s t r a i n t that the sum  entered was  incorporated  The trial  the  minimization  of  the  d i f f e r e n c e s of  q u a l i t y parameters from the t a r g e t q u a l i t y v a l u e s . proximate composition,  of the  i n t o the c o n s t r a i n t s u b r o u t i n e .  considered  quality.  of the FORPLEX computer  of the products of the s t a n d a r d i z e d  the  cost  different  q u a l i t y p r e d i c t i o n equations r e q u i r e d i n t o the  as  frankfurter  formulations'  were entered  by  possible  q u a l i t y parameters as measures of the  program.  (b)  Scheffe's  were used to  the FORPLEX program. "Best q u a l i t y " formulations  to  used  s p e c i a l c u b i c model to the experimental data c o l l e c t e d  the p o i n t s of an extreme v e r t i c e s design  those  was  (c) c o s t .  q u a l i t y p r e d i c t i o n equations obtained  canonical at  formulations  as  a  c o n s t r a i n t the  i n the c o n s t r a i n t  i n g r e d i e n t s equal When a q u a l i t y prediction  subroutine.  -97-  cost 1 were  parameter  equation  was  This  new  programming  formula  for formula  optimum f o r m u l a t i o n s formulations  o p t i m i z a t i o n method was  found  optimization.  found  was  by l i n e a r programming. The  composition by  the  of  and  by  formulations  used  to  t h a t met  quality.  program  The  find  seven  the  specifications same  were  used.  for  ingredient  The  least-cost  total  proximate  limits  and  formulations bind  used to account for the q u a l i t y r e q u i r e d  value by  the  i n terms of the amount of f a t bound per u n i t weight  formulation.  provided  formulations  programming  c o n s t r a i n t s used for the o p t i m i z a t i o n of  was  formulations  was  formulations  FORPLEX  constraint  made  formulations.  programming  composition  optimum  of the models to p r e d i c t  3. Formula o p t i m i z a t i o n using l i n e a r  frankfurter  computed  formulations.  computed optimum  performed to t e s t the accuracy  Linear  5  linear  comparison was  c o s t of the computed  v e r i f i c a t i o n of two  q u a l i t y of these  In a d d i t i o n ,  by FORPLEX were compared with  i n terms of the q u a l i t y and Experimental  compared with  Bind  Sebastian were  value constants (1989).  compared  with  The  of the seven  ingredients computed  optimum f o r m u l a t i o n s  were  optimum found  by  FORPLEX as d i s c u s s e d above. All  l i n e a r programming c a l c u l a t i o n s were performed using LINDO  software  (LINDO Systems, I n c . ) .  -98-  RESULTS AND  A. OPTIMIZATION  DISCUSSION  OF CONSTRAINED MATHEMATICAL MODELS  The performance of the m o d i f i e d Complex method computer program)  in  handling  linear  (i.e.,  FORPLEX  and n o n l i n e a r o b j e c t i v e  f u n c t i o n problems that were l i n e a r l y c o n s t r a i n e d was t e s t e d . effect  of u s i n g d i f f e r e n t s t a r t i n g p o i n t s ,  random number  and |3 values on the e f f e c t i v e n e s s of l o c a t i n g the optimum Complex was e v a l u a t e d . formulated  Published  The  parameters  follows:  In the f o l l o w i n g  r e f e r s to the Complex method,  "complex" r e f e r s to the f l e x i b l e specified  seeds by the  mathematical problems and some  by the author were used.  the term "Complex"  The  unit  discussion  while the term  ( i . e . search  unit).  a t each o p t i m i z a t i o n run  were  as  (a) the convergence parameters ALPHA ( a ) , BETA (|3) and  GAMMA  ( r ) were s e t a t 1.3, 0.1 and 5 r e s p e c t i v e l y . In some cases,  BETA  was  accuracy  set of  a t 0.001  the r e s u l t s ;  complex was s e t random number  to K=2N,  seed used  i n order  to demonstrate  and (b)  the number  where N i s the number was  3  -99-  the  of p o i n t s  improved i n the  of f a c t o r s .  unless otherwise s p e c i f i e d .  The  Test problem 1 Test  problem  constrained  1 is a  nonlinear  objective  function-linearly  problem.  Objective  f u n c t i o n : Maximize  Y= Xl*X2*X3  Subject t o : Explicit constraints:  0 < XI < 42 0 < X2 < 42 0 < X3 < 42  Implicit This  constraint:  problem was t e s t e d by Kuester  performance equal  of  reported  problem  was  by Kuester  random number  (Table  6).  locate  the  generator  the  (1973) global  (1973) with  regardless  random  number  4 different  sequence  random number  5 the Complex was  of the i n i t i a l  starting  seeds  able  to  point.  of i t e r a t i o n s needed to a r r i v e at the optimum starting point.  Table  6 shows the e f f e c t  seeds on the performance of the  All  out  carried  of i t e r a t i o n s  of  Complex.  using the same s t a r t i n g p o i n t ( X l = l ,  X2=l, X3=l). Convergence to the optimum was achieved The number  maximum  (Table 5) as w e l l as seeding  d i f f e r e n t random number runs were  f o r the  X2=12.007 and X3=12.014.  s o l v e d using the  and Mize  optimum  on  /  As can be seen i n Table  However, the number depended  and Mize  the Complex a l g o r i t h m and has a  to 3455.99 a t Xl=23.958  This  the  0 < XI + 2X2 + 2X3 < 72  needed  to  i n f l u e n c e d by the random number seed.  -100-  arrive  at  i n a l l runs.  the optimum was  Table  Run No.  5. O p t i m i z a t i o n  results  of t e s t problem  Starting point  l . * " ' B  c  Computed optimum  XI  X2  X3  1  1  1  1  3455.93  23.95  11.97  12 .06  106  2  10  10  10  3455.98  24.03  12.00  11.98  118  3  20  10  10  3455.91  24.08  11.94  12 . 02  4  15  10  10  3455.94  24 .06  11.97  12.00  111  5  5  10  10  3455.99  23.98  12.00  12.02  90  Ymax  X2  XI  *• Convergence parameters: ct = 1.3, (3 = 0.1, r = 5 Number of p o i n t s i n complex K=6 Random number sequence of Kuester and Mize Number of i t e r a t i o n s B  c D  -101-  (1973)  X3  IT  D  54  Table  Run No.  6. O p t i m i z a t i o n r e s u l t s random number seeds  Random number seed  of t e s t problem 1 using ' B  different  c  Computed optimum Ymax  XI  X2  X3  IT  1  3  3455.97  24 . 05 12.00  11.97  112  2  300  3455.97  24 . 00 12.00  12.00  54  3  547  3455.98  24.03  12.01  11.97  85  4  183  3455.97  23.99  12.03  11.98  73  *• S t a r t i n g f e a s i b l e p o i n t : X l = l,X2 = l X 3 = l Convergence parameters: a = 1.3, (3 = 0.1, r=5 Number of p o i n t s i n complex K=6 Number of i t e r a t i o n s /  B  e  D  -102-  D  Test problem 7 Test  problem  constrained Objective Subject  2  is  a  linear  function:  Maximize  is  optimum Y=19  a  starting  +  5X2  0 < XI  < 4  0 < X2  < 3  i s at Xl=2 and  problem,  X2=3 (Wolfe and  i n Table  7.  The  X2=2  and  was  X2 = l  allowed small  as s t a r t i n g p o i n t .  values was upon the  (0.001)  attained, exact  between  for the  thus i n c r e a s i n g  location  found  i n run  the  Koelling 3  and  5  stalled optimum. 2 using  optimum u n t i l  best and  0.001 a  worst response  the chance of convergence  of the optimum.  improve the r e s u l t s obtained  towards the  4  were  S e t t i n g the |3 parameter to  the Complex to keep s e a r c h i n g difference  in runs 1,  In run 6 the complex was  By s e t t i n g (3 to 0.001, the true optimum was  with  s t a r t i n g points  using (3 = 0.1  not able to progress  true  K o e l l i n g , 1983).  l o o k i n g at the f i g u r e r e p o r t e d by Wolfe and  were c l o s e to the t r u e optimum.  Xl = l ,  whose  r e s u l t s of t e s t problem 2 obtained  Optimum r e s u l t s obtained  at Xl=4,  <8  l i n e a r programming  p o i n t s are given by  (1983).  2X1  0 < X l + 2X2  simple  optimization  chosen  Y=  to:  Implicit constraint:  The  function-linearly  problem.  Explicit constraints:  This  objective  However t h i s  d i d not  when the s t a r t i n g p o i n t was  Xl=3 and  X2 = 2. The  number  optimum depended  of f u n c t i o n e v a l u a t i o n s needed to a r r i v e on  the s t a r t i n g p o i n t used, -103-  and  at  the  as expected,  Table 7. O p t i m i z a t i o n  Run No.  D  X2  1  Computed optimum e  1  18.91  2 .14  2.93  21  1  1  19 . 00  2.00  3.00  65  3  2  18.96  2.07  2.96  19  3  2  18.96  2.07  2.96  25  5  1  3  18 . 89  2.18  2.91  15  6=  4  0  18 .00  4 . 00  2.00  39  D  4  B  XI  B  X2  3  B  S t a r t i n g point  A  XI  2  c  t e s t problem 2 . '  Ymax  1  A  r e s u l t s of  D  I T  Convergence parameters: a=1.3, (3 = 0.1, r = 5, unless otherwise indicated Number of p o i n t s i n complex K=4 Number of i t e r a t i o n s Convergence parameters: a = 1.3, 13 = 0.001, r = 5 Search was s t a l l e d  -104-  decreasing  (3  the  function  value  evaluations  caused  (i.e.  an  increase  iterations)  i n the  number  needed to  reach  of the  opt imum. Test  problem 3  Test  problem  constrained  is  a  linear  function:  Maximize  Y=  2:  i s l o c a t e d at the  c o n s t r a i n t s at Xl=44 and Convergence  when  11X2  0 < XI  < 60  0 < X2  < 40  2X1  X2  <  104  XI + 2X2  <  76  +  (Table (3 = 0.001  to 8), was  the  i n t e r s e c t i o n of the  X2=16 (Nakai and optimum was  two  implicit  four  starting  however accuracy of the r e s u l t s was  improved  used i n runs 2 and  i t e r a t i o n s to a r r i v e at the  needed 27  and  caused an  increase  23  whose true  Arteaga, 1990).  achieved using  needed 30  the  +  i s another simple l i n e a r programming problem,  optimum Y=440  points  6X1  constraints: 1:  This  function-linearly  to:  Explicit constraints:  Implicit  objective  problem.  Objective Subject  3  iterations,  5.  At |3 = 0.1  runs 3 and  optimum while runs 1 and  respectively.  S e t t i n g (3 to  i n the number of i t e r a t i o n s needed to  optimum.  -105-  4 6  0.001 locate  Table  Run No.  8. O p t i m i z a t i o n r e s u l t s  of t e s t problem  Starting point  3.*"  B  Computed optimum  XI  X2  Ymax  20  20  439 . 47  44.03  15.93  27  20  20  440.00  44.00  16.00  60  3  0  0  439 .97  43.94  16.03  30  4  15  20  439.99  43.97  16.01  30  15  20  440.00  44.00  16.00  48  30  10  439.92  43.85  16 .08  23  1 2  5 6  B  D  XI  X2  IT  C  Convergence parameters: a=1.3, 13 = 0.1, r=5, unless otherwise indicated Number of p o i n t s i n complex K=4 ° Number of i t e r a t i o n s Convergence parameters: a = 1.3, (3 = 0.001, r=5 B  D  -106-  Test problem 4 Test problem 4 i s a simple frankfurter Arteaga  formulation.  (1990)  formulation  to  l e a s t - c o s t f o r m u l a t i o n problem of a  This  explain  problems.  problem was used by  the  The  application  original  of  problem has  Nakai  and  in  meat  LP  been s l i g h t l y  modified. The problem i s the f o l l o w i n g : formulation X2=beef  is  flanks,  formulation  sought  using  X3=pork l o i n ,  the most economical f r a n k f u r t e r 5  ingredients  meat content  line  i s 100 kg/day  must be a t l e a s t 65%  may be no more than 28%  4. p r o t e i n content  must be g r e a t e r or equal  5. mutton meat content 6. lean pork content Mathematically  to 11%  may be no more than 9%  must be g r e a t e r or equal  to 30%  the problem i s s t a t e d as f o l l o w s :  function:  Minimize  Y= 2.5X1 + 1.5X2  Subject t o : Explicit  The  i s s u b j e c t to the f o l l o w i n g c o n s t r a i n t s :  3. the f a t content  Objective  fronts,  X4=pork f a t and X5=mutton).  1. the c a p a c i t y of the prod u c t i o n 2. t o t a l  (Xl=beef  constraints:  0 < XI < 100 0 < X2 < 100 30 < X3 < 100 0 < X4 < 100 0 < X5 <  -107-  9  + 4X3 + 0.7X4 + X5  Implicit  constraints:  1. c a p a c i t y :  XI + X2 + X3 + X4 + X5 = 100  2. meat c o n t e n t : 3. f a t content:  65 < XI + X2 + X3 + X5 .047X1 + .221X2 + .075X3 + .947X4 + .198X5 < 28  4. p r o t e i n c o n t e n t : l l < .221X1 + .16X2 + .207X3 + .009X4 + .176X5 As mentioned Complex  i n Materials  cannot work  constraint  with  and Methods s e c t i o n G.2.2.I.,  equality constraints.  was removed by r e p l a c i n g  The  the  equality  the v a r i a b l e X5  with  the  express ion 100 - XI - X2 - X3 -X4 i n the o b j e c t i v e implicit and  constraint  upper  algorithm and  f u n c t i o n and i n c o r p o r a t i n g (replacing  l i m i t s of 0  implicit  l i m i t s of c o n s t r a i n t s , was  set  a t 100,  constraint  3  was  set  a t 0 and the  linear  the lower  limit  upper  limit  the  Complex  both lower  for  implicit  for  implicit  of  implicit  4 was s e t at 100.  optimal s o l u t i o n of t h i s problem was found using programming computer program.  LINDO  The minimum value of Y  208.94 a t X1=0, X2=46.56, X3=30, X4=14.44 and X5=9. lies  with lower  requires  the upper l i m i t  2  The  1)  Since  program)  constraint  constraint  constraint  and 9 r e s p e c t i v e l y .  ( i . e . FORPLEX computer  upper  t h i s e x p r e s s i o n as an  a t the i n t e r s e c t i o n of three c o n s t r a i n t s  a is  This optimum  (X3=30,  X5=9  and  f a t content=28 ) . The Table  optimization 9.  significant  The  r e s u l t s of t h i s t e s t problem are  factor  values  have been  rounded  f i g u r e f o r ease of p r e s e n t a t i o n . -108-  given  off  The four  to  in one  starting  Table 9 . O p t i m i z a t i o n  Run No,  Starting  r e s u l t s of t e s t problem 4 . * "  point  B  Computed optimum  XI  X2  X3  1  30.6  30.6  30.6  0.0  208.95  2  43.5  6.7  33.7  15.8  3  0.1  49.5  30.1  4  7.4  39.3  36.8  X4  XI  e  X2  X3  X4  0.0  46.6  30.0  14 . 4  307  208.98  0.0  46.5  30.0  14. 5  377  12.3  208.94  0 .0  46.6  30.0  14 . 4  226  7.5  208.95  0.0  46.6  30.0  14.5  271  Ymin  *• Convergence parameters: a = 1 . 3 , 3 = 0 . 0 0 1 , Number of p o i n t s i n complex K=8 Number of i t e r a t i o n s B  c  -109-  r=5  I T  points  used  order  to  were found using  o b t a i n accurate r e s u l t s a l l runs were  3=0.001.  The  regardless  Complex  was  able  of the s t a r t i n g p o i n t ,  were more a c c u r a t e . the  the FPOINT computer  to  locate  program. performed  the  true  in with  optimum  however the r e s u l t s of run  3  The number of i t e r a t i o n s needed to a r r i v e at  optimum was i n f l u e n c e d  by the s t a r t i n g p o i n t used.  Compared  to t e s t problem 2 and 3 the number of i t e r a t i o n s needed to l o c a t e the  optimum increased Test  problem 5  Test  problem  constrained  5  due to increased  is  a  linear  problem. Figure  number  of f a c t o r s .  objective  function-linearly  7 shows the contour p l o t of t h i s t e s t  problem. Objective  function:  Maximize  Y= 6X1 + 11X2  Subject t o : Explicit  c o n s t r a i n t s . Two l i m i t s on the f a c t o r s were used:  Broad l i m i t s  Narrow l i m i t s  0 < XI < 100  11 < XI < 45  0 < X2 < 100  0 < X2 < 24  Implicit  constraints: 1: 2:  XI + 2X2 < 76 20 < XI + X2  3: The The  optimal s o l u t i o n was found using LINDO computer  maximum value of Y i s 440  located limit  0 < 0.667X1 - X2 < 13.34  a t Xl=44,  at the i n t e r s e c t i o n of i m p l i c i t  of c o n s t r a i n t 3. -110-  X2=16.  program.  The optimum i s  constraint 1  and  upper  Figure  7 . Contour p l o t of t e s t problem 5 . I m p l i c i t are represented by dotted l i n e s .  -Ill-  constraints  Since requires limit of  the  Complex  both  lover  for  implicit  implicit The  on  using  p l o t of  was and  converged  close  a c c u r a c y of and  8.  the  of  Figure  7),  factors  was  The was It X2=8  able  the  set  of set  constraints, 0 and  to  the  the  10.  i n Table  5 using  The  p r o g r a m and 7).  (Figure  starting points.  The  In r u n s  respectively.  i n runs  improved  of  limit  function  3  using  limits  to  2  the  Xl=34.02,  at The  8.  at  obtained  and  (3 = 0 . 0 0 1  evaluations  used  looking  1  1  X2=20.07  r e s u l t s was  upper  results  Xl=35.85,  optimum  lower  broad  by  constraint  the  the  s t a r t i n g points  implicit  to  program)  100.  to  on  Complex  However  the  i n runs  4,6,  depended  on  the  limit  the  (3 v a l u e .  implicit  constraints  factors within the  e f f e c t of  used  in this  a narrower  using  problem  range  than  more r e a s o n a b l e  0-100  limits  (see on  the  studied.  to  locate  i s important was  1 was  problem  r e s u l t s presented  optimum, were  this  by  and  the  limits  FPOINT computer  number  point  Since values  the  The  starting  given  stalled  X2=20.98  ( i . e . FORPLEX computer  r e s u l t s f o r t e s t problem  the  influenced  complex  2 was  constraint  contour  were  upper  constraint  f a c t o r s are  were f o u n d the  and  optimization  the  algorithm  used  the  at  (Table  number  of  10,  both not  11  demonstrate that  optimum r e g a r d l e s s  to point  w h i c h was  used  i n Table  out  (3 v a l u e s  the  run  that  2).  iterations  when t h e the  needed -112-  the  to  arrived  limits  d e f i n i t e trend locate  Complex  starting point.  s t a r t i n g point  Complex  c a s e when b r o a d No  of  the  on was  the  the  Xl=12, at  the  factors  observed  on  optimum  by  Table 1 0 . O p t i m i z a t i o n r e s u l t s of t e s t problem 5 with broad l i m i t s on the independent variables.*-  Run No.  Starting point  Computed optimum  XI  X2  Ymax  25  15  434.94  34.02  20.98  20  XI  X2  IT  2  C D  12  8  435.93  35.85  20.07  17  3  d  20  0  4 3 9 .91  43.94  16.03  27  4=  20  0  440.00  43.99  16.00  57  5  D  25  10  4 3 9 .96  43.96  16.02  32  6  s  25  10  439 . 99  43.99  16 .00  53  7  D  35  10.1  439.90  43.80  16 .10  27  8=  35  10.1  440.00  43.99  16.00  53  *• Number of p o i n t s i n complex K = 4 Number of i t e r a t i o n s Search was s t a l l e d Convergence parameters: 01 = 1 . 3 , 0 = 0 . 1 , r = 5 Convergence parameters: ct = 1 . 3 , (3 = 0 . 0 0 1 , r = 5 B  c  D a  -113-  B  Table 11. O p t i m i z a t i o n r e s u l t s of t e s t problem 5 with narrow l i m i t s on the independent v a r i a b l e s  Run No.  1 2  D  3 4  D  Starting point  Computed optimum  XI  X2  12  8  439.59  43.95  15.99  27  12  8  439 .99  43.99  16 .00  72  20  0  439.94  43.93  16.03  30  20  0  439 .99  . 43.99  16 .00  49  Ymax  XI  X2  5  25  10  439.77  43.54  16.23  34  6  22. 5  15  439 .92  43.87  16 .06  22  7  35  10.1  439.95  43.91  16.04  39  *• Convergence parameters: a=1.3, 0 = 0.1, r=5, unless otherwise indicated Number of p o i n t s i n complex K=4 Number of i t e r a t i o n s Convergence parameters: 01 = 1.3, 0 = 0.001, r=5 s  c  D  -114-  narrowing the r e s u l t s was  l i m i t s on  the  factors,  observed when 13 was  increased accuracy  o£  the  s e t at 0.001.  Test problem 6 Test problem 6 was  developed to analyze  the performance of the  Complex using a q u a d r a t i c o b j e c t i v e f u n c t i o n - l i n e a r l y problem.  The  method f o r c r e a t i n g symmetric  reported by Nakai and form  for  Arteaga  m u l t i f a c t o r equations  assume the minimum value constraints was  (1990)  Objective Subject  followed. 2  of 0 when Xl=a and  r e q u i r e d to be at Xl=25 and  response  surfaces  The  i s Y=(Xl-a) +(X2-b) ;  as t e s t problem 5,  p l o t of t h i s t e s t  was  constrained  Y  2  X2=b.  general  Using  will  the same  the minimum response value X2=10. Figure  8 shows the  of  0  contour  problem.  function:  Minimize  Y=  (Xl-25)  2  + (X2-10)  2  to:  Explicit constraints: 0 < XI  <  100  0 < X2  <  100  Implicit constraints: 1: 2: 3: Table The  XI + 2X2 20  < 76  < XI + X2  0 < 0.667X1 - X2  <  13.34  12 c o n t a i n s the o p t i m i z a t i o n r e s u l t s of t e s t problem  Complex v a r i e d i n i t s success  depending on the s t a r t i n g  used. Note t h a t the optimum obtained  i n runs 1, 2, 3 and  0 = 0.1)  13 to 0.001  was  not e x a c t l y 0. Decreasing  the accuracy  of the r e s u l t s .  The -115-  6  6.  point (when  i n run 4 improved  number of i t e r a t i o n s needed to  10  0  20  30  40  50  X1  Figure  8.  Contour plot of t e s t problems 6 and constraints are r e p r e s e n t e d by s t r a i g h t  -116-  7. Implicit lines.  Table  Run No.  12. O p t i m i z a t i o n r e s u l t s  of t e s t problem 6 . * "  S t a r t i n g point  B  Computed optimum  XI  X2  Ymin  XI  1  25  15  0.0194  24.91  10 .10  31  2  34  20  0.0126  25.11  10. 03  37  3  20  0  0.0073  25.08  9.98  34  20  0  0.0006  24 .98  10.01  50  4  D  X2  IT  C  5  22. 5  15  0.1457  25.25  10.29  75  6  35  10 .1  0 .0777  24.74  9.89  26  *• Convergence parameters: ct = 1.3, 0 = 0 . 1, r = 5, unless otherwise indicated Number of p o i n t s i n complex K=4 Number of i t e r a t i o n s Convergence parameters: a=1.3, 0=0.001, r=5 B  c  D  -117-  locate  the  optimum  Increased number Test  was  influenced  by  the  starting  points.  of i t e r a t i o n s were needed when (3 = 0.001.  problem 7  Maximization  of the o b j e c t i v e  f u n c t i o n of t e s t problem 6  performed using  the same c o n s t r a i n t s . The contour p l o t  clearly  that  shows  feasible  area  is  the maximum of located  at  the  this  function  intersection  (Figure within  of  starting  points  presented used  i n Table 13 demonstrate that  with  runs 3 and 4 were c l o s e to the optimum,  optimum was improved by s e t t i n g (3 to 0.001 problem 8  Test  problem  8  was developed  to be l o c a t e d  Objective  function:  outside  in a similar  the f e a s i b l e area  manner  constraints:  (Xl=50,  to (Y=0)  0 < XI < 100 0 < X2 < 100  -118-  test was  X2=25).  Y= (XI - 50)* + (X2-25)*  Minimize  the  i n run 4 .  Subject t o : Explicit  results  the accuracy of  problem 6 except that the minimum value of the f u n c t i o n required  the  p a r t i c u l a r l y i n run 5 where the complex  was s t a l l e d at the boundary of c o n s t r a i n t 1. O p t i m i z a t i o n  Test  X2=16.  i n runs 1, 2, 5 and 6, the Complex f a i l e d to  converge to the optimum,  of  the  i s Y=397.  At t h i s point the f u n c t i o n value results  8)  implicit  c o n s t r a i n t 1 and the upper l i m i t of c o n s t r a i n t 3 at X l = 4 4 ,  The  was  Table 13. O p t i m i z a t i o n r e s u l t s  Run No.  B G D E  s  S t a r t ing p o i n t  Computed optimum  XI  X2  Ymax  1  25  15  392.22  43.89  15.94  44  2  20  0  383.48  43 . 70  15.81  53  3  34  20  395.17  43.96  15.98  91  34  20  396.77  43.99  16.00  134  5*  22.5  15  206.33  34.56  20.72  22  6  35  10.1  393.95  43.93  15.96  24  4  x  of t e s t problem 7.*-'  D  XI  X2  IT  C  Convergence parameters: 01=1.3, 3 = 0.1, r=5, unless otherwise indicated Number of p o i n t s i n complex K=4 Number of i t e r a t i o n s Convergence parameters: ct = 1.3, 3 = 0.001, r=5 Search was s t a l l e d  -119-  Implicit  constraints: 1:  XI + 2X2 < 76  2:  20 < XI + X2  3: As  0 < 0.667X1 - X2 < 13.34  can be seen i n Figure  9 the minimum value  f u n c t i o n w i t h i n the f e a s i b l e area this  optimum  point  lying  at  i s Y=117 the  of the o b j e c t i v e  a t Xl=44  intersection  and  X2=16,  of  implicit  c o n s t r a i n t 1 and the upper l i m i t of c o n s t r a i n t 3. The  optimization  r e s u l t s f o r t e s t problem 8 obtained  d i f f e r e n t s t a r t i n g p o i n t s are given A,  i n Table 14.  using  5  Except f o r run  where the complex was s t a l l e d at the boundary of c o n s t r a i n t 1,  the  Complex  was able t o l o c a t e the optimum.  No d i f f e r e n c e  observed i n the accuracy of the r e s u l t s using d i f f e r e n t 3 when  the s t a r t i n g p o i n t was s e t at Xl=25,  iterations  X2=10.  was  values  The number of  needed to l o c a t e the optimum depended on the s t a r t i n g  p o i n t and 3 v a l u e . Test  problem 9 of the o b j e c t i v e f u n c t i o n of t e s t problem 8  Maximization performed  using  Figure  reveals  9  feasible of  area  implicit  the same c o n s t r a i n t s .  The contour p l o t shown i n  that the maximum f u n c t i o n  value  within  i s l o c a t e d a t the i n t e r s e c t i o n of the lower constraints  3  and 2 a t  was  Xl=12  and  X2=8  the limit where  Y=1733.10. Table 15 Convergence The  contains  the o p t i m i z a t i o n r e s u l t s of t e s t problem 9.  to the optimum depended on the s t a r t i n g p o i n t  Complex f a i l e d  to converge to the optimum i n runs 3 -120-  used. and  4,  -121-  Table 14. O p t i m i z a t i o n  Run No.  Starting  r e s u l t s of  t e s t problem 8,*"»  point  Computed optimum  X2  Ymin  25  10  117.03  43.99  16.00  38  25  10  117.02  43.99  16.00  55  3  20  0  117.09  43.97  16.01  31  4=  22.5  15  256.64  34.56  20.72  22  20  117.06  43.98  16 .01  71  10.1  117.11  43.99  16 . 00  41  1 2  D  5  - 34  6  35  XI  IT  XI  X2  C  *• Convergence parameters: ot = 1.3, (3 = 0.1, r = 5, unless otherwise ind i c a t e d Number of p o i n t s i n complex K=4 Number of i t e r a t i o n s Convergence parameters: a=1.3, 13 = 0.001, r=5 * Search was s t a l l e d B c  D  -122-  Table  Run No.  15. O p t i m i z a t i o n r e s u l t s  of t e s t problem  S t a r t i n g point X2  1  25  10  1732.89  2  20  0  3  22 . 5  4 5 D  B  Computed optimum  XI  6  9.*'  X2  IT  12.00  8.00  67  1732.98  12 . 00  8.00  99  15  1713.22  12.48  7 . 52  52  34  20  1725.36  12. 08  8.05  70  35  10.1  1732.91  12.00  8.00  60  35  10.1  1733.08  12.00  8.00  102  Ymax  XI  C  *• Convergence parameters: ot = 1.3, 0 = 0.1, r=5, unless otherwise indicated Number of p o i n t s i n complex K=4 Number of i t e r a t i o n s Convergence parameters: a=1.3, 0=0.001, r=5 B  c  D  -123-  however  the  Complex was s u c c e s s f u l  i n l o c a t i n g the  optimum  in  runs 1, 2, 5 and 6. S e t t i n g 3 to 0.001 with s t a r t i n g p o i n t Xl=35, X2=10.1  increased  the accuracy of the r e s u l t s .  i t e r a t i o n s needed to l o c a t e  The number  of  the optimum depended on the s t a r t i n g  p o i n t and p value. Test  problem 10  Test the  problem 10  Complex  when  was developed t o analyze the performance  of  a  an  quadratic  objective  i n t e r a c t i o n term and l i n e a r i m p l i c i t Objective  function  with  c o n s t r a i n t s are used.  f u n c t i o n : Minimize Y= 725 - 50X1 - 20X2 +X1*  + X2*-X1X2  Subject t o : Explicit constraints:  0 < XI < 100 0 < X2 < 100  Implicit  constraints: 1:  XI + 2X2 < 76  2:  20 < XI -I- X2  3: It within  0 < 0.667X1 - X2 < 13.34  is difficult  to l o c a l i z e the true optimum of t h i s f u n c t i o n  the f e a s i b l e area j u s t by a n a l y z i n g  Figure  10. The optimum  should l i e a t the boundary of i m p l i c i t c o n s t r a i n t 1 c l o s e r to the corner Xl=32.56, X2=21.72 where the f u n c t i o n value i s Y=-512.695. As the r e s u l t s i n Table 16 to a s i n g l e p o i n t . However, expected the  to be greater  Complex  found  Y=-513.29. R e s u l t s  reveal,  the Complex d i d not converge  s i n c e the accuracy of the r e s u l t s i s  when 3=0.001,  i t could  the minimum a t Xl=33.13,  be that  i n run  X2=21.44  2  where  of runs 1, 3 and 5 were c l o s e to t h i s minimum. -124-  Figure 10. Contour plot of t e s t problems 10 and 11. I m p l i c i t c o n s t r a i n t s are represented by s t r a i g h t l i n e s .  -125-  Table 16. O p t i m i z a t i o n r e s u l t s  Run No.  of t e s t problem 10.*"  S t a r t i n g point  B  Computed optimum  XI  X2  Ymin  25  10  -513 .28  33 .17  21. 42  15  25  10  -513.29  33.13  21.44  24  3  25  15  -513.28  33 .17  21.41  17  4  32  20  -513.19  32.93  21.53  11  5  35  10 .1  -513.27  33.05  21.47  20  1 2  D  XI  X2  IT  e  *• Convergence parameters: a = 1.3, 3 = 0.1, r = 5, unless otherwise indicated Number of p o i n t s i n complex K=4 Number of i t e r a t i o n s Convergence parameters: <x = 1.3, 3 = 0.001, r=5 B c D  -126-  The  number  of  function evaluations  needed  at  the  o b j e c t i v e f u n c t i o n of t e s t problem 10  was  optimum depended on the s t a r t i n g p o i n t and Test problem  to  arrive  (3 v a l u e .  11  Maximization of the  performed. It appears from the contour p l o t i n Figure maximum  of  this function  constraint  2  and  (Y=125)  l i e s at the  10  that  intersection  the upper l i m i t of c o n s t r a i n t 3 at Xl=20  the of and  X2 = 0. The  o p t i m i z a t i o n r e s u l t s f o r t e s t problem 11 obtained  d i f f e r e n t s t a r t i n g p o i n t s are given 2,  where  the  constraint  complex  2 and  lower l i m i t  able to l o c a t e the using The  13 = 0.001 number  was  optimum.  i n Table 17.  stalled of  at  As  can  be  is  able  to  linear  However, this  locate  was  function  the  problems  (1965)  improved  set at X l = 32,  3  to  X2 = 20.  arrive  at  the  value.  optimum that  this  modified  of l i n e a r were  and  linearly  s t a t e d that the Complex method  optimum i n  nonlinear  problems  whose  i s convex, the Complex seems to work s a t i s f a c t o r i l y  problems  whose f e a s i b l e area  the Complex method i s not  type of  the Complex  needed  able to l o c a t e the  Although Box  f e a s i b l e area in  was  objective  constrained.  3,  observed from the r e s u l t s obtained,  Complex a l g o r i t h m nonlinear  of  Accuracy of the r e s u l t s was  optimum depended on the s t a r t i n g point and  run  intersection  constraint  function evaluations  for  4  the  when the s t a r t i n g p o i n t was of  Except  using  problem.  Linear -127-  is  a  planar  surface.  an e f f i c i e n t method to  programming  solve  i s d e f i n i t e l y more  Table 17. O p t i m i z a t i o n  Run No.  r e s u l t s of  t e s t problem l l . * - '  S t a r t ing point  Computed optimum  XI  X2  Ymax  25  10  124.89  25  15  3  40  4  1 2  5  D  s  B  XI  X2  IT  20 .00  0 .002  58  76.94  12.00  8.004  38  15  124.89  20.00  0.003  74  32  20  124.87  20.00  0. 003  62  32  20  125.00  20.00  0.000  109  C  *• Convergence parameters: ct = 1.3, (3 = 0.1, r = 5, unless otherwise indicated Number of p o i n t s i n complex K=4 Number of i t e r a t i o n s Search was s t a l l e d Convergence parameters: a = 1.3, (3 = 0.001, r=5 B  c D  E  -128-  efficient  t h a n t h e Complex method  nonlinear  objective  constrained  the  function  Complex  method  method and t h e r e f o r e  optimization  applications.  converged  on  parameter  0  increase  an  area  Weighing  of  FORPLEX computer program  use  several  order only  generator several been  a specific  starting  t o check one  the  s u i t a b l e t o be u s e d  f o r formula  that  starting had  t o a s i n g l e p o i n t , but  optimum. of  t o be  was  reseeded  t o take  r u n s had t o be p e r f o r m e d  seed,  formula i s not somewhat  weights given  by t h e  i t was n e c e s s a r y t o values  of 0  been l o c a t e d . the  random  optimization  t o check  -129-  is  to  place.  available,  found.  For  exactly  as d i f f e r e n t  i n each  order  i n the r e s u l t s  t h e optimum had i n f a c t point  convergence in  results.  of i n g r e d i e n t s  as w e l l  The  0.001  ingredients  random number  points  be  linearly effective  accuracy  is likely  are  For  an  to the  high  thus t h e rounding  When u s i n g  to  of the o p t i m i z a t i o n  applications  essential.  seems  t o be s e t t o a v a l u e  the accuracy  difficult,  close  of problem.  that  t h e Complex d i d n o t c o n v e r g e  had  optimization  type  problems  optimization  Generally,  for this  search  in  When number and  whether t h e optimum had  B. Development of i n g r e d i e n t - q u a l i t y r e l a t i o n s h i p s f o r a 3 - i n g r e d i e n t model f r a n k f u r t e r f o r m u l a t i o n 1. Proximate a n a l y s i s The  proximate  composition  deboned p o u l t r y meat (MDPM) Protein content  and was  moisture  corresponding  of the  content  analysis  the moisture,  vertices  and  meat,  mechanically  pork f a t are shown i n Table  18.  were g r e a t e s t i n  fat  beef  to  design  of the raw p r o t e i n and  the  10  i n g r e d i e n t s was f a t content  (Tables 2 and  3).  