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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 Benitez Biochemical Engineer Manager in Food Processing, The Institute of Technology and Higher Studies 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 thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA AUGUST 1990 Maria C e c i l i a Vazquez Benitez In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of FOOD SCIENCE The University of British Columbia Vancouver, Canada Date s r P T F M R F R t DE-6 (2/88) ABSTRACT The purpose of t h i s r e s e a r c h p r o j e c t was to e s t a b l i s h a formula o p t i m i z a t i o n computer program to be used f o r q u a l i t y c o n t r o l i n the meat p r o c e s s i n g i n d u s t r y . In c o n t r a s t to l i n e a r programming, such a program would search f o r the best q u a l i t y f o r m u l a t i o n s t h a t meet predetermined product s p e c i f i c a t i o n s w i t h i n a l l o w a b l e c o s t ranges. Since q u a l i t y as a f u n c t i o n of the i n g r e d i e n t s has been found to be e x p l a i n e d b e t t e r by n o n l i n e a r equations, the program had t o be a b l e to handle n o n l i n e a r 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 to make i t an e f f e c t i v e formula o p t i m i z a t i o n method. The f i r s t p a r t of the study e s t a b l i s h e d the IBM BASIC formula o p t i m i z a t i o n computer program (FORPLEX). The FORPLEX i s based on the m o d i f i e d v e r s i o n of the Complex method of Box. The FORPLEX was found to be e f f e c t i v e i n the o p t i m i z a t i o n of 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 t h a t were l i n e a r l y c o n s t r a i n e d , making 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 purposes. The second p a r t of t h i s study i n v o l v e d the development of s t a t i s t i c a l l y s i g n i f i c a n t q u a l i t y p r e d i c t i o n equations 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 . The three i n g r e d i e n t s were: pork f a t , m e c h a n i c a l l y deboned p o u l t r y meat and beef meat. I n g r e d i e n t - q u a l i t y equations were generated through mixture e x p e r i m e n t a t i o n . S p e c i f i c q u a l i t y parameters were ev a l u a t e d a t o b s e r v a t i o n p o i n t s given by an extreme v e r t i c e s d e s i g n . S c h e f f e ' s c a n o n i c a l s p e c i a l c u b i c model f o r three components was f i t t e d to the experimental data u s i n g 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 . The s t a t i s t i c a l v a l i d i t y of the equations f o r p r e d i c t i o n purposes was assessed by a n a l y s i s of v a r i a n c e , a d j u s t e d m u l t i p l e c o e f f i c i e n t of d e t e r m i n a t i o n , standard e r r o r of the estimate and a n a l y s i s of r e s i d u a l s . Fourteen of 17 r e g r e s s i o n models developed were co n s i d e r e d adequate t o be used f o r p r e d i c t i o n purposes. In order to have a b e t t e r understanding of the r e l a t i o n s h i p between i n g r e d i e n t p r o p o r t i o n s and the q u a l i t y parameters, three d i f f e r e n t techniques were used: (a) response s u r f a c e contour a n a l y s i s , (b) c o r r e l a t i o n a n a l y s i s and (c) s c a t t e r p l o t matrices a n a l y s i s . The t h i r d p a r t of t h i s study c o n s i s t e d of the computational 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 the FORPLEX program. S e v e r a l f r a n k f u r t e r f o r m u l a t i o n o p t i m i z a t i o n t r i a l s were performed. In each t r i a l , d i f f e r e n t combinations of 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 e i t h e r s e l e c t e d based on a t a r g e t f o r m u l a t i o n or were i n d i v i d u a l l y s e l e c t e d . In both cases the FORPLEX was able to f i n d best q u a l i t y f o r m u l a t i o n s that met the c o n s t r a i n t s imposed on them. D i f f e r e n c e s between p r e d i c t e d and t a r g e t q u a l i t y values e x i s t e d i n a l l the computed optimum f o r m u l a t i o n s when the t a r g e t v a l u e s were i n d i v i d u a l l y s e l e c t e d . D i f f e r e n c e s e x i s t e d because i t was d i f f i c u l t f o r the for m u l a t i o n s to meet a l l the t a r g e t q u a l i t y v a l u e s . Target q u a l i t y values should be s e l e c t e d c a r e f u l l y s i n c e f a i l u r e to o b t a i n f o r m u l a t i o n s t h a t meet the t a r g e t q u a l i t y as c l o s e l y as p o s s i b l e l a y not with the performance of the FORPLEX but with the s e l e c t i o n of the t a r g e t q u a l i t y v a l u e s . F i v e optimum f o r m u l a t i o n s found by FORPLEX were compared with seven l e a s t - c o s t f o r m u l a t i o n s which were found by i n c r e a s i n g the lower l i m i t of the f a t b i n d i n g c o n s t r a i n t . The p r e d i c t e d q u a l i t y of each FORPLEX optimum f o r m u l a t i o n was c l o s e t o i t s r e s p e c t i v e t a r g e t q u a l i t y . The l e a s t - c o s t f o r m u l a t i o n s showed, i n g e n e r a l , 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 v a l u e s s e t i n the FORPLEX f o r m u l a t i o n s . The adequacy of the models f o r 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 f o r m u l a t i o n s c o u l d not be eva l u a t e d s i n c e the meat i n g r e d i e n t s had been s t o r e d f r o z e n f o r 6 months. The models d i d not account f o r the e f f e c t of extended f r o z e n storage on the q u a l i t y of the f o r m u l a t i o n s . R e s u l t s of t h i s study i n d i c a t e d that formula o p t i m i z a t i o n based on the Complex method (FORPLEX) i s the more s u i t a b l e technique f o r food f o r m u l a t i o n . The FORPLEX may be able to r e p l a c e l i n e a r programming computer programs c u r r e n t l y being used i n the processed meat i n d u s t r y . - i v -TABLE OF CONTENTS Page ABSTRACT 1 i TABLE OF CONTENTS V LIST OF TABLES Ix LIST OF FIGURES x i i ACKNOWLEDGMENTS X V i i INTRODUCTION 1 LITERATURE REVIEW 5 A. Formula o p t i m i z a t i o n 5 1. Experimental formula o p t i m i z a t i o n 6 2. Computational formula o p t i m i z a t i o n 8 2.1. L i n e a r programming 9 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 the food i n d u s t r y 11 2.1.2. A p p l i c a t i o n s i n food f o r m u l a t i o n 12 2.1.3. A p p l i c a t i o n s i n the meat p r o c e s s i n g i n d u s t r y 15 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 formula o p t i m i z a t i o n method 23 B. Q u a l i t y p r e d i c t i o n models 24 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) approach 25 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 approach 26 2.1. Mixture designs 31 C. Nonl 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 33 1. Nonlinear c o n s t r a i n e d o p t i m i z a t i o n techniques 33 2. The Complex method 35 2.1. The g e n e r a l o p t i m i z a t i o n problem 35 2.2. The Complex method a l g o r i t h m . . 36 2.3. M o d i f i c a t i o n s to the Complex method 39 2.4. A p p l i c a t i o n s of the Complex method 45 D. Comminuted meat products 46 1. Product d e s c r i p t i o n 46 2. P r o c e s s i n g steps 49 3. Some f a c t o r s t h a t a f f e c t f i n a l product c h a r a c t e r i s t i c s 51 3.1. Compositional f a c t o r s 51 3.2. P r o c e s s i n g f a c t o r s 56 MATERIALS AND METHODS 58 A. Experimental methodology 58 B. I n g r e d i e n t s 60 C. Proximate a n a l y s i s 62 -v-Page 1. Determination o£ moisture 62 2. Determination of crude fat 62 3. Determination of protein 62 D. Experimental design 63 E. Frankfurter preparation 65 F. Quality parameters evaluated 71 1. Determination of pH 71 2. Emulsion s t a b i l i t y analysis 71 3. Per cent weight loss a f t e r processing and storage 74 4. Consumer cook test 74 5. Juiciness evaluation 75 6. Texture evaluation 76 6.1. Texture p r o f i l e analysis 76 6.2. Shear force 76 G. S t a t i s t i c a l analysis 77 1. Regression analysis 77 2. Correlation analysis 83 3. Response surface contour analysis 83 H. Optimization methods 83 1. Feasible point computer program (FPOINT) 83 2. Formula optimization computer program (FORPLEX) 86 2.1. Program description 86 2.1.1. General description 86 2.1.2. Description of parameters 91 2.1.3. Summary of user requirements 93 2.1.4. New routines 94 2.1.4.1. Generation of random numbers 94 2.1.4.2. Reflection through best point 95 2.2. Limitations of the formula optimization algorithm • . 95 2.2.1. Equality constraints 95 2.2.2. Endless contraction due to i m p l i c i t constraint v i o l a t i o n 96 2.3. Optimization of constrained mathematical models 96 2.4. Optimization of frankfurter formulations 97 3. Formula optimization using linear programming 98 RESULTS AND DISCUSSION 99 A. Optimization of constrained mathematical models 99 B. Development of ingredient-quality relationships for a 3-ingredient model frankfurter formulation 130 1. Proximate analysis 130 2. Quality parameters evaluated 133 2.1. Product weight loss at d i f f e r e n t stages of the frankfurter preparation process 134 2.2. Emulsion s t a b i l i t y analysis 137 2.3. Juiciness c h a r a c t e r i s t i c s of the cooked frankfurters 141 2.4. Textural parameters of the cooked frankfurters 144 - v i -Page 2.5 Determination of pH 150 3. Q u a l i t y p r e d i c t i o n models . 153 3.1. Regression a n a l y s i s 153 3.2. Response s u r f a c e contour a n a l y s i s 189 3.3. C o r r e l a t i o n a n a l y s i s 211 3.4. S c a t t e r p l o t matrices a n a l y s i s 221 C. Computational 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 236 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 the new formula o p t i m i z a t i o n computer program (FORPLEX) 236 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 245 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 246 1.2.1. 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 where combinations of two to 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 s 248 1.2.2. 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 where combinations of f i v e q u a l i t y parameters were con 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 s e t i n d i v i d u a l l y 255 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 f o r m u l a t i o n s when 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 274 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 u s i n g l i n e a r programming 285 3. Comparison of FORPLEX and l i n e a r programming computed optimum f o r m u l a t i o n s 291 3.1. Comparison i n terms of p r e d i c t e d q u a l i t y 291 3.2. Comparison i n terms of c o s t 302 4. Comparison of FORPLEX with l i n e a r programming f o r meat formula o p t i m i z a t i o n 304 5. 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 v a l u e s of two computed optimum fo r m u l a t i o n s 307 SUMMARY AND CONCLUSIONS 313 REFERENCES 319 APPENDIX A. E x p l a n a t i o n of how to read the i n g r e d i e n t p r o p o r t i o n s i n t r i a n g u l a r graphs 333 APPENDIX B. Lotus 1-2-3 template f o r t e x t u r e p r o f i l e a n a l y s i s 337 APPENDIX C. L i s t i n g of the FPOINT computer program 341 - v i i -Page APPENDIX D. L i s t i n g of the FORPLEX computer program 344 APPENDIX E. Nomenclature and d e f i n i t i o n of the q u a l i t y parameters 350 - v i i i -LIST OF TABLES Page Table 1. Ingredients and their l i m i t s used for the extreme verti c e s design 64 Table 2. Extreme vertices experimental design 66 Table 3. Weights of ingredients used in each formulation(g) 68 Table 4. D e f i n i t i o n a l and working formulas for multiple regression analysis of variance for mixture models 81 Table 5. Optimization results of test problem 1 101 Table 6. Optimization r e s u l t s of test problem 1 using d i f f e r e n t random number seeds 102 Table 7. Optimization r e s u l t s of t e s t problem 2 104 Table 8. Optimization results of test problem 3 106 Table 9. Optimization re s u l t s of test problem 4 109 Table 10. Optimization results of test problem 5 with broad l i m i t s on the independent variables 113 Table 11. Optimization re s u l t s of test problem 5 with narrow l i m i t s on the independent variables 114 Table 12. Optimization results of test problem 6 117 Table 13. Optimization re s u l t s of test problem 7 119 Table 14. Optimization re s u l t s of test problem 8 122 Table 15. Optimization re s u l t s of test problem 9 123 Table 16. Optimization results of test problem 10 126 Table 17. Optimization r e s u l t s of test problem 11 128 Table 18. Proximate composition and pH of raw ingredients 131 Table 19. Experimental data for product weight loss at d i f f e r e n t stages of the frankfurter preparation process 135 Table 20. Experimental data for emulsion s t a b i l i t y analysis 138 Table 21. Experimental data for juiciness c h a r a c t e r i s t i c s of the cooked frankfurters 143 Table 22. Experimental data for textural parameters of the cooked frankfurters 147 Table 23. 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 loss after processing (Shrink) data 158 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 loss after 13 days under vacuum packaged storage (Vacuum shrink) data 160 Table 25. 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 loss a f t e r the consumer cook test (Cook shrink) data 161 Table 26. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to emulsion s t a b i l i t y (Tmloss) data 162 - i x -Page Table 27. 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 "best" model f i t t e d t o emulsion s t a b i l i t y (Twloss) data 164 Table 28. Reg 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 the "best" model f i t t e d to emulsion s t a b i l i t y (ES) data 166 Table 29. 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 "be s t " model f i t t e d t o 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 167 Table 30. Reg 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 the "best" model f i t t e d to per cent e x p r e s s i b l e water (Exwater) d a t a . . . . . 169 Table 31. 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 "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 (E x f a t ) data 171 Table 32. Regression s t a t i s t i c s corresponding t o the "best" model f i t t e d to pH data 173 Table 33. Regression 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 the "best" model f i t t e d to hardness a t f i r s t compression (Hardl) data 175 Table 34. Regression s t a t i s t i c s c o rresponding to the "best" model f i t t e d t o hardness a t second compression (Hard2) data 177 Table 35. Regression 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 the "best" model f i t t e d to maximum shear f o r c e (Shear) data 179 Table 36. Regression s t a t i s t i c s c o rresponding to the "best" model f i t t e d to firmness (Firm) data 181 Table 37. Regression s t a t i s t i c s c o rresponding to the "best" model f i t t e d to cohesiveness (Cohes) data 182 Table 38. Regression s t a t i s t i c s c o rresponding t o the "best" model f i t t e d to gumminess (Gummy) data 184 Table 39. Regression 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 the "best" model f i t t e d to chewiness (Chewy) data 186 Table 40. Q u a l i t y p r e d i c t i o n models 188 Table 41. C o r r e l a t i o n s between proximate composition of the meat b l o c k s and raw emulsions and the q u a l i t y parameters ev a l u a t e d 212 Table 42. C o r r e l a t i o n s between the q u a l i t y parameters e v a l u a t e d 217 Table 43. C o r r e l a t i o n s between t e x t u r a l parameters 220 Table 44. Optimum combinations of 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 valu e s of the q u a l i t y parameters and t a r g e t d i f f e r e n c e values 247 Table 45. 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 p a i r s of 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 ' 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 1 (Xx=0.250, X 2 = 0.200, X»=0.550) 250 -x-Page Table 46. Frankfurter formula optimization t r i a l s where three and four q u a l i t y parameters were considered measures of the formulations' qu a l i t y . Target q u a l i t y values were calculated from target point 2 (Xr=0.150, Xa=0.100, X3=0.750) 252 Table 47. Frankfurter formula optimization t r i a l s where fi v e q u a l i t y parameters were considered measures of the formulations' q u a l i t y . Target q u a l i t y values were calculated from target point 2 (X1=0.150/ X 2 = 0.100, Xs 0.750) 254 Table 48. Frankfurter formula optimization t r i a l s where five q u a l i t y parameters were considered measures of the formulations' q u a l i t y . Target q u a l i t y values were set i n d i v i d u a l l y 259 Table 49. Frankfurter formula optimization t r i a l s where a qu a l i t y parameter was considered a constraint 276 Table 50. Objective function and constraint equations used in the optimization of frankfurter formulations using linear programming 288 Table 51. Least-cost frankfurter formulations 289 Table 52. Comparison of FORPLEX and linear programming for the optimization of meat formulations 305 Table 53. Experimental v e r i f i c a t i o n of the predicted q u a l i t y values of Formula.. 309 Table 54. Experimental v e r i f i c a t i o n of the predicted q u a l i t y values of LP1 310 - x i -LIST OF FIGURES Page F i g u r e 1. A two-dimensional case of the Complex method search f o r the optimum 40 F i g u r e 2. Flow c h a r t of the Complex method a l g o r i t h m 41 F i g u r e 3. Extreme v e r t i c e s experimental d e s i g n 67 F i g u r e 4. 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 steps and q u a l i t y parameters e v a l u a t e d 72 F i g u r e 5. Flow c h a r t of the F u n c t i o n s u b r o u t i n e of FPOINT computer program 85 F i g u r e 6. Flow c h a r t of the FORPLEX a l g o r i t h m 87 F i g u r e 7. Contour p l o t of t e s t problem 5. I m p l i c i t c o n s t r a i n t s are represented by d o t t e d l i n e s I l l F i g u r e 8. Contour p l o t of t e s t problems 6 and 7. 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 116 F i g u r e 9. Contour p l o t of t e s t problems 8 and 9. 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 121 F i g u r e 10. Contour p l o t 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 F i g u r e 11. Mean pH valu e s of the raw emulsions. Repl and Rep2 are 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 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 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 Fi 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 165 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 F i g u r e 16. P l o t of r e s i d u a l s f o r the Exwater model 170 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 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 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 F i g u r e 20. P l o t of r e s i d u a l s f o r the Hard2 model 178 F i g u r e 21. P l o t of r e s i d u a l s f o r the Shear model 180 F i g u r e 22. P l o t of r e s i d u a l s f o r the Cohes model 183 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 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 F i g u r e 25. Response s u r f a c e contour p l o t f o r the Shrink model 192 F i g u r e 26. Response s u r f a c e contour p l o t f o r the Tmloss model 193 F i g u r e 27. Response s u r f a c e contour p l o t f o r the Twloss model 19 5 F i g u r e 28. Response s u r f a c e contour p l o t f o r the ES model 196 F i g u r e 29. Response s u r f a c e contour p l o t f o r the E x f l u i d model 198 F i g u r e 30. Response s u r f a c e contour p l o t f o r the Exwater model 199 F i g u r e 31. Response s u r f a c e contour p l o t f o r the E x f a t model 200 - x i i -Page Figure 32. Response surface contour plot for the pH model 202 Figure 33. Response surface contour plot for the Hardl model 203 Figure 34. Response surface contour plot for the Hard2 model 204 Figure 35. Response surface contour plot for the Gummy model 205 Figure 36. Response surface contour plot for the Shear model 207 Figure 37. Response surface contour plot for the Cohes model 208 Figure 38. Response surface contour plot for the Chewy model 209 Figure 39. Relationships between the ingredients proportions and the proximate composition of the meat blocks and raw emulsions, added ice and pH of the raw emulsions. Pork f a t , XI; MDPM, X2; Beef meat, X3. Proximate composition of meat block: moisture, f a t , protein and f a t -to-protein r a t i o (FP); composition of raw emulsion: moisture (moistem) 223 Figure 40. Relationships between the ingredients proportions and the q u a l i t y parameters that describe product weight loss, emulsion s t a b i l i t y and juicine s s c h a r a c t e r i s t i c s . Pork fa 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 given in Appendix E 225 Figure 41. Relationships 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 that describe product weight loss, emulsion s t a b i l i t y and juiciness c h a r a c t e r i s t i c s . Proximate composition of meat block: moisture, f a t , protein and fa t - t o -protein 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 in Appendix E 226 Figure 42. Relationships between the ingredients proportions and the textural parameters. Pork fa 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 given in Appendix E 229 - x i i i -Page Fi g u r e 43. 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 t e x t u r a l parameters. Proximate composition of meat b l o c k : 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 give n i n Appendix E 230 Fi g u r e 44. 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 gi v e n i n Appendix E 232 Fi g u r e 45. R e l a t i o n s h i p s between 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 . 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 Fi g u r e 46. R e l a t i o n s h i p s 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 Fi g u r e 47. Flow c h a r t of the Fu n c t i o n s u b r o u t i n e of the FORPLEX program 243 Fi g u r e 48. Response s u r f a c e contour l i n e s c o r r e s p o n d i n g to the t a r g e t q u a l i t y v a l u e s of Twloss, Exwater, E x f a t , H a r d l , 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 i s giv 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 Fi g u r e 49. Response s u r f a c e contour l i n e s c o r r e s p o n d i n g 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 l u i d , Shear, Cohes and Gummy 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 r e g i o n which i s giv 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 (Form2) i s represented by a c l o s e d symbol 263 Fi g u r e 50. Response s u r f a c e contour l i n e s c o r r e s p o n d i n g to the t a r g e t q u a l i t y v a l u e s 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 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 giv 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 (Form3) i s represented by a c l o s e d symbol 266 - x i v -Page F i g u r e 51. Response s u r f a c e contour l i n e s c o r r e s p o n d i n g t o 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 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 r e g i o n which i s 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 (Form4) i s represented by a c l o s e d symbol 269 F i g u r e 52. Response s u r f a c e contour l i n e s c o r r e s p o n d i n g to the t a r g e t q u a l i t y values of Twloss, ES, E x f a t , H a r d l 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 r e g i o n which i s given 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 (Form4*) i s represented by a c l o s e d symbol 272 F i g u r e 53. Response s u r f a c e contour l i n e s c o r r e s p o n d i n g to the t a r g e t q u a l i t y 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 g i v e n 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 F i g u r e 54. Response s u r f a c e contour l i n e s c orresponding to the t a r g e t q u a l i t y values of Exwater, E x f a t , H a r d l 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 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) 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 280 F i g u r e 55. Response s u r f a c e contour l i n e s c orresponding 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 i n 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 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 (Hardl) 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 282 F i g u r e 56. Response s u r f a c e contour p l o t f o r the TBV e q u a t i o n . 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 n s t r a i n t s 287 F i g u r e 57. 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 the p r e d i c t e d q u a l i t y v a l u e s of Formula and the l e a s t - c o s t f o r m u l a t i o n s 292 F i g u r e 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 the 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 the l e a s t - c o s t f o r m u l a t i o n s 293 F i g u r e 59. 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 the p r e d i c t e d q u a l i t y v a l u e s of Form3 and the l e a s t - c o s t f o r m u l a t i o n s 294 -xv-Page Figure 60. Differences between s p e c i f i e d target q u a l i t y values and the predicted q u a l i t y values of Form4 and the least-cost formulations 295 Figure 61. Differences between s p e c i f i e d target q u a l i t y values and the predicted q u a l i t y values of Form4* and the least-cost formulations 296 Figure 62. Cost ($/kg of meat block) of the optimum formulations found by FORPLEX and the l e a s t -cost formulations 303 Figure A l . General mixture problem for three ingredients 334 Figure A2. Experimental feasible mixture space for three ingredients 335 Figure A3. Optimization feasible mixture space for three ingredients 336 Figure B l . Lotus 1-2-3 template for texture p r o f i l e analysis 338 - x v i -ACKNOWLEDGEMENTS I wish t o express my g r a t e f u l thanks to Dr. S. Nakai f o r h i s encouragement and guidance d u r i n g the course of t h i s work. I t has been an i n t e r e s t i n g and rewarding experience. I would a l s o l i k e t o express my a p p r e c i a t i o n t o the members of my committee, Drs. E. Li-Chan, J . Vanderstoep, T. Durance and S.T. Chen, f o r t h e i r h e l p f u l suggestions and encouragement. My s i n c e r e thanks go to Ray Seb a s t i a n and the s t a f f of the q u a l i t y c o n t r o l l a b of I n t e r c o n t i n e n t a l Packers, Vancouver, B.C., fo r a l l t h e i r c o o p e r a t i o n with t h i s p r o j e c t . Many thanks are a l s o due to Eleanore Wellwood, who m e t i c u l o u s l y p r o o f - r e a d my t h e s i s . Most i m p o r t a n t l y , I want t o thank my husband, G u i l l e r m o , f o r h i s u n f a i l i n g support and f a i t h i n me. Gu i l l e r m o , e s p e c i a l l y , has shared my joys and sorrows, f a i l u r e s ans successes, and h i s b e l i e f i n my a b i l i t y t o overcome any problem has giv e n me the courage to f i n i s h t h i s work. I t i s to him and to our son E m i l i o t h a t I d e d i c a t e t h i s t h e s i s . - x v i i -I N T R O D U C T I O N The food industry cannot work with fixed blending schemes because of constantly varying composition of the raw ingredients. Implementation of optimization methods in day-to-day formulation is e ssential i f the product i s to survive in the market. Computational optimization techniques are less time consuming than experimental optimization techniques, and thus are suitable for routine q u a l i t y control purposes. Currently, linear programming i s the most popular computer-aided technique being used for food formulation. The meat processing industry uses linear programming to determine least-cost meat formulations that w i l l meet predetermined product s p e c i f i c a t i o n s with the available ingredients. The product s p e c i f i c a t i o n s , or constraints imposed on the formulations, are set by government regulations, product label claims, 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 in order to market the product (Rust and Olson, 1987). In sausage formulations, the constraints are placed to r e s t r i c t : (a) ingredient contents (e.g. beef meat content), (b) proximate composition (e.g. fat content) and (c) qu a l i t y (e.g. fat binding capacity). Advantages of using linear programming in formulating low-cost meat products have been seen in more accurate control of costs, uniformity in composition and maintenance of q u a l i t y (Pearson and Tauber, 1984a). Although linear programming has been proven useful as an optimization technique for meat product formulation, -1-i t presents several l i m i t a t i o n s : (A) Linear programming places heavy emphasis on cost reduction but far less emphasis on f i n a l product quali t y , by dealing with the q u a l i t y parameters as constraints rather than as objective functions. Because of t h i s , l inear programming does not always provide the best q u a l i t y product. (B) Linear relationships have been assumed between ingredients and q u a l i t y parameters, such as bind value constants. Quality parameters are usually not described by simple linear functions; instead nonlinear functions have been found to explain q u a l i t y more accurately (Nakai and Arteaga, 1990). (C) Bind value constants (a measure of the fat binding q u a l i t y of the meat ingredients) have been used by meat processors as indicators of the functional performance of the raw meat ingredients (Parks et a l . 1985; Pearson and Tauber 1984a). However, i t has been reported that the parameters commonly incorporated into bind value constants, e.g., emulsifying capacity and protein or s a l t - s o l u b l e protein content, do not accurately predict the q u a l i t y of the finished products (Parks et a l . 1985; Comer and Dempster 1981). Furthermore bind value constants consider only the a b i l i t y of s a l t - s o l u b l e proteins to emulsify f a t . Water-binding properties and gelation a b i l i t y of meat ingredients, which are important properties in forming stable comminuted meat products, have not been considered. -2-since q u a l i t y assurance should be emphasized at least equally with cost reduction, a better approach may be to search within allowable cost l i m i t s for the best q u a l i t y formulation. However, accurate q u a l i t y prediction equations are required to define the objective functions. Ingredient-quality relationships generated through mixture experimentation have been proven useful in describing the q u a l i t y of food products as a function of their composition. These relationships have been found to be nonlinear. Since linear programming cannot manipulate nonlinear objective functions another optimization technique should replace i t in order to optimize a meat product formulation e f f e c t i v e l y . The Complex method of Box i s a d i r e c t search optimization method that has been used to optimize nonlinear functions subject to linear and nonlinear constraints. This makes i t suitable for formula optimization purposes. Nonlinear q u a l i t y prediction equations can be used as objective functions and the linear equations that describe product s p e c i f i c a t i o n s and cost, and, i f necessary, q u a l i t y prediction equations can be used as constraints. The main objective of t h i s thesis was to e s t a b l i s h a formula optimization computer program to be used for q u a l i t y control in the meat processing industry. Such a program would search for the best q u a l i t y formulations that meet predetermined product s p e c i f i c a t i o n s within allowable cost ranges. -3-To f u l f i l l t his objective, t h i s study was divided in three main parts: (1) Establishment of the formula optimization computer program, (2) Development of ingredient-quality relationships for a 3-ingredient model frankfurter formulation through mixture experimentation, (3) Optimization of several hypothetical frankfurter formulations using the formula optimization computer program. The second objective was to compare t h i s formula optimization program with linear programming for the optimization of frankfurter formulations. -4-LITERATURE REVIEW A. FORMULA OPTIMIZATION In the food industry the blending of d i f f e r e n t raw ingredients is usually required to produce a food product. The choice of the ingredients and the level s to be used are important factors that a f f e c t the f i n a l food product c h a r a c t e r i s t i c s . The t r a d i t i o n a l methods of finding the best combination of ingredients to give the best food product are the " t r i a l and error" and "one variable at a time" approaches. These methods involve numerous experimental t r i a l s u n t i l the appropriate proportion of ingredients i s found to y i e l d an acceptable product (Smith et a l . , 1984). These two approaches present several disadvantages. They are subjective, labour intensive, time consuming, expensive and often the best formulation i s not determined. S t a t i s t i c a l l y and mathematically founded procedures for optimization have become available and are replacing the t r a d i t i o n a l methods for formula optimization. Formula optimization can be described as a set of a c t i v i t i e s that leads to the choice of the best feasible product formulation (Gordon and Norback, 1985). To obtain the best product, d i f f e r e n t formulations must be evaluated and compared in terms of an established c r i t e r i o n . The choice of the c r i t e r i o n to use should be based on the optimization objectives, and these would be: (a) maximum consumer a c c e p t a b i l i t y , (b) maximum quality , and (c) minimum cost. -5-Formula optimization methods can be divided into experimental and computational optimization methods. The former methods are those that require experimentation in order to determine the optimum formulation. The l a t t e r methods are those in which the optimization is computed through the use of established mathematical models that describe the product c h a r a c t e r i s t i c s . 1. Experimental formula optimization One of the most popular experimental optimization techniques used for formula optimization is Response Surface Methodology (RSM) (Norback and Evans, 1983). RSM can be defined as a s t a t i s t i c a l method that uses quantitative data from appropiate experimental designs to determine mathematical models by multiple linear regression analysis (Giovanni, 1983). These mathematical models are useful for describing the effects of changes in the ingredient l e v e l s , the independent variables on the evaluated response, the dependent variable, ( i . e . product c h a r a c t e r i s t i c s ) . The models can be used to predict responses for any combination of ingredients within the range of ingredients studied. The models can be derivatized and/or response surfaces can be generated to determine optimum ingredient combinations that w i l l y i e l d the most desired response. RSM is useful in cases when only a r e s t r i c t e d number of food ingredients have an influence on the c h a r a c t e r i s t i c s of the food product (Nakai and Arteaga, 1990). RSM provides an e f f i c i e n t and systematic procedure for developing food products with s p e c i f i c c h a r a c t e r i s t i c s , so i t - 6 -has found extensive application in the food product development-area. The food industry has been a major user of RSM since the early 1970's (Myers et a l . , 1989). Several publications have discussed the use of RSM in food formulation studies. K i s s e l l (1967), for example, optimized the formulation of a white layer cake by varying the level s of seven ingredients. Min and Thomas (1980) used RSM to determine the relat i o n s h i p between three ingredients ( f a t , s t a b i l i z e r and corn syrup s o l i d s ) and the physical c h a r a c t e r i s t i c s of a dairy whipped topping. Henika (1982) described the application of RSM in developing food products based on sensory evaluation data. Tong et a l . (1984) developed an ice cream formulation containing blends of safflower o i l and milk f a t . Flshken (1983) outlined the procedure he followed to develop cost-reduced formulations using RSM. Mixture response surface methodology, a subset of RSM, has also enjoyed extensive applications in the food area. This technique i s p a r t i c u l a r l y useful in the optimization of food formulations when a l l the ingredients have a strong influence on the product c h a r a c t e r i s t i c s . In th i s case, the response of interest ( i . e . product c h a r a c t e r i s t i c s ) depends upon the proportions of the d i f f e r e n t ingredients. Hence, the response is a function of the composition of the mixture (Cornell, 1981). Several studies have been published regarding the use of mixture designs to optimize food formulations. Huor et a l . (1980) optimized the proportions of watermelon, pineapple and orange juice in f r u i t punches. Johnson and Zabik (1981) found ranges of -7-albumen proteins that optimized foaming and volume of an angel food cake formulation. Rockower et a l . (1983) optimized the proportions of turbot, sodium alginate, soy flour and soy protein concentrate in minced f i s h p a t t i e s . 2. Computational formula optimization When optimum formulations must be determined in the most rapid manner, computational optimization methods must be used. In this case, mathematical models that describe the optimization objectives and, i f necessary, the constraints on the formulation must be known in advance. Two computational optimization techniques have been used to optimize ingredient blending: computational simplex optimization and linear programming. Computational simplex optimization was used to optimize the blending r a t i o of three strawberry essences with reconstituted juice concentrate to simulate the aroma of fresh strawberry juice (Aishima et a l . , 1987). Datta (1989) applied t h i s technique to optimize the blending r a t i o s of wines to standarize aroma qua l i t y . The development of t h i s technique for blending optimization based on the aroma c h a r a c t e r i s t i c s of the food product is quite recent. Applications of t h i s technique as a q u a l i t y control method appear to hold promise for food products which currently r e l y on blending procedures based on sensory testing, for example, wines, processed cheese, coffee, and f r u i t juices (Nakai -8-and Arteaga, 1990). The second and most popular computational optimization technique used for formula optimization is linear programming. A d e t a i l discussion i s given in the following section. 2.1. Linear programming Linear programming (LP) i s a mathematical programming technique. The term "programming" comes from the f i e l d of economics, where a "program" refers to a plan of action, that i s , a s p e c i f i e d , or programmed, solution procedure to be used (Wolfe and Koelling, 1983; Evans, 1982). For a problem to be solved by LP i t must be described by linear equations both in the objective function and in the constraints. The term "linear equation" has been used to describe an equation of f i r s t degree in a l l of the variables involved and in which no products of variables occur (Skinner and Debling, 1969). Before describing the general linear programming problem, there are three optimization components that must be defined (Norback and Evans, 1983): (1) The objective function i s the sum of constant multiples of the variables of interest, also c a l l e d decision variables. This l i n e a r function represents the desired objective of the problem that can be maximized or minimized (Wolfe and Koelling, 1983). This function i s used to compare the possible solutions q u a n t i t a t i v e l y and select the optimum. (2) Decision variables 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 be adjusted to achieve the -9-objective (Evans, 1982) and are also known as resources, input variables, variables of interest or independent variables. (3) Constraints are linear functions which place r e s t r i c t i o n s on the system and the decision variables, r e s t r i c t i n g the value of the objective function. The constraint functions are always compared to some constant. The comparison i s in terms of less than or equal to (<), equal to (=), and greater than or equal to (>) expressions (Norback and Evans, 1983). The general LP problem may be stated as follows: "Given a set of m linear i n e q u a l i t i e s or equations in n decision variables, find non-negative values of the decision variables which s a t i s f y a l l constraints and maximize or minimize some lin e a r combination of the decision variables" (Harper and Wanninger J r . , 1970b). Mathematically, t h i s can be written n Maximize (or minimize) = E c±X± (1) i = l subject to m E a i j X i < b-, (2) j = l x i > 0 ( 3 ) i=l,2, . . .,n j=1,2,...,m where X i = decision variable C i = c o e f f i c i e n t s of the decision variables in the objective function a t j = c o e f f i c i e n t of the i t h decision variable in the jth - 1 0 -constraint Linear programming problems with two decision variables can be solved by graphical so l u t i o n . Linear algebra can also be used to solve LP problems, however i t has been described by Wolfe and Koelling (1983) as an impractical method. The simplex method is an algorithm that i s more commonly used to solve LP problems. Although LP problems can be solved by t h i s method manually, LP computer programs are widely used since the time required to solve these problems is s i g n i f i c a n t l y reduced. Standard computer programs based on the simplex method can be found at many computer centers, and several software packages are available for personal computers (Nakai and Arteaga 1990). The Computing Services of the University of B r i t i s h Columbia has several LP software: LIP, MINOS, LINDO, XMP, MPSX, LPXMP (UBC Computing Center, 1988). Details on LP theory can be found in Nakai and Arteaga (1990), Restrepo (1987), Bender and Kramer (1983), Bender et a l . (1976). 2.1.1. Applications in industries related to the food industry As mentioned above, LP can solve problems in which the objective and a l l the r e s t r i c t i o n s are expressed using linear functions. Since many food industry related problems can be expressed in t h i s way, LP has been applied in determining optimum transportation schedules (Nakai and Arteaga, 1990; Bender et a l . 1982; Skinner and Debling, 1969); in the food service industry to minimize the manufacturing cost of menu items while maintaining -11-q u a l i t y and composition constraints (Norback, 1982); in the feed mixing industry to select the amount of ingredients to be mixed to meet nutrient s p e c i f i c a t i o n s at minimum cost (Pesti and M i l l e r , 1988); in the f i s h farming industry for determining the best combination of production options that maximize p r o f i t (Varvarigos and Home, 1986); among d i e t i c i a n s and n u t r i t i o n i s t s to find the most economical combination of food items in a meal to ensure proper n u t r i t i o n a l balance (Nakai and Arteaga, 1990). 