The  performed  by  the  computed to  i n the raw  was  ratio  to  moisture  the  raw  emulsions equal to f o u r .  emulsions  rather  to avoid  and and  formulations  p r o t e i n contents protein  having  not  moisture-  than  only  using  content.  not exceed four times f i n i s h e d weight  T h i s r a t i o was  moisture  and  content  weight  different  based on American content  the percentage of p r o t e i n plus  (Pearson  a  but a l s o d i f f e r e n t p r o p o r t i o n s of  government r e g u l a t i o n s which s p e c i f y that moisture  Total  maintain  per cent of added water based on t o t a l meat block  chosen  moisture  in  ratio  determine  Varying the amount of added water to make up a constant to-protein  extreme  moisture-to-protein  amount of water to be added i n the form of i c e  the m o i s t u r e - t o - p r o t e i n r a t i o  to  of the meat blocks  f o r m u l a t i o n s given  the meat block of each f o r m u l a t i o n was  constant  and  g r e a t e s t i n pork f a t .  Proximate obtain  of the beef  10%  should of the  Tauber, 1984a). of  -130-  each  formulation  was  computed  Table 18. Proximate  Ingredient  composition and pH of raw ingredients.*-  Moisture  8  (%)  Protein  0  (%)  Beef meat  73.64+0.64  22.49±0.17  MDPM"  65.69±1.42  15.51+0.12  Pork f a t  14.58+1.22  4.96±0.02  5  Fat (%)  pH  1  D  3.72  Before After f r o z e n storage 5.51  5.21  18.67  6 .69  6 . 49  80.37  n. d.  n. d.  *• Values are mean + standard d e v i a t i o n Mean of s i x samples of beef meat and MDPM and of four samples of pork f a t Mean of three samples Mean of two samples pH of meat i n g r e d i e n t s was determined before and a f t e r s i x months f r o z e n storage a t -30°C. Values are means of two samples *" M e c h a n i c a l l y deboned p o u l t r y meat n.d.= not determined B  c  D  =  -131-  (moisture  content  determine  the amount of s a l t to be added.  constant  effective  frankfurter rather  of meat block plus weight of added water)  salt  concentration  f o r m u l a t i o n s . The  use  meat  comparing  block weight) was f o r m u l a t i o n s of  aqueous  would show  phase.  It  is  was  an  (i.e. %  important  well  functionality.  Trout  is  documented  that  generally  (1983)  (1986) had  meat i n g r e d i e n t s was  a  r e p o r t e d t h a t mechanical deboning y i e l d s a  has a pH  to  1988).  The  to 7.4  that  changes  in  effective good  r e p o r t e d that  using  no b e n e f i c i a l  effect  products.  determined before  product  and  Mechanically I t has  been  with a higher marrow  (Harding Thomsen and  pH of the meats a p p a r e n t l y decreased  time i n f r o z e n s t o r a g e . Powrie  Acton et  ensure  to the presence of bone  i n the range of 6.8  functional  4%  deboned p o u l t r y meat had a higher pH than beef meat.  than hand deboning due  when  concentration  a f t e r s i x months f r o z e n storage at -30OC (Table 18).  pH  on  the meat  Schmidt, 1987;  necessary  Schmidt  in  the  by the  on the water b i n d i n g a b i l i t y of comminuted meat pH of the raw  based  consideration  concentrations  e f f e c t i v e s a l t c o n c e n t r a t i o n s above 2.9%  The  salt  levels  are determined  and  the  f o r m u l a t i o n s having higher  1983). As mentioned by Acton et a l . concentration  in  overall  of s a l t s i n the aqueous phase (Trout and  salt  2.5%  If an  high s a l t  p r o p e r t i e s of meat products  al.,  maintained  d i f f e r e n t composition.  s a l t c o n c e n t r a t i o n were used, of f a t content  In t h i s study a  of e f f e c t i v e s a l t c o n c e n t r a t i o n  than o v e r a l l s a l t c o n c e n t r a t i o n  total  to  which  Zeuthen,  with l e n g t h  of  (1973) and Fennema (1973) r e p o r t e d  the pH of meat -132-  during  frozen  storage  occur  because phase.  of  increasing  c o n c e n t r a t i o n of s a l t s i n  These changes are  enzyme a c t i o n  2. Q u a l i t y  unfrozen  dependent on storage temperature,  composition, p h y s i o l o g i c a l and  the  state,  buffering  capacity  of  salt  proteins  (Powrie, 1973).  parameters evaluated  S p e c i f i c q u a l i t y parameters were evaluated as a measure of q u a l i t y of the  frankfurter  formulations.  evaluated can  be  i n t o : (a)  processing thermal  and  divided  storage,  treatment,  frankfurters, frankfurters, Weight  (c)  (d) and  loss  (b)  of the  processed  economic importance but  be  regarded as an  on the  measurements conditions  are that  raw  loss during emulsions  meats has  always  only from the  e.g.  storage.  actual  meat  Weight  product  pilot  the  can  be  lost  by the  of  great  quality  and  attributes and  1957 ). Weight l o s s  can  of the  meat product  l o s s measurements while emulsion  samples  plant  been  or  under  amount of  fat  and  are  stability processing  commercial  p r o d u c t i o n procedures. Emulsion s t a b i l i t y methods y i e l d v a l u a b l e i n f o r m a t i o n r e g a r d i n g the  cooked  j u i c i n e s s , flavour  stability  performed on s m a l l simulate  cooked  standpoint of y i e l d  texture,  i n d i c a t o r of the  of  to  emulsions.  (Wierbicki. et a l . ,  to thermal treatment and performed  raw  characteristics  because i t a f f e c t s the  f i n i s h e d products,  overall acceptability.  parameters  product weight  s t a b i l i t y of the  textural  concern to meat packers not  of the  quality  j u i c i n e s s c h a r a c t e r i s t i c s of the  (e) pH of  The  the  scale  rapid water  and that  a c t u a l meat product d u r i n g thermal treatment. -133-  Texture and j u i c i n e s s c h a r a c t e r i s t i c s of meat products p l a y vital  role  firmness  i n consumer acceptance.  are  preference  attributes  cheviness  that consumers use to  f o r commercial  frankfurters  Voisey et a l . ,  1975).  determine  acceptability  the  Juiciness,  a  and  describe  their  Patel,  1984;  (Lee and  Although sensory e v a l u a t i o n i s needed or p r e f e r e n c e of a  meat  to  product,  o b j e c t i v e methods f o r t e x t u r e and j u i c i n e s s e v a l u a t i o n were used in t h i s  2.1.  study.  Product weight l o s s at d i f f e r e n t stages of the f r a n k f u r t e r p r e p a r a t i o n process  As mentioned by Brown and Ledward i s synonymous with measure  of  the  product s t a b i l i t y . unbound  p r o c e s s i n g and s t o r a g e . are  a f f e c t e d by the  meat p r o t e i n s , i . e .  Data  play a v i t a l f o r product  frankfurter reported weight  f a t and  l o s s of 7  conditions.  lost  loss is a  during  stability, by the  F u n c t i o n a l p r o p e r t i e s of the  water- and f a t - b i n d i n g c a p a c i t y and g e l a t i o n role  i n s t a b i l i z i n g the cooked  weight  sausages  loss at  different  were  product.  stages  obtained  of the Results from  the  per f o r m u l a t i o n .  l o s s a f t e r thermal treatment  replication  i n the second The  solids  loss  composition of the f o r m u l a t i o n and  f o r these q u a l i t y parameters  i n the f i r s t  8.99%.  T o t a l weight  p r e p a r a t i o n process are given i n Table 19.  Per cent weight  and  water,  product weight  Weight l o s s , and thus product  p r o c e s s i n g and storage  ability,  (1987),  (Shrink)  varied  from 7.94 to 10.45% with a mean of 9.12%  r e p l i c a t i o n from 8.15  o v e r a l l mean f o r Shrink -134-  t o 9.89% f o r the  20  with a mean of formulations  Table 19. Experimental data f o r product weight l o s s a t d i f f e r e n t stages of the f r a n k f u r t e r p r e p a r a t i o n process.  Replication No. 1  Formulation No.  A  1 2 3 4 5 6 7 8 9 10  mean 2  mean x B c  D  1 2 3 4 5 6 7 8 9 10  Shrink (%)  Vacuum shrink  B  Cook shrink (%)  D  8.30 9 .78 8 .11 7.94 10.23 9.65 8.62 8.41 9 . 71 10. 45  1.68 1.92 1. 87 1.38 1. 87 1.92 2.01 1.73 2.39 2.04  4.12 5. 36 4.93 5.21 6 .22 5.74 14 . 59 7.65 6.47 7.10  9 .12  1.88  6.74  9 . 39 9 .04 8 . 31 8.65 9 . 57 9 .48 8 .23 8 .15 9 . 89 9.15  1.05 1.56 1.94 1.05 2.02 2 . 25 1.44 1. 39 1.77 2.38  4.95 4 . 01 7 . 32 13 .98 9.36 8 . 54 7.27 7.20 8.90 6 .65  8.99  1.69  7 . 82  I n g r e d i e n t s p r o p o r t i o n s are given i n Table 2 Per cent weight l o s s a f t e r p r o c e s s i n g (equation 20) Per cent weight l o s s a f t e r 13 days under vacuum packaged storage (equation 21) Per cent weight l o s s a f t e r the consumer cook t e s t (equation 22)  -135-  evaluated  was  9.06%.  As mentioned by Trout  (1988)  to  obtain  a c c u r a t e r e s u l t s , c o n t r o l of cooking temperature i s c r i t i c a l . this  study,  a c c u r a t e c o n t r o l of cooking  second stage of the thermal difficult used.  to  achieve due  normal  to the  variability  mentioned by  Per  cent  storage to  2.39%  1.05  between the  Shrink  was  weight l o s s a f t e r 13  cent  weight  loss  do  under  not  as  reflect  yield  and  from  addition and  The  o v e r a l l mean for Vacuum  the  cook  test  (Cook  to 14.59% with from  4.01  to  o v e r a l l mean f o r Cook s h r i n k f o r 7.28%.  t h a t the r e p l i c a b i l i t y of Vacuum shrink  i s poor. to  1.78%.  i n the second r e p l i c a t i o n  i s r e a d i l y apparent  packaged  i n the second r e p l i c a t i o n  r e p l i c a t i o n from 4.12  the 20 f o r m u l a t i o n s evaluated was  Cook s h r i n k data  vacuum  1.38  a f t e r the consumer  13.98% with a mean of 7.82%. The  ingredients  the  Furthermore,  considered product  f o r m u l a t i o n s evaluated was  of 6.74%,  in  replications  r e p l i c a t i o n from  with a mean of 1.69%.  a  mean  utensils  i n a d d i t i o n to  values  days  with a mean of 1.88%, and  v a r i e d i n the f i r s t  that,  of the cooking  ingredients.  the c a s i n g s ,  shrink)  and  was  loss.  s h r i n k f o r the 20  It  i . e . steam cooking,  (Vacuum s h r i n k ) v a r i e d i n the f i r s t  to 2.38%  Per  the  treatment  the meat (1987a)  temperature d u r i n g  weight l o s s s i n c e f a t cookout, l o c a t e d between  f r a n k f u r t e r s and not product  thermal  of  Whiting  a c c u r a t e l y product  and  to the nature  T h e r e f o r e , the v a r i a t i o n observed  can be a t t r i b u t e d  the  treatment,  In  the  A  p o s s i b l e e x p l a n a t i o n could  inherent  comminution -136-  and  variability stuffing  of  be  the meat  steps,  the  variability  introduced by the  thermal  treatment a f f e c t e d  s h r i n k r e s u l t s , and these f a c t o r s a f f e c t e d Cook s h r i n k  2.2. Emulsion s t a b i l i t y The  stability  of  number of i n t e r r e l a t e d f a c t o r s such as meat  ratios;  (b)  temperature quantity and  meat emulsions seems to be a f f e c t e d factors.  These  include:  by  of  amount and type of f a t and  nonmeat a d d i t i v e s ,  comminution  and  fat:protein:moisture  and heat treatment,  and q u a l i t y of p r o t e i n capacity,  Ledward, 1987; Asghar stability  of  and  gelation  et a l . 1985; Sofos, raw emulsions  evaluated using two emulsion s t a b i l i t y  to  ability  point  (c)  i n terms of s o l u b i l i t y ,  and  a  (a) c o m p o s i t i o n a l  p r o c e s s i n g f a c t o r s such as s e v e r i t y and end  fat-binding  The  results.  analysis  the amount of s a l t s ,  ingredients,  vacuum  the  vater-  (Brown  and  treatment  was  1983a). thermal  methods.  The s t a b i l i t y of  the  meat emulsions e v a l u a t e d by the modified method of S a f f l e et  al.  ( 1967 ) determined  the extent of f a t r e l e a s e d by h e a t i n g the  emulsions a t 80°C. As mentioned fat  released during  the  by Whiting (1987a)  emulsion s t a b i l i t y  the amount of  tests indicates  whether  f i n i s h e d product would present f a t caps i n s i d e the c a s i n g , or  would cent  result fat  i n f a t l o s s e s when reheated by the  released  (ES)  data obtained by the  consumer.  modified  Per Saffle  method are presented i n Table 20. ES v a l u e s are the mean from two o b s e r v a t i o n s . ES values v a r i e d to  from  0.00  2.45% with a mean of 0.51%, and i n the second r e p l i c a t i o n  from  0.00  i n the f i r s t  to 1.45% with a mean of 0.37%. -137-  replication  The o v e r a l l mean f o r ES f o r  Table 20. Experimental  Replication No. 1  data f o r emulsion  Formulation No.  B  mean 2  1 2 3 4 5 6 7 8 9 10  mean  analysis.*-  ES (%)  Tmloss (%)  0.25 0.20 0.60 0.45 0.05 0.00 2.45 0.80 0.25 0.00  31.27 32.21 36.77 38.02 37.01 32.66 46.86 45.03 42.49 42.21  20.19 21.75 21.86 20.66 23.71 22.52 25.86 25.18 26.90 29.84  0.51  38.46  23.85  0.00 0.00 1.45 0.45 0.30 0.00 0.90 0.55 0.00 0.00  34.74 30.02 37.30 34.17 39.54 37.66 39.28 40.90 42.13 40.46  22.42 20.26 22.17 18.57 25.33 25.97 21.68 22.87 26.67 28.60  0.37  37.62  23.45  C  1 2 3 4 5 6 7 8 9 10  stability  D  Twloss* (%)  *• R e s u l t s r e p o r t e d are the mean from 2 o b s e r v a t i o n s I n g r e d i e n t s p r o p o r t i o n s are given i n Table 2 By the modified method of S a f f l e e t a l . (1967) and expressed as per cent f a t r e l e a s e d a f t e r thermal treatment By the modified method of Townsend e t a l . (1968) and expressed as per cent water l o s s per moisture content of the meat block (equation 28) By the modified method of Townsend e t a l . (1968 ) and expressed as per cent weight l o s s a f t e r thermal treatment (equation 29) B  c  D  85  -138-  the  20  formulations  out  that  low  ES  evaluated values,  was  0.44%. I t i s important to  indicate  high  R e l i a b i l i t y of ES data i s q u e s t i o n a b l e results  of d u p l i c a t e observations  among r e p l i c a t i o n s was  poor.  reproducibility variations  of  that  and  occur  w i t h i n a given  tests  during  and  the  the  formulation  and  reported  poor when more than 0.8%  Bawa et a l . (1988)  stability  stability.  s i n c e r e p e a t a b i l i t y of  S a f f l e et a l . (1967)  r e p e a t a b i l i t y of the method was r e l e a s e d . Comer (1979)  emulsion  point  that  fat  a l s o reported  attributed  preparation  low  this  of  was  to  the  raw  emuls i ons. The  second  method  emulsion s t a b i l i t y  of Townsend et a l .  (1968).  method used was  the  modified  As mentioned i n the  Material  and  Methods s e c t i o n , t h i s method aims at e v a l u a t i n g the  of  the emulsions when held at 80OC by determining the extent  f a t and of  water r e l e a s e d . The  the  amount of water r e l e a s e d  stability  i s a measure  tendency of a f r a n k f u r t e r emulsion to lose water  thermal treatment and  i s r e l a t e d to the water h o l d i n g  of  during  capacity  of  the meat emulsion. As mentioned above, the amount of f a t r e l e a s e d indicates  whether a product would present  f a t caps  fat  l o s s e s when reheated by the consumer and  fat  binding  capacity  of  the meat emulsion  Results  could  not  (1968)  since  p r e l i m i n a r y experiments  released during layer  that  released  i t i s r e l a t e d to (Whiting,  of  showed  the  l e s s than -139-  The  amount  the amount r e l e a s e d  al. fluid  the emulsions never developed a  be measured a c c u r a t e l y .  by t h i s method was  that  the  1987a).  be expressed as suggested by Townsend et  heating  could  or r e s u l t i n  lipid  of by  fat the  modified method of s a f f l e et a l . (1967). T h i s d i f f e r e n c e could be explained heating  by  the  f a c t that the l a t t e r method,  in addition  the raw emulsions, makes use of c e n t r i f u g a l f o r c e .  e x t e r n a l f o r c e helps r e l e a s e l o o s e l y bound f a t . preliminary  experiments  emulsions appeared the  the  loss  of  the  expressed cooked A)  modified  method of  from  the  al.  l e a d i n g to results  (1968)  i n two d i f f e r e n t ways based on the weight  by  cooked  Emulsion s t a b i l i t y  Townsend et  This  As i n d i c a t e d  to be the major c o n t r i b u t i n g f a c t o r  decrease i n emulsion s t a b i l i t y .  using  water  to  lost  were  from the  emulsions: Assuming  emulsion percentage  the  stability of  weight values  original  (Tmloss). Tmloss  (%) was  l o s s was  due s o l e l y  were r e p o r t e d as  moisture defined  water  content has been used as  after  x  1957).  be c o n s i d e r e d a l s o a measure of the water  W i e r b i c k i et a l .  (1957)  the Townsend et a l . the water  (1987),  followed the  (1968)  loss,  loss  as  meat  a  block,  100  thermal treatment  a measure of the water  M i t t a l et a l .  the  water  as  of meat products ( W i e r b i c k i et a l .  meat emulsions.  water  content of  water l o s s a f t e r thermal treatment moisture content of meat block The r a t i o of unbound  to  (28) to moisture  holding  capacity  T h e r e f o r e , Tmloss  can  h o l d i n g c a p a c i t y of the Mast et a l .  (1982)  and  same procedure as that of  emulsion s t a b i l i t y t e s t t o determine  h o l d i n g c a p a c i t y of comminuted meat p r o d u c t s .  B) As proposed by Sofos (1983a) emulsion s t a b i l i t y values were a l s o expressed as weight  l o s s as a percentage of the raw -140-  emulsion  weight,  (Twloss). Twloss  (%) was d e f i n e d as  emulsion weight l o s s a f t e r thermal treatment weight of emulsion before thermal treatment Emulsion  stability  x  data obtained by the m o d i f i e d  100 method  Townsend e t a l . (1968) i s presented i n Table 20. R e s u l t s are  the  mean from 2 o b s e r v a t i o n s .  r e p l i c a t i o n from 31.27 the  second  37.62%.  to 46.86%  Tmloss v a r i e d  The  in  to 42.13%  o v e r a l l mean f o r Tmloss  with  f o r the  the  a  20  of  reported  with a mean of 38.46%,  r e p l i c a t i o n from 30.02  (29)  first and i n  mean  of  formulations  evaluated was 38.04%. Twloss values v a r i e d  i n the f i r s t  29.84%  with a mean of 23.85%  18.57  to  28.60%  r e p l i c a t i o n from  20.19  and i n the second r e p l i c a t i o n  with a mean of 23.45%.  to from  The o v e r a l l mean f o r  Twloss f o r the 20 f o r m u l a t i o n s evaluated was 23.65%. It  i s important to p o i n t out that low Tmloss and Twloss v a l u e s  i n d i c a t e high emulsion  stability.  2.3. J u i c i n e s s c h a r a c t e r i s t i c s of the cooked As mentioned  by Lee and P a t e l  (1984),  frankfurters  j u i c i n e s s appears to be  the  q u a l i t y a t t r i b u t e which determines the o v e r a l l  of  commercial  frankfurters.  Juiciness  as  acceptability  perceived  consumer i s b e l i e v e d t o be i n f l u e n c e d by the q u a n t i t y , of  release  mastication reported  by  measurement  and  the  (Lee and Lee of  and  the  composition of the Patel, Patel  1984). (1984)  juiciness  An  fluid  the  rate  expressed  upon  instrumental  was used  characteristics  -141-  by the  as  an  of  method  objective the  cooked  frankfurters. %  expressible f l u i d  and  c h a r a c t e r i s t i c s measured were:  The j u i c i n e s s (Exfluid)  (b) %  e x p r e s s i b l e water  (a)  (Exwater)  (c) % e x p r e s s i b l e f a t ( E x f a t ) of the cooked f r a n k f u r t e r s .  et a l .  (1987)  juiciness juiciness Data  and  Lee and P a t e l  characteristics  correlated  f o r the  juiciness  within  first  from 8.66  replication  The  variation  Exwater  of v a r i a t i o n  from 7.62  varied  In the f i r s t from  cooked  were computed f o r varied  i n the  to 13.04%  3.1  f o r the 20  replication  to  with a  13.1%  of  formulations  the  and  mean  coefficients  in  the  second  from 2.7 to 12.4%. varied  i n the  with a mean of 7.00%,  first  replication  from 4.59  and i n the second r e p l i c a t i o n  10.83% with a mean of 6.44%. The o v e r a l l  of v a r i a t i o n  second r e p l i c a t i o n Exfat varied  varied  to  9.70%  from 3.47 to  mean f o r Exwater f o r the  20 f o r m u l a t i o n s evaluated was 6.72%. In the f i r s t coefficients  the  t o 14.75% with a mean of 11.01%, and  o v e r a l l mean f o r E x f l u i d  evaluated was 10.53%.  from 4.5  r e p l i c a t i o n the  to 13.1%  and i n the  from 4.2 to 14.5%.  i n the f i r s t  replication  from 2.02  a mean of 4.02%, and i n the second r e p l i c a t i o n with  instrumental  of  each r e p l i c a t i o n . E x f l u i d  the second r e p l i c a t i o n  replication  characteristics  Coefficients  each f o r m u l a t i o n  of  w e l l with these  sensory  are given i n Table 21. R e s u l t s r e p o r t e d are the mean  from 4 o b s e r v a t i o n s .  10.04%.  r e p o r t e d that  characteristics.  frankfurters  in  (1984)  Lee  a mean of 3.60%  from 1.71 t o 6.71%  The o v e r a l l mean f o r the 20  evaluated was 3.81%. In the f i r s t -142-  replication  t o 9.44% with  formulations  the c o e f f i c i e n t s of  Table  21. Experimental data f o r j u i c i n e s s c h a r a c t e r i s t i c s of the cooked f r a n k f u r t e r s . * -  Replication No. 1  Formulation No. 1 2 3 4 5 6 7 8 9 10  mean 2  mean  1 2 3 4 5 6 7 8 9 10  13  Exfluid (%)  0  Exwater (%)  0  Exfat * (%) 1  10. 41(13 .1) 10. 06(3. 9) 10.19(3. 1) 9.19(3. 4) 8.66( 5. 2) 9.78(8. 0) 14.75(3. 1) 13.42(6. 8) 11.87(6. 0) 11. 80(12 .3)  6 .88(8. 5) 7 .38(4. 5) 5 .94(7. 0) 4 .59(7. 4) 6 .28(5. 7) 7 .76(8. 5) 5 .31(8. 6) 7 .50(9 . 2) 8 .65(5. 1) 9 .70(13 .1)  3 .53(31 .1) 2 .68(8. 8) 4 .25(6. 3) 4 .59(3. 2) 2 .38(5. 6) 2 .02(9. 8) 9 .44(3. 5) 5 .93(4. 1) 3 .23(10 .3) 2 .10(9. 3)  11. 01  7 .00  4 .02  7. 62(5. 0) 5 .37(4. 2) 7. 65(11 .2) 5 .78(11 .2) 7.68(6. 9) 4 .32(8. 3) 10.18(4. 5) 3 .47(14 .5) 9.75(6. 0) 6 .18(6. 9) 9 .21(6. 3) 7 .04(8. 2) 10.97(2. 7) 5 .66(7. 0) 11.72(4. 1) 6 .79(8 . 1) 13.04(8. 4) 8 .92(10 .8) 12. 54 (12 .4)10 .83(11 .8)  2 .25(7. 4) 1 .87(12 .6) 3 .36(7. 4) 6 .71(2. 2) 3 .57(4. 5) 2 .17(5. 2) 5 .31(6. 9) 4 .93(8. 7) 4 .12(9. 2) 1 .71(20 .6)  10. 04  3 .60  6 . 44  * R e s u l t s r e p o r t e d are the mean ( c o e f f i c i e n t of v a r i a t i o n ) from 4 observations I n g r e d i e n t s p r o p o r t i o n s are given i n Table 2 Per cent e x p r e s s i b l e f l u i d Per cent e x p r e s s i b l e water Per cent e x p r e s s i b l e f a t B  c  B  B  -143-  v a r i a t i o n v a r i e d from 3.2 from 2.2  large v a r i a t i o n i n replications.  it  the  Manual  treatments,  the  and  i n the second  replication  to 20.6%.  From these r e s u l t s  i n the  to 31.1%  storage  is  evident  t h a t there  is a relatively  j u i c i n e s s c h a r a c t e r i s t i c s measured w i t h i n processing  of  c o n d i t i o n s and  the  emulsions,  thermal  the d i f f e r e n t steps  involved  e v a l u a t i o n of j u i c i n e s s c h a r a c t e r i s t i c s a l l c o n t r i b u t e to  variability  formulations (1984)  observed.  The  variability  observed  cannot be compared with the r e s u l t s of Lee  in and  the Patel  s i n c e the authors d i d not r e p o r t the r e p r o d u c i b i l i t y of  t h i s method.  2.4.  T e x t u r a l parameters of the cooked f r a n k f u r t e r s  Thermal  processing  transformation like water  of f r a n k f u r t e r emulsions r e s u l t s  from a v i s c o u s s o l to a r i g i d and  s t r u c t u r e t h a t can  Acton  actomyosin and  et a l . ,  chemically  1983).  1985).  gelation  ability  temperature-time al.,  stabilized  I t i s the  solid-  Frankfurter texture of meat p r o t e i n s ,  1987;  i s a f f e c t e d by method of  ( S i r i p u r a p u et a l . , -144-  proteins,  Montejano composition,  (Singh  and et  important f a c t o r s i n the  play a v i t a l  1987).  al.,  comminution  c o n d i t i o n s during thermal p r o c e s s i n g  o v e r a l l q u a l i t y of f r a n k f u r t e r s and  et  structure building  ( S a l i b a et a l . ,  1985). T e x t u r a l c h a r a c t e r i s t i c s are  acceptance  (Saliba  gel-forming  myosin, t h a t act as t e x t u r e and  components i n the meat products et a l . ,  elastic  the  be viewed as a p r o t e i n g e l where f a t and  are p h y s i c a l l y and  1987;  in  r o l e i n consumer  Although humans are  the  best  instrument  1985)  f o r e v a l u a t i n g food t e x t u r e  researchers  the  textural  properties  difficulties 1985;  have used i n s t r u m e n t a l  evaluation  Montejano et a l . , 1985).  important  Brady  and  Hunecke,  methods for  of f r a n k f u r t e r s due  i n v o l v e d i n sensory  food t e x t u r e ,  (Brady and  to  measuring  the  cost  (Brady and  and  Hunecke,  Because of the complex nature  Hunecke (1985)  suggested  that  of  it  is  to measure as many of a food's t e x t u r a l parameters  as  p o s s i b l e , e i t h e r by using s e v e r a l d i f f e r e n t t e s t s , each d e t e c t i n g one  or more parameters,  or by using one  of parameters may  be c h a r a c t e r i z e d .  have  extensively  been  used  properties  of  In  twice;  given  parameters Bourne  compression  were  (1978)  at f i r s t  f o r c e d u r i n g the  first  (Hard2)  of  several  positive  force  the  tests  textural  compression  defined  test  was  as  i n the was  under  the  of  definitions evaluated. peak  Hardness at second  the peak  (Cohes)  (Spring) was  compressed  d e f i n e d as the  cycle.  was  a number  were  F r a c t u r a b i 1 i t y (Fract)  areas  is  curves  parameters  (Hardl)  compression. Cohesiveness the  of  shear  F o l l o w i n g the  s i g n i f i c a n t break  compressions. S p r i n g i n e s s the  extracted.  compression  was  f o r c e at the f i r s t  first  evaluation  force-deformation  compression  second compression c y c l e . the  for  t h i s method a b i t e - s i z e piece of food  by  Hardness  Compression and  known as t e x t u r e p r o f i l e a n a l y s i s  from the r e s u l t i n g  textural  from which a number  f r a n k f u r t e r s . In t h i s study the  d e s c r i b e d by Bourne (1978) used.  test  force during was  defined  curve  on  d e f i n e d as the second  the  and  d e f i n e d as the height  the ratio first that  food recovers d u r i n g the time t h a t elapses between the end -145-  as  of  the Two  first  compression and  other  parameters  the  were  start  of the second  derived  by  compression.  calculation  from  the  measured parameters: Gumminess (Gummy) was d e f i n e d as the product of  Hardl  x  Cohes,  and Chewiness (Chewy)  product of Gummy x Spring as the slope of the  was d e f i n e d  (Bourne, 1975). Firmness  l i n e a r r e g i o n of the f i r s t  as the  (Firm) d e f i n e d  compression curve  (Segars and K a p s a l i s , 1987), was a l s o measured. Shear  testing  products the  has  been w i d e l y performed  and p r o t e i n g e l s .  samples  tenderness  has and  measurements  been  meat,  meat  The maximum f o r c e r e q u i r e d t o  shear  taken  as  gel strength.  a  with  quantitative  In  this  measure  study  shear  were performed on f r a n k f u r t e r samples  force  without  s k i n to determine the s t r e n g t h of the f r a n k f u r t e r contents Maximum  shear  force  (Shear)  was  extracted  of  from  the only.  the  force-  deformation curves o b t a i n e d . Data are  f o r the t e x t u r a l parameters of the cooked  g i v e n i n Table 22.  parameters  are  the  Results reported  mean  from  4  o b s e r v a t i o n s f o r maximum shear f o r c e .  f o r the t e x t u r e  observations  i n the f i r s t  and  profile from  3  C o e f f i c i e n t s of v a r i a t i o n s  were computed f o r each f o r m u l a t i o n w i t h i n each Hardl v a r i e d  frankfurters  replication  replication.  from 78.70  to 208.38  N  with a mean of 155.00 N, and i n the second r e p l i c a t i o n from 83.88 to for  218.13 the 20  replication  N with a mean of 156.67 formulations evaluated the  N. The o v e r a l l mean f o r Hardl was 155.84  N.  c o e f f i c i e n t s of v a r i a t i o n v a r i e d  In the f i r s t from  18.1% and i n the second r e p l i c a t i o n from 4.1 t o 15.2%. -146-  5.6  to  Table  22. E x p e r i m e n t a l  Replication No. 1  Formulation No." 1 2 3 4 5 6 7 8 9 10  mean 2  mean  k  • ° ° * " ° H 1 J  data  1 2 3 4 5 6 7 8 9 10  tot  textural  Hardl (N)°  parameters  Hard2 (N) D  ot the cooked crank f u r t e r s . *•  Firm {U/mm)  Cones" K  Spring (mm)°  Gummy (N)  Chewy (Nmm)  H  1  Shear (N)  J  161.32(5.6) 208.38(8.6) 113.41(7.7) 78.70(9.6) 186.87(13.2) 141.85(18.1) 149.56(8.1) 161.83(8.0) 158.72(13.6) 189.42(6.9)  99.44(13.1) 126.72(16.6) 73.58(4.7) 55.46(18.6) 135.16(12.4) 99.98(8.6) 98.48(15.5) 105.52(7.3) 104.89(12.4) 147.99(19.1)  19.14(14.4) 18.92(11.9) 15.56(8.2) 13.84(5.6) 19.41(11.3) 18.60(12.4) 12.78(16.1) 14.64(12.0) 19.24(8.6) 23.11(5.0)  0,.259(15.2) 0.,271(14.1) 0.,248(6.0) 0,,238(11.2) 0.258(12.0) 0.,293(4.8) 0,,266(12.0) 0,,259(19.5) 0..264(7.2) 0,,318(10.3)  3 .76(23 , .4) 3.,59(19 .2) 3,,67(7. 4) 3,,67(12 .9) 3,,34(8. 2) 4.,34(16 .6) 3,,42(16 .7) 3,,59(20 .6) 4.,34(6. 3) 4.76(14 .5)  41,.54(11 .6) 56,,24(14 .5) 28,,06(3. 8) 18,,70(12 .4) 48,,55(22 .8) 41,,70(21 .0) 39,,93(19 .4) 41,,74(18 .7) 41.,94(15 .5) 60,,48(15 .7)  159.,02(34 .9) 205,.79(32 .2) 102,,93(5. 0) 68,,83(18 .4) 163,.79(29 .9) 179 ,,15(19 .3) 138,.29(30 .5) 153,.63(38 .6) 182,.86(20 .1) 290,.67(25 .3)  5. 44(18 .4) 5. 93(22 .2) 5. 80( 24 .7) 3. 61( 11 .3) 6 .46(12 . 4 ) 5. 6 9 ( 3 . 3) 3. 28(17 .3) 5. 24(14 .3) 6. 50(11 • 8) 7.02(12 • 2)  155.00  104.72  17.52  0.,267  3,,85  41..89  164,.50  5. 50  131.70(9.8) 151.90(5.2) 125.44(9.3) 83.88(9.0) 201.21(4.1) 191.68(15.2) 126.36(14.7) 159.21(11.5) 177.22(8.9) 218.13(9.4)  86.79(4.0) 101.16(18.4) 105.02(13.4) 63.23(14.8) 120.78(9.3) 133.25(4.1) 86.15(4.3) 91.35(13.0) 129.98(5.4) 166.17(13.9)  13.12(3.3) 19.31(12.4) 17.85(7.9) 11.75(4.0) 18.99(8.6) 17.94(8.5) 15.29(7.9) 16.25(2.2) 14.19(4.3) 19.40(14.9)  0,,269(13.8) 0,,286(8.6) 0, 274(5.6) 0.,236(6.2) 0,,274(7.6) 0,.285(6.6) 0,,245(6.6) 0.,237(9.5) 0,,272(5.9) 0.,311(9.4)  3 ,26(9. , 8) 2.,84(6. 8) 3 ,26(5. , 1) 3..34(8. 2) 3,,42(16 .7) 3 .26(15 .4) 2 ,92(10 , .9) 2,,84(6. 8) 3 .67(7. , 4) 3,,34(8. 2)  35.71(22 .7) 4 3 . 3 8 ( 6 . 0) 34.44(14 .2) 19.,85(13 • 2) 54,,98(6. 3) 54.,33(9. 7) 30,,76(11 .9) 37.,94(19 .1) 48,.28(14 .1) 67,,79(12 .4)  115.48(19 .4) 1 2 3 . 1 7 ( 9 . 6) 112.05(14 .7) 65,,93(10 .3) 188,,33(17 .6) 176,.27(15 .4) 89 , ,08(4. 0) 107,.54(19 .2) 177,.66(16 .8) 227,,19(18 .2)  4 .44(18 .3) 6. 53(11 .9) 4 .14(4. 4) 4 .60(2. 1) 5. 22(17 .6) 6 .28(8. 0) 4 .14(6. 2) 4 .71 ( 49) . 5 .30(8. 2) 6 .53(10 .4)  156.67  108.39  16.97  0.269  3,.22  42 , .75  138 .27  5 .19  R e s u l t s r e p o r t e d a r e t h e mean ( c o e f f i c i e n t o f v a r i a t i o n ) from 4 o b s e r v a t i o n s f o r t h e t e x t u r e p r o f i l e p a r a m e t e r s and from 3 o b s e r v a t i o n s f o r maximum s h e a r f o r c e Ingredients proportions are reported i n Table 2 Hardness a t f i r s t c o m p r e s s i o n , Nevtons Hardness a t second c o m p r e s s i o n , Nevtons F i r m n e s s , Newtons/ml11imeter Cohesiveness Springiness, millimeters Gumminess, N e v t o n s C h e v l n e s s , Nevtons m i l l i m e t e r s Maximum s h e a r f o r c e , N e v t o n s  Hard2  varied  in  the f i r s t  from 55.46 to 147.99 N  replication  with a mean of 104.72 N, and i n the second r e p l i c a t i o n from 86.15 to 166.17 N for  with a mean of 108.39  the  N. The o v e r a l l mean f o r Hard2  20 f o r m u l a t i o n s evaluated was 106.56 N. In  replication  the c o e f f i c i e n t s  of v a r i a t i o n  varied  the f i r s t  from  4.7  to  19.1% and i n the second r e p l i c a t i o n from 4.0 to 18.4%. Fracturabi1ity  data i s not reported due  to  the  encountered i n measuring t h i s parameter.  In some  samples  frankfurter  obtained  from  a  particular  c o n t r a d i c t o r y r e s u l t s , that i s , a one  sample  fracture  point  while i n the other t h i s point was  difficulties  cases,  the 2  presented  was present i n  absent  from  the  curve. Firm v a r i e d  i n the f i r s t  with  a mean of 17.52  11.75  to 19.40 N/mm  replication  N/mm,  from 12.78 to 23.11  and i n the second r e p l i c a t i o n  with a mean of 16.97 N/mm.  replication  the  coefficients  of v a r i a t i o n  to 16.1% and i n the second r e p l i c a t i o n Cohes v a r i e d with a  mean  in  the  of 0.267,  first  from  The o v e r a l l mean  for Firm f o r the 20 f o r m u l a t i o n s e v a l u a t e d was 17.25 N/mm. first  N/mm  varied  In the  from 5.0  from 2.2 to 14.9%.  replication  from 0.238 to 0.318  and i n the second r e p l i c a t i o n  from 0.236  to 0.311 with a mean of 0.269. The o v e r a l l mean f o r Cohes f o r the 20 f o r m u l a t i o n s evaluated was 0.268. In the f i r s t coefficients  of v a r i a t i o n  second r e p l i c a t i o n Spring  varied  from  varied  mm,  t o 19.5%  and i n the  5.6 to 13.8%.  i n the f i r s t  with a mean of 3.85  from 4.8  r e p l i c a t i o n the  r e p l i c a t i o n from 3.34  to 4.76  and i n the second r e p l i c a t i o n -148-  mm  from 2.84  to  3.67 mm with a mean of 3.22  for  the 20  replication  mm.  The o v e r a l l  f o r m u l a t i o n s evaluated was 3.54 the c o e f f i c i e n t s  of v a r i a t i o n  23.4% and i n the second r e p l i c a t i o n Gummy v a r i e d  i n the f i r s t  mean f o r Spring  mm.  In  varied  the  first  from 6.3  to  from 5.1 t o 16.7%.  replication  from 18.70  t o 60.48  with a mean of 41.89 N, and i n the second r e p l i c a t i o n  N  from 19.85  to 67.79 N with a mean of 42.75 N. The o v e r a l l mean f o r Gummy f o r the  20  formulations  replication  evaluated  the c o e f f i c i e n t s  was 42.32  of v a r i a t i o n  22.8% and i n the second r e p l i c a t i o n Chewy v a r i e d with  i n the f i r s t  a mean of 164.50  65.93  first  replication  replication  and i n the second r e p l i c a t i o n  the c o e f f i c i e n t s  N with a mean of 5.19 N.  replication  second r e p l i c a t i o n As  noted  textural  among  different  addition  varied  of v a r i a t i o n  Nmm.  varied  In from  from 4 t o 19.4%.  from 3.28 to 7.02 N with from 4.14 t o 6.53  In the f i r s t  from 3.3  r e p l i c a t i o n the  t o 24.7%  and i n the  from 2.1 t o 18.3%  by Breene (1975),  their  from  The o v e r a l l mean f o r Shear f o r the 20  f o r m u l a t i o n s evaluated was 5.35 N. of v a r i a t i o n  to  The o v e r a l l mean  f o r m u l a t i o n s evaluated was 151.39  i n the f i r s t  3.8  from 68.83 to 290.67 Nmm  a mean of 5.50 N, and i n the second r e p l i c a t i o n  coefficients  from  first  from 6.0 t o 22.7%.  5.0 t o 38.6% and i n the second r e p l i c a t i o n Shear v a r i e d  In the  varied  t o 227.19 Nmm with a mean of 138.27 Nmm.  for Chewy f o r the 20 the  Nmm,  N.  meat and meat products  homogeneity with meat  a i r pockets,  variations  samples but a l s o  observed  within  which are more p r e v a l e n t -149-  a  vary i n not only  sample.  In  i n laboratory  prepared  sausages than i n commercial products,  more heterogeneous  (Whiting,  1987b).  make the product-  Variability  of t e x t u r a l  parameters of the f r a n k f u r t e r s were w i t h i n the normal found i n  other s t u d i e s .  parameters  were  researchers.  variability  C o e f f i c i e n t s of v a r i a t i o n s of  calculated  For example,  from  data  i n Montejano  reported  textural  by  et a l .  several  (1985), maximum  c o e f f i c i e n t s of v a r i a t i o n f o r beef, pork and t u r k e y g e l s obtained were  8.6%  f o r hardness,  26.2%  for cohesiveness,  5.8%  for  s p r i n g i n e s s , 27.0% f o r gumminess, 30.0% f o r chewiness. R e s u l t s of Park  et  a l . (1989)  coefficients  of  springiness, 26.0%  working with f r a n k f u r t e r s  variation  13.6%  with  f o r hardness,  f o r cohesiveness,  f o r chewiness.  working  of 15.6%  Similarly,  frankfurters  showed  22.0%  maximum  12.5%  for  f o r gumminess and  Sofos and A l l e n  obtained  maximum  (1977)  also  coefficients  of  v a r i a t i o n of 22.0% f o r hardness. As  can  Montejano  be noted i n t h i s study as w e l l as those et a l .  parameters of coefficients both  (1985)  and Park e t a l .  (1989)  gumminess and chewiness present of v a r i a t i o n .  parameters  This v a r i a b i l i t y  are c a l c u l a t e d  reported the  by  textural  r e l a t i v e l y higher i s expected  since  from the measured parameters  of  hardness, cohesiveness and s p r i n g i n e s s .  2.5. D e t e r m i n a t i o n o f pH The  pH of meat i s considered to be one of the most  f a c t o r s a f f e c t i n g meat q u a l i t y . functional  important  pH i s a s s o c i a t e d with changes i n  p r o p e r t i e s of the meat p r o t e i n s , -150-  i . e . water and f a t  b i n d i n g c a p a c i t i e s and g e l a t i o n ,  thus a f f e c t i n g y i e l d ,  s t a b i l i t y and t e x t u r e of meat products (Solomon, Measurements duplicate. first  of  pH of the raw emulsions  R e s u l t s are shown i n F i g u r e 11.  r e p l i c a t i o n from 5.53  emulsion  1987).  were  performed  The pH v a r i e d  to 5.93 with a mean of 5.74,  in  i n the and i n  the second r e p l i c a t i o n from 5.41 to 5.80 with a mean of 5.60. The o v e r a l l mean f o r pH f o r the 20  emulsions e v a l u a t e d was 5.67. I t  i s r e a d i l y apparent  pH  replication  that  the  of  d i f f e r e d q u i t e markedly.  the  raw emulsions w i t h i n  In a d d i t i o n the pH of the  raw emulsions of the second r e p l i c a t i o n were lower than the pH of the emulsions of the f i r s t  r e p l i c a t i o n . The meat samples  used f o r  the p r e p a r a t i o n of the f r a n k f u r t e r s of the second r e p l i c a t i o n had been frozen earlier,  frozen storage  ingredients. frozen protein  a t -30°C f o r approximately two months.  Changes  had an e f f e c t on the i n the pH of meats  pH  during  storage have been r e p o r t e d to c o n t r i b u t e t o alterations  variability  of  the  (Powrie, quality  1973). of  the  Therefore, frankfurter  As mentioned of  the  meat  freezing  and  myofibrillar the  formulations  between r e p l i c a t i o n s can be a t t r i b u t e d a l s o to changes of the meat i n g r e d i e n t s due to f r o z e n s t o r a g e .  -151-  observed  i n the  pH  •  REP2  H REP1 FORMULATION No.  Figure 11. Mean pH values of the raw emulsions. Repl and Rep2 a r e r e p l i c a t i o n 1 and 2 r e s p e c t i v e l y .  -152-  3. Q u a l i t y p r e d i c t i o n m o d e l s 3.1.  Regression a n a l y s i s  M u l t i p l e r e g r e s s i o n analyses were c a r r i e d out f o r each q u a l i t y parameter  evaluated  as  the  dependent  variable  to  obtain  mathematical r e l a t i o n s h i p s between the q u a l i t y parameters and p r o p o r t i o n of the three i n g r e d i e n t s : pork f a t ( X i ) , deboned p o u l t r y meat ( X ) , 2  and  beef meat ( X ) .  The  MGLH module of SYSTAT (Wilkinson,  Scheffe's  canonical  (equation 23) and  to  39  performed  w e l l as R A ,  on  standard  variance  table  (1974).  Residuals  the  data  of  Seventeen  indicated  three  components 19  to  22  each  the r e g r e s s i o n  quality  parameter  regression c o e f f i c i e n t s  t - t e s t on the c o e f f i c i e n t s ,  e r r o r of the estimate  and  were  analyzed  quality  by examining  models  the  a n a l y s i s of v a r i a n c e of the  that  the  F-values  for  regression  plots  24).  were  of  of Snee  residual  ( F i g u r e s 12 to  prediction  as  the a n a l y s i s  which f o l l o w s the approach i n Marquadt and  ( r e s i d u a l s versus p r e d i c t e d values)  Examination  used to f i t  summarize the r e s u l t s of  the r e s u l t s of the Student's J  was  data shown i n Tables  e v a l u a t e d . Each t a b l e shows the estimated and  1988a)  11. 23  analyses  and  i n Appendix E.  c u b i c model for  to the experimental  Figure Tables  special  mechanically  Nomenclature  3  d e f i n i t i o n of the q u a l i t y parameters are given  the  developed.  fitted  were  models  significant  (p<0.05),  meaning that the q u a l i t y parameters were dependent  the mixture  components.  the models possessed  With the exception of the ES model,  no s i g n i f i c a n t -153-  on  all  lack of f i t (p>0.05). Further  d i s c u s s i o n on the ES model i s given below. R*A.  The  (Table 25) values shrink  values v a r i e d  to 0.88  from 0.23  f o r the Cook  f o r the Exwater model (Table 30).  were found  f o r Cook s h r i n k (0.23) (Table 25)  (0.41) (Table 24) models.  The  poor f i t was  the r e p l i c a b i 1 i t y of Vacuum s h r i n k and In  addition,  Vacuum the  s h r i n k data the  (Table 19) model.  The  presented  a low R * * . value  accounted  for  data,  i t was  prediction models  purposes.  were  proportion  greater  the  The than  For  Shrink,  found  for  Firm  these  data  on  also  Since these models in  the  observed  models were not adequate f o r  R * . values for the 2  0.60,  which  other  implied  i n the responses  fourteen  that was  a good  explained  these  responses  by SYSTAT i n d i c a t e d  Exfluid the  and  Exfat  observations  that  presented  from the  two  to the o u t l i e r s were removed from  the  the remaining data were r e a n a l y z e d .  Analysis  of  residuals,  r e s i d u a l s versus  performed  1981). The  an approximation  by examining  the  predicted values, i s a q u a l i t a t i v e  a s s e s s i n g the adequacy of the f i t t e d Smith,  to  of the v a r i a t i o n  Tmloss,  r e p l i c a t i o n s corresponding data and  values  models.  from  outliers.  poor.  have had a negative e f f e c t  a n a l y s i s of r e s i d u a l s performed  data  and Vacuum  Cook s h r i n k data was  model f i t t e d  of the t o t a l v a r i a t i o n  by the f i t t e d The  that  R * *  Low  since  (0.56) (Table 36).  l e s s than 60% concluded  may  model  expected  the r e l a t i v e l y s m a l l range of  f i t of  shrink  model (Zar, 1984;  p l o t s of  measure for Draper and  r e s i d u a l p l o t s presented must be considered as  of the d i s t r i b u t i o n of the r e s i d u a l s , -154-  s i n c e , as  reported  by  distribution  C o r n e l l (1981), of  the  it  residuals  is  d i f f i c u l t to see  about  a random  zero with l e s s  than  30  observations. Analysis (Table  of  24),  r e s i d u a l s was  Cook shrink  not performed  (Table  25), and  due  to the poor f i t of these models,  due  to the m o d i f i c a t i o n s  below.  The  Hard2 (Figure 20),  Chewy (Figure  24)  Shear  Exfat  above and  the adequacy  of the  model  discussed  (Figure  12),  (Figure 17),  pH  that  Gummy (Figure the  residuals  below zero. For  f i t t e d models.  the r e s i d u a l p l o t s for Twloss (Figure 14), (Figure 19), and  Cohes (Figure 22)  r e s i d u a l s formed nonrandom  patterns.  model, i n c r e a s i n g v a r i a b i l i t y i n c r e a s i n g Hardl observed  of  (Figures 14 and  the  these  leaving  a  assumptions  gap of  (Figure  15),  models showed that  their  In the case of the r e s i d u a l s was  (Figure 19).  Cohes models,  22 r e s p e c t i v e l y ) .  expressible  Exfluid  The  that  Exfluid  be seen i n F i g u r e  15  constant  variance -155-  For  Hardl  observed with  The  opposite  is,  was  decreasing values  model  formed  a  values  between  10  f l u i d were not p r e d i c t e d by the model,  i n between these v a l u e s .  be  However  r e s i d u a l s with i n c r e a s i n g p r e d i c t e d  p e c u l i a r p a t t e r n . As can 12%  of the  p r e d i c t e d values  for the Twloss and  variability  and  models  28)  as  (Figure 21),  models showed  36)  (Table  Shrink  shrink  the assumptions about the r e s i d u a l s d i d not appear to  v i o l a t e d , suggesting  Hardl  (Table  f o r ES  (Figure 16),  appeared to be d i s t r i b u t e d evenly models  and  a n a l y s i s of r e s i d u a l p l o t s of  (Figure 18), and  Firm  Vacuum  performed on the ES data,  Tmloss (Figure 13), Exwater  23),  for  these models,  appeared to be v i o l a t e d  thus, the (Zar,  1984;  Draper and  Smith,  the  observations  and  Draper  and  1981). The  of these responses as suggested Smith (1981)  could be  assumptions about the r e s i d u a l s . not  performed  logarithmic transformation  and  the  by Zar  the  However the t r a n s f o r m a t i o n  was  were  to  (1984)  meet  models  performed  on  c o n s i d e r e d adequate  for  p r e d i c t i o n purposes. A the  model with an R**. g r e a t e r than data  (Table  obtained  20)  from S a f f l e ' s  T h i s was  repeatability  could not be f i t t e d  emulsion  expected,  of the r e s u l t s  0.39  since  was  poor.  stability  test  (ES)  noted  earlier  the  as The  a n a l y s i s of r e s i d u a l s  of t h i s model i n d i c a t e d 2 o u t l i e r s . However, r a t h e r than the  observations  outliers  (3  and  Exfat  models,  using  the  replicate,  value i.e.  of  to  be  Exfluid  and  the values of the o b s e r v a t i o n s were modified  by  7) as performed for Shrink, Tmloss,  of the f o r m u l a t i o n number  and  from  2.45  0.90%.  for  (b) f o r m u l a t i o n 7 Table  performed on t h i s new  accounted f o r 72% regression  28  of the v a r i a t i o n was  significant  lack of f i t (p<0.05).  model  not  performed.  The  performed because a model was account  f o r the s t a b i l i t y  in terms of r e l e a s e d f a t .  the  2 was  replication  in the data  (p<0.05),  changed from 1 was  the  the f i t t e d model and model  m o d i f i c a t i o n s to the formula  the  F-value  presented  ES model was  data  this were  o p t i m i z a t i o n to  of the emulsions to thermal  -156-  changed  A n a l y s i s of r e s i d u a l s of  needed f o r  The  opposite  shows the r e g r e s s i o n a n a l y s i s  data s e t . Although  significant was  from  (a) f o r m u l a t i o n 3 r e p l i c a t i o n  to 0.60%,  results  removing  the 2 f o r m u l a t i o n s that were found  1.45  to  to  considered  treatment only as an  approximation  of  the amount  of  f a t lost  during  thermal  processing. Table  40  t a b u l a t e s the r e g r e s s i o n models i n t h i s study  that  were considered  t o be adequate i n r e p r e s e n t i n g the e x p e r i m e n t a l l y  observed data,  and thus t o be used f o r response s u r f a c e a n a l y s i s  and  formula  optimization.  Shear and Chewy  As can be observed,  were explained  ES, E x f a t ,  s o l e l y by l i n e a r b l e n d i n g  of the i n g r e d i e n t s ( i . e . l i n e a r terms).  Tmloss,  and Cohes were e x p l a i n e d  nonlinear  b l e n d i n g e f f e c t s of the i n g r e d i e n t s ( i . e . Shrink,  Hardl,  effects  Twloss, E x f l u i d ,  Exwater,  q u a d r a t i c terms).  pH,  by l i n e a r blending and b i n a r y  Hard2  linear  and  and Gummy were e x p l a i n e d  by l i n e a r b l e n d i n g , b i n a r y and t e r n a r y n o n l i n e a r blending  effects  of  terms).  the i n g r e d i e n t s  Synergistic  (i.e  linear,  q u a d r a t i c and c u b i c  and a n t a g o n i s t i c e f f e c t s of the b i n a r y mixtures  s y n e r g i s t i c e f f e c t s of the t e r n a r y fat  and m e c h a n i c a l l y  synergistic  effects  mixture were observed.  deboned p o u l t r y meat on  Tmloss  had  (XiX ) 2  and Twloss,  and Pork  binary  and a n t a g o n i s t i c  e f f e c t s on Shrink, E x f l u i d , Exwater, H a r d l , Hard2 and Gummy. Pork f a t and beef meat ( X x X ) had b i n a r y s y n e r g i s t i c e f f e c t s on Tmloss 3  and  Twloss,  Mechanically  and a n t a g o n i s t i c  effects  Gummy.  on Shrink,  Cohes  and Gummy.  deboned p o u l t r y meat and beef meat ( X X ) had b i n a r y 2  a n t a g o n i s t i c e f f e c t s on Shrink, and  on  The t e r n a r y mixture  Exfluid, (XxX X )  H a r d l , Hard2, and Gummy.  -157-  2  3  3  Exwater, H a r d l , Hard2, had a s y n e r g i s t i c  effect  Table 23. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o per cent weight l o s s a f t e r p r o c e s s i n g (Shrink) data. x  Est imated regression coeff i c i e n t (bi)  Var i a b l e  X Xo.X  Standard e r r o r of coeff i c i e n t ( S b i  2  X Xa 2  Xa.X X3 2  )  6.11 0.27 55.81 10.99 105.94  17.13 11.83 -104.50 -21.18 231.70  2  t-value  2.81* 43.16* -1.87n.s. -1.93n.s. 2.19"  R**= 0.669 Standard e r r o r of estimate= 0.427 * s i g n i f i c a n t l y d i f f e r e n t from zero (p<0.05) n.s.= not s i g n i f i c a n t l y d i f f e r e n t from zero (p>0.05)  A n a l y s i s of v a r i a n c e Source of variation  Degrees of freedom  Sum of squares  Mean square  F-value  Regress i on  4  6.98  1.74  9 . 58*  2 . 37  0 . 18  Residual  13  Lack of f i t  4  0.87  0.22  Pure  9  1. 50  0.17  17  9.35  Total  error  1. 31n  " s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05) *• Formulation 10 was found t o be an o u t l i e r and the corresponding o b s e r v a t i o n s of the two r e p l i c a t i o n s were not considered i n the r e g r e s s i on  -158-  0.76  0.38 GO  < ZD Q  0.00  GO UJ  rr  -0.38 H  -0.76  8  7  9  PREDICTED VALUES  F i g u r e 12. P l o t  of r e s i d u a l s  f o r the Shrink model.  -159-  10  Table 24. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to per cent weight l o s s a f t e r 13 days under vacuum packaged storage (Vacuum s h r i n k ) data.  Variable  X X  2 3  Est imated regression coefficient  Standard e r r o r of coefficient  (bi)  (Sbi  1.07 2 .44  0.37 0.13  R**= 0.414 Standard e r r o r iof estimate= 0. 292 * s i g n i f i c a n t l y d i f f e r e n t from zero  t-value  )  2.91* 18.83*  (p<0.05)  A n a l y s i s of v a r i a n c e Source of variation  Degrees of freedom  Sum of squares  Regression  1  1.230  1.230  18  1.536  0.085  8  0.680  0.085  10  0.856  0.086  19  2.77  Res i d u a l Lack of f i t Pure e r r o r Total  " s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05)  -160-  Mean square  F-value  14.41*  0.99n.s  Table 25. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o per cent weight l o s s a f t e r the consumer cook t e s t (Cook s h r i n k ) data.  Variable  Xa. X 3  Estimated regression coefficient  Standard e r r o r of coefficient  (bi )  (Sbi)  16.95 6.77  4.77 1.39  R**= 0.2 26 Standard e r r o r of estimate= 2. 481 * s i g n i f i c a n t l y d i f f e r e n t from zero  t-value  3.56" 4 .'89*  (p<0.05)  Analys i s of v a r i a n c e Source of var i a t ion Regress ion Res i d u a l Lack of f i t Pure e r r o r Total  Degrees of freedom  Sum of squares  Mean square  1  40.40  40.40  18  110.79  6.16  8  29.43  3.68  10  81.36  8.14  19  151.19  " s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05)  -161-  F-value  6.56*  0.4 5n.s .  Table 26. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o emulsion s t a b i l i t y (Tmloss) d a t a . *  Variable  Estimated regression coefficient  Standard e r r o r of coefficient  (bi)  X X X1X2 2  3  X3.X3  t-value  (Sbi)  16.43 39.69 68.15 69.65  3.58* 24.412 .12" 7. 41-  4 . 59 1.63 32.22 9.40  R**= 0.789 Standard e r r o r of estimate= 1.979 - s i g n i f i c a n t l y d i f f e r e n t from zero (p<0. 05)  Analys i s of v a r i a n c e Sum of squares  Mean square  3  260.89  86.96  14  54 .84  3 .92  Lack of f i t  5  13.04  2.61  Pure e r r o r  9  41.80  4.64  17  315.73  Source of var i a t ion Regress ion Res i d u a l  Total  Degrees of freedom  F-value  22 .20"  0 . 56n.s.  " s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05) *• Formulation 7 was found to be an o u t l i e r and the corresponding o b s e r v a t i o n s of the two r e p l i c a t i o n s were not c o n s i d e r e d i n the r e g r e s s i o n  -162-  3.6  2.4 f  1.2 h GO _J  <  Q  0.0  GO UJ DC  1.2  -2.4 h -3.6  30  40  35  PREDICTED VALUES  F i g u r e 13. P l o t of r e s i d u a l s  f o r the Tmloss model  -163-  45  Table 27. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o emulsion s t a b i l i t y (Twloss) data.  Variable  Xa Xa.Xa XiXa  Estimated regression coefficient (bi)  Standard e r r o r of coefficient (s i)  t-value  b  3.45 1.21 24 .25 6.58  10.11 29.75 29 . 1 1 16.48  2.93* 24.69* 1. 20n.s 2.51*  R**.= 0.755 Standard e r r o r of estimate= 1.490 * s i g n i f i c a n t l y d i f f e r e n t from zero (p<0.05) n.s. not s i g n i f i c a n t l y d i f f e r e n t from zero (p>0.05)  A n a l y s i s of v a r i a n c e Source of variation  Degrees of freedom  Regression  3  136.94  45.65  16  35.51  2 .22  6  10.20  1.70  10  25. 31  2.53  19  172.45  Residual Lack of f i t Pure e r r o r Total  Sum of squares  * s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05)  -164-  Mean square  F-value  20.57*  0.67n.s  3.2  1.6 f-  < Q CO LU  0.0  cr  -1.6 h  -3.2  20  22  24  26  28  PREDICTED VALUES  F i g u r e 14. P l o t of r e s i d u a l s  f o r the Twloss model.  - 1 6 5 -  30  Table 28. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o emulsion s t a b i l i t y (ES) data.*-  Var i a b l e  Estimated r e g r e s s ion coefficient  Standard e r r o r of coeff i c i e n t  (bi)  (Stoi )  2.61 -0.19  Xo. X 3  t-value  0.32 0.09  8.05" -1.96n.s.  R**= 0.724 Standard e r r o r of estimate= 0.169 * s i g n i f i c a n t l y d i f f e r e n t from zero (p<0.05) n.s.= not s i g n i f i c a n t l y d i f f e r e n t from zero (p>0.05)  Analys i s of v a r i a n c e Source of var i a t ion  Degrees of freedom  Sum of squares  Mean square  F-value  Regression  1  1.45  1. 45  50.83"  18  0.51  0 .03  8  0.37  0.05  10  0.15  0 . 01  19  1.96  Res i d u a l Lack of f i t Pure e r r o r Total  3.17"-  " s i g n i f i c a n t (p<0.05) -- s i g n i f i c a n t lack of f i t (p<0.05) *• response values f o r f o r m u l a t i o n s 3 and 7 were m o d i f i e d  -166-  Table 29. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to per cent e x p r e s s i b l e f l u i d ( E x f l u i d ) data.*-  Estimated regression coefficient (b )  Variable  Standard e r r o r of coefficient (s )  t  X X  XiX X X 2  D l  13.67 20.64 12.13 -31.93 -27.43  Xa. 2 3  2 3  t-value  3 . 31 9.37 0.82 19.97 16.24  4 .13* 2.20* 14.83* -1.60n.s. -1.69n.s.  0.678 Standard e r r o r of estimate= 1.029 * s i g n i f i c a n t l y d i f f e r e n t form zero (p<0.05) n.s.= not s i g n i f i c a n t l y d i f f e r e n t from zero (p>0.05)  R*A=  A n a l y s i s of v a r i a n c e Source of variation  Degrees of freedom  Sum of squares  Mean square  F-value  4  42. 06  10.52  9.94*  13  13.76  1.06  Lack of f i t  4  0.16  0.04  Pure e r r o r  9  13.60  1.51  Regress i on Res i d u a l  Total  0.03n  17  * s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05) *• Formulation 7 was found to be an o u t l i e r and the corresponding o b s e r v a t i o n s of the two r e p l i c a t i o n s were not c o n s i d e r e d i n the r e g r e s s i o n  -167-  1.6  0.8  h  _j <  Q  0.0 h  CO UJ cr  -0.8  -1.6  8  9  10  11  12  PREDICTED VALUES  F i g u r e 15. P l o t of r e s i d u a l s  f o r the E x f l u i d model.  -168-  13  Table 30. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o per cent e x p r e s s i b l e water (Exwater) data.  Var i a b l e  X X Xa.X X X 2  3  2  2  3  Estimated regression coeff i c i e n t (bi)  Standard e r r o r of coeff i c i e n t  t-value  (Sbi )  18.15 10. 58 -35.57 -25.31  5.63 0.31 8.69 9.67  3.22' 34 .66' -4.09' -2.62'  R**= 0.882 Standard e r r o r of estimate= 0. 632 * s i g n i f i c a n t l y d i f f e r e n t from zero (p<0. 05)  A n a l y s i s of v a r i a n c e Source of var i a t ion Regression Res i d u a l Lack of f i t Pure e r r o r Total  Degrees of freedom 3 16  Sum of squares 57 .98  Mean square 19 . 33  6 . 39  0.40  6  0.78  0 .13  10  5. 61  0.56  19  64 . 37  " s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05)  -169-  F-value  48.41"  0.23n.s  1.2  0.6 CO _i <  ZD Q  0.0 \-  CO  111  DC  -0.6  -1.2  4  5  6  7  8  9  PREDICTED VALUES  Figure  16.  Plot  of  residuals  for  -170-  the  Exwater  model.  10  11  Table 31. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o per cent e x p r e s s i b l e f a t (Exfat) data.*-  Variable  Xa. X X 2  3  Estimated regression coefficient (bi)  Standard e r r o r of coefficient (s )  t-value  D i  15.01 1.72 1.24  1.36 0.80 0.39  R' = 0.811 Standard e r r o r of estimate= 0.629 * s i g n i f i c a n t l y d i f f e r e n t from zero  11.05* 2.16* 3.16*  A  (p<0.05)  A n a l y s i s of v a r i a n c e Source of variation  Degrees of freedom  Sum of squares  Mean square  F-value  Regress ion  2  29 .69  14.84  37 .55*  15  5.93  0.40  Lack of f i t  6  0.45  0.07  Pure e r r o r  9  5.48  0.61  17  35.62  Res i d u a l  Total  0.12n  * s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05) *- Formulation 7 was found to be an o u t l i e r and the corresponding o b s e r v a t i o n s of the two r e p l i c a t i o n s were not considered i n the r e g r e s s i o n  -171-  1.28  I  i  i  0  0.64 \-  o 0  co < 3 Q  o  1  o  o 0  0.00 -  CO UJ DC  -  0  ° 0  o -0.64  1.28  o  o h  o  2  o  o  1  1  1  3  4  5  PREDICTED VALUES  F i g u r e 17. P l o t of r e s i d u a l s f o r the E x f a t model.  -172-  6  Table 32. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o pH d a t a .  Estimated regression coefficient  Var i a b l e  Standard e r r o r of coefficient  (bi)  x  X  (Sbi  5.82 6.34 5.45  2 3  t - value  )  0.17 0.11 0.05  35 .01* 58 .30* 101 .69*  R**.= 0.6 69 Standard e r r o r of estimate= 0. 086 * s i g n i f i c a n t l y d i f f e r e n t from zero (p<0.05)  A n a l y s i s of v a r i a n c e Source of var i a t i o n  Degrees of freedom  Sum of squares  Mean square  F-value  Regression  2  0.302  0.151  20.22*  17  0.127  0.007  7  0.023  0.003  10  0.104  0.010  19  0.429  Res i d u a l Lack of f i t Pure e r r o r Total  * s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05)  -173-  0.31n.s.  0.16  0.08 CO _i <  Q  0.00  CO LU  DC  -0.08 H  -0.16  5.4  5.5  5.6  5.7  PREDICTED VALUES  F i g u r e 18. P l o t of r e s i d u a l s  f o r the pH model.  -174-  5.8  5.9  Table 33. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o hardness at f i r s t compression (Hardl) data.  Variable  Estimated regression coefficient (b )  Standard e r r o r of coefficient (s )  t  X X XxXa X X Xo.XaX 2  3  2  3  3  t-value  b l  795.99 212.25 -8308.69 -1131.69 1 4 4 1 0 . 51  281.44 9.60 2574.38 505.29 4 8 7 5 . 70  R«*.= 0.737 Standard e r r o r of estimate= 19.738 * s i g n i f i c a n t l y d i f f e r e n t from zero  2.83* 22.12* -3.23* -2.24* 2.96*  (p<0.05)  A n a l y s i s of v a r i a n c e Source of variation  Degrees of freedom  Regression  4  Res i d u a l Lack of f i t Pure e r r o r Total  Sum of squares  Mean square  F-value  22328. 86  5582 .22  14.33*  15  5844. 13  389 .61  5  1524 .08  304 .82  10  4320. 05  432. 01  19  28172. 99  * s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05)  -175-  0.71n  40  I  I  I  1  20 CO  10  o o  ZD  Q CO HI  DC  o —  o  0  0 -10  —  o  o 0  _  _J  <  -  0  30 (-  o  0 0  o o  o o  -20 o  o o  -30 -40 80  0 I  1  1  1  106  132  158  184  PREDICTED VALUES  F i g u r e 19. P l o t of r e s i d u a l s  f o r the Hardl model.  -176-  210  Table 34. Regression s t a t i s t i c s c o r r e s p o n d i n g to the " b e s t " model f i t t e d to hardness at second compression (Hard2) d a t a .  Estimated regression coefficient (bi)  Variable  Xo. X X  -85. 73 431.91 167.61 -4397.06 -704.61 8456.16  a  3  Xa.X X X Xa.X X 2  2  Standard e r r o r of coefficient (s*i)  2 3 3  43.11 206.52 11.12 1883.81 380.12 3651.66  t-value  -1.99n.s. 2.09n.s. 15.07* -2.33" -1.85n.s. 2.32*  R'x= 0.739 Standard e r r o r of estimate= 14.245 * s i g n i f i c a n t l y d i f f e r e n t from zero (p<0.05) n.s.= not s i g n i f i c a n t l y d i f f e r e n t from zero (p>0.05)  A n a l y s i s of v a r i a n c e Source of variation  Degrees of freedom  Sum of squares  Regression  5  11925.08  2385 .02  14  2841.03  202 .93  4  596.87  149 .22  10  2244.16  224 .42  19  14766.11  Res i d u a l Lack of f i t Pure Total  error  * s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05)  -177-  Mean square  F-value  11.75*  0.67n  30 20 10 h CO _J  <  0  ZD Q  F-  CO UJ DC  10 -20 h -30  60  80  100  120  PREDICTED VALUES  Figure  20. P l o t  of r e s i d u a l s  f o r t h e Hard2 model  -178-  140  160  Table 35. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o maximum shear f o r c e (Shear) data.  Estimated regression coefficient (b )  Variable  Standard e r r o r of coefficient (s )  t  Xo. X X  b t  -1.79 4.71 7.33  2  3  t-value  1.26 0.82 0.41  -1.42n.s. 5.74* 18.12*  R'A.= 0.628 Standard e r r o r of estimate= 0.653 * s i g n i f i c a n t l y d i f f e r e n t from zero (p<0.05) n.s.= not s i g n i f i c a n t l y d i f f e r e n t from zero (p>0.05)  A n a l y s i s of v a r i a n c e Source of variation  Degrees of freedom  Regress ion Res i d u a l Lack of f i t Pure e r r o r Total  Sum of squares  Mean square  2  14.49  7.25  17  7.24  0.43  7  2. 39  0.34  10  4 .85  0.49  19  21.73  " s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05)  -179-  F-value  17.02*  0.70n  1.2  0.6 h  co _J <  ZD  Q  0.0 h  CO UJ DC  -0.6 \-  -1.2  6 PREDICTED VALUES  F i g u r e 2 1 . P l o t of r e s i d u a l s  f o r the Shear model.  -180-  7  Table 36. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o firmness (Firm) d a t a .  Variable  Estimated regression coefficient (b )  Standard e r r o r of coefficient (s )  t  X= X 3  t-value  b l  16.36 21.49  2.45 0.86  R**= 0.561 Standard e r r o r of estimate= 1.952 - s i g n i f i c a n t l y d i f f e r e n t from zero  6.6724.86"  (p<0.05)  A n a l y s i s of v a r i a n c e Source of variation Regress ion Res i d u a l Lack of f i t Pure e r r o r Total  Degrees of freedom 1  Sum of squares  Mean square  96 .,44  96 .44 .  18  68 . 58  3 .81 .  8  21., 20  2..65  10  47..38  4 .74 ,  19  165. .02  - s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05)  -181-  F-value  25,. 31-  0., 56n.s  Table 37. Regression s t a t i s t i c s corresponding t o the "best" model f i t t e d t o cohesiveness (Cohes) data.  Estimated regression coefficient (bi)  Variable  2 3 3  t-value  b i  0.110 0.020 0.010 0.195  0.291 0.222 0.328 -0.336  Xo.  X X XiX  Standard e r r o r of coefficient (s )  2.64*11.02* 32.54" -1.72n.s.  R**= 0.792 Standard e r r o r of estimate= 0.010 * s i g n i f i c a n t l y d i f f e r e n t form zero (p<0.05) n.s.= not s i g n i f i c a n t l y d i f f e r e n t from zero (p>0.05)  Analys i s of v a r i a n c e Source of variation  Degrees of freedom  Regress ion Res i d u a l Lack of f i t Pure e r r o r Total  Sum of squares  Mean square  F-value  3  0.0080  0.0030  25.05*  16  0.0020  0.0001  6  0.0008  0.0001  10  0.0012  0.0001  19  0.0100  * s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05)  -182-  1.18n.s.  0.02  0.01 CO  <  ZD Q  CO HI  0.00  h  -0.01  h  cr  -0.02 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 PREDICTED VALUES  F i g u r e 2 2 . P l o t of r e s i d u a l s  f o r the Cohes model.  -183-  Table 38. Regression s t a t i s t i c s corresponding t o the "best" model f i t t e d t o gumminess (Gummy) data.  Estimated regression coefficient  Variable  Standard e r r o r of coefficient  (soi)  (bi)  X X XiXa XiX XaX ,XaX  220.27 68.79 -2266.94 -56.08 -339.77 4015.16  a  3  3  3  3  t-value  81.38 4.84 745.24 25.98 150.29 1440.25  2.71* 14.22* -3.04* -2.16* -2.26* 2.79*  R**= 0.809 Standard e r r o r of estimate= 5.563 * s i g n i f i c a n t l y d i f f e r e n t from zero (p<0.05)  A n a l y s i s of v a r i a n c e Source of variation  Degrees of freedom  Regress ion Residual Lack of f i t Pure e r r o r Total  Sum of squares  Mean square  F-value  5  2651.14  530 . 