2.1.2. Applications in food formulation Linear programming has been used extensively in solving food formulation problems to determine the optimum quantity or per-cent of an ingredient to be used, subject to necessary constraints, to either maximize or minimize a s p e c i f i e d objective (Smith et a l . , 1984). The constraints are the r e s t r i c t i o n s imposed on the formulation and are generally based on product s p e c i f i c a t i o n s and l e g a l , compositional and functional requirements (Smith et a l . , 1984). The objective is usually to produce the best possible product at a minimum cost. P r o f i t maximization and maximization of consumer a c c e p t a b i l i t y have also been used as objective functions (Beausire et a l . , 1988). Linear programming has been used for formulating various types of foods: (A) Cereal-based foods Linear programming has been u t i l i z e d to develop low-cost, high-protein infant foods. Valencia et a l . (1988) used LP to - 1 2 -develop three low-cost chickpea based infant food formulations o£ high n u t r i t i v e value. Constraints were used to r e s t r i c t the ingredient levels and to f u l f i l l minimum requirements for lysine and sulfur amino acids. Cavins et a l . (1972) used LP to develop several least-cost cereal-based foods. In t h i s study protein q u a l i t y was controlled by r e s t r i c t i n g the percentage of each esse n t i a l amino acid to certain l i m i t s . Hsu et a l . (1977 a;b) optimized the blending of several plant and animal protein sources to obtain low-cost formulations for bread, pasta, cookies and an extruded corn-meal snack. Restrictions were placed to meet functional standards and n u t r i t i o n a l q u a l i t y . Valencia et a l . (1988) commented on the application of LP for the formulation of low-cost, high protein bakery products using combinations of cereals and legumes. Alraeida-Dominguez et a l . (1990) used LP to formulate nutritionally-improved corn-based snacks. Smith et a l . (1984) applied LP to the reformulation of English-style crumpets. The objectives were to develop a least-cost, shelf-stable formulation; constraints were placed on batch s i z e , water a c t i v i t y and moisture, carbohydrate and protein content. (B) Meat products The formulation of meat products i s a problem which the meat industry has commonly solved by linear programming (Norback and Evans 1983). Many publications have reported the use of LP in formulating meat products, where the objective is to minimize the formulation cost while meeting a set of composition and q u a l i t y r e s t r i c t i o n s with the available ingredients (Norback and Evans -13-1983). Some simple examples of sausage formulations can be found in Nakai and Arteaga (1990), Norback and Evans (1983), and Skinner and Debling (1969). Beausire et a l . (1988) formulated a low-cost, fresh turkey bratwurst, a coarse ground type sausage, using LP. An a c c e p t a b i l i t y constraint function was used to at t a i n maximum consumer a c c e p t a b i l i t y of the product. Restrictions on protein, fat and moisture were also used. A detailed description of the app l i c a t i o n of LP in the formulation of a low-cholesterol, low-fat beef stew was given by Bender et a l . (1976). The objective in thi s case was to formulate a low-cost beef stew that met the n u t r i t i o n a l recommendations for low fat and low cholesterol d i e t s . Details of the application of LP in the meat processing industry w i l l be reviewed in the following section. (C) Other food products Ice cream formulations and beer blends have been succesfully formulated using linear programming (Norback and Evans, 1983). Norback and Evans (1983) describe an example where four beer stocks and water must be blended to give a low-cost beer while meeting product s p e c i f i c a t i o n s for alcohol content, s p e c i f i c gravity, color and hop resin content. Other examples of the application of linear programming for food formulation optimization can be found in Bender and Kramer (1983), Norback and Evans (1983), Bender et a l . (1982), Bender et a l . (1976), and Harper and Wanninger (1970b). -14-2.1.3. Applications In the meat processing Industry Due to continuous variations in the composition of the raw ingredients, the food industry cannot work with fixed blending schemes. Therefore, the implementation of optimization methods in day-to-day food formulation i s esse n t i a l i f the product is to survive in the market. Computational optimization techniques such as linear programming are less time consuming than experimental optimization techniques, and thus are suitable for routine qu a l i t y control purposes. In the meat processing industry there i s a large choice of species and cuts of meats that can be used as raw ingredients, each with their own composition and c h a r a c t e r i s t i c s . Moreover, composition of the raw ingredients is highly variable, e s p e c i a l l y in terms of fat and moisture content (Pearson and Tauber, 1984a) Since processed meat products are s t r i c t l y regulated on the basis of their composition, the processor must implement techniques to formulate meat products that meet product standards. Moreover, the cost of meat has increased, exerting pressure on processors to reduce the proportion of expensive meat ingredients in the formulations (Rust and Olson, 1987). This has led to replacing c o s t l y meat ingredients with less c o s t l y ingredients (Rust and Olson, 1987). The term "least-cost formulation" has been used in the meat processing industry to define the use of linear programming to formulate meat products at the lowest possible cost while meeting a l l product s p e c i f i c a t i o n s with the available ingredients. Since 1958 the -15-meat processing Industry has been using computerized linear programming to determine least-cost formulations with more emphasis being placed on emulsion-type products due to their large production volume (Pearson and Tauber, 1984a). Advantages of using linear programming in formulating low-cost products have been seen in more accurate control of costs, uniformity in composition and maintenance of q u a l i t y (Pearson and Tauber, 1984a). Linear programming has been used in meat formulation problems because the components of the problem can be s p e c i f i e d in the way required by linear programming (Norback and Evans, 1983). Both the objective function and the constraints can be expressed as linear functions of the ingredients. To formulate the problem, the processor must have the following information: (1) l i s t of available ingredients and t h e i r costs, (2) composition of each ingredient, and (3) s p e c i f i c a t i o n s of the finished product (Pearson and Tauber, 1984a). As in a l l optimization techniques, the problem must be mathematically defined in terms of: (A) objective function and (B) constraints. (A) Objective function In least-cost formulation problems, the t o t a l cost of the meat formulation is minimized. Therefore, the objective function to minimize i s a linear function that comprises a l l the ingredients (decision variables) that may be used in the formulation and the cost of each ingredient. This function takes the following form, assuming there are n d i f f e r e n t ingredients -16-n minimize cost = E C 1 X 1 (4) i = l where c± = i s the cost of the i t h ingredient X i = is the weight or per cent of the i t h ingredient to be computed Since several r e s t r i c t i o n s are placed in the formulation, the least-cost formula w i l l probably not contain a l l possible ingredients. (B) Constraints The constraints are product s p e c i f i c a t i o n s set by government regulations, product label claims, market demands, company p o l i c i e s and standards that must be s a t i s f i e d for each product produced in order to maintain q u a l i t y (Rust and Olson, 1987). In sausage formulations the constraints are placed to control (a) ingredient content, (b) proximate composition, and (c) quali t y . Mathematically the constraints are the equations or ine 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 formulation, (a) Ingredient constraints Ingredient constraints are used to control the amount of an ingredient or a combination of ingredients in the formulation. That i s , the use of an ingredient or ingredients combination can be limited either to a minimum or maximum, to a fixed l e v e l or to a s p e c i f i e d range (Pearson and Tauber, 1984a). These l i m i t s are imposed by the meat processor in order to a t t a i n 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 ingredients. The most common ingredient constraint is the one that -17-states that the sum of the Ingredients must equal unity or 100 units of weight or some other s p e c i f i e d batch s i z e . Constraints can be developed by the meat processor to r e s t r i c t the use of frozen meat ingredients, since freezing decreases the qu a l i t y of the raw meats in terms of binding, flavour and colour q u a l i t y (Pearson and Tauber, 1984a). Constraints on the use of mechanically deboned meat in sausages have also be developed. American federal regulations r e s t r i c t the use of deboned red meat to a l e v e l of 20% (Pearson and Tauber, 1984a). The use of mechanically 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 frankfurters at leve l s higher than 20% (Pearson and Tauber, 1984a). Constraints are also used to l i m i t the use of non-meat ingredients such as f i l l e r s and binders since their levels in sausages are s t r i c t l y regulated (Rakosky, 1989; Long et a l . , 1982). (b) Composition constraints Constraints on composition are used to specify the required f i n a l composition of the product (Pearson and Tauber, 1984a). In products such as cooked sausages American and Canadian government regulations specify the l i m i t s for f a t , protein and moisture content of the finished product. In the USA, fat content is limited to 28%, minimum protein content i s set at 11% and moisture content should not exceed four times the percentage of protein plus 10% of the finished weight (Pearson and Tauber, 1984a). In Canada, fat content is limited to 25%, minimum protein -18-content is set at 11% and moisture content should not exceed 60% (Meat Inspection Regulations, C.R.C. 1978). (c) Quality constraints Using only ingredient and composition constraints the processor can be confident of producing products with uniform composition but not necessarily uniform q u a l i t y (Pearson and Tauber, 1984a). In order to meet certain q u a l i t y a t t r i b u t e s , the processors have developed q u a l i t y constraints. The most widely used q u a l i t y constraints have been the color and binding constraints (Pearson and Tauber, 1984a). Color constraints have been used to maintain the color i n t e n s i t y of the meat products since the color i s diluted by blending the d i f f e r e n t ingredients (Pearson and Tauber, 1984a). Binding constraints have been used to maintain the fat in a bound state to ensure stable meat products. These constraints are linear functions of the ingredients where the c o e f f i c i e n t s of these functions are the so c a l l e d color and bind values that have been developed as indicators of the q u a l i t y and functional performance of the raw meat ingredients (Parks et a l . 1985; Pearson and Tauber, 1984a). Bind values have been used as a measure of fat binding q u a l i t y of meat Ingredients since S a f f l e introduced t h i s concept in 1964 for least-cost sausage formulations (Porteus, 1979). Sa f f l e ' s bind values are based on the emulsifying capacity of s a l t soluble proteins (Parks and Carpenter, 1987; Comer and Dempster, 1981) and are expressed in grams of fat emulsified per gram of meat (Comer, 1979). Sa f f l e ' s bind value system is based on the theory -19-of emulsion formation in f i n e l y comminuted meat products. This theory holds that during chopping, s a l t - s o l u b l e muscle proteins and water form the continuous phase of the emulsion in which the fat globules are dispersed. The soluble protein coats the fat globules which are then s t a b i l i z e d upon thermal processing (Parks and Carpenter, 1987). Bind values for meat ingredients following Saffle's system have been obtained by the following procedure (Lauck, 1975): (1) Fat binding capacity of sa l t - s o l u b l e proteins is determined by t i t r a t i n g l i q u i d fat into saline extract of meat proteins (Carpenter and S a f f l e , 1964). (2) The f r a c t i o n of the t o t a l protein which is salt - s o l u b l e is determined (Carpenter and S a f f l e , 1964). ( 3 ) M u l t i p l i c a t i o n of fat binding per unit weight of s a l t -soluble protein by the fr a c t i o n percent of salt - s o l u b l e protein results in a product c a l l e d bind value constant. This, value characterizes a unit weight of t o t a l protein for fat binding capacity. (4) For each meat ingredient, the f r a c t i o n of t o t a l protein multiplied by the corresponding bind value constant results in the bind value per unit weight of meat ingredient. The bind value multiplied by the corresponding amount of ingredient in the meat formulation and summed for a l l meat ingredients results in the t o t a l bind value for the formulation (Lauck, 1975). In addition to Saffle's bind value system other bind value -20-systems have been developed by Anderson and C l i f t o n (1967), Kramlich et a l . (1973) and Porteus (1979). In addition, meat processors have developed ranking systems based on their own experience (Kramlich et a l . 1973). The bind value systems of Anderson and C l i f t o n (1967) and Kramlich et a l . (1973) are based on r e l a t i v e bind value scales, e.g. 0 to .1.0 while Porteous (1979) bind value constant system is based on s a l t - s o l u b l e protein content, emulsion capacity and emulsion s t a b i l i t y of the meat ingredients. Although bind values have been proven useful in least-cost formulation, several authors have indicated l i m i t a t i o n s to their use. Parks et a l . (1985) stated that the bind values reported by S a f f l e are: (1) not accurate in predicting emulsion s t a b i l i t y of emulsion-type products and (2) were determined for meat ingredients with s p e c i f i c protein, water and fat composition. Porteous (1979) questioned the accuracy of Saffle's bind values. Comer and Dempster (1981) mentioned that the functional property tests e.g. emulsion capacity test, are of limited use in predicting functional performance ( y i e l d , cook s t a b i l i t y and textural performance) of meat ingredients in comminuted meat products. The question of whether or not a comminuted meat system can be viewed as an emulsion has increased doubt about the r e l i a b i l i t y of the bind value constant systems (Li-Chan et a l . , 1987) . The use of an unreliable bind value as an indicator of product q u a l i t y can r e s u l t in either poor product q u a l i t y due to -21-underestimation of the requirement of functional ingredients, or needlessly high product cost due to overestimation of the need for expensive ingredients ( G i l l e t t et a l . , 1977). In addition, bind value constants consider only the a b i l i t y of s a l t - s o l u b l e meat proteins to emulsify f a t . Water binding properties and gelation a b i l i t y of meat ingredients, which are important properties in forming stable comminuted meat products, have not been considered. Furthermore, the functional performance of nonraeat ingredients has been overlooked, and bind constants for these products have not been established. It has been widely recognized that nonmeat ingredients also play an important role in the q u a l i t y of meat products. Parks et a l . (1985) and Comer and Dempster (1981) emphasized that there i s a need for new concepts for estimating the functional contribution of meat and nonmeat ingredients, e s p e c i a l l y water holding capacity and gelation phenomena. Comer and Dempster (1981) mentioned that using a bind value scale that r e f l e c t s the gelation properties of ingredients would perhaps be more e f f e c t i v e in estimating the functional behavior of the ingredients. Based on t h i s need, they developed a hypothetical bind value scale based on t o t a l protein content and on a r b i t r a r y q u a l i t y factors that r e f l e c t the o v e r a l l functional effects of the ingredients in the meat products. Although the bind values were subjectively determined, th e i r work was the f i r s t attempt to develop a bind value system that includes the functional performance of both meat and nonmeat ingredients in comminuted - 2 2 -meat products. Comer and Allan-Wojtas (1988) stated that r e l a t i v e functional performance of ingredients is dependent upon the composition of the t o t a l system. This indicates that the use of bind values or any other ranking system that uses singular numerical values to estimate individual functional performance in meat products should be questioned. 2.1.3.1. Limitations of linear programming as a meat formula  optimization method Linear programming places strong emphasis on cost reduction but far less emphasis on f i n a l product quali t y , by dealing with the q u a l i t y parameters only as constraints. Because of t h i s , LP does not always provide the best q u a l i t y product. (Nakai and Arteaga 1990). Furthermore, the use of current bind value constant systems has been questioned and no objective consideration has been given to the contribution of ingredient f u n c t i o n a l i t y to the q u a l i t y of the finished product. Since q u a l i t y assurance should be emphasized at least equally with cost reduction, a better approach may be to search for the best q u a l i t y formulation. Quality prediction equations may be used as the objective functions to be maximized while r e s t r i c t i n g the processing costs within allowable ranges. However, there are two problems in attempting t h i s . F i r s t l y , accurate q u a l i t y prediction equations are needed. Quality parameters are usually -23-not described by simple linear functions; instead quadratic functions have been found to explain q u a l i t y more accurately (Nakai and Arteaga, 1990). For example, q u a l i t y parameters such as rheological properties are nonlinearly related to product composition (Nakai, 1987) and product a c c e p t a b i l i t y is nonlinearly related to ingredient levels (Fishken, 1983). Secondly, linear programming cannot manipulate nonlinear objective functions, and therefore another optimization method should replace i t in order to optimize a meat product formulation e f f e c t i v e l y . B. QUALITY PREDICTION MODELS To understand the behavior of the system under study, mathematical models are usually used. Generally, the objective i s to describe the cause-effect 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 possibly manipulating the system (Harper and Wanninger 1969). Experimenters may attempt to i d e n t i f y and describe the response of a system as a function of the variables of interest (independent v a r i a b l e s ) . It is then assumed that changes in the independent variables are responsible for changes in the response. A mathematical description implying how t h i s r e l a t i o n s h i p takes place is c a l l e d a model and the process for deriving the s p e c i f i c model i s c a l l e d modelling (Deming, 1989; Harper and Wanninger, 1970a). It is important to bear in mind that the model i s simply an approximation of the system (Deming, 1989 ) . - 2 4 -Mathematical models can be divided into two types (Denting, 1989; Harper and Wanninger, 1970a): (1) mechanistic models are based on some known mechanism responsible for the relat i o n s h i p between factors and responses. These models are based on chemical and physical theory, and, (2) empirical models are developed when the r e l a t i o n s h i p between the factors and responses i s not known. An assumption is made that the true function f, r e l a t i n g a response y with n number of factors, X i (i=l,...n) y = f ( X i , . . . X „ ) (5) can be approximated by a known and simple mathematical model. Two approaches of empirical modeling have been helpful in describing food product c h a r a c t e r i s t i c s . l . 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) approach The need to predict b i o l o g i c a l a c t i v i t y of compounds requires knowledge of the relat i o n s h i p between molecular structure and b i o l o g i c a l a c t i v i t y . Generation of quantitative structure-a c t i v i t y relationships has received attention in areas such as a n a l y t i c a l chemistry, pharmacology , biology and biochemistry (Stuper et a l . , 1979; Brown at a l . , 1988). Studies on structure-a c t i v i t y relationships are based on the formation of empirical models that use linear free-energy related parameters ( i . e . hydrophobic, ele c t r o n i c and s t e r i c ) as the independent variables and a c t i v i t y as the dependent variable (Stuper et a l . 1979). St 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 have been performed in the food science area. The QSAR approach in this area is based on - 2 5 -the development of empirical models ( i . e . polynomials) where the v a r i a t i o n in a c t i v i t y ( i . e . protein f u n c t i o n a l i t y ) can be predicted from well defined physicochemical properties (e.g. s o l u b i l i t y , d i s p e r s i b i l i t y , hydrophobicity) (Nakai and Li-Chan, 1988) . Li-Chan et a l . (1987) developed equations to predict f u n c t i o n a l i t y of meat proteins, e.g. gel strength, cookloss, water binding properties and fat binding capacity from their physicochemical properties including the contents of f a t , protein and moisture, pH, protein s o l u b i l i t y , hydrophobicity and sulfhydryl group content. These multivariate prediction equations included squared terms of the independent variables and cross product terms. Nonlinearity of the independent variables suggests that an optimum balance of the physicochemical properties is necessary for the best f u n c t i o n a l i t y ; below or above this value the f u n c t i o n a l i t y is i n f e r i o r (Li-Chan et a l . , 1987; Nakai, 1987 ) . 2. Inqredient-aualitv relationships approach The need to predict food product c h a r a c t e r i s t i c s requires knowledge of the r e l a t i o n s h i p between ingredient composition and/or processing conditions and product c h a r a c t e r i s t i c s . For example, in food formulation studies an experimenter might be interested in studying the e f f e c t of product composition on the y i e l d 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 a food product. These relationships need to be quantified to have a better -26-understanding of the magnitude of the relationship between the factors that a f f e c t s the system (ingredients) and the responses ( y i e l d , product q u a l i t y c h a r a c t e r i s t i c s ) (Harper and Wanninger, 1969). Generally, these relationships are not known and empirical models are postulated with the hope that they can describe the system accurately. It i s assumed that the response as a function of the ingredients can be described by a polynomial model (Deming, 1989; Myers et a l . 1989; Floros and Chinnan, 1988; Thompson, 1982; Harper and Wanninger, 1970a). The most commonly used are the f i r s t - o r d e r polynomial: n Y = |3o + E |3iX± (6) i = l and the second-order polynomial n n n-1 n Y = 13= + E (3iX± + E p n X i * + E E P^XiX-i (7) i=l i=l i=l j=i+l In order to approximate the relationship between ingredients and responses with a polynomial, or with any form of model, some preselected number of experimental runs, given by an appropriate experimental design, needs to be performed. Once the observations are c o l l e c t e d , the parameters in the model can be estimated by the method of least squares (Cornell, 1981). The equations derived from t h i s general procedure can y i e l d valuable information about the product. They can be used for d i f f e r e n t purposes such as prediction, optimization of the formulation, and response surface analysis. The experimental strategy and analysis of response surface -27-methodology revolves around the procedure stated above. Response surface experiments attempt to i d e n t i f y and/or evaluate the response of a system as a function of the variables of interest using d i f f e r e n t mathematical models and d i f f e r e n t classes of experimental designs (Myers et a l . , 1989; Thompson, 1982). In the case when the factors are completely independent of one another the most popular response surface experimental design is the central composite rotatable design (CCRD). In order to generate ingredient-quality relationships several meat product formulation studies (Vazquez-Arteaga and Nakai, 1989; Bawa et a l . , 1988; M i t t a l and Usborne, 1986) have used this class of designs. For example, Vazquez-Arteaga and Nakai (1989) studied the e f f e c t s of four ingredients (beef, mechanically deboned poultry meat, soy protein i s o l a t e and wheat flour) on cook y i e l d , emulsion s t a b i l i t y and textural c h a r a c t e r i s t i c s of frankfurter formulations. Graphical representation of the equations developed helped to gain an understanding of the product qua l i t y as the levels of the ingredients were changed. The equations were also used to optimize the q u a l i t y of the formulation. In the case of mixture problems where the variables are interdependent, the response of the system depends only on the proportions of the components of the mixture and not on the t o t a l amount of the ingredients (Agreda and Agreda, 1989; Cornell, 1981). These type of problems c a l l for canonical polynomials and for a s p e c i f i c class of response surface designs known as mixture designs (Agreda and Agreda,1989; Cornell, 1981). In a mixture -28-problem, the factors, X i , (l=l,...k) represent proportionate amounts of the mixture, and, i f expressed as fractions of the mixture, must sum to unity. It is t h i s constraint that causes the variables to be interdependent (Agreda and Agreda, 1989; Cornell, 1981). Food formulations can be considered mixtures or blends of more than one ingredient. Therefore product q u a l i t y can be related to the ingredient proportions using canonical polynomials. Johnson and Zabik (1981) reported prediction equations for cake volume and cake tenderness as a function of a mixture of six egg albumen proteins. Rockower et a l . (1983) reported prediction equations for several textural attributes of minced f i s h patties as a function of f i v e ingredients (two f i s h species, soy protein concentrate, soy flour and sodium al g i n a t e ) . Huor et a l . (1980) reported prediction equations for a c c e p t a b i l i t y of a f r u i t punch formulation containing watermelon, pineapple and orange juices. Generation of ingredient-quality relationships through mixture experimentation could be the most appropriate approach to the study of functional performance of meat and nonmeat ingredients in comminuted meat products. As Parks et a l . (1985) and Comer and Dempster (1981) suggested, new concepts are needed to estimate the functional contributions of ingredients. Comer and A l l a n -Wojtas (1988) emphasized that functional performance of ingredients is dependent on the composition of the meat system as a whole, and suggested that the functional e f f e c t of the ingredients should be evaluated in the actual meat products. -29-Researchers at the Food Research Institute in B r i s t o l have used t h i s approach to predict 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 comminuted sausages as a function of three ingredients: lean meat, fat and added water. The prediction equations found for color, shear and cooking loss helped to understand the ef f e c t of the ingredient levels on each q u a l i t y parameter evaluated (Anonymous, 1985). As mentioned before, equations derived through response surface experiments can be used to optimize the system under study. Agreda and Agreda (1989) commented on the use of computerized optimization algorithms with models developed through mixture designs to optimize mixture responses. Myers et a l . (1989) reviewed the applications of optimization methods to optimize a set of response functions derived through RSM simultaneously. Khuri and Conlon (1981) developed an optimization method for the simultaneous optimization of several response functions represented by polynomial models. They suggested that t h i s optimization method could be applied for the optimization of mixture models. - 3 0 -2.1 M i x t u r e d e s i g n s In mixture systems, the components are stated as proportions of the t o t a l ( i . e . unity), and therefore the proportion of each component X± in the mixture must l i e between 0 and 1.0 (Cornell, 1981) . k E X i = 1.0 (8) i = l 0 < X i < 1.0 (9) As mentioned before, a special class of experimental designs (e.g. mixture designs) i s needed to c o l l e c t observations of mixture systems so that a maximum amount of information 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 s p e c i f i c purposes. Simplex-lattice and simplex-centroid designs can be used when the proportions of a l l the components in the mixture can take values from zero to unity and a l l blends among the ingredients are possible (Cornell, 1981). However, frequently situations exist where some or a l l of the component proportions are r e s t r i c t e d by either a lower bound and/or an upper bound (Nakai and Arteaga, 1990; Cornell, 1981). Such situations are encountered in meat product formulations, where ingredient proportions are constrained between a lower l i m i t a± and an upper l i m i t b± 0 < at < X i < bt < 1 (10) This constraint i s often caused by economic, technical or regulatory considerations. -31-McLean and Anderson (1966) developed the extreme vertices design which was found to be useful for constrained mixture problems. Their procedure permits exploration of the region of interest by using the extreme v e r t i c e s , edge and face centroids of the hyperpolyhedron defined by the constraints. A major drawback of t h i s design i s that the number of experimental points is quite large when the number of components is more than 5 (Snee and Marquardt, 1974). Saxena and Nigam (1977) used the symmetric-simplex design approach reported by Murty and Das (1968) for constrained mixture problems. By using these designs the experimental points uniformly cover the constrained region (Saxena and Nigam, 1977). However, Cornell (1981) had doubts about the e f f i c i e n c y of using a symmetric-simplex design over an extreme vertices design with additional boundary points. - 3 2 -C. NONLINEAR CONSTRAINED OPTIMIZATION 1• Nonlinear constrained optimization techniques Constrained optimization consists of maximizing or minimizing a known objective function while s a t i s f y i n g a set of constraints. When the objective function and/or constraints are nonlinear functions of the independent variables then nonlinear constrained optimization is required. Special techniques have been developed for handling these problems. G i l l and Murray (1974) edited a book on numerical methods for l i n e a r l y and nonlinearly constrained optimization. Schwefel (1981) c l a s s i f i e s the techniques for handling constraints into two main groups: mathematical programming and transformation methods. Mathematical programming is c l a s s i f i e d as linear (e.g. linear programming) and nonlinear programming (e.g. quadratic programming). Transformation methods are those that transform a constrained optimization problem into an unconstrained problem. The objective functions are modified by applying a penalty to the objective function at nonfeaslble points (Saguy, 1983). Examples of these methods are the penalty and barrier function methods and the sequential unconstrained minimization technique (SUMT) (Saguy, 1983). A d i f f e r e n t c l a s s i f i c a t i o n scheme for constrained optimization methods was proposed by Umeda (1969). He c l a s s i f i e d the methods in three groups: search methods, mathematical programming and v a r i a t i o n a l methods. Search methods include pattern search method, ridge analysis, generalized Newton-Raphson method and gradient methods -33-(Umeda, 1969 ). V a r i a t i o n a l methods consist of those related to Pontryagin's maximum p r i n c i p l e (Umeda, 1969). Gradient methods are those that select the d i r e c t i o n of the search by using the values of p a r t i a l derivatives of the objective function with respect to the independent variables. Hemistitching and " r i d i n g the constraints" method (Schwefel, 1981) and Newton-type and Quasi-Newton methods ( G i l l and Murray, 1974) are some examples. The majority of the methods for handling constraints make use of the f i r s t and sometimes the second derivatives of the objective functions and in certain cases of the constraints (Swann, 1974). These methods are l i k e l y to lead to the solution most quickly. However there are a number of drawbacks in the implementation of these techniques. The most important is that the functions should be continuous and d i f f e r e n t i a b l e , and i f so, evaluation of the function and constraints can involve a lengthy and complicated c a l c u l a t i o n (Swann, 1974). An alternative approach is to employ an optimization procedure which does not c a l l for derivative values. Such methods are known as d i r e c t -search methods. Direct search methods have proven useful in problems for which d i f f e r e n t i a t i o n i s d i f f i c u l t , or when the derivatives are discontinuous or when the function values are subject to error (Swann, 1974). The strategy of direct-search methods is based on the progress towards the optimum by evaluation and comparison of the objective function values at a sequence of t r i a l points. One of the most successful of the unconstrained direct-search procedures i s the simplex method -34-o r i g i n a l l y proposed by Spendley et a l . (1962). Box (1965) modified t h i s method to handle constraints and termed his procedure the Complex method (constrained simplex). Box's Complex method has had wide acceptance because of the simple way in which the constraints are handled and the high p r o b a b i l i t y of locating the global optimum (Saguy et a l . , 1984; Kuester and Mize, 1973; Ghani, 1972). 2. The Complex method 2.1. The general optimization problem The optimization problem consists of maximizing (or minimizing) some chosen nonlinear objective function of n independent variables (equation 11) subject to e x p l i c i t constraints on the independent variables (equation 12) and to m nonlinear i m p l i c i t constraints (constraints on functions of the independent variables) (equation 13). y = fo(Xi,X 2,...X„) (11) L i < X i < U i (12) i=1,2,...,n L* < g* (Xo.,X 2, . . .X„) < U* (13) k=l,2,..., m Where L t and U i represent the lower and upper l i m i t s placed on the independent variables, and L* and U* represent the lower and upper l i m i t s placed on the nonlinear functions of the independent variables and are either constants or functions of the independent variables (Box, 1965). Mathematically speaking, i t is important to point out that the -35-term "nonlinear model" refers to a model in which the parameters appear nonlinearly (Ratkowsky, 1983; Draper and Smith, 1981), for example y ex te-St - e"8it] (14) (Si - e 2) where 8 i and 0 2 are the parameters to be estimated. However, in this study the term "nonlinear model" w i l l be used to denote models that are not f i r s t - o r d e r in the independent variables such as f i r s t - and second-order polynomials. 2.2. The Complex method algorithm The description that follows is taken from Box (1965). The Complex method handles the constraints by the use of a f l e x i b l e figure ( i . e . search unit) c a l l e d the complex. This figure consists of more than n + 1 vertices (n i s the number of independent variables) which contracts when constraints are vi o l a t e d . The Complex method tends to find the global optimum by evaluating and comparing the objective function value at a l l feasible vertices of the complex and replacing the worst vertex by a new feasible point. In contrast to Spendley's Simplex method, there i s no attempt to preserve the regular figure in which each vertex is equidistant from a l l other points (Friedman, 1971). The f i r s t step is to generate an i n i t i a l complex in the feasible region, with a number of vertices K > (n+1). i t is required that a feasible s t a r t i n g point ( i . e . does not v i o l a t e any constraint), X n , be known. The additional K-l points, X u , -36-are obtained one at a time by the use o£ pseudo-random numbers and the ranges for each of the independent variables by means of X i * = L t + r i ( U i - L i ) (15) i = l , 2 , . . .,n j = 2, • ••, K where r t i s a pseudo-random number deviate rectangularly d i s t r i b u t e d over the i n t e r v a l (0,1). L± and U± are the lower and upper l i m i t , respectively, of X i . Each point generated in t h i s way is feasible with respect to the e x p l i c i t constraints, but may v i o l a t e one or more of the i m p l i c i t constraints. If an i m p l i c i t constraint is violated, the t r i a l point, Xi^(bad), is moved halfway towards the centroid of the remaining vertices X i 3 ( n e w ) = (X i d(bad) + X4,=)/2 (16) where the coordinates of the centroid of the remaining points, X i o , are defined by K Xi= = E X i a - X i S(bad)) (17) K-l j-1 The centroid i s the point each of whose coordinates i s the numerical average of the corresponding coordinates of the points of the complex, except the worst. This contraction step is repeated as necessary u n t i l a l l the i m p l i c i t constraints are s a t i s f i e d . Box (1965) states that by assuming the feasible region is convex, ultimately, a s a t i s f a c t o r y point w i l l be found. After generating the i n i t i a l complex the following procedure -37-is repeated; the objective function is evaluated at each vertex of the complex. The vertex having the worst response value, X i d(worst), i s replaced by a point a>l times as far from the centroid of the remaining points as the r e f l e c t i o n of the worst point in the centroid, the new point, Xi->(new), being colinear with the rejected point, Xi-j (worst), and the centroid, Xt = , of the retained vertices Xi.-,(new) = o ( X i c - X i 3 (worst) ) + X i e (18) If t h i s new point, which is replacing the previous worst point, is a feasible point the function i s evaluated and the above described expansion step is repeated. I f , however, the point does not s a t i s f y one of the e x p l i c i t constraints on an independent variable, that variable is reset to a value Delta (0.000001) inside the violated l i m i t to y i e l d a feasible point. I f , the function value of the new point repeats in giving the worst function value or i f an i m p l i c i t constraint is violated,. the vertex is moved halfway towards the centroid of the remaining points to give a new t r i a l point. The contraction step i s repeated as necessary u n t i l a better response value or a feasible point i s obtained. Thus as long as the complex has not collapsed into the centroid, progress w i l l continue. Convergence i s assumed when five successive evaluations of the objective function y i e l d equal values to within a spec i f i e d c r i t e r i a . This usually occurs at the optimum when the complex collapses into i t s centroid (Box, 1965). The usual method for checking that the global rather than a -38-l o c a l maximum or minimum has been found Is to r e s t a r t the search from d i f f e r e n t s t a r t i n g points and to infer that, i f they a l l converge on the same r e s u l t , a global optimum has been found. With the Complex method, there i s no d i f f i c u l t y in using the same st a r t i n g point since d i f f e r e n t pseudo-random number i n i t i a t o r s w i l l i n i t i a t e the problem with a d i f f e r e n t complex (Box, 1965). The use of the r e f l e c t i o n factor a>l tends to cause a continual enlargement of the complex and thus to compensate for the contractions towards the centroid. It also enables rapid progress to be made when the i n i t i a l point i s far away from the optimum (Box, 1965). The use of K>n+1 points aids in maintaining the f u l l dimensionality of the complex, since, with K=n+1 points, the complex tends to collapse and f l a t t e n along the f i r s t constraint which is encountered (Box, 1965). Box (1965) recommends a value of «=1.3 and K=2n. However i f n i s greater than 5, less points may be used. For the two dimensional case, the procedure is shown schematically in Figure 1. A flow chart I l l u s t r a t i n g the above procedure i s given in Figure 2. 2.3. Modifications to the Complex method A number of modifications have been made to the o r i g i n a l Complex method to overcome the l i m i t a t i o n s of the algorithm. In the Complex method, a r e f l e c t i o n point must be moved in halfway towards the centroid of the remaining points when the new - 3 9 -F i g u r e 1 . A two-dimensional case of the Complex method search f o r the optimum (Umeda, T. 1 9 6 9 ) . -40-(^Start) \ r — F e a s i b l e s t a r t i n g p o i n t / Generate i n i t i a l complex (• E x p l i c i t c o n s t r a i n t s v i o l a t e d No Yes Move p o i n t i n > v a d i s t a n c e 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 m p l i c i t c o n s t r a i n t s v i o l a t e d Yes Move 1/2 way towards t h e c e n t r o i d _. No I n i t i a l complex g e n e r a t e d No Yes E v a l u a t e o b j e c t i v e f u n c t i o n a t complex p o i n t s F i n d worst p o i n t |t i n complex Check convergence c r i t e r i a Yes No C a l c u l a t e c e n t r o i d of a l l p o i n t s e x c e p t worst R e f l e c t worst p o i n t t h r o u g h c e n t r o i d F i g u r e 2. Flow c h a r t of the Complex method a l g o r i t h m -41-® Yes 1 • Move p o i n t i n a d i s t a n c e 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 E x p l i c i t c o n s t r a i n t s v i o l a t e d No Move 1 / 2 way towards the c e n t r o i d I m p l i c i t c o n s t r a i n t s v i o l a t e d Yes No E v a l u a t e o b j e c t i v e f u n c t i o n New p o i n t r e p e a t s as worst p o i n t Yes No - 4 2 -point repeats as being the worst. The contraction is carried out u n t i l a point better than the rejected one is found. If by chance a l l points on the li n e from the centroid to the projected point are worse than the o r i g i n a l point, application of t h i s rule causes the projected point eventually to coincide with the centroid, thus terminating the search. To overcome t h i s s i t u a t i o n Guin (1968) and Mitchell and Kaplan (1968) recommend that i f the r e f l e c t i o n factor a i s found to have been reduced below a certain quantity without obtaining a better function value, then this t r i a l point should be returned to i t s o r i g i n a l position (worst point) and the second worst point rejected instead. Friedman (1971) and Friedman and Pinder (1972) recommend that a fixed number of contractions toward the centroid should be allowed after which a new point i s generated by r e f l e c t i n g the centroid through the best point. These modifications allow the search to continue in situations where i t would otherwise have stopped (Guin, 1968; Friedman and Pinder, 1972; Friedman, 1971). The Complex f a i l s in finding the optimum when the centroid f a l l s into a non-feasible region; this often occurs when searching non-convex spaces (Guin, 1968). Guin (1968) suggests checking i f the centroid is found to be not f e a s i b l e . If i t i s , a l l but the best point of the complex should be discarded and a new complex constructed. Mitchell and Kaplan (1968) suggest that i f the centroid i s not feasible the search be stopped, and restarted by generating a new i n i t i a l complex. - 4 3 -M i t c h e l l and Kaplan (1968) were concerned with the random nature o£ the i n i t i a l complex, since i t does not take into account a s t a r t i n g point that is in close proximity to the optimum. They developed a non-random i n i t i a l i z a t i o n procedure where the i n i t i a l complex is influenced by the s t a r t i n g point. However, Umeda (1969) stated that the i n i t i a l configuration seemed to have no major e f f e c t on the results and the rate of convergence. One disadvantage of the Complex method is the r e l a t i v e l y slow convergence to an optimum point and the many i t e r a t i v e computations needed to arrive at the optimum (Saguy et a l . , 1984; Saguy, 1983; Umeda and Ichikawa, 1971). Umeda and Ichikawa (1971) proposed that in order to improve the convergence rate the determination of the r e f l e c t i o n point must take into account the function values at each vertex of the complex. Ghani (1972) introduced the expansion and contraction moves into the Complex algorithm based 7 on Nelder and Mead's Simplex algorithm. He compared his Complex method to that of Box's Complex. Fewer function evaluations were needed to reach the optimum responses in a number of examples. The stopping c r i t e r i o n was modified by Umeda (1969) and Umeda and Ichikawa (1971) to be ABS (worst response - best response) < 0,001 (19) worst response - 4 4 -2.4. Applications of the Complex method The Complex method has been found to be applicable to many broad and d i f f e r e n t optimization problems (Saguy et a l . , 1984). In the chemical engineering area, the Complex method has been used for the optimal design of chemical processes and for equipment optimization. Umeda (1969) used the Box Complex method for the optimal design of an absorber-stripper system. Adelman and Stevens (1972) maximized per cent return in investment of a chemical plant. Friedman (1971) optimized a c a t a l y t i c -polymerization process. In the area of food science and technology the Complex method has been used to optimize several food processes (Saguy et a l . , 1984). Mishkin et a l . (1984) used the Complex method to derive optimal temperature p r o f i l e s during food dehydration. Sullivan et a l . (1981) optimized a carrot dehydration process to achieve high q u a l i t y low moisture carrot pieces with l i t t l e loss of vitamin A and color. Saguy (1982) used the Complex method together with a simulation program to optimize a fermentation process. The objective of that study was to maximize the p r o f i t of the fermentation process. The Complex method has also been used in the optimization of food formulations. Moskowitz and Jacobs (1987) ci t e d the applications of the Complex to maximize texture preference of a pie crust formulation while maintaining an acceptable l e v e l of storage s t a b i l i t y . The Complex method was also used to find pie crust formulations with s p e c i f i c sensory p r o f i l e s (Moskowitz and Jacobs, 1987). -45-D. COMMINUTED MEAT PRODUCTS 1. Product description Comminuted meat products such as wieners, frankfurters and bologna are f i n e l y comminuted, cooked, semisolid sausages, either smoked or unsmoked (Pearson and Tauber, 1984b). The basic ingredients are raw sk e l e t a l meats, fat tissue, water, s a l t , n i t r i t e s and possibly f i l l e r s , extenders and binders. Comminuted meat products are produced by the dispersion of fine p a r t i c l e s of fat within a matrix obtained while chopping the meat with s a l t and water, and further s t a b i l i z e d by thermal processing (Lacroix and Castaigne, 1985; Morrissey et a l . , 1982). Comminuted meat products are often referred to as "fine emulsion products" or "emulsion-type products" (Comer and A l l a n -Wojtas, 1988) since Hansen (1960) and Saffle (1968) reported that the physical structures and properties of the raw batter resembled true emulsions. An emulsion is a biphase system consisting of two immiscible l i q u i d s , one being dispersed as f i n i t e droplets having diameters between 0.1 and 10 )im (discontinuous phase) within the continuous phase (Comer and Allan-Wojtas, 1988; Asghar et a l . , 1985). Hansen (1960) and Saff l e (1968) described the basic structure of a meat emulsion as a mixture of f i n e l y divided meat constituents dispersed as a fat-in-water emulsion, in which the discontinuous phase is fat and the continuous phase is water containing s o l u b i l i z e d protein components. The salt- s o l u b l e - 4 6 -proteins of the meat act as emulsifying agents by forming a protein membrane surrounding fat globules ( S a f f l e , 1968). This emulsion theory has been supported by numerous studies regarding the emulsifying capacity of meat proteins in model systems (S a f f l e , 1968; Carpenter and S a f f l e , 1964; Swift and Sulzbacher, 1963; Swift et a l . , 1961). S a f f l e (1968) considered emulsification as the primary factor responsible for the s t a b i l i t y observed in finished products. Several authors have commented that comminued meat products are not true emulsion systems in the c l a s s i c a l sense (Comer and Allan-Wojtas, 1988; Asghar et a l . , 1985; Acton et a l . , 1983; Morrissey et a l . , 1982). However, as pointed out by Morrissey et a l . (1982) and Acton et a l . (1983), they do r e t a i n some of the c h a r a c t e r i s t i c s of an emulsion. Photomicrographs of raw and cooked emulsions show a proteinaceous membrane surrounding the fat globules (Barbut, 1988; Jones and Mandigo, 1982; Swasdee et a l . , 1982; Borchert et a l . , 1967; Helmer and S a f f l e , 1963). However, not a l l fat droplets are of uniform size or are uniformly surrounded by protein membranes (Swasdee et a l . , 1982; Borchert et a l . , 1967). Morrissey et a l . (1982) proposed a model for the i n t e r f a c i a l protein membrane in comminuted meat products. They suggest that the hydrophobic heads of myosin molecules extend into the l i q u i d fat layer at the surface of the fat globule, while actomyosin concentrates in a multilayer region around the fat globule in association with other muscle constituents. The protein multilayer entraps water and possesses - 4 7 -s u f f i c i e n t v i s c o s i t y , coheslveness and e l a s t i c i t y to s t a b i l i z e the raw batter (Morrissey et a l . , 1982). Asghar et a l . (1985) redefined the so-called meat emulsions as meat suspensions, and described them as consisting of a multiphase system in which the continuous phase (called matrix) is a complex hydrophilic c o l l o i d a l aqueous solution of s a l t s and soluble proteins with s o l i d compounds, such as insoluble proteins, fat p a r t i c l e s and other insoluble components of muscle tissue dispersed and immobilized within the matrix. This matrix functions both chemically and mechanically to s t a b i l i z e the dispersed phase in the raw and cooked products (Morrissey et a l . , 1982). It gives the product i t s c h a r a c t e r i s t i c texture, b i t e , moistness, appearance, o v e r a l l q u a l i t y and product i d e n t i t y (Schmidt et a l . , 1981). Several researchers (Comer and Allan-Wojtas, 1988; Ziegler and Acton, 1984; Lee et a l . , 1981) have suggested that the protein matrix formation, in contrast to the i n t e r f a c i a l protein membrane, is the p r i n c i p a l factor that prevents fat coalescence by diminishing the mobility once the fat d i s t r i b u t i o n i s completed. Comer and Allan-Wojtas (1988) and Acton et a l . (1983) pointed out that emulsion formation i s not the primary factor responsible for the s t a b i l i t y of the finished products. Water binding and gelation a b i l i t y of meat proteins play a major role in s t a b i l i z i n g the cooked product. - 4 8 -2 . Processing steps Excellent descriptions of comminuted meat products processing steps have been written by Pearson and Tauber (1984b), Long et a l . (1982) and Saff l e (1968). The operational processing of comminuted meat products usually begins with grinding of the meat ingredients through a 1/8 in grinder plate and pork trimmings and fa t t y tissue through a 3/8 in grinder plate. The lean ground meats are mixed to give a uniform d i s t r i b u t i o n of the meat ingredients. The mix is transferred to a chopper (also c a l l e d s i l e n t c u t t e r ) . Temperature control during chopping is esse n t i a l to prevent emulsion breakdown (Townsend et a l . , 1968). One t h i r d of the ice or cold water, salt-cure mix and seasoning are added to the meats and chopped. The remainder of the ice i s slowly added to keep the meat temperature near 4<>C. When a l l the ice i s added f a t t y tissue i s added and chopping i s continued u n t i l the meat emulsion temperature reaches 14.4-15.6°C. The completion of the comminution depends on temperature rather than time (Whiting, 1988). The product from the chopper is then passed through an emulsifier i f t h i s equipment is avai l a b l e . The temperature of the finished emulsion should not be over 18<>C. If a vacuum chopper has not been used, i t i s recommended that the emulsion be transferred to a pr e c h i l l e d vacuum mixer. Continuous use of a vacuum during emulsion formation has been found to increase product s t a b i l i t y , decrease product shrinkage and promote uniform textural strength (Whiting, 1988; Tantikarnjathep et a l . , 1983). -49-The emulsion is transferred to s t u f f e r s for extruding into natural or synthetic casings. Frankfurters are stuffed into long casings and linked according to the desired length. Stuffing under vacuum produces a denser product with better binding and texture than does nonvacuum s t u f f i n g (Whiting, 1988). The next step is thermal processing of the product. The stages of thermal processing can be divided into (Mittal et a l . , 1987): (A) Drying. Drying i s used to promote skin formation as well as to condition the surface of the product for smoke deposition and color development. (B) Smoking. The purposes of smoking are development of flavor and color, protection from oxidation and preservation. (C) Resting. The resting period i s used to allow smoke penetration into the product. (D) Cooking. The purposes of cooking are: (a) destruction of microorganisms and improvement of storage l i f e , (b) denaturation and gelation of the meat proteins, (c) improvement of the p e e l a b i l i t y of the f i n a l product, (d) s t a b i l i z a t i o n of the red color in cured products and (e) modification of the product texture. (E) C h i l l i n g . C h i l l i n g i s performed by spraying the product with cold water followed by r e f r i g e r a t i o n . The purposes are to wash off excess smoke and to make the product mechanically stable. There are no uniform thermal processing schedules because d i f f e r e n t types of smokehouses are used (Long et a l . , 1982; - 5 0 -S a f f l e , 1968), Saf f l e (1968) mentions a common thermal processing schedule where the smokehouse i s raised 5 <>C/15 min from 60°C to 82°C. The r e l a t i v e humidity may range between 30 to greater than 80%. When the product reaches an internal temperature of 63°C the re l a t i v e humidity should be high. This steam cook operation can be done either in the smokehouse or in a steam cooker (Long et a l . , 1982). The product should be held under these conditions u n t i l i t reaches an internal temperature of 68°C (Long et a l . , 1982; S a f f l e , 1968). 3. Some factors that a f f e c t f i n a l product c h a r a c t e r i s t i c s 3.1. Compositional factors Salt (NaCl) i s essential for the manufacture of meat products (Sofos, 1986). It functions as a preservative, enhances product texture, s t a b i l i t y , and cooking y i e l d and influences product flavor (Lacroix and Castaigne, 1985; Sofos, 1983b). It extracts and s o l u b i l i z e s m y o f i b r i l l a r proteins (Morrissey et a l . , 1982; Comer and Allan-Wojtas, 1988; Acton et a l . , 1983; Sofos, 1983a;b). During extraction an increase in water-binding capacity of the my o f i b r i l l a r proteins occurs, due to the " s a l t i n g - i n " e f f e c t of sodium chloride and tissue d i s i n t e g r a t i o n (Schut, 1976; Acton et a l . , 1983). Swelling of the myofibrils occurs, which af f e c t s several c h a r a c t e r i s t i c s of the meat emulsion such as body, thickening and v i s c o s i t y (Morrissey et a l . , 1982) which in turn - 5 1 -s t a b i l i z e s the meat emulsion (Smith, 1988; s a f f l e , 1 9 6 8 ) . These s o l u b i l i z e d proteins can also emulsify and bind fat and other ingredients (Sofos, 1983 a;b). After extraction, these s a l t soluble proteins form the matrix, which upon coagulation during thermal processing, w i l l hold the dispersed phase within i t . S u f f i c i e n t s o l u b i l i z a t i o n i s needed to have enough protein available to form the gel (Whiting, 1987b). Frankfurter formulations t y p i c a l l y contain 2.5-3.0% s a l 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 (or brine) concentration, a 4% brine is generally necessary to ensure a good batter (Acton et a l . , 1983). Flaked or granulated s a l t can be used. Sofos (1983a;b) found no difference between them in his frankfurter formulations. Several studies have demonstrated that reducing NaCl levels below 2.0% results in increased cooking loses, reduced emulsion s t a b i l i t y , poor p e e l a b i l i t y , unacceptable flavor, reduced shelf l i f e and softer products. (Barbut, 1988; Hand et a l . , 1987; Clarke et a l . , 1987; Sofos 1983a;b). Water addition increases y i e l d (Comer and Allan-Wojtas, 1988), has an e f f e c t on product juiciness and tenderness (Uram et a l . , 1984; Swift et a l . , 1954), and texture (Comer and Allan-Wojtas, 1988), and participates in the s o l u b i l i z a t i o n and extraction of m y o f i b r i l l a r proteins (Lacroix and Castaigne, 1985; Acton et a l . , 1983). Incorporation of water into the meat batter at 15 to 25% gives a good s t a b i l i t y to the meat emulsion during cooking (Lacroix and Castaigne, 1985; Acton et a l . , 1983). At lower water le v e l s , extraction of meat proteins is poor and the meat emulsion - 5 2 -cannot reach i t s optimal s t a b i l i t y , thus y i e l d i n g an unacceptable product. On the other hand, addition of higher levels of water causes d i l u t i o n of the proteins and s a l t s which results in a decrease in water holding capacity (Schut, 1976) and an increase in product shrinkage (Uram et a l . , 1984). Ice and/or cold water are used to absorb the heat generated during comminution in order to prevent coagulation of the proteins (Pearson and Tauber, 1984b). Fat in comminuted meat products contributes to texture, mouthfeel, juiciness and flavor (Townsend et a l . , 1968). Fat reduction increases the toughness of the finished product (Barbut and M i t t a l , 1989; Hand et a l . , 1987) and reduces meat flavor (Anonymous, 1989). A substantial difference in behaviour between the animal fats exists (Schut, 1976). General fat c h a r a c t e r i s t i c s such as the s o l i d - l i q u i d r a t i o , s o l i d consistency and l i q u i d v i s c o s i t y , heats of fusion and t r a n s i t i o n and melting temperature ranges have an ef f e c t on the behaviour of fats during processing of comminuted meat products (Acton et a l . , 1983; Schut, 1976). Schut (1976) c i t e s a study where sausages made with pork suet and beef tallow were unstable, while those made with pork back fat and b e l l y fat resulted in a stable product. Lee et a l . (1981) found that the incorporation of soft fat dest a b i l i z e d the meat emulsion due to a non-uniform dispersion of f a t . It i s recommended that the end point chopping temperature for formulations having pork and beef fats be between 16 to 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 products (Acton et a l . , 1983). Comminuted meat products may be prepared from one or more kinds of raw sk e l e t a l muscle meats and/or raw or cooked poultry meat. Raw or cooked poultry meat s h a l l not comprise more than 15% of the t o t a l ingredients. These products may also contain by-products and mechanically separated meats. Mechanically deboned red meats are r e s t r i c t e d to a maximum of 20% (Pearson and Tauber, 1984a). Limits must be placed by the meat processor on ingredients such as by-products and mechanically separated meats. At high levels of incorporation these ingredients can a l t e r the product y i e l d , texture, color, flavor and s t a b i l i t y (Harding Thomsen and Zeuthen, 1988; Pearson and Tauber, 1984a; Comer and Dempster, 1981; Dhillon and Maurer, 1975). Frozen meats are also used. Frozen meat has poorer functional properties than fresh meat but can s t i l l have adequate f u n c t i o n a l i t y i f fast freezing rates, low storage temperature and slow thawing rates are used (Whiting, 1988; Schut, 1976). M i l l e r et a l . (1980) showed that s i g n i f i c a n t losses in the functional properties of meats occur during extended frozen storage. Powrie (1973) made an extensive review of the protein changes in meat during frozen storage. Conformational changes of m y o f i b r i l l a r proteins take place during frozen storage. These changes are manifested as an increase in protein-protein interaction, decrease in s o l u b i l i t y and reduced water holding capacity (Powrie, 1973). One of the major considerations in sele c t i n g the meat - 5 4 -ingredients for a comminuted meat product has been the a b i l i t y of the meat to "bind" or emulsify fat and r e t a i n moisture (Sa f f l e , 1968). It is common in the meat processing industry to refer to "high binding", "medium binding", and "low binding meats" (Acton et a l . , 1983; S a f f l e , 1968). Such a c l a s s i f i c a t i o n of meats follows the bind values established by S a f f l e very c l o s e l y as well as the m y o f i b r i l l a r protein content present in the meats (Acton et a l . , 1983). Examples of "high binding meats" include s k e l e t a l muscle meat from b u l l , cows and mutton, of "medium binding meats" include cheek meat, veal and pork trimmings and of "low binding meats" ox l i p s , t r i p e , hearts and tongue trimmings (Rakosky, 1989; Acton et a l . , 1983; S a f f l e , 1968). In " a l l meat systems" ( i . e . no f i l l e r s , extenders or binders added), i t is the meat proteins that are r e l i e d upon to impart a suitable texture and s t a b i l i t y to the comminuted meat products (Deng et a l . , 1981). The m y o f i b r i l l a r proteins, actomyosin and myosin, are considered to provide the most f u n c t i o n a l i t y in processed meat products (Smith, 1988; Nakai and Li-Chan, 1988; Whiting, 1987b). The functional properties important in raw batter include water binding and fat holding capacity while during the thermal process and finished product, gelation and fat and water binding capacity are the more important properties (Smith, 1988; Acton et a l . , 1983). Nakai and Li-Chan (1988), Smith (1988), Ziegler and Acton (1984), Acton et a l . , (1983), K i n s e l l a (1983), Morrissey et a l . , (1982), and Schut (1976) have extensively reviewed the functional properties of meat proteins - 5 5 -in the processing of meat products. 3.2. Processing factors The most studied processing variables have been the end point chopping temperature and the temperature and humidity during thermal processing. Helmer and Sa f f l e (1963) reported that emulsion breakdown occurs at comminution temperatures above 16°C and was not due to protein denaturation. Townsend et a l . (1968) noted that the i n s t a b i l i t y of emulsions comminuted at temperatures above 18.5<>C coincided with the onset of melting of the high melting portions of f a t s . Deng et a l . (1981) explained that during comminution the extent of protein-protein interaction became greater as temperature exceeded 15°C. This protein-protein interaction int e r f e r r e d with water and fat binding by the protein leading to an increase in water and fat separation. Lee et a l . (1981) and C a r r o l l and Lee (1981) sustain that the s t a b i l i t y of the product depends on the r i g i d i t y of the gel and the patterns of fat d i s t r i b u t i o n at the beginning of the gel matrix formation. Jones and Mandigo (1982) also found that optimal s t a b i l i t y was achieved at a f i n a l comminution temperature of 16°C. During thermal processing of comminuted meat products a s o l -to-gel transformation of the meat proteins occurs due to an increase in temperature which results in a three-dimensional protein network where fat and water are p h y s i c a l l y and chemically s t a b i l i z e d (Acton et a l . , 1983). Heat induced gelation is a two--56-step process which involves p a r t i a l unfolding of protein followed by aggregation into a continuous network (Kin s e l l a , 1983). For the formation of a highly ordered g e l , i t is necessary that the aggregation step proceeds at a rate slower than the unfolding step (Whiting, 1987b; K i n s e l l a , 1983). Extreme conditions of temperature and rapid heating lead to formation of precipitates which lack the c h a r a c t e r i s t i c s of a gel (Ki n s e l l a , 1983). During thermal processing, temperature and humidity are processing factors that a f f e c t the s t a b i l i t y and texture of the finished product. There is general agreement that high temperature and high humidity have an adverse e f f e c t on the emulsion s t a b i l i t y , texture and color of the finished product (Monagle et a l . , 1974; Saf f l e et a l . , 1967). Cooking temperatures should be increased stepwise. This procedure allows rapid surface dehydration in the i n i t i a l stages of cooking which, together with low humidity, allows skin formation (Mittal and B l a i s d e l , 1983; Saffle et a l . , 1967). Monagle et a l . (1974) reported that low humidity and a steady rate of increase in the smokehouse temperature resulted in a more acceptable product. M i t t a l and B l a i s d e l l (1983) developed a model to predict the weight loss of frankfurters during thermal processing under various conditions and with various product compositions. M i t t a l et a l . (1987) modeled the effects of various rates of increase of smokehouse temperature and r e l a t i v e humidity on the water holding capacity, emulsion s t a b i l i t y , shrinkage, textural parameters and sensory attributes of frankfurters. -57-MATERIALS AND METHODS A. EXPERIMENTAL METHODOLOGY The main objective of t h i s study was to est a b l i s h a formula optimization program to be used for qua l i t y control in the meat processing industry. Such a new formula optimization program would search for the best q u a l i t y formulations that meet predetermined product s p e c i f i c a t i o n s within allowable cost ranges. This new method may replace linear programming computer programs currently being used in the meat processing industry for finding least-cost formulations. To f u l f i l l the main objective, t h i s study was divided in three main parts: (1) Establishment of the Formula Optimization computer program A computer program based on Box's Complex method was written in IBM BASIC. For the purpose of thi s study the computer program was named "FORPLEX". The Complex method has been used to optimize nonlinear functions subject to linear and nonlinear constraints making i t suitable for formula optimization purposes. Nonlinear q u a l i t y prediction equations, which have been found to explain q u a l i t y better, can be used as objective functions and the linear equations that 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 prediction equations can be used as constraints. ( 2 ) Development of ingredient-quality relationships for a 3-ingredient model frankfurter formulation A three ingredient model frankfurter formulation was chosen to - 5 8 -test the s u i t a b i l i t y of the Complex method for formula optimization. Quality prediction models as a function of the ingredients in the formulation were developed. As mentioned in the l i t e r a t u r e review the process of modelling i s carried out in a set fashion. An empirical model should be postulated to represent the r e l a t i o n s h i p between the ingredients and quality c h a r a c t e r i s t i c s of the food product. An adequate experimental design should then be used which provides observation points from which the data can be col l e c t e d to which the model can be f i t t e d . After testing the s t a t i s t i c a l v a l i d i t y of the equations these equations can be used for prediction and optimization purposes. In this study, generation of qu a l i t y prediction models was performed through mixture experimentation. This approach was taken since the rel a t i o n s h i p between the ingredient proportions and the qu a l i t y of the formulation could be quantified. As Comer and Allan-Wojtas (1988) suggested, functional performance of the ingredients is dependent on the composition of the meat formulation as a whole, making mixture experimentation a suitable approach to study the ingredient-quality relationships of sausage formulations. Scheffe's canonical special cubic for three components was the postulated model that was f i t t e d to experimental data collected at each experimental point ( i . e . formulation) of an extreme vertices design using multiple regression analysis. The term "experimental data" refers to several q u a l i t y parameters evaluated at each experimental formulation. The quality parameters evaluated were: (a) product -59-weight loss at d i f f e r e n t stages of the frankfurter preparation process, (b) s t a b i l i t y of the raw emulsions to thermal processing, (c) juiciness of the cooked sausages, (d) s p e c i f i c textural c h a r a c t e r i s t i c s of the cooked sausages and (e) pH. The s i g n i f i c a n c e and adequacy of each q u a l i t y prediction model was assessed by analysis of variance, adjusted multiple c o e f f i c i e n t of determination and analysis of residuals. The models developed were useful in predicting the q u a l i t y of frankfurter formulations as a function of the 3 ingredients used only within the mixture space studied. (3 ) Optimization of frankfurter formulations using the FORPLEX program The q u a l i t y prediction equations obtained were used to optimize several hypothetical frankfurter formulations using the FORPLEX computer program. These equations were used as part of the multi-objective function and were also used as constraints. The second objective was to compare linear programming and t h i s new optimization method for formula optimization. The FORPLEX was compared with linear programming for the optimization of frankfurter formulations. In addition, computed optimum formulations found by the FORPLEX program were compared to the ones obtained from linear programming. B.INGREDIENTS Lean beef meat, frozen mechanically deboned poultry meat, (MDPM), and skinless pork fat were obtained from Intercontinental - 6 0 -Packers Limited, Vancouver,B.C. The lean beef meat was trimmed of v i s i b l e fat and connective tissue, cut into cubes and ground twice through a 6.3 mm plate using a Hobart (Model 84142) grinder (Hobart Manufacturing Company Limited, Ont) and mixed by hand to obtain a uniform blend. The pork fat was trimmed of meat traces, cut in pieces and ground once through a 9.4 mm plate and mixed by hand. The frozen MDPM was p a r t i a l l y thawed at 1 oc for 16 hrs and was used d i r e c t l y without further grinding. These ingredients were analyzed for proximate composition. For the purpose of thi s study the three ingredients: beef meat, MDPM and pork f a t , w i l l be referred to as the "meat block" of the frankfurter formulations. The meats and fat were weighed into the r e l a t i v e proportions s p e c i f i e d in the experimental plan to t o t a l 250 g of meat block as described in sections D and E. Each of these three components was packaged i n d i v i d u a l l y in Z i p l o c R freezer bags (DCP Canada Inc. Paris, Ont). A fourth freezer bag was used to hold the three ingredient bags of each formulation. The bags were placed in cardboard boxes, frozen and stored at -30°C u n t i l required. Table s a l t (NaCl) ("Windsor" brand, The Canadian Salt Company Ltd., Ont) was purchased at a lo c a l r e t a i l store. Sodium n i t r i t e was from BDH Chemicals (Toronto, Ont). -61-C . PROXIMATE ANALYSIS 1. Determination of moisture Moisture content was determined using the AOAC (1980) method (24.002). Samples of beef meat, MDPM and porkfat (ca 2 g) were placed in aluminum sample pans and p a r t i a l l y covered with aluminum f o i l . Drying was performed in a vacuum oven at 90°C with a vacuum of 27 inches u n t i l constant weight (5 hrs). The moisture contents were calculated as a percentage (wet basi s ) . 2. D e t e r m i n a t i o n of crude f a t The crude fat content was determined by petroleum ether extraction with a Goldfish apparatus (Laconco, Kansas City, MO) following the method described by Pomeranz and Meloan (1978). Samples of dried beef meat, MDPM and pork fat (ca 1 gr) were used. The crude fat contents were calculated as a percentage (wet basis ) . 3. Determination of protein Protein content was determined on dried-defatted beef meat, MDPM and pork fat samples. Samples (5 to 8 mg) were digested using the rapid Micro-Kjeldahl method of Concon and Soltess (1973). Digested samples were analyzed for t o t a l nitrogen content using an Auto Analyzer II system (Technicon Instruments Co., Tarrytown, NY). The crude protein content was then calculated by - 6 2 -multiplying the t o t a l nitrogen content by a factor of 6.25. The protein contents were calculated as a percentage (wet basis). D. EXPERIMENTAL DESIGN A three-component model frankfurter was chosen for easy v i s u a l i z a t i o n of the factor space and contour response surfaces. The three ingredients used were: pork f a t , ( X i ) ; MDPM, ( X a ) and beef meat, ( X 3 ) . Table 1 shows the lower and upper l i m i t s within which each ingredient was allowed to vary. As in a l l mixture experiments the constraint for the sum of the ingredients was unity. The model used was Scheffes's canonical special cubic for three components f i t t e d to data collected at the points of an extreme vertices design, as described by McLean and Anderson (1966). Selection of this design was based on i t s c a p a b i l i t y of maintaining the sum of the mixture components equal to unity, while allowing for v a r i a t i o n of the component levels within the constra i n t s . The design consisted of 10 experimental points: fiv e extreme ve r t i c e s , four edge centroids and one central point. A computer program was used to compute the extreme vertices (Nakai and Arteaga, 1990). The edge centroids were found by averaging the leve l s of extreme vertices having a common ingredient l e v e l . The central point of the hyper-polyhedron represented the average of the extreme v e r t i c e s . The experimental design was replicated in order to estimate the experimental error adequately. In addition, the design provided a measure of the lack of f i t of the model, -63-Table 1. Ingredients and their l i m i t s used for the extreme vertices design. Limits Ingredient Coded variable Lower Upper Pork fat X i 0.05 0.30 Mechanically deboned poultry X 2 0.00 0.40 meat Beef meat X 3 0.50 0.95 -64-since the number of experimental points was greater than the number of parameters (Snee, 1971). The ingredient proportions for the 10 formulations are reported in Table 2 and the experimental design i s i l l u s t r a t e d in Figure 3. Refer to Appendix A Figures Al and A2 for an explanation on how to read the proportions of components in triangular graphs. E. FRANKFURTER PREPARATION The experimental design was replicated, within each r e p l i c a t e the 10 formulations were prepared in a random order. For the preparation of the frankfurters, the meat ingredients and pork fat were thawed overnight at 1°C prior to use. Frankfurter emulsions were formulated to contain a 4:1 moisture to protein r a t i o , 2.5% e f f e c t i v e s a l t concentration and 0.01% sodium n i t r i t e of the weight of the meat ingredients. Table 3 shows the ingredient levels used in each formulation. Chopping was performed in a food processor (Cuisinart, France) in a 1°C cold room. Total chopping time was 105 sec and the f i n a l temperature of the 20 formulations did not exceed 16°C. In order to maintain constant chopping conditions, chopping time rather than chopping temperature was the processing factor that controlled the completion of the chopping operation. High levels of ice were needed in some formulations to maintain the required moisture to protein r a t i o . This caused lower emulsion temperatures, making chopping temperature impractical as the -65-Table 2. Extreme vertices experimental design. Formulation Ingredient proportions 1 3 No.*-Xa. X 2 X 3 1 0 .10 0 .40 0 . 50 2 0.05 0.40 0.55 3 0.20 0.30 0 . 50 4 0.30 0.20 0.50 5 0.13 0.22 0.65 6 0.05 0.22 0 . 73 7 0 . 30 0.09 0.61 8 0 . 30 0.00 0.70 9 0 .18 0.00 0.82 10 0 .05 0 . 00 0.95 *• Each formulation was replicated for a t o t a l of 20 formulations B Xi=Pork fat; X 2=Mechanically deboned poultry meat; X3=Beef meat -66-MDPM F A T 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 d e s i g n - 6 7 -Table 3. Weights of ingredients used in each formulation (g). Meat block (250 g)-Formulation No. Ice Salt N i t r i t e 1 25.0 100.0 125.0 18 .1 4 . 5 0.023 2 12.5 100. 0 137.5 19.5 4 . 7 0.024 3 50.0 75.0 125.0 20.3 4 . 2 0 .020 4 75.0 50 . 0 125.0 22.5 4.0 0 . 018 5 33 . 3 54 . 3 162.5 26 . 3 4.7 0 . 022 6 12.5 54.3 183 . 3 28.6 5.0 0.024 7 75.0 23 . 0 152.0 27.9 4 . 2 0.018 8 75.0 0.0 175. 0 32 . 5 4.3 0.018 9 43 . 8 0.0 206 . 3 36.0 4.9 0 .021 10 12.5 0 . 0 237 . 5 39 . 4 5.4 0.024 Xi=Pork fat; X 2=Mechanically deboned poultry meat; X 3=Beef meat -68-c o n t r o l l i n g factor. The s a l t and sodium n i t r i t e were weighed and mixed together. The meat Ingredients were transferred to a c h i l l e d chopping bowl. The mixed dry ingredients and half the crushed ice were uniformly di s t r i b u t e d over the meat ingredients and chopped for 45 sec. Chopping was performed with b r i e f interruptions to scrape the sides of the bowl, mix the emulsion and measure the temperature at each period. The remainder of the ice was added and chopped for 10 sec, after which pork fat was added and chopped for 10 sec. Four additional 10 sec consecutive chopping periods were done. F i n a l emulsion temperatures were recorded. Immediately after chopping the raw emulsions were stuffed into 1 . 5 cm diameter, 10 cm long c e l l u l o s e casings (Viskase, Lindsey,Ont), using an e l e c t r i c food gun (Proctor-Silex Canada Inc., Picton, Ont.). These were tie d with a s t r i n g at both ends. Eight frankfurters of 5 to 7 cm in length were produced per formulation. The s t u f f i n g operation was performed in a lO^C cold room. The frankfurters were transferred to p l a s t i c bags and placed in a 1°C cold room for less than 2 h u n t i l thermal processing. Emulsion samples from each formulation were collected in p l a s t i c bags from the food gun and kept in ice for emulsion s t a b i l i t y analysis and pH measurement. The raw frankfurters were hung on a cooking rack and placed in a preheated, mechanical convection, horizontal air-flow e l e c t r i c oven model OV - 4 9 0 A - 2 (Blue M E l e c t r i c Co., Blue Island, IL) and cooked at 70<>C for 20 min. The frankfurters were then transferred - 6 9 -to a steam pot. For each formulation, one T-type thermocouple probe was inserted into the center of one frankfurter. A thermocouple-thermometer model 450-ATT (Omega Engineering, Inc., Stamford, CT) was used to record the frankfurter temperature. The frankfurters were steam-cooked u n t i l an internal temperature of 68-69°C was reached. After thermal processing, the frankfurters were cold-water showered, placed in polyethylene bags and stored in a IOC cold room overnight. Weights were recorded before thermal processing and after overnight cold storage to determine processing weight loss as a percentage (Shrink). Shrink(%) was defined as: (weight before processing - weight after cold storage) x 100 weight before processing (20) The next day frankfurters were hand peeled and surface f a t t y material was c a r e f u l l y wiped off and weighed. Peeled frankfurters were packaged in polyester/lacquer lam./polyethylene pouches (DRG Packaging, Toronto, Ont.) and vacuum sealed with a Multivac vacuum sealer (Sepp Haggenmuller KG, Allgau, W. Germany). The vacuum packaged frankfurters were stored for 13 days at 1°C u n t i l the consumer cook test was performed. Percentage weight loss after 13 days under vacuum packaged storage (Vacuum shrink, %) was defined as: (weight before - weight after vacuum storage) x 100 weight before vacuum storage (21) -70-Figure 4 shows the flow chart of the frankfurter preparation. F. QUALITY PARAMETERS EVALUATED As a measure of the qu a l i t y of the formulations the following q u a l i t y parameters were evaluated at each experimental point. These parameters are commonly used to evaluate the qu a l i t y of comminuted meat products. 1. Determination of pH For the determination of pH of raw emulsions, a 10 g sample was blended ("blend" setting) in an Osterizer Galaxie VII blender (Sunbeam Corporation Limited, Toronto, Ont.) for 10 sec with 100 mL d e i o n i z e d - d i s t i l l e d water. A Fisher model 420 pH-meter (Fisher S c i e n t i f i c Co., Pittsburgh, PA) with a combined glass/reference electrode was used to test the pH of each formulation in duplicate. 2. Emulsion s t a b i l i t y analysis The s t a b i l i t y of the emulsions to thermal treatment was determined by two methods. The emulsion s t a b i l i t y test reported by Saffle et a l . (1967) was followed after some modifications. Eighteen g emulsion samples were placed in 20% d i v i s i o n Paley fat bottles. The bottles were placed in a water bath held at 80°C. One control bottle in each batch was equipped with a T-type thermocouple probe. The bottles were l e f t in the water bath for 30 min after -71-Composition of the f o r m u l a t i o n g i v e n by the experimental plan Water was added to maintain a 4:1 moisture to p r o t e i n r a t i o . S a l t was added a t a 2.5% e f f e c t i v e c o n c e n t r a t i o n . N i t r i t e was added a t 0.01% of meats. Mixing and chopping i n a food processor f o r 105 sec. S t u f f i n g i n c e l l u l o s e c a s i n g s Emuls ion s t a b i l i t y t e s t s Dry and steam cooking. I n t e r n a l temperature 68°C Showering Overnight storage @1°C P e e l i n g Shrink Vacuum packaging T Storage @1<>C f o r 13 days Consumer cook t e s t Vacuum s h r i n k Cook s h r i n k Texture and j u i c i n e s s t e s t s F i g u r e 4. 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 steps and q u a l i t y parameters evaluated -72-the control reached an internal temperature o£ 60°c. This cooking schedule resulted in an internal emulsion temperatures of approximately 79°c. The bottles were f i l l e d with 70<>C water, centrifuged for 5 min at "speed No. 2" setting in a heated (70<>C) Babcock centrifuge (Garver Manufacturing Co. Union City, Indiana). Immediately after centrifugation the per cent fat released was read d i r e c t l y from the stem of the Paley fat b o t t l e . Since the bottles used are designed for 9 gr samples the per cent fat reading was divided by two. This test was performed in duplicate for each formulation. The emulsion s t a b i l i t y was also determined using the procedure developed by Townsend et a l . (1968) with some modifications. Twenty g emulsion samples were placed in 2.5x10.3cm polypropylene centrifuge tubes. The tubes were centrifuged at 4°C for 5 min at lOOOxG in a Sorvall RC-2B automatic refrigerated centrifuge (Ivan S o r v a l l , Inc., Norwalk, CT). The tubes were t i g h t l y covered with parafilm and aluminum f o i l and were placed in a water bath held at 80°C. One control tube in each batch was equipped with a T-type thermocouple probe. The tubes were l e f t in the water bath for 30 min after the control reached an internal temperature of 60oc. This cooking schedule resulted in an internal emulsion temperature of approximately 79°C. This method requires the volume of fa t , water and proteinaceous s o l i d s released during thermal treatment to be measured in order to express emulsion s t a b i l i t y as %v/w unbound fat, water and s o l i d s . However, preliminary experiments showed that the f l u i d released during - 7 3 -thermal treatment never developed a l i p i d layer and the volume of the s o l i d material was d i f f i c u l t to measure accurately. Therefore, the emulsion s t a b i l i t y results could not be expressed as suggested by Townsend et a l . (1968). Instead, the f i n a l weights of the cooked emulsions after draining and cooling to room temperature were recorded. Assuming the weight loss during thermal treatment was due s o l e l y to water loss, results were expressed as per cent water loss of o r i g i n a l moisture content of the meat block and as per cent weight loss. This test was performed in duplicate for each formulation. After preliminary experiments the above cooking schedules were found to produce the least v a r i a t i o n within r e p l i c a t e samples. 3. Per cent weight loss after processing and storage Processing weight loss as a percentage (Shrink) and per cent weight loss after 13 days under vacuum packaged storage (Vacuum shrink) were calculated using equation 20 and 21, respectively. 4. Consumer cook test Frankfurters were weighed and placed in 500 mL b o i l i n g water for 5 min. The frankfurters were removed, drained well and cooled to room temperature i n p l a s t i c boats. These samples were then weighed and placed in beakers covered with saran wrap and l e f t at room temperature for less than 3 h u n t i l j u i c i n e s s and texture evaluation tests were performed. Per cent weight loss after the consumer cook test (Cook - 7 4 -s h r ink, %) was de £ i ned as : (weight before - weight after cooking) x 100 (22) weight before cooking 5. Juiciness evaluation Juiciness of the frankfurters was measured by the method of Lee and Patel (1984) with some modifications. Samples from the consumer cook test were used for th i s determination. Two s l i c e s of 1 cm in length per frankfurter and two frankfurters per formulation were used. An Instron (Model 1122) Universal Testing Machine (Instron Corp. Canton, MA) was used to measure the expressible f l u i d under single compression of the samples using a 5 cm diameter uniaxial f l a t plate. A deformation rate of 50 mm/min was used. The sample was weighed and compressed along i t s axis to 70% deformation, the compression head was reversed automatically. The amount of f l u i d expressed during compression was determined by weight gain of two sheets of f i l t e r paper (Whatman No. 41) and expressed in terms of % expressible f l u i d in r e l a t i o n to sample weight. The f l u i d collected in the f i l t e r papers was analyzed for moisture and fat content. Moisture was determined by weight loss of the f i l t e r papers after 5 hrs at 70°C in a mechanical convection horizontal a i r flow e l e c t r i c oven. Results were expressed as % expressible water in r e l a t i o n to sample weight. Fat was estimated by the weight of dry matter remaining in the f i l t e r paper after drying. Results were expressed as % expressible fat in r e l a t i o n to sample weight. -75-6. Texture evaluation 6.1. Texture p r o f i l e analysis Texture p r o f i l e analysis (TPA) as described by Bourne (1978) was performed using an Instron (Model 1122) Universal Testing Machine equipped with a 500 kg load c e l l . Two s l i c e s of 1 cm in length per frankfurter and two frankfurters per formulation from the consumer cook test were used. A cross-head speed of 100 mm/min and chart speed of 200 mm/min were used. Each sample was compressed twice using a 5 cm diameter uniaxial f l a t plate to 75% of the frankfurter i n i t i a l height. The output from the Instron was transmitted d i r e c t l y to an IBM-compatible PC through a data ac q u i s i t i o n system and software (PC-LabCard Model PCL-718 and Labtech Acquire software package. Laboratory Technologies Corporation, Wilmington, MA). The raw data (voltage readings) were i n i t i a l l y stored as ASCII f i l e s and transferred for data manipulation to the spreadsheet software Lotus 1-2-3" (Lotus Development Co., Cambridge, MA). The textural parameters of hardness ( f i r s t and second peaks), f r a c t u r a b i l i t y , firmness, springiness, cohesiveness, gumminess and chewiness were calculated from the textural p r o f i l e s using a Lotus 1-2-3 template (see Appendix B). 6.2 Shear force Shear force was measured using a single shear compression c e l l mounted on an Instron (Model 1122) Universal Testing Machine -76-equipped with a 500 kg load c e l l . Three frankfurters from the consumer cook test were used. The ends were removed from each frankfurter and a 3 cm long center core was removed using a No. 7 (1.3 cm internal diameter) cork borer. A cross-head speed of 100 mm/min and chart speed of 200 mm/min were used. Shear was expressed as the maximum force in Newtons required to shear through the cross-section of the frankfurters. The ac q u i s i t i o n system, software and the Lotus 1-2-3 template used for TPA were also used in t h i s t e s t . G. STATISTICAL ANALYSIS S t a t i s t i c a l analysis 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 ignificance were made at the pr o b a b i l i t y l e v e l of ct = 0.05. 1. Regression analysis The average of n determinations per qua l i t y parameter evaluated per rep l i c a t e was used in the regression analysis. The data obtained were treated by multiple regression analysis using least squares methodology. The MGLH module of SYSTAT was used. Scheffe's canonical special cubic for 3 components was the model f i t t e d to data col l e c t e d at each experimental point. Y = 0 i X i + 0 2 X a + OaXa + B a . 2 X 3 . X 2 + 0a. 3XxXa + BaaXsXa + GxaaXiXaXa (23) The (3 parameters are the p a r t i a l regression c o e f f i c i e n t s which when estimated indicate the effects of the ingredients on the - 7 7 -dependent v a r i a b l e (response). The 13 terms with s i n g l e s u b s c r i p t s (e.g. 0a.) r e p r e s e n t s the l i n e a r b l e n d i n g e f f e c t of a s i n g l e mixture component (e.g. Xi) . The 0 terms with double s u b s c r i p t s (e.g. (3 i a) r e p r e s e n t the n o n l i n e a r b l e n d i n g e f f e c t of two components (e.g. XxXa). The (3 term with the t r i p l e s u b s c r i p t ( i . e . 0X23) r e p r e s e n t s the n o n l i n e a r b l e n d i n g e f f e c t caused by b l e n d i n g the three components ( i . e . XiXsXa) ( C o r n e l l , 1981). When 0±d or 0 u * are p o s i t i v e the b i n a r y or t e r n a r y mixture has a s y n e r g i s t i c e f f e c t on the reponse. If i t i s negative the b i n a r y or t e r n a r y mixture has an a n t a g o n i s t i c e f f e c t on the response ( C o r n e l l , 1981). Not a l l 7 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 were r e q u i r e d to adequately d e s c r i b e the e f f e c t of the 3 components on each of the responses ( q u a l i t y parameters e v a l u a t e d ) . A number of s t a t i s t i c a l procedures can be used to determine which of the components do not have a s i g n i f i c a n t e f f e c t on the response and t h e r e f o r e can be e l i m i n a t e d from the model. In t h i s study r e d u c t i o n of the f u l l model f o r each response evaluated was performed i n the f o l l o w i n g manner. A Student's t -t e s t on the i n d i v i d u a l parameters i n each model, using the r a t i o of the estimated 0 c o e f f i c i e n t value and i t s standard e r r o r was computed using SYSTAT. T h i s r a t i o t e s t s the n u l l hypothesis H 0: the parameter i 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 0i. = 0 the a l t e r n a t e hypothesis i s -78-H«: the parameter is s i g n i f i c a n t l y d i f f e r e n t from zero 0 i * 0 If the calculated value of | t | for the estimated 0 c o e f f i c i e n t was equal to or greater than the c r i t i c a l value (to.o»,m,««-P>) / where N was the t o t a l number of observations and p the number of parameters in the model, the n u l l hypothesis was rejected. Thus, i t was concluded that the component associated with the p a r t i a l regression c o e f f i c i e n t tested had a s i g n i f i c a n t e f f e c t on the response and therefore should not be removed from the model (Zar, 1984). The approach that was taken to reduce the f u l l model in t h i s study to arrive at the "best" equation was to remove the least s i g n i f i c a n t higher-degree terms f i r s t s t a r t i n g with the three component term ( i . e . 0123X1X2X3) followed by the two component terms (e.g. 012X1X2) and l a s t l y the one component terms (e.g. 0 i X i ) . After one term was dropped from the model a new regression equation was computed u t i l i z i n g the remaining terms. The n u l l hypothesis was then tested for each p a r t i a l regression c o e f f i c i e n t in the new model and the reduction procedure was repeated u n t i l a l l the estimated 0 c o e f f i c i e n t s in the model were concluded to be s i g n i f i c a n t l y d i f f e r e n t from zero (Zar, 1984). The si g n i f i c a n c e and adequacy of the qu a l i t y prediction models found were assessed by analysis of variance, adjusted multiple c o e f f i c i e n t of determination (R«x) and analysis of residuals (Draper arid Smith, 1981). The n u l l hypothesis to be tested for regression analysis for -79-mixture models is (Marquadt and Snee, 1974; Cornell, 1981): Ho.