23  17 .13*  14  433.32  30 .95  4  115.99  29 .00  10  317.33  31 .73  19  3084.46  * s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05)  -184-  0 .91n  10  5 r CO _J  <  ZD Q  0 f-  GO til DC  -5 H  -10  20  29  38  47  PREDICTED VALUES  F i g u r e 23. P l o t of r e s i d u a l s  f o r the Gummy model.  -185-  56  65  Table 39. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d t o chewiness (Chewy) data.  Estimated regression coefficient (b*)  Variable  Xi X X  Standard e r r o r of coefficient (s ) t o t  -219.77 46.17 275.30  2  3  t-value  50.03 32.74 16.13  -4.39* 1.41n.s. 17.07"  R'*= 0.781 Standard e r r o r of estimate= 26.013 * s i g n i f i c a n t l y d i f f e r e n t from zero (p<0.05) n.s. not s i g n i f i c a n t l y d i f f e r e n t from zero (p>0.05)  A n a l y s i s of v a r i a n c e Source of variation  Degrees of freedom  Sum of squares  Regression  2  47331.31  23665.66  17  11503.65  676 .68  7  2490.40  355.77  10  9013.25  901.33  Residual Lack of f i t Pure e r r o r Total  * s i g n i f i c a n t (p<0.05) n.s.= not s i g n i f i c a n t lack of f i t (p>0.05)  -186-  Mean square  F-value  34.97'  0.39n.s  50  30 -  co _ j <  10 f-  ZD Q  CO UJ DC  -10 k  -30 -  -50 80  110  140  170  200  PREDICTED VALUES  F i g u r e 24. P l o t of r e s i d u a l s  f o r the Chewy model.  -187-  230  260  Table 40. Q u a l i t y p r e d i c t i o n models. * " Shrlnk= 17.13X  + 11.83X  2  231.70XiX=X  Tmloss= 16.43X  B  - 104.50Xa.Xa - 21.18X X  3  2  + 39.69X  3  + 68.15XiX  2  + 69.65XxX  Tvloss= l O . H X a  + 29.75X  a  + 29.1lXxX  2  +16.48XxX  3  ES  -  - 31.93XxX  2  2  = 2.61Xx  0.19X  Exfat  + 10.58X  2  = 15.01X:L  + 1.72X2 + 1 . 2 4 X  Hardl = 795.99X  + 212.25X 2 14410.51XxX X 2  2  3  2  3  3  3  3  3  - 8308.69XxX  2  + 167.61X  2  - 1131.69X X 2  3  - 4397.06XxX  3  2  -  3  -  704.61X X 2  3  3  Shear = -1.79Xx + 4 . 7 1 X  2  + 7.33X  3  Cohes = 0.291X1. + 0.222X2 + 0.328X  3  - 0.336XxX  Gummy = 2 2 0 . 2 7 X + 68.79X - 2266.94XxX 339.77X2X + 4015.16X1X2X3 3  2  - 27.43X X  3  Hard2 = -85.73Xx + 4 3 1 . 9 1 X +8456.16XxX X  3  - 35.57XxX 2 - 25.31X=X  3  pH = 5.82Xi. + 6.34X2 + 5.45X +  +  3  E x f l u i d = 1 3 . 6 7 X i + 20.64X2 + 1 2 . 1 3 X Exvater= 1 8 . 1 5 X  3  3  2  3  - 56.08XxX  3  Chewy = - 2 1 9 . 7 7 X i + 4 6 . 1 7 X  2  +  275.30X  3  *• Nomenclature and d e f i n i t i o n of the q u a l i t y parameters are given i n Appendix E Xi=pork f a t ; X =MDPM; X =beef meat  s  2  3  -188-  3.2.  Response s u r f a c e contour  The  examination  reveal the  little  analysis  of the r e g r e s s i o n models l i s t e d  proportions  and  v i s u a l i z e these r e l a t i o n s h i p s , g r a p h i c a l l y as response of  equal  proportions. shape  These  understanding  of of  the  e f f e c t s of  response  surface  surface  Hunter,  e f f e c t s of changes the responses  fat, X ) 3  showing  forms  1966).  parameters studied how  proportions. three-  p a r t of any  response  Zabik,  1989;  1981;  Rockower  Huor et a l .  These p l o t s are used to study the ( i n g r e d i e n t s ) on  (Thompson, 1982).  38 i l l u s t r a t e the response  s u r f a c e contour  e f f e c t s of the i n g r e d i e n t p r o p o r t i o n s  on each q u a l i t y parameter. plots  an  and  (Agreda and Agreda,  Johnson and  the  provided  plots  X i ; m e c h a n i c a l l y deboned p o u l t r y meat,  these  on  the  in elucidating  i n the l e v e l s of the f a c t o r s  through  ingredient  ingredient  contour  plots  p l o t s are  different  evaluated ( q u a l i t y parameters)  F i g u r e s 25 plots  the  To  expressed  Contour  s u r f a c e s of these models and  C o r n e l l , 1981;  H i l l and  for  40  between  parameters.  the f i t t e d models were  values  s u r f a c e experimentation study  1980;  quality  p l o t s were most h e l p f u l  dimensional response  et a l . 1983;  the  s u r f a c e contour p l o t s .  response  of the response  Generation  Table  d i r e c t understanding of the r e l a t i o n s h i p  ingredient  lines  in  X ; 2  (pork  and beef meat,  I t i s important to p o i n t out that  r e f l e c t the p r e d i c t e d  responses  f o r i n g r e d i e n t combinations  within  for  the  quality  the mixture  space  (Table 1 ) . Refer to Appendix A F i g u r e A2 f o r i n s t r u c t i o n s to  Nomenclature  read  the i n g r e d i e n t  proportions  on  these  and d e f i n i t i o n of the q u a l i t y parameters -189-  are  plots. given  i n Appendix  E.  Straight models  p a r a l l e l contours r e s u l t e d from  that  included  parameters  that were  e f f e c t of the  Shear  effect  for those linear  (Figure 28), E x f a t  (Figure  and  by  ingredients  on  the response  of the  (i.e.  quality blending 31),  Chewy (Figure 38).  on the response;  towards the vertex  prediction  the  (Figure 36)  only  is,  s l a n t i n g of the contour l i n e s provides  e f f e c t of the slanted  described  i n g r e d i e n t s : ES  (Figure 32), models,  l i n e a r terms, that  quality  In these  a measure of  the  the contour l i n e s  are  i n g r e d i e n t with the  largest  pH  strongest  regression  coefficient  value). Curvilinear that  included  quality binary  contours r e s u l t e d from q u a l i t y p r e d i c t i o n l i n e a r and  parameters nonlinear  quadratic  terms,  that were d e s c r i b e d blending  that  by l i n e a r  e f f e c t s of the  (Figure 29),  Exwater  (Figure  30),  and  present  i n these models t h a t generate the q u a d r a t i c  I t i s the q u a d r a t i c  lines.  included  those  linear,  quadratic  and  c u b i c terms,  q u a l i t y parameters that were d e s c r i b e d  b i n a r y and  t e r n a r y nonlinear  Shrink  (Figure 25),  Gummy  (Figure  deviations  terms  d e v i a t i o n of  More complex contours r e s u l t e d from q u a l i t y p r e d i c t i o n that  and  Tmloss  Twloss (Figure  37).  those  ingredients:  26),  Cohes (Figure  Exfluid  for  blending  (Figure  the contour  27),  is,  models  35).  Hardl  blending  by l i n e a r  e f f e c t s of the  (Figure 33), Hard2  In these models the  of the contour l i n e s from the -190-  that  cubic  models  is,  blending,  ingredients:  (Figure 34), term  f i r s t and  for  and  generated  second  order  approximations A  (Snee, 1971).  description  proportions  on  of  the q u a l i t y parameters w i l l  s i n c e the p r o p o r t i o n s other,  the e f f e c t s of changes i n the i n g r e d i e n t  of  the i n g r e d i e n t s  be g i v e n .  are dependent on  s i m p l i f i c a t i o n of the d e s c r i p t i o n w i l l  maintaining  each  ingredient  at  a  However,  be  each  performed  constant  proportion.  D e s c r i p t i o n of the e f f e c t s of the i n g r e d i e n t s on Shrink, Hard2 of  Hardl,  and Gummy i s somewhat d i f f i c u l t due t o the complex  the contour  plots  generated  generalized d e s c r i p t i o n w i l l The  Shrink  maintaining  a  plots.  model (Figure  the p r o p o r t i o n of MDPM constant,  as the p r o p o r t i o n of pork  the p r o p o r t i o n  nature  However  contour  p l o t f o r the Shrink  (a) m a i n t a i n i n g  decreased  the models.  be given f o r these  response s u r f a c e contour  25) shows t h a t :  by  by  of  the p r o p o r t i o n  beef  of  meat  pork  f a t increased; constant,  decreased  as  maintaining  the p r o p o r t i o n of pork f a t constant,  Shrink  f a t increased; Shrink  (b)  (c)  decreased  as the p r o p o r t i o n of MDPM i n c r e a s e d . The r e g i o n of minimum  Shrink  values was l o c a l i z e d a t the vertex d e f i n e d by 0.30 pork f a t , 0.20 MDPM and 0.50 beef meat;  the r e g i o n of maximum shrink values was  l o c a l i z e d a t the v e r t e x d e f i n e d by 0.05 pork f a t , 0.00 MDPM and 0.95 beef meat. The  response s u r f a c e contour  26) shows t h a t : the  p l o t f o r the Tmloss model (Figure  (a) m a i n t a i n i n g  the p r o p o r t i o n of MDPM constant,  p r o p o r t i o n of beef meat had l i t t l e  maintaining decreased  the p r o p o r t i o n  of  beef  effect meat  on  constant,  as the p r o p o r t i o n of MDPM i n c r e a s e d ; -191-  Tmloss;  (b) Tmloss  (c) m a i n t a i n i n g  MDPM  Figure 25. Response surface contour  p l o t f o r the Shrink  model.  MDPM  Figure 26. Response surface contour p l o t f o r the Tmloss model.  the  proportion  of  pork f a t constant,  Tmloss decreased  as the  p r o p o r t i o n of MDPM i n c r e a s e d . The r e g i o n of minimum Tmloss values was l o c a l i z e d and  a t the vertex d e f i n e d by 0.05 pork f a t ,  0.55 beef meat;  localized  along  the r e g i o n of maximum  the edge  defined  Tmloss  0.40 MDPM, values  was  by 0.00 MDPM, between 0.20 and  0.30 pork f a t . The  response s u r f a c e contour  27) shows t h a t :  plot  (a) m a i n t a i n i n g  f o r the Twloss model (Figure  the p r o p o r t i o n of MDPM constant,  Twloss i n c r e a s e d as the p r o p o r t i o n of beef maintaining  the p r o p o r t i o n of beef meat constant,  MDPM had l i t t l e  effect  of pork f a t constant, increased.  on Twloss;  (c) m a i n t a i n i n g  Twloss decreased  (b)  the i n c r e a s e i n the p r o p o r t i o n  as the p r o p o r t i o n of MDPM  The r e g i o n of minimum Twloss values was l o c a l i z e d a t  the vertex d e f i n e d by 0.10 meat;  meat i n c r e a s e d ;  the r e g i o n  pork f a t ,  0.40  MDPM and 0.50  of maximum Twloss values was l o c a l i z e d  beef  a t the  vertex d e f i n e d by 0.05 pork f a t , 0.00 MDPM and 0.95 beef meat. The  response s u r f a c e contour  shows t h a t :  (a) m a i n t a i n i n g  plot  f o r the ES model (Figure 28)  the p r o p o r t i o n of MDPM constant, ES  increased  as  the p r o p o r t i o n  maintaining  the p r o p o r t i o n of beef meat constant,  the p r o p o r t i o n MDPM i n c r e a s e d ;  of pork  (c) m a i n t a i n i n g  pork f a t constant,  the increase  little  ES. The r e g i o n  effect  localized 0.95 along  on  f a t increased;  ES decreased as  the p r o p o r t i o n of  i n the p r o p o r t i o n of MDPM had of minimum  ES  a t the vertex d e f i n e d by 0.05 pork f a t ,  beef meat;  values  the edge d e f i n e d by 0.30 pork f a t .  was  0.00 MDPM and  the r e g i o n of maximum ES values was  -194-  (b)  localized  MDPM  Figure 27. Response surface contour p l o t  f o r the Twloss model.  MDPM  Figure 28. Response surface contour p l o t f o r the ES model.  The  response  surface  contour  plot  f o r the E x f l u i d  (Figure 29) showed t h a t : (a) m a i n t a i n i n g constant, effect  on E x f l u i d ;  (b) m a i n t a i n i n g  E x f l u i d decreased  maintaining  decreased  the p r o p o r t i o n of beef meat  as the p r o p o r t i o n of MDPM i n c r e a s e d ;  the p r o p o r t i o n of pork f a t constant,  Exfluid  as the p r o p o r t i o n of MDPM i n c r e a s e d . Although  difficult  to observe i n t h i s f i g u r e , was l o c a l i z e d along maximum  the p r o p o r t i o n of MDPM  the i n c r e a s e i n the p r o p o r t i o n of beef meat had l i t t l e  constant, (c)  model  Exfluid  the r e g i o n of minimum E x f l u i d  values  the l i n e d e f i n e d by 0.35 MDPM; the region of  values was l o c a l i z e d a t the vertex  defined  by  0.30 pork f a t , 0.00 MDPM and 0.70 beef meat. The  response  surface  contour  plot  f o r the Exwater  (Figure 30) showed t h a t : (a) m a i n t a i n i n g constant,  Exwater  increased;  (b) m a i n t a i n i n g  the  increase  Exwater;  increased  Exwater decreased  maximum  as the p r o p o r t i o n  pork f a t , Exwater  of beef  the p r o p o r t i o n of beef meat little  constant, effect  the p r o p o r t i o n of pork f a t  meat  on  constant,  as the p r o p o r t i o n of MDPM i n c r e a s e d . The r e g i o n  of minimum Exwater values 0.30  the p r o p o r t i o n of MDPM  i n the p r o p o r t i o n of MDPM had  (c) m a i n t a i n i n g  model  was l o c a l i z e d a t the  vertex d e f i n e d by  0.20 MDPM and 0.50 beef meat;  the r e g i o n of  values was l o c a l i z e d a t the v e r t e x  defined  by  0.05 pork f a t , 0.00 MDPM and 0.95 beef meat. The  response s u r f a c e contour  31) showed t h a t : (a) m a i n t a i n i n g Exfat  plot  the p r o p o r t i o n of MDPM constant,  i n c r e a s e d as the p r o p o r t i o n of  maintaining  f o r the E x f a t model (Figure  pork  f a t increased;  the p r o p o r t i o n of beef meat constant, -197-  Exfat  (b)  decreased  MDPM  MDPM  Figure 30. Response surface contour p l o t f o r the Exwater model.  MDPM  Figure 31. Response surface contour p l o t  f o r the E x f a t  model  as  the p r o p o r t i o n  proportion  of MDPM i n c r e a s e d ;  of pork f a t constant,  of MDPM had  little  effect  values was l o c a l i z e d MDPM and 0.95 localized The  (c)  the  the i n c r e a s e i n the p r o p o r t i o n  on E x f a t .  The r e g i o n of minimum  Exfat  a t the v e r t e x d e f i n e d by 0.05 pork f a t , 0.00  beef meat;  the r e g i o n of maximum E x f a t values was  along the edge d e f i n e d by 0.30 pork f a t .  response s u r f a c e contour  showed t h a t : decreased maintaining  (a) m a i n t a i n i n g  as  plot  the p r o p o r t i o n  of  beef  meat  (b)  pH i n c r e a s e d as  (c) m a i n t a i n i n g  the p r o p o r t i o n  pH i n c r e a s e d as the the p r o p o r t i o n of MDPM  The r e g i o n of minimum  pH values was l o c a l i z e d  v e r t e x d e f i n e d by 0.05 pork f a t , 0.00 MDPM region  increased;  the p r o p o r t i o n of beef meat constant,  of pork f a t constant, increased.  f o r the pH model (Figure 32)  the p r o p o r t i o n of MDPM constant, pH  the p r o p o r t i o n of MDPM i n c r e a s e d ;  the  maintaining  of maximum pH values  and  was l o c a l i z e d  a t the  0.95 beef meat; at  the  vertex  d e f i n e d by 0.10 pork f a t , 0.40 MDPM and 0.50 beef meat. The  response s u r f a c e contour  p l o t s f o r the H a r d l ,  Gummy models (Figure 33, 34 and 35) showed t h a t : the p r o p o r t i o n of MDPM constant, as the p r o p o r t i o n of pork  (a) m a i n t a i n i n g  H a r d l , Hard2 and Gummy  f a t increased;  meat constant,  of  beef  decreased  as  the p r o p o r t i o n  maintaining  the p r o p o r t i o n of pork f a t constant,  of  Hardl, pork  Hard2  v e r t e x d e f i n e d by 0.30  and Gummy values pork  were  the  and Gummy  f a t increased;  (c)  Hardl Hard2 and  as the p r o p o r t i o n of MDPM i n c r e a s e d .  of minimum H a r d l , Hard2  decreased  (b) m a i n t a i n i n g  proportion  Gummy decreased  Hard2 and  The regions  localized  a t the  f a t , 0.20 MDPM and 0.50 beef meat; -201-  MDPM  F i g u r e 33. Response s u r f a c e contour  p l o t f o r the H a r d l model.  MDPM  Figure  34. Response s u r f a c e  contour  plot  f o r t h e Hard2  model.  MDPM  Figure 3 5 . Response surface contour p l o t  f o r the Gummy model.  the  regions  localized  of maximum H a r d l ,  Har<32  a t the v e r t e x d e f i n e d by  and Gummy  values  were  0.05 pork f a t , 0.00 MDPM and  0.95 beef meat. The response s u r f a c e contour  plot  36) showed t h a t : (a) m a i n t a i n i n g Shear  increased  maintaining as  of MDPM i n c r e a s e d ;  pork  f a t constant,  p r o p o r t i o n of MDPM i n c r e a s e d . was l o c a l i z e d and  0.50  localized  the p r o p o r t i o n of MDPM constant,  the p r o p o r t i o n of beef meat constant,  of  (c)  Shear  Shear  maintaining  decreased  the  as the values  pork f a t , 0.20 MDPM  the r e g i o n of maximum Shear  the vertex d e f i n e d by 0.05  (b)  increased  The r e g i o n of minimum Shear  a t the vertex d e f i n e d by 0.30  beef meat; at  (Figure  as the p r o p o r t i o n of beef meat i n c r e a s e d ;  the p r o p o r t i o n  proportion  f o r the Shear model  values  was  pork f a t , 0.00 MDPM and  0.95 beef meat. The response s u r f a c e contour 37) showed t h a t : (a) m a i n t a i n i n g Cohes  increased  maintaining as  plot  f o r the Cohes model  (Figure  the p r o p o r t i o n of MDPM constant,  as the p r o p o r t i o n of beef meat i n c r e a s e d ;  the p r o p o r t i o n of beef meat constant,  the p r o p o r t i o n  decreased  the  proportion  of MDPM i n c r e a s e d . The r e g i o n of minimum Cohes values  region  Cohes  maintaining  of  localized  f a t constant,  (c)  Cohes increased  proportion  was  pork  of MDPM i n c r e a s e d ;  (b)  along the edge d e f i n e d by 0.30  of maximum  Cohes  values was l o c a l i z e d  pork  as the  f a t ; the  a t the vertex  d e f i n e d by 0.05 pork f a t , 0.00 MDPM and 0.95 beef meat. The response s u r f a c e contour 38) showed t h a t : (a) m a i n t a i n i n g  plot  f o r the Chewy model  (Figure  the p r o p o r t i o n of MDPM constant,  -206-  MDPM  FAT  Figure 36. Response surface contour p l o t f o r the Shear model.  MDPM  F i g u r e 37. Response s u r f a c e contour p l o t f o r the Cohes model.  MDPM  Chewy  increased as the p r o p o r t i o n of beef meat  maintaining  the p r o p o r t i o n of beef meat constant,  as the p r o p o r t i o n proportion  of  pork  of  MDPM  increased;  f a t constant,  p r o p o r t i o n of beef meat i n c r e a s e d . v a l u e s was l o c a l i z e d MDPM and 0.50 localized  increased;  (c)  Chewy  (b)  Chewy i n c r e a s e d maintaining  increased  The r e g i o n of  as  minimum  the the Chewy  a t the v e r t e x d e f i n e d by 0.30 pork f a t , 0.20  beef meat;  the r e g i o n of maximum Chewy values was  a t the vertex d e f i n e d by 0.05 pork f a t , 0.00  0.95 beef meat.  -210-  MDPM and  3.3. C o r r e l a t i o n The 11  analysis  experimental data r e p o r t e d i n Tables 19  were  pooled  for correlation analysis.  coefficients  (r)  relationships  between  and  of  were  proximate  the raw emulsions  (Table 41), between the and  computed  between the  textural  and  to  to 22  and F i g u r e  Pearson's evaluate  composition  correlation the  of the meat  linear blocks  the q u a l i t y parameters  evaluated  parameters e v a l u a t e d  (Table 42)  quality  parameters (Table 43). Only  correlation  c o e f f i c i e n t s s i g n i f i c a n t a t p<0.05 are r e p o r t e d . Nomenclature and definition The  of the q u a l i t y parameters are given i n Appendix E.  c o r r e l a t i o n c o e f f i c i e n t s between proximate  the meat  b l o c k s and raw emulsions  and raw emulsions'  q u a l i t y parameters evaluated are presented Positive meat  blocks  moisture found  correlations with  for protein  Negative  f o r f a t content and f a t - t o - p r o t e i n  moisture  and  protein  pH  of  and the  i n Table 41.  content of the  content of the raw emulsions.  with moisture  content of the meat b l o c k s and  correlations  were  r a t i o of the meat blocks  content of the meat blocks and  with  content of the raw emulsions.  Moisture, p r o t e i n  and f a t content and f a t - p r o t e i n  meat b l o c k s and moisture correlations Moisture with  were found  both moisture  composition  Shrink,  content of the raw emulsions  with most of the q u a l i t y  Exwater and with a l l the t e x t u r a l with ES and E x f a t .  correlated  parameters,  and  Moisture content of the  followed the same r e l a t i o n s h i p s -211-  showed high  parameters.  content of the meat blocks was p o s i t i v e l y  negatively correlated raw emulsions  r a t i o of the  as moisture  content  Table  41. C o r r e l a t i o n s between proximate composition of the meat blocks and raw emulsions and the q u a l i t y parameters e v a l u a t e d . * " 3  Meat block Moisture Prote i n Fat F/P Moisture pH Shr ink Tmloss Twloss ES Exfluid Exwater Exfat Hardl Hard2 Shear Cohes Gummy Chewy  0  Raw  Protein  Fat  -0..950 -0,.954 0,.965  0.,997 -0,.998  -0,.998  -0,.755  -0 .. 770  0..765  -0..859  0 ., 699 -0,,769  -0,. 496 0,. 851  -0 .519 , 0,. 844  0,, 532 -0 .842 ,  0 ,651 , -0.. 839 0 .692 , 0,.740 0 ,. 793 0,. 854 0.,793 0,.784  0 ,, 857 -0,. 710 0 .766 , 0,.850 0 ,. 805 0,. 883 0 .870 , 0,. 893  -0,,709 0,.821 -0 .720 , -0,.777 -0 , 808 -0,.873 -0 ,823 . -0 ., 821  -0 .730 , 0 ., 816 -0 .749 , -0,.792 -0,.816 -0,. 860 -0 ,838 . -0,.831  0,,741 -0,, 808 0 ,739 . 0,, 796 0 ,814 . 0., 877 0.,840 0,.839  0.,916 -0.,996 -0 ,990 , 0,.989 s 0.. 736  0 ., 769  F/P  emulsion  c  Moisture  pH  -0,. 561 -0..620 -0..454 -0.,516 -0,,465  * Nomenclature and d e f i n i t i o n of the q u a l i t y parameters are g i v e n i n Appendix E C o r r e l a t i o n c o e f f i c i e n t s s i g n i f i c a n t at p<0.05 Fat-to-protein ratio Moisture content of the raw emulsions not s i g n i f i c a n t a t p<0.05 B  c  D  B  -212-  of the meat b l o c k s ,  the e x c e p t i o n being Twloss.  of the meat b l o c k s was Exwater and  P r o t e i n content  p o s i t i v e l y c o r r e l a t e d with Shrink,  with a l l the  textural  parameters,  Twloss,  and n e g a t i v e l y  c o r r e l a t e d with ES and E x f a t . Formulations  having  moisture  content  thermal  treatment  high  lead  to:  (Shrink,  e x p r e s s i b l e water i n the greater  (a)  content  greater  Twloss),  and  thus  weight l o s s e s  (b)  greater  cooked f r a n k f u r t e r s  high during  amounts  (Exwater),  protein-high  moisture  Marquez et a l .  (1989)  f r a n k f u r t e r s had  than low p r o t e i n - l o w moisture Singh et a l . (1985)  (c)  found  found t h a t  lower smokehouse  frankfurters.  high yields  Simon et a l .  r e p o r t e d that t e x t u r e p r o f i l e  of  and  t e x t u r a l s t r e n g t h . S i m i l a r r e l a t i o n s h i p s have been  by s e v e r a l r e s e a r c h e r s .  and  protein  (1965)  parameters  were higher f o r samples with higher p r o t e i n c o n t e n t s . St. John et al.  (1986)  chewiness  indicated  were  that g r e a t e r values f o r  presumably  content.  Marquez et a l .  textural  parameter  i n t e r n a l molecular expected  Negative and  (1989)  result  of  increased  and  protein  i n d i c a t e d t h a t cohesiveness,  h i g h l y dependent  on  the  strength  bonds making up the body of the  to be higher i n h i g h - p r o t e i n f r a n k f u r t e r s .  M o r r i s e y et a l . increases  the  gumminess  of  a the  product,  is  In a d d i t i o n ,  (1982) r e p o r t e d t h a t the s t r e n g t h of myosin g e l s  p r o p o r t i o n a l l y to the square correlations  of myosin  between both moisture and  both E x f a t and ES were expected,  p r o t e i n content of the raw  emulsions  content decreased. T h i s decrease -213-  concentration. protein  content  s i n c e as the moisture increased,  and  the r e l a t i v e f a t  c o n t r i b u t e d to the reduced  level  of f a t test  (ES)  Fat content and r e l a t i o n s h i p to  moisture ratio  stability  and to l o v e r l e v e l s of f a t to be expressed from  cooked f r a n k f u r t e r s  a  l o s t d u r i n g s a f f l e ' s emulsion  a v a i l a b l e to be  and vere  (Exfat). f a t - t o - p r o t e i n r a t i o of the meat b l o c k s shoved a l l q u a l i t y parameters opposite to  p r o t e i n content. positively  Fat content  correlated  n e g a t i v e l y c o r r e l a t e d v i t h Shrink, the t e x t u r a l  the  with  and ES  Tvloss,  that  of  fat-to-protein and  Exfat  and  Exvater and v i t h a l l  parameters.  Positive  r e l a t i o n s h i p s betveen both ES and E x f a t and both f a t  content and  f a t - t o - p r o t e i n r a t i o of the meat blocks vere expected  since  as  the f a t content of the emulsions  p r o t e i n matrix vas a v a i l a b l e to s t a b i l i z e l e v e l s of f a t vere l o s t (ES)  loss  (Exfat).  fat-to-protein (Shrink and  r a t i o of the meat blocks and Tvloss)  increased  fat levels  reduced  from  has  and  also  Maurer increased  (1989) found t h a t l o v - f a t Mittal  been  due  resistance  (1975) and  found  and  the  veight  by s e v e r a l that  yields  a t t r i b u t e d t h i s to the Park  et  f r a n k f u r t e r s l o s t more v a t e r d u r i n g  Blaisdell  to the hydrophobic to  test cooked  product  observed  (1983)  hypothesized  f r a n k f u r t e r s having high f a t - t o - p r o t e i n r a t i o s had losses  greater  the  moisture l e v e l s that vere l o s t d u r i n g cooking.  cooking.  a  The negative r e l a t i o n s h i p betveen both f a t  Dhillon  as  Thus  of  d u r i n g S a f f l e ' s emulsion s t a b i l i t y  researchers.  al.  the f a t .  and g r e a t e r l e v e l s of f a t vere expressed  frankfurters and  increased less  diffusion -214-  nature of  of  lover  the f a t that  moisture.  The  that  moisture offers negative  r e l a t i o n s h i p s between Exwater and r a t i o of the meat of  water  water  in  to  blocks  and  fat-to-protein  i n d i c a t e d that the r e l a t i v e p r o p o r t i o n  the f o r m u l a t i o n s  be  f a t content  decreased and thus the amounts of  expressed from the  frankfurters  negative  relationships  parameters  has been documented by other r e s e a r c h e r s .  (1987) force  demonstrated  between  that  fat  decreased.  content  found that r e d u c t i o n i n f a t content caused  Mittal  and  parameters The pH  showed  parameters. stability  (1989)  i n c r e a s e d firmness and (1987) and  found negative c o r r e l a t i o n s the  between  texture p r o f i l e  sausages.  the  poorest  correlations  with  the t e x t u r a l  pH showed negative c o r r e l a t i o n s with parameters  greater  Park et a l .  r a t i o s of raw emulsions and  of cooked  Hand et a l .  f r a n k f u r t e r s . S i r i p u r a p u et a l .  Usborne (1986)  fat-to-protein  textural  low-fat f r a n k f u r t e r s r e q u i r e d  t o shear than h i g h - f a t f r a n k f u r t e r s .  s p r i n g i n e s s i n cooked  and  The  the  emulsion  obtained by the method of Townsend et  al.  (1968) (Tmloss and Twloss).  T h i s i n d i c a t e s that as the pH of the  raw  g r e a t e r weight  emulsions  greater  water l o s s e s  (Tmloss)  Tmloss  h o l d i n g c a p a c i t y of  be  (Twloss) of  meat  and block  thus c o n t r i b u t i n g to the d e c l i n e of the  emulsions can  losses  per u n i t moisture content  were observed,  s t a b i l i t y of the before,  decreased,  to  thermal treatment.  considered  As  as a measure of  the meat emulsions.  mentioned the  water  I t i s widely recognized  t h a t water h o l d i n g c a p a c i t y i s s t r o n g l y dependent on the meat increasing  with i n c r e a s i n g pH  Whiting, 1988;  M i l l e r et a l . ,  (Harding Thomsen and 1968). The pH was  -215-  Zeuthen  also  pH,  1988;  negatively  correlated  to E x f l u i d , Exwater and  Correlation stability,  Hard2.  c o e f f i c i e n t s between product weight l o s s , emulsion  j u i c i n e s s and  textural  parameters are  shown i n Table  42. Shrink  correlated  Shrink was textural A  parameters; and  Exwater i n d i c a t e s  The  that  positive  greater  weight  loss  Trout and  concentration  protein  frankfurters textural both ES  and  weight  losses,  emulsion from the  textural  Exfat  may  have  Schmidt  (1987)  occurs  stability cooked  negative  indicated  released  was  Exfat.  expected  due  lower  t e s t and  to  of  to  leading  that  greater that  loss  of  to g r e a t e r  between Shrink exhibiting  fat  of  been  reported  weight  thus  and  positive  hypothesized  formulations,  amounts  The  led  relationship  that  after  parameters has  et a l . (1987)  d u r i n g thermal treatment,  s t r e n g t h . The  and  greater l e v e l s  cooked f r a n k f u r t e r s .  compression f o r c e s . of  Twloss  at higher weight l o s s e s ,  r e s e a r c h e r s . Lee  smokehouse  with ES  a l l the  r e l a t i o n s h i p between Shrink  between weight l o s s and  reported by s e v e r a l  parameters.  measures of product weight l o s s  water were expressed from the relationship  correlated  c o r r e l a t i o n between Shrink and  treatment.  quality  with Twloss, Exwater and  negatively  both parameters are  thermal  with most of the  p o s i t i v e l y correlated  positive  since  well  during  and high  Saffle's  lower amounts of f a t were expressed  frankfurters.  Tmloss c o r r e l a t e d  with few  quality  positive  correlations  with Twloss,  positive  relationships  indicate -216-  parameters. Exfluid  and  Tmloss showed Exfat.  that at higher water l o s s e s  These per  Table 42. C o r r e l a t i o n s between the q u a l i t y  Shrink Tmloss Twloss ES Exfluid Exwater Exfat Hardl Hard2 Shear Cohes Gummy Chewy  Tmloss  Twloss  parameters evaluated*-'  ES  Exfluid  -0.481 0.823 -0.462 -0.532 -0.687 -0.621 -0.551 -0.548  0.482 0.564  Exwater  Exfat  -0.451 0.734 0.772 0.681 0.665 0.786 0.842  -0.501 -0.718 -0.594 -0.515 -0.487  e  0 .551 -0 .739 0 .541 -0 .541 0 .640 0 .695 0 .652 0 .632 0 .688 0 .761  0 .710 0 . 768 0 .500  0.556 0.772 0.662 0.757 0.642 0.724 0.777  * Nomenclature and d e f i n i t i o n of the q u a l i t y parameters are given in Appendix E C o r r e l a t i o n c o e f f i c i e n t s s i g n i f i c a n t a t p<0.05 not s i g n i f i c a n t a t p<0.05 B  c  B  u n i t moisture  content' i n the meat block  l o s s e s d u r i n g thermal fat  were expressed  treatment  and  from the cooked  (Tmloss),  g r e a t e r weight  g r e a t e r amounts of f l u i d and frankfurters.  Twloss was p o s i t i v e l y c o r r e l a t e d with Shrink, Tmloss, E x f l u i d , Exwater,  and  relationship  with the t e x t u r e p r o f i l e parameters. with  both E x f l u i d and Exwater  The p o s i t i v e  indicates  that  higher weight l o s s e s (Twloss), g r e a t e r amounts of f l u i d were  expressed  from the cooked f r a n k f u r t e r s .  As  ES  was  positively  correlated  with  parameters.  c o r r e l a t e d with  Shrink,  The  Exwater,  positive at  higher  Exfat,  and with  and  ES  levels  fat  that  emulsions  during  expressed  from the cooked f r a n k f u r t e r s . The negative  thermal  negatively  a l l the  indicates  treatment,  released  textural  and  thermal  treatment,  Exfluid  relationship  (r=0.56)  emulsions  a  slightly  than with Exwater  indicating  relationships that  at  higher  (r=0.48).  correlation As expected,  with  a l l the  from the  higher l e v e l s of  expressed  effect. -218-  with a  Exfat  negative  Exwater showed  textural  t e x t u r a l s t r e n g t h of the f r a n k f u r t e r s was observed. an opposite  during  f r a n k f u r t e r s were obtained.  r e l a t i o n s h i p was found between E x f a t and Exwater. positive  indicates  lower amounts of water were expressed  showed  the  higher amounts of f a t were  of r e l e a s e d f a t from the  cooked f r a n k f u r t e r s and s o f t e r  Exfat  from  between ES and Exwater and with the t e x t u r a l parameters that a t higher l e v e l s  Shrink,  parameters.  r e l a t i o n s h i p between of  and water  with  Twloss c o r r e l a t e d p o s i t i v e l y with the t e x t u r e p r o f i l e  at  parameters  water  greater  E x f a t showed  The  correlation  coefficients  indicating  between the t e x t u r a l parameters are l i s t e d The  textural  parameters  were h i g h l y  o t h e r . I t i s important t o point computed Gummy  f o r Hardl  and Chewy,  out that  and Gummy with  s i g n i f i c a n t since  and  Positive  hardness  parameters  measures  of  frankfurters. researchers.  Similar Siripurapu  and hardness a t second  both  gumminess found second  hardness  and chewiness.  between  Cohes with  cannot  first  cohesiveness  both  were expected between the these  been  parameters are of  reported  (1987) working with  the  by s e v e r a l frankfurters compression  compression, between  gumminess  and second compression,  However,  negative  On the other hand Z i e g l e r e t a l .  