* the response does not depend on the mixture components (3d = B = > j = 1,2,...,q (linear terms) B J = 0 j = q+l,...,p (other terms) and the alternate hypothesis is H » : the response does depend on the mixture components. D e f i n i t i o n a l and working formulas for multiple regression analysis of variance for mixture models are shown in Table 4. The Fisher F r a t i o for lack of f i t (mean square lack o£ fit/mean square pure error) was used to test the models for s t a t i s t i c a l l y s i g n i f i c a n t lack of f i t (Deming, 1989; Draper and Smith, 1981). This F r a t i o tests the n u l l hypothesis Ho: there is no s i g n i f i c a n t lack of f i t S 1 l»c=K o £ f i t ~ S !pur« a r c o x = 0 where s* stands for variance, and the alternate hypothesis is H « : there is s i g n i f i c a n t lack of f i t If the calculated value of F lack of f i t was equal to or greater than the c r i t i c a l value (Fo.o»,ii),«- P-i.i), the n u l l hypothesis was rejected and i t was concluded that the model appeared to be inadequate. If the F r a t i o for lack of f i t was not s i g n i f i c a n t the model appeared to be adequate and the n u l l hypothesis for regression for the mixture model was tested based on the F - r a t i o of mean square regression to mean square residual ( F-test for regression). The n u l l hypothesis was rejected at <* = 0.05 i f the value of the F - r a t i o was equal to or greater than the c r i t i c a l value (Fa.o»,(u,p-i,ii-p) and i t was concluded that the response - 8 0 -Table 4. D e f i n i t i o n a l and working formulas for multiple regression analysis of variance for mixture models.*-Source of var i a t i o n Degrees of freedom Sum of squares(SS) Mean F square(MS) test Regression p-1 Residual N-p Lack of N-p-1 f i t Pure error 1 Total N-1 SSR= E(y - y )* SSE= Z(y^ - & ) * SSLF= SSE-SSPE SSR/(p-l) MSR/MSE SSE/(N-p) SSLF/(N-p-l) MSLF/MSPE SSPE= El/2(yj. - y^)* SSPE/1 SST= E(y K - y ) « *• p = number of parameters in the model N = t o t a l number of observations ya = predicted response value at the uth observation y = overal l response average of N observations y„ = observed response value at the uth observation y.( and y^ = observed reponse value at the jth design point for replicate 1 and 2 respectively Source: Deming (1989), Draper and.Smith (1981), Cornell (1981) and Marquadt and Snee (1974) -81-depended on the mixture components (Cornell, 1981). The adjusted multiple c o e f f i c i e n t of determination was determined as follows: R**. = 1 - [(SSE/(N-p))/(SST/(N-l))] (24) This s t a t i s t i c has been used for canonical polynomials (models that do not contain a Bo term) as a measure of how much of the va r i a t i o n in the response values is explained by the model. This s t a t i s t i c takes into account the number of parameters (p) in the f i t t e d model. The R 2 A. and the standard error of the estimate were also taken into consideration in deciding the "best" equation. The closer a model's adjusted multiple c o e f f i c i e n t of determination i s to 1, the better the model f i t s the observed responses. The standard error of estimate, S, is an o v e r a l l indication of the accuracy with which the f i t t e d regression equation predicts the dependence of Y on X. The smaller the value of S, the more accurate w i l l be the predictions (Zar, 1984). The models were further checked using the analysis of residuals. SYSTAT performs the analysis of residuals and indicates i f there are o u t l i e r s . When o u t l i e r s were found, the corresponding observations from the 2 replicates were removed from the data, after which the data were reanalyzed without these observations. Residuals were analyzed by examining residual plots (residuals vs. predicted values) with the GRAPH module of SYSTAT. The assumption for residual analysis is that the residuals are independent and follow a normal d i s t r i b u t i o n (Cornell, 1981). The -82-residuals should be seen to be randomly d i s t r i b u t e d about zero. However, i t is d i f f i c u l t to see such d i s t r i b u t i o n with N<30 (Cornell, 1981). If the plots show a deterministic pattern or i f the dispersion of the residuals changes with the predicted values then the assumption underlying the analysis of residuals is violated (Joglekar and May, 1987). 2. Correlation analysis Pearson's c o r r e l a t i o n c o e f f i c i e n t s (r) were computed to compare the relat i o n s h i p between the proximate composition of the formulations and the qual i t y parameters evaluated and between each pair of qu a l i t y parameters evaluated. CORR module of S Y S T A T was used. 3. Response surface contour analysis Contour plots of estimated response values were constructed using the prediction models found. The contour plots were obtained using SYGRAPH a graphics program package (Wilkinson, 1988b). H. OPTIMIZATION METHODS I. Feasible point computer program (FPOINT) To s t a r t the search for the optimum, the Complex method requires an i n i t i a l point that does not vi o l a t e the constraints. In problems with complicated and numerous constraints i t is often -83-very d i f f i c u l t to e s t a b l i s h a feasible point. Box (1965) and Vaessen (1984) described the objective function to be used to determine a feasible point for constraint optimization problems. This function is the sum of the amounts by which each constraint is violated by a given point. Minimization of t h i s function should be performed so that when the function has a value of zero, a feasible point is found. Any optimization technique which does not make use of the continuity of the f i r s t derivatives of the function can be used (Box, 1965). In t h i s study computational simplex optimization (CSO) (Nakai and Arteaga, 1990) which i s based on the Simplex algorithm of Morgan and Deming (1974) was used with the following changes: 1) Lines 38 and 39 were modified to allow for the simplex to stop the search when a response function of zero was found. 2) The function subroutine which st a r t s in li n e 46 was modified. Only the i m p l i c i t constraints are involved in the objective function computation, since the Simplex algorithm takes care of not v i o l a t i n g the e x p l i c i t constraints (upper and lower l i m i t s of the factors) through the prohibit-range-trespassing routine (Nakai and Arteaga, 1990). When more than one s t a r t i n g point was required, either the lower and upper l i m i t s of the factors or i m p l i c i t constraints were narrowed. A flow chart of the function subroutine i s shown in Figure 5. The computer program is l i s t e d in Appendix C. -84-Subroutine s t a r t s 1 i F u n c t i o n = 0 Compute c o n s t r a i n t * 3 value at a c t u a l v e r t e x Is c o n s t r a i n t value higher than upper l i m i t of c o n s t r a i n t i No. Yes F u n c t i o n = F u n c t i o n + c o n s t r a i n t value - upper l i m i t Is c o n s t r a i n t value lower than lower l i m i t of c o n s t r a i n t i , No Yes st_ F u n c t i o n = F u n c t i o n + lower l i m i t - c o n s t r a i n t value No L a s t c o n s t r a i n t Yes * Minimize F u n c t i o n F i g u r e 5. Flow c h a r t of the F u n c t i o n s u b r o u t i n e of FPOINT computer program - 8 5 -2. Formula optimization computer program (FORPLEX) A computer program of the Complex method of Box was written in IBM BASIC based on the FORTRAN program of Kuester and Mize (1973). Modifications to t h i s program were performed to improve i t s e f f i c i e n c y and i t s output and to overcome l i m i t a t i o n s 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 (section C.2.) gives complete description of the Complex method. The formula optimization computer program (FORPLEX) finds the maximum of a nonlinear function of one or more variables, subject to e x p l i c i t and i m p l i c i t (linear or nonlinear) inequality constraints. If minimization of the objective function is required, maximization of the negative of the function (max. of -f(x)) should be performed. Figure 6 shows a detailed flow chart of the FORPLEX algorithm. The computer program is l i s t e d in Appendix D. 2.1. Program description 2.1.1. General description The program consists of a main program, three general subroutines (CONSX, CHECK, CENTR) and two user supplied subroutines (FUNCTION and CONSTRAINT). (A) Main program functions (a) Establishment of the dimension of the following parameters which can be modified according to the requirements of each part 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  / F e a s i b l e s t a r t i n g p o i n t / 1 Generate i n i t i a l complex E x p l i c i t c o n s t r a i n t s v i o l a t e d Yes No K — Move p o i n t i n a d i s t a n c e DELTA i n s i d e t h e v i o l a t e d c o n s t r a i n t J I m p l i c i t c o n s t r a i n t s v i o l a t e d Yes Move 1/2 way towards t h e c e n t r o i d No I n i t i a l complex g e n e r a t e d No X Yes E v a l u a t e o b j e c t i v e f u n c t i o n a t complex p o i n t s F i n d worst and b e s t p o i n t s i n complex Check convergence c r i t e r i a Yes No C a l c u l a t e c e n t r o i d of a l l p o i n t s e x c e p t worst I R e f l e c t worst p o i n t t h r o u g h c e n t r o i d F i g u r e 6. Flow c h a r t of t h e FORPLEX a l g o r i t h m -87-Yes 1 E x p l i c i t c o n s t r a i n t s \ . v i o l a t e d Move p o i n t i n a d i s t a n c e 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 NO > * I m p l i c i t c o n s t r a i n t s v i o l a t e d Move 1/2 way towards the c e n t r o i d Yes No r E v a l u a t e o b j e c t i v e f u n c t i o n No New p o i n t r e p e a t s as worst p o i n t No Has i t been c o n t r a c t e d 5 or more times Yes > Generate a 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 t h r o u g h the b e s t p o i n t - 8 8 -(b) interaction with the user to obtain the following information to st a r t the optimization: (1) number of independent variables (factors) involved in the objective function computation. (2) t o t a l number of constraints (3) complex size (4) maximum number of it e r a t i o n s to be performed. An it e r a t i o n i s defined as the calculations required to select a feasible point that does not repeat in y i e l d i n g the worst function value. Iterations are counted after i n i t i a l complex i s computed. (5) p r i n t i n g procedure (6) convergence c r i t e r i a values: ALPHA, BETA and GAMMA (7) lower and upper l i m i t s of factors (8) lower and upper l i m i t s of i m p l i c i t constraints (9) feasible s t a r t i n g point (c) Generation of random numbers (d) Printout of information entered by the user (e) A l l information gathered i s transferred to the other subroutines (f) Printout of the f i n a l value of the function, the factors coordinates and the t o t a l number of ite r a t i o n s when the solution has converged to within the allowable range (BETA), or when the maximum number of iterati o n s is exceeded The parameters defined in main program are: N, M, K, ITMAX, IC, IPRINT, ALPHA, BETA, GAMMA, DELTA, I, G, H, X, R -89-(B) Subroutine CONSX CONSX is the primary subroutine. It is ca l l e d from the main program and coordinates a l l other subroutines (CHECK, CENTR, FUNCTION, CONSTRAINTS). The computations performed in t h i s subroutine are as follows: (a) computes i n i t i a l complex (b) printout of i n i t i a l complex (c) finds points with worst and best function value (d) checks convergence c r i t e r i a (e) computes r e f l e c t i o n point (f) computes contraction point (g) finds i f r e f l e c t i o n or contraction points repeat as worst points (h) computes new point by r e f l e c t i n g the centroid through the best point (i) printout of function values, factors and centroid coordinates at each i t e r a t i o n i f user desires (j) printout of function value, factors coordinates and a message when no improvement is observed by r e f l e c t i n g the centroid through the best point The parameters used in CONSX are: N, M, K, ITMAX, IT, ALPHA, BETA, GAMMA, DELTA, X, I, R, F, IEVl, IEV2, G, H, XC, IPRINT, LLY, CC, CNT, TT, KODE, K l , KOUNT (C) Subroutine CHECK This subroutine checks against e x p l i c i t and i m p l i c i t constraints and applies correction i f vi o l a t i o n s are found. - 9 0 -(a) i£ e x p l i c i t constraints are violated, moves points in a distance DELTA inside the violated constraint (b) i f i m p l i c i t constraints are violated, contracts the point halfway towards the centroid. The parameter used in CHECK are: N, M, K, X, G, H, I, IEV1, KODE, XC, DELTA, KT, TT, CC (D) Subroutine CENTR This subroutine calculates the centroid of a l l points except the worst point. The parameters used in CENTR are: N, IEV1, XC, X, Kl (E) Subroutine FUNCTION This is a user supplied subroutine where the objective function is s p e c i f i e d . The parameters used in FUNCTION are: X, F, I. (F) Subroutine CONSTRAINT This is a user supplied subroutine where the i m p l i c i t constraints are s p e c i f i e d . The parameters used in CONSTRAINT are: X, I. 2.1.2. Description of parameters N Number of independent variables (optimization factors) M Total number of constraints K Number of points (vertices) in the complex ITMAX Maximum number of ite r a t i o n s IC Number of i m p l i c i t constraints ALPHA Reflection factor -91-BETA Convergence parameter. Maximum allowed difference between best and worst response values GAMMA Convergence parameter. Consecutive it e r a t i o n s with "same" response value DELTA E x p l i c i t constraint v i o l a t i o n correction IPRINT Code used to control 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 printed, IPRINT=0 suppresses printing u n t i l f i n a l solution is obtained X Optimization factors and constraints depended variables R Pseudo-random numbers between 0 and 1 F Objective function IT Iteration number IEVl Point number with worst function value IEV2 Point number with best function value G Lower l i m i t of constraints H Upper l i m i t of constraints XC Centroid I Point number KODE Code used to determine i f i m p l i c i t constraints are provided. KODE=0 when no i m p l i c i t constraints are provided, KODE=l when i m p l i c i t constraints are provided Kl Do loop l i m i t KT Code used to describe i f i m p l i c i t constraints are v i o l a t e d . KT=0 no v i o l a t i o n occurs, KT=1 v i o l a t i o n - 9 2 -occurs TT Code used to determine i f centroid needs to be computed. TT=1 to be computed, TT=0 not to be computed CC Number of contractions performed in each i t e r a t i o n LLY Number of contractions performed when point repeats as worst point CNT Code used to determine i f r e f l e c t i o n through the best point has been performed KOUNT code used to check convergence c r i t e r i a 2.1.3. Summary of user requirements (A) Determine values for N, M, K, ITMAX, IC, ALPHA, BETA, GAMMA, DELTA, IPRINT Recommendations for specifying the following parameters K = 2N ALPHA =1.3 BETA = small number, magnitude of function value times 1 0 - 4 GAMMA = 5 DELTA = small number, magnitude of factor values times 10 ~* (B) Specify upper and lower l i m i t s for e x p l i c i t and i m p l i c i t constraints. If a par t i c u l a r factor or constraint has no upper or lower l i m i t , the user should provide a reasonable estimate of the 1imit. (C) Determine i n i t i a l feasible s t a r t i n g point using FPOINT computer program. (D) Adjust DIM (dimension) statement as necessary (line 20) - 9 3 -(E) Specify objective function in FUNCTION subroutine. The optimization factors should be described as X(I f factor number) and the dependent variable being optimized as F ( l ) . (F) Specify i m p l i c i t constraint functions in CONSTRAINT subroutine. Implicit constraints are functions of the optimization factors. The optimization factors should be described as X (I,factor number) and the i m p l i c i t constraint dependent variables as X(I f factor numbers + 1) 2.1.4. New routines 2.1.4.1. Generation of random numbers Every pseudo-random number generator produces a series of random draws that eventually repeats (Modianos et a l . , 1984). Seeding must always be done to generate a par t i c u l a r sequence of numbers. In IBM BASIC seeding can be done using two functions: (1) RND(A) where A is a negative number and (2) RANDOMIZE A where A is a number between -32768 and 32767. RANDOMIZE function was used since seeding with RND(A) gave i d e n t i c a l sequences no matter what numbers were used. In th i s study RANDOMIZE 3 was used in order to generate the same sequence of pseudo-random numbers in each run. This allowed for mistakes to be followed more e a s i l y and thus correction to the program to be made i f required. For future applications, RANDOMIZE and RANDOMIZE TIMER functions can be used to reseed the random number generator each time the program i s run. In th i s study the term "random number" was used -94-to denote pseudo-random numbers. 2.1.4.2. Reflection through best point The modification to the o r i g i n a l Complex method done by Friedman (1971) and Friedman and Pinder (1972) was added to the Kuester and Mize Complex algorithm. This modification consists of r e f l e c t i n g the centroid, X i c , through the best point, X u ( b e s t ) , after a fixed number of contractions (5) towards the centroid are performed when the r e f l e c t i o n and subsequent contraction points repeat as worst points. This new r e f l e c t i o n p o i n t , X i j , is defined by: X I H = Xi.d(best) + oi(Xi.d (best) - X l o ) (25) 2.2. Limitations of the formula optimization algorithm 2.2.1. Equality constraints Equality constraints cannot be handled as such by this algorithm; instead, they have to be removed. Simple linear equality constraints can be removed by using them to remove a variable (Vaessen, 1984). This can be better explained with the following example. The linear constraint X i + X 2 + X 3 = 1 (26) can be incorporated by replacing in the objective function a l l occurrences of the variable Xa with the expression 1 - Xa. - Xa (27) -95-2.2.2. Endless c o n t r a c t i o n due to i m p l i c i t c o n s t r a i n t s v i o l a t i o n Box (1965) commented that the c o n t r a c t i o n s t e p performed due to v i o l a t i o n of i m p l i c i t c o n s t r a i n t s should be repeated as necessary u n t i l a f e a s i b l e p o i n t i s found. However, i f by chance a l l the p o i n t s on the l i n e from the c e n t r o i d to the r e f l e c t i o n p o i n t v i o l a t e the i m p l i c i t c o n s t r a i n t s , a p p l i c a t i o n of t h i s r u l e causes the p r o j e c t e d p o i n t e v e n t u a l l y to c o i n c i d e with the c e n t r o i d , thus t e r m i n a t i n g the s e a r c h . The formula o p t i m i z a t i o n a l g o r i t h m does not provide a s o l u t i o n to t h i s problem. If t h i s problem occurs endless computation of the c o n t r a c t i o n p o i n t w i l l be done. In t h i s case the search should be terminated by "breaking" ( p r e s s i n g C t r l C or C t r l Break keys) the program. 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 The s u i t a b i l i t y of the modified Complex method ( i . e . , FORPLEX computer program) to l o c a t e the optimum of l i n e a r 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 t h a t were l i n e a r l y c o n s t r a i n e d was f i r s t t e s t e d using c o n s t r a i n e d mathematical models. Mathematical problems r e p o r t e d i n the l i t e r a t u r e and some formulated by the author were used. The r e s u l t s obtained with the FORPLEX program were v e r i f i e d with optimum r e s u l t s given by l i n e a r programming (LINDO software (LINDO Systems, Inc.)) i n the case of l i n e a r 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 c o n s t r a i n e d problems, or by response s u r f a c e a n a l y s i s when 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 - l i n e a r l y c o n s t r a i n e d problems were used. -96-2.4. Optimization of f rankfurter formulations The formula optimization computer program (FORPLEX) was used to obtain "best q u a l i t y " formulations that met predetermined product s p e c i f i c a t i o n s for: (a) proximate composition, (b) ingredient levels and (c) cost. The q u a l i t y prediction equations obtained by f i t t i n g Scheffe's canonical special cubic model to the experimental data co l l e c t e d at the points of an extreme vertices design were used to predict the q u a l i t y of frankfurter formulations at each point computed by the FORPLEX program. "Best q u a l i t y " formulations were defined as those formulations that met the product s p e c i f i c a t i o n s and cost constraints and whose predicted quality was as close as possible to a predetermined target q u a l i t y . Several frankfurter formulation optimization t r i a l s were performed using d i f f e r e n t q u a l i t y parameters as measures of the formulations' q u a l i t y . The q u a l i t y prediction equations required for each optimization t r i a l were entered into the function subroutine of the FORPLEX computer program. The overall response was the minimization of the absolute value of the products of the standardized differences of the predicted q u a l i t y parameters from the target quality values. The linear equations that described proximate composition, cost and the constraint that the sum of the ingredients equal 1 were entered into the constraint subroutine. When a q u a l i t y parameter was considered as a constraint the prediction equation was incorporated in the constraint subroutine. -97-This new formula optimization method was compared with linear programming for formula optimization. In addition, 5 computed optimum formulations found by FORPLEX were compared with optimum formulations found by linear programming. The comparison was made in terms of the q u a l i t y and cost of the computed formulations. Experimental v e r i f i c a t i o n of two computed optimum formulations was performed to test the accuracy of the models to predict the q u a l i t y of these formulations. 3. Formula optimization using linear programming Linear programming was used to find seven least-cost frankfurter formulations that met s p e c i f i c a t i o n s for proximate composition and qu a l i t y . The same ingredient l i m i t s and composition constraints used for the optimization of formulations by the FORPLEX program were used. The t o t a l bind value constraint was used to account for the qu a l i t y required by the formulations in terms of the amount of fat bound per unit weight of formulation. Bind value constants of the ingredients were provided by Sebastian ( 1 9 8 9 ) . The seven computed optimum formulations were compared with optimum formulations found by FORPLEX as discussed above. A l l linear programming calculations were performed using LINDO software (LINDO Systems, Inc.). - 9 8 -RESULTS AND DISCUSSION A. OPTIMIZATION OF CONSTRAINED MATHEMATICAL MODELS The performance of the modified Complex method ( i . e . , FORPLEX computer program) in handling linear and nonlinear objective function problems that were l i n e a r l y constrained was tested. The effe c t of using d i f f e r e n t s t a r t i n g points, random number seeds and |3 values on the effectiveness of locating the optimum by the Complex was evaluated. Published mathematical problems and some formulated by the author were used. In the following discussion the term "Complex" refers to the Complex method, while the term "complex" refers to the f l e x i b l e unit ( i . e . search u n i t ) . The parameters spe c i f i e d at each optimization run were as follows: (a) the convergence parameters ALPHA (a), BETA (|3) and GAMMA (r) were set at 1.3, 0.1 and 5 respectively. In some cases, BETA was set at 0.001 in order to demonstrate the improved accuracy of the re s u l t s ; and (b) the number of points in the complex was set to K=2N, where N is the number of factors. The random number seed used was 3 unless otherwise s p e c i f i e d . - 9 9 -Test problem 1 Test problem 1 i s a nonlinear objective f u n c t i o n - l i n e a r l y constrained problem. Objective function: Maximize Y= Xl*X2*X3 Subject to: E x p l i c i t constraints: 0 < XI < 42 0 < X2 < 42 0 < X3 < 42 Implicit constraint: 0 < XI + 2X2 + 2X3 < 72 This problem was tested by Kuester and Mize (1973) for the performance of the Complex algorithm and has a global maximum equal to 3455.99 at Xl=23.958/ X2=12.007 and X3=12.014. This problem was solved using the random number sequence reported by Kuester and Mize (1973) (Table 5) as well as seeding the random number generator with 4 d i f f e r e n t random number seeds (Table 6). As can be seen in Table 5 the Complex was able to locate the optimum regardless of the i n i t i a l s t a r t i n g point. However, the number of it e r a t i o n s needed to arrive at the optimum depended on the s t a r t i n g point. Table 6 shows the eff e c t of d i f f e r e n t random number seeds on the performance of the Complex. A l l runs were carried out using the same st a r t i n g point (Xl=l, X2=l, X3=l). Convergence to the optimum was achieved in a l l runs. The number of iterati o n s needed to arrive at the optimum was influenced by the random number seed. -100-Table 5. Optimization results of test problem l . * " B ' c Run No. Starting point Computed optimum XI X2 X3 Ymax XI X2 X3 IT D 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 54 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 *• Convergence parameters: ct = 1.3, (3 = 0.1, r = 5 B Number of points in complex K=6 c Random number sequence of Kuester and Mize (1973) D Number of it e r a t i o n s -101-Table 6. Optimization results of test problem 1 using d i f f e r e n t random number seeds B ' c Run Random Computed optimum No. number seed Ymax XI X2 X3 I T D 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 *• Starting feasible point: Xl = l,X2 = l /X3 = l B Convergence parameters: a = 1.3, (3 = 0.1, r=5 e Number of points in complex K=6 D Number of ite r a t i o n s -102-Test problem 7 Test problem 2 is a linear objective f u n c t i o n - l i n e a r l y constrained problem. Objective function: Maximize Y= 2X1 + 5X2 Subject to: E x p l i c i t constraints: 0 < XI < 4 0 < X2 < 3 Implicit constraint: 0 < Xl + 2X2 <8 This i s a simple linear programming problem, whose true optimum Y=19 is at Xl=2 and X2=3 (Wolfe and Koelling, 1983). The optimization results of test problem 2 obtained with 4 s t a r t i n g points are given in Table 7. The s t a r t i n g points were chosen by looking at the figure reported by Wolfe and Koelling (1983). Optimum results obtained using (3 = 0.1 in runs 1, 3 and 5 were close to the true optimum. In run 6 the complex was s t a l l e d at Xl=4, X2=2 and was not able to progress towards the optimum. By se t t i n g (3 to 0.001, the true optimum was found in run 2 using Xl = l , X2 = l as s t a r t i n g point. Setting the |3 parameter to 0.001 allowed the Complex to keep searching for the optimum u n t i l a small difference (0.001) between the best and worst response values was attained, thus increasing the chance of convergence upon the exact location of the optimum. However this did not improve the results obtained when the s t a r t i n g point was Xl=3 and X2 = 2. The number of function evaluations needed to arrive at the optimum depended on the s t a r t i n g point used, and as expected, -103-Table 7. Optimization results of test problem 2. A' B Run No. Starting point Computed optimum XI X2 Ymax XI X2 I T e 1 1 1 18.91 2 .14 2.93 21 2 D 1 1 19 . 00 2.00 3.00 65 3 3 2 18.96 2.07 2.96 19 4 D 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 A Convergence parameters: a=1.3, (3 = 0.1, r = 5, unless otherwise indicated B Number of points in complex K=4 c Number of iterati o n s D Convergence parameters: a = 1.3, 13 = 0.001, r = 5 B Search was s t a l l e d -104-decreasing the (3 value caused an increase in the number of function evaluations ( i . e . iterations) needed to reach the opt imum. Test problem 3 Test problem 3 is a linear objective f u n c t i o n - l i n e a r l y constrained problem. Objective function: Maximize Y= 6X1 + 11X2 Subject to: E x p l i c i t constraints: 0 < XI < 60 0 < X2 < 40 Implicit constraints: 1: 2X1 + X2 < 104 2: XI + 2X2 < 76 This is another simple linear programming problem, whose true optimum Y=440 is located at the intersection of the two i m p l i c i t constraints at Xl=44 and X2=16 (Nakai and Arteaga, 1990). Convergence to the optimum was achieved using four s t a r t i n g points (Table 8), however accuracy of the results was improved when (3 = 0.001 was used in runs 2 and 5 . At |3 = 0.1 runs 3 and 4 needed 30 i t e r a t i o n s to arrive at the optimum while runs 1 and 6 needed 27 and 23 i t e r a t i o n s , respectively. Setting (3 to 0.001 caused an increase in the number of i t e r a t i o n s needed to locate the optimum. -105-Table 8. Optimization results of test problem 3.*"B Run No. Starting point Computed optimum XI X2 Ymax XI X2 IT C 1 20 20 439 . 47 44.03 15.93 27 2 B 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 5 D 15 20 440.00 44.00 16.00 48 6 30 10 439.92 43.85 16 .08 23 Convergence parameters: a=1.3, 13 = 0.1, r=5, unless otherwise indicated B Number of points in complex K=4 ° Number of iterati o n s D Convergence parameters: a = 1.3, (3 = 0.001, r=5 -106-T e s t problem 4 Test problem 4 is a simple least-cost formulation problem of a frankfurter formulation. This problem was used by Nakai and Arteaga (1990) to explain the application of LP in meat formulation problems. The o r i g i n a l problem has been s l i g h t l y modified. The problem i s the following: the most economical frankfurter formulation is sought using 5 ingredients (Xl=beef fronts, X2=beef flanks, X3=pork l o i n , X4=pork fat and X5=mutton). The formulation is subject to the following constraints: 1. the capacity of the production l i n e is 100 kg/day 2. t o t a l meat content must be at least 65% 3. the fat content may be no more than 28% 4. protein content must be greater or equal to 11% 5. mutton meat content may be no more than 9% 6. lean pork content must be greater or equal to 30% Mathematically the problem is stated as follows: Objective function: Minimize Y= 2.5X1 + 1.5X2 + 4X3 + 0.7X4 + X5 Subject to: E x p l i c i t constraints: 0 < XI < 100 0 < X2 < 100 30 < X3 < 100 0 < X4 < 100 0 < X5 < 9 -107-Implicit constraints: 1. capacity: XI + X2 + X3 + X4 + X5 = 100 2. meat content: 65 < XI + X2 + X3 + X5 3. fat content: .047X1 + .221X2 + .075X3 + .947X4 + .198X5 < 28 4. protein c o n t e n t : l l < .221X1 + .16X2 + .207X3 + .009X4 + .176X5 As mentioned in Materials and Methods section G.2.2.I., the Complex cannot work with equality constraints. The equality constraint was removed by replacing the variable X5 with the express ion 100 - XI - X2 - X3 -X4 in the objective function and incorporating this expression as an i m p l i c i t constraint (replacing i m p l i c i t constraint 1) with lower and upper l i m i t s of 0 and 9 respectively. Since the Complex algorithm ( i . e . FORPLEX computer program) requires both lower and upper l i m i t s of constraints, the upper l i m i t for i m p l i c i t constraint 2 was set at 100, the lower l i m i t for i m p l i c i t constraint 3 was set at 0 and the upper l i m i t of i m p l i c i t constraint 4 was set at 100. The optimal solution of this problem was found using LINDO a linear programming computer program. The minimum value of Y i s 208.94 at X1=0, X2=46.56, X3=30, X4=14.44 and X5=9. This optimum l i e s at the intersection of three constraints (X3=30, X5=9 and fat content=28 ) . The optimization results of this test problem are given in Table 9. The factor values have been rounded off to one s i g n i f i c a n t figure for ease of presentation. The four s t a r t i n g -108-Table 9 . Optimization results of test problem 4 . * " B Run Starting point Computed optimum No, X I X2 X3 X4 Ymin X I X2 X3 X4 I T e 1 3 0 . 6 3 0 . 6 3 0 . 6 0 . 0 2 0 8 . 9 5 0 . 0 4 6 . 6 3 0 . 0 14 . 4 307 2 4 3 . 5 6 . 7 3 3 . 7 1 5 . 8 2 0 8 . 9 8 0 . 0 4 6 . 5 3 0 . 0 1 4 . 5 377 3 0 . 1 4 9 . 5 3 0 . 1 1 2 . 3 2 0 8 . 9 4 0 . 0 4 6 . 6 3 0 . 0 14 . 4 226 4 7 . 4 3 9 . 3 3 6 . 8 7 . 5 2 0 8 . 9 5 0 . 0 4 6 . 6 3 0 . 0 1 4 . 5 271 *• Convergence parameters: a = 1 . 3 , 3 = 0 . 0 0 1 , r=5 B Number of points in complex K=8 c Number of ite r a t i o n s - 1 0 9 -points used were found using the FPOINT computer program. in order to obtain accurate results a l l runs were performed with 3=0.001. The Complex was able to locate the true optimum regardless of the s t a r t i n g point, however the results of run 3 were more accurate. The number of iterat i o n s needed to arrive at the optimum was influenced by the s t a r t i n g point used. Compared to test problem 2 and 3 the number of iterations needed to locate the optimum increased due to increased number of factors. Test problem 5 Test problem 5 i s a linear objective f u n c t i o n - l i n e a r l y constrained problem. Figure 7 shows the contour plot of this test problem. Objective function: Maximize Y= 6X1 + 11X2 Subject to: E x p l i c i t constraints. Two l i m i t s on the factors 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: XI + 2X2 < 76 2: 20 < XI + X2 3: 0 < 0.667X1 - X2 < 13.34 The optimal solution was found using LINDO computer program. The maximum value of Y is 440 at Xl=44, X2=16. The optimum is located at the intersection of i m p l i c i t constraint 1 and upper l i m i t of constraint 3. -110-Figure 7 . Contour plot of test problem 5 . Implicit constraints are represented by dotted l i n e s . - I l l -Since the Complex a l g o r i t h m ( i . e . FORPLEX computer program) r e q u i r e s both l o v e r and upper l i m i t s of c o n s t r a i n t s , the lower l i m i t f o r i m p l i c i t c o n s t r a i n t 1 was s e t to 0 and the upper l i m i t of i m p l i c i t c o n s t r a i n t 2 was s e t to 1 0 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 5 using broad l i m i t s on the f a c t o r s are given i n Table 1 0 . The s t a r t i n g p o i n t s used were found using the FPOINT computer program and by l o o k i n g at the contour p l o t of t h i s problem (Figure 7 ) . The r e s u l t s obtained were 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 . In runs 1 and 2 the complex was s t a l l e d on i m p l i c i t c o n s t r a i n t 1 at Xl=34 . 0 2 , X2=20.98 and Xl=35.85, X 2 = 2 0 . 0 7 r e s p e c t i v e l y . The Complex converged c l o s e to the optimum i n runs 3 to 8. However the accuracy of the r e s u l t s was improved using (3 = 0 . 0 0 1 i n runs 4,6, and 8. The number of f u n c t i o n e v a l u a t i o n s depended on the s t a r t i n g p o i n t and (3 v a l u e . Since the i m p l i c i t c o n s t r a i n t s used i n t h i s problem l i m i t the values of the f a c t o r s w i t h i n a narrower range than 0 - 1 0 0 (see F i g u r e 7 ) , the e f f e c t of using more reasonable l i m i t s on the f a c t o r s was s t u d i e d . The r e s u l t s presented i n Table 11 demonstrate t h a t the Complex was able to l o c a t e the optimum r e g a r d l e s s of the s t a r t i n g p o i n t . I t i s important to p o i n t out t h a t when the s t a r t i n g p o i n t Xl = 1 2 , X2=8 was used at both (3 values the Complex a r r i v e d at the optimum, which was not the case when broad l i m i t s on the f a c t o r s were used (Table 1 0 , run 2 ) . No d e f i n i t e trend was observed on the number of i t e r a t i o n s needed to l o c a t e the optimum by - 1 1 2 -Table 1 0 . Optimization results of test problem 5 with broad l i m i t s on the independent variables.*-Run No. Starting point Computed optimum X I X2 Ymax X I X2 I T B 25 15 4 3 4 . 9 4 3 4 . 0 2 2 0 . 9 8 20 2 C D 12 8 4 3 5 . 9 3 3 5 . 8 5 2 0 . 0 7 17 3 d 20 0 439 .91 4 3 . 9 4 1 6 . 0 3 27 4 = 20 0 4 4 0 . 0 0 4 3 . 9 9 1 6 . 0 0 57 5 D 25 10 439 .96 4 3 . 9 6 1 6 . 0 2 32 6 s 25 10 439 . 99 4 3 . 9 9 16 .00 53 7 D 3 5 1 0 . 1 4 3 9 . 9 0 4 3 . 8 0 16 .10 27 8= 35 1 0 . 1 4 4 0 . 0 0 4 3 . 9 9 1 6 . 0 0 53 *• Number of points in complex K = 4 B Number of ite r a t i o n s c Search was s t a l l e d D Convergence parameters: 01 = 1 . 3 , 0 = 0 . 1 , r=5 a Convergence parameters: ct = 1.3, (3 = 0 . 0 0 1 , r=5 - 1 1 3 -Table 11. Optimization results of test problem 5 with narrow l i m i t s on the independent variables Run No. Starting point Computed optimum XI X2 Ymax XI X2 1 12 8 439.59 43.95 15.99 27 2 D 12 8 439 .99 43.99 16 .00 72 3 20 0 439.94 43.93 16.03 30 4 D 20 0 439 .99 . 43.99 16 .00 49 5 2 5 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 s Number of points in complex K=4 c Number of ite r a t i o n s D Convergence parameters: 01 = 1.3, 0 = 0.001, r=5 -114-narrowing the l i m i t s on the factors, increased accuracy o£ the re s u l t s was observed when 13 was set at 0.001. Test problem 6 Test problem 6 was developed to analyze the performance of the Complex using a quadratic objective f u n c t i o n - l i n e a r l y constrained problem. The method for creating symmetric response surfaces reported by Nakai and Arteaga (1990) was followed. The general form for multifactor equations i s Y=(Xl-a) 2+(X2-b) 2; Y w i l l assume the minimum value of 0 when Xl=a and X2=b. Using the same constraints as test problem 5, the minimum response value of 0 was required to be at Xl=25 and X2=10. Figure 8 shows the contour plot of t h i s test problem. Objective function: Minimize Y= (Xl-25) 2 + (X2-10) 2 Subject to: E x p l i c i t constraints: 0 < XI < 100 0 < X2 < 100 Implicit constraints: 1: XI + 2X2 < 76 2: 20 < XI + X2 3: 0 < 0.667X1 - X2 < 13.34 Table 12 contains the optimization results of test problem 6. The Complex varied in i t s success depending on the s t a r t i n g point used. Note that the optimum obtained in runs 1, 2, 3 and 6 (when 0 = 0.1) was not exactly 0. Decreasing 13 to 0.001 in run 4 improved the accuracy of the r e s u l t s . The number of itera t i o n s needed to -115-0 10 20 30 40 50 X1 F i g u r e 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 a n d 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 b y s t r a i g h t l i n e s . -116-Table 12. Optimization results of test problem 6 . * " B Run No. Starting point Computed optimum XI X2 Ymin XI X2 I T C 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 4 D 20 0 0.0006 24 .98 10.01 50 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 B Number of points in complex K=4 c Number of ite r a t i o n s D Convergence parameters: a=1.3, 0=0.001, r=5 -117-locate the optimum was influenced by the s t a r t i n g points. Increased number of it e r a t i o n s were needed when (3 = 0.001. Test problem 7 Maximization of the objective function of test problem 6 was performed using the same constraints. The contour plot (Figure 8) c l e a r l y shows that the maximum of th i s function within the feasible area is located at the intersection of i m p l i c i t constraint 1 and the upper l i m i t of constraint 3 at Xl= 4 4 , X2=16. At this point the function value is Y=397. The results presented in Table 13 demonstrate that with the st a r t i n g points used in runs 1, 2, 5 and 6, the Complex f a i l e d to converge to the optimum, p a r t i c u l a r l y in run 5 where the complex was s t a l l e d at the boundary of constraint 1. Optimization results of runs 3 and 4 were close to the optimum, the accuracy of the optimum was improved by setting (3 to 0.001 in run 4 . Test problem 8 Test problem 8 was developed in a similar manner to test problem 6 except that the minimum value of the function (Y=0) was required to be located outside the feasible area (Xl=50 , X2=25). Objective function: Minimize Y= (XI - 50)* + (X2-25)* Subject to: E x p l i c i t constraints: 0 < XI < 100 0 < X2 < 100 -118-Table 13. Optimization results of test problem 7.*-'s Run No. Start ing point Computed optimum XI X2 Ymax XI X2 IT C 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 4 D 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 x Convergence parameters: 01=1.3, 3 = 0.1, r=5, unless otherwise indicated B Number of points in complex K=4 G Number of ite r a t i o n s D Convergence parameters: ct = 1.3, 3 = 0.001, r=5 E Search was s t a l l e d -119-Implicit constraints: 1: XI + 2X2 < 76 2: 20 < XI + X2 3: 0 < 0.667X1 - X2 < 13.34 As can be seen in Figure 9 the minimum value of the objective function within the feasible area is Y=117 at Xl=44 and X2=16, this optimum point lying at the intersection of i m p l i c i t constraint 1 and the upper l i m i t of constraint 3. The optimization results for test problem 8 obtained using 5 di f f e r e n t s t a r t i n g points are given in Table 14. Except for run A, where the complex was s t a l l e d at the boundary of constraint 1, the Complex was able to locate the optimum. No difference was observed in the accuracy of the results using d i f f e r e n t 3 values when the s t a r t i n g point was set at Xl=25, X2=10. The number of iteratio n s needed to locate the optimum depended on the s t a r t i n g point and 3 value. Test problem 9 Maximization of the objective function of test problem 8 was performed using the same constraints. The contour plot shown in Figure 9 reveals that the maximum function value within the feasible area is located at the intersection of the lower l i m i t of i m p l i c i t constraints 3 and 2 at Xl=12 and X2=8 where Y=1733.10. Table 15 contains the optimization results of test problem 9. Convergence to the optimum depended on the s t a r t i n g point used. The Complex f a i l e d to converge to the optimum in runs 3 and 4, -120--121-Table 14. Optimization results of test problem 8,*"» Run No. Starting point Computed optimum XI X2 Ymin XI X2 IT C 1 25 10 117.03 43.99 16.00 38 2 D 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 5 - 34 20 117.06 43.98 16 .01 71 6 35 10.1 117.11 43.99 16 . 00 41 *• Convergence parameters: ot = 1.3, (3 = 0.1, r = 5, unless otherwise ind icated B Number of points in complex K=4 c Number of iterati o n s D Convergence parameters: a=1.3, 13 = 0.001, r=5 * Search was s t a l l e d -122-Table 15. Optimization results of test problem 9.*'B Run No. Starting point Computed optimum XI X2 Ymax XI X2 IT C 1 25 10 1732.89 12.00 8.00 67 2 20 0 1732.98 12 . 00 8.00 99 3 22 . 5 15 1713.22 12.48 7 . 52 52 4 34 20 1725.36 12. 08 8.05 70 5 35 10.1 1732.91 12.00 8.00 60 6 D 35 10.1 1733.08 12.00 8.00 102 *• Convergence parameters: ot = 1.3, 0 = 0.1, r=5, unless otherwise indicated B Number of points in complex K=4 c Number of ite r a t i o n s D Convergence parameters: a=1.3, 0=0.001, r=5 -123-however the Complex was successful in locating the optimum in runs 1, 2, 5 and 6. Setting 3 to 0.001 with s t a r t i n g point Xl=35, X2=10.1 increased the accuracy of the r e s u l t s . The number of iter a t i o n s needed to locate the optimum depended on the s t a r t i n g point and p value. Test problem 10 Test problem 10 was developed to analyze the performance of the Complex when a quadratic objective function with an interaction term and linear i m p l i c i t constraints are used. Objective function: Minimize Y= 725 - 50X1 - 20X2 +X1* + X2*-X1X2 Subject to: E x p l i c i t constraints: 0 < XI < 100 0 < X2 < 100 Implicit constraints: 1: XI + 2X2 < 76 2: 20 < XI -I- X2 3: 0 < 0.667X1 - X2 < 13.34 It is d i f f i c u l t to l o c a l i z e the true optimum of this function within the feasible area just by analyzing Figure 10. The optimum should l i e at the boundary of i m p l i c i t constraint 1 closer to the corner Xl=32.56, X2=21.72 where the function value is Y=-512.695. As the results in Table 16 reveal, the Complex did not converge to a single point. However, since the accuracy of the results is expected to be greater when 3=0.001, i t could be that in run 2 the Complex found the minimum at Xl=33.13, X2=21.44 where Y=-513.29. Results of runs 1, 3 and 5 were close to t h i s minimum. -124-Figure 10. Contour plot of test problems 10 and 11. Implicit constraints are represented by straight l i n e s . -125-Table 16. Optimization results of test problem 10.*" B Run No. Starting point Computed optimum XI X2 Ymin XI X2 IT e 1 25 10 -513 .28 33 .17 21. 42 15 2 D 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 *• Convergence parameters: a = 1.3, 3 = 0.1, r = 5, unless otherwise indicated B Number of points in complex K=4 c Number of it e r a t i o n s D Convergence parameters: <x = 1.3, 3 = 0.001, r=5 -126-The number of function evaluations needed to arrive at the optimum depended on the s t a r t i n g point and (3 value. Test problem 11 Maximization of the objective function of test problem 10 was performed. It appears from the contour plot in Figure 10 that the maximum of t h i s function (Y=125) l i e s at the intersection of constraint 2 and the upper l i m i t of constraint 3 at Xl=20 and X2 = 0. The optimization results for test problem 11 obtained using 4 d i f f e r e n t s t a r t i n g points are given in Table 17. Except for run 2, where the complex was s t a l l e d at the intersection of constraint 2 and lower l i m i t of constraint 3, the Complex was able to locate the optimum. Accuracy of the results was improved using 13 = 0.001 when the s t a r t i n g point was set at Xl = 32, X2 = 20. The number of function evaluations needed to arrive at the optimum depended on the s t a r t i n g point and 3 value. As can be observed from the results obtained, t h i s modified Complex algorithm was able to locate the optimum of linear and nonlinear objective function problems that were l i n e a r l y constrained. Although Box (1965) stated that the Complex method is able to locate the optimum in nonlinear problems whose feasible area is convex, the Complex seems to work s a t i s f a c t o r i l y in linear problems whose feasible area is a planar surface. However, the Complex method is not an e f f i c i e n t method to solve t h i s type of problem. Linear programming is d e f i n i t e l y more -127-Table 17. Optimization results of test problem l l . * - ' B Run No. Start ing point Computed optimum XI X2 Ymax XI X2 IT C 1 25 10 124.89 20 .00 0 .002 58 2 D 25 15 76.94 12.00 8.004 38 3 40 15 124.89 20.00 0.003 74 4 32 20 124.87 20.00 0. 003 62 5 s 32 20 125.00 20.00 0.000 109 *• Convergence parameters: ct = 1.3, (3 = 0.1, r = 5, unless otherwise indicated B Number of points in complex K=4 c Number of ite r a t i o n s D Search was s t a l l e d E Convergence parameters: a = 1.3, (3 = 0.001, r=5 -128-e f f i c i e n t than the Complex method f o r t h i s type of problem. For no 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 are l i n e a r l y c o n s t r a i n e d the Complex method seems to be an e f f e c t i v e o p t i m i z a t i o n method and t h e r e f o r e s u i t a b l e to be used f o r formula o p t i m i z a t i o n a p p l i c a t i o n s . G e n e r a l l y , the Complex d i d not converge to a s i n g l e p o i n t , but converged on an area c l o s e to the optimum. The convergence parameter 0 had to be s e t to a value of 0.001 i n order to incr e a s e the accuracy of the o p t i m i z a t i o n r e s u l t s . For formula o p t i m i z a t i o n a p p l i c a t i o n s high accuracy i n the r e s u l t s i s not e s s e n t i a l . Weighing of the i n g r e d i e n t s e x a c t l y i s somewhat d i f f i c u l t , thus the rounding of i n g r e d i e n t s weights given by the FORPLEX computer program i s l i k e l y to take p l a c e . When using a s p e c i f i c random number seed, i t was necessary to use s e v e r a l s t a r t i n g p o i n t s as we l l as d i f f e r e n t values of 0 in order t o check t h a t the optimum had i n f a c t been l o c a t e d . When only one s t a r t i n g p o i n t was a v a i l a b l e , the random number generator had to be reseeded i n each o p t i m i z a t i o n search and s e v e r a l runs had to be performed to check whether the optimum had been found. -129-B. Development of ingredient-quality relationships for a 3-ingredient model frankfurter formulation 1. Proximate analysis The proximate composition of the beef meat, mechanically deboned poultry meat (MDPM) and pork fat are shown in Table 18. Protein and moisture content were greatest in beef and fat content was greatest in pork f a t . Proximate analysis of the raw ingredients was performed to obtain the moisture, protein and fat content of the meat blocks corresponding to the 10 formulations given by the extreme vertices design (Tables 2 and 3). The moisture-to-protein r a t i o of the meat block of each formulation was computed to determine the amount of water to be added in the form of ice to maintain the moisture-to-protein r a t i o in the raw emulsions equal to four. Varying the amount of added water to make up a constant moisture-to-protein r a t i o in the raw emulsions rather than using a constant per cent of added water based on t o t a l meat block weight was chosen to avoid formulations having not only d i f f e r e n t moisture and protein contents but also d i f f e r e n t proportions of moisture and protein content. This r a t i o was based on American government regulations which specify that moisture content should not exceed four times the percentage of protein plus 10% of the finished weight (Pearson and Tauber, 1984a). Total moisture content of each formulation was computed -130-Table 18. Proximate composition and pH of raw ingredients.*-Ingredient Moisture 8 P r o t e i n 0 ( % ) ( % ) F a t D (%) pH1 Before After frozen storage Beef meat 73.64+0.64 MDPM5" 65.69±1.42 Pork fat 14.58+1.22 22.49±0.17 15.51+0.12 4.96±0.02 3.72 18.67 80.37 5.51 6 .69 n. d. 5.21 6 . 49 n. d. *• Values are mean + standard deviation B Mean of six samples of beef meat and MDPM and of four samples of pork fat c Mean of three samples D Mean of two samples = pH of meat ingredients was determined before and after six months frozen storage at -30°C. Values are means of two samples *" Mechanically deboned poultry meat n.d.= not determined -131-(moisture content of meat block plus weight of added water) to determine the amount of s a l t to be added. In t h i s study a 2.5% constant e f f e c t i v e s a l t concentration was maintained in the frankfurter formulations. The use of e f f e c t i v e s a l t concentration rather than overa l l s a l t concentration ( i . e . % s a l t based on t o t a l meat block weight) was an important consideration when comparing formulations of d i f f e r e n t composition. If an overall s a l t concentration were used, formulations having higher levels of fat content would show high s a l t concentrations in the meat aqueous phase. It is well documented that the functional properties of meat products are determined by the concentration of s a l t s in the aqueous phase (Trout and Schmidt, 1987; Acton et a l . , 1983). As mentioned by Acton et a l . (1983) a 4% e f f e c t i v e s a l t concentration is generally necessary to ensure good f u n c t i o n a l i t y . Trout and Schmidt (1986) reported that using e f f e c t i v e s a l t concentrations above 2.9% had no b e n e f i c i a l e f f e c t on the water binding a b i l i t y of comminuted meat products. The pH of the raw meat ingredients was determined before and after six months frozen storage at -30OC (Table 18). Mechanically deboned poultry meat had a higher pH than beef meat. It has been reported that mechanical deboning yie l d s a product with a higher pH than hand deboning due to the presence of bone marrow which has a pH in the range of 6.8 to 7.4 (Harding Thomsen and Zeuthen, 1988). The pH of the meats apparently decreased with length of time in frozen storage. Powrie (1973) and Fennema (1973) reported that changes in the pH of meat during frozen storage occur -132-because of increasing concentration of s a l t s in the unfrozen phase. These changes are dependent on storage temperature, s a l t composition, physiological state, buffering capacity of proteins and enzyme action (Powrie, 1973). 2. Quality parameters evaluated S p e c i f i c q u a l i t y parameters were evaluated as a measure of the q u a l i t y of the frankfurter formulations. The q u a l i t y parameters evaluated can be divided into: (a) product weight loss during processing and storage, (b) s t a b i l i t y of the raw emulsions to thermal treatment, (c) juiciness c h a r a c t e r i s t i c s of the cooked frankfurters, (d) textural c h a r a c t e r i s t i c s of the cooked frankfurters, and (e) pH of the raw emulsions. Weight loss of processed meats has always been of great concern to meat packers not only from the standpoint of y i e l d and economic importance but because i t affects the q u a l i t y attributes of the finished products, e.g. texture, j u i c i n e s s , flavour and over a l l a c c e p t a b i l i t y . (Wierbicki. et a l . , 1957 ). Weight loss can be regarded as an indicator of the s t a b i l i t y of the meat product to thermal treatment and storage. Weight loss measurements are performed on the actual meat product while emulsion s t a b i l i t y measurements are performed on small samples under processing conditions that simulate p i l o t plant or commercial scale production procedures. Emulsion s t a b i l i t y methods y i e l d rapid and valuable information regarding the amount of fat and water that can be l o s t by the actual meat product during thermal treatment. -133-Texture and juiciness c h a r a c t e r i s t i c s of meat products play a v i t a l role in consumer acceptance. Juiciness, cheviness and firmness are attributes that consumers use to describe their preference for commercial frankfurters (Lee and Patel, 1984; Voisey et a l . , 1975). Although sensory evaluation is needed to determine the a c c e p t a b i l i t y or preference of a meat product, objective methods for texture and juiciness evaluation were used in t h i s study. 2.1. Product weight loss at d i f f e r e n t stages of the frankfurter  preparation process As mentioned by Brown and Ledward (1987), product weight loss is synonymous with product s t a b i l i t y . Total weight loss is a measure of the unbound water, fat and so l i d s l o s t during processing and storage. Weight loss, and thus product s t a b i l i t y , are affected by the composition of the formulation and by the processing and storage conditions. Functional properties of the meat proteins, i . e . water- and fat-binding capacity and gelation a b i l i t y , play a v i t a l role in s t a b i l i z i n g the cooked product. Data for product weight loss at d i f f e r e n t stages of the frankfurter preparation process are given in Table 19. Results reported for these qu a l i t y parameters were obtained from the weight loss of 7 sausages per formulation. Per cent weight loss after thermal treatment (Shrink) varied in the f i r s t r e p l i c a t i o n from 7.94 to 10.45% with a mean of 9.12% and in the second r e p l i c a t i o n from 8.15 to 9.89% with a mean of 8.99%. The overall mean for Shrink for the 20 formulations -134-Table 19. Experimental data for product weight loss at d i f f e r e n t stages of the frankfurter preparation process. Replication Formulation Shrink Vacuum shrink Cook shrink No. No. A (%) B (%) D 1 1 8.30 1.68 4.12 2 9 .78 1.92 5. 36 3 8 .11 1. 87 4.93 4 7.94 1.38 5.21 5 10.23 1. 87 6 .22 6 9.65 1.92 5.74 7 8.62 2.01 14 . 59 8 8.41 1.73 7.65 9 9 . 71 2.39 6.47 10 10. 45 2.04 7.10 mean 9 .12 1.88 6.74 2 1 9 . 39 1.05 4.95 2 9 .04 1.56 4 . 01 3 8 . 31 1.94 7 . 32 4 8.65 1.05 13 .98 5 9 . 57 2.02 9.36 6 9 .48 2 . 25 8 . 54 7 8 .23 1.44 7.27 8 8 .15 1. 39 7.20 9 9 . 89 1.77 8.90 10 9.15 2.38 6 .65 mean 8.99 1.69 7 . 82 x Ingredients proportions are given in Table 2 B Per cent weight loss after processing (equation 20) c Per cent weight loss after 13 days under vacuum packaged storage (equation 21) D Per cent weight loss after the consumer cook test (equation 22) -135-evaluated was 9.06%. As mentioned by Trout (1988) to obtain accurate r e s u l t s , control of cooking temperature is c r i t i c a l . In th i s study, accurate control of cooking temperature during the second stage of the thermal treatment, i . e . steam cooking, was d i f f i c u l t to achieve due to the nature of the cooking utensils used. Therefore, the v a r i a t i o n observed between the r e p l i c a t i o n s can be attributed to the thermal treatment in addition to the normal v a r i a b i l i t y of the meat ingredients. Furthermore, as mentioned by Whiting (1987a) Shrink values do not r e f l e c t accurately product weight loss since fat cookout, located between the frankfurters and the casings, was considered product y i e l d and not product loss. Per cent weight loss after 13 days under vacuum packaged storage (Vacuum shrink) varied in the f i r s t r e p l i c a t i o n from 1.38 to 2.39% with a mean of 1.88%, and in the second r e p l i c a t i o n from 1.05 to 2.38% with a mean of 1.69%. The overa l l mean for Vacuum shrink for the 20 formulations evaluated was 1.78%. Per cent weight loss after the consumer cook test (Cook shrink) varied in the f i r s t r e p l i c a t i o n from 4.12 to 14.59% with a mean of 6.74%, and in the second r e p l i c a t i o n from 4.01 to 13.98% with a mean of 7.82%. The overa l l mean for Cook shrink for the 20 formulations evaluated was 7.28%. It i s r e a d i l y apparent that the r e p l i c a b i l i t y of Vacuum shrink and Cook shrink data is poor. A possible explanation could be that, in addition to the inherent v a r i a b i l i t y of the meat ingredients and the comminution and s t u f f i n g steps, the -136-v a r i a b i l i t y introduced by the thermal treatment affected vacuum shrink r e s u l t s , and these factors affected Cook shrink r e s u l t s . 2.2. Emulsion s t a b i l i t y analysis The s t a b i l i t y of meat emulsions seems to be affected by a number of interrelated factors. These include: (a) compositional factors such as the amount of s a l t s , amount and type of fat and meat ingredients, nonmeat additives, and fat:protein:moisture r a t i o s ; (b) processing factors such as severity and end point temperature of comminution and heat treatment, and (c) the quantity and qual i t y of protein in terms of s o l u b i l i t y , vater-and fat-binding capacity, and gelation a b i l i t y (Brown and Ledward, 1987; Asghar et a l . 1985; Sofos, 1983a). The s t a b i l i t y of raw emulsions to thermal treatment was evaluated using two emulsion s t a b i l i t y methods. The s t a b i l i t y of the meat emulsions evaluated by the modified method of Saf f l e et a l . ( 1967 ) determined the extent of fat released by heating the emulsions at 80°C. As mentioned by Whiting (1987a) the amount of fat released during emulsion s t a b i l i t y tests indicates whether the finished product would present fat caps inside the casing, or would r e s u l t in fat losses when reheated by the consumer. Per cent fat released (ES) data obtained by the modified Saffle method are presented in Table 20. ES values are the mean from two observations. ES values varied in the f i r s t r e p l i c a t i o n from 0.00 to 2.45% with a mean of 0.51%, and in the second r e p l i c a t i o n from 0.00 to 1.45% with a mean of 0.37%. The ov e r a l l mean for ES for -137-Table 20. Experimental data for emulsion s t a b i l i t y analysis.*-Replication Formulation B ES C Tmloss D Twloss* No. No. (%) (%) (%) 1 1 0.25 31.27 20.19 2 0.20 32.21 21.75 3 0.60 36.77 21.86 4 0.45 38.02 20.66 5 0.05 37.01 23.71 6 0.00 32.66 22.52 7 2.45 46.86 25.86 8 0.80 45.03 25.18 9 0.25 42.49 26.90 10 0.00 42.21 29.84 mean 0.51 38.46 23.85 2 1 0.00 34.74 22.42 2 0.00 30.02 20.26 3 1.45 37.30 22.17 4 0.45 34.17 18.57 5 0.30 39.54 25.33 6 0.00 37.66 25.97 7 0.90 39.28 21.68 8 0.55 40.90 22.87 9 0.00 42.13 26.67 10 0.00 40.46 28.60 mean 0.37 37.62 23.45 *• Results reported are the mean from 2 observations B Ingredients proportions are given in Table 2 c By the modified method of Saf f l e et a l . (1967) and expressed as per cent fat released after thermal treatment D By the modified method of Townsend et a l . (1968) and expressed as per cent water loss per moisture content of the meat block (equation 28) 8 5 By the modified method of Townsend et a l . (1968 ) and expressed as per cent weight loss after thermal treatment (equation 29) -138-the 20 formulations evaluated was 0.44%. It is important to point out that low ES values, indicate high emulsion s t a b i l i t y . R e l i a b i l i t y of ES data is questionable since r e p e a t a b i l i t y of the r e s u l t s of duplicate observations within a given formulation and among re p l i c a t i o n s was poor. Saffle et a l . (1967) reported that r e p e a t a b i l i t y of the method was poor when more than 0.8% fat was released. Comer (1979) and Bawa et a l . (1988) also reported low r e p r o d u c i b i l i t y of s t a b i l i t y tests and attributed t h i s to variations that occur during the preparation of the raw emuls i ons. The second emulsion s t a b i l i t y method used was the modified method of Townsend et a l . (1968). As mentioned in the Material and Methods section, t h i s method aims at evaluating the s t a b i l i t y of the emulsions when held at 80OC by determining the extent of fat and water released. The amount of water released i s a measure of the tendency of a frankfurter emulsion to lose water during thermal treatment and is related to the water holding capacity of the meat emulsion. As mentioned above, the amount of fat released indicates whether a product would present fat caps or r e s u l t in fat losses when reheated by the consumer and i t is related to the fat binding capacity of the meat emulsion (Whiting, 1987a). Results could not be expressed as suggested by Townsend et a l . (1968) since preliminary experiments showed that the f l u i d released during heating of the emulsions never developed a l i p i d layer that could be measured accurately. The amount of fat released by this method was less than the amount released by the -139-modified method of s a f f l e et a l . (1967). This difference could be explained by the fact that the l a t t e r method, in addition to heating the raw emulsions, makes use of c e n t r i f u g a l force. This external force helps release loosely bound f a t . As indicated by preliminary experiments the loss of water from the cooked emulsions appeared to be the major contributing factor leading to the decrease in emulsion s t a b i l i t y . Emulsion s t a b i l i t y results using the modified method of Townsend et a l . (1968) were expressed in two d i f f e r e n t ways based on the weight lost from the cooked emulsions: A) Assuming the weight loss was due s o l e l y to water loss, emulsion s t a b i l i t y values were reported as water loss as a percentage of o r i g i n a l moisture content of the meat block, (Tmloss). Tmloss (%) was defined as water loss after thermal treatment x 100 (28) moisture content of meat block The r a t i o of unbound water after thermal treatment to moisture content has been used as a measure of the water holding capacity of meat products (Wierbicki et a l . 1957). Therefore, Tmloss can be considered also a measure of the water holding capacity of the meat emulsions. M i t t a l et a l . (1987), Mast et a l . (1982) and Wierbicki et a l . (1957) followed the same procedure as that of the Townsend et a l . (1968) emulsion s t a b i l i t y test to determine the water holding capacity of comminuted meat products. B) As proposed by Sofos (1983a) emulsion s t a b i l i t y values were also expressed as weight loss as a percentage of the raw emulsion -140-weight, (Twloss). Twloss (%) was defined as emulsion weight loss after thermal treatment x 100 (29) weight of emulsion before thermal treatment Emulsion s t a b i l i t y data obtained by the modified method of Townsend et a l . (1968) is presented in Table 20. Results reported are the mean from 2 observations. Tmloss varied in the f i r s t r e p l i c a t i o n from 31.27 to 46.86% with a mean of 38.46%, and in the second r e p l i c a t i o n from 30.02 to 42.13% with a mean of 37.62%. The ove r a l l mean for Tmloss for the 20 formulations evaluated was 38.04%. Twloss values varied in the f i r s t r e p l i c a t i o n from 20.19 to 29.84% with a mean of 23.85% and in the second r e p l i c a t i o n from 18.57 to 28.60% with a mean of 23.45%. The overall mean for Twloss for the 20 formulations evaluated was 23.65%. It is important to point out that low Tmloss and Twloss values indicate high emulsion s t a b i l i t y . 2.3. Juiciness c h a r a c t e r i s t i c s of the cooked frankfurters As mentioned by Lee and Patel (1984), juiciness appears to be the q u a l i t y attribute which determines the overall a c c e p t a b i l i t y of commercial frankfurters. Juiciness as perceived by the consumer i s believed to be influenced by the quantity, the rate of release and the composition of the f l u i d expressed upon mastication (Lee and Patel, 1984). An instrumental method reported by Lee and Patel (1984) was used as an objective measurement of the juiciness c h a r a c t e r i s t i c s of the cooked -141-frankfurters. The juiciness c h a r a c t e r i s t i c s measured were: (a) % expressible f l u i d (Exfluid) (b) % expressible water (Exwater) and (c) % expressible fat (Exfat) of the cooked frankfurters. Lee et a l . (1987) and Lee and Patel (1984) reported that sensory juiciness c h a r a c t e r i s t i c s correlated well with these instrumental juiciness c h a r a c t e r i s t i c s . Data for the juiciness c h a r a c t e r i s t i c s of the cooked frankfurters are given in Table 21. Results reported are the mean from 4 observations. Coefficients of va r i a t i o n were computed for each formulation within each r e p l i c a t i o n . E x f l u i d varied in the f i r s t r e p l i c a t i o n from 8.66 to 14.75% with a mean of 11.01%, and in the second r e p l i c a t i o n from 7.62 to 13.04% with a mean of 10.04%. The ove r a l l mean for E x f l u i d for the 20 formulations evaluated was 10.53%. In the f i r s t r e p l i c a t i o n the c o e f f i c i e n t s of v a r i a t i o n varied from 3.1 to 13.1% and in the second r e p l i c a t i o n from 2.7 to 12.4%. Exwater varied in the f i r s t r e p l i c a t i o n from 4.59 to 9.70% with a mean of 7.00%, and in the second r e p l i c a t i o n from 3.47 to 10.83% with a mean of 6.44%. The overall mean for Exwater for the 20 formulations evaluated was 6.72%. In the f i r s t r e p l i c a t i o n the c o e f f i c i e n t s of va r i a t i o n varied from 4.5 to 13.1% and in the second r e p l i c a t i o n from 4.2 to 14.5%. Exfat varied in the f i r s t r e p l i c a t i o n from 2.02 to 9.44% with a mean of 4.02%, and in the second r e p l i c a t i o n from 1.71 to 6.71% with a mean of 3.60% The overall mean for the 20 formulations evaluated was 3.81%. In the f i r s t r e p l i c a t i o n the c o e f f i c i e n t s of -142-Table 21. Experimental data for juiciness c h a r a c t e r i s t i c s of the cooked frankfurters.*-Replication Formulation 1 3 E x f l u i d 0 Exwater 0 Exfat 1* No. No. (%) (%) (%) 1 1 10. 2 10. 3 10. 4 9. 5 8. 6 9. 7 14. 8 13. 9 11. 10 11. mean 11. 2 1 7. 2 7. 3 7. 4 10. 5 9. 6 9 . 7 10. 8 11. 9 13. 10 12. mean 10. 41(13 .1) 6 .88(8. 5) 3 .53(31 .1) 06(3. 9) 7 .38(4. 5) 2 .68(8. 8) 19(3. 1) 5 .94(7. 0) 4 .25(6. 3) 19(3. 4) 4 .59(7. 4) 4 .59(3. 2) 66( 5. 2) 6 .28(5. 7) 2 .38(5. 6) 78(8. 0) 7 .76(8. 5) 2 .02(9. 8) 75(3. 1) 5 .31(8. 6) 9 .44(3. 5) 42(6. 8) 7 .50(9 . 2) 5 .93(4. 1) 87(6. 0) 8 .65(5. 1) 3 .23(10 .3) 80(12 .3) 9 .70(13 .1) 2 .10(9. 3) 01 7 .00 4 .02 62(5. 0) 5 .37(4. 2) 2 .25(7. 4) 65(11 .2) 5 .78(11 .2) 1 .87(12 .6) 68(6. 9) 4 .32(8. 3) 3 .36(7. 4) 18(4. 5) 3 .47(14 .5) 6 .71(2. 2) 75(6. 0) 6 .18(6. 9) 3 .57(4. 5) 21(6. 3) 7 .04(8. 2) 2 .17(5. 2) 97(2. 7) 5 .66(7. 0) 5 .31(6. 9) 72(4. 1) 6 .79(8 . 1) 4 .93(8. 7) 04(8. 4) 8 .92(10 .8) 4 .12(9. 2) 54 (12 .4)10 .83(11 .8) 1 .71(20 .6) 04 6 . 44 3 .60 * Results reported are the mean ( c o e f f i c i e n t of variation) from 4 observations B Ingredients proportions are given in Table 2 c Per cent expressible f l u i d B Per cent expressible water B Per cent expressible fat -143-v a r i a t i o n varied from 3.2 to 31.1% and in the second r e p l i c a t i o n from 2.2 to 20.6%. From these results i t is evident that there i s a r e l a t i v e l y large v a r i a t i o n in the juiciness c h a r a c t e r i s t i c s measured within r e p l i c a t i o n s . Manual processing of the emulsions, thermal treatments, storage conditions and the d i f f e r e n t steps involved in the evaluation of juiciness c h a r a c t e r i s t i c s a l l contribute to the v a r i a b i l i t y observed. The v a r i a b i l i t y observed in the formulations cannot be compared with the results of Lee and Patel (1984) since the authors did not report the r e p r o d u c i b i l i t y of th i s method. 2.4. Textural parameters of the cooked frankfurters Thermal processing of frankfurter emulsions results in the transformation from a viscous sol to a r i g i d and e l a s t i c s o l i d -l i k e structure that can be viewed as a protein gel where fat and water are p h y s i c a l l y and chemically s t a b i l i z e d (Saliba et a l . , 1987; Acton et a l . , 1983). It is the gel-forming proteins, actomyosin and myosin, that act as texture and structure building components in the meat products (Saliba et a l . , 1987; Montejano et a l . , 1985). Frankfurter texture i s affected by composition, gelation a b i l i t y of meat proteins, method of comminution and temperature-time conditions during thermal processing (Singh et a l . , 1985). Textural c h a r a c t e r i s t i c s are important factors in the o v e r a l l q u a l i t y of frankfurters and play a v i t a l role in consumer acceptance (Siripurapu et a l . , 1987). Although humans are the -144-best instrument for evaluating food texture (Brady and Hunecke, 1985) researchers have used instrumental methods for measuring the textural properties of frankfurters due to the cost and d i f f i c u l t i e s involved in sensory evaluation (Brady and Hunecke, 1985; Montejano et a l . , 1985). Because of the complex nature of food texture, Brady and Hunecke (1985) suggested that i t is important to measure as many of a food's textural parameters as possible, either by using several d i f f e r e n t t e s t s , each detecting one or more parameters, or by using one test from which a number of parameters may be characterized. Compression and shear tests have been used extensively for evaluation of the textural properties of frankfurters. In t h i s study the compression test described by Bourne (1978) known as texture p r o f i l e analysis was used. In t h i s method a b i t e - s i z e piece of food is compressed twice; from the r e s u l t i n g force-deformation curves a number of textural parameters were extracted. Following the d e f i n i t i o n s given by Bourne (1978) several parameters were evaluated. Hardness at f i r s t compression (Hardl) was defined as the peak force during the f i r s t compression cycle. Hardness at second compression (Hard2) was defined as the peak force during the second compression cycle. Fracturabi1ity (Fract) was defined as the force at the f i r s t s i g n i f i c a n t break in the curve on the f i r s t compression. Cohesiveness (Cohes) was defined as the r a t i o of the positive force areas under the second and f i r s t compressions. Springiness (Spring) was defined as the height that the food recovers during the time that elapses between the end of -145-the f i r s t compression and the st a r t of the second compression. Two other parameters were derived by c a l c u l a t i o n from the measured parameters: Gumminess (Gummy) was defined as the product of Hardl x Cohes, and Chewiness (Chewy) was defined as the product of Gummy x Spring (Bourne, 1975). Firmness (Firm) defined as the slope of the linear region of the f i r s t compression curve (Segars and Kapsalis, 1987), was also measured. Shear testing has been widely performed with meat, meat products and protein gels. The maximum force required to shear the samples has been taken as a quantitative measure of tenderness and gel strength. In t h i s study shear force measurements were performed on frankfurter samples without the skin to determine the strength of the frankfurter contents only. Maximum shear force (Shear) was extracted from the force-deformation curves obtained. Data for the textural parameters of the cooked frankfurters are given in Table 22. Results reported for the texture p r o f i l e parameters are the mean from 4 observations and from 3 observations for maximum shear force. C o e f f i c i e n t s of variations were computed for each formulation within each r e p l i c a t i o n . Hardl varied in the f i r s t r e p l i c a t i o n from 78.70 to 208.38 N with a mean of 155.00 N, and in the second r e p l i c a t i o n from 83.88 to 218.13 N with a mean of 156.67 N. The ove r a l l mean for Hardl for the 20 formulations evaluated was 155.84 N. In the f i r s t r e p l i c a t i o n the c o e f f i c i e n t s of var i a t i o n varied from 5.6 to 18.1% and in the second r e p l i c a t i o n from 4.1 to 15.2%. -146-Table 22. Experimental data tot t e x t u r a l parameters ot the cooked crank f u r t e r s . *• R e p l i c a t i o n Formulation Hardl Hard2 Firm Cones" S p r i n g Gummy Chewy Shear No. No." (N)° ( N ) D {U/mm)K (mm)° ( N ) H (Nmm)1 ( N ) J 1 1 161.32(5.6) 99.44(13.1) 19.14(14.4) 2 208.38(8.6) 126.72(16.6) 18.92(11.9) 3 113.41(7.7) 73.58(4.7) 15.56(8.2) 4 78.70(9.6) 55.46(18.6) 13.84(5.6) 5 186.87(13.2) 135.16(12.4) 19.41(11.3) 6 141.85(18.1) 99.98(8.6) 18.60(12.4) 7 149.56(8.1) 98.48(15.5) 12.78(16.1) 8 161.83(8.0) 105.52(7.3) 14.64(12.0) 9 158.72(13.6) 104.89(12.4) 19.24(8.6) 10 189.42(6.9) 147.99(19.1) 23.11(5.0) mean 155.00 104.72 17.52 2 1 131.70(9.8) 86.79(4.0) 13.12(3.3) 2 151.90(5.2) 101.16(18.4) 19.31(12.4) 3 125.44(9.3) 105.02(13.4) 17.85(7.9) 4 83.88(9.0) 63.23(14.8) 11.75(4.0) 5 201.21(4.1) 120.78(9.3) 18.99(8.6) 6 191.68(15.2) 133.25(4.1) 17.94(8.5) 7 126.36(14.7) 86.15(4.3) 15.29(7.9) 8 159.21(11.5) 91.35(13.0) 16.25(2.2) 9 177.22(8.9) 129.98(5.4) 14.19(4.3) 10 218.13(9.4) 166.17(13.9) 19.40(14.9) mean 156.67 108.39 16.97 0, .259(15.2) 3 , .76(23 .4) 41, .54(11 .6) 159. ,02(34 .9) 5. 44(18 .4) 0. ,271(14.1) 3. ,59(19 .2) 56, ,24(14 .5) 205, .79(32 .2) 5. 93(22 .2) 0. ,248(6.0) 3, ,67(7. 4) 28, ,06(3. 8) 102, ,93(5. 0) 5. 80( 24 .7) 0, ,238(11.2) 3, ,67(12 .9) 18, ,70(12 .4) 68, ,83(18 .4) 3. 61( 11 .3) 0.258(12.0) 3, ,34(8. 2) 48, ,55(22 .8) 163, .79(29 .9) 6 . 46(12 . 4 ) 0. ,293(4.8) 4. ,34(16 .6) 41, ,70(21 .0) 179 , 15(19 .3) 5. 69(3. 3) 0, ,266(12.0) 3, ,42(16 .7) 39, ,93(19 .4) 138, .29(30 .5) 3. 28(17 .3) 0, ,259(19.5) 3, ,59(20 .6) 41, ,74(18 .7) 153, .63(38 .6) 5. 24(14 .3) 0. .264(7.2) 4. ,34(6. 3) 41. ,94(15 .5) 182, .86(20 .1) 6. 50(11 • 8) 0, ,318(10.3) 4.76(14 .5) 60, ,48(15 .7) 290, .67(25 .3) 7.02(12 • 2) 0. ,267 3, ,85 41. .89 164, .50 5. 50 0, ,269(13.8) 3 , 26(9. 8) 35.71(22 .7) 115.48(19 .4) 4 . 44(18 .3) 0, ,286(8.6) 2. ,84(6. 8) 43.38(6. 0) 123.17(9. 6) 6. 53(11 .9) 0, 274(5.6) 3 , 26(5. 1) 34.44(14 .2) 112.05(14 .7) 4 . 14(4. 4) 0. ,236(6.2) 3. .34(8. 2) 19. ,85(13 • 2) 65, ,93(10 .3) 4 . 60(2. 1) 0, ,274(7.6) 3, ,42(16 .7) 54, ,98(6. 3) 188, ,33(17 .6) 5. 22(17 .6) 0, .285(6.6) 3 .26(15 .4) 54. ,33(9. 7) 176, .27(15 .4) 6 . 28(8. 0) 0, ,245(6.6) 2 , 92(10 .9) 30, ,76(11 .9) 89 , 08(4. 0) 4 . 14(6. 2) 0. ,237(9.5) 2, ,84(6. 8) 37. ,94(19 .1) 107, .54(19 .2) 4 . 71 ( 4 . 9)0, ,272(5.9) 3 , .67(7. 4) 48, .28(14 .1) 177, .66(16 .8) 5 . 30(8. 2) 0. ,311(9.4) 3, ,34(8. 2) 67, ,79(12 .4) 227, ,19(18 .2) 6 . 53(10 .4) 0.269 3, .22 42 , .75 138 .27 5 . 19 k R e s u l t s reported 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 o b s e r v a t i o n s for the tex t u r e p r o f i l e parameters and from 3 ob s e r v a t i o n s for maximum shear force • I n g r e d i e n t s p r o p o r t i o n s are r e p o r t e d i n Table 2 ° Hardness at f i r s t compression, Nevtons ° Hardness at second compression, Nevtons * Firmness, Newtons/ml11imeter " Cohesiveness ° S p r i n g i n e s s , m i l l i m e t e r s H Gumminess, Nevtons 1 Chevlness, Nevtons m i l l i m e t e r s J Maximum shear f o r c e , Nevtons Hard2 varied in the f i r s t r e p l i c a t i o n from 55.46 to 147.99 N with a mean of 104.72 N, and in the second r e p l i c a t i o n from 86.15 to 166.17 N with a mean of 108.39 N. The overall mean for Hard2 for the 20 formulations evaluated was 106.56 N. In the f i r s t r e p l i c a t i o n the c o e f f i c i e n t s of var i a t i o n varied from 4.7 to 19.1% and in the second r e p l i c a t i o n from 4.0 to 18.4%. Fracturabi1ity data is not reported due to the d i f f i c u l t i e s encountered in measuring this parameter. In some cases, the 2 samples obtained from a pa r t i c u l a r frankfurter presented contradictory r e s u l t s , that i s , a fracture point was present in one sample while in the other this point was absent from the curve. Firm varied in the f i r s t r e p l i c a t i o n from 12.78 to 23.11 N/mm with a mean of 17.52 N/mm, and in the second r e p l i c a t i o n from 11.75 to 19.40 N/mm with a mean of 16.97 N/mm. The ove r a l l mean for Firm for the 20 formulations evaluated was 17.25 N/mm. In the f i r s t r e p l i c a t i o n the c o e f f i c i e n t s of va r i a t i o n varied from 5.0 to 16.1% and in the second r e p l i c a t i o n from 2.2 to 14.9%. Cohes varied in the f i r s t r e p l i c a t i o n from 0.238 to 0.318 with a mean of 0.267, and in 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 for Cohes for the 20 formulations evaluated was 0.268. In the f i r s t r e p l i c a t i o n the c o e f f i c i e n t s of va r i a t i o n varied from 4.8 to 19.5% and in the second r e p l i c a t i o n from 5.6 to 13.8%. Spring varied in the f i r s t r e p l i c a t i o n from 3.34 to 4.76 mm with a mean of 3.85 mm, and in the second r e p l i c a t i o n from 2.84 -148-to 3.67 mm with a mean of 3.22 mm. The overall mean for Spring for the 20 formulations evaluated was 3.54 mm. In the f i r s t r e p l i c a t i o n the c o e f f i c i e n t s of variation varied from 6.3 to 23.4% and in the second r e p l i c a t i o n from 5.1 to 16.7%. Gummy varied in the f i r s t r e p l i c a t i o n from 18.70 to 60.48 N with a mean of 41.89 N, and in the second r e p l i c a t i o n from 19.85 to 67.79 N with a mean of 42.75 N. The overall mean for Gummy for the 20 formulations evaluated was 42.32 N. In the f i r s t r e p l i c a t i o n the c o e f f i c i e n t s of vari a t i o n varied from 3.8 to 22.8% and in the second r e p l i c a t i o n from 6.0 to 22.7%. Chewy varied in the f i r s t r e p l i c a t i o n from 68.83 to 290.67 Nmm with a mean of 164.50 Nmm, and in the second r e p l i c a t i o n from 65.93 to 227.19 Nmm with a mean of 138.27 Nmm. The overall mean for Chewy for the 20 formulations evaluated was 151.39 Nmm. In the f i r s t r e p l i c a t i o n the c o e f f i c i e n t s of variation varied from 5.0 to 38.6% and in the second r e p l i c a t i o n from 4 to 19.4%. Shear varied in the f i r s t r e p l i c a t i o n from 3.28 to 7.02 N with a mean of 5.50 N, and in the second r e p l i c a t i o n from 4.14 to 6.53 N with a mean of 5.19 N. The overall mean for Shear for the 20 formulations evaluated was 5.35 N. In the f i r s t r e p l i c a t i o n the co e f f i c i e n t s of vari a t i o n varied from 3.3 to 24.7% and in the second r e p l i c a t i o n from 2.1 to 18.3% As noted by Breene (1975), meat and meat products vary in the i r textural homogeneity with variations observed not only among d i f f e r e n t meat samples but also within a sample. In addition a i r pockets, which are more prevalent in laboratory -149-prepared sausages than in commercial products, make the product-more heterogeneous (Whiting, 1987b). V a r i a b i l i t y of textural parameters of the frankfurters were within the normal v a r i a b i l i t y found in other studies. Coefficients of variations of textural parameters were calculated from data reported by several researchers. For example, in Montejano et a l . (1985), maximum c o e f f i c i e n t s of va r i a t i o n for beef, pork and turkey gels obtained were 8.6% for hardness, 26.2% for cohesiveness, 5.8% for springiness, 27.0% for gumminess, 30.0% for chewiness. Results of Park et a l . (1989) working with frankfurters showed maximum c o e f f i c i e n t s of variation of 15.6% for hardness, 12.5% for springiness, 13.6% for cohesiveness, 22.0% for gumminess and 26.0% for chewiness. S i m i l a r l y , Sofos and Allen (1977) also working with frankfurters obtained maximum c o e f f i c i e n t s of vari a t i o n of 22.0% for hardness. As can be noted in thi s study as well as those reported by Montejano et a l . (1985) and Park et a l . (1989) the textural parameters of gumminess and chewiness present r e l a t i v e l y higher c o e f f i c i e n t s of v a r i a t i o n . This v a r i a b i l i t y is expected since both parameters are calculated from the measured parameters of hardness, cohesiveness and springiness. 2.5. D e t e r m i n a t i o n of pH The pH of meat i s considered to be one of the most important factors a f f e c t i n g meat qual i t y . pH is associated with changes in functional properties of the meat proteins, i . e . water and fat -150-binding capacities and gelation, thus a f f e c t i n g y i e l d , emulsion s t a b i l i t y and texture of meat products (Solomon, 1987). Measurements of pH of the raw emulsions were performed in duplicate. Results are shown in Figure 11. The pH varied in the f i r s t r e p l i c a t i o n from 5.53 to 5.93 with a mean of 5.74, and in 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 overall mean for pH for the 20 emulsions evaluated was 5.67. It is r e a d i l y apparent that the pH of the raw emulsions within r e p l i c a t i o n d i f f e r e d quite markedly. In addition 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 for the preparation of the frankfurters of the second r e p l i c a t i o n had been frozen at -30°C for approximately two months. As mentioned e a r l i e r , frozen storage had an eff e c t on the pH of the meat ingredients. Changes in the pH of meats during freezing and frozen storage have been reported to contribute to m y o f i b r i l l a r protein a l t e r a t i o n s (Powrie, 1973). Therefore, the observed v a r i a b i l i t y of the qual i t y of the frankfurter formulations between r e p l i c a t i o n s can be attributed also to changes in the pH of the meat ingredients due to frozen storage. -151-• REP2 H REP1 FORMULATION No. Figure 11. Mean pH values of the raw emulsions. Repl and Rep2 are r e p l i c a t i o n 1 and 2 respectively. -152-3. Quality prediction m o d e l s 3.1. Regression analysis Multiple regression analyses were carried out for each quality parameter evaluated as the dependent variable to obtain mathematical relationships between the quality parameters and the proportion of the three ingredients: pork fat ( X i ) , mechanically deboned poultry meat (X 2 ) , and beef meat ( X 3 ) . Nomenclature and d e f i n i t i o n of the qu a l i t y parameters are given in Appendix E. The MGLH module of SYSTAT (Wilkinson, 1988a) was used to f i t Scheffe's canonical special cubic model for three components (equation 23) to the experimental data shown in Tables 19 to 22 and Figure 11. Tables 23 to 39 summarize the results of the regression analyses performed on the data of each qua l i t y parameter evaluated. Each table shows the estimated regression c o e f f i c i e n t s and the results of the Student's t-test on the c o e f f i c i e n t s , as well as R J A , standard error of the estimate and the analysis of variance table which follows the approach in Marquadt and Snee (1974). Residuals were analyzed by examining residual plots (residuals versus predicted values) (Figures 12 to 24). Seventeen q u a l i t y prediction models were developed. Examination of the analysis of variance of the f i t t e d models indicated that the F-values for regression were s i g n i f i c a n t (p<0.05), meaning that the q u a l i t y parameters were dependent on the mixture components. With the exception of the ES model, a l l the models possessed no s i g n i f i c a n t lack of f i t (p>0.05). Further -153-discussion on the ES model is given below. The R * A . values varied from 0.23 for the Cook shrink model (Table 25) to 0.88 for the Exwater model (Table 30). Low R * * values were found for Cook shrink (0.23) (Table 25) and Vacuum shrink (0.41) (Table 24) models. The poor f i t was expected since the r e p l i c a b i 1 i t y of Vacuum shrink and Cook shrink data was poor. In addition, the r e l a t i v e l y small range of values found for Vacuum shrink data (Table 19) may have had a negative ef f e c t on the f i t of the model. The model f i t t e d to Firm data also presented a low R * * . value (0.56) (Table 36). Since these models accounted for less than 60% of the variation in the observed data, i t was concluded that these models were not adequate for prediction purposes. The R 2 * . values for the other fourteen models were greater than 0.60, which implied that a good proportion of the t o t a l v a r i a t i o n in the responses was explained by the f i t t e d models. The analysis of residuals performed by SYSTAT indicated that the data from Shrink, Tmloss, Ex f l u i d and Exfat presented o u t l i e r s . For these responses the observations from the two r e p l i c a t i o n s corresponding to the o u t l i e r s were removed from the data and the remaining data were reanalyzed. Analysis of residuals, performed by examining the plots of residuals versus predicted values, is a q u a l i t a t i v e measure for assessing the adequacy of the f i t t e d model (Zar, 1984; Draper and Smith, 1981). The residual plots presented must be considered as an approximation of the d i s t r i b u t i o n of the residuals, since, as -154-reported by Cornell (1981), i t is d i f f i c u l t to see a random d i s t r i b u t i o n of the residuals about zero with less than 30 observations. Analysis of residuals was not performed for Vacuum shrink (Table 24), Cook shrink (Table 25), and Firm (Table 36) models due to the poor f i t of these models, and for ES (Table 28) model due to the modifications performed on the ES data, as discussed below. The analysis of residual plots of Shrink (Figure 12), Tmloss (Figure 13), Exwater (Figure 16), Exfat (Figure 17), pH (Figure 18), Hard2 (Figure 20), Shear (Figure 21), Gummy (Figure 23), and Chewy (Figure 24) models showed that the residuals appeared to be d i s t r i b u t e d evenly above and below zero. For these models the assumptions about the residuals did not appear to be violated, suggesting the adequacy of the f i t t e d models. However the residual plots for Twloss (Figure 14), E x f l u i d (Figure 15), Hardl (Figure 19), and Cohes (Figure 22) models showed that their residuals formed nonrandom patterns. In the case of the Hardl model, increasing v a r i a b i l i t y of the residuals was observed with increasing Hardl predicted values (Figure 19). The opposite was observed for the Twloss and Cohes models, that i s , decreasing v a r i a b i l i t y of the residuals with increasing predicted values (Figures 14 and 22 r e s p e c t i v e l y ) . The E x f l u i d model formed a peculiar pattern. As can be seen in Figure 15 values between 10 and 12% expressible f l u i d were not predicted by the model, thus, leaving a gap in between these values. For these models, the assumptions of constant variance appeared to be violated (Zar, -155-1984; Draper and Smith, 1981). The logarithmic transformation on the observations of these responses as suggested by Zar (1984) and Draper and Smith (1981) could be performed to meet the assumptions about the residuals. However the transformation was not performed and the models were considered adequate for prediction purposes. A model with an R**. greater than 0.39 could not be f i t t e d to the data obtained from Saffle's emulsion s t a b i l i t y test (ES) (Table 20) This was expected, since as noted e a r l i e r the r e p e a t a b i l i t y of the results was poor. The analysis of residuals of t h i s model indicated 2 o u t l i e r s . However, rather than removing the observations of the 2 formulations that were found to be o u t l i e r s (3 and 7) as performed for Shrink, Tmloss, E x f l u i d and Exfat models, the values of the observations were modified by using the value of the formulation number from the opposite r e p l i c a t e , i . e . (a) formulation 3 r e p l i c a t i o n 2 was changed from 1.45 to 0.60%, and (b) formulation 7 r e p l i c a t i o n 1 was changed from 2.45 to 0.90%. Table 28 shows the regression analysis results performed on this new data set. Although the f i t t e d model accounted for 72% of the v a r i a t i o n in the data and the F-value for regression was s i g n i f i c a n t (p<0.05), the model presented s i g n i f i c a n t lack of f i t (p<0.05). Analysis of residuals of this model was not performed. The modifications to the data were performed because a model was needed for formula optimization to account for the s t a b i l i t y of the emulsions to thermal treatment in terms of released f a t . The ES model was considered only as an -156-approximation of the amount of fat lost during thermal processing. Table 40 tabulates the regression models in thi s study that were considered to be adequate in representing the experimentally observed data, and thus to be used for response surface analysis and formula optimization. As can be observed, ES, Exfat, pH, Shear and Chewy were explained s o l e l y by linear blending effects of the ingredients ( i . e . linear terms). Tmloss, Twloss, E x f l u i d , Exwater, and Cohes were explained by linear blending and binary nonlinear blending effects of the ingredients ( i . e . linear and quadratic terms). Shrink, Hardl, Hard2 and Gummy were explained by linear blending, binary and ternary nonlinear blending e f f e c t s of the ingredients (i.e l i n e a r , quadratic and cubic terms). Synergistic and antagonistic effects of the binary mixtures and syner g i s t i c effects of the ternary mixture were observed. Pork fat and mechanically deboned poultry meat ( X i X 2 ) had binary sy n e r g i s t i c effects on Tmloss and Twloss, and antagonistic effects on Shrink, E x f l u i d , Exwater, Hardl, Hard2 and Gummy. Pork fat and beef meat ( X x X 3 ) had binary synergistic effects on Tmloss and Twloss, and antagonistic effects on Cohes and Gummy. Mechanically deboned poultry meat and beef meat ( X 2 X 3 ) had binary antagonistic effects on Shrink, E x f l u i d , Exwater, Hardl, Hard2, and Gummy. The ternary mixture ( X x X 2 X 3 ) had a syn e r g i s t i c e f f e c t on Shrink, Hardl, Hard2, and Gummy. -157-Table 23. 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 loss after processing (Shrink) data. x Var iable Est imated regression coeff i c i e n t (bi) Standard error of coeff i c i e n t ( S b i ) t-value X 2 X o . X 2 X 2 X a X a . X 2 X 3 17.13 11.83 -104.50 -21.18 231.70 6.11 0.27 55.81 10.99 105.94 2.81* 43.16* -1.87n.s. -1.93n.s. 2.19" R**= 0.669 Standard error 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) Analysis of variance Source of Degrees of Sum of Mean F-value v a r i a t i o n freedom squares square Regress i on 4 6.98 1.74 9 . 58* Residual 13 2 . 37 0 . 18 Lack of f i t 4 0.87 0.22 1. 31n Pure error 9 1. 