and both hardness a t f i r s t  -219-  between  correlations  and both hardness a t f i r s t  compression.  cooked  between chewiness and both  working with dry and semidry sausages found p o s i t i v e between  each  be c o n s i d e r e d  between hardness a t f i r s t  cohesiveness  compression.  have  compression,  at  Chewy  because  et a l .  and a t second  with  correlation coefficients  characteristics  findings  found p o s i t i v e c o r r e l a t i o n s  and  Shear  the s t r e n g t h  hardness a t f i r s t  correlated  these parameters depend on Hardl  correlations  and  relationship  i n Table 43.  with both Gummy and Chewy,  statistically Cohes.  the  were  and a t (1987)  correlations  and a t  second  Table 43. C o r r e l a t i o n s  Hard2 Shear Cohes Gummy Chewy * B  between t e x t u r a l  parameters*"  Hardl  Hard2  Shear  Cohes  Gummy  0.922 0.625 0.627 0.969 0.836  0.655 0.776 0.962 0.873  0.602 0.672 0.702  0.793 0.819  0.906  3  Nomenclature and d e f i n i t i o n of the q u a l i t y parameters are given i n Appendix E C o r r e l a t i o n c o e f f i c i e n t s s i g n i f i c a n t a t p<0.05  -220-  3.4.  Scatterplot  Analysis visualize the  of  the  analysis  response  surface  contour  plots  the e f f e c t of changes i n the i n g r e d i e n t  quality  parameters.  significant and  matrices  Correlation analysis  l i n e a r r e l a t i o n s h i p s between  helped  proportions identified  proximate  on the  composition  q u a l i t y parameters and between q u a l i t y parameters.  However,  to understand the r e l a t i o n s h i p between the i n g r e d i e n t  proportions  and  to pool a l l  the  the q u a l i t y parameters f u r t h e r information  (i.e.  q u a l i t y parameters) matrices  i t was  ingredients,  necessary  proximate  and t o d i s p l a y  composition  and  i t i n a s e r i e s of s c a t t e r p l o t  (SPLOM).  SPLOM and c o r r e l a t i o n a n a l y s i s are techniques that can be used to a r r i v e at s i m i l a r c o n c l u s i o n s . graphical  method  displaying  information  of  the  f o r data  statistical  basis  between independent  (quality where  thus,  a basic  studied  data f o r the (Table  models (Table  2) 40).  be  assumption  in  variables  is  SPLOM are e a s i e r to  used  SPLOM to  (ingredients)  parameters)  should be dependent v a r i a b l e s The  can  By  understanding  different  v a r i a b l e s are involved  and,  1985).  form, f u r t h e r  a c o r r e l a t i o n matrix of the d a t a .  relationships  analysis  SPLOM i s b a s i c a l l y a  (Cleveland,  i n t e r r e l a t i o n s h i p s between the  than  variables  analysis  in a graphical  o b t a i n e d . When s e v e r a l analyze  However,  contrast  i s that a l l  depict  and to  have  no the  dependent  correlation  the  variables  (Bender e t a l . , 1982).  q u a l i t y parameters was generated using  f o r the 10 the q u a l i t y  Nomenclature and d e f i n i t i o n -221-  formulations prediction  of the q u a l i t y  parameters are given refer 10.  i n Appendix E. The numbers w i t h i n each  t o the f o r m u l a t i o n number.  Each panel  i s a miniature  the maximum value F i g u r e 39  XY p l o t ranging  emulsions,  and  protein  of the meat b l o c k s , moisture content  added i c e and pH of the raw emulsions. content  of  the meat  p r o p o r t i o n of pork f a t decreased of beef meat i n c r e a s e d . the meat  blocks  Fat content  of  p r o t e i n content,  the p r o p o r t i o n dependent  r a t i o s . Therefore, the e s t a b l i s h e d  r a t i o of 4 i n the raw emulsions. As  and  thus  the moisture  increased.  The  pH  the raw  p r o p o r t i o n of MDPM i n c r e a s e d , beef meat i n c r e a s e d .  reported  pork f a t  the  original  meat blocks  having  of beef meat, had  higher  l e v e l s of i c e  moisture-to-protein i n Figure  39  of  the raw  emulsions  and decreased  emulsions  the  increased  also as the  as the p r o p o r t i o n of  than t h a t of beef meat (Table 18).  t h a t the a d d i t i o n of m e c h a n i c a l l y  i n c r e a s e s the pH of meat (Harding  observed  of  T h i s r e l a t i o n s h i p was expected s i n c e the pH  of the MDPM was higher been  on  ratio  the p r o p o r t i o n of beef meat i n c r e a s e d  content  of  as the  increased as the p r o p o r t i o n  i . e . higher p r o p o r t i o n s  l e v e l s of i c e i n c r e a s e d as  Moisture  and the f a t - t o - p r o t e i n  i c e was  needed t o make up f o r  of the  increased  r a t i o of the meat b l o c k s ;  lower m o i s t u r e - t o - p r o t e i n were  blocks  and  increased as  Addition  moisture-to-protein higher  from the minimum to  shows the r e l a t i o n s h i p s between the i n g r e d i e n t s and  raw  increased.  Number 0 r e f e r s t o f o r m u l a t i o n  of each parameter.  proximate composition  of  panel  deboned  I t has meat  emulsions c o n t a i n i n g hand deboned meat  Thomsen and Zeuthen, 1988). -222-  X1  U  MOISTURE  ^  15  3  g  1 6  6  9  9 3  FAT  8  5 7  » 9  9  7  5  FP  MOISTEM  3  !* e  0  150  9  (1 U  ° 15 9 3  ICE  °  8  9  2 1 ' 6  n  F i g u r e 39.  6  1  9  8  7  3  a  e 8  5  1  7  8  ^  7  1  7  S  3 3  \ 0  fl 0 9  8 9  fl  6  2 12 t  9 ^  6  5  6  , 7 e  4  2  4  2  i1  8  8  8  5  9  Z  5  6  5  i  i 1  g  8  1  2 ,2  a  7  1  5  2  PH  8  1  O  R6  2  17  ^ 3  8  a  3  ? T  5  n 0 PHOT EN  X3  X2  7  9  8  6  5  8  fl 9  0  R e l a t i o n s h i p s between the i n g r e d i e n t s p r o p o r t i o n s and the proximate composition of the meat b l o c k s and raw emulsions, added i c e and pH of the raw emulsions. Pork fat, XI; MDPM, X2; Beef meat, X3. Proximate composition of meat b l o c k : moisture, f a t , p r o t e i n and fat-to-protein ratio (FP); composition of raw emulsion: moisture (moistem).  -223-  Shrink decreased Relating  i n c r e a s e d as the p r o p o r t i o n of. beef meat i n c r e a s e d and as the p r o p o r t i o n of pork f a t i n c r e a s e d  t h i s q u a l i t y parameter t o the proximate composition  the meat blocks and raw emulsions as  the moisture  moisture the  content  f a t content  and  (Figure 41),  p r o t e i n content  of the raw emulsions and  blocks and  i n c r e a s e d and decreased as blocks  (1989), M i t t a l and  (1983), and D h i l l o n and Maurer (1975) r e p o r t e d i t can be concluded  of beef meat were d e t r i m e n t a l t o the y i e l d the f a c t that higher moisture  of  increased  f a t - t o - p r o t e i n r a t i o of the meat  f i n d i n g s . From these r e s u l t s  observed  Shrink  of the meat  i n c r e a s e d . Marquez e t a l . (1989), Park e t a l . Blaisdell  (Figure 40).  contents  and  similar  t h a t high  levels  of the product lower  due to  pH values were  f o r f o r m u l a t i o n s c o n t a i n i n g higher p r o p o r t i o n s of  beef  meat. Tmloss 40).  decreased  as the p r o p o r t i o n of MDPM i n c r e a s e d  (Figure  Tmloss was not r e l a t e d t o the proximate composition  meat blocks and emulsions, pH of the raw emulsions  of the  however i t r e l a t e d n e g a t i v e l y with the  (Figure 41). As mentioned before,  Tmloss  could be considered as a measure of the water h o l d i n g c a p a c i t y of the  meat  ability  emulsions  (i.e.  of the emulsions  the lower  the value  t o r e t a i n water),  p r o p e r t y i s known t o be a f f e c t e d by the pH the  pH of the raw emulsions  the p r o p o r t i o n of MDPM decrease  (Figure  with i n c r e a s i n g  of t h i s meat. Harding  was found 39)  the higher  and t h i s  functional  of the meats.  to relate  the  positively  Since to  the tendency f o r Tmloss to  MDPM may be a t t r i b u t e d t o the pH e f f e c t  Thomsen  and Zeuthen (1986)  -224-  r e p o r t e d that  TMLOSS  TWLOSS  | 0 6 ? 0  59  3  5  2  2  0  4  1  u  6  5  9  3  9  I  9 8  5 6  9  Q  B 0  9  EXFLUD  6 i5 *  g  S  EXWATER 1 6  EXFAT  e  3  IS"  0  7 4  y  3  u  2  4/8 3  i 1 2 2  3  K5  5  fi  9  6 n « y o  7 4Q 5 6  7  I 4 8 / 4 3 9 0  y  §  ?  4 i 8/4  ES  40.  5 6 9 6 g 7 . 3 ? | 7 8 7 0 t 3 19  9 8 \  y  2 1  Figure  U  u  U 8HRMK  X3  X2  X1  9  U  3  4 / 8 3  2  1 5 9 '2 fi  n  R e l a t i o n s h i p s between the i n g r e d i e n t s p r o p o r t i o n s and the q u a l i t y parameters that d e s c r i b e product weight loss, emulsion stability and juiciness characteristics. P o r k f a t , XI; MDPM, X2; B e e f meat, X3. Nomenclature and definition of the quality parameters are given i n Appendix E.  -225-  pncnat  FATt  MOBTUC  m e  508  0 • «  8 7  8  0  "TMLOSS  3  4  12 moss  ES  0  0  6  3  6  n 8  4  4  3  BAUD  EXMMTEn  ^ M 0 0 0  8  II 2°  *\  *  a  4  0 0  6  0  0,  2 , 3 4 4 3,2 874 478 3 3  43^ 8 7 4 a  20  0  3 0  4  8  3  8  iS« 8 8 . 0  «\  0 0 0 8 0 7  6  M  0 ,  a  a^  8 7 4  V  2 0 00  fl  2 002  4  a f 8  *  O  0  7 4  0  0  8  7  7  7 4  y 478 »•  200 0 02  0  0  ' ' « 1 2 0 o 0 8 2 8 0 8 8 0 0  8  4B EM7CT  a  4  8  2  0<J 8 o 0  7 3  0  8  o  7 " " 2 j 3 4 43 l 2 478 a 3 JT0  7 '  7  7 0^3  °fl  0  2  a  8 0  0 • 2 1  fl  0  4 4* 3 , 2 a  05  0  1a  a  «  0  0  60  t  —  0 0  60 —  PH  0  fl 2  0  ,0  7  3  4 874 478  '*  200  F i g u r e 41. R e l a t i o n s h i p s between the proximate composition of the meat b l o c k s and raw emulsions and the pH of the raw emulsions, and the q u a l i t y parameters t h a t d e s c r i b e product weight l o s s , emulsion s t a b i l i t y and j u i c i n e s s characteristics. Proximate composition of meat block: moisture, f a t , p r o t e i n and f a t - t o - p r o t e i n r a t i o (FP); composition of raw emulsion: moisture (moistem). Nomenclature and d e f i n i t i o n of the q u a l i t y parameters are given i n Appendix E.  -226-  addition  of  increased  mechanically  the pH  and  deboned meat i n sausage  the  water h o l d i n g c a p a c i t y of  emulsions and reduced t h e i r cooking As  with  Shrink,  thermal treatment. meat increased  Twloss  (Figure 41).  stability  test  contributing tendency  to  pH  of  weight  loss  after of beef  I t increased as the p r o t e i n  content  results that  pH from  water  of the raw emulsions Townsend's  c o n t a i n i n g higher (i.e.  water)  these emulsions which  emulsion  l o s s was a major  weight l o s s of the cooked emulsions.  f o r formulations  meat  as the p r o p o r t i o n  The  indicated  meat to l o s e more weight lower  increased  increased and as the  decreased  the  losses.  Twloss i s a measure of  (Figure 40).  of the meat blocks  formulations  factor  Thus  the  of  beef  proportions  may be a t t r i b u t e d to lowered  the  water  the  binding  abi1ity. ES  increased as the p r o p o r t i o n  40).  of pork f a t increased  I t decreased as moisture and p r o t e i n content  blocks  and  moisture content  increased as the f a t content blocks  increased  the  meat  of the raw emulsions increased  and  and f a t - t o - p r o t e i n r a t i o of the meat  (Figure 41).  These r e l a t i o n s h i p s suggest that  high  l e v e l s of pork f a t and thus high  block  have a negative  thermal  f a t contents  e f f e c t on the s t a b i l i t y  treatment s i n c e  of  (Figure  there  i n the  meat  of the emulsions to  i s not enough lean meat and  thus  not enough p r o t e i n t o s t a b i l i z e the f a t . As Asghar et a l .  (1985)  pointed  has  out  the  amount of f a t i n a sausage f o r m u l a t i o n  d i r e c t e f f e c t on the s t a b i l i t y  of the meat emulsion.  -227-  a  MDPM  and  (Figures  pH  40  expressed  had a s i m i l a r e f f e c t on E x f l u i d as  and 41),  that i s ,  l e s s e r amounts of  from f r a n k f u r t e r s e x h i b i t i n g higher  thus higher  on  Tmloss  fluid  were  l e v e l s of MDPM and  pH v a l u e s .  Exwater  increased  as the p r o p o r t i o n  of beef  meat  increased  (Figure 40). I t increased as the moisture and p r o t e i n contents of the meat  blocks  increased  and as the f a t content  meat  blocks  emulsions in  decreased.  decreased  formulations  been  and moisture  content  of  the raw  and f a t - t o - p r o t e i n r a t i o  I t increased  as the pH  with higher  proportions  lower pH found i n these f o r m u l a t i o n s binding  of  of the the raw  (Figure 41). Higher moisture contents of beef meat  the cause of increased e x p r e s s i b l e water.  water  emulsions  properties  giving  may  have  In a d d i t i o n the  account  higher  may  found  f o r the lower  l e v e l s of e x p r e s s i b l e  water i n the cooked f r a n k f u r t e r s . Exfat (Figure  increased  as  of  pork  40). I t increased as the f a t content  r a t i o of the meat blocks and  the p r o p o r t i o n  p r o t e i n contents  the  raw  high  l e v e l s of  of  f a t increased  and f a t - t o - p r o t e i n  increased and decreased as the moisture the meat blocks and moisture content of  emulsions i n c r e a s e d .  These r e l a t i o n s h i p s suggest  pork f a t and thus high  f a t contents  that  l e d t o higher  l e v e l s of e x p r e s s i b l e f a t i n the cooked f r a n k f u r t e r s . A l l the t e x t u r a l fashion  to  composition 43).  parameters  the p r o p o r t i o n s  appeared t o r e l a t e i n a s i m i l a r of  the i n g r e d i e n t s  of the meat blocks and raw emulsions  and  to the  (Figures 42 and  The t e x t u r a l parameters decreased as the p r o p o r t i o n of pork -228-  HARD1  y  5Q . 3  u  It 4  HARM  3 8HEAR  GUMMY  u 6 2 -,59 u a -so  CHEWY  u 6  COHES  5 1  3  u  8 7  «  8  5  2  6 X  2 3 7 R 2  5  2  7  4  U  ,2 5 6 9 7  4  9  3  6 7  9 U  9  i  7  3  9 0 8 7 4  Q  2  4  6 5  u 9  5 6 9  9  2 i  |  8 7 U  8  Figure 42.  <  U  8  *••.  ?  §  9  §15 9 3  X3  X2  X1  u  8  0  3  6  6 9 3  1  Relationships between the ingredients proportions and the textural parameters. Pork f a t , XI; MDPM, X2; Beef meat, X3. Nomenclature and d e f i n i t i o n of the quality parameters are given in Appendix E.  -229-  VWTURE  FK7TEN  FAT  0 0  0 0  4  02 6 i  a  4  0 0  0 a  6  6  0  67447%  & as 62 0 1  1  a 0 0  fl  C  8 2  8  0  9  98  a  0  8  /  0  0  0 874  a 478  f  2  87 4 0 0 028 0 85 2 820 87 87 78 a a 4 a 0 0 4 0 4. A 9 fl 9 6 2 25 a? 4 4 3  1  1  4 4  44  6  8  0  V V  0  0 0  000  €  f  8 0 9 6^  78 a  / Figure  00  2  4  0 0  00  CHBVY  0  fl  0  l  Moarai  9 05 2 02$ §20 1 67 87 1 78 1 a a a 4 4 0 0 4  tt>8  00  OOHEB  % a 4  0  4  0 0  6  r  a 4  V4y  1  SHEAR  CUMY  °-90 9  a  HAWM  *a 4  FP  PH  1  1  0  8  1  8  43. R e l a t i o n s h i p s between t h e p r o x i m a t e c o m p o s i t i o n o f t h e meat blocks a n d raw e m u l s i o n s and t h e pH o f t h e raw emulsions, and t h e t e x t u r a l parameters. Proximate c o m p o s i t i o n o f meat b l o c k : m o i s t u r e , f a t , p r o t e i n and fat-to-protein ratio (FP); composition of raw emulsion: moisture (moistem). Nomenclature and definition of the q u a l i t y parameters are given i n Appendix E. -230-  fat  increased  and  increased.  It  tenderizing  effect  has  (Park et a l . 1989; et a l .  i n c r e a s e d as the  (1989),  been  proportion  r e p o r t e d that f a t  while lean meats have Lyon et a l . 1981;  St.  John et a l .  M o r r i s e y et a l . (1982) and  of  beef  tends  a  to  meat  have  a  toughening  effect  S w i f t et a l . 1954).  Marquez  (1986),  Singh et a l .  (1985)  Simon et a l . (1965) have r e p o r t e d that  t e x t u r a l parameter values are higher f o r meat samples with higher protein  contents.  observed  i n sausage f o r m u l a t i o n s with high f a t  fat-to-protein  Lower  ratios  (Park et a l .  S i r i p u r a p u et a l . 1987; for  contents and  Hand et  al.  high 1987;  the t e x t u r a l parameters to i n c r e a s e with i n c r e a s i n g  moisture  loss)  attributed  (Figure 44)  contents  observed  Usborne,  been  tendency  be  M i t t a l and  1989;  have  1986). The  contents may  to  t e x t u r a l parameter values  to i n c r e a s e s i n Shrink in  f o r m u l a t i o n s with higher  compression  f o r c e s (Lee et a l . 1987)  c o n c e n t r a t i o n of p r o t e i n that leads (Trout and  positive  these q u a l i t y  parameters.  Twloss and Exwater  contents and  parameters. Exfluid  of  between Shrink  to  Twloss, reflected  thus the e f f e c t of  the pH of the raw emulsions  The p o s i t i v e r e l a t i o n observed  (Figure 45)  r e f l e c t e d the  MDPM and thus the e f f e c t of pH on While  and  (Figure 45)  p r o p o r t i o n of beef meat and  moisture and p r o t e i n  proportion  p o s s i b l y due  to g r e a t e r t e x t u r a l s t r e n g t h  r e l a t i o n s observed  the e f f e c t of the  and  leads  Schmidt, 1987).  Shrink and Exwater, and  Tmloss  moisture  (Figure 41). I t has been reported that weight l o s s  greater  The  ( i . e . weight  the p o s i t i v e r e l a t i o n observed -231-  effect these  on  between of  the  quality  between ES  and  8HRNK  4  3  TMLO88  fi  9  0  0  9 TWLO88  7 ° 43£ 2§ 871 4  6  0  5  21 34 2 6 1  7  8  9 2 5 g 78 1 78 3 3 4 4  3 0  5  0  6  0  KAR01  0  0 1^  4 '  0 2  Figure 4 4 .  5  t  3  0  6  6  6 48?  0  0  9  f  SHEAR  0  6  6 COHES  5  9  4  478 §78  3 3  8  I"  47c?  R e l a t i o n s h i p s between some q u a l i t y parameters. Nomenclature and d e f i n i t i o n of the q u a l i t y parameters are given i n Appendix E.  -232-  F T tvcoas  IT  9  TWU388  6 43g  *i 7  8  2| 3 4 478  478 3  1 ° 26  2  1 2  8  7  00  7  4  EXRJUD  58  3£l 0 2L  4  9  EB  5 6 "8"  7J9  e a  e  f  83  7  4  9  5  *  7  4  1  78  478  "7J9 8  "OCT  9  3 1 ° 2g  Figure  1 Q 2 6  84 7 9  1 28 5  6  0  2  6  4L  EXWATB*  JL  8  f  EMWT  23  45. R e l a t i o n s h i p s between the q u a l i t y parameters that describe product weight l o s s , emulsion s t a b i l i t y and j u i c i n e s s c h a r a c t e r i s t i c s . Nomenclature and d e f i n i t i o n of the q u a l i t y parameters are given i n Appendix E.  -233-  Exfat  (Figure  f a t and thus  45) r e f l e c t e d the e f f e c t of the p r o p o r t i o n the  effect  of  of pork  f a t content and the f a t - t o - p r o t e i n  r a t i o on these q u a l i t y parameters. All  the t e x t u r a l  (Figure  46).  analysis As  parameters r e l a t e d  These r e s u l t s  confirm  the f i n d i n g s  of c o r r e l a t i o n  performed on the experimental data.  can be observed, not only the i n g r e d i e n t  e f f e c t on the q u a l i t y parameters, fat  p o s i t i v e l y t o each other  p r o p o r t i o n s had an  but the moisture,  p r o t e i n and  contents of the meat blocks and moisture content of the  emulsions, affected emulsions  though dependent on the i n g r e d i e n t s the q u a l i t y parameters. and  affected  several  quality  of  interrelated  In a d d i t i o n ,  the weight l o s s a f t e r  the  thermal  proportions  raw also  the pH of the raw treatment  further  q u a l i t y parameters. I t can be concluded that the formulations  was  factors.  -234-  affected  by  a  number  of  HAFC1  0 HAFD2  4  3  8  6 1* 4  3  0  8  0 8 HEAR  8 Y  0  0 6  0 6  6 3 4 e7 l  0  2  8  3  0  471?  OOHE8  0  0 3  29 4  3 0  4  3  78 3 4  B  1  0  4  6  GUMMY  1  3 0 CHBVY  F i g u r e 46.  Relationships between the t e x t u r a l parameters. Nomenclature and d e f i n i t i o n of the q u a l i t y parameters are given i n Appendix E.  -235-  C. COMPUTATIONAL OPTIMIZATION OF FRANKFURTER FORMULATIONS 1.  Optimization of f r a n k f u r t e r formulations using formula o p t i m i z a t i o n computer program (FORPLEX) The  main  o b j e c t i v e of t h i s study was t o e s t a b l i s h a  optimization the  computer program to  meat p r o c e s s i n g  to  search  predetermined ranges.  of  product  Unlike  least-cost  quality  search  specifications  v e r s i o n of Box's Complex  optimization  nonlinear  objective  formulation  formulations within  T h i s formula o p t i m i z a t i o n program  the modified  formula  t h i s new formula o p t i m i z a t i o n  f o r best  method  new  be used f o r q u a l i t y c o n t r o l i n  industry.  programs, the approach is  the  that  has  functions subject  that  meet  allowable  cost  (FORPLEX)  method, been  program  i s based on  which i s a d i r e c t used  to l i n e a r  to  optimize  and  nonlinear  program  f o r meat  constra i n t s . To  test  formula  the  s u i t a b i l i t y of the  optimization,  optimization  trials  problems there defined:  were  were three  under  optimization  (MDPM), below,  X ) 2  X  3  i n a l l optimization  o p t i m i z a t i o n components that had to be f a c t o r s , (B) the c o n s t r a i n t s ,  factors  In t h i s study,  (pork f a t ,  are  the  variables,  the o p t i m i z a t i o n  i . e . the  Xa., and mechanically  not be considered  product  f a c t o r s were  two  deboned p o u l t r y meat  of the model f r a n k f u r t e r f o r m u l a t i o n . could  and  factors  that a f f e c t the c h a r a c t e r i s t i c s of the food  study.  ingredients  As  formulation  function.  (A) The o p t i m i z a t i o n  ingredients,  frankfurter  performed.  (A) the o p t i m i z a t i o n  (C) the o b j e c t i v e  The  several  FORPLEX  an o p t i m i z a t i o n  As explained f a c t o r due to  the c o n s t r a i n t of the t o t a l of the Ingredients p r o p o r t i o n s  having  to equal one. (B) The c o n s t r a i n t s The  constraints  restrictions  on  restricting  the  Evans,  selected.  The  constraints  that  optimization  value of the o b j e c t i v e  were The  the mathematical f u n c t i o n s  the f o r m u l a t i o n s and the  1983).  formulations  are  function  imposed  kept r e l a t i v e l y simple  on and  place  factors,  (Norback  and  the  frankfurter  were  arbitrarily  f o l l o w i n g c o n s t r a i n t s were used  throughout  the  study: 1. C o n s t r a i n t s on the i n g r e d i e n t p r o p o r t i o n s The  l i m i t s s e t f o r the p r o p o r t i o n s of the  were the ones used i n the extreme v e r t i c e s l i m i t s s e t f o r the experimental quality  prediction  tested.  These  models  limits  three  design  (Table 1 ) . The  design had to be used,  can only be used w i t h i n  are  commonly  referred  ingredients  to  s i n c e the the  limits  as e x p l i c i t  constra i n t s . 0.05 < X i < 0.30  2. this  0.00 < X  2  < 0.40  0.50 < X  3  < 0.95  C o n s t r a i n t on the t o t a l of the i n g r e d i e n t p r o p o r t i o n s , case  the  sum of the i n g r e d i e n t s  proportions  must  in equal  unity. Xi + X 3. C o n s t r a i n t s on the (a) f a t content  2  + X  3  = 1  (30)  proximate composition  had to be greater than -237-  of the meat block  or equal to 8.0% but l e s s  than or equal to 2 8 . 0 % . 8.0 <  80.37X1  + 18.67X  + 3.72X  2  < 28.0  3  (31)  (b) p r o t e i n content had to be g r e a t e r than or equal to 1 6 . 0 % but l e s s than or equal t o 2 1 . 0 % . 16.0 < 4.96Xa. + 1 5 . 5 1 X (c)  + 22.49X  2  < 21.0  3  (32)  moisture content had to be g r e a t e r than or equal  to 5 4 . 3 %  but l e s s than or equal t o 8 9 . 0 % 54.3 < 14.58X3. +  + 73.64X  65.69X2  < 89.0  3  (33)  4. C o n s t r a i n t on the c o s t of the meat block The c o s t of the meat block had t o be $1.9/Kg  but  l e s s than  or equal to $2.8/kg.  i n g r e d i e n t s were provided by Sebastian 1.9 < 0 . 2 0 X i + 0 . 8 0 X As  can  reduced  be  i n Appendix A  linear  cost and  i.e.  the  implicit  with e q u a l i t y  of  < 2.8  3  F i g u r e A3  the  (34)  these  proximate  However,  constraints  constraints,  this  ( r e p l a c i n g the e q u a l i t y  composition,  ingredients  were  entered  the 3  expression  -238-  cannot  constraint  was  with the e x p r e s s i o n (35)  a  as  constraint)  of 0.50 and 0.95 r e s p e c t i v e l y .  equality  equal  i n the  s i n c e the Complex method  o b j e c t i v e f u n c t i o n and the  incorporating  the  constraints,  1 - Xa. - X the  + 3.20 X  the c o n s t r a i n t that the sum of  removed by r e p l a c i n g the f a c t o r X  in  cost  (1989).  equations that d e s c r i b e d  Constraint subroutine. work  2  The  the f e a s i b l e area.  The  unity  seen  g r e a t e r than or equal t o  implicit an  constraints,  implicit  and  constraint  with lower and upper  limits  (c) The o b j e c t i v e The  function  objective  represents  the  function  i s the  the  objective  ingredient  function  d e s i r e d o b j e c t i v e of the problem,  e i t h e r maximized or minimized study,  mathematical  was  proportions  and  that  can  be  (Wolfe and K o e l l i n g , 1983). In t h i s to  f i n d the  optimal  combination of  which gave best q u a l i t y f o r m u l a t i o n s  that  met the product s p e c i f i c a t i o n s and c o s t c o n s t r a i n t s . Best q u a l i t y formulations quality  were d e f i n e d  was  quality.  as  as those  c l o s e as p o s s i b l e  Nakai and Arteaga  quality  values  parameter,  formulations to  (1990)  a  whose  predetermined  recommend  to unacceptable products,  target  the use of  s i n c e maximization or m i n i m i z a t i o n  e.g. hardness a t f i r s t  predicted  compression  of  target  a quality  (Hardl),  may  lead  i . e . extremely hard or extremely s o f t  frankfurters. Several trials  hypothetical  were  measures  performed  (Table  subroutine  of  frankfurter  40)  the  were  formulation  more  than  different quality.  optimization  q u a l i t y parameters The q u a l i t y  incorporated  into  the  the  it  is  Function  quality  a t each p o i n t computed by the studies  t r a i l s were  f i v e q u a l i t y p r e d i c t i o n models. into  one  f u n c t i o n , was performed. -239-  of  program.  f r e q u e n t l y necessary to  performed using  Therefore, equation,  as  prediction  one q u a l i t y parameter s i m u l t a n e o u s l y ,  computational o p t i m i z a t i o n  objective functions  formulation  FORPLEX program to p r e d i c t  formulations  i n food  optimize  using  of the f o r m u l a t i o n s '  equations  Since  frankfurter  two  the to  combination of s i n g l e i.e. a multi-objective  Target  values and t a r g e t d i f f e r e n c e values were s e t f o r  q u a l i t y parameter. quality  Target values are the optimum values  f o r the  parameters and a t a r g e t d i f f e r e n c e value i s the  minimum  difference  in  noticeable  difference  a d d i t i o n , the objective  the  quality  quality  resulting  parameter  i n dimensions of q u a l i t y parameter function  that  was  value of the products of s t a n d a r d i z e d  predicted  Subtraction  respective  target  optimization  of the p r e d i c t e d quality  objectives,  between the p r e d i c t e d (optimal)  values  value.  the  Mathematically,  a In  single  e f f e c t s of  minimized  was  the  d i f f e r e n c e s of the  was performed  q u a l i t y of a formulation  quality.  each  target q u a l i t y  q u a l i t y values  i . e . minimization  in  values.  q u a l i t y parameters from t h e i r r e s p e c t i v e  values.  target  the  value  which i s u s e f u l to prevent  multi-objective  absolute  in  parameter  t a r g e t d i f f e r e n c e values s t a n d a r d i z e  function,  difference The  each  from to  of the  their  meet  the  difference  and i t s r e s p e c t i v e the m u l t i - o b j e c t i v e  f u n c t i o n can be w r i t t e n N R = TI i=l  Abs  [ target quality* - predicted target d i f f e r e n c e *  quality* ]  (36)  i= 1, . . .N where N = number  of q u a l i t y parameters being optimized  target q u a l i t y i predicted  = t a r g e t value s e t f o r the i t h q u a l i t y parameter  qualityi  = predicted  value of the i t h q u a l i t y  parameter t a r g e t d i f f e r e n c e i = t a r g e t d i f f e r e n c e value s e t f o r the i t h q u a l i t y parameter -240-  It should the  maximum  be pointed value  minimization  of  out that the FORPLEX program searches f o r  of a n o n l i n e a r the  multi-objective  maximization of the negative The  multi-objective  subroutine  which  r e p l a c e each than  also  standardized  value  included  conditional  i n the  required,  unity;  the  functions  minimum  i s unity.  Function  instructions  single objective function  t h i s r e s t r i c t i o n was to a v o i d :  zero.  is  Since  of t h i s f u n c t i o n was performed.  which can be expected  individual objective  function.  function  f u n c t i o n was i n c o r p o r a t e d  or equal to one with  function  objective  value  to less  multi-objective The  purpose  of  (a) overemphasizing or o v e r l o o k i n g 1  and (b) having response values of  2  For example, i f two q u a l i t y parameters Shrink and Hardl are being optimized, and t h e i r t a r g e t values are 8.70 and 160.00 r e s p e c t i v e l y , t h e i r t a r g e t d i f f e r e n c e values are 0.022 and 0.803 respectively, and t h e i r p r e d i c t e d values f o r a p a r t i c u l a r p o i n t are 8.71 and 180.00 r e s p e c t i v e l y , the m u l t i - o b j e c t i v e f u n c t i o n for t h i s p a r t i c u l a r p o i n t would be 1  Abs  [8.70 - 8.71] 0.022  x Abs  [160.00 -180.003 0.803  =  0.45 x 24.91  The response value would be 11.21 i f the f u n c t i o n value f o r Shrink i s not r e p l a c e d by u n i t y . In t h i s case the FORPLEX program may emphasize i t s search towards the minimum value of Shrink and may overlook the Hardl response value, which i s f a r from i t s target. For example, i f the p r e d i c t e d Shrink value i s equal to i t s target, the m u l t i - o b j e c t i v e f u n c t i o n f o r t h i s p a r t i c u l a r point would be 2  Abs  [8.70 - 8.701 0.022  X Abs  [160.00 - 180.001 0.803  = 0.00 X 24.91  The response value would be zero i f the f u n c t i o n value f o r Shrink i s not r e p l a c e d by u n i t y . In t h i s case the search w i l l be ended and Hardl w i l l be overlooked.  -241-  The 47.  flow c h a r t of the F u n c t i o n  When the computed value  function  is  noticeable quality it  equal  or  difference  of  than 1 ( r i <  between and  the  target  thus a value  ( r i =1). When the m u l t i - o b j e c t i v e  all  the p r e d i c t e d q u a l i t y values  targets.  For  product establish value  a  l o s s and  in  target  some  (ri.)  no  the  and  trials  standardized  were greater  followed  the  several  predicted  from  any  b e t t e r q u a l i t y . For  single objective  frankfurter  given  this  incorporated  If the  t h e i r target values,  the  these  function values  predicted  computation  47. by  Nakai  formula o p t i m i z a t i o n  and  Arteaga  trials  performed having a s i n g l e q u a l i t y parameter as a c o n s t r a i n t . q u a l i t y p r e d i c t i o n models were subroutine. functions  The to  simultaneously  conversion  constraints  incorporated  i n t o the  of r e l a t i v e l y unimportant simplifies  to  predicted  with u n i t y when the p r e d i c t e d  recommendations  to  their  necessary  but  that  the procedure o u t l i n e d i n Figure  Following (1990)  than  no  i s 1 (R = 1 ) ,  i t was  were equal to or l e s s than t h e i r t a r g e t v a l u e s . values  the  is  such as those d e s c r i b i n g  considered  were r e p l a c e d  there  different  t a r g e t value  optimization  q u a l i t y parameters, values  was  1),  of u n i t y i s assigned  emulsion s t a b i l i t y  maximum allowable  below t h i s  reason,  are  Figure  single objective  f u n c t i o n value  some q u a l i t y parameters,  weight  i s shown i n  a standardized  less  parameter value  subroutine  the  were The  Constraint objective  optimization  emphasizes more important q u a l i t y items (Nakai  and and  Arteaga, 1 9 9 0 ) . In a d d i t i o n to s p e c i f y i n g the -242-  quality prediction  models,  the  Computation of the q u a l i t y p r e d i c t i o n models ( N )  r  t  = Abs  [target q u a l i t y ^ - predicted target d i f f e r e n c e !  