50 0.17 Total 17 9.35 " 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 to be an o u t l i e r and the corresponding observations of the two re p l i c a t i o n s were not considered in the regress i on -158-0.76 0.38 -GO < ZD Q GO UJ rr 0.00 -0.38 H -0.76 7 8 9 10 PREDICTED VALUES Figure 12. Plot of residuals for the Shrink model. -159-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 loss after 13 days under vacuum packaged storage (Vacuum shrink) data. Est imated Standard Variable regression error of t-value c o e f f i c i e n t c o e f f i c i e n t (bi) (Sbi ) X 2 1.07 0.37 2.91* X 3 2 .44 0.13 18.83* R**= 0.414 Standard error i of 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 (p<0.05) Analysis of variance Source of Degrees of Sum of Mean F-value variation freedom squares square Regression 1 1.230 1.230 14.41* Res idual 18 1.536 0.085 Lack of f i t 8 0.680 0.085 0.99n.s Pure error 10 0.856 0.086 Total 19 2.77 " 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 -Table 25. 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 loss after the consumer cook test (Cook shrink) data. Estimated Standard Variable regression error of t-value c o e f f i c i e n t c o e f f i c i e n t ( b i ) ( S b i ) Xa. 16.95 4.77 3.56" X 3 6.77 1.39 4 .'89* R**= 0.2 26 Standard error 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 (p<0.05) Analys is of variance Source of Degrees of Sum of Mean F-value var iat ion freedom squares square Regress ion 1 40.40 40.40 6.56* Res idual 18 110.79 6.16 Lack of f i t 8 29.43 3.68 0.4 5n.s . Pure error 10 81.36 8.14 Total 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-Table 26. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to emulsion s t a b i l i t y (Tmloss) data.* Estimated Standard Variable regression error of t-value c o e f f i c i e n t c o e f f i c i e n t (bi) (Sbi) X 2 X 3 X1X2 X 3 . X 3 16.43 39.69 68.15 69.65 4 . 59 1.63 32.22 9.40 3.58* 24.41-2 .12" 7. 41-R**= 0.789 Standard error 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 Analys is zero (p<0. of variance 05) Source of var iat ion Degrees of freedom Sum of squares Mean square F-value Regress ion 3 260.89 86.96 22 .20" Res idual 14 54 .84 3 .92 Lack of f i t 5 13.04 2.61 0 . 56n.s. Pure error 9 41.80 4.64 Total 17 315.73 " 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) *• Formulation 7 was found to be an o u t l i e r and the corresponding observations of the two re p l i c a t i o n s were not considered in the regression -162-GO _ J < Q GO UJ DC 3.6 2.4 f 1.2 h 0.0 1.2 -2.4 h -3.6 30 35 40 45 PREDICTED VALUES Figure 13. Plot of residuals for the Tmloss model -163-Table 27. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to emulsion s t a b i l i t y (Twloss) data. Estimated Standard Variable regression error of t-value c o e f f i c i e n t c o e f f i c i e n t ( b i ) ( s b i ) Xa Xa.Xa XiXa 1 0 . 1 1 2 9 . 7 5 29 . 1 1 1 6 . 4 8 3.45 1.21 24 .25 6.58 2.93* 24.69* 1. 20n.s 2.51* R**.= 0.755 Standard error 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) Analysis of variance Source of var i a t i o n Degrees of freedom Sum of squares Mean square F-value Regression 3 Residual 16 Lack of f i t 6 Pure error 10 Total 19 136.94 35.51 10.20 25. 31 172.45 45.65 20.57* 2 .22 1.70 0.67n.s 2.53 * 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) -164-3.2 1.6 f-< Q CO L U cr 0.0 -1.6 h -3.2 20 22 24 26 28 PREDICTED VALUES 30 Figure 14. Plot of residuals for the Twloss model. - 1 6 5 -Table 28. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to emulsion s t a b i l i t y (ES) data.*-Estimated Standard Var iable regress ion c o e f f i c i e n t ( b i ) error of coeff i c i e n t ( S t o i ) t-value Xo. X 3 2.61 -0.19 0.32 0.09 8.05" -1.96n.s. R**= 0.724 Standard error 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 is of variance Source of var iat ion Degrees of freedom Sum of squares Mean square F-value Regression 1 1.45 1. 45 50.83" Res idual 18 0.51 0 .03 Lack of f i t 8 0.37 0.05 3.17"-Pure error 10 0.15 0 . 01 Total 19 1.96 " 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 for formulations 3 and 7 were modified -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 expressible f l u i d (Exfluid) data.*-Estimated Standard Variable regression error of t-value c o e f f i c i e n t c o e f f i c i e n t (b t) ( s D l ) Xa. 1 3 . 6 7 3 . 31 4 . 1 3 * X 2 2 0 . 6 4 9 . 3 7 2 . 2 0 * X 3 1 2 . 1 3 0 . 8 2 1 4 . 8 3 * X i X 2 - 3 1 . 9 3 1 9 . 9 7 - 1 . 6 0 n . s . X 2X 3 - 2 7 . 4 3 1 6 . 2 4 - 1 . 6 9 n . s . R*A= 0.678 Standard error 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) Analysis of variance Source of Degrees of Sum of Mean F-value va r i a t i o n freedom squares square Regress i on 4 42. 06 10.52 9.94* Res idual 13 13.76 1.06 Lack of f i t 4 0.16 0.04 0.03n Pure error 9 13.60 1.51 Total 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 ou t l i e r and the corresponding observations of the two re p l i c a t i o n s were not considered in the regression - 1 6 7 -1.6 0.8 h _ j < Q CO UJ cr 0.0 h -0.8 -1.6 8 9 10 11 12 PREDICTED VALUES 13 Figure 15. Plot of residuals for the Exfluid model. -168-Table 30. 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 expressible water (Exwater) data. Var iable Estimated regression coeff i c i e n t (bi) Standard error of coeff i c i e n t ( S b i ) t-value X 2 X 3 Xa.X2 X 2 X 3 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 error 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) Analysis of variance Source of Degrees of Sum of Mean F-value var iat ion freedom squares square Regression 3 57 .98 19 . 33 48.41" Res idual 16 6 . 39 0.40 Lack of f i t 6 0.78 0 .13 0.23n.s Pure error 10 5. 61 0.56 Total 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) -169-1.2 0.6 CO _i < ZD Q CO 111 DC 0.0 \--0.6 -1.2 4 5 6 7 8 9 10 11 PREDICTED VALUES F i g u r e 1 6 . 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 m o d e l . -170-Table 31. 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 expressible fat (Exfat) data.*-Estimated Standard Variable regression error of t-value c o e f f i c i e n t c o e f f i c i e n t (bi) ( s D i ) Xa. 15.01 1.36 11.05* X 2 1.72 0.80 2.16* X 3 1.24 0.39 3.16* R'A= 0.811 Standard error 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 (p<0.05) Analysis of variance Source of Degrees of Sum of Mean F-value var i a t i o n freedom squares square Regress ion 2 29 .69 14.84 37 .55* Res idual 15 5.93 0.40 Lack of f i t 6 0.45 0.07 0.12n Pure error 9 5.48 0.61 Total 17 35.62 * 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 observations of the two r e p l i c a t i o n s were not considered in the regression - 1 7 1 -1.28 0.64 \-co < 3 Q CO UJ DC 0.00 --0.64 h 1.28 I i i 1 o 0 o o o 0 0 0 -° 0 o o o o o 1 1 o 1 2 3 4 5 6 PREDICTED VALUES Figure 17. Plot of residuals for the Exfat model. -172-Table 32. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to pH data. Estimated Standard Var iable regression error of t- value c o e f f i c i e n t c o e f f i c i e n t (bi) (Sbi ) 5.82 0.17 35 .01* x 2 6.34 0.11 58 .30* X 3 5.45 0.05 101 .69* R**.= 0.6 69 Standard error 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) Analysis of variance Source of Degrees of Sum of Mean F-value var i a t i o n freedom squares square Regression 2 0.302 0.151 20.22* Res idual 17 0.127 0.007 Lack of f i t 7 0.023 0.003 0.31n.s. Pure error 10 0.104 0.010 Total 19 0.429 * 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) -173-0.16 0.08 -CO _i < Q CO LU DC 0.00 -0.08 H -0.16 5.4 5.5 5.6 5.7 5.8 5.9 PREDICTED VALUES Figure 18. Plot of residuals for the pH model. -174-Table 33. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to hardness at f i r s t compression (Hardl) data. Estimated Standard Variable regression error of t-value c o e f f i c i e n t c o e f f i c i e n t (b t) ( s b l ) X 2 795.99 281.44 2.83* X 3 212.25 9.60 22.12* XxXa -8308.69 2574.38 -3.23* X 2 X 3 -1131.69 505.29 -2.24* Xo.XaX3 14410 . 51 4875. 70 2.96* R«*.= 0.737 Standard error 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 (p<0.05) Analysis of variance Source of Degrees of Sum of Mean F-value va r i a t i o n freedom squares square Regression 4 22328. 86 5582 . 22 14.33* Res idual 15 5844. 13 389 . 61 Lack of f i t 5 1524 . 08 304 . 82 0.71n Pure error 10 4320. 05 432. 01 Total 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-CO _ J < ZD Q CO H I DC 40 30 (-20 10 0 -10 -20 -30 -40 I I I 1 0 -o o — o o o _ 0 — o 0 o 0 o 0 o o o o o o 0 I 1 1 1 80 106 132 158 PREDICTED VALUES 184 210 Figure 19. Plot of residuals for the Hardl model. -176-Table 34. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to hardness at second compression (Hard2) data. Estimated Standard Variable regression error of t-value c o e f f i c i e n t c o e f f i c i e n t (bi) ( s * i ) Xo. -85. 73 43.11 -1.99n.s. X a 431.91 206.52 2.09n.s. X 3 167.61 11.12 15.07* X a . X 2 -4397.06 1883.81 -2.33" X 2 X 3 -704.61 380.12 -1.85n.s. X a . X 2 X 3 8456.16 3651.66 2.32* R'x= 0.739 Standard error 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) Analysis of variance Source of Degrees of Sum of Mean F-value va r i a t i o n freedom squares square Regression 5 11925.08 2385 .02 11.75* Res idual 14 2841.03 202 .93 Lack of f i t 4 596.87 149 .22 0.67n Pure error 10 2244.16 224 .42 Total 19 14766.11 * 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-CO _ J < ZD Q CO UJ DC 30 20 -10 h 0 F-10 -20 h -30 60 80 100 120 140 160 PREDICTED VALUES F i g u r e 20. P l o t of r e s i d u a l s f o r the Hard2 model -178-Table 35. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to maximum shear force (Shear) data. Estimated Standard Variable regression error of t-value c o e f f i c i e n t c o e f f i c i e n t (b t) ( s b t ) Xo. -1.79 1.26 -1.42n.s. X 2 4.71 0.82 5.74* X 3 7.33 0.41 18.12* R'A.= 0.628 Standard error 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) Analysis of variance Source of Degrees of Sum of Mean F-value va r i a t i o n freedom squares square Regress ion 2 14.49 7.25 17.02* Res idual 17 7.24 0.43 Lack of f i t 7 2. 39 0.34 0.70n Pure error 10 4 .85 0.49 Total 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) -179-1.2 0.6 h c o _ J < ZD Q CO UJ DC 0.0 h -0.6 \--1.2 6 7 PREDICTED VALUES Figure 2 1 . Plot of residuals for the Shear model. -180-Table 36. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to firmness (Firm) data. Estimated Standard Variable regression error of t-value c o e f f i c i e n t c o e f f i c i e n t (b t) ( s b l ) X= 16.36 2.45 6.67-X 3 21.49 0.86 24.86" R**= 0.561 Standard error 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 (p<0.05) Analysis of variance Source of Degrees of Sum of Mean F-value var i a t i o n freedom squares square Regress ion 1 96 . ,44 96 . 44 25, . 31-Res idual 18 68 . 58 3 . 81 Lack of f i t 8 21. , 20 2. .65 0. , 56n.s Pure error 10 47. .38 4 , .74 Total 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-Table 37. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to cohesiveness (Cohes) data. Estimated Standard Variable regression error of t-value c o e f f i c i e n t c o e f f i c i e n t (bi) ( s b i ) Xo. 0.291 X 2 0.222 X 3 0.328 X i X 3 -0.336 0.110 2.64*-0.020 11.02* 0.010 32.54" 0.195 -1.72n.s. R**= 0.792 Standard error 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 is of variance Source of Degrees of Sum of Mean F-value v a r i a t i o n freedom squares square Regress ion 3 0.0080 0.0030 25.05* Res idual 16 0.0020 0.0001 Lack of f i t 6 0.0008 0.0001 1.18n.s. Pure error 10 0.0012 0.0001 Total 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) -182-0.02 0.01 CO < ZD Q CO HI cr 0.00 h -0.01 h -0.02 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 PREDICTED VALUES Figure 2 2 . Plot of residuals for the Cohes model. -183-Table 38. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to gumminess (Gummy) data. Estimated Standard Variable regression error of t-value c o e f f i c i e n t c o e f f i c i e n t (bi) ( s o i ) X a 220.27 81.38 2.71* X 3 68.79 4.84 14.22* X i X a -2266.94 745.24 -3.04* X i X 3 -56.08 25.98 -2.16* X a X 3 -339.77 150.29 -2.26* ,XaX 3 4015.16 1440.25 2.79* R**= 0.809 Standard error 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) Analysis of variance Source of Degrees of Sum of Mean F-value va r i a t i o n freedom squares square Regress ion 5 2651.14 530 . 23 17 .13* Residual 14 433.32 30 .95 Lack of f i t 4 115.99 29 .00 0 .91n Pure error 10 317.33 31 .73 Total 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) - 1 8 4 -10 5 r CO _ J < ZD Q GO t i l DC 0 f--5 H -10 20 29 38 47 56 65 PREDICTED VALUES Figure 23. Plot of residuals for the Gummy model. -185-Table 39. Regression s t a t i s t i c s corresponding to the "best" model f i t t e d to chewiness (Chewy) data. Estimated Standard Variable regression error of t-value c o e f f i c i e n t c o e f f i c i e n t (b*) (s t o t) X i -219.77 50.03 -4.39* X 2 46.17 32.74 1.41n.s. X 3 275.30 16.13 17.07" R'*= 0.781 Standard error 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) Analysis of variance Source of vari a t i o n Degrees of freedom Sum of squares Mean square F-value Regression 2 Residual 17 Lack of f i t 7 Pure error 10 Total 47331.31 11503.65 2490.40 9013.25 23665.66 676 .68 355.77 901.33 34.97' 0.39n.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) -186-50 30 -c o _ j < ZD Q CO UJ DC 10 f--10 k -30 --50 80 110 140 170 200 230 260 PREDICTED VALUES Figure 24. Plot of residuals for the Chewy model. -187-Table 40. Quality prediction models. *" B Shrlnk= 17.13X2 + 11.83X3 - 104.50Xa.Xa - 21.18X 2X 3 + 231.70XiX=X 3 Tmloss= 16.43X 2 + 39.69X 3 + 6 8 . 1 5 X i X 2 + 69.65XxX 3 Tvloss= l O . H X a + 29.75X a + 29.1lXxX 2 +16.48XxX 3 ES = 2.61Xx - 0.19X 3 Exfluid= 13.67Xi + 20.64X2 + 12.13X 3 - 31.93XxX 2 - 27.43X 2X 3 Exvater= 18.15X 2 + 10.58X 3 - 35.57XxX 2 - 25.31X=X3 Exfat = 15.01X:L + 1.72X2 + 1.24X 3 pH = 5.82Xi. + 6.34X2 + 5.45X 3 Hardl = 795.99X 2 + 212.25X 3 - 8308.69XxX 2 - 1131.69X 2X 3 + 14410.51XxX 2X 3 Hard2 = -85.73Xx + 431.91X 2 + 167.61X 3 - 4397.06XxX 2 - 704.61X 2X 3 +8456.16XxX 2X 3 Shear = -1.79Xx + 4.71X 2 + 7.33X 3 Cohes = 0.291X1. + 0.222X2 + 0.328X3 - 0.336XxX 3 Gummy = 220.27X 2 + 68.79X 3 - 2266.94XxX2 - 56.08XxX 3 -339.77X2X 3 + 4015.16X1X2X3 Chewy = -219.77Xi + 46.17X 2 + 275.30X 3 *• Nomenclature and d e f i n i t i o n of the qua l i t y parameters are given in Appendix E s Xi=pork f a t ; X2=MDPM; X 3=beef meat -188-3.2. Response surface contour analysis The examination of the regression models l i s t e d in Table 40 reveal l i t t l e d i r e c t understanding of the relationship between the ingredient proportions and the q u a l i t y parameters. To v i s u a l i z e these relationships, the f i t t e d models were expressed graphically as response surface contour pl o t s . Contour plots are lines of equal response values for d i f f e r e n t ingredient proportions. These plots were most helpful in elucidating the shape of the response surfaces of these models and provided an understanding of the effects of the ingredient proportions. Generation of response surface contour plots and three-dimensional response surface plots forms part of any response surface experimentation study (Agreda and Agreda, 1989; Rockower et a l . 1983; Cornell, 1981; Johnson and Zabik, 1981; Huor et a l . 1980; H i l l and Hunter, 1966). These plots are used to study the effects of changes in the levels of the factors (ingredients) on the responses evaluated (quality parameters) (Thompson, 1982). Figures 25 through 38 i l l u s t r a t e the response surface contour plots showing the effects of the ingredient proportions (pork fa t , X i ; mechanically deboned poultry meat, X 2; and beef meat, X 3) on each q u a l i t y parameter. It is important to point out that these plots r e f l e c t the predicted responses for the qu a l i t y parameters for ingredient combinations within the mixture space studied (Table 1). Refer to Appendix A Figure A2 for instructions on how to read the ingredient proportions on these plots. Nomenclature and d e f i n i t i o n of the qu a l i t y parameters are given - 1 8 9 -in Appendix E. Straight p a r a l l e l contours resulted from q u a l i t y prediction models that included linear terms, that i s , for those qu a l i t y parameters that were described only by the linear blending e f f e c t of the ingredients: ES (Figure 28), Exfat (Figure 31), pH (Figure 32), Shear (Figure 36) and Chewy (Figure 38). In these models, slanting of the contour lines provides a measure of the e f f e c t of the ingredients on the response; the contour lines are slanted towards the vertex of the ingredient with the strongest e f f e c t on the response ( i . e . largest regression c o e f f i c i e n t value). Curvilinear contours resulted from qu a l i t y prediction models that included linear and quadratic terms, that i s , for those qual i t y parameters that were described by linear blending and binary nonlinear blending e f f e c t s of the ingredients: Tmloss (Figure 26), Twloss (Figure 27), E x f l u i d (Figure 29), Exwater (Figure 30), and Cohes (Figure 37). It is the quadratic terms present in these models that generate the quadratic deviation of the contour l i n e s . More complex contours resulted from qu a l i t y prediction models that included l i n e a r , quadratic and cubic terms, that i s , for those q u a l i t y parameters that were described by linear blending, binary and ternary nonlinear blending effects of the ingredients: Shrink (Figure 25), Hardl (Figure 33), Hard2 (Figure 34), and Gummy (Figure 35). In these models the cubic term generated deviations of the contour lines from the f i r s t and second order - 1 9 0 -approximations (Snee, 1971). A description of the effects of changes in the ingredient proportions on the qua l i t y parameters w i l l be given. However, since the proportions of the ingredients are dependent on each other, s i m p l i f i c a t i o n of the description w i l l be performed by maintaining each ingredient at a constant proportion. Description of the effects of the ingredients on Shrink, Hardl, Hard2 and Gummy i s somewhat d i f f i c u l t due to the complex nature of the contour plots generated by the models. However a generalized description w i l l be given for these contour p l o t s . The response surface contour plot for the Shrink model (Figure 25) shows that: (a) maintaining the proportion of MDPM constant, Shrink decreased as the proportion of pork fat increased; (b) maintaining the proportion of beef meat constant, Shrink decreased as the proportion of pork fat increased; (c) maintaining the proportion of pork fat constant, Shrink decreased as the proportion of MDPM increased. The region of minimum Shrink values was lo c a l i z e d at the vertex defined by 0.30 pork f a t , 0.20 MDPM and 0.50 beef meat; the region of maximum shrink values was loc a l i z e d at the vertex defined by 0.05 pork fat, 0.00 MDPM and 0.95 beef meat. The response surface contour plot for the Tmloss model (Figure 26) shows that: (a) maintaining the proportion of MDPM constant, the proportion of beef meat had l i t t l e e f f e c t on Tmloss; (b) maintaining the proportion of beef meat constant, Tmloss decreased as the proportion of MDPM increased; (c) maintaining -191-MDPM Figure 25. Response surface contour plot for the Shrink model. MDPM Figure 26. Response surface contour plot for the Tmloss model. the proportion of pork fat constant, Tmloss decreased as the proportion of MDPM increased. The region of minimum Tmloss values was lo c a l i z e d at the vertex defined by 0.05 pork fat, 0.40 MDPM, and 0.55 beef meat; the region of maximum Tmloss values was loc a l i z e d along the edge defined by 0.00 MDPM, between 0.20 and 0.30 pork f a t . The response surface contour plot for the Twloss model (Figure 27) shows that: (a) maintaining the proportion of MDPM constant, Twloss increased as the proportion of beef meat increased; (b) maintaining the proportion of beef meat constant, the increase in MDPM had l i t t l e e f f e c t on Twloss; (c) maintaining the proportion of pork fat constant, Twloss decreased as the proportion of MDPM increased. The region of minimum Twloss values was lo c a l i z e d at the vertex defined by 0.10 pork fat, 0.40 MDPM and 0.50 beef meat; the region of maximum Twloss values was lo c a l i z e d at the vertex defined by 0.05 pork f a t , 0.00 MDPM and 0.95 beef meat. The response surface contour plot for the ES model (Figure 28) shows that: (a) maintaining the proportion of MDPM constant, ES increased as the proportion of pork fat increased; (b) maintaining the proportion of beef meat constant, ES decreased as the proportion MDPM increased; (c) maintaining the proportion of pork fat constant, the increase in the proportion of MDPM had l i t t l e e f f e c t on ES. The region of minimum ES values was loc a l i z e d at the vertex defined by 0.05 pork f a t , 0.00 MDPM and 0.95 beef meat; the region of maximum ES values was loca l i z e d along the edge defined by 0.30 pork f a t . -194-MDPM Figure 27. Response surface contour plot for the Twloss model. MDPM Figure 28. Response surface contour plot for the ES model. The response surface contour plot for the Ex f l u i d model (Figure 29) showed that: (a) maintaining the proportion of MDPM constant, the increase in the proportion of beef meat had l i t t l e e f f e c t on E x f l u i d ; (b) maintaining the proportion of beef meat constant, E x f l u i d decreased as the proportion of MDPM increased; (c) maintaining the proportion of pork fat constant, E x f l u i d decreased as the proportion of MDPM increased. Although d i f f i c u l t to observe in thi s figure, the region of minimum Exf l u i d values was lo c a l i z e d along the line defined by 0.35 MDPM; the region of maximum Ex f l u i d values was lo c a l i z e d at the vertex defined by 0.30 pork fat, 0.00 MDPM and 0.70 beef meat. The response surface contour plot for the Exwater model (Figure 30) showed that: (a) maintaining the proportion of MDPM constant, Exwater increased as the proportion of beef meat increased; (b) maintaining the proportion of beef meat constant, the increase in the proportion of MDPM had l i t t l e e f f e c t on Exwater; (c) maintaining the proportion of pork fat constant, Exwater decreased as the proportion of MDPM increased. The region of minimum Exwater values was loc a l i z e d at the vertex defined by 0.30 pork f a t , 0.20 MDPM and 0.50 beef meat; the region of maximum Exwater values was lo c a l i z e d at the vertex defined by 0.05 pork fat, 0.00 MDPM and 0.95 beef meat. The response surface contour plot for the Exfat model (Figure 31) showed that: (a) maintaining the proportion of MDPM constant, Exfat increased as the proportion of pork fat increased; (b) maintaining the proportion of beef meat constant, Exfat decreased -197-MDPM MDPM Figure 30. Response surface contour plot for the Exwater model. MDPM Figure 31. Response surface contour plot for the Exfat model as the proportion of MDPM increased; (c) maintaining the proportion of pork fat constant, the increase in the proportion of MDPM had l i t t l e e f f e c t on Exfat. The region of minimum Exfat values was lo c a l i z e d at the vertex defined by 0.05 pork f a t , 0.00 MDPM and 0.95 beef meat; the region of maximum Exfat values was loc a l i z e d along the edge defined by 0.30 pork f a t . The response surface contour plot for the pH model (Figure 32) showed that: (a) maintaining the proportion of MDPM constant, pH decreased as the proportion of beef meat increased; (b) maintaining the proportion of beef meat constant, pH increased as the proportion of MDPM increased; (c) maintaining the proportion of pork fat constant, pH increased as the the proportion of MDPM increased. The region of minimum pH values was loc a l i z e d at the vertex defined by 0.05 pork fat, 0.00 MDPM and 0.95 beef meat; the region of maximum pH values was localized at the vertex defined by 0.10 pork fat, 0.40 MDPM and 0.50 beef meat. The response surface contour plots for the Hardl, Hard2 and Gummy models (Figure 33, 34 and 35) showed that: (a) maintaining the proportion of MDPM constant, Hardl, Hard2 and Gummy decreased as the proportion of pork fat increased; (b) maintaining the proportion of beef meat constant, Hardl, Hard2 and Gummy decreased as the proportion of pork fat increased; (c) maintaining the proportion of pork fat constant, Hardl Hard2 and Gummy decreased as the proportion of MDPM increased. The regions of minimum Hardl, Hard2 and Gummy values were lo c a l i z e d at the vertex defined by 0.30 pork f a t , 0.20 MDPM and 0.50 beef meat; -201-MDPM Figure 33. Response surface contour p l o t for the Hardl model. MDPM Figure 34. Response s u r f a c e contour p l o t f o r the Hard2 model. MDPM Figure 3 5 . Response surface contour plot for the Gummy model. the regions of maximum Hardl, Har<32 and Gummy values were lo c a l i z e d at the vertex defined by 0.05 pork f a t , 0.00 MDPM and 0.95 beef meat. The response surface contour plot for the Shear model (Figure 36) showed that: (a) maintaining the proportion of MDPM constant, Shear increased as the proportion of beef meat increased; (b) maintaining the proportion of beef meat constant, Shear increased as the proportion of MDPM increased; (c) maintaining the proportion of pork fat constant, Shear decreased as the proportion of MDPM increased. The region of minimum Shear values was l o c a l i z e d at the vertex defined by 0.30 pork fat, 0.20 MDPM and 0.50 beef meat; the region of maximum Shear values was loc a l i z e d at the vertex defined by 0.05 pork fat, 0.00 MDPM and 0.95 beef meat. The response surface contour plot for the Cohes model (Figure 37) showed that: (a) maintaining the proportion of MDPM constant, Cohes increased as the proportion of beef meat increased; (b) maintaining the proportion of beef meat constant, Cohes increased as the proportion of MDPM increased; (c) maintaining the proportion of pork fat constant, Cohes decreased as the proportion of MDPM increased. The region of minimum Cohes values was l o c a l i z e d along the edge defined by 0.30 pork fat ; the region of maximum Cohes values was lo c a l i z e d at the vertex defined by 0.05 pork fat, 0.00 MDPM and 0.95 beef meat. The response surface contour plot for the Chewy model (Figure 38) showed that: (a) maintaining the proportion of MDPM constant, -206-MDPM FAT Figure 36. Response surface contour plot for the Shear model. MDPM Figure 37. Response surface contour p l o t for the Cohes model. MDPM Chewy increased as the proportion of beef meat increased; (b) maintaining the proportion of beef meat constant, Chewy increased as the proportion of MDPM increased; (c) maintaining the proportion of pork fat constant, Chewy increased as the proportion of beef meat increased. The region of minimum Chewy values was lo c a l i z e d at the vertex defined by 0.30 pork f a t , 0.20 MDPM and 0.50 beef meat; the region of maximum Chewy values was loc a l i z e d at the vertex defined by 0.05 pork f a t , 0.00 MDPM and 0.95 beef meat. -210-3.3. Correlation analysis The experimental data reported in Tables 19 to 22 and Figure 11 were pooled for co r r e l a t i o n analysis. Pearson's c o r r e l a t i o n c o e f f i c i e n t s (r) were computed to evaluate the linear relationships between proximate composition of the meat blocks and of the raw emulsions and the qual i t y parameters evaluated (Table 41), between the qu a l i t y parameters evaluated (Table 42) and between the textural parameters (Table 43). Only co 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 are reported. Nomenclature and d e f i n i t i o n of the qu a l i t y parameters are given in Appendix E. The 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 composition of the meat blocks and raw emulsions and raw emulsions' pH and the quali t y parameters evaluated are presented in Table 41. Positive correlations were found for protein content of the meat blocks with both moisture content of the meat blocks and moisture content of the raw emulsions. Negative correlations were found for fat content and fat-to-protein r a t i o of the meat blocks with moisture and protein content of the meat blocks and with moisture content of the raw emulsions. Moisture, protein and fat content and fat-protein r a t i o of the meat blocks and moisture content of the raw emulsions showed high correlations with most of the qu a l i t y parameters. Moisture content of the meat blocks was p o s i t i v e l y correlated with Shrink, Exwater and with a l l the textural parameters, and negatively correlated with ES and Exfat. Moisture content of the raw emulsions followed the same relationships as moisture content -211-Table 41. Correlations between proximate composition of the meat blocks and raw emulsions and the qu a l i t y parameters evaluated.*" 3 Meat block Raw emulsion Moisture Protein Fat F/P c Moisture pH Prote in 0. ,916 Fat -0. ,996 -0. .950 F/P -0 , 990 -0, .954 0. ,997 Moisture 0 0, .989 0, .965 -0, .998 -0, .998 pH s Shr ink 0. . 736 0 , . 769 -0, .755 -0 . 770 0. .765 Tmloss -0, . 561 Twloss 0 , . 699 -0, . 496 -0 , .519 0, , 532 -0. .620 ES -0. .859 -0, ,769 0, . 851 0, . 844 -0 , .842 E x f l u i d -0. .454 Exwater 0 , 651 0 , 857 -0, ,709 -0 , .730 0, ,741 -0. ,516 Exfat -0. . 839 -0, . 710 0, .821 0 , . 816 -0, , 808 Hardl 0 , .692 0 , .766 -0 , .720 -0 , .749 0 . ,739 Hard2 0, .740 0, .850 -0, .777 -0, .792 0, , 796 -0, ,465 Shear 0 . , 793 0 . , 805 -0 , 808 -0, .816 0 . ,814 Cohes 0, . 854 0, . 883 -0, .873 -0, . 860 0. , 877 Gummy 0. ,793 0 , .870 -0 . ,823 -0 . ,838 0. ,840 Chewy 0, .784 0, . 893 -0 , . 821 -0, .831 0, .839 * Nomenclature and d e f i n i t i o n of the qu a l i t y parameters are given in Appendix E B Correlation 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 c Fat-to-protein r a t i o D Moisture content of the raw emulsions B not s i g n i f i c a n t at p<0.05 -212-of the meat blocks, the exception being Twloss. Protein content of the meat blocks was p o s i t i v e l y correlated with Shrink, Twloss, Exwater and with a l l the textural parameters, and negatively correlated with ES and Exfat. Formulations having high protein content and thus high moisture content lead to: (a) greater weight losses during thermal treatment (Shrink, Twloss), (b) greater amounts of expressible water in the cooked frankfurters (Exwater), and (c) greater textural strength. Similar relationships have been found by several researchers. Marquez et a l . (1989) found that high protein-high moisture frankfurters had lower smokehouse yields than low protein-low moisture frankfurters. Simon et a l . (1965) and Singh et a l . (1985) reported that texture p r o f i l e parameters were higher for samples with higher protein contents. St. John et a l . (1986) indicated that greater values for gumminess and chewiness were presumably the res u l t of increased protein content. Marquez et a l . (1989) indicated that cohesiveness, a textural parameter highly dependent on the strength of the internal molecular bonds making up the body of the product, is expected to be higher in high-protein frankfurters. In addition, Morrisey et a l . (1982) reported that the strength of myosin gels increases proportionally to the square of myosin concentration. Negative correlations between both moisture and protein content and both Exfat and ES were expected, since as the moisture and protein content of the raw emulsions increased, the r e l a t i v e fat content decreased. This decrease contributed to the reduced l e v e l -213-of fat available to be lost during s a f f l e's emulsion s t a b i l i t y test (ES) and to lover levels of fat to be expressed from the cooked frankfurters (Exfat). Fat content and fat-to-protein r a t i o of the meat blocks shoved a rela t i o n s h i p to a l l q u a l i t y parameters opposite to that of moisture and protein content. Fat content and fat-to-protein r a t i o vere p o s i t i v e l y correlated with ES and Exfat and negatively correlated v i t h Shrink, Tvloss, Exvater and v i t h a l l the textural parameters. Positive relationships betveen both ES and Exfat and both fat content and fat-to-protein r a t i o of the meat blocks vere expected since as the fat content of the emulsions increased less of a protein matrix vas available to s t a b i l i z e the f a t . Thus greater levels of fat vere l o s t during Saffle's emulsion s t a b i l i t y test (ES) and greater levels of fat vere expressed from the cooked frankfurters (Exfat). The negative relationship betveen both fat and fat-to-protein r a t i o of the meat blocks and product veight loss (Shrink and Tvloss) has also been observed by several researchers. Dhillon and Maurer (1975) found that yields increased as fat levels increased and attributed t h i s to the reduced moisture levels that vere lost during cooking. Park et a l . (1989) found that lov-fat frankfurters l o s t more vater during cooking. M i t t a l and B l a i s d e l l (1983) hypothesized that frankfurters having high fat-to-protein r a t i o s had lover moisture losses due to the hydrophobic nature of the fat that offers resistance to the d i f f u s i o n of moisture. The negative -214-relationships between Exwater and fat content and fat-to-protein r a t i o of the meat blocks indicated that the r e l a t i v e proportion of water in the formulations decreased and thus the amounts of water to be expressed from the frankfurters decreased. The negative relationships between fat content and textural parameters has been documented by other researchers. Hand et a l . (1987) demonstrated that low-fat frankfurters required greater force to shear than high-fat frankfurters. Park et a l . (1989) found that reduction in fat content caused increased firmness and springiness in cooked frankfurters. Siripurapu et a l . (1987) and Mi t t a l and Usborne (1986) found negative correlations between fat-to-protein r a t i o s of raw emulsions and the texture p r o f i l e parameters of cooked sausages. The pH showed the poorest correlations with the textural parameters. pH showed negative correlations with the emulsion s t a b i l i t y parameters obtained by the method of Townsend et a l . (1968) (Tmloss and Twloss). This indicates that as the pH of the raw emulsions decreased, greater weight losses (Twloss) and greater water losses per unit moisture content of meat block (Tmloss) were observed, thus contributing to the decline of the s t a b i l i t y of the emulsions to thermal treatment. As mentioned before, Tmloss can be considered as a measure of the water holding capacity of the meat emulsions. It is widely recognized that water holding capacity i s strongly dependent on the meat pH, increasing with increasing pH (Harding Thomsen and Zeuthen 1988; Whiting, 1988; M i l l e r et a l . , 1968). The pH was also negatively -215-correlated to E x f l u i d , Exwater and Hard2. Correlation c o e f f i c i e n t s between product weight loss, emulsion s t a b i l i t y , juiciness and textural parameters are shown in Table 42. Shrink correlated well with most of the q u a l i t y parameters. Shrink was p o s i t i v e l y correlated with Twloss, Exwater and a l l the textural parameters; and negatively correlated with ES and Exfat. A positive c o r r e l a t i o n between Shrink and Twloss was expected since both parameters are measures of product weight loss after thermal treatment. The positive relationship between Shrink and Exwater indicates that at higher weight losses, greater levels of water were expressed from the cooked frankfurters. The positive r e l a t i o n s h i p between weight loss and textural parameters has been reported by several researchers. Lee et a l . (1987) reported that greater smokehouse weight loss may have led to greater compression forces. Trout and Schmidt (1987) hypothesized that concentration of protein occurs due to weight loss of frankfurters during thermal treatment, thus leading to greater textural strength. The negative relationship between Shrink and both ES and Exfat indicated that formulations, exhibiting high weight losses, released lower amounts of fat during Saffle's emulsion s t a b i l i t y test and lower amounts of fat were expressed from the cooked frankfurters. Tmloss correlated with few q u a l i t y parameters. Tmloss showed positive correlations with Twloss, E x f l u i d and Exfat. These positive relationships indicate that at higher water losses per -216-Table 42. Correlations between the qual i t y parameters evaluated*-' B Shrink Tmloss Twloss ES Ex f l u i d Exwater Exfat Tmloss e Twloss 0 .551 0 .710 ES -0 .739 Exfluid 0 . 768 0.556 Exwater 0 .541 0.772 Exfat -0 .541 0 .500 Hardl 0 .640 0.662 Hard2 0 .695 0.757 Shear 0 .652 Cohes 0 .632 0.642 Gummy 0 .688 0.724 Chewy 0 .761 0.777 -0.481 0.482 0.823 0.564 -0.451 -0.462 0.734 -0.532 0.772 -0.501 -0.687 0.681 -0.718 -0.621 0.665 -0.594 -0.551 0.786 -0.515 -0.548 0.842 -0.487 * Nomenclature and d e f i n i t i o n of the qual i t y parameters are given in Appendix E B Correlation 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 c not s i g n i f i c a n t at p<0.05 unit moisture content' in the meat block (Tmloss), greater weight losses during thermal treatment and greater amounts of f l u i d and fat were expressed from the cooked frankfurters. Twloss was p o s i t i v e l y correlated with Shrink, Tmloss, E x f l u i d , Exwater, and with the texture p r o f i l e parameters. The positive r e l a t i o n s h i p with both E x f l u i d and Exwater indicates that at higher weight losses (Twloss), greater amounts of f l u i d and water were expressed from the cooked frankfurters. As with Shrink, Twloss correlated p o s i t i v e l y with the texture p r o f i l e parameters. ES was p o s i t i v e l y correlated with Exfat, and negatively correlated with Shrink, Exwater, and with a l l the textural parameters. The positive r e l a t i o n s h i p between ES and Exfat indicates that at higher levels of released fat from the emulsions during thermal treatment, higher amounts of fat were expressed from the cooked frankfurters. The negative relationship between ES and Exwater and with the textural parameters indicates that at higher levels of released fat from the emulsions during thermal treatment, lower amounts of water were expressed from the cooked frankfurters and softer frankfurters were obtained. E x f l u i d showed a s l i g h t l y higher co r r e l a t i o n with Exfat (r=0.56) than with Exwater (r=0.48). As expected, a negative r e l a t i o n s h i p was found between Exfat and Exwater. Exwater showed positive relationships with a l l the textural parameters indicating that at higher levels of expressed water greater textural strength of the frankfurters was observed. Exfat showed an opposite e f f e c t . -218-The c o r r e l a t i o n c o e f f i c i e n t s indicating the relationship between the textural parameters are l i s t e d in Table 43. The textural parameters were highly correlated with each other. It i s important to point out that c o r r e l a t i o n c o e f f i c i e n t s computed for Hardl with both Gummy and Chewy, Cohes with both Gummy and Chewy, and Gummy with Chewy cannot be considered s t a t i s t i c a l l y s i g n i f i c a n t since these parameters depend on Hardl and Cohes. Positive correlations were expected between the hardness parameters and Shear because these parameters are measures of the strength c h a r a c t e r i s t i c s of the cooked frankfurters. Similar findings have been reported by several researchers. Siripurapu et a l . (1987) working with frankfurters found positive correlations between hardness at f i r s t compression and hardness at second compression, between chewiness and both hardness at f i r s t and at second compression, between gumminess and both hardness at f i r s t and second compression, between gumminess and chewiness. However, negative correlations were found between cohesiveness and both hardness at f i r s t and at second compression. On the other hand Ziegler et a l . (1987) working with dry and semidry sausages found positive correlations between cohesiveness and both hardness at f i r s t and at second compression. -219-Table 43. Correlations between textural parameters*" 3 Hardl Hard2 Shear Cohes Gummy Hard2 0.922 Shear 0.625 0.655 Cohes 0.627 0.776 0.602 Gummy 0.969 0.962 0.672 0.793 Chewy 0.836 0.873 0.702 0.819 0.906 * Nomenclature and d e f i n i t i o n of the quality parameters are given in Appendix E B Correlation 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 -220-3.4. Scatterplot matrices analysis Analysis of the response surface contour plots helped v i s u a l i z e the e f f e c t of changes in the ingredient proportions on the q u a l i t y parameters. Correlation analysis i d e n t i f i e d the s i g n i f i c a n t linear relationships between proximate composition and q u a l i t y parameters and between q u a l i t y parameters. However, to understand the rel a t i o n s h i p between the ingredient proportions and the qu a l i t y parameters further i t was necessary to pool a l l the information ( i . e . ingredients, proximate composition and quali t y parameters) and to display i t in a series of scatterplot matrices (SPLOM). SPLOM and co r r e l a t i o n analysis are techniques that can be used to arrive at similar conclusions. However, SPLOM is b a s i c a l l y a graphical method for data analysis (Cleveland, 1985). By displaying information in a graphical form, further understanding of the int e r r e l a t i o n s h i p s between the d i f f e r e n t variables i s obtained. When several variables are involved SPLOM are easier to analyze than a c o r r e l a t i o n matrix of the data. SPLOM have no s t a t i s t i c a l basis and, thus, can be used to depict the relationships between independent (ingredients) and dependent variables (quality parameters) in contrast to co r r e l a t i o n analysis where a basic assumption is that a l l the variables should be dependent variables (Bender et a l . , 1982). The data for the qual i t y parameters for the 10 formulations studied (Table 2) was generated using the qu a l i t y prediction models (Table 40). Nomenclature and d e f i n i t i o n of the qual i t y -221-parameters are given in Appendix E. The numbers within each panel refer to the formulation number. Number 0 refers to formulation 10. Each panel is a miniature XY plot ranging from the minimum to the maximum value of each parameter. Figure 39 shows the relationships between the ingredients and proximate composition of the meat blocks, moisture content of the raw emulsions, added ice and pH of the raw emulsions. Moisture and protein content of the meat blocks increased as the proportion of pork fat decreased and increased as the proportion of beef meat increased. Fat content and the fat-to-protein r a t i o of the meat blocks increased as the proportion of pork fat increased. Addition of ice was dependent on the o r i g i n a l moisture-to-protein r a t i o of the meat blocks; meat blocks having higher protein content, i . e . higher proportions of beef meat, had lower moisture-to-protein r a t i o s . Therefore, higher levels of ice were needed to make up for the established moisture-to-protein r a t i o of 4 in the raw emulsions. As observed in Figure 39 the levels of ice increased as the proportion of beef meat increased and thus the moisture content of the raw emulsions also increased. The pH of the raw emulsions increased as the proportion of MDPM increased, and decreased as the proportion of beef meat increased. This r e l a t i o n s h i p was expected since the pH of the MDPM was higher than that of beef meat (Table 18). It has been reported that the addition of mechanically deboned meat increases the pH of meat emulsions containing hand deboned meat (Harding Thomsen and Zeuthen, 1988). - 2 2 2 -X1 X2 X3 MOISTURE ^ 1 5 9 3 3 U 6 ? 9 5 T 3 8 7 5 R O 2 6 1 5 g 1 7 8 FAT g 1 6 9 » 7 ^ 3 9 5 1 n 6 17 8 ^ 5 6 8 fl PHOT EN ! 0 8 1 2 a 7 a 1 0 ,2 S 8 9 i 7 8 FP 3 * 0 1 5 0 e 7 3 a 9 e i (1 1 7 8 1 9 5 9 6 fl MOISTEM ° 1 5 9 3 8 U 8 2 9 5 t 2 6 ^ 1 Z 5 9 ICE ° 9 8 2 1 3 4 8 6 9 , 7 5 8 e 2 PH 2 ' 3 4 6 \ i n 0 1 2 7 5 6 8 fl 9 0 Figure 39. Relationships between the ingredients proportions and the proximate composition of the meat blocks and raw emulsions, added ice and pH of the raw emulsions. Pork fat, XI; MDPM, X2; Beef meat, X3. Proximate composition of meat block: moisture, f a t , protein and fat-to-protein r a t i o (FP); composition of raw emulsion: moisture (moistem). - 2 2 3 -Shrink increased as the proportion of. beef meat increased and decreased as the proportion of pork fat increased (Figure 40). Relating this q u a l i t y parameter to the proximate composition of the meat blocks and raw emulsions (Figure 41), Shrink increased as the moisture and protein content of the meat blocks and moisture content of the raw emulsions increased and decreased as the fat content and fat-to-protein r a t i o of the meat blocks increased. Marquez et a l . (1989), Park et a l . (1989), M i t t a l and B l a i s d e l l (1983), and Dhillon and Maurer (1975) reported similar findings. From these results i t can be concluded that high levels of beef meat were detrimental to the y i e l d of the product due to the fact that higher moisture contents and lower pH values were observed for formulations containing higher proportions of beef meat. Tmloss decreased as the proportion of MDPM increased (Figure 40). Tmloss was not related to the proximate composition of the meat blocks and emulsions, however i t related negatively with the pH of the raw emulsions (Figure 41). As mentioned before, Tmloss could be considered as a measure of the water holding capacity of the meat emulsions ( i . e . the lower the value the higher the a b i l i t y of the emulsions to reta i n water), and th i s functional property is known to be affected by the pH of the meats. Since the pH of the raw emulsions was found to relate p o s i t i v e l y to the proportion of MDPM (Figure 39) the tendency for Tmloss to decrease with increasing MDPM may be attributed to the pH effe c t of t h i s meat. Harding Thomsen and Zeuthen (1986) reported that -224-X1 X2 X3 8HRMK U | 5 9 u 9 6 g 8 7 . 3 ? U 2 5 6 9 | 27 8 T M L O S S 0 y \ 6 5 3 4 ? 1 t 3 7 0 y 0 19 T W L O S S 0 6 5 9 9 2 1 3 I u 9 8 ? § 4 3 2 u ES i B 8 / 4 9 5 3 i Q 6 2 4 / 8 3 K 9 1 5 2 6 n E X F L U D 0 9 0 7 6 i5 * 4 y 7 « y o 7 4Q 5 6 EXWATER S g 1 6 3 I 4 3 9 U EXFAT 3 I S " e 8 / 4 9 5 3 0 fi 2 4 / 8 1 5 9 '2 fi n F i g u r e 40. R e l a t i o n s h i p s between t h e i n g r e d i e n t s p r o p o r t i o n s and t h e q u a l i t y p a r a m e t e r s t h a t d e s c r i b e p r o d u c t w e i g h t l o s s , e m u l s i o n 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 . P o r k f a t , XI; MDPM, X2; B e e f meat, X3. 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. - 2 2 5 -MOBTUC FATt pncnat PH — — 6 m e 0 0 • 0 508 0 0 05 0 0 • 2 1 8 0 "TMLOSS 4 3 6 0 1 2 « 8 8 0 7 0 6 3 4 4 3 ° f l 1a n f l8 0 7 0^3 0 8 a 4 78 * 0 4 a 6 0 moss 6 a t 4 3 , 2 0 0 2 j 3 4 0 7 " " 43 l 2 o 43^ 0 2 , 3 4 0 0 4 3 , 2 ES 4* a 3 0<J2 478 a JT0 2 0 o 8 7 4 a ' ' « 1 0 8 2 874 3 0 8 2 478 3 2 0 O BAUD « a o 7 ' 3 ^ 0 8 8 7 M 3 4 8 0 0 7 43 i S « 7 «\ 0 8 8 7 M a 4 8 0 0 7 4 a f 2 f l EXMMTEn 0 0 8 II 2° 0 0 0, 8 0 8 8 . 0 y 0 0 8 0 , 7 6 a^ 0 0 * 7 3 4 0 0 8 ,0 EM7CT 4B *\ 200 » • 0 02 478 V 2 0 0 8 7 4 0 fl 2 874 002 478 '* 200 Figure 41. Relationships 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 that describe product weight loss, emulsion s t a b i l i t y and juicine 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 , protein and fat-to-protein 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 qu a l i t y parameters are given in Appendix E. -226-addition of mechanically deboned meat in sausage formulations increased the pH and the water holding capacity of the meat emulsions and reduced t h e i r cooking losses. As with Shrink, Twloss is a measure of weight loss after thermal treatment. Twloss increased as the proportion of beef meat increased (Figure 40). It increased as the protein content of the meat blocks increased and as the pH of the raw emulsions decreased (Figure 41). The results from Townsend's emulsion s t a b i l i t y test indicated that water loss was a major factor contributing to weight loss of the cooked emulsions. Thus the tendency for formulations containing higher proportions of beef meat to lose more weight ( i . e . water) may be attributed to the lower pH of these emulsions which lowered the water binding ab i 1 i t y . ES increased as the proportion of pork fat increased (Figure 40). It decreased as moisture and protein content of the meat blocks and moisture content of the raw emulsions increased and increased as the fat content and fat-to-protein r a t i o of the meat blocks increased (Figure 41). These relationships suggest that high levels of pork fat and thus high fat contents in the meat block have a negative e f f e c t on the s t a b i l i t y of the emulsions to thermal treatment since there is not enough lean meat and thus not enough protein to s t a b i l i z e the f a t . As Asghar et a l . (1985) pointed out the amount of fat in a sausage formulation has a 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. - 2 2 7 -MDPM and pH had a similar e f f e c t on Ex f l u i d as on Tmloss (Figures 40 and 41), that i s , lesser amounts of f l u i d were expressed from frankfurters exhibiting higher levels of MDPM and thus higher pH values. Exwater increased as the proportion of beef meat increased (Figure 40). It increased as the moisture and protein contents of the meat blocks and moisture content of the raw emulsions increased and as the fat content and fat-to-protein r a t i o of the meat blocks decreased. It increased as the pH of the raw emulsions decreased (Figure 41). Higher moisture contents found in formulations with higher proportions of beef meat may have been the cause of increased expressible water. In addition the lower pH found in these formulations may account for the lower water binding properties giving higher levels of expressible water in the cooked frankfurters. Exfat increased as the proportion of pork fat increased (Figure 40). It increased as the fat content and fat-to-protein r a t i o of the meat blocks increased and decreased as the moisture and protein contents of the meat blocks and moisture content of the raw emulsions increased. These relationships suggest that high levels of pork fat and thus high fat contents led to higher levels of expressible fat in the cooked frankfurters. A l l the textural parameters appeared to relate in a similar fashion to the proportions of the ingredients and to the composition of the meat blocks and raw emulsions (Figures 42 and 43). The textural parameters decreased as the proportion of pork -228-X1 X2 X3 HARD1 y 5 Q . 3 4 I t § ? < 3 H A R M u 3 5 U 8 7 « u 9 5 6 9 2 7 8 8HEAR § 1 5 9 3 1 9 | 2 8 7 4 3 i 2 5 9 COHES u 6 2 -,59 U 6 9 5 2 8 7 4 3 U 6 X2 5 9 3 7 R GUMMY u a -so 09 6 2 8 7 4 3 U ,2 5 6 9 7 8 0 CHEWY u 6 Q * • • . i u 9 6 8 7 4 3 1 u 6 9 Figure 42. Relationships between the ingredients proportions and the textural parameters. Pork fat, XI; MDPM, X2; Beef meat, X3. Nomenclature and definition of the quality parameters are given in Appendix E. -229-VWTURE FAT FK7TEN PH FP Moarai 0 a 4 0 02 6 i 9 a 4 0 °-90 a 4 0 9 05 2 8 7 1 a 4 0 02$ 1 67 a 4 0 §20 78 1 a 4 HAWM 0 *a 1 4 0 V 4 0 tt>8 y 4 0 4 0 fl2f V 4 V 4 SHEAR 0 6 r 0 0 a l 0 0 0 8 0 6 0 8 / OOHEB 0 6 0 0 674 0 6 47% 0 0 9 6 ^ 8 7 43 0 8 874 0 0 a f 2 478 CUMY 0 & as % 1 a 4 0 62 0 1 € a 4 0 000 781 a 4 0 0 85 2 87 1 a 4 0 820 1 87 a 4 0 028 78 1 a 4. CHBVY 0 fl 8 C 2 0 8 9 0 98 a / 0 9 0 6 2 0 A 9 25 1 a8? 4 0 fl 8 4 F i g u r e 43. 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 t e x t u r a l parameters. Proximate composition of meat b l o c k : 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. -230-fat increased and increased as the proportion of beef meat increased. It has been reported that fat tends to have a tenderizing e f f e c t while lean meats have a toughening ef f e c t (Park et a l . 1989; Lyon et a l . 1981; Swift et a l . 1954). Marquez et a l . (1989), St. John et a l . (1986), Singh et a l . (1985) Morrisey et a l . (1982) and Simon et a l . (1965) have reported that textural parameter values are higher for meat samples with higher protein contents. Lower textural parameter values have been observed in sausage formulations with high fat contents and high fat-to-protein r a t i o s (Park et a l . 1989; Hand et a l . 1987; Siripurapu et a l . 1987; Mittal and Usborne, 1986). The tendency for the textural parameters to increase with increasing moisture contents may be attributed to increases in Shrink ( i . e . weight loss) (Figure 44) observed in formulations with higher moisture contents (Figure 41). It has been reported that weight loss leads to greater compression forces (Lee et a l . 1987) possibly due to concentration of protein that leads to greater textural strength (Trout and Schmidt, 1987). The positive relations observed between Shrink and Twloss, Shrink and Exwater, and Twloss and Exwater (Figure 45) re f l e c t e d the e f f e c t of the proportion of beef meat and thus the ef f e c t of moisture and protein contents and the pH of the raw emulsions on these q u a l i t y parameters. The positive r e l a t i o n observed between Tmloss and E x f l u i d (Figure 45) re f l e c t e d the e f f e c t of the proportion of MDPM and thus the ef f e c t of pH on these quality parameters. While the positive r e l a t i o n observed between ES and -231-8HRNK 4 3 fi TMLO88 0 9 7 ° 4 3 £ 0 9 6 5 7 8 21 34 TWLO88 0 2 § 871 3 4 2 6 5 9 1 78 3 4 0 2 5 6 g 1 78 3 4 KAR01 0 4 ' 0 0 0 1^ SHEAR 0 6 48? 0 6 2t 5 9 3 4 7 8 0 6 f 5 9 § 7 8 0 6 3 I" 4 3 8 0 6 47c? COHES Figure 4 4 . Relationships 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 in Appendix E. -232-F T 9 4 3 g 1 ° 2 6 7 43£l 0 tvcoas 6 i 2| 3 4 IT * 7 8 4 7 8 2 8 2L 7J -9 9 e e a 4 TWU388 4 7 8 3 1 5 2 6 "8" 4 58 0 0 9 "OCT f 7 5 83 EB 7 9 * 7 EXRJUD "7J-9 8 EXWATB* 4 1 8 4 7 8 1 ° 2 g Q 4 7 8 1 6  2 6 0 7 8 2 6 4L 3 1 5 28 9 JL f 23 EMWT Figure 4 5 . Relationships between the qu a l i t y parameters that describe product weight loss, emulsion s t a b i l i t y and juiciness 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 in Appendix E. - 2 3 3 -Exfat (Figure 45) reflected the eff e c t of the proportion of pork fat and thus the eff e c t of fat content and the fat-to-protein r a t i o on these qua l i t y parameters. A l l the textural parameters related p o s i t i v e l y to each other (Figure 46). These results confirm the findings of co r r e l a t i o n analysis performed on the experimental data. As can be observed, not only the ingredient proportions had an effe c t on the qual i t y parameters, but the moisture, protein and fat contents of the meat blocks and moisture content of the raw emulsions, though dependent on the ingredients proportions also affected the qual i t y parameters. In addition, the pH of the raw emulsions and the weight loss after thermal treatment further affected several q u a l i t y parameters. It can be concluded that the quali t y of the formulations was affected by a number of interrelated factors. -234-HAFC1 0 3 8 4 HAFD2 0 6 1 * 3 8 4 Y 0 8 8 HEAR 0 6 0 6 3 l 2 8 4 3 e 7 0 6 471? OOHE8 0 3 4 0 29 3 4 0 78 1 3 4 0 3 6 B 1 3 4 GUMMY 0 0 0 CHBVY Figure 46. Relationships between the textural 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 in Appendix E. -235-C. COMPUTATIONAL OPTIMIZATION OF FRANKFURTER FORMULATIONS 1 . Optimization of frankfurter formulations using the new  formula optimization computer program (FORPLEX) The main objective of thi s study was to establish a formula optimization computer program to be used for q u a l i t y control in the meat processing industry. Unlike least-cost formulation programs, the approach of thi s new formula optimization program is to search for best q u a l i t y formulations that meet predetermined product s p e c i f i c a t i o n s within allowable cost ranges. This formula optimization program (FORPLEX) is based on the modified version of Box's Complex method, which is a di r e c t search optimization method that has been used to optimize nonlinear objective functions subject to linear and nonlinear constra i n t s . To test the s u i t a b i l i t y of the FORPLEX program for meat formula optimization, several frankfurter formulation optimization t r i a l s were performed. As in a l l optimization problems there were three optimization components that had to be defined: (A) the optimization factors, (B) the constraints, and (C) the objective function. (A) The optimization factors The optimization factors are the variables, i . e . the ingredients, 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 product under study. In this study, the optimization factors were two ingredients (pork fat, Xa., and mechanically deboned poultry meat (MDPM), X 2) of the model frankfurter formulation. As explained below, X 3 could not be considered an optimization factor due to the constraint of the t o t a l of the Ingredients proportions having to equal one. (B) The constraints The constraints are the mathematical functions that place r e s t r i c t i o n s on the formulations and the optimization factors, r e s t r i c t i n g the value of the objective function (Norback and Evans, 1983). The constraints imposed on the frankfurter formulations were kept r e l a t i v e l y simple and were a r b i t r a r i l y selected. The following constraints were used throughout the study: 1. Constraints on the ingredient proportions The l i m i t s set for the proportions of the three ingredients were the ones used in the extreme vertices design (Table 1). The l i m i t s set for the experimental design had to be used, since the quali t y prediction models can only be used within the l i m i t s tested. These l i m i t s are commonly referred to as e x p l i c i t constra i n t s . 0.05 < X i < 0.30 0.00 < X 2 < 0.40 0.50 < X 3 < 0.95 2. Constraint on the t o t a l of the ingredient proportions, in th i s case the sum of the ingredients proportions must equal unity. X i + X 2 + X 3 = 1 (30) 3. Constraints on the proximate composition of the meat block (a) fat content had to be greater than or equal to 8.0% but less -237-than or equal to 28.0%. 8.0 < 8 0 . 3 7 X 1 + 18.67X 2 + 3.72X 3 < 28.0 (31) (b) protein content had to be greater than or equal to 16.0% but less than or equal to 21.0%. 16.0 < 4.96Xa. + 15.51X 2 + 22.49X 3 < 21.0 (32) (c) moisture content had to be greater than or equal to 54.3% but less than or equal to 89.0% 54.3 < 14.58X3. + 6 5 . 6 9 X 2 + 73.64X 3 < 89.0 (33) 4. Constraint on the cost of the meat block The cost of the meat block had to be greater than or equal to $1.9/Kg but less than or equal to $2.8/kg. The cost of the ingredients were provided by Sebastian (1989). 1.9 < 0.20Xi + 0.80X 2 + 3.20 X 3 < 2.8 (34) As can be seen in Appendix A Figure A3 these constraints reduced the feasible area. The linear equations that described proximate composition, cost and the constraint that the sum of the ingredients equal unity i . e . the i m p l i c i t constraints, were entered in the Constraint subroutine. However, since the Complex method cannot work with equality constraints, the equality constraint was removed by replacing the factor X 3 with the expression 1 - Xa. - X a (35) in the objective function and the i m p l i c i t constraints, and incorporating t h i s expression as an i m p l i c i t constraint (replacing the equality constraint) with lower and upper l i m i t s of 0.50 and 0.95 respectively. -238-(c) The objective function The objective function is the mathematical function that represents the desired objective of the problem, and can be either maximized or minimized (Wolfe and Koelling, 1983). In th i s study, the objective was to find the optimal combination of ingredient proportions which gave best q u a l i t y formulations that met the product s p e c i f i c a t i o n s and cost constraints. Best qu a l i t y formulations were defined as those formulations whose predicted qu a l i t y was as close as possible to a predetermined target quality. Nakai and Arteaga (1990) recommend the use of target q u a l i t y values since maximization or minimization of a qua l i t y parameter, e.g. hardness at f i r s t compression (Hardl), may lead to unacceptable products, i . e . extremely hard or extremely soft frankfurters. Several hypothetical frankfurter formulation optimization t r i a l s were performed using d i f f e r e n t q u a l i t y parameters as measures of the formulations' q u a l i t y . The q u a l i t y prediction equations (Table 40) were incorporated into the Function subroutine of the FORPLEX program to predict the qua l i t y of frankfurter formulations at each point computed by the program. Since in food formulation studies i t is frequently necessary to optimize more than one qua l i t y parameter simultaneously, the computational optimization t r a i l s were performed using two to five q u a l i t y prediction models. Therefore, combination of single objective functions into one equation, i . e . a multi-objective function, was performed. -239-Target values and target difference values were set for each qu a l i t y parameter. Target values are the optimum values for the quali t y parameters and a target difference value is the minimum difference in the qua l i t y parameter value r e s u l t i n g in a noticeable difference in the qua l i t y parameter value. In addition, the target difference values standardize each single objective function, which is useful to prevent the effects of difference in dimensions of qua l i t y parameter values. The multi-objective function that was minimized was the absolute value of the products of standardized differences of the predicted q u a l i t y parameters from their respective target q u a l i t y values. Subtraction of the predicted q u a l i t y values from their respective target quality values was performed to meet the optimization objectives, i . e . minimization of the difference between the predicted q u a l i t y of a formulation and i t s respective target (optimal) qu a l i t y . Mathematically, the multi-objective function can be written N R = TI Abs [ target quality* - predicted quality* ] (36) i=l target difference* i= 1, . . .N where N = number of qu a l i t y parameters being optimized target q u a l i t y i = target value set for the i t h qua l i t y parameter predicted q u a l i t y i = predicted value of the i t h qua l i t y parameter target d i f f e r e n c e i = target difference value set for the i t h quali t y parameter -240-It should be pointed out that the FORPLEX program searches for the maximum value of a nonlinear objective function. Since minimization of the multi-objective function is required, maximization of the negative of t h i s function was performed. The multi-objective function was incorporated in the Function subroutine which also included conditional instructions to replace each standardized single objective function value less than or equal to one with unity; the minimum multi-objective function value which can be expected is unity. The purpose of t h i s r e s t r i c t i o n was to avoid: (a) overemphasizing or overlooking individual objective functions 1 and (b) having response values of zero. 2 1 For example, i f two qual i t y parameters Shrink and Hardl are being optimized, and their target values are 8.70 and 160.00 respectively, their target difference values are 0.022 and 0.803 respectively, and their predicted values for a pa r t i c u l a r point are 8.71 and 180.00 respectively, the multi-objective function for this p a r t i c u l a r point would be Abs [8.70 - 8.71] x Abs [160.00 -180.003 = 0.45 x 24.91 0.022 0.803 The response value would be 11.21 i f the function value for Shrink is not replaced by unity. In this case the FORPLEX program may emphasize i t s search towards the minimum value of Shrink and may overlook the Hardl response value, which is far from i t s target. 2 For example, i f the predicted Shrink value is equal to i t s target, the multi-objective function for this p a r t i c u l a r point would be Abs [8.70 - 8.701 X Abs [160.00 - 180.001 = 0.00 X 24.91 0.022 0.803 The response value would be zero i f the function value for Shrink is not replaced by unity. In this case the search w i l l be ended and Hardl w i l l be overlooked. -241-The flow chart of the Function subroutine is shown in Figure 47. When the computed value of a standardized single objective function is equal or less than 1 ( r i < 1 ) , there is no noticeable difference between the target and the predicted q u a l i t y parameter value and thus a value of unity is assigned to i t ( r i =1). When the multi-objective function value is 1 (R = 1 ) , a l l the predicted q u a l i t y values are no d i f f e r e n t from their targets. For some qual i t y parameters, such as those describing product weight loss and emulsion s t a b i l i t y i t was necessary to establish a maximum allowable target value but any predicted value below this target was considered better quality. For this reason, in some optimization t r i a l s that incorporated these qual i t y parameters, the standardized single objective function values (ri.) were replaced with unity when the predicted values were equal to or less than their target values. If the predicted values were greater than their target values, the computation followed the procedure outlined in Figure 47. Following the recommendations given by Nakai and Arteaga (1990) several frankfurter formula optimization t r i a l s were performed having a single q u a l i t y parameter as a constraint. The q u a l i t y prediction models were incorporated into the Constraint subroutine. The conversion of r e l a t i v e l y unimportant objective functions to constraints s i m p l i f i e s the optimization and simultaneously emphasizes more important qu a l i t y items (Nakai and Arteaga, 1990). In addition to specifying the quality prediction models, the -242-Computation of the q u a l i t y prediction models (N) r t = Abs [target quality^ - predicted quality^] target d i f f e r e n c e ! No N R = it r i ^ i = l Figure 47. Flow chart of the Function subroutine of the FORPLEX computer program. - 2 4 3 -multi-objective function and the i m p l i c i t constraint functions, the FORPLEX program required the following for each frankfurter formulation optimization t r i a l : (A) S p e c i f i c a t i o n of the convergence parameters The convergence parameters ALPHA, BETA and GAMMA were set at 1.3, 0.0001 and 5 respectively, unless otherwise s p e c i f i e d . (B) Number of points in the complex As recommended by Box (1965) the number of points in the complex should be double the number of optimization factors. Therefore the number of points in the complex was set 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 for the e x p l i c i t constraints (on the optimization factors, X i and X 2) and i m p l i c i t constraints (proximate composition, cost and X 3, and on the qu a l i t y parameters i f they were considered constraints). (D) S p e c i f i c a t i o n of the Random number seed (line number 270) As mentioned in Materials and Methods section H.2.1.4.1. the random number seed was set to 3, thus, the sequence of random numbers was the same for a l l the optimization t r i a l s . Because one random number seed was used, several s t a r t i n g points had to be used in each optimization t r i a l to check the convergence to the global optimum. (E) S p e c i f i c a t i o n of the i n i t i a l feasible point To s t a r t a search the FORPLEX program requires an i n i t i a l point that does not v i o l a t e the constraints. Four s t a r t i n g points were generated using the FPOINT computer program. These were: (a) X i = 0.127, X 2 = 0.123; (b) X i = 0.291, X 2 = 0.104; (c) Xa. = 0.100, -244-X=> = 0.220; (d) Xi=0.100, X2=0.380. Appendix C l i s t s the FPOINT computer program. The Function subroutine used starts in line 46. As mentioned in Materials and Methods section H.I., the lower and upper l i m i t s of the optimization factors and/or constraints were narrowed to obtain the four s t a r t i n g points. In some optimization t r i a l s additional points were needed. These were obtained by examining the corresponding feasible mixture spaces. The following sections show the results of the computational optimization t r i a l s performed using the FORPLEX program. In addition to searching for best q u a l i t y formulations that met the constraints for product s p e c i f i c a t i o n s and cost, the FORPLEX program was used for finding the minimum and maximum values of each quality parameter within the feasible area. 1.1. Single objective optimization The FORPLEX program was used for finding the global maximum and minimum values of the qual i t y parameters, except pH and Hard2, within the feasible area, that i s , meeting the constraints for: (a) proximate composition, (b) ingredient levels and (c) cost. The purpose of these t r i a l s was to evaluate the performance of the FORPLEX program in handling each linear and nonlinear q u a l i t y prediction equations i n d i v i d u a l l y (Table 40). In t h i s section the convergence parameter BETA was set at 0.1 for the optimization of a l l q u a l i t y parameters except Cohes, where BETA was set at 0.001. As mentioned before, four s t a r t i n g points were used in each optimization t r i a l , thus, four optimization -245-runs were performed for each q u a l i t y parameter. The reason for r e s t a r t i n g each search from d i f f e r e n t s t a r t i n g points was to check whether the global rather than a l o c a l maximum or minimum had been found. Further v e r i f i c a t i o n of the optimum results was performed by examining the response surface contour lines of each quality prediction model within the feasible area (figures not shown). Table 44 contains the global maximum and minimum values of each q u a l i t y parameter and the corresponding optimal combinations of ingredient proportions. As expected, for the optimization of each qu a l i t y parameter, convergence to the optimum depended on the s t a r t i n g points. In general, the FORPLEX did not converge to a single point with a l l four s t a r t i n g points, but converged on areas close to the global optimum. In some runs the FORPLEX f a i l e d to converge to the optimum since the search was s t a l l e d at the boundaries of the constraints. 1.2. Multi-objective optimization As mentioned e a r l i e r , the objective of the optimization of frankfurter formulations using the FORPLEX program was to find optimal combinations of ingredient proportions which gave best q u a l i t y formulations that met the product s p e c i f i c a t i o n s and cost constraints. Several hypothetical frankfurter formulation optimization t r i a l s were performed using two to fiv e quality parameters as measures of the formulations' q u a l i t y . In addition, optimization of frankfurter formulations where a quality parameter was considered a constraint was performed. Target -246-T a b l e 4 4 . opt imum 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 the 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 . Maximum Minimum T a r g e t Q u a l i t y d i f f e r e n c e p a r a m e t e r * v a l u e 0 X i X 2 X a V a l u e X i X 2 Xa V a l u e S h r i n k 0, .08 0, .07 0, .85 10, . 53d» T m l o s s 0, , 21 0. .00 0, .79 42 ,91c T w l o s s 0. .13 0. , 00 0. , 87 27, . 6 8abc ES 0 , .30 0, ,08 0 , 62 0 ,67d E x f l u i d 0, , 30 0. , 00 0, , 70 12, . 59a-d E x w a t e r 0 , .13 0 , 00 0, .87 9, ,15a E x f a t 0, , 30 0. .08 0. ,62 5, ,4 Id H a r d l 0 . ,13 0 , 10 0, ,77 192, ,12a S h e a r 0. ,05 0. ,10 0. ,85 6. . 60ab Cohes 0. ,05 0. ,10 0. ,85 0, . 30ab Gummy 0. .09 0. ,06 0. , 85 56. ,46c Chewy 0. ,05 0. , 10 0. ,85 226, , 66b-i) 0 .28 0. 19 0. 53 8.37c 0 .022 0 .05 0. 40 0. ,55 31.69a-d 0 .112 0 .10 0 . 40 0. ,50 20.91ac 0 .068 0 .05 0. 10 0. ,85 0 .0 3ac 0 .006 0 .14 0. 36 0, . 50 8.86a 0 .037 0 .26 0. 22 0, .52 4.60a-d 0 .046 0 .05 0. 11 0 . , 84 1.98a 0 .034 0 .25 0. 23 0, , 52 111.87bc 0 .803 0 .29 0. 15 0, .56 4.32b 0 .023 0 .30 0. 08 0, .62 0 .25M 0 .0006 0 .28 0 . 20 0, .52 28 .07c 0 .284 0 .28 0. 19 0, .53 92.97a 1 .337 *• 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 same g l o b a l opt imum 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 . 2 9 1 , 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 . 3 8 0 ° 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 - minimum v a l u e ) ( 3 7 ) values and target difference values were set for each quality parameter. As discussed below, target values for each qu a l i t y parameter were chosen according to the objective of the optimization section. The target difference value set in th i s study for each q u a l i t y parameter was equal to 1% of the range of each q u a l i t y parameter value within the feasible area (Table 44). 1 . 2 . 1 . O p t i m i z a t i o n p j frankfurter formulations. where combinations of two to five quality parameters were  considered measures of the formulations' q u a l i t y . Target  qu a l i t y values were calculated from target points. The objective of th i s section was to test the s u i t a b i l i t y of the FORPLEX program for obtaining frankfurter formulations whose predicted q u a l i t y was not d i f f e r e n t from their target q u a l i t y . To meet th i s objective a l l the target quality values had to be attained in the same combination of ingredient proportions, thus, two feasible target points ( i . e . , two formulations) were a r b i t r a r i l y chosen: point 1 (X:L = 0.250, X 2 = 0.200, X 3=0.550) and point 2 ( X i = 0.150, X 2 = 0.100, X 3 = 0.*750 ). The values of the quality parameters for these two combinations of ingredients proportions were predicted using the qu a l i t y prediction models (Table 40). The predicted values of the q u a l i t y parameters of point 1 were used as targets in the frankfurter formulation optimization t r i a l s that incorporated pairs of q u a l i t y parameters as measures of the formulations' quality. The predicted values of the qu a l i t y parameters of point 2 were used as targets in the frankfurter optimization t r i a l s that incorporated three to five q u a l i t y parameters as measures of the formulations' q u a l i t y . The -248-combination of the q u a l i t y parameters used in each optimisation t r i a l was a r b i t r a r i l y selected. The the o r e t i c a l minimum response value of these optimization t r i a l s was expected to be unity and the optimum combinations of ingredient proportions were expected to be close to the proportions of the two target points. The four s t a r t i n g points (a to d) mentioned before were used in each optimization t r i a l , thus, four optimization runs were performed in each t r i a l . The use of d i f f e r e n t s t a r t i n g points allowed not only a check of whether the the o r e t i c a l minimum had been found but also proof that more than one combination of ingredient proportions ( i . e . formulations) could be found whose predicted qu a l i t y was not d i f f e r e n t from the target quality. The reason why di f f e r e n t combinations of ingredient proportions can give a multi-objective function value equal to one is based on the conditional instructions incorporated in the Function subroutine that replaced each standardized single objective function value less than or equal to one with unity. Table 45 shows the optimization results for frankfurter formulations whose required q u a l i t y was given by the combination of pairs of quality parameters (six optimization t r i a l s ) . The convergence parameter BETA was set at 0.01 for t r i a l s 1 and 2, and at 0.0001 for a l l other t r i a l s . The Function subroutine of the FORPLEX program for t r i a l s 1 and 2 had conditional instructions to replace the standardized function value of Shrink and Twloss with unity when the predicted values of these quality parameters were equal to or less than their target values. This -249-Table 45. 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 p a i r s of q u a l i t y parameters were co 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 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= 0.250, Xa = 0.200,X.= 0 . 5 50 ) * . Optimum I n g r e d i e n t s P r e d i c t e d q u a l i t y p r o p o r t i o n s v a l u e s T r i a l Q u a l i t y Target S t a r t i n g IT° No. parameters" values p o i n t s 0 X i Xa X j 1 2 1 1. Hardl 132. 30 a 22 0.250 0.201 0 . 549 131, ,85 8.75 2. S h r i n k " 8 . 75 b 20 0.282 0 .160 0 . 558 132, , 37 8 . 57 2 1. Twloss* 22. ,10 a 13 0.238 0.242 0. 520 21, .64 4.53 2. Shear 4. ,53 b 22 0.241 0 .233 0. 526 21, ,73 4 .53 c 14 0.237 0.245 0. ,518 21 .60 4 . 53 d 15 0.237 0.242 0. .521 21 .64 4.53 3 1. E x f l u i d 9. 60 b 59 0.246 0.196 0, ,557 9 .63 0.25 2. Cohes 0. ,25 c 17 0.251 0.204 0.545 9 .57 0.25 d 22 0.247 0.199 0, ,554 9 .61 0.25 4 1. Tmloss 38, ,10 a 20 0.250 0.201 0, ,549 38 .07 0.55 2. ES 0. .55 b 30 0.250 0.202 0, .548 38 .06 0.55 c 99 0.249 0.205 0, ,547 38 .00 0.55 d 27 0.249 0.198 0, .552 38 .14 0.55 5 1. E x f a t 4.78 b 72 0.252 0.199 0, .549 4 .81 33.66 2. Gummy 33 .87 c 23 0.251 0.198 0, ,552 4 .79 34.09 d 30 0.252 0.198 0. , 550 4 .81 33.78 6 1. Exwater 4 .88 a 24 0. 252 0.198 0, . 550 4 .88 105.22 2. Chewy 105 .71 b 18 0.243 0.210 0, .547 4 .87 106.80 c 23 0.250 0.199 0 . 551 4 .90 106.14 d 23 0.253 0 .197 0 . 550 4 .88 104.79 k Mult 1-objectWe f u n c t i o n value = 1 • 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. ° S t a r t i n g p o i n t s where convergence to the g l o b a l minimum was achiev e d . (a) Xi= 0.127, X,= 0.123; (b) Xi= 0.291, Xj= 0.104; (c) Xi= 0.100, X 2= 0.220; (d) Xi= 0.100, Xi= 0.380. ° Number of I t e r a t i o n s * I f the p r e d i c t e d q u a l i t y value was l e s s than or equal t o the 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 value was r e p l a c e d w i t h u n i t y was performed since formulations with lower predicted product weight loss than the predetermined target had better than optimal weight loss. Results reported are only for those optimization runs where the FORPLEX converged at the the o r e t i c a l minimum, that i s , at those computed formulations whose predicted q u a l i t y was not d i f f e r e n t from their target quality. The the o r e t i c a l minimum of 1 was found in a l l the optimization t r i a l s . The FORPLEX was able to locate the th e o r e t i c a l minimum in t r i a l s 2, 4 and 6 regardless of the s t a r t i n g points used. However in t r i a l s 1, 3, and 5 convergence to the minimum depended on the star t i n g points. In a l l the optimization t r i a l s the FORPLEX converged to the theo r e t i c a l minimum when st a r t i n g point b was used. With the exception of t r i a l 1, the FORPLEX also converged to the minimum when st a r t i n g point c was used. The number of iterati o n s needed to locate the optimum in each optimization t r i a l was influenced by the s t a r t i n g points. As can be seen (Table 45), d i f f e r e n t optimum combinations of ingredients proportions were found in each successful run in each optimization t r i a l . These optimum points ( i . e . , formulations) were located in the v i c i n i t y of target point 1 (Xi=0.250, X2=0.200, X3=0.550). The optimization results for frankfurter formulations whose required q u a l i t y was given by the combination of three (two optimization t r i a l s ) and four q u a l i t y parameters (one optimization t r i a l ) are given in Table 46. Results reported are only for those optimization runs where the FORPLEX converged at the t h e o r e t i c a l minimum, that i s , at those computed formulations -251-T a b l e 46. 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 t h r e e and f o u r 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 s ' 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 f r o m t a r g e t p o i n t 2 ( X x = 0 . 1 5 0 , X 2 = 0 . 1 0 0 , X,= 0 . 7 5 0 ) * . T r i a l No. Q u a l i t y P a r a m e t e r s ' T a r g e t v a l u e s S t a r t i n g P o i n t s 0 I T 1 Optimum I n g r e d i e n t s p r o p o r t i o n s P r e d i c t e d q u a l i t y v a l u e s I to to I 1 1. E x w a t e r 7.32 a 31 0 .148 0. ,101 0 . 752 7 .33 179 . 11 53 .11 2. Chewy 178 .13 c 17 0 .154 0. ,094 0 . 752 . 7 .35 1 7 7 . 43 52 .80 3. Gummy 53 .00 2 1. S h r i n k 10 .04 a 17 0 .152 0. .089 0 . 759 1 0 . 0 4 0 . 25 1 9 1 . 0 4 2. ES 0 .25 b 47 0 .148 0, .117 0. 735 1 0 . 0 2 0 . 25 191 .33 3. H a r d l 191 .40 c 20 0.149 0, .092 0. 759 1 0 . 0 5 0 . 25 1 9 1 . 3 5 d 43 0 .151 0 , .101 0 . 748 1 0 . 0 3 0 . 25 1 0 1 . 3 2 3 1. T m l o s s 40 .27 a 50 0 .150 0. ,100 0 . 750 4 0 . 2 7 2 5 . 61 1 0 . 6 7 2 . T w l o s s 25 .61 b 40 0 .150 0, ,101 0. 749 40 .24 2 5 . 60 1 0 . 6 6 3 . E x f l u i d 10.68 d 52 0 .150 0. .100 0. 750 4 0 . 2 7 2 5 . 51 10 .68 4. E x f a t 3.35 3 .35 3.35 3 .35 M u l t i - o b j e c t i v e f u n c t i o n v a l u e = 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 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 a c h i e v e d : (a) X i= 0 . 1 2 7 , X a = 0 . 1 2 3 ; (b) Xx= 0 . 2 9 1 , X a = 0 . 1 0 4 ; ( c ) X i = 0 . 1 0 0 , Xa= 0 . 2 2 0 ; (d) X i= 0 . 1 0 0 , X 3 = 0 . 3 8 0 . Number o f i t e r a t i o n s whose predicted q u a l i t y was not d i f f e r e n t from their target qu a l i t y . The th e o r e t i c a l minimum of 1 was found in a l l the optimization t r i a l s . The FORPLEX was able to locate the the o r e t i c a l minimum in t r i a l 2 regardless of the s t a r t i n g point used. However in t r i a l s 1 and 3 convergence to the minimum depended on the s t a r t i n g points. In a l l the optimization t r i a l s the FORPLEX converged to the t h e o r e t i c a l minimum when st a r t i n g point a was used. This may be due to the fact that this point was located in the v i c i n i t y of target point 2 (Xi=0.150/ X2=0.100, X3=0.750) and therefore i t was easier for the FORPLEX to locate the optimum. The number of iterati o n s needed to locate the optimum in each optimization t r i a l was influenced by the s t a r t i n g points. In t r i a l s 1 and 2 d i f f e r e n t combinations of ingredient proportions were found in each successful run. These optimum points ( i . e . formulations) were located in the v i c i n i t y of target point 2. However, in t r i a l 3 the FORPLEX converged to the same combination of ingredient proportions in runs of st a r t i n g points a and d; thi s combination being the target point 2. The run of st a r t i n g point b converged in close proximity of the target point. Four frankfurter formula optimization t r i a l s were performed where five q u a l i t y parameters were considered measures of the formulations quality. The optimization results are given in Table 47. Results reported are only for those optimization runs where the FORPLEX converged at the the o r e t i c a l minimum, that i s , at those computed formulations whose predicted q u a l i t y was not -253-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 vhere f i v e q u a l i t y parameters vere 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 vere c a l c u l a t e d from t a r g e t p o i n t 2 (Xi= 0.150, X*= 0.100, X,= 0 .750 ) * . Optimum I n g r e d i e n t s P r e d i c t e d q u a l i t y p r o p o r t i o n s v a l u e s T r i a l Q u a l i t y Target S t a r t i n g I T D  No. parameters* values p o l n t s a Xx X» X, 1 2 3 ' 4 5 1. Twloss 25. Gl a 16 0, ,150 0.100 0 , 750 25, .61 7 . ,32 3.35 191, ,43 0,27 2 . Exwater 7 . ,32 c 56 0, ,152 0.101 0, .747 25, .57 7 .28 3.38 191, ,28 0 . 27 3 . E x f a t 3. ,35 4 . H a r d l 191. 40 5. Cohes 0, ,27 1. Shrink 10, .04 a 95 0, ,149 0.098 0, ,753 10, ,05 10, .70 5.72 0, ,27 53.08 2. E x f l u i d 10, ,68 b 30 0, ,150 0.101 0, ,749 10, ,03 10, ,67 5.70 0, ,27 52.98 3 . Shear 5, .70 d 72 0, ,151 0.101 0 , 748 10, ,03 10 , .67 5.69 0, ,27 52.93 4 . Cohes 0, .27 5. Gummy 52 .99 1. Tmloss 40. .27 a 27 0. ,150 0.101 0, , 749 40. ,23 3, ,35 191.43 53. ,00 177.96 2 . E x f a t 3, , 35 b 39 0. ,150 0 .097 0 , 753 40 . ,33 3 , 35 191.40 53 . 05 179.04 3 . H a r d l 191, .40 c 41 0. ,149 0.101 0 , 749 40. ,23 3, ,35 191.45 53, ,02 178.16 4 . Gummy 53 , 00 d 40 0, , 150 0 .098 0, ,752 40. , 31 3 . ,36 191.38 53. ,00 178.45 5. Chewy 178, .13 1. Twloss 25, ,61 a 30 0.149 0.102 0, ,749 25. , 59 0, .25 3.35 191, ,45 0.27 2. ES 0 .25 c 35 0 , 150 0.098 0 , 752 25. ,65 0 , 25 3.35 191. ,38 0.27 3 . E x f a t 3, , 35 4 . H a r d l 191 .40 5. Cohes 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 * 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 ° S t a r t i n g p o i n t s where convergence to the g l o b a l minimum was ac h i e v e d : (a) Xx= 0.127, X a= 0.123; (b) Xx = 0.291, Xa = 0.104; (c) Xx= 0.100, Xa= 0.220; (d) Xx= 0.100, Xa= 0.380. ° Number of i t e r a t i o n s d i f f e r e n t from their target q u a l i t y . The th e o r e t i c a l minimum of 1 vas found in a l l the optimization t r i a l s . The FORPLEX was able to locate the th e o r e t i c a l minimum in t r i a l 3 regardless of the st a r t i n g point used. However in t r i a l s 1, 2 and 4 convergence to the minimum depended on the s t a r t i n g points. As in the case of the frankfurter optimization t r i a l s reported in Table 46, the FORPLEX converged to the th 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 it e r a t i o n s needed to locate the optimum in each optimization t r i a l was influenced by the s t a r t i n g points. With the exception of t r i a l 2, fewer it e r a t i o n s were needed to locate the optimum with s t a r t i n g point a. In each optimization t r i a l d i f f e r e n t combinations of ingredient proportions were found in each successful run. These optimum points ( i . e . formulations) were located in close proximity of target point 2 (Xi=0.150, X 2=0.100, X 3=0.750). combinat i ons of five d u a l i t y parameters were considered measures of the formulations' 1 quality. Tarqet Quality values were set i n d i v i d u a l l y Unlike the previous section, the objective of th i s section was to test the s u i t a b i l i t y of the FORPLEX to obtain frankfurter formulations whose predicted q u a l i t y was as close as possible to a predetermined target q u a l i t y . To meet this objective the target qu a l i t y values were i n d i v i d u a l l y selected. This complicated the optimization procedure, since in th i s case the optimal combinations of ingredient proportions for one standardized single objective function could be far from optimal for the -255-other functions. Thus, the FORPLEX had to arrive at a combination of ingredient proportions where a "compromised" optimum was obtained, that i s , where the difference between each predicted q u a l i t y parameter and i t s target value was as small as possible. The multi-objective function value was expected to be higher than unity. The lower the function value the closer the predicted q u a l i t y values were from their target q u a l i t y values. The target quality values of each quality parameter were chosen i n d i v i d u a l l y by examining the response surface contour plots of each quality prediction model (Figures 25 to 38) and taking into consideration that: (a) low weight loss ( i . e . low values of Shrink) and high emulsion s t a b i l i t y ( i . e . low values of ES, Twloss, and Tmloss) are required in the manufacture of frankfurters, and (b) extremely hard or soft frankfurters are not acceptable to the consumer. Thus values in the middle range of Hardl, Shear, Cohes, Gummy and Chewy were chosen. It is recommended that in future studies objective selection of the target values should be made based on previous experience and consumer sensory tests. Five frankfurter formula optimization t r i a l s were performed using the four combinations of five q u a l i t y parameters reported in Table 4 7 . Each optimization t r i a l considered measures of either product weight loss (Shrink) or one emulsion s t a b i l i t y parameter (ES, Tmloss or Twloss), one or two juiciness parameters (Exfluid , Exwater, Exfat) and two or three texture parameters (Hardl, Shear, Cohes, Gummy, Chewy). The Function subroutine of the FORPLEX had conditional instructions to replace the -256-standardized function value of Shrink, ES, Twloss or Tmloss with unity when the predicted values of these qu a l i t y parameters were equal to or less than their target values. This was performed because formulations with lower predicted product weight loss (Shri in each optimization t r i a l . However, additional points were needed in some t r i a l s in order to v e r i f y the convergence to the optimum, these were obtained by examining Figures 48 to 52. These points were: (e) X : L = 0.100, X 2=0.100; (f) Xx = 0.250, X 2=0.050; (g) X i = 0.280, X 2=0.100; (h) Xj. = 0.200, X 2=0.250; (i) X i = 0.280, X 2=0.120. The per cent difference from target was calculated for each qu a l i t y parameter considered in each optimization t r i a l using the followin in each optimization t r i a l . However, additional points were needed in some t r i a l s in order to v e r i f y the convergence to the optimum, these were obtained by examining Figures 48 to 52. These points were: (e) Xa. = 0.100, X 2=0.100; (f) X i = 0.250, X 2=0.050; (g) Xi=0.280, X 2=0.100; (h) X!=0.200, X 2=0.250; (i) Xi=0.280, X 2 = 0 .120. The per cent difference from target was calculated for each qu a l i t y parameter considered in each optimization t r i a l using the following equation % difference = predicted q u a l i t y value - target quality value from target target q u a l i t y value (38) This percentage gave a measure of how far the predicted q u a l i t y value was from i t s target. A positive per cent difference from target value for a par t i c u l a r q u a l i t y parameter meant that the -257-predicted q u a l i t y value was greater than the target q u a l i t y value. For a negative value, the predicted quality value was lower. A zero per cent difference from target meant that the predicted q u a l i t y parameter was not d i f f e r e n t from i t s target. A zero per cent difference from target was assigned to Shrink, Twloss, Tmloss and ES when the predicted values of these quality parameters were less than their target values. This was done to avoid large per cent difference values when in fact the predicted qu a l i t y was better than the required target q u a l i t y . It should be noted that the target difference value was not considered in the computation of t h i s percentage, thus a predicted q u a l i t y value considered by the FORPLEX to be not d i f f e r e n t from i t s target (equation 36) did appear as being d i f f e r e n t from i t s target using equation 38. The optimization results of the fiv e optimization t r i a l s are shown in Table 48. In each optimization t r i a l the FORPLEX varied in i t s success at locating the optimum depending on the s t a r t i n g points used. In addition, the number of i t e r a t i o n s to locate the global minimum depended on the s t a r t i n g points used. In t r i a l 1 the q u a l i t y parameters that were optimized were: Twloss, Exwater, Exfat, Hardl and Cohes. The s t a r t i n g points used in t h i s t r i a l were: a, b, c, d, e and f. The FORPLEX converged to a single point (Xi=0.228, X2=0.176, X3=0.596) with minimum function value of 77.98 in four of six runs (starting points: a, b, e, and f ) . The number of i t e r a t i o n s needed to locate t h i s point were 93, 79, 86, and 55 respectively. Since the majority of -258-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 6 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 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 M u l t i - p r o p o r t i o n s P r e d i c t e d % D i f f e r e n c e T r i a l Q u a l i t y Target o b j e c t i v e q u a l i t y from No. parameters* values f u n c t i o n v a l u e s t a r g e t value X i X* X j 1 1. Twloss" 22.50 Formula 0 2. Exwater 6.00 3. E x f a t 4.50 4 . Hardl 160.00 5. Cohes 0.255 2 1. S h r i n k " 8 .70 Form2 2. E x f l u i d 11.00 3. Shear 4.80 4. Cohes 0.255 5. Gummy 40.00 3 1. Tmloss" 35.00 Form3 2. Exwater 6.00 3. Har d l 160.00 4. Gummy 40.00 5. Chewy 130.00 4 1. Twl03S» 22.50 Form4 2. ES» 0.20 3. E x f a t 4.50 4. Hardl 160.00 5. Cohes 0.255 5 1. Twloss" 22.50 Form4* 2. ES" 0.20 3. E x f a t 3.50 4 . Hardl 160.00 5. Cohes 0.270 77.98 0.228 0.176 334.10 0.288 0.086 21.94 0.107 0.393 293.32 0.228 0.176 70.42 0.118 0.326 0. 