quality^]  No N R = it  i =l  Figure  r i  ^  47. Flow c h a r t of the F u n c t i o n subroutine of the FORPLEX computer program.  -243-  multi-objective the  function  and the i m p l i c i t c o n s t r a i n t  FORPLEX program r e q u i r e d  formulation  optimization  the f o l l o w i n g f o r each  functions, frankfurter  trial:  (A) S p e c i f i c a t i o n of the convergence parameters The  convergence parameters ALPHA,  1.3, 0.0001 and 5 r e s p e c t i v e l y , unless (B) Number of p o i n t s As  recommended  complex  should  Therefore  BETA and GAMMA were s e t at otherwise s p e c i f i e d .  i n the complex  by  Box (1965)  the number of p o i n t s  be double the number  the number of p o i n t s  in  of  optimization  constraints  (on the o p t i m i z a t i o n (proximate  factors.  the complex was s e t at 4.  (C) S p e c i f i c a t i o n of the upper and lower l i m i t s constraints  i n the  f o r the e x p l i c i t  f a c t o r s , X i and X ) and i m p l i c i t 2  composition,  c o s t and X , 3  q u a l i t y parameters i f they were considered  and  on  the  constraints).  (D) S p e c i f i c a t i o n of the Random number seed ( l i n e number 270) As random  mentioned  i n M a t e r i a l s and Methods s e c t i o n H.2.1.4.1.  number seed was s e t to 3,  thus,  the sequence of  numbers was the same f o r a l l the o p t i m i z a t i o n random used  number  trials.  random  Because one  seed was used, s e v e r a l s t a r t i n g p o i n t s had  i n each o p t i m i z a t i o n t r i a l  the  to be  to check the convergence t o  the  g l o b a l optimum. (E) S p e c i f i c a t i o n of the i n i t i a l To s t a r t a search  the  FORPLEX  f e a s i b l e point program r e q u i r e s  an  initial  p o i n t that does not v i o l a t e the c o n s t r a i n t s . Four s t a r t i n g p o i n t s were generated using the FPOINT computer X i = 0.127,  X = 0.123; 2  (b)  X i = 0.291, -244-  program. These were: (a)  X = 0.104; 2  (c)  Xa. = 0.100,  X=> = 0.220;  (d)  Xi=0.100,  X =0.380. 2  computer program. The Function As mentioned  Appendix C l i s t s the FPOINT  subroutine  used s t a r t s i n l i n e 46.  i n M a t e r i a l s and Methods s e c t i o n H.I., the lower and  upper l i m i t s of the o p t i m i z a t i o n  f a c t o r s and/or c o n s t r a i n t s  were  narrowed t o o b t a i n the four s t a r t i n g p o i n t s . In some  optimization  trials  obtained  additional  p o i n t s were needed.  These were  by  examining the corresponding f e a s i b l e mixture spaces. The  f o l l o w i n g s e c t i o n s show the r e s u l t s of the  optimization addition  trials  to s e a r c h i n g  constraints  for  computational  performed using the FORPLEX f o r best  program.  q u a l i t y formulations  product s p e c i f i c a t i o n s and c o s t ,  program was used f o r f i n d i n g  the minimum  and  In  that met the the  FORPLEX  maximum values of  each q u a l i t y parameter w i t h i n the f e a s i b l e area.  1.1.  Single objective  The and  FORPLEX  minimum  optimization  program was used f o r f i n d i n g the g l o b a l  values  of the q u a l i t y parameters,  Hard2, w i t h i n the f e a s i b l e area, for:  that  (a) proximate composition,  cost.  The  performance nonlinear  purpose  of  (b)  these  except  maximum pH  and  i s , meeting the c o n s t r a i n t s ingredient  trials  was  l e v e l s and  (c)  evaluate  the  each l i n e a r  and  to  of the FORPLEX program i n handling  q u a l i t y p r e d i c t i o n equations i n d i v i d u a l l y (Table  40).  In t h i s s e c t i o n the convergence parameter BETA was s e t a t 0.1 f o r the  optimization  of a l l q u a l i t y parameters except Cohes,  BETA was s e t at 0.001. As mentioned before, were  used i n each o p t i m i z a t i o n  trial,  -245-  thus,  where  four s t a r t i n g p o i n t s four  optimization  runs  were  performed f o r each q u a l i t y parameter.  restarting check had  each search  from d i f f e r e n t  whether the g l o b a l r a t h e r been found.  Further  starting  prediction  shown).  Table 44  of  each  combinations  of  optimization  of  contains  was  to  minimum  and  f e a s i b l e area  the  (figures minimum  corresponding  ingredient proportions.  was  contour l i n e s of each  the g l o b a l maximum and  parameter  each  points  v e r i f i c a t i o n of the optimum r e s u l t s  model w i t h i n the  quality  reason for  than a l o c a l maximum or  performed by examining the response s u r f a c e quality  The  convergence  optimum depended on the s t a r t i n g p o i n t s . In g e n e r a l ,  values optimal  As expected,  q u a l i t y parameter,  not  for  the  to  the  the FORPLEX  d i d not converge to a s i n g l e p o i n t with a l l four s t a r t i n g p o i n t s , but converged on areas c l o s e to the g l o b a l optimum. the  FORPLEX f a i l e d  was  stalled  1.2.  to converge to the optimum s i n c e  at the boundaries of the  Multi-objective  As  mentioned  frankfurter optimal  formulations  combinations  constraints. optimization  the  parameter  was  of  optimization  using the FORPLEX program was  t h a t met  proportions  hypothetical  were  performed  using two  -246-  find  cost  formulation  to f i v e  quality  q u a l i t y . In a d d i t i o n ,  formulations  a constraint  of  which gave best  frankfurter  formulations'  frankfurter considered  to  the product s p e c i f i c a t i o n s and  parameters as measures of the optimization  search  constraints.  o b j e c t i v e of the  of i n g r e d i e n t  Several trials  the  optimization  earlier,  q u a l i t y formulations  In some runs  was  where  a  performed.  quality Target  Table  44.  optimum  combinations  quality  parameters  of  and  Ingredients target  proportions  difference  f o r . maximum  and  minimum  Maximum  Minimum  0  Xi  0,.08 0,, 21 0..13 0 .30 , 0,, 30 0 .13 , 0,, 30 0 ,13 . 0.,05 0.,05 0..09 0.,05  *• N o m e n c l a t u r e a n d in Appendix E •  °  of  Target difference value  Quality parameter*  Shrink Tmloss Twloss ES Exfluid Exwater Exfat Hardl Shear Cohes Gummy Chewy  values  values.  X  X  2  0,.07 0..00 0., 00 0,,08 0., 00 0 , 00 0..08 0 , 10 0.,10 0.,10 0.,06 0., 10  0,.85 0,.79 0., 87 0 ,62 , 0,, 70 0,.87 0.,62 0,,77 0.,85 0.,85 0., 85 0.,85  definition  Value  Xi  1 0 , . 53d» 42 ,91c 2 7 , . 6 8abc 0 ,67d 1 2 , . 59a-d 9,,15a 5,,4 Id 1 9 2 , ,12a 6.. 60ab 0,. 30ab 5 6 . ,46c 2 2 6 , , 66b-i)  0 .28 0 .05 0 .10 0 .05 0 .14 0 .26 0 .05 0 .25 0 .29 0 .30 0 .28 0 .28  a  of  the  quality  parameters  X  0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.  are  Xa  2  0 . 53 0 .,55 0.,50 0 .,85 0,. 50 0,.52 0 ., 84 0,, 52 0,.56 0,.62 0,. 5 2 0,. 5 3  19 40 40 10 36 22 11 23 15 08 20 19  given  S t a r t i n g p o i n t s w h e r e c o n v e r g e n c e t o t h e same g l o b a l optimum was a c h i e v e d : ( a ) X i = 0 . 1 2 7 , Xa= 0 . 1 2 3 ; ( b ) X i = 0.291, Xa= 0 . 1 0 4 ; ( c ) X i = 0 . 1 0 0 , Xa 0 . 2 2 0 ; ( d ) XJ.= 0 . 1 0 0 , X , = 0.380 T a r g e t d i f f e r e n c e v a l u e = 0 . 0 1 x (maximum - m i n i m u m v a l u e ) (37)  Value  8.37c 31.69a-d 20.91ac 0 . 0 3ac 8.86a 4.60a-d 1.98a 111.87bc 4.32b 0.25M 28.07c 92.97a  0 .022 0 .112 0 .068 0 .006 0 .037 0 .046 0 .034 0 .803 0 .023 0 .0006 0 .284 1 .337  the  values  and t a r g e t  parameter.  d i f f e r e n c e values  As d i s c u s s e d  parameter  were  optimization  chosen  section.  below,  were s e t f o r each  t a r g e t values  according  to  the  f o r each  study f o r each q u a l i t y parameter was equal to 1%  1.2.1.  the  set in  this  of the range of  w i t h i n the f e a s i b l e area  o b j e c t i v e of t h i s s e c t i o n was  the FORPLEX program f o r o b t a i n i n g p r e d i c t e d q u a l i t y was meet  this  attained  (Table 4 4 ) .  objective  a l l the t a r g e t q u a l i t y values  target  points point  two  trials the  quality  had  to  3  targets  that  the  incorporated  frankfurter  quality.  parameters  point  frankfurter  proportions  models (Table  of  optimization  formulation  1  trials  The p r e d i c t e d values  2 were used that  as  targets  incorporated  q u a l i t y parameters as measures of the f o r m u l a t i o n s ' -248-  40). were  optimization  p a i r s of q u a l i t y parameters as  formulations'  were  of the q u a l i t y  of the q u a l i t y parameters of p o i n t  in  thus,  3  of i n g r e d i e n t s  using the q u a l i t y p r e d i c t i o n  be  X = 0 . 5 5 0 ) and  2  for these two combinations  The p r e d i c t e d values as  whose  formulations)  1 (X:L = 0.250, X = 0.200,  2  predicted  used  (i.e.,  of  t a r g e t q u a l i t y . To  2 ( X i = 0.150, X = 0.100, X = 0.*750 ). The values  parameters were  f r a n k f u r t e r formulations  i n the same combination of i n g r e d i e n t p r o p o r t i o n s ,  feasible  point  to t e s t the s u i t a b i l i t y  not d i f f e r e n t from t h e i r  a r b i t r a r i l y chosen:  of  of  Optimization pj frankfurter formulations. where combinations of two to f i v e q u a l i t y parameters were considered measures of the f o r m u l a t i o n s ' q u a l i t y . Target q u a l i t y values were c a l c u l a t e d from t a r g e t p o i n t s .  The  two  quality  objective  The t a r g e t d i f f e r e n c e value  each q u a l i t y parameter value  quality  three  measures of  the  in  the  to f i v e  quality.  The  combination  of the q u a l i t y parameters used i n each  trial  was a r b i t r a r i l y  value  of these o p t i m i z a t i o n  the  selected.  optimum combinations  starting  points  optimization  i n each t r i a l . only but  were  used  in  each  four o p t i m i z a t i o n runs were performed  The use of d i f f e r e n t s t a r t i n g p o i n t s allowed  proof  proportions  were expected  of the two t a r g e t p o i n t s . The four  mentioned before  thus,  a check of whether also  was expected to be u n i t y and  of i n g r e d i e n t p r o p o r t i o n s  (a to d)  trial,  The t h e o r e t i c a l minimum response  trials  to be c l o s e to the p r o p o r t i o n s  optimisation  that  the  not  t h e o r e t i c a l minimum had been found  more than one combination  ( i . e . formulations)  could  of  ingredient  be found whose p r e d i c t e d  q u a l i t y was not d i f f e r e n t from the t a r g e t q u a l i t y . The reason why different  combinations  multi-objective conditional  of  ingredient proportions  f u n c t i o n value  equal  i n s t r u c t i o n s incorporated  t h a t replaced  each s t a n d a r d i z e d  to in  one  is  can  give  based  the Function  a  on the  subroutine  single objective function  value  l e s s than or equal to one with u n i t y . Table  45  formulations of  pairs  shows  the o p t i m i z a t i o n  whose r e q u i r e d q u a l i t y was given  for  and  a t 0.0001  the  FORPLEX  BETA was s e t at 0.01  f o r a l l other program  i n s t r u c t i o n s to replace Twloss with u n i t y  trials.  for t r i a l s  the s t a n d a r d i z e d  for t r i a l s  and  2  parameters were equal to or l e s s than t h e i r  The  1 and  2,  subroutine  of  had c o n d i t i o n a l  f u n c t i o n value  when the p r e d i c t e d values  -249-  trials).  The F u n c t i o n 1  frankfurter  by the combination  of q u a l i t y parameters ( s i x o p t i m i z a t i o n  convergence parameter  and  results  of Shrink  of these q u a l i t y  target values.  This  T a b l e 45. F r a n k f u r t e r f o r m u l a o p t i m i z a t i o n t r i a l s where p a i r s o f q u a l i t y p a r a m e t e r s were considered measures o f t h e f o r m u l a t i o n s ' quality. Target q u a l i t y values w c a l c u l a t e d from t a r g e t p o i n t 1 (Xi= 0 . 2 5 0 , Xa = 0 . 2 0 0 , X . = 0 . 5 5 0 ) * .  Trial No.  Quality parameters"  Target values  1  1. H a r d l 2. S h r i n k "  132. 30 8 . 75  a b  2  1. T w l o s s * 2. Shear  22. ,10 4.,53  3  1. E x f l u i d 2. Cohes 1. Tmloss 2. ES  4  5  1. E x f a t 2. Gummy  6  1. Exwater 2. Chewy  Starting points  IT°  Optimum I n g r e d i e n t s proportions  0  Xi  Xa  22 20  0.250 0.282  0.201 0 .160  0 . 549 0 . 558  131, ,85 132, , 37  8.75 8 . 57  a b c d  13 22 14 15  0.238 0.241 0.237 0.237  0.242 0 .233 0.245 0.242  0. 520 0. 526 0. ,518 0. .521  21, .64 21, ,73 21 .60 21 .64  4.53 4 .53 4 . 53 4.53  9. 60 0.,25  b  59 17 22  0.246 0.251 0.247  0.196 0.204 0.199  0, ,557 0.545 0,,554  9 .63 9 .57 9 .61  0.25 0.25 0.25  38, ,10 0..55  a b d  20 30 99 27  0.250 0.250 0.249 0.249  0.201 0.202 0.205 0.198  0,,549 0,.548 0,,547 0,.552  .07 .06 .00 .14  0.55 0.55 0.55 0.55  4.78 33 .87  b c d  72 23 30  0.252 0.251 0.252  0.199 0.198 0.198  0,.549 0,,552 0., 550  4 .81 4 .79 4 .81  33.66 34.09 33.78  4 .88 105 .71  a b c d  24 18 23 23  0. 252 0.243 0.250 0.253  0.198 0.210 0.199 0 .197  0,. 550 0,.547 0 . 551 0 . 550  4 .88 4 .87 4 .90 4 .88  105.22 106.80 106.14 104.79  c  d  c  Xj  Mult 1-objectWe f u n c t i o n value = 1 • N o m e n c l a t u r e and d e f i n i t i o n o f t h e q u a l i t y p a r a m e t e r s a r e g i v e n i n A p p e n d i x E. ° S t a r t i n g p o i n t s where c o n v e r g e n c e t o t h e g l o b a l minimum was achieved. (a) Xi= 0.127, X,= 0 . 1 2 3 ; (b) Xi= 0 . 2 9 1 , Xj= 0 . 1 0 4 ; (c) Xi= 0 . 1 0 0 , X = 0 . 2 2 0 ; (d) Xi= 0 . 1 0 0 , X i = 0 . 3 8 0 . ° Number o f I t e r a t i o n s * I f t h e p r e d i c t e d q u a l i t y v a l u e was l e s s t h a n o r e q u a l t o t h e t a r g e t value then the s t a n d a r d i z e d s i n g l e o b j e c t i v e f u n c t i o n v a l u e was r e p l a c e d w i t h u n i t y k  2  Predicted quality values 1  38 38 38 38  2  was  performed  since  formulations  with lower p r e d i c t e d  product  weight l o s s than the predetermined t a r g e t had b e t t e r than weight  loss.  Results  reported  are only f o r those  optimal  optimization  runs where the FORPLEX converged at the t h e o r e t i c a l minimum, that is,  a t those computed formulations  not d i f f e r e n t of  to  of the s t a r t i n g p o i n t s used.  optimization  t h e o r e t i c a l minimum exception  of t r i a l  trials  the  The FORPLEX 2,  4  1,  1,  when s t a r t i n g p o i n t c  the FORPLEX a l s o was used.  optimum  starting  points.  As can be seen  trial  (Table  combinations of i n g r e d i e n t s p r o p o r t i o n s  (i.e.,  formulations)  to the With the  The number of i t e r a t i o n s needed  each s u c c e s s f u l run i n each o p t i m i z a t i o n points  3,  converged to the minimum  l o c a t e the optimum i n each o p t i m i z a t i o n  by the  used.  6  points.  FORPLEX converged was  was  and  However i n t r i a l s  when s t a r t i n g p o i n t b  was  minimum  5 convergence to the minimum depended on the s t a r t i n g  In a l l the  to  trials.  l o c a t e the t h e o r e t i c a l minimum i n t r i a l s  regardless  quality  from t h e i r t a r g e t q u a l i t y . The t h e o r e t i c a l  1 was found i n a l l the o p t i m i z a t i o n  able  and  whose p r e d i c t e d  was  influenced  45),  different  were  trial.  were l o c a t e d  in  found  These the  in  optimum  vicinity  of  t a r g e t p o i n t 1 (Xi=0.250, X =0.200, X =0.550). 2  The required  optimization results for frankfurter quality  optimization optimization  was  trials)  given  four  t r i a l ) are given  t h e o r e t i c a l minimum,  formulations  by the combination of  and  o n l y f o r those o p t i m i z a t i o n the  3  quality  i n Table 46. runs  that  parameters  Results  reported  (two (one are  where the FORPLEX converged at  i s , at those computed  -251-  three  whose  formulations  Table  46.  Frankfurter of  the  (X = x  formula  optimization  formulations'  0.150,  X  2  =  quality.  0.100,  X,=  trials  where  Target  three  quality  and f o u r values  quality  were  parameters  calculated  1  I to  Quality Parameters'  Starting Points  Target values  point  measures  2  Predicted quality values  IT  1  0  7.33 7.35  1. 2. 3.  Exwater Chewy Gummy  7.32 178.13 53.00  a c  31 17  0.148 0.154  0.,101 0.,094  0 . 752 0 . 752  2  1. 2. 3.  Shrink ES Hardl  10.04 0.25 191.40  a b c d  17 47 20 43  0.152 0.148 0.149 0.151  0..089 0,.117 0,.092 0 .101 ,  0. 0. 0. 0.  759 735 759 748  10.04 10.02 10.05 10.03  3  1. T m l o s s 2 . Twloss 3. Exfluid 4. E x f a t  40.27 25.61 10.68 3.35  a b d  50 40 52  0.150 0.150 0.150  0.,100 0,,101 0..100  0 . 750 0 . 749 0 . 750  40.27 40.24 40.27  to I  considered  target  0.750)*. Optimum I n g r e d i e n t s proportions  Trial No.  were  from  M u l t i - o b j e c t i v e function value = 1 N o m e n c l a t u r e and d e f i n i t i o n of the q u a l i t y p a r a m e t e r s a r e g i v e n in Appendix E S t a r t i n g p o i n t s w h e r e c o n v e r g e n c e t o t h e g l o b a l m i n i m u m was a c h i e v e d : ( a ) X i = 0 . 1 2 7 , X = 0 . 1 2 3 ; ( b ) Xx= 0 . 2 9 1 , X = 0.104; ( c ) X i = 0 . 1 0 0 , Xa= 0 . 2 2 0 ; ( d ) X i = 0 . 1 0 0 , X = 0.380. Number o f iterations a  a  3  .  179 . 11 1 7 7 . 43  53.11 52.80  25 25 25 25  191.04 191.33 191.35 101.32  2 5 . 61 2 5 . 60 2 5 . 51  10.67 10.66 10.68  0. 0. 0. 0.  3.35 3.35 3.35  whose p r e d i c t e d quality.  was  The t h e o r e t i c a l  optimization theoretical used.  quality  trials.  not  minimum  The  in  of  FORPLEX  minimum i n t r i a l  However  d i f f e r e n t from  trials  1 was found  was  converged  able  2 regardless  to  in  to  target a l l the  locate  the  of the s t a r t i n g  1 and 3 convergence  depended on the s t a r t i n g p o i n t s . In a l l the FORPLEX  their  the  to  the  point  minimum  optimization  trials  the t h e o r e t i c a l minimum when s t a r t i n g  p o i n t a was used. T h i s may be due to the f a c t that t h i s p o i n t was located  i n the  X =0.750)  and t h e r e f o r e  3  the  vicinity  optimum.  The  of  In t r i a l s  proportions points point  were  1 and  trial  and d;  starting  b  2  locate  2 d i f f e r e n t combinations of  were l o c a t e d  in t r i a l  converged  the  ingredient  These  optimum  i n the v i c i n i t y of t a r g e t  3 the FORPLEX converged to  t h i s combination being point  to  found i n each s u c c e s s f u l r u n .  However,  X =0.100,  /  was i n f l u e n c e d by the s t a r t i n g  combination of i n g r e d i e n t p r o p o r t i o n s a  i  number of i t e r a t i o n s needed  ( i . e . formulations) 2.  2 (X =0.150  i t was e a s i e r f o r the FORPLEX t o l o c a t e  optimum i n each o p t i m i z a t i o n points.  target point  the same  i n runs of s t a r t i n g  the t a r g e t p o i n t  in close proximity  2.  points  The run of  of  the  target  point. Four where  f r a n k f u r t e r formula o p t i m i z a t i o n five  trials  performed  measures  of the  formulations  q u a l i t y . The o p t i m i z a t i o n r e s u l t s are given  i n Table  47.  reported  Results  q u a l i t y parameters were considered  were  the FORPLEX converged those  computed  are only at  f o r those o p t i m i z a t i o n  the t h e o r e t i c a l minimum,  formulations  whose  -253-  predicted  runs where  that  i s , at  q u a l i t y was not  T a b l e 47.' F r a n k f u r t e r formula optimization t r i a l s vhere formulations' quality. Target q u a l i t y values X,= 0 . 7 5 0 ) * .  Trial No.  Quality parameters*  Target values  Starting polntsa  IT  f i v e q u a l i t y parameters v e r e c o n s i d e r e d measures of t h e v e r e c a l c u l a t e d from t a r g e t p o i n t 2 ( X i = 0 . 1 5 0 , X*= 0.100,  Optimum I n g r e d i e n t s proportions  Predicted quality values  D  Xx  X»  X,  1  2  3  '  4  5  1. 2. 3. 4. 5.  Twloss Exwater Exfat Hardl Cohes  25. Gl 7 .,32 3.,35 191. 40 0,,27  a c  16 56  0,,150 0,,152  0.100 0.101  0 ,,750 0,.747  25, .61 25, .57  7 .,32 7 .28  3.35 3.38  191, ,43 191, ,28  1. 2. 3. 4. 5.  Shrink Exfluid Shear Cohes Gummy  10, .04 10, ,68 5,.70 0,.27 52 .99  a b d  95 30 72  0,,149 0,,150 0,,151  0.098 0.101 0.101  0,,753 0,,749 0 ,,748  10, ,05 10, ,03 10, ,03  10, .70 10, ,67 10 ,.67  5.72 5.70 5.69  0,,27 0,,27 0,,27  53.08 52.98 52.93  1. 2. 3. 4. 5.  Tmloss Exfat Hardl Gummy Chewy  40. .27 3,, 35 191, .40 53 ,,00 178, .13  a b c d  27 39 41 40  0.,150 0.,150 0.,149 0,, 150  0.101 0 .097 0.101 0 .098  0,, 749 0 ,,753 0 ,,749 0,,752  40. ,23 40 .,33 40. ,23 40. , 31  3,,35 3 ,35 , 3,,35 3 ,36 .  191.43 191.40 191.45 191.38  53. ,00 53 . 05 53, ,02 53. ,00  177.96 179.04 178.16 178.45  1. 2. 3. 4. 5.  Twloss ES Exfat Hardl Cohes  25, ,61 0 .25 3,, 35 191 .40 0 .27  a c  30 35  0.149 0 ,,150  0.102 0.098  0,,749 0 ,,752  25. , 59 25. ,65  0,.25 0 ,,25  3.35 3.35  191, ,45 191. ,38  0.27 0.27  * M u l t i - o b j e c t i v e f u n c t i o n value = 1 * N o m e n c l a t u r e and d e f i n i t i o n o f t h e q u a l i t y p a r a m e t e r s a r e g i v e n i n Appendix E ° S t a r t i n g p o i n t s where c o n v e r g e n c e t o t h e g l o b a l minimum was a c h i e v e d : (a) Xx= 0 . 1 2 7 , X = 0 . 1 2 3 ; (b) Xx = 0 . 2 9 1 , Xa = 0 . 1 0 4 ; ( c ) Xx= 0 . 1 0 0 , Xa= 0 . 2 2 0 ; (d) Xx= 0.100, Xa= 0 . 3 8 0 . ° Number o f i t e r a t i o n s a  0,27 0 . 27  different vas  from t h e i r  t a r g e t q u a l i t y . The t h e o r e t i c a l minimum of 1  found i n a l l the o p t i m i z a t i o n  locate  the  theoretical  s t a r t i n g p o i n t used.  trials.  minimum  in t r i a l  However i n t r i a l s  the minimum depended on the s t a r t i n g the  frankfurter optimization  FORPLEX  trials  The FORPLEX was able to 3 regardless  of  the  1, 2 and 4 convergence to  points.  As i n the case of  reported  in  Table 46,  the  converged to the t h e o r e t i c a l minimum when s t a r t i n g point  a was used. The number of i t e r a t i o n s needed to l o c a t e the optimum in each o p t i m i z a t i o n  trial  With  of t r i a l  the  exception  l o c a t e the optimum with trial  2, fewer i t e r a t i o n s were needed  s t a r t i n g point  a.  In each  located  in  These optimum p o i n t s  close proximity  (i.e.  of t a r g e t p o i n t  to  optimization  d i f f e r e n t combinations of i n g r e d i e n t p r o p o r t i o n s  in each s u c c e s s f u l run. were  was i n f l u e n c e d by the s t a r t i n g p o i n t s .  were found  formulations) 2  (Xi=0.150,  X =0.100, X =0.750). 2  3  combinat i ons of f i v e d u a l i t y parameters were considered measures of the f o r m u l a t i o n s ' q u a l i t y . Tarqet Quality values were s e t i n d i v i d u a l l y 1  Unlike to  test  the previous the  formulations  s e c t i o n , the o b j e c t i v e of t h i s s e c t i o n was  suitability  of the FORPLEX to o b t a i n  frankfurter  whose p r e d i c t e d q u a l i t y was as c l o s e as p o s s i b l e  to  a predetermined t a r g e t q u a l i t y . To meet t h i s o b j e c t i v e the t a r g e t q u a l i t y values  were i n d i v i d u a l l y s e l e c t e d .  optimization  procedure,  combinations single  of  objective  ingredient function  since  in  this  proportions could -255-  This complicated case  for  be f a r from  one  the  the  optimal  standardized  optimal  f o r the  other of  f u n c t i o n s . Thus, the FORPLEX had t o a r r i v e a t a combination  ingredient proportions  obtained,  that  is,  where  a  "compromised"  where the d i f f e r e n c e between each  q u a l i t y parameter and i t s t a r g e t value The  multi-objective  unity.  The  optimum was  f u n c t i o n value  was as  predicted  small as p o s s i b l e .  was expected t o be higher  lower the f u n c t i o n value  than  the c l o s e r the p r e d i c t e d  q u a l i t y values  were from t h e i r t a r g e t q u a l i t y v a l u e s .  q u a l i t y values  of each q u a l i t y parameter were chosen i n d i v i d u a l l y  by examining the response s u r f a c e p r e d i c t i o n model (Figures that:  are  contour p l o t s of each  25 t o 38) and t a k i n g  (a) low weight l o s s ( i . e .  emulsion s t a b i l i t y required  extremely hard  (i.e.  low values  low values  i n the manufacture  based on previous  I t i s recommended values  47.  parameter  t o the  Shear, Cohes, that i n f u t u r e  should  formula o p t i m i z a t i o n  t r i a l s were  the four combinations of f i v e q u a l i t y parameters  Table  and (b)  be  made  experience and consumer sensory t e s t s .  frankfurter  e i t h e r product  Each o p t i m i z a t i o n  trial  weight l o s s (Shrink)  considered  performed reported  measures  or one emulsion  of  stability  (ES, Tmloss or Twloss), one or two j u i c i n e s s parameters  ( E x f l u i d , Exwater, (Hardl, Shear, the  and Tmloss)  frankfurters,  s t u d i e s o b j e c t i v e s e l e c t i o n of the t a r g e t  in  consideration  of ES, Twloss, of  quality  of Shrink) and high  i n the middle range of Hardl,  Gummy and Chewy were chosen.  using  into  or s o f t f r a n k f u r t e r s are not acceptable  consumer. Thus values  Five  The t a r g e t  FORPLEX  Exfat)  Cohes,  and two or three  texture  Gummy, Chewy). The F u n c t i o n  had c o n d i t i o n a l -256-  instructions  to  parameters  subroutine of replace  the  standardized  f u n c t i o n value  of Shrink,  u n i t y when the p r e d i c t e d values equal to or l e s s than t h e i r because (Shri  formulations  with  target  X i = 0.280,  values.  i n order  However,  to v e r i f y  2  X =0.100; 2  (h)  product  Xj. = 0.200,  weight l o s s  a d d i t i o n a l p o i n t s were the convergence to the  by examining F i g u r e s  X:L = 0.100, X = 0 . 1 0 0 ;  were  T h i s was performed  lower p r e d i c t e d  optimum, these were obtained p o i n t s were: (e)  Twloss or Tmloss with  of these q u a l i t y parameters  i n each o p t i m i z a t i o n t r i a l .  needed i n some t r i a l s  ES,  48 to 5 2 . These  ( f ) Xx = 0.250, X = 0 . 0 5 0 ; (g) 2  X =0.250; 2  (i)  X i = 0.280,  X =0.120. 2  The  per cent d i f f e r e n c e  from t a r g e t was c a l c u l a t e d f o r  q u a l i t y parameter considered followin  i n each o p t i m i z a t i o n  i n each o p t i m i z a t i o n  needed i n some t r i a l s  trial.  i n order  However,  to v e r i f y  the convergence to the  by examining F i g u r e s  p o i n t s were: (e)  2  Xi=0.280,  X =0.100; 2  (h)  X!=0.200,  using the  a d d i t i o n a l p o i n t s were  optimum, these were obtained  Xa. = 0.100, X = 0 . 1 0 0 ;  trial  each  48 to 5 2 . These  ( f ) X i = 0.250, X = 0 . 0 5 0 ; (g) 2  X =0.250; 2  (i)  Xi=0.280,  X = 0 .120. 2  The  per cent d i f f e r e n c e  from t a r g e t was c a l c u l a t e d f o r  q u a l i t y parameter considered  i n each o p t i m i z a t i o n  trial  each  using the  f o l l o w i n g equation % d i f f e r e n c e = p r e d i c t e d q u a l i t y value - t a r g e t q u a l i t y value from t a r g e t t a r g e t q u a l i t y value (38) T h i s percentage value target  gave a measure of how f a r the p r e d i c t e d  was from i t s t a r g e t . value  A  quality  p o s i t i v e per cent d i f f e r e n c e  f o r a p a r t i c u l a r q u a l i t y parameter meant that -257-  from the  predicted value.  quality For  lower.  A  predicted zero  value  a negative  zero  per  was  greater  value,  cent  cent  the  d i f f e r e n c e from t a r g e t  l a r g e per  ES when the p r e d i c t e d values  q u a l i t y was noted  d i f f e r e n c e values  b e t t e r than the r e q u i r e d  of t h i s percentage,  considered  by  (equation  36)  using equation The  optimization  was  not  r e s u l t s of the  In each o p t i m i z a t i o n  trial  its  be  i n the value target  trials  are  the FORPLEX v a r i e d  optimum depending on the s t a r t i n g  In a d d i t i o n , the number of i t e r a t i o n s to l o c a t e  1 the  quality  e and  were: a,  to a s i n g l e p o i n t f u n c t i o n value  (Xi=0.228,  of 77.98  f).  p o i n t were 93,  b, c, d,  79,  The 86,  f.  s t a r t i n g points The  were: used  FORPLEX converged  X =0.176, X =0.596) 2  the  used.  parameters that were optimized  in t h i s t r i a l  and  to  predicted  considered  five optimization  Cohes. The  e,  done  d i f f e r e n t from i t s t a r g e t  Twloss, Exwater, E x f a t , Hardl and  b,  T h i s was  thus a p r e d i c t e d q u a l i t y  g l o b a l minimum depended on the s t a r t i n g p o i n t s In t r i a l  A  Shrink,  t a r g e t q u a l i t y . I t should  as being  i t s success at l o c a t i n g the  p o i n t s used.  the  38.  i n Table 48.  shown  was  of these q u a l i t y  the FORPLEX to be not d i f f e r e n t from d i d appear  to  when i n f a c t the  t h a t the t a r g e t d i f f e r e n c e value  computation  in  cent  value  meant that  assigned  parameters were l e s s than t h e i r t a r g e t v a l u e s . avoid  quality  not d i f f e r e n t from i t s t a r g e t .  d i f f e r e n c e from t a r g e t was  Twloss, Tmloss and  target  the p r e d i c t e d q u a l i t y  q u a l i t y parameter was  per  than  3  with minimum  i n four of s i x runs ( s t a r t i n g p o i n t s :  number of i t e r a t i o n s needed to l o c a t e and  55 r e s p e c t i v e l y . Since -258-  a,  this  the m a j o r i t y  of  T a b l e 48. F r a n k f u r t e r formula optimization t r i a l s measures o f t h e f 6 r m u l a t i o n s ' quality.  Trial No.  Quality parameters*  Target values  Multiobjective function value  where f i v e q u a l i t y p a r a m e t e r s were considered Target q u a l i t y values were s e t i n d i v i d u a l l y .  Optimum I n g r e d i e n t s proportions Xi  X*  Xj  Predicted quality values  %Difference from target  1. 2. 3. 4. 5.  Twloss" Exwater Exfat Hardl Cohes  22.50 6.00 4.50 160.00 0.255  77.98  0.228  0.176  0. 596  22. 92 5.,42 4.,47 159.19 0.255  1.87 -9.67 -0.67 -0.51 0.00  2 Form2  1. 2. 3. 4. 5.  Shrink" Exfluid Shear Cohes Gummy  8 .70 11.00 4.80 0.255 40.00  334.10  0.288  0.086  0. 626  8.,74 11,.04 4,,48 0.,248 39, ,71  0.46 0.36 -6.67 -2.75 -0.72  3 Form3  1. 2. 3. 4. 5.  Tmloss" Exwater Hardl Gummy Chewy  35.00 6.00 160.00 40.00 130.00  21.94  0.107  0.393  0,,500  32, ,90 5.95 150, ,17 40.29 132, ,28  4 Form4  1. 2. 3. 4. 5.  Twl03S» ES» Exfat Hardl Cohes  22.50 0.20 4.50 160.00 0.255  293.32  0.228  0.176  0,,596  22, ,92 0,,49 4.,47 159, .19 0,.255  1.87 145.00 -0.67 -0.51 0.00  5 Form4*  1. 2. 3. 4. 5.  Twloss" ES" Exfat Hardl Cohes  22.50 0.20 3.50 160.00 0.270  70.42  0.118  0.326  0..556  22, .03 0 .21 3 .02 160, .75 0,, 267  0.00 5.00 -13.43 0.47 -1.11  1 Formula  0  * Nomenclature and d e f i n i t i o n o f t h e q u a l i t y p a r a m e t e r s a r e g i v e n in Appendix E • I f t h e p r e d i c t e d q u a l i t y v a l u e was l e s s t h a n o r e q u a l t o t h e target value, then the s t a n d a r d i z e d s i n g l e o b j e c t i v e f u n c t i o n v a l u e was r e p l a c e d w i t h u n i t y See t e x t f o r d e s c r i p t i o n o f names a s s i g n e d t o optimum f o r m u l a t i o n s o f t r i a l s 1 t o 5.  c  0.00 -0.83 -6.14 0.72 1.75  the runs lead to t h i s s i n g l e s o l u t i o n , considered difficult there  to a t t a i n a multi-objective  the p r e d i c t e d  (Formula)  was q u i t e  value  of  p r e d i c t e d Twloss,  quality  close  Shrink,  2  it is  of one s i n c e  intersect.  the optimum  In  formulation  t o the r e q u i r e d t a r g e t q u a l i t y . The  Cohes was a c h i e v e d .  The d i f f e r e n c e  E x f a t and Hardl values  between  and t h e i r t a r g e t  values  the q u a l i t y parameters  that were optimized  were:  FORPLEX  converged  optimal  point  Xi=0.288, Xi=0.290,  a,  b, c, d, e, f and g.  to an optimal  X =0.086,  X =0.626  2  X =0.086, 2  X =0.624  t o be 49).  (run of  3  and 343.03 the p o i n t  (Form2)  (Figure  minimum  value  more than once  optimum  point  was l o c a l i z e d  the FORPLEX t o converge  to a  These  points  the  single were: and  The optimum  encountered  may be due t o the  point  b  i n l o c a t i n g the fact  that the  c l o s e t o the boundary between the  to a  r e g i o n making i t d i f f i c u l t f o r  single point.  The r e l a t i v e l y  i n d i c a t e d that the t a r g e t q u a l i t y values  and t h e i r  used  s t a r t i n g point g) with  respectively.  met. The d i f f e r e n c e between p r e d i c t e d values  than  found using s t a r t i n g  The d i f f i c u l t y  area and a c o n s t r a i n e d  f u n c t i o n value  rather  In t h i s t r i a l  (run of s t a r t i n g p o i n t b)  3  of 334.10  considered  feasible  area  i n two of the seven runs.  f u n c t i o n values  Gummy  were:  E x f l u i d , Shear, Cohes and Gummy. The s t a r t i n g p o i n t s  in t h i s t r i a l  be  of  48  was  l e s s than ±2%, while t h a t of Exwater was -9.7%. In t r i a l  was  f u n c t i o n value  i s no common p o i n t where a l l contour l i n e s  target  (Formula)  the g l o b a l minimum. As can be seen i n F i g u r e  general,  was  t h i s point  t a r g e t values -260-  Shrink,  high  could not  E x f l u i d and  was l e s s than ±1%,  while  F i g u r e 48. Response s u r f a c e contour l i n e s corresponding to the target quality values of Twloss, Exwater, Exfat, Hardl, and Cohes s e t i n t r i a l 1. The black area r e p r e s e n t s the c o n s t r a i n e d r e g i o n which Is g i v e n by the proximate composition and c o s t c o n s t r a i n t s . The optimum f o r m u l a t i o n (Formula) i s represented by a c l o s e d symbol •  -261-  MDPM  F i g u r e 49. Response s u r f a c e contour l i n e s corresponding to the t a r g e t q u a l i t y values of Shrink, E x f l u i d , Shear, Cohes and Gummy s e t i n t r i a l 2. The black area represents the c o n s t r a i n e d r e g i o n which i s g i v e n by the proximate composition and cost constraints. The optimum f o r m u l a t i o n (Form2) i s represented by a c l o s e d symbol  -263-  >  Q  CO  ^  LU DC -T  <  2  J  LU L DC  ^  ZD -»  ^  3 O i x X CD O CD UJ 03 I  1  I  I  •  and -6.7%  that of Cohes and Shear was -2.8% In  trial  3 the q u a l i t y parameters  respectively.  t h a t were optimized  Tmloss, Exwater, Hardl, Gummy and Chewy. The s t a r t i n g p o i n t s in  value  (Xi=0.107, X =0.393, X =0.500) with minimum f u n c t i o n 2  3  of 21.94 i n two of f i v e  runs ( s t a r t i n g p o i n t s :  b and h ) .  number of i t e r a t i o n s needed t o l o c a t e t h i s p o i n t were 63 and  191  respectively.  This point  g l o b a l minimum. In t h i s t r i a l reasonably  close  t o each  Tmloss as i s shown i n Figure quality  of  required  target  predicted set  used  t h i s t r i a l were: a, b, c, d, and h. The FORPLEX converged t o a  single point  The  were:  (Form3)  Tmloss value  t o be the  the t a r g e t q u a l i t y values other  were set  and below the t a r g e t value  (Form3)  low m u l t i - o b j e c t i v e was b e t t e r  (i.e.  t o be c l o s e t o function value).  lower)  than  and t h e i r t a r g e t values  its The  the t a r g e t  for t h i s parameter. The d i f f e r e n c e between p r e d i c t e d  Gummy and Chewy values  of  50. This allowed for the p r e d i c t e d  the optimum f o r m u l a t i o n (i.e.  was considered  Exwater,  was l e s s than  ±2%,  while that of Hardl was -6.1%. In  trial  Twloss,  4  the q u a l i t y parameters that were optimized  ES,  considered  Exfat,  quality.  ( s e c t i o n B.3.I.),  the  model  experimental data. released  The ES trial  parameter  As mentioned i n R e s u l t s  appropriate found  and  fat.  for o p t i m i z a t i o n  d i d not adequately  In t h i s t r i a l  Discussion  -265-  f o r the  purposes even describe  the ES t a r g e t value  T h i s low t a r g e t value  was  as a measure of the  the q u a l i t y p r e d i c t i o n model obtained  ES data was considered  0.20%  and Cohes.  i n t h i s formula o p t i m i z a t i o n  formulation's  though  Hardl  were:  meant  the  was set that  at  high  F i g u r e 50. Response surface contour l i n e s corresponding to the t a r g e t q u a l i t y values of Tmloss, Exwater, H a r d l , Gummy and Chewy s e t i n t r i a l 3. The black area represents the c o n s t r a i n e d r e g i o n which i s given by the proximate composition and cost c o n s t r a i n t s . The optimum f o r m u l a t i o n (Form3) i s represented by a c l o s e d symbol  -266-  emulsion s t a b i l i t y was r e q u i r e d The and  i n the optimum  s t a r t i n g p o i n t s used i n t h i s t r i a l  i .  The  FORPLEX  were: a, b, c, d,  single  X =0.176  /  two  the seven runs ( s t a r t i n g p o i n t s b and  2  of  X =0.596)  converged to a  3  iterations  needed  respectively.  to  locate  This point  this  target  point  was d i f f i c u l t  line  corresponding  to the  where  while  60  and  meeting  could  not be found.  (i.e.  51.  the  selection  predicted  of  ES value  T h i s can  and Cohes are c l o s e to each the ES t a r g e t  ES values  i s located with  far high  equal to or l e s s than 0.20%)  the performance  that met the  of the FORPLEX  the t a r g e t v a l u e s .  The  and i t s t a r g e t value  was 145%.  d i f f e r e n c e between p r e d i c t e d Twloss, E x f a t t h e i r t a r g e t values  this  As can be observed,  F a i l u r e to o b t a i n a f o r m u l a t i o n  ES t a r g e t l a y not with  the  the contour l i n e s corresponding to the  the contour l i n e of  stability  72  of maximum convergence of  from t h i s r e g i o n . This c l e a r l y shows that a f o r m u l a t i o n emulsion  in  to be the g l o b a l  four q u a l i t y parameters.  t a r g e t s s e t f o r Twloss, E x f a t , Hardl other,  was  formulations  was s e t too f a r from the r e g i o n  i s a region  of 293.32  found was expected s i n c e the ES  be b e t t e r understood by examining F i g u r e there  (Xi=0.228,  to achieve due to the f a c t that  the contour l i n e s of the other  e, g  i ) . The number of  (Form4) was considered  value  contour  point  with a minimum f u n c t i o n value  minimum. The l a r g e f u n c t i o n value target  formulation.  but with  difference  between  However,  and Hardl  the  values and  was l e s s than ±2%, while the t a r g e t value of  Cohes was met. In F i g u r e  51 i t can a l s o be  observed  -268-  that the contour  lines  F i g u r e 51. Response s u r f a c e contour l i n e s corresponding to the t a r g e t q u a l i t y values o£ Twloss, ES, E x f a t , Hardl and Cohes s e t i n t r i a l 4. The black area r e p r e s e n t s the constrained r e g i o n which i s given by the proximate composition and cost constraints. The optimum formulation (Form4) i s represented by a c l o s e d symbol  -269-  CO CO  CO  LU T^ DC Q  ,< 11  O I UJ •  1  1  o  Sh 1  1  corresponding far  to the t a r g e t s s e t f o r E x f a t and Cohes vere  from the contour l i n e of the ES t a r g e t .  contour l i n e s of the ES, each  other  contour  i n an area  Twloss and Hardl l o c a t e d around  l i n e s of the E x f a t and  In  located  addition,  the  t a r g e t s were c l o s e to  X =0.30,  far  2  Cohes t a r g e t s .  from  T h i s meant  the that  the t a r g e t s s e t f o r E x f a t and Cohes were a f f e c t i n g the search f o r a formulation  t h a t met  the  ES t a r g e t .  If  it  i s of paramount  importance to meet the ES t a r g e t , i t may be necessary to c o n s i d e r modifying the t a r g e t values the t a r g e t values  s e t f o r E x f a t and Cohes.  more r e a l i s t i c a l l y ,  By choosing  optimum f o r m u l a t i o n s  found  where the d i f f e r e n c e between t a r g e t and p r e d i c t e d  values  i s smaller.  The  t a r g e t values  3.50  and 0.255  Figure  52,  target,  that  to 0.270,  a,  (Xi=0.118,  respectively.  most  could b,  the contour  be found. c  and d.  values  a as  i n an  closely  The s t a r t i n g p o i n t s used i n  Xa=0.556)  with minimum f u n c t i o n value  5  This p o i n t  compared to t r i a l  of  ( s t a r t i n g p o i n t s a, b and c ) . The  (Form4*)  was considered  g l o b a l minimum. The lower m u l t i - o b j e c t i v e this t r i a l  trial  as  The FORPLEX converged to a s i n g l e p o i n t  of the four runs  respectively.  area  formulation  number of i t e r a t i o n s needed to l o c a t e t h i s p o i n t were 65, 73  in  l i n e of the E x f a t  T h i s c l e a r l y shows that  of i t s t a r g e t q u a l i t y  Xs=0.326,  70.42 i n three  of  5. to  As can be observed  contour l i n e s were c l o s e t o each other 2  possible were:  of E x f a t and Cohes were changed from 4.50  around X =0.30.  meets  quality  m o d i f i c a t i o n was performed i n t r i a l  with the exception the  located  This  can be  4 meant that  -271-  70 and  to be  f u n c t i o n value  the  found i n  the o v e r a l l d i f f e r e n c e  F i g u r e 52. Response s u r f a c e contour l i n e s corresponding to the t a r g e t q u a l i t y v a l u e s of Twloss, ES, E x f a t , Hardl and Cohes s e t i n t r i a l 5. The black area r e p r e s e n t s the constrained r e g i o n which i s given by the proximate composition and cost constraints. The optimum f o r m u l a t i o n (Form4*) i s represented by a c l o s e d symbol •  -272-  between p r e d i c t e d and was  smaller.  The  t a r g e t value -1.1%. than  The  was  t a r g e t q u a l i t y of  difference 5%,  the  between p r e d i c t e d  f o r Hardl was  E x f a t value  and  was  better  o b j e c t i v e of t h i s s e c t i o n was  the FORPLEX to o b t a i n best met  a  (i.e.  quality  the  lower)  predicted  when a q u a l i t y  to t e s t the s u i t a b i l i t y  q u a l i t y frankfurter formulations  restriction  in  addition  to  the  s p e c i f i c a t i o n s f o r proximate composition,  ingredient  cost.  were  Three formula o p t i m i z a t i o n  trials  quality  parameter  formulation.  As  in  was  considered  the previous included  a  section  product  levels,  performed  quality  parameters  (Shrink)  or an emulsion s t a b i l i t y parameter  j u i c i n e s s parameters texture of  parameters  measures of  (Exfluid,  (Hardl,  The  values  lower  restricted In  and the  order  results,  product  Exwater, Exfat)  and  a r b i t r a r i l y selected.  formulations  quality  the  two  or  of loss  or  two  three  combination The  target section.  parameters  that  were a r b i t r a r i l y chosen.  to have a b e t t e r understanding of the  contour  on  one  were the same ones used i n the previous upper l i m i t s f o r those  while  weight  (Twloss),  and  where  combination  Shear, Cohes, Gummy). The  the q u a l i t y parameters was  quality  restriction the  of that  d i f f e r e n t combinations of q u a l i t y parameters were optimized one  was  -13.7%..  1.2.3. O p t i m i z a t i o n of f r a n k f u r t e r formulations parameter was considered a c o n s t r a i n t The  and i t s  for Cohes  however the d i f f e r e n c e between  i t s t a r g e t value  formulation  ES value  l e s s than 0.5%,  p r e d i c t e d q u a l i t y of Twloss was  i t s target,  optimum  l i n e s of the t a r g e t q u a l i t y values -274-  optimization within  the  f e a s i b l e areas vere drawn f o r each o p t i m i s a t i o n t r i a l  (Figures 53  to 55). Two s t a r t i n g p o i n t s were used i n each o p t i m i z a t i o n These p o i n t s were s e l e c t e d by examining the f e a s i b l e area optimization  trial.  minimum  each  points  in  3  o p t i m i z a t i o n t r i a l depended  on  the  starting  r e s u l t s of t h i s s e c t i o n are shown  subroutine  quality  of Shrink  in  Table  of the FORPLEX program f o r t r i a l s  had c o n d i t i o n a l i n s t r u c t i o n s to r e p l a c e  f u n c t i o n values this  The number of i t e r a t i o n s to l o c a t e the g l o b a l  optimization  The F u n c t i o n  and  of each  used.  The 49.  trial.  the  1  standardized  with u n i t y when the p r e d i c t e d values of  parameter were equal to or l e s s  than  the  target  values s e t . In  trial  Shrink,  Shear,  constraint. formulation shows the The  the  the q u a l i t y parameters that were  optimized  were  Cohes and Gummy while E x f l u i d was considered  The  predicted E x f l u i d  was r e s t r i c t e d  value  between 9.50  of  the  and 10.50%.  a  optimum  Figure  53  f e a s i b l e area and the contour l i n e s corresponding  to  t a r g e t values  of the four q u a l i t y parameters being  two s t a r t i n g p o i n t s used i n t h i s t r i a l  X =0.120 2  1  were:  (a)  optimized. Xx=0.100,  and (b) Xi=0.270 and X =0.120. The FORPLEX converged i n 2  both runs to a s i n g l e p o i n t minimum f u n c t i o n value  (Xi=0.223, X =0.198, 2  of 18.98.  X =0.579) 3  T h i s point was considered  with to be  the g l o b a l minimum. The number of i t e r a t i o n s needed to l o c a t e the optimum was 86 and 55 f o r both runs r e s p e c t i v e l y . low  function  value  found i n d i c a t e d  between p r e d i c t e d and t a r g e t q u a l i t y -275-  The r e l a t i v e l y  that the o v e r a l l d i f f e r e n c e of  the optimum  formulation  Table  Trial No.  1  1 0 J n  49.  Frankfurter  Quality parameters*  Target values  1. 2. 3. 4.  Shrink Shear Cohes Gummy  2  1. 2. 3. 4.  Exwater Exfat Hardl Cohes  3  1: 2. 3.  Shrink Exfat Cohes  i i  *  formula  0  0  8.70 4 . 80 0.255 40.00 6.00 4.50 160.00 0.255 9.00 4.50 0. 255  optimization  trials  Q u a l i t y parameter as a c o n s t r a i n t "  Exfluid  9.50  - 10. 5  Twloss  22.00  -  24.00  Hardl  150.00  -  where  a  quality  Multiobjective function value  parameter  was  considered  Optimum ingredients proportions  Xi  Xa  Xj  a  constraint.  Predicted quality values  %Difference from target  4 . 71 - 0 . , 42 0 .00 . - 0 . ,73  18.98  0.223  0.198  0.579  9.11 4.78 0.255 39.71  11.25  0.233  0.164  0.602  5.48 4.53 160.81 0.255  - 8 .,67 0,.67 0,.51 0,,00  2.92  0.230  0.194  0.576  9.06 4.50 0.254  0,,67 0,.00 - 0 ,.39  180.00  N o m e n c l a t u r e and d e f i n i t i o n of the q u a l i t y p a r a m e t e r s a r e given in Appendix E • L o w e r a n d u p p e r l i m i t s p l a c e d on t h e q u a l i t y p a r a m e t e r considered a constraint ° If t h e p r e d i c t e d q u a l i t y v a l u e was l e s s t h a n o r e q u a l t o t h e t a r g e t v a l u e , then the s t a n d a r d i z e d s i n g l e o b j e c t i v e f u n c t i o n v a l u e was r e p l a c e d w i t h u n i t y  Figure 5 3 .  Response s u r f a c e contour l i n e s corresponding to the target quality values of Shrink, Shear, Cohes and Gummy s e t i n t r i a l 1. The black area r e p r e s e n t s the c o n s t r a i n e d r e g i o n which i s given by the proximate composition, c o s t and q u a l i t y ( E x f l u i d ) c o n s t r a i n t s . The optimum f o r m u l a t i o n i s represented by a c l o s e d symbol •  -277-  CO  T-  1  1  DC LU H  (—  LjJ Q 5 < X QC LL S o < X X O I UJ LU 1  1  was  small.  The d i f f e r e n c e between p r e d i c t e d  values and t h e i r t a r g e t values i t was 4.7%,  and  was l e s s than -1.0%,  In t r i a l  for  f o r m u l a t i o n was  2 the q u a l i t y parameters t h a t were  Exfat,  restricted  f e a s i b l e area values  Hardl  between 22.00 and  to  optimized  and 24.00%.  the contour  F i g u r e 54  l i n e s corresponding  trial  were:  single  p o i n t (Xi=0.233,  value of 11.25. minimum.  X =0.164, 2  function  The  found  X =0.602)  between p r e d i c t e d and t a r g e t q u a l i t y small.  The  of  the optimum  the  locate  the  formulation  d i f f e r e n c e between p r e d i c t e d E x f a t  and for  Hardl Exwater  while the t a r g e t value of Cohes was achieved.  The  23.01%.  3 the q u a l i t y parameters optimized were Shrink, E x f a t  Cohes while Hardl  was considered a  p r e d i c t e d value of the optimum 150.00  be  t h a t the o v e r a l l d i f f e r e n c e  Twloss value of t h i s optimum f o r m u l a t i o n was In t r i a l  to  The r e l a t i v e l y  values and t h e i r t a r g e t values was l e s s than -1.0%, i t was -8.7%,  runs  with minimum  3  This p o i n t was considered  indicated  two  2  The number of i t e r a t i o n s needed to  value  the  to the t a r g e t  2  a  a  (a) Xi=0.050, X =0.300  optimum was 50 and 66 f o r both runs r e s p e c t i v e l y .  and  shows  (b) Xa. = 0.290 and X = 0.050. The FORPLEX converged i n both  global  was  were:  formulation  of the four q u a l i t y parameters being o p t i m i z e d .  function  low  The  and Cohes while Twloss was c o n s i d e r e d  s t a r t i n g p o i n t s used i n t h i s and  Shrink  9.6%.  c o n s t r a i n t . The p r e d i c t e d Twloss value of the optimum was  Gummy  while the t a r g e t value of Cohes was achieved.  E x f l u i d value of t h i s optimum  Exwater,  Shear  and 180.00N.  F i g u r e 55  constraint.  The Hardl  f o r m u l a t i o n was r e s t r i c t e d  between  shows the f e a s i b l e area and  -279-  the  F i g u r e 54. Response s u r f a c e contour l i n e s corresponding to the target q u a l i t y values of Exwater, E x f a t , Hardl and Cohes s e t i n t r i a l 2. The black area r e p r e s e n t s the constrained r e g i o n which i s given by the proximate composition, c o s t and q u a l i t y (Twloss) constraints. The optimum f o r m u l a t i o n i s represented by a closed symbol•  -280-  >  CO  ZD  O  ]> LU  o o I  LU OC coco i  1  F i g u r e 5 5 . Response s u r f a c e contour l i n e s corresponding to the t a r g e t q u a l i t y v a l u e s of Shrink, E x f a t , and Cohes s e t in t r i a l 3. The black area r e p r e s e n t s the c o n s t r a i n e d region which i s g i v e n by the proximate composition, c o s t and q u a l i t y (Hardl) constraints. The optimum f o r m u l a t i o n i s represented by a c l o s e d symbol*  -282-  contour  lines  corresponding  to the t a r g e t values o£  q u a l i t y parameters being optimized. The in t h i s t r i a l X =0.000.  were: (a)  Xx=0.050,  two  (Xi=0.230,  X =0.194 2  X =0.576)  /  three  s t a r t i n g p o i n t s used  X =0.400 and 2  (b) X =0.200 and x  The FORPLEX converged i n both runs to  2  the  a  s i n g l e point  with minimum f u n c t i o n value  3  2.92.  T h i s p o i n t was  considered to be the g l o b a l minimum.  number  of i t e r a t i o n s needed to l o c a t e the optimum was  94  of The  and  66  r e s p e c t i v e l y . This t r i a l  gave the lowest m u l t i - o b j e c t i v e f u n c t i o n  value  that  which  existed  between  difference target  predicted  a  very  and  small o v e r a l l  target  quality  achieved.  l e s s than ±1.0%, The  Hardl  while the  difference  values.  between p r e d i c t e d Shrink and Cohes values  values was  E x f a t was was  indicates  target  and  The their  value  of  value of t h i s optimum f o r m u l a t i o n  150.00N.  -284-  2. O p t i m i z a t i o n of f r a n k f u r t e r f o r m u l a t i o n s using programming The  meat p r o c e s s i n g i n d u s t r y has been using l i n e a r  computer  programs  possible  cost while meeting a l l  available  ingredients.  to  f i n d seven  to  for  composition  and  (c)  constraint  on the sum  the  formulate  meat products product  In t h i s study  (a)  of  the composition  optimization  program were used.  of  the  account f o r the q u a l i t y  levels,  same  (b)  ingredient  limits,  the  formulations  for a f r a n k f u r t e r f o r m u l a t i o n was TBV  The  the  FORPLEX used to  i n terms of ( i . e . fat  t o t a l bind value  given by equation  n = E (bind value* x i =l  met  (equation  bind value c o n s t r a i n t was  formulation).  used  31 to 33) used f o r  f o r m u l a t i o n s by  r e q u i r e d by the  the  proximate  the amount of f a t bound per u n i t weight of f o r m u l a t i o n b i n d i n g c a p a c i t y of the  lowest  that  ingredients proportions  frankfurter total  the  s p e c i f i c a t i o n s with  c o n s t r a i n t s (equations  The  at  formulations  ingredient  q u a l i t y . The  programming  l i n e a r programming was  least-cost frankfurter  specifications  30) and  linear  (TBV)  (39):  X*)  (39)  where bind v a l u e * = bind value constant* x p r o t e i n content* i = l, . . . n  -285-  (40)  where n = number Xi  of. I n g r e d i e n t s  i n the f o r m u l a t i o n  = p r o p o r t i o n o f t h e i t h meat i n g r e d i e n t  bind  valuei  bind  value  protein  = bind value constant*  contents  per u n i t  weight  = bind value  = fraction  of i t h i n g r e d i e n t  constant  of the i t h i n g r e d i e n t  of p r o t e i n c o n t e n t  of the i t h  ingred ient The  bind  value  (MDPM), X ,  constants  and b e e f  2  of m e c h a n i c a l l y  meat, X ,  deboned p o u l t r y  were 15 and 24 r e s p e c t i v e l y .  3  constants  were p r o v i d e d  these  i n g r e d i e n t s were computed u s i n g e q u a t i o n  two  were 2.327 value  by S e b a s t i a n  and 5.398 f o r X  of beef  meat  (X )  2  and X  meant  3  fat  b i n d i n g c a p a c i t y than  this  study  s u r f a c e contour  within  the  corresponding as lower  cost  objective  equations  as  least-cost  (40) and  these  The h i g h e r  i n g r e d i e n t had a  The TBV e q u a t i o n  2  + 5.398X  2  lines  bind  higher  used  in  feasible  limits  set  (41)  3  were g e n e r a t e d  r e g i o n . - The lines  function  to  shown  be m i n i m i z e d  t h e y were e n t e r e d  into  p r o p o r t i o n s of beef  the  fat  binding  equation  bind  in Figure Table  and  50  the  values 56  were  shows t h e constraint The  seven  in  Table  meat were r e q u i r e d as t h e  lower  found  computer. are reported  f o r t h e TBV c o n s t r a i n t i n c r e a s e d ,  minimum  using  total  o f t h e TBV c o n s t r a i n t .  frankfurter formulations  Higher  required  (X ).  to the seven contour  used  limit  The b i n d v a l u e s f o r  respectively.  3  These  was  Response  51.  (1989).  that this  MDPM  TBV = 2.327X  (41)  meat  capacity  -286-  of  that the  is,  as  the  formulations  MDPM  Figure 56. Response surface contour p l o t f o r the TBV equation. The black area represents the c o n s t r a i n e d r e g i o n which i s given by the proximate composition c o n s t r a i n t s .  Table 50. O b j e c t i v e f u n c t i o n and c o n s t r a i n t equations used optimization of f r a n k f u r t e r f o r m u l a t i o n s u s i n g programming.  i n the linear  O b j e c t i v e f u n c t i o n to be minimized (cost of the f o r m u l a t i o n ) 0 . 2 0 X i + 0 . 8 0 X + 3.20Xa a  Subject t o : (1) f a t content of the meat block 8 < 8 0 . 3 7 X i + 1 8 . 6 7 X + 3.72X < 28 (2) p r o t e i n content of the meat block 16 < 4.96X1 + 1 5 . 5 1 X + 2 2 . 4 9 X < 21 (3) moisture content of the meat block 54.3 < 1 4 . 5 8 X i + 6 5 . 6 9 X + 7 3 . 6 4 X < 89 (4) t o t a l bind value (TBV) of the formulation*' 3.428 < 2 . 3 2 7 X + 5 . 3 9 8 X (5) i n g r e d i e n t contents 0.05 < X i < 0.30 0.00 < X < 0.40 0.50 < X < 0.95 Xi + X + X = 1 2  3  2  3  2  2  3  3  2  3  2  3  *• The v a l u e s corresponding to the seven contour l i n e s shown F i g u r e 56 were used as lower l i m i t s of the TBV c o n s t r a i n t .  -288-  in  Table 51. L e a s t - c o s t f r a n k f u r t e r Formulation  Ingredients Xa.  LP1 LP2 LP3 LP4 LP5 LP6 LP7  0.187 0.098 0.058 0.050 0.050 0.050 0.050  X  formulations.  proportions X  2  0.313 0.400 0.400 0.345 0.275 0.205 0.136  -289-  3  0.500 0.502 0.542 0.606 0.675 0.745 0.815  Cost $/kg 1.89 1.95 2.07 2.22 2.39 2.56 2.72  Lower l i m i t TBV 3.428 3.642 3.856 4.070 4.284 4.498 4.712  increased. The  T h i s can be b e t t e r understood by examining F i g u r e 56.  feasible  constraint  is  corresponding the  area the  after  s e t t i n g a lower  one  to  the  t o the p a r t i c u l a r  right  limit of  the  meat were needed t o meet the higher  the  contour  lower l i m i t chosen;  lower l i m i t of the TBV c o n s t r a i n t , higher  capacity.  for  TBV line  increasing  proportions  of beef  requirements f o r f a t  binding  The c o s t per kg of the f r a n k f u r t e r f o r m u l a t i o n s  also  i n c r e a s e d as the lower l i m i t s e t f o r the TBV c o n s t r a i n t increased (Table  48)  due to the need f o r higher  proportions  of the  most  c o s t l y meat i n g r e d i e n t ,  X . In other words as the q u a l i t y of the  formulations  of f a t b i n d i n g c a p a c i t y  in  terms  f o r m u l a t i o n LP1 to LP7,  3  the cost  increased.  -290-  per  kg  of  increased  from  formulation  also  3. Comparison of FORPLEX and l i n e a r programming computed optimum formulations The 48)  f i v e computed optimum f o r m u l a t i o n s found  were compared with the seven  l i n e a r programming (Table 51).  by FORPLEX  (Table  optimum f o r m u l a t i o n s found  The comparison was made  by  i n terms  of the p r e d i c t e d q u a l i t y and c o s t of the computed f o r m u l a t i o n s .  3.1. Comparison i n terms of p r e d i c t e d q u a l i t y Each of the f i v e  FORPLEX  computed  optimum f o r m u l a t i o n s were  compared with the seven l e a s t - c o s t f o r m u l a t i o n s . was  based  quality  on  the  values  per cent d i f f e r e n c e  of  comparison among  the  t o the  x-axis  and the  (equation 38) to the y - a x i s . target  value f o r  predicted  formulation  formulations.  to 61).  As can be observed  of  A p o s i t i v e per cent d i f f e r e n c e from  greater  than  the the  f o r m u l a t i o n . The opposite  value.  i n F i g u r e s 57  the optimum f o r m u l a t i o n s Form4,  the  To f a c i l i t a t e the  a p a r t i c u l a r q u a l i t y parameter meant t h a t  was true f o r a negative  Form2, Form3,  and  The f o r m u l a t i o n s were  q u a l i t y value of a f o r m u l a t i o n was  of  predicted  per cent d i f f e r e n c e from t a r g e t  t a r g e t q u a l i t y value s e t on the FORPLEX  values  the  comparison  computed optimum f o r m u l a t i o n s a s e r i e s  histograms were made ( F i g u r e s 57 assigned  between  each computed optimum  t a r g e t values s e t i n the FORPLEX  The  t o 61 found  the p r e d i c t e d q u a l i t y by  FORPLEX  (Formula,  Form4*) were c l o s e t o t h e i r t a r g e t q u a l i t y  values  as seen by the low per cent d i f f e r e n c e from t a r g e t values  found.  The only e x c e p t i o n was Form4  c o u l d not be  achieved  (Figure 60) whose ES t a r g e t  because f o r m u l a t i o n s which approached the -291-  40  20  -  0  -  -20  -  -40  -  -60  o>  sf  1  v?  3  O?  6  \J?  6  O?  •  COHES  S  HARD1  0  EXFAT  •  EXWATER  •  TWLOSS  1  FORMULATIONS  Figure  57. D i f f e r e n c e s b e t w e e n s p e c i f i e d t a r g e t q u a l i t y v a l u e s and t h e p r e d i c t e d q u a l i t y v a l u e s o f F o r m u l a and t h e least-cost formulations.  40 30 UJ  CD  O  OC  i  Ui I  u_ LU  o  Z LU  20 10  CL  -10  Q  -20  LU LOLL  m  L  0  F  -30  ^  o?  A  sf  1  v ? * vJ?  6  • 0 • • •  GUMMY COHES SHEAR EXFLUID SHRINK  0?" 1  FORMULATIONS Figure  58. D i f f e r e n c e s between s p e c i f i e d t a r g e t q u a l i t y v a l u e s and t h e p r e d i c t e d q u a l i t y v a l u e s of Form2 and t h e least-cost formulations.  -{•63% DIFFERENCE FROM TARGET cn O  C H  O  CD  Ul  fl) 3 OJ  r—  D J r-h  hh  01  fl) rt H o rr ft) rt I O  tn cr  CP  3  CD  o  CD  o  Co  J3  l-h l-<  o n> cr n a (i S w- r r c o < t - r t fl) 0)  CD  CD  r r a s o iO tn  3  C  CP  o CD  D  tn at CD • I— o 1  M-  r—  r t i-h  »< t— <  fl) a  QJ r-> C 0> CD t<  cr  tn  O l-h  CP  in  CD  rt  .Q * l C O Oi  3  OJ  i rt  OJ 3 < Q i OJ rt 3" CD  •  •  •  C CD  a  tn  D3  Q  •  O  O O  150  CD  o  100  50  cn  i  to  <o cn I  LL LL)  O  Z LU DC  j  0  j  j  y  i  111  LL LL Q  3^  -50  -100  v^  A  OP*- 0? 2  s  0?*  OP  6  v?  6  _ •  COHES  0  HARD1  •  EXFAT  •  ES  •  TWLOSS  0?  1  FORMULATIONS Figure 60. D i f f e r e n c e s between s p e c i f i e d t a r g e t q u a l i t y values and the p r e d i c t e d q u a l i t y values of Form4 and the least-cost formulations.  100  CD  50  -  O cr u_ LU I I  O z  LU OC LU LL LL  0  Q  -50  ^* \JP  A  ^  ^  0?^ v ? 0? vJ? 6  6  •  COHES  •  HARD1  •  EXFAT  •  ES  •  TWLOSS  1  FORMULATIONS Figure 61. D i f f e r e n c e s between s p e c i f i e d t a r g e t q u a l i t y values and the p r e d i c t e d q u a l i t y values of Form4* and the l e a s t - c o s t formulations.  t a r g e t of  t h i s q u a l i t y parameter were l o c a t e d f a r from the  target q u a l i t y values (LP1-LP7)  showed,  (Figure  51).  in general,  The  least-cost  other  formulations  c o n s i d e r a b l e departure  from  the  t a r g e t q u a l i t y s e t t o the FORPLEX f o r m u l a t i o n s . These d i f f e r e n c e s observed optimum  between  the l e a s t - c o s t  formulations  were  o p t i m i z a t i o n techniques FORPLEX searched close  expected  work  based  since  and  the  FORPLEX  these  two  formula  on d i f f e r e n t p r i n c i p l e s .  f o r f o r m u l a t i o n s whose p r e d i c t e d q u a l i t y  as p o s s i b l e to  parameters  formulations  could  a  predetermined  was  target quality;  be optimized because there  were  The as  quality  mathematical  models t h a t p r e d i c t e d the q u a l i t y parameters as a f u n c t i o n of the i n g r e d i e n t s . On the other hand, l i n e a r programming d i d not for maximum q u a l i t y and  search  c o u l d not c o n s i d e r t a r g e t q u a l i t y v a l u e s .  Furthermore, the o p t i m i z a t i o n performed by l i n e a r programming d i d not c o n s i d e r , that  much  less control,  were c o n s i d e r e d  formula  o p t i m i z a t i o n using FORPLEX.  However, i n some cases the l e a s t - c o s t  f o r m u l a t i o n s found d i d meet  some of  but t h i s was  matter quality  the  i n the  any of the q u a l i t y parameters  target q u a l i t y values,  of chance. values  The d i f f e r e n c e observed  essentially  a  between the p r e d i c t e d  of the l e a s t - c o s t f o r m u l a t i o n s  and  the  target  q u a l i t y values s e t on the FORPLEX f o r m u l a t i o n s w i l l be d i s c u s s e d . (A)  Quality  losses  (Shrink,  A zero per cent d i f f e r e n c e from t a r g e t f o r Shrink was  assigned  Twloss and  to  LP1  parameters  r e l a t e d to weight  Tmloss).  (Figure 58)  ( F i g u r e s 57, 60 and  61)  and  f o r Twloss to L P l , because -297-  LP2,  LP3  the p r e d i c t e d values  and  LP4  of these  quality  p a r a m e t e r s were l e s s  FORPLEX  formulations  mentioned b e f o r e , (i.e. set  product  f o r these  and  Twloss  greater  Twloss by  said  f r o m LP1  t o LP7 The  increased  the  and F o r m 4 * ) .  As  from 3  though  increased  weight  l o s s moved  FORPLEX  formulations.  than  target value  formulations moisture value  Tmloss v a l u e s  and  related to  X  0.500 i n LP1  increased  in  3  the  t o 0.815  least-cost  i n LP7. Thus,  i n the l e a s t - c o s t  formulations  were p r e d i c t e d and hence  target values the  could  Shrink  and T w l o s s were p o s i t i v e l y  quality, t o LP7,  farther  from  were o b t a i n e d . i n terms  of  I t c a n be  fat  the q u a l i t y the  greater  in  targets  binding terms of  s e t on t h e  d i f f e r e n c e f r o m t a r g e t f o r T m l o s s was  s e t on Form3  (Figure 59).  o f meat b l o c k )  set for this  for  explained  whose p r e d i c t e d T m l o s s  content  were  f o r m u l a t i o n s and  t o LP7  t o LP2 t o LP5 b e c a u s e t h e p r e d i c t e d T m l o s s v a l u e s the  Shrink  formulations  FORPLEX  observed  the t a r g e t s  be  from LP1  product  A zero per cent  the  and T w l o s s  The p r e d i c t e d  least-cost  trend  and T w l o s s v a l u e s  even  s e t on  were lower  The p r o p o r t i o n o f  d i f f e r e n c e from  capacity,  Form4,  and f r o m LP2  that Shrink  40).  Shrink  that  values  t a r g e t s s e t on  the p r o p o r t i o n of X  cent  Form2,  of the other  the  (Figure  per  loss)  respectively.  higher  target values  f o r m u l a t i o n s whose p r e d i c t e d S h r i n k  values  formulations as  the  p a r a m e t e r s had b e t t e r q u a l i t y .  the f i n d i n g  Xa  (Formula,  weight  than  increased  than  (i.e.  v a l u e was  assigned  were lower  As m e n t i o n e d water  lower  p a r a m e t e r had b e t t e r q u a l i t y .  loss than The  before,  per the  unit target  predicted  f o r f o r m u l a t i o n s L P 1 , LP6 and LP7 were g r e a t e r -298-  than  than  the  target set (B)  Q u a l i t y parameters r e l a t e d to  (Exfluid, The by  E x w a t e r and  Exfluid  any  values The  i n Form3.  of  than the  predicted Exfluid  observed  could  between  LP7.  formulations  (Figure  X  per  the  (Figure  2  of  Exfluid cent  the  FORPLEX  negative The  from  Xa  t o LP7.  met  Exfluid  trend  relationship  found  of X a i n  0.400 i n LP3  to  i n these  from  be  The  were p r e d i c t e d ,  difference  not  formulation.  proportion  decreased  values  could  predicted  from LP4  40).  decreased  proportion  higher  negative  by  58)  The  on  increased  explained and  the  target set  values  formulations  Thus a s  lower  be  Exfluid  least-cost  i n Form2  least-cost formulations.  were l o w e r  characteristics  Exfat).  target set  the  juiciness  0.136  in  least-cost and  target  the  hence  values  were  obtained. The  Exwater  (Figure  target  59)  could  formulations. values  were  formulations. LP7,  as  set not  With the greater The  than the  3  proportion  of X  the  Exwater  in  Thus,  values  The  Exfat  as  (Figure  any  of LP1  target  were  40).  of  the set  As  57)  the  proportion  of  and  the  i n the  two  increased was  X  and  greater  per  Formula  (Figure  57),  FORPLEX  to  to be  above,  the  increased  from  increased  3  Exwater  f r o m LP1  found  mentioned  Form3  least-cost  predicted  least-cost formulations  were p r e d i c t e d ,  from t a r g e t v a l u e s  by  c o n s i d e r i n g t h a t Exwater  r e l a t e d to X  t o LP7.  met  exception  positively  LP1  be  (Figure  p r e d i c t e d Exwater v a l u e s  expected  3  i n Formula  cent  higher  difference  obtained.  t a r g e t s s e t on  -299-  Form4  (Figure  60)  and  cost  Form4*  (Figure  formulations. 61),  with  values  were  lower and  could  be  between  Exfat  and  LP4  and  proportion Exfat  of  values  difference (C)  could  be  textural the  at  values  of  and  to  With the  explained  by  the  p a r a m e t e r s and  values  the  However t h e not  Xa  Exfat FORPLEX  strong.  decrease  positive relationship  found  proportion  0.187  i n LP4  o f Xa. i n  in LPl  to  t o LP7.  the  0.050  Thus,  negative  in  as  least-cost formulations  the  lower  per  cent  obtained. texture  characteristics  targets set  61)  could  be  the  exception  met  i n a l l the by any  of L P l  of  from L P l  t o LP7.  and  The  (Figure  42).  increased  were p r e d i c t e d ,  and  d i f f e r e n c e s from t a r g e t v a l u e s  -300-  As  the  were  obtained.  the  observed  between  t o LP7  hence g r e a t e r  the  than  proportion  from L P l  least-  LP2  trend  p o s i t i v e r e l a t i o n s h i p found  FORPLEX  the  t e x t u r a l p a r a m e t e r s were g r e a t e r  increased  Form4*  Chewy).  t e x t u r a l parameter 57  in  trend  parameters r e l a t e d to  (Figures  on  hence g r e a t e r  were  least-  The  from  i n the  of t h e  predicted  set  was  The  0.050  least-cost formulations  textural cent  decreased  decreased  formulations.  values  the  any  the  t o LP7.  40).  S h e a r , C o h e s , Gummy and  predicted  in  Xa. ( F i g u r e  by  target set  of LP1,  t o LP7  by  from t a r g e t v a l u e s  formulations  target  i n LP4  explained  met  targets  were p r e d i c t e d , and  None o f t h e  cost  Xi  the  be  to the  from L P l  constant  Quality  (Hardl,  than  formulations  remained  not  exception  values  observed  least-cost  the  decreased  in predicted Exfat  could  When compared  (Figure  formulations  61)  of  the X  3  higher  p o s i t i v e per  (D) A LP2  ES  zero t o LP7  these the  The  (per cent  per  cent  were  better  quality.  formulations  lower  formulations predicted  formulations  the  The  higher  ES,  decreased  from  during  It  from  thermal  is  quality linear  the As  of X  which  by  B  the  between  the  the  The  be  in  this  parameter  was  concluded  the  The  TBV  released)  reason fact  of  that  this  TBV  to Xi  and  ES  t o LP7,  predicted to  t o make an  overall  the  on d i f f e r e n t t h a t the  The  TBV  capacity  of  therefore X*  were  (Figure  40),  parameters  were  cent  comparison found  techniques  required  to  fat released  in  by  search  principles.  formulations -301-  these  decrease.  optimum f o r m u l a t i o n s two  per  lower  constraint  r e q u i r e d and  proportions  related  why  constraint.  limit  had  least-cost  f a t b i n d i n g c a p a c i t y was  programming s i n c e t h e s e based  lower  before,  fat  i n the  the  lower  and  f o r m u l a t i o n LP1 heating  mentioned  for  of  target set in  r e q u i r e d minimum f a t b i n d i n g  t o LP7. as  the  to  values  cent  values  lies  assigned  per  t o LP7.  is positively  LP1  difficult  formulations cannot  LP1  than  f a t b i n d i n g c a p a c i t y was  negatively related; increase  (i.e.  was  p r e d i c t e d ES  F o r m 4 * ) . As  target set  restricted  proportions  required.  the  the  less  were f o u n d  controlled  higher  ES  from  formulations.  increased,  vere  p r e d i c t e d ES  values were  because  (Form4 and  than  decreased  ES  constraint  61)  whose p r e d i c t e d  values  of  and  FORPLEX f o r m u l a t i o n s  formulations  parameter  d i f f e r e n c e f r o m t a r g e t f o r ES  ( F i g u r e s 60  least-cost  fat released)  found  terms  FORPLEX  of and  f o r optimum  Furthermore, by FORPLEX  it have  better  quality  than  the  optimum  found  However,  whose q u a l i t y  i s as c l o s e as p o s s i b l e t o a predetermined  the  objective  optimum  The found  cost  least-cost  mentioned  before  of  while  minimized  in  programming.  formulations range.  the  The  l o w e r and upper  can  found  by FORPLEX  this  meet  FORPLEX  using  fell  limits  ( F i g u r e 62) within this  requirements s e t on  the  block,  permissible  increased  limits  linear  o f meat  cost  is effective  within  of the expensive  be  a l l t h e optimum  formulations  lower  to  using  program  also f e l l  the  FORPLEX  t h a t were s e t f o r  However,  increasing  the  62.  the  formulations  formulations.  proportions  ( i . e . higher  i n Figure  2.8 p e r k i l o g r a m  formulations  the  and  t h a t meet t h e c o s t r e s t r i c t i o n .  the c o s t of these  to the higher  formulations  used as a c o n s t r a i n t i n  shows t h a t t h e FORPLEX  formulations  the  and  of  be o b s e r v e d  before,  capacity  meet  the o b j e c t i v e function  optimization  As  optimum  formulations  the  least-cost  to  of the  i t was c o n s i d e r e d  on  needed  quality,  formulations.  t h e c o s t was  imposed  due  FORPLEX  (LP1 t o LP7) a r e shown  frankfurter  This c l e a r l y  finding  by  formulations  Form2, Form3, Form4, Form4*)  c o n s t r a i n t were $1.9  respectively.  of  (Formula,  formulations  program,  in  the l e a s t - c o s t  ( $ / k g o f meat b l o c k )  optimization  cost  found  linear  i n terms of c o s t  by FORPLEX  As  formulations  b e t t e r than  3.2. C o m p a r i s o n  the o b j e c t i v e i s to f i n d  by  programming.  then  if  formulations  The c o s t  the as  limits  mentioned  f r o m LP1 t o LP7  meat of  the t o t a l  ingredient fat bind  X  3  binding value  FORMULATIONS  F i g u r e 62. Cost ($/kg of meat block) of the optimum f o r m u l a t i o n s found by FORPLEX and the l e a s t - c o s t f o r m u l a t i o n s .  -303-  c o n s t r a i n t ) . The cost of formulations  LP4 to LP7 were higher than  the cost of the FORPLEX optimum formulations. The lower l i m i t set on  the t o t a l bind value constraint should be c a r e f u l l y  if  the  with  objective  acceptable  probably  to  be  frankfurter  formulations  The processed meat industry  moderate lower l i m i t s for the t o t a l  in  formulations; have  q u a l i t y scores.  using  constraint  i s to find low-cost  order  to  control  otherwise high used,  selected  the  cost  of  quality-expensive  increasing  the  cost  bind the  is  value optimum  meat ingredients  of  the  frankfurter  formulations.  4. Comparison of FORPLEX with linear programming for meat formula optimization One  of the objectives of t h i s thesis was to compare  the  new  formula optimization method (FORPLEX) with linear programming for meat formula optimization. Table 52 these  two  techniques.  The  summarizes the comparison of  main  difference  between  techniques i s based on the optimization objectives. searches  for best q u a l i t y formulations  that meet  The  these FORPLEX  predetermined  product s p e c i f i c a t i o n s within allowable cost ranges, while linear programming  searches  predetermined heavy  least-cost  formulations  that  meet  product s p e c i f i c a t i o n s . Linear programming places  emphasis  quality.  for  in cost  reduction but  FORPLEX considers  far  q u a l i t y to be  less  as  emphasis  on  important as cost  reduction. Linear  programming  optimization  problems  can  only  in  which -304-  be the  used  to  objective  solve  formula  function  and  Table  52. Comparison of FORPLEX and l i n e a r programming o p t i m i z a t i o n of meat f o r m u l a t i o n s . FORPLEX  Optimization objective Constraints •Product specifications •Quality *Cost Objective function •Linear •Nonlinear •Multiple objective function Constraints •Linear •Nonlinear  Maximize quality  yes yes  Linear  programming Minimize cost  yes yes o  yes  n  yes yes  y  e  s  no  yes  no  yes yes  yes no  -305-  f o r the  c o n s t r a i n t s are expressed hand,  FORPLEX  linear  or n o n l i n e a r ,  Linear  using  c a n accommodate  linear  uses  functions  linear  the q u a l i t y of the formulations.  quality  constraints  (Pearson  and  Tauber,  functions  of  functions  are  the  color  1984a).  the ingredients the  color  bind  nonlinear  functions  ( N a k a i and A r t e a g a ,  of b i n d  has been q u e s t i o n e d  been  values  found  finished ability  to  products. of  the  optimization not  be i n a c c u r a t e  of  considered  meat  formulations  water b i n d i n g  are  coefficients  found  used  linear  linear  of  However  by s i m p l e  these quality  functions;  to explain  quality  1990). Furthermore,  t h e use  since  bind  ingredients  and  constraints  these  parameters  i n p r e d i c t i n g the q u a l i t y  In a d d i t i o n ,  meat  binding  values.  have been  more a c c u r a t e l y  describe  constraints  where t h e  and  to  The most w i d e l y  and  These  parameters are u s u a l l y not described instead  other  and c o n s t r a i n t s .  equations  restrict  are  On t h e  a n y f o r m o f r e l a t i o n s h i p s , whether  as o b j e c t i v e  programming  functions.  to  values bind  using  consider fat.  linear  have  of  the  only  the  Until  now,  programming has  and g e l a t i o n p r o p e r t i e s  of the  meat  ingredients. The  FORPLEX  can  accommodate  prediction  equations  constraints,  thus a l l o w i n g  that  describe  quality  FORPLEX  can optimize  yield,  juiciness,  prediction  as  linear  objective  and  functions  t h e use o f more a c c u r a t e  as  a  function  of  any q u a l i t y parameter, texture,  equations  roust  nonlinear  or c o l o r . be -306-  as  quality well  relationships  the i n g r e d i e n t s . f o r example,  advance  in  The  product  However, r e l i a b l e  known i n  as  quality  order  to  optimize  a meat  target  quality  formulations  values  target  t o be s e t  for  q u a l i t y i s as c l o s e as p o s s i b l e  to a  This  therefore  multiple  objective  optimization functions.  for multi-objective  accommodating combined  option  In g e n e r a l , suitable  While  c a n n o t be p e r f o r m e d  by  linear  This  optimization  programming  allows  of s e v e r a l  for  i t would seem t h a t  f o r food  The a d e q u a c y o f t h e values  formulations,  Formula X =0.313,  formulations  vere  procedure  tested  described  for  prepared  Before d i s c u s s i n g  functions  simultaneous  FORPLEX i s t h e more  in  q u a l i t y values  computed  X =0.176, 2  These  duplicate  f o r the p r e p a r a t i o n  in Materials  of  X =0.596) two  the  same given  section E). the  methods  F.  r e s u l t s i t i s important  t h e models found a r e o n l y -307-  frankfurter  following  following  section  the v e r i f i c a t i o n  and L P l  3  ( M a t e r i a l s and Methods  and Methods  optimum  of the f o r m u l a t i o n s  vere evaluated  of  p r e d i c t i n g the q u a l i t y  two  3  parameters  k e e p i n mind t h a t  for  X =0.500).  by t h e extreme v e r t i c e s d e s i g n quality  found  (Xi=0.228,  3  followed  the  be  formulation.  models  was  (Xi=0.187,  cannot  t h e FORPLEX i s c a p a b l e  5. E x p e r i m e n t a l v e r i f i c a t i o n o f t h e p r e d i c t e d two computed optimum formulations.  parameter  of  q u a l i t y parameters.  therefore,  technique  the  f u n c t i o n composed  one e q u a t i o n .  of s e v e r a l  means  optimization,  an o b j e c t i v e  into  optimization  The  for  searches  quality.  and  allows  The FORPLEX  pro gramming.  Multi-objective  used  effectively.  whose p r e d i c t e d  predetermined linear  formulation  estimates  to  of the t r u e  relationships  between  i n g r e d i e n t s and q u a l i t y p a r a m e t e r s .  this  reason  when p r e d i c t i n g a q u a l i t y p a r a m e t e r  into  account  not only the v a r i a t i o n  the  variation  predicting given  study  predicted using  the  95%  the  their  The each  confidence  equations  quality  reported  data  t h e two f o r m u l a t i o n s  and  54.  I t should  section  probability  level  were c o n s t r u c t e d  for  In each  ( F o r m u l a and L P l ) In  order  formulations  had  to  to the fall  intervals. confidence  and t h e e x p e r i m e n t a l  i n t e r v a l of q u a l i t y data  the confidence  i n Tables  interval  s i n c e t h e ES  o f f i t ( R e s u l t s and  53  was n o t model was  Discussions  B.3.1.).  Variability  of the q u a l i t y data  was o b s e r v e d . As m e n t i o n e d B.2.)  (also  1982).  t h e 95%  lack  for a  (Bender e t a l .  f o r t h e p r e d i c t e d ES v a l u e s ,  t o have a s i g n i f i c a n t  when  f o r p r e d i c t i o n purposes  of these  that  The  interval  ( F o r m u l a and L P l ) a r e g i v e n  be n o t e d  but  1982).  q u a l i t y parameter  by C o r n e l l (1981).  q u a l i t y value,  take  variations  o f t h e two f o r m u l a t i o n s  for  found  intervals  respective confidence  predicted  these  a t a predetermined  predicted q u a l i t y values,  constructed  (i.e.  m o d e l s t o be a d e q u a t e  experimental within  consider  s i n g l e f u t u r e event  q u a l i t y value  consider  to  (Bender e t a l .  i s t o c a l c u l a t e the confidence  prediction interval)  the p a r t i c u l a r  this  used  a s i n g l e f u t u r e event  formulation)  called for  method  must  In t h e m a t e r i a l s t u d i e d  i n t h e r e g r e s s i o n model  statistical  one  For  the  processing  Inherent steps  before  variability  involved  of the d u p l i c a t e  ( R e s u l t s and D i s c u s s i o n o f t h e meat  i n the preparation  -308-  formulations section  ingredients,  the  of the formulations  Table 53. Experimental v e r i f i c a t i o n of the p r e d i c t e d q u a l i t y values of Formula.*-  Quality parameters*  Predicted quality values  Confidence interval  Experimental q u a l i t y values 0  1 Shrink Tmloss Twloss ES PH Exfluid Exwater Exfat Hardl Shear Cohes Gummy Chewy  9..19 38.,75 22..92 0,,49 5..69 9 ,82 . 5,.42 4., 47 159. ,19 4 ,79 . 0.,255 41., 58 122. .12  8,.01 34.,17 19 ..54  --  10..37 43.,33 26,,30  -- _E>  5,.44 5,.94 6 ,24 . 13., 35 4 ,01 . 6,.83 2,, 55 6,, 39 141, ,14 177. .24 2.,83 6.,75 0., 219 0..291 25., 67 57 ,49 . 43,.99 200 ,.25  8..33 49 ..66 29..19 2 .20 , 5,. 52 7 ,78 . 4,.00 3 ,78 . 67.,56 3.,72 0.,302 20.,33 76,.61  2 10..49 46 .,11 27,.10 0.,90 5,.53 6.,78 3..51 3.,27 66,,94 4 ,95 . 0,.295 19 ,,75 71,.61  *• X = 0.228 X =0.176, X = 0.596 Nomenclature and d e f i n i t i o n of the q u a l i t y parameters are given i n Appendix E ° of d u p l i c a t e formulations not c a l c u l a t e d i  /  2  3  B  D  -309-  Table  54. Experimental v e r i f i c a t i o n of the p r e d i c t e d q u a l i t y values of LPl.*-  Quality parameters*  Predicted quality values  Confidence interval  Experimental q u a l i t y values c  1  Shrink Tmloss Twloss ES PH Exfluid Exwater Exfat Hardl Shear Cohes Gummy Chewy *• Xx = 0 . 1 8 7 , B  a D  8,. 6 3 35, . 48 2 1 , .28 0. ,40 5..79 8..92 4,.93 3,.96 1 1 3 , .69 4 .81 , 0, .257 29 ,, 77 1 1 1 , .09  X =0.313, 2  7,.69 33, ,01 1 9 , ,45  --  5.,60 6.,22 4.,08 2., 53 70, .97 3., 34 0, .230 1 7 . ,23 52. .65  9 .57 , 37, .95 23, .11  --  5,.98 1 1 , ,62 5.,78 5,, 39 1 5 6 , .41 6,.28 0, .284 42. . 31 1 6 9 , .53  10, .29 49 ,.73 29, .90 2,,40 5,.67 8.,28 4,.18 4 ,10 , 84, .30 4 ., 7 5 0, . 2 8 5 24 ,.16 9 1 , .41  2  1 5 , .06 57, .32 34, .46 4,,75 5,, 69 8 ,13 . 2.,92 5,,21 1 0 6 , .57 4 ,53 , 0. .326 34. ,60 1 3 8 . .48  X = 0.500 a  Nomenclature and d e f i n i t i o n of the q u a l i t y parameters are given i n Appendix E of d u p l i c a t e f o r m u l a t i o n s not c a l c u l a t e d  -310-  and  the d i f f e r e n t steps  Involved  i n the e v a l u a t i o n of the q u a l i t y  parameters a l l c o n t r i b u t e t o the v a r i a b i l i t y observed. As  can be observed not a l l the experimental q u a l i t y data  within  t h e i r r e s p e c t i v e confidence  q u a l i t y data of one d u p l i c a t e and  the  other  adequacy  of  considered  d i d not.  the models  fell  Intervals.  w i t h i n the confidence  Interval  Before making any c o n c l u s i o n s  on the  for predicting quality,  models were obtained  predicted  quality  performed  using  at  storage  parameter values meat  -30°C.  water-  this  i n turn  (Lanier,  The v e r i f i c a t i o n of the  of the two f o r m u l a t i o n s  i n g r e d i e n t s t h a t had been  I t has been reported  that  was  stored  for 6  extended  frozen  proteins,  and f a t - b i n d i n g c a p a c i t y and g e l a t i o n a b i l i t y , and affects  the q u a l i t y  1990; Whiting, 1988; Smith,  1984a; M o r r i s s e y  of  the  of  finished  product.  1988; Pearson and Tauber,  e t a l . , 1982; Powrie, 1973; Schut, 1976; S a f f l e ,  Although the e f f e c t of f r o z e n storage  properties  be  the q u a l i t y  a f f e c t s the f u n c t i o n a l p r o p e r t i e s of the meat  i.e.  1968).  i t must  with meat i n g r e d i e n t s s t o r e d f o r  a p e r i o d of l e s s than 2 months a t -30<>C.  months  In some cases the  t h a t the q u a l i t y data used f o r g e n e r a t i n g  prediction  fell  on the  functional  the meat i n g r e d i e n t s was not assessed  in  this  study, i t may be reasonable t o suggest that the length of time i n f r o z e n storage and  a f f e c t e d the f u n c t i o n a l i t y of the meat  thus the q u a l i t y of  adequacy  of  formulations ingredients  the  f i n i s h e d products.  the models  for predicting  could  have  only  Therefore,  the q u a l i t y  been evaluated  using  s t o r e d a t -30<>C f o r l e s s than 2 months, -311-  ingredients the  of the the  meat  since  the  models on  the  d i d not quality  account of the  f o r the  effect  formulations.  -312-  of extended  frozen  storage  SUMMARY AND  A  computer p r o g r a m  modified  v e r s i o n of  BASIC.  For  the  CONCLUSIONS  for constrained  the  o p t i m i z a t i o n based  Complex method o f Box  purpose of t h i s  study  named "FORPLEX".  The  suitability  of  optimum o f  and  nonlinear  linear  the  constrained  was  mathematical  models.  FORPLEX was  regardless  of  were u s e d .  However,  solving For  linear  these  effective  these To the  or  On  the  other  able  found  for  constrained  FORPLEX  hand,  solving  the  objective t o be  cannot  FORPLEX was  nonlinear  functions  inefficient  replace found  objective  suitability of  of t h e  frankfurter  FORPLEX computer formulations,  q u a l i t y p r e d i c t i o n equations  ingredlent  model fat,  functionuse  for solve  of  mechanically the  deboned  three  an  equations  vas  for a  vere  3-  ingredients beef  meat.  performed  extreme v e r t i c e s  q u a l i t y parameters evaluated -313-  The  for  statistically  p o u l t r y meat and  ingredient-quality using  program  were d e v e l o p e d  frankfurter formulation.  through mixture experimentation eighteen  an  problems. the  Generation  linear  t o be  programming c a n n o t  pork  in  problems.  Linear  test  that  optimum  o p t i m i z a t i o n purposes.  significant  The  l o c a t e the  i t s u i t a b l e for  optimization  vere:  to  making  of  was  constrained  problems,  type  IBM  program  tested using  nonlinear  FORPLEX was  of problems the  method  linearly  the  in  FORPLEX f o r l o c a t i n g t h e  initially  linear  computer  objective function-linearly constrained  types  programming.  formula  whether  written  the  o b j e c t i v e f u n c t i o n problems  were l i n e a r l y  The  the  was  on  classified  design. into 5  groups:  (a)  product  weight  frankfurter  preparation  emulsions  thermal  the  to  cooked  cooked  frankfurters,  model f o r  seven  (d)  and  three  partial  (e)  the  quality  parameters evaluated.  each  by  a  model.  prediction multiple  Student's The  model was  coefficient  e s t i m a t e and  analysis  of  the  and  determination,  significant the  ES  values  for  a probability level  model, a l l the  (p>0.05).  found  at  of to  The  t h r e e models be  l e s s than  prediction  Tmloss,  of  multiple  0.60,  purposes. pH,  no  of  be  Twloss, E x f l u i d ,  -314-  adjusted of  the  prediction  of v a r i a n c e  of  m o d e l s were exception  of  of f i t  determination and  F i r m ) were  considered  residuals  H a r d 2 , S h e a r , Gummy and to  in  quality  error  of  Vacuum s h r i n k  analysis  residuals  was  s i g n i f i c a n t lack  The  analysis  model  each  With the  t h u s were n o t  t h e s e models a p p e a r e d  a l l of  the  regression  0.05.  the  each of  variance,  analysis  and  showed t h a t  of  of  the  to  full  of  coefficient  (Cook s h r i n k ,  Exwater, E x f a t ,  of  the  models p o s s e s s e d  adjusted  raw  required  Seventeen q u a l i t y  a l l the  that  Not  on  standard  the  indicated  the  adequacy  analysis  residuals.  were  to  i n d i v i d u a l parameters  of  models  analysis.  of  m o d e l s were d e v e l o p e d . E x a m i n a t i o n fitted  the  special  model f i t t e d  proportions  Reduction  t - t e s t on  of  the  coefficients  by  of  Scheffe's canonical  Ingredient  assessed  the  characteristics  regression  significance  of  of  j u i c i n e s s c h a r a c t e r i s t i c s of  pH.  multiple  the  stability  textural  describe  performed  (b)  (c)  regression  e f f e c t of  d i f f e r e n t stages  components was  experimental data using the  at  process,  treatment,  frankfurters,  cubic  loss  adequate. H a r d l and  of  adequate Shrink,  Chewy m o d e l s Although  the  Cohes models  suggested the  that  the  assumptions about  m o d e l s were c o n s i d e r e d  quality  prediction  only  both  or  Three  and  different  the  helped  models d e v e l o p e d  l i n e a r and  u n d e r s t a n d i n g of  nonlinear  the  visualize on  the the  quality  proximate  composition  of  the  parameters.  and  between  moisture, content of  the  protein  of  only and  raw  thermal  the  pH  p r o v i d e d an  treatment  of  the  most  the  the  analysis  found  ingredients of  had  the  an  raw  further important  affected  e f f e c t s of  following:  -315-  the  that  displaying  most h e l p f u l on  the  effect,  the  the  in  quality but  the  moisture  composition  q u a l i t y parameters.  e m u l s i o n s and affected  of  proximate  meat b l o c k s and  the  and  parameters.  proportions, I t was  between  emulsions  o v e r a l l view  different factors  fat content  the  ingredient  quality  s c a t t e r p l o t m a t r i c e s was  also  of  analysis  Correlation  raw  better  proportions  the  e m u l s i o n s , t h o u g h d e p e n d e n t on  formulations,  addition,  Some  the  of  The terms  a  contour  in  the  ingredient  e f f e c t s of  Not  linear  provide  meat b l o c k s and  q u a l i t y parameters.  the  to  changes  c o m p o s i t i o n and  explaining  purposes.  either  parameters.  analysis  in a series  violated,  significant linear relationships  parameters, matrices  of  between  data  included  were u s e d  relationship  the  for prediction  Response s u r f a c e  effect  statistically  Scatterplot  were  r e l a t i o n s h i p between i n g r e d i e n t  identified  quality  residuals  terms.  techniques  q u a l i t y parameters.  proportions  the  adequate  the  weight  loss  other  quality  the  ingredients  In  after  parameters. were  the  (A)  high  loss, and  high  high  (B)  o£ b e e f  f o rthe textural o f pork  high  emulsions  product  parameters.  f a t caused  low p r o d u c t  weight  treatment  loss,  frankfurters,  p a r a m e t e r s a n d low e m u l s i o n  t o thermal  weight  i n t h e cooked f r a n k f u r t e r s  o f e x p r e s s i b l e f a t i n t h e cooked  f o r the textural  raw  meat c a u s e d  o f e x p r e s s i b l e water  proportions  levels  values the  levels  values  high  high  proportions  ( i . e . high  low  s t a b i l i t y of  levels  of f a t  released). (C) h i g h caused  proportions  low  water  block,  thus  thermal  treatment.  (D)  both  emulsions. and  beef  decreased  meat  content  of  performed  f o r meat  combinations  of quality  t h e pH  these  objective  proportions  was  which  the  of  vere  optimal  "best and  d e f i n e d as those  -316-  t h e raw  optimization  of  t h e FORPLEX different  measures  incorporated  into  of the  vere  considered  i n theConstraint  subroutine.  combinations  quality" cost  to  increased  In each t r i a l ,  q u a l i t y parameters  incorporated  gave  formulation  were  meat  increased.  p a r a m e t e r s were c o n s i d e r e d  t o find  specifications  formulations  optimization.  When  were  meat  the s u i t a b i l i t y  q u a l i t y and thus  subroutine.  constraints  to test  formula  formulations'  frankfurter  (MDPM)  of the emulsions  a n d MDPM a f f e c t e d  hypothetical  program  product  moisture  as t h e p r o p o r t i o n o f beef  were  Function  per unit  contributing to the s t a b i l i t y  trials  The  loss  deboned p o u l t r y meat  The pH i n c r e a s e d a s t h e p r o p o r t i o n o f MDPM  Several  the  of mechanically  of  ingredient  formulations  constraints. formulations  "Best  that  met  quality"  vhose p r e d i c t e d  quality  was  quality.  as close  to a predetermined  target  Target q u a l i t y values were either selected based  target formulation  predicted  on a  or were i n d i v i d u a l l y selected. When the former  procedure was used whose  as possible  the FORPLEX quality  was  was  able  to find  no d i f f e r e n t  from  formulations their  target  q u a l i t y . Using the l a t t e r procedure, the FORPLEX was able to find formulations  whose predicted q u a l i t y was as  the target q u a l i t y .  In both cases  the  constraints imposed on them.  and  target  formulations because  quality using  the optimum formulations met Differences between predicted  values existed i n a l l the computed the l a t t e r procedure.  Care  target q u a l i t y values.  from the other four formulation target  to meet  existed all  the  should be taken in the s e l e c t i o n of Setting the target value for ES far  target q u a l i t y values resulted in an optimum  (Form4)  quality  optimum  Differences  i t was d i f f i c u l t for the formulations  target q u a l i t y values. the  close as possible to  that d i d not meet the ES target.  value i s not met by a formulation,  necessary to consider modifying the target  values  When a  i t may be set on other  q u a l i t y parameters. Five optimum formulations seven  found by FORPLEX were compared  least-cost formulations  which were found by increasing the  lower l i m i t of the f a t binding constraint. of  each  FORPLEX  optimum formulation  was  The predicted quality very  close  respective target q u a l i t y . The least-cost formulations general,  considerable  set i n the FORPLEX  with  -317-  The cost  its  showed, i n  departure from the target q u a l i t y  formulations.  to  values  of the least-cost  formulations  fell  formulations.  However t h e i r c o s t s depended on the lower l i m i t s e t  on the f a t b i n d i n g The  adequacy  within  the l i m i t s  of  the models f o r p r e d i c t i n g  Formula linear  FORPLEX  quality  since  f r o z e n f o r 6 months.  f o r the e f f e c t of extended f r o z e n  q u a l i t y of the  the  the  c o u l d not be evaluated  i n g r e d i e n t s had been s t o r e d account  on  constraint.  f r a n k f u r t e r formulations  not  Imposed  the  of meat  The models d i d storage  on  the  formulations.  o p t i m i z a t i o n based on the Complex method can  programing  programs  processing  industry.  formulations  t h a t meet  presently  The  latter  being  used i n  searches  for  replace the  meat  least-cost  predetermined product s p e c i f i c a t i o n s  quality.  The new formula o p t i m i z a t i o n method  quality  formulations  o p t i m i z a t i o n method can  searches  within  allowable  cost  handle  nonlinear  quality  and  f o r best  ranges.  This  equations as  o b j e c t i v e f u n c t i o n s as w e l l as c o n s t r a i n t s , thus a l l o w i n g the use of  more  accurate  relationships  that  describe  quality  as  a  f u n c t i o n of the i n g r e d i e n t s . This  new  applicable processed  computerized to  formulation  meats.  formula o p t i m i z a t i o n  method  may  of any type of food product not  I t i s simple t o use,  routine q u a l i t y control.  -318-  thus being  suitable  be only for  REFERENCES  Acton, J . C , Z i e g l e r , G.R.,and Burge, J r . D.L. 1983. F u n c t i o n a l i t y of muscle c o n s t i t u e n t s i n the p r o c e s s i n g of comminuted meat p r o d u c t s . CRC C r i t . Rev. Food S c i . Nutr. 18:99. Adelman, A., and Stevens, W.F. 1972. 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