596 22. 92 1.87 5. ,42 -9.67 4. ,47 -0.67 159.19 -0.51 0.255 0.00 0. 626 8. ,74 0.46 11, .04 0.36 4, ,48 -6.67 0. ,248 -2.75 39, ,71 -0.72 0, ,500 32, ,90 0.00 5.95 -0.83 150, ,17 -6.14 40.29 0.72 132, ,28 1.75 0, ,596 22, ,92 1.87 0, ,49 145.00 4. ,47 -0.67 159, .19 -0.51 0, .255 0.00 0. .556 22, .03 0.00 0 .21 5.00 3 .02 -13.43 160, .75 0.47 0, , 267 -1.11 * 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 • I f the p r e d i c t e d q u a l i t y value was l e s s than or equal t o the ta 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 value was rep l a c e d w i t h u n i t y c See t e x t f o r d e s c r i p t i o n of names assigned to optimum fo r m u l a t i o n s of t r i a l s 1 to 5. the runs lead to thi s single solution, t h i s point (Formula) was considered the global minimum. As can be seen in Figure 48 i t i s d i f f i c u l t to a t t a i n a multi-objective function value of one since there i s no common point where a l l contour lines intersect. In general, the predicted q u a l i t y of the optimum formulation (Formula) was quite close to the required target q u a l i t y . The target value of Cohes was achieved. The difference between predicted Twloss, Exfat and Hardl values and their target values was less than ±2%, while that of Exwater was -9.7%. In t r i a l 2 the qual i t y parameters that were optimized were: Shrink, E x f l u i d , Shear, Cohes and Gummy. The st a r t i n g points used in t h i s t r i a l were: a, b, c, d, e, f and g. In thi s t r i a l the FORPLEX converged to an optimal area rather than to a single optimal point in two of the seven runs. These points were: Xi=0.288, X2=0.086, X3=0.626 (run of st a r t i n g point b) and Xi=0.290, X2=0.086, X3=0.624 (run of st a r t i n g point g) with function values of 334.10 and 343.03 respectively. The optimum was considered to be the point found using s t a r t i n g point b (Form2) (Figure 49). The d i f f i c u l t y encountered in locating the minimum value more than once may be due to the fact that the optimum point was lo c a l i z e d close to the boundary between the feasible area and a constrained region making i t d i f f i c u l t for the FORPLEX to converge to a single point. The r e l a t i v e l y high function value indicated that the target quality values could not be met. The difference between predicted Shrink, E x f l u i d and Gummy values and their target values was less than ±1%, while -260-Figure 48. Response surface contour l i n e s corresponding to the target q u a l i t y values of Twloss, Exwater, Exfat, Hardl, and Cohes set in t r i a l 1. The black area represents the constrained region which Is given by the proximate composition and cost constraints. The optimum formulation (Formula) i s represented by a closed symbol • -261-MDPM Figure 49. Response surface contour lines corresponding to the target q u a l i t y values of Shrink, E x f l u i d , Shear, Cohes and Gummy set in t r i a l 2. The black area represents the constrained region which i s given by the proximate composition and cost constraints. The optimum formulation (Form2) i s represented by a closed symbol -263-> CO ^ LU Q DC ZD ^ - T < - » ^ 2 J LU L DC 3 O i x X CD O CD UJ 03 I 1 I I • that of Cohes and Shear was -2.8% and -6.7% respectively. In t r i a l 3 the qua l i t y parameters that were optimized were: Tmloss, Exwater, Hardl, Gummy and Chewy. The st a r t i n g points used in this t r i a l were: a, b, c, d, and h. The FORPLEX converged to a single point (Xi=0.107, X2=0.393, X3=0.500) with minimum function value of 21.94 in two of fi v e runs (starting points: b and h). The number of ite r a t i o n s needed to locate t h i s point were 63 and 191 respectively. This point (Form3) was considered to be the global minimum. In thi s t r i a l the target q u a l i t y values were set reasonably close to each other and below the target value of Tmloss as is shown in Figure 50. This allowed for the predicted qu a l i t y of the optimum formulation (Form3) to be close to i t s required target ( i . e . low multi-objective function value). The predicted Tmloss value was better ( i . e . lower) than the target set for t h i s parameter. The difference between predicted Exwater, Gummy and Chewy values and their target values was less than ±2%, while that of Hardl was -6.1%. In t r i a l 4 the quality parameters that were optimized were: Twloss, ES, Exfat, Hardl and Cohes. The ES parameter was considered in th i s formula optimization t r i a l as a measure of the formulation's quality. As mentioned in Results and Discussion (section B.3.I.), the qual i t y prediction model obtained for the ES data was considered appropriate for optimization purposes even though the model found did not adequately describe the experimental data. In th i s t r i a l the ES target value was set at 0.20% released f a t . This low target value meant that high -265-Figure 50. Response surface contour lines corresponding to the target q u a l i t y values of Tmloss, Exwater, Hardl, Gummy and Chewy set in t r i a l 3. The black area represents the constrained region which i s given by the proximate composition and cost constraints. The optimum formulation (Form3) i s represented by a closed symbol -266-emulsion s t a b i l i t y was required in the optimum formulation. The s t a r t i n g points used in this t r i a l were: a, b, c, d, e, g and i . The FORPLEX converged to a single point (Xi=0.228, X2=0.176/ X3=0.596) with a minimum function value of 293.32 in two of the seven runs (starting points b and i ) . The number of ite r a t i o n s needed to locate t h i s point was 60 and 72 respectively. This point (Form4) was considered to be the global minimum. The large function value found was expected since the ES target value was d i f f i c u l t to achieve due to the fact that the contour l i n e corresponding to the formulations meeting this target was set too far from the region of maximum convergence of the contour lines of the other four quality parameters. This can be better understood by examining Figure 51. As can be observed, there is a region where the contour lines corresponding to the targets set for Twloss, Exfat, Hardl and Cohes are close to each other, while the contour lin e of the ES target is located far from th i s region. This c l e a r l y shows that a formulation with high emulsion s t a b i l i t y ( i . e . ES values equal to or less than 0.20%) could not be found. Failure to obtain a formulation that met the ES target lay not with the performance of the FORPLEX but with the s e l e c t i o n of the target values. The difference between predicted ES value and i t s target value was 145%. However, the difference between predicted Twloss, Exfat and Hardl values and their target values was less than ±2%, while the target value of Cohes was met. In Figure 51 i t can also be observed that the contour lines -268-Figure 51. Response surface contour l i n e s corresponding to the target q u a l i t y values o£ Twloss, ES, Exfat, Hardl and Cohes set in t r i a l 4. The black area represents the constrained region which i s given by the proximate composition and cost constraints. The optimum formulation (Form4) i s represented by a closed symbol -269-CO LU CO CO o T - Q ,< ^ DC 11 O I UJ S h • 1 1 1 1 corresponding to the targets set for Exfat and Cohes vere located far from the contour l i n e of the ES target. In addition, the contour lines of the ES, Twloss and Hardl targets were close to each other in an area located around X2=0.30, far from the contour lines of the Exfat and Cohes targets. This meant that the targets set for Exfat and Cohes were a f f e c t i n g the search for a formulation that met the ES target. If i t is of paramount importance to meet the ES target, i t may be necessary to consider modifying the target values set for Exfat and Cohes. By choosing the target values more r e a l i s t i c a l l y , optimum formulations can be found where the difference between target and predicted q u a l i t y values is smaller. This modification was performed in t r i a l 5. The target values of Exfat and Cohes were changed from 4.50 to 3.50 and 0.255 to 0.270, respectively. As can be observed in Figure 52, with the exception of the contour line of the Exfat target, the contour lines were close to each other in an area located around X2=0.30. This c l e a r l y shows that a formulation that meets most of i t s target quality values as clo s e l y as possible could be found. The s t a r t i n g points used in t r i a l 5 were: a, b, c and d. The FORPLEX converged to a single point (Xi=0.118, Xs=0.326, Xa=0.556) with minimum function value of 70.42 in three of the four runs (starting points a, b and c ) . The number of ite r a t i o n s needed to locate t h i s point were 65, 70 and 73 respectively. This point (Form4*) was considered to be the global minimum. The lower multi-objective function value found in thi s t r i a l compared to t r i a l 4 meant that the overall difference -271-Figure 52. Response surface contour l i n e s corresponding to the target q u a l i t y values of Twloss, ES, Exfat, Hardl and Cohes set in t r i a l 5. The black area represents the constrained region which i s given by the proximate composition and cost constraints. The optimum formulation (Form4*) i s represented by a closed symbol • -272-between predicted and target q u a l i t y of the optimum formulation was smaller. The difference between predicted ES value and i t s target value was 5%, for Hardl was less than 0.5%, for Cohes was -1.1%. The predicted q u a l i t y of Twloss was better ( i . e . lower) than i t s target, however the difference between the predicted Exfat value and i t s target value was -13.7%.. 1.2.3. Optimization of frankfurter formulations when a qu a l i t y  parameter was considered a constraint The objective of this section was to test the s u i t a b i l i t y of the FORPLEX to obtain best quality frankfurter formulations that met a q u a l i t y r e s t r i c t i o n in addition to the product s p e c i f i c a t i o n s for proximate composition, ingredient l e v e l s , and cost. Three formula optimization t r i a l s were performed where d i f f e r e n t combinations of q u a l i t y parameters were optimized while one q u a l i t y parameter was considered a r e s t r i c t i o n on the formulation. As in the previous section the combination of qu a l i t y parameters included measures of product weight loss (Shrink) or an emulsion s t a b i l i t y parameter (Twloss), one or two juiciness parameters (Exfluid, Exwater, Exfat) and two or three texture parameters (Hardl, Shear, Cohes, Gummy). The combination of the q u a l i t y parameters was a r b i t r a r i l y selected. The target q u a l i t y values were the same ones used in the previous section. The lower and upper l i m i t s for those qu a l i t y parameters that r e s t r i c t e d the formulations were a r b i t r a r i l y chosen. In order to have a better understanding of the optimization r e s u l t s , contour lines of the target q u a l i t y values within the -274-feasible areas vere drawn for 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 points were used in each optimization t r i a l . These points were selected by examining the feasible area of each optimization t r i a l . The number of it e r a t i o n s to locate the global minimum in each optimization t r i a l depended on the s t a r t i n g points used. The optimization results of t h i s section are shown in Table 49. The Function subroutine of the FORPLEX program for t r i a l s 1 and 3 had conditional instructions to replace the standardized function values of Shrink with unity when the predicted values of this q u a l i t y parameter were equal to or less than the target values set. In t r i a l 1 the qua l i t y parameters that were optimized were Shrink, Shear, Cohes and Gummy while Exflu i d was considered a constraint. The predicted E x f l u i d value of the optimum formulation was r e s t r i c t e d between 9.50 and 10.50%. Figure 53 shows the feasible area and the contour lines corresponding to the target values of the four q u a l i t y parameters being optimized. The two s t a r t i n g points used in th i s t r i a l were: (a) Xx=0.100, X2=0.120 and (b) Xi=0.270 and X2=0.120. The FORPLEX converged in both runs to a single point (Xi=0.223, X2=0.198, X3=0.579) with minimum function value of 18.98. This point was considered to be the global minimum. The number of it e r a t i o n s needed to locate the optimum was 86 and 55 for both runs respectively. The r e l a t i v e l y low function value found indicated that the overal l difference between predicted and target q u a l i t y of the optimum formulation -275-T a b l e 4 9 . 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 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 c o n s t r a i n t . T r i a l No. Q u a l i t y p a r a m e t e r s * T a r g e t v a l u e s Q u a l i t y p a r a m e t e r as a c o n s t r a i n t " M u l t i -o b j e c t i v e f u n c t i o n v a l u e Opt imum i n g r e d i e n t s p r o p o r t i o n s X i X a X j P r e d i c t e d q u a l i t y v a l u e s % D i f f e r e n c e f r o m t a r g e t 1 1 . S h r i n k 0 8.70 E x f l u i d 18.98 0.223 0.198 0.579 9.11 4 . 71 2 . S h e a r 4 . 80 9.50 - 10. 5 4.78 -0 . , 42 3 . Cohe s 0.255 0.255 0 . 00 1 4 . Gummy 40.00 39.71 -0 . ,73 0 J 2 1 . E x w a t e r 6.00 T w l o s s 11.25 0.233 0.164 0.602 5.48 -8. ,67 n i 2 . E x f a t 4.50 22.00 - 24.00 4.53 0, .67 i 3 . H a r d l 160.00 160.81 0, .51 4 . Cohe s 0.255 0.255 0, ,00 3 1 : S h r i n k 0 9.00 H a r d l 2.92 0.230 0.194 0.576 9.06 0, ,67 2 . E x f a t 4.50 150.00 - 180.00 4.50 0, .00 3 . Cohe s 0. 255 0.254 -0, .39 * 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 • Lower and 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 c o n s i d e r e d a c o n s t r a i n t ° 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 or e q u a l t o t h e t a r g e t v a l u e , t h e n t h e 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 surface contour lines corresponding to the target q u a l i t y values of Shrink, Shear, Cohes and Gummy set in t r i a l 1. The black area represents the constrained region which i s given by the proximate composition, cost and q u a l i t y (Exfluid) constraints. The optimum formulation i s represented by a closed symbol • - 2 7 7 -DC LU CO T- (— H LjJ Q 5 < X QC LL S o < X X O I UJ LU 1 1 1 1 was small. The difference between predicted Shear and Gummy values and their target values was less than -1.0%, for Shrink i t was 4.7%, while the target value of Cohes was achieved. The Exf l u i d value of th i s optimum formulation was 9.6%. In t r i a l 2 the q u a l i t y parameters that were optimized were: Exwater, Exfat, Hardl and Cohes while Twloss was considered a constraint. The predicted Twloss value of the optimum formulation was r e s t r i c t e d between 22.00 and 24.00%. Figure 54 shows the feasible area and the contour lines corresponding to the target values of the four qu a l i t y parameters being optimized. The two st a r t i n g points used in this t r i a l were: (a) Xi=0.050, X2=0.300 and (b) Xa. = 0.290 and X 2 = 0.050. The FORPLEX converged in both runs to a single point (Xi=0.233, X2=0.164, X3=0.602) with minimum function value of 11.25. This point was considered to be the global minimum. The number of ite r a t i o n s needed to locate the optimum was 50 and 66 for both runs respectively. The r e l a t i v e l y low function value found indicated that the overal l difference between predicted and target quality of the optimum formulation was small. The difference between predicted Exfat and Hardl values and their target values was less than -1.0%, for Exwater i t was -8.7%, while the target value of Cohes was achieved. The Twloss value of th i s optimum formulation was 23.01%. In t r i a l 3 the q u a l i t y parameters optimized were Shrink, Exfat and Cohes while Hardl was considered a constraint. The Hardl predicted value of the optimum formulation was r e s t r i c t e d between 150.00 and 180.00N. Figure 55 shows the feasible area and the -279-Figure 54. Response surface contour l i n e s corresponding to the target q u a l i t y values of Exwater, Exfat, Hardl and Cohes set in t r i a l 2. The black area represents the constrained region which i s given by the proximate composition, cost and q u a l i t y (Twloss) constraints. The optimum formulation i s represented by a closed symbol• -280-> CO ]> LU ZD O o o LU OC coco I i 1 Figure 5 5 . Response surface contour l i n e s corresponding to the target q u a l i t y values of Shrink, Exfat, and Cohes set in t r i a l 3. The black area represents the constrained region which i s given by the proximate composition, cost and q u a l i t y (Hardl) constraints. The optimum formulation i s represented by a closed symbol* - 2 8 2 -contour lines corresponding to the target values o£ the three q u a l i t y parameters being optimized. The two s t a r t i n g points used in t h i s t r i a l were: (a) Xx=0.050, X2=0.400 and (b) Xx=0.200 and X2=0.000. The FORPLEX converged in both runs to a single point (Xi=0.230, X2=0.194/ X3=0.576) with minimum function value of 2.92. This point was considered to be the global minimum. The number of i t e r a t i o n s needed to locate the optimum was 94 and 66 respectively. This t r i a l gave the lowest multi-objective function value which indicates that a very small o v e r a l l difference existed between predicted and target q u a l i t y values. The difference between predicted Shrink and Cohes values and their target values was less than ±1.0%, while the target value of Exfat was achieved. The Hardl value of t h i s optimum formulation was 150.00N. -284-2. Optimization of frankfurter formulations using linear  programming The meat processing industry has been using linear programming computer programs to formulate meat products at the lowest possible cost while meeting a l l product s p e c i f i c a t i o n s with the available ingredients. In t h i s study linear programming was used to find seven least-cost frankfurter formulations that met s p e c i f i c a t i o n s for (a) ingredient l e v e l s , (b) proximate composition and (c) qu a l i t y . The same ingredient l i m i t s , the constraint on the sum of the ingredients proportions (equation 30) and the composition constraints (equations 31 to 33) used for the optimization of frankfurter formulations by the FORPLEX program were used. The t o t a l bind value constraint was used to account for the qu a l i t y required by the formulations in terms of the amount of fat bound per unit weight of formulation ( i . e . fat binding capacity of the formulation). The t o t a l bind value (TBV) for a frankfurter formulation was given by equation (39): n TBV = E (bind value* x X*) i = l (39) where bind value* = bind value constant* x protein content* (40) i = l , . . . n -285-where n = number 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 X i = p r o p o r t i o n of the i t h meat i n g r e d i e n t bind v a l u e i = bind value per u n i t weight of i t h i n g r e d i e n t bind value c o n s t a n t * = bind value constant of the i t h i n g r e d i e n t p r o t e i n contents = f r a c t i o n of p r o t e i n content of the i t h ingred i e n t The bind value constants of m e c h a n i c a l l y deboned p o u l t r y meat (MDPM), X 2, and beef meat, X 3, were 15 and 24 r e s p e c t i v e l y . These constants were provided by Sebasti a n (1989). The bind v a l u e s f o r these two i n g r e d i e n t s were computed using equation (40) and these were 2.327 and 5.398 f o r X 2 and X 3 r e s p e c t i v e l y . The higher bind value of beef meat (X 3) meant that t h i s i n g r e d i e n t had a higher f a t b i n d i n g c a p a c i t y than MDPM ( X 2 ) . The TBV equation used i n t h i s study was TBV = 2.327X 2 + 5.398X 3 (41) Response s u r f a c e contour l i n e s were generated using equation (41) w i t h i n the f e a s i b l e r e g i o n . - The t o t a l bind values corresponding to the seven contour l i n e s shown i n 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 . Table 50 shows the c o s t o b j e c t i v e f u n c t i o n to be minimized and the c o n s t r a i n t equations as they were entered i n t o the computer. The seven 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 found are r e p o r t e d i n Table 51. Higher p r o p o r t i o n s of beef meat were r e q u i r 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 i n c r e a s e d , t h a t i s , as the r e q u i r e d minimum f a t b i n d i n g c a p a c i t y of the f o r m u l a t i o n s - 2 8 6 -MDPM Figure 56. Response surface contour plot for the TBV equation. The black area represents the constrained region which is given by the proximate composition constraints. Table 50. Objective function and constraint equations used in the optimization of frankfurter formulations using linear programming. Objective function to be minimized (cost of the formulation) 0.20Xi + 0.80X a + 3.20Xa Subject to: (1) fat content of the meat block 8 < 80.37Xi + 18.67X 2 + 3.72X 3 < 28 (2) protein content of the meat block 16 < 4.96X1 + 15.51X 2 + 22.49X 3 < 21 (3) moisture content of the meat block 54.3 < 14.58Xi + 65.69X 2 + 73.64X 3 < 89 (4) t o t a l bind value (TBV) of the formulation*' 3.428 < 2.327X 2 + 5.398X 3 (5) ingredient contents 0.05 < X i < 0.30 0.00 < X 2 < 0.40 0.50 < X 3 < 0.95 X i + X 2 + X 3 = 1 *• The values corresponding to the seven contour lines shown in Figure 56 were used as lower l i m i t s of the TBV constraint. -288-Table 51. Least-cost frankfurter formulations. Formulation Ingredients proportions Cost Lower l i m i t Xa. X 2 X 3 $/kg TBV LP1 0.187 0.313 0.500 1.89 3.428 LP2 0.098 0.400 0.502 1.95 3.642 LP3 0.058 0.400 0.542 2.07 3.856 LP4 0.050 0.345 0.606 2.22 4.070 LP5 0.050 0.275 0.675 2.39 4.284 LP6 0.050 0.205 0.745 2.56 4.498 LP7 0.050 0.136 0.815 2.72 4.712 -289-increased. This can be better understood by examining Figure 56. The feasible area after setting a lower l i m i t for the TBV constraint i s the one to the right of the contour lin e corresponding to the p a r t i c u l a r lower l i m i t chosen; increasing the lower l i m i t of the TBV constraint, higher proportions of beef meat were needed to meet the higher requirements for fat binding capacity. The cost per kg of the frankfurter formulations also increased as the lower l i m i t set for the TBV constraint increased (Table 48) due to the need for higher proportions of the most costly meat ingredient, X 3. In other words as the quality of the formulations in terms of fat binding capacity increased from formulation LP1 to LP7, the cost per kg of formulation also increased. - 2 9 0 -3. Comparison of FORPLEX and linear programming computed optimum formulations The fiv e computed optimum formulations found by FORPLEX (Table 48) were compared with the seven optimum formulations found by line a r programming (Table 51). The comparison was made in terms of the predicted q u a l i t y and cost of the computed formulations. 3.1. Comparison in terms of predicted q u a l i t y Each of the five FORPLEX computed optimum formulations were compared with the seven least-cost formulations. The comparison was based on the per cent difference between the predicted q u a l i t y values of each computed optimum formulation and the target values set in the FORPLEX formulations. To f a c i l i t a t e the comparison among the computed optimum formulations a series of histograms were made (Figures 57 to 61). The formulations were assigned to the x-axis and the per cent difference from target (equation 38) to the y-axis. A positive per cent difference from target value for a par t i c u l a r q u a l i t y parameter meant that the predicted q u a l i t y value of a formulation was greater than the target q u a l i t y value set on the FORPLEX formulation. The opposite was true for a negative value. As can be observed in Figures 57 to 61 the predicted q u a l i t y values of the optimum formulations found by FORPLEX (Formula, Form2, Form3, Form4, Form4*) were close to the i r target q u a l i t y values as seen by the low per cent difference from target values found. The only exception was Form4 (Figure 60) whose ES target could not be achieved because formulations which approached the -291-4 0 2 0 -0 -- 2 0 -- 4 0 -- 6 0 • C O H E S S HARD1 0 E X F A T • E X W A T E R • T W L O S S o> s f 1 v ? 3 O ? 6 \J?6 O ? 1 FORMULATIONS F i g u r e 5 7 . 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 a n d 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 a n d t h e l e a s t - c o s t f o r m u l a t i o n s . 4 0 i Ui I UJ CD O OC u_ LU o Z LU CL LU LO-LL Q 3 0 2 0 1 0 0 - 1 0 - 2 0 - 3 0 L m F ^ o?A sf1 v?* vJ?6 0?"1 • GUMMY 0 COHES • SHEAR • EXFLUID • SHRINK 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 values and the p r e d i c t e d q u a l i t y values of Form2 and the l e a s t - c o s t f o r m u l a t i o n s . -{•63-% DIFFERENCE F R O M T A R G E T C H CD U l fl) 3 r — OJ DJ r-h 01 h h r t fl) I r t H o rr ft) O CD 3 tn o cr 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 CD o iO tn 3 C D tn at CD • I—1 o M - r— r t i - h »< t— fl) < a QJ r-> cr C 0> CD t< tn i n CD O r t l -h . Q * l C O Oi 3 i -O J r t OJ 3 < Q i OJ r t C 3 " CD CD tn o J3 o CD CP Co CP CP cn O O O O O • • • Q • D3 a 1 5 0 i to <o cn I CD o cn LL LL) O Z LU DC 111 LL LL Q 3^  1 0 0 5 0 0 - 5 0 - 1 0 0 j j j y i _ v^ A OP*2- 0?s 0?* OP6 v ? 6 0?1 • C O H E S 0 HARD1 • E X F A T • E S • T W L O S S FORMULATIONS Figure 60. Differences between specified target q u a l i t y values and the predicted quality values of Form4 and the least-cost formulations. 1 0 0 I I CD O cr u_ LU O z LU OC LU LL LL Q 5 0 -0 - 5 0 ^* \JP A ^ ^ 0?^  v ? 6 0?6 vJ? 1 • C O H E S • HARD1 • E X F A T • E S • T W L O S S FORMULATIONS Figure 61. Differences between specified target q u a l i t y values and the predicted quality values of Form4* and the least-cost formulations. target of t h i s q u a l i t y parameter were located far from the other target q u a l i t y values (Figure 51). The least-cost formulations (LP1-LP7) showed, in general, considerable departure from the target q u a l i t y set to the FORPLEX formulations. These differences observed between the least-cost formulations and the FORPLEX optimum formulations were expected since these two formula optimization techniques work based on d i f f e r e n t p r i n c i p l e s . The FORPLEX searched for formulations whose predicted q u a l i t y was as close as possible to a predetermined target quality; q u a l i t y parameters could be optimized because there were mathematical models that predicted the q u a l i t y parameters as a function of the ingredients. On the other hand, linear programming did not search for maximum qu a l i t y and could not consider target q u a l i t y values. Furthermore, the optimization performed by linear programming did not consider, much less control, any of the q u a l i t y parameters that were considered in the formula optimization using FORPLEX. However, in some cases the least-cost formulations found did meet some of the target q u a l i t y values, but t h i s was e s s e n t i a l l y a matter of chance. The difference observed between the predicted q u a l i t y values of the least-cost formulations and the target q u a l i t y values set on the FORPLEX formulations w i l l be discussed. (A) Quality parameters related to weight losses (Shrink, Twloss and Tmloss). A zero per cent difference from target for Shrink was assigned to LP1 (Figure 58) and for Twloss to LPl, LP2, LP3 and LP4 (Figures 57, 60 and 61) because the predicted values of these -297-q u a l i t y parameters were l e s s than the t a r g e t v a l u e s s e t on the FORPLEX f o r m u l a t i o n s (Formula, Form2, Form4, and Form4*). As mentioned b e f o r e , f o r m u l a t i o n s whose p r e d i c t e d Shrink and Twloss ( i . e . product weight l o s s ) values were lower than the t a r g e t s s e t f o r these parameters had b e t t e r q u a l i t y . The p r e d i c t e d Shrink and Twloss val u e s of the other l e a s t - c o s t f o r m u l a t i o n s were g r e a t e r than the t a r g e t s s e t on the FORPLEX fo r m u l a t i o n s and i n c r e a s e d from LP1 to LP7 and from LP2 to LP7 f o r Shrink and Twloss r e s p e c t i v e l y . The t r e n d observed c o u l d be e x p l a i n e d by the f i n d i n g t h a t Shrink and Twloss were p o s i t i v e l y r e l a t e d to X a (Figure 40). The p r o p o r t i o n of X 3 i n the l e a s t - c o s t f o r m u l a t i o n s i n c r e a s e d from 0.500 i n LP1 to 0.815 i n LP7. Thus, as the p r o p o r t i o n of X 3 i n c r e a s e d i n the l e a s t - c o s t f o r m u l a t i o n s higher Shrink and Twloss values were p r e d i c t e d and hence g r e a t e r per cent d i f f e r e n c e from t a r g e t values were obt a i n e d . I t can be s a i d t h a t even though the q u a l i t y , i n terms of f a t b i n d i n g c a p a c i t y , i n c r e a s e d from LP1 to LP7, the q u a l i t y i n terms of product weight l o s s moved f a r t h e r from the t a r g e t s s e t on the FORPLEX f o r m u l a t i o n s . A zero per cent d i f f e r e n c e from t a r g e t f o r Tmloss was assig n e d to LP2 to LP5 because the p r e d i c t e d Tmloss values were lower than the t a r g e t value s e t on Form3 ( F i g u r e 59). As mentioned b e f o r e , f o r m u l a t i o n s whose p r e d i c t e d Tmloss ( i . e . water l o s s per u n i t moisture content of meat block) value was lower than the t a r g e t value s e t f o r t h i s parameter had b e t t e r q u a l i t y . The p r e d i c t e d Tmloss val u e s f o r f o r m u l a t i o n s LP1, LP6 and LP7 were g r e a t e r than -298-the t a r g e t s e t i n Form3. (B) Q u a l i t y parameters r e l a t e d to j u i c i n e s s c h a r a c t e r i s t i c s ( E x f l u i d , Exwater and E x f a t ) . The E x f l u i d t a r g e t s e t i n Form2 (Figure 58) c o u l d not be met by any of the l e a s t - c o s t f o r m u l a t i o n s . The p r e d i c t e d E x f l u i d values were lower than the t a r g e t s e t on the FORPLEX f o r m u l a t i o n . The p r e d i c t e d E x f l u i d values i n c r e a s e d from LP4 to LP7. The trend observed c o u l d be e x p l a i n e d by the negative r e l a t i o n s h i p found between E x f l u i d and X 2 (Figure 40). The p r o p o r t i o n of X a i n the l e a s t - c o s t f o r m u l a t i o n s decreased from 0.400 i n LP3 to 0.136 i n LP7. Thus as the p r o p o r t i o n of X a decreased i n these l e a s t - c o s t f o r m u l a t i o n s higher E x f l u i d values were p r e d i c t e d , and hence lower negative per cent d i f f e r e n c e from t a r g e t v a l u e s were obta i n e d . The Exwater t a r g e t s e t i n Formula (Figure 57) and Form3 (Figure 59) could not be met by any of the l e a s t - c o s t f o r m u l a t i o n s . With the e x c e p t i o n of LP1 the p r e d i c t e d Exwater value s were g r e a t e r than the t a r g e t s e t i n the two FORPLEX f o r m u l a t i o n s . The p r e d i c t e d Exwater values i n c r e a s e d from LP1 to LP7, as expected c o n s i d e r i n g t h a t Exwater was found to be p o s i t i v e l y r e l a t e d to X 3 (Figure 40). As mentioned above, the p r o p o r t i o n of X 3 i n the l e a s t - c o s t f o r m u l a t i o n s i n c r e a s e d from LP1 to LP7. Thus, as the p r o p o r t i o n of X 3 i n c r e a s e d higher Exwater values were p r e d i c t e d , and g r e a t e r per cent d i f f e r e n c e from t a r g e t v a l u e s were obtained. The E x f a t t a r g e t s s e t on Formula (Figure 57), Form4 (Figure -299-60) and Form4* (F i g u r e 61) c o u l d not be met by any of the l e a s t -c o s t f o r m u l a t i o n s . When compared to the t a r g e t s e t i n Form4* (Fi g u r e 61), with the e x c e p t i o n of LP1, the p r e d i c t e d E x f a t v a l u e s were lower than the t a r g e t s s e t on the FORPLEX fo r m u l a t i o n s and decreased from L P l to LP7. However the decrease i n p r e d i c t e d E x f a t values i n LP4 to LP7 was not s t r o n g . The trend observed c o u l d be e x p l a i n e d by the p o s i t i v e r e l a t i o n s h i p found between E x f a t and Xa. (Figure 40). The p r o p o r t i o n of Xa. i n the l e a s t - c o s t f o r m u l a t i o n s decreased from 0.187 i n L P l to 0.050 i n LP4 and remained constant at 0.050 i n LP4 to LP7. Thus, as the p r o p o r t i o n of X i decreased i n the l e a s t - c o s t f o r m u l a t i o n s lower E x f a t v a l u e s were p r e d i c t e d , and hence g r e a t e r negative per cent d i f f e r e n c e from t a r g e t values were obt a i n e d . (C) Q u a l i t y parameters r e l a t e d to t e x t u r e c h a r a c t e r i s t i c s ( H a r d l , Shear, Cohes, Gummy and Chewy). None of the t e x t u r a l parameter t a r g e t s s e t i n a l l the FORPLEX fo r m u l a t i o n s ( F i g u r e s 57 to 61) c o u l d be met by any of the l e a s t -c o s t f o r m u l a t i o n s . With the e x c e p t i o n of L P l and LP2 the p r e d i c t e d values of the t e x t u r a l parameters were g r e a t e r than the t a r g e t v a l u e s and i n c r e a s e d from L P l to LP7. The trend observed c o u l d be e x p l a i n e d by the p o s i t i v e r e l a t i o n s h i p found between the t e x t u r a l parameters and Xa (Figure 42). As the p r o p o r t i o n of X 3 i n the l e a s t - c o s t f o r m u l a t i o n s i n c r e a s e d from L P l to LP7 higher t e x t u r a l v a l u e s were p r e d i c t e d , and hence g r e a t e r p o s i t i v e per cent d i f f e r e n c e s from t a r g e t v a l u e s were obtained. -300-(D) The ES (per cent f a t r e l e a s e d ) parameter A zero per cent d i f f e r e n c e from t a r g e t f o r ES was assigned to LP2 to LP7 ( F i g u r e s 60 and 61) because the p r e d i c t e d ES values of these l e a s t - c o s t f o r m u l a t i o n s vere l e s s than the t a r g e t s e t i n the FORPLEX f o r m u l a t i o n s (Form4 and Form4*). As mentioned before, f o r m u l a t i o n s whose p r e d i c t e d ES ( i . e . per cent f a t r e l e a s e d ) v a l u e s were lower than the t a r g e t s e t i n t h i s parameter had b e t t e r q u a l i t y . The p r e d i c t e d ES v a l u e s f o r the l e a s t - c o s t f o r m u l a t i o n s decreased from LP1 to LP7. The reason why lower p r e d i c t e d ES v a l u e s were found l i e s i n the f a c t t h a t these f o r m u l a t i o n s were r e s t r i c t e d by the TBV c o n s t r a i n t . The TBV c o n s t r a i n t c o n t r o l l e d the r e q u i r e d minimum f a t b i n d i n g c a p a c i t y of the f o r m u l a t i o n s . As the lower l i m i t of t h i s c o n s t r a i n t i n c r e a s e d , higher f a t b i n d i n g c a p a c i t y was r e q u i r e d and t h e r e f o r e higher p r o p o r t i o n s of X B and lower p r o p o r t i o n s of X* were r e q u i r e d . ES, which i s p o s i t i v e l y r e l a t e d t o Xi (Figure 40), decreased from LP1 to LP7. The TBV and ES parameters were n e g a t i v e l y r e l a t e d ; as the f a t b i n d i n g c a p a c i t y was r e q u i r e d to i n c r e a s e from f o r m u l a t i o n LP1 to LP7, the per cent f a t r e l e a s e d d u r i n g thermal h e a t i n g was p r e d i c t e d to decrease. I t i s d i f f i c u l t to make an o v e r a l l comparison i n terms of q u a l i t y between the optimum f o r m u l a t i o n s found by FORPLEX and l i n e a r programming s i n c e these two techniques search f o r optimum f o r m u l a t i o n s based on d i f f e r e n t p r i n c i p l e s . Furthermore, i t cannot be concluded t h a t the f o r m u l a t i o n s found by FORPLEX have -301-b e t t e r q u a l i t y than the optimum fo r m u l a t i o n s found by l i n e a r programming. However, i f the o b j e c t i v e i s to f i n d f o r m u l a t i o n s 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 to a predetermined q u a l i t y , then the optimum f o r m u l a t i o n s found by FORPLEX meet t h i s o b j e c t i v e b e t t e r than the l e a s t - c o s t f o r m u l a t i o n s . 3.2. Comparison i n terms of c o s t The c o s t ($/kg of meat block) of the optimum fo r m u l a t i o n s found by FORPLEX (Formula, Form2, Form3, Form4, Form4*) and the l e a s t - c o s t f o r m u l a t i o n s (LP1 to LP7) are shown i n F i g u r e 62. As mentioned before the c o s t was used as a c o n s t r a i n t 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 the FORPLEX program, while i t was c o n s i d e r e d the o b j e c t i v e f u n c t i o n t o be minimized i n the o p t i m i z a t i o n of f o r m u l a t i o n s u s i n g l i n e a r programming. The lower and upper l i m i t s t h a t were s e t f o r the c o s t c o n s t r a i n t were $1.9 and 2.8 per kilo g r a m of meat block, r e s p e c t i v e l y . As can be observed (Figure 62) a l l the optimum fo r m u l a t i o n s found by FORPLEX f e l l w i t h i n t h i s p e r m i s s i b l e c o s t range. T h i s c l e a r l y shows t h a t the FORPLEX program i s e f f e c t i v e i n f i n d i n g f o r m u l a t i o n s that meet the c o s t r e s t r i c t i o n . The c o s t of the l e a s t - c o s t f o r m u l a t i o n s a l s o f e l l w i t h i n the l i m i t s imposed on the FORPLEX f o r m u l a t i o n s . However, as mentioned b e f o r e , the c o s t of these f o r m u l a t i o n s i n c r e a s e d from LP1 to LP7 due to the higher p r o p o r t i o n s of the expensive meat i n g r e d i e n t X3 needed to meet the i n c r e a s i n g requirements of f a t b i n d i n g c a p a c i t y ( i . e . higher lower l i m i t s s e t on the t o t a l bind value FORMULATIONS Figure 62. Cost ($/kg of meat block) of the optimum formulations found by FORPLEX and the least-cost formulations. -303-constraint). The cost of formulations LP4 to LP7 were higher than the cost of the FORPLEX optimum formulations. The lower limit set on the total bind value constraint should be carefully selected i f the objective is to find low-cost frankfurter formulations with acceptable quality scores. The processed meat industry is probably using moderate lower limits for the total bind value constraint in order to control the cost of the optimum formulations; otherwise high quality-expensive meat ingredients have to be used, increasing the cost of the frankfurter formulations. 4. Comparison of FORPLEX with linear programming for meat  formula optimization One of the objectives of this thesis was to compare the new formula optimization method (FORPLEX) with linear programming for meat formula optimization. Table 52 summarizes the comparison of these two techniques. The main difference between these techniques is based on the optimization objectives. The FORPLEX searches for best quality formulations that meet predetermined product specifications within allowable cost ranges, while linear programming searches for least-cost formulations that meet predetermined product specifications. Linear programming places heavy emphasis in cost reduction but far less emphasis on quality. FORPLEX considers quality to be as important as cost reduction. Linear programming can only be used to solve formula optimization problems in which the objective function and -304-Table 52. Comparison of FORPLEX and linear programming for the optimization of meat formulations. FORPLEX Linear programming Optimization Maximize Minimize objective q u a l i t y cost Constraints •Product s p e c i f i c a t i o n s yes •Quality yes *Cost yes yes yes n o Objective function •Linear •Nonlinear yes •Multiple objective function yes yes y e s no no Constraints •Linear •Nonlinear yes yes yes no -305-c o n s t r a i n t s are expressed u s i n g l i n e a r f u n c t i o n s . On the other hand, FORPLEX can accommodate any form of r e l a t i o n s h i p s , whether l i n e a r or n o n l i n e a r , as o b j e c t i v e f u n c t i o n s and c o n s t r a i n t s . L i n e a r programming uses l i n e a r equations t o d e s c r i b e and r e s t r i c t the q u a l i t y of the f o r m u l a t i o n s . The most w i d e l y used q u a l i t y c o n s t r a i n t s are the c o l o r and b i n d i n g c o n s t r a i n t s (Pearson and Tauber, 1984a). These 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 of the i n g r e d i e n t s 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 c o l o r and bind v a l u e s . However q u a l i t y parameters are u s u a l l y not d e s c r i b e d by simple l i n e a r f u n c t i o n s ; i n s t e a d n o n l i n e a r f u n c t i o n s have been found to e x p l a i n q u a l i t y more a c c u r a t e l y (Nakai and Arteaga, 1990). Furthermore, the use of bind values has been questioned s i n c e these parameters have been found t o be i n a c c u r a t e i n p r e d i c t i n g the q u a l i t y of the f i n i s h e d products. In a d d i t i o n , bind values c o n s i d e r o n l y the a b i l i t y of the meat i n g r e d i e n t s t o bind f a t . U n t i l now, 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 using l i n e a r programming has not c o n s i d e r e d water b i n d i n g and g e l a t i o n p r o p e r t i e s of the meat i n g r e d i e n t s . The FORPLEX can accommodate l i n e a r and n o n l i n e a r q u a l i t y p r e d i c t i o n 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 r e l a t i o n s h i p s t h a t d e s c r i b e q u a l i t y as a f u n c t i o n of the i n g r e d i e n t s . The FORPLEX can optimize any q u a l i t y parameter, f o r example, product y i e l d , j u i c i n e s s , t e x t u r e , or c o l o r . However, r e l i a b l e q u a l i t y p r e d i c t i o n equations roust be known i n advance i n order to -306-optimize a meat f o r m u l a t i o n e f f e c t i v e l y . The FORPLEX a l l o w s f o r t a r g e t q u a l i t y values t o be s e t and t h e r e f o r e searches 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 i s 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 . T h i s o p t i o n cannot be performed by l i n e a r programming. 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 means the o p t i m i z a t i o n of m u l t i p l e o b j e c t i v e f u n c t i o n s . While l i n e a r programming cannot be used f o r 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 , the FORPLEX i s capable of accommodating an o b j e c t i v e f u n c t i o n composed of s e v e r a l f u n c t i o n s combined i n t o one eq u a t i o n . T h i s a l l o w s f o r the simultaneous o p t i m i z a t i o n of s e v e r a l q u a l i t y parameters. In g e n e r a l , t h e r e f o r e , i t would seem t h a t FORPLEX i s the more s u i t a b l e technique f o r food f o r m u l a t i o n . 5. 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 v a l u e s of two computed optimum f o r m u l a t i o n s . The adequacy of the models found f o r p r e d i c t i n g the q u a l i t y parameter valu e s was t e s t e d f o r two computed optimum f o r m u l a t i o n s , Formula (Xi=0.228, X 2=0.176, X 3=0.596) and L P l (Xi=0.187, X 3=0.313, X 3=0.500). These two f r a n k f u r t e r f o r m u l a t i o n s vere prepared i n d u p l i c a t e f o l l o w i n g the same procedure followed f o r the p r e p a r a t i o n of the f o r m u l a t i o n s given by the extreme v e r t i c e s d e s i g n ( M a t e r i a l s and Methods s e c t i o n E ) . The q u a l i t y parameters vere ev a l u a t e d f o l l o w i n g the methods d e s c r i b e d i n M a t e r i a l s and Methods s e c t i o n F. Before d i s c u s s i n g the v e r i f i c a t i o n r e s u l t s i t i s important to keep i n mind t h a t the models found are onl y estimates of the t r u e -307-r e l a t i o n s h i p s between i n g r e d i e n t s and q u a l i t y parameters. For t h i s reason when p r e d i c t i n g a q u a l i t y parameter one must take i n t o account not o n l y the v a r i a t i o n In the m a t e r i a l s t u d i e d but the v a r i a t i o n i n the r e g r e s s i o n model (Bender e t a l . 1982). The s t a t i s t i c a l method used to c o n s i d e r these v a r i a t i o n s when p r e d i c t i n g a s i n g l e f u t u r e event ( i . e . q u a l i t y parameter f o r a give n f o r m u l a t i o n ) i s to c a l c u l a t e the co n f i d e n c e i n t e r v a l ( a l s o c a l l e d p r e d i c t i o n i n t e r v a l ) a t a predetermined p r o b a b i l i t y l e v e l f o r the p a r t i c u l a r s i n g l e f u t u r e event (Bender e t a l . 1982). In t h i s study 95% conf i d e n c e i n t e r v a l s were c o n s t r u c t e d f o r each p r e d i c t e d q u a l i t y value of the two f o r m u l a t i o n s (Formula and L P l ) usi n g the equations r e p o r t e d by C o r n e l l (1981). In order t o co n s i d e r the models to be adequate f o r p r e d i c t i o n purposes the experimental q u a l i t y data of these f o r m u l a t i o n s had to f a l l w i t h i n t h e i r r e s p e c t i v e c o n f i d e n c e i n t e r v a l s . The p r e d i c t e d q u a l i t y v a l u e s , the 95% confidence i n t e r v a l of each p r e d i c t e d q u a l i t y v a l u e , and the experimental q u a l i t y data f o r the two f o r m u l a t i o n s (Formula and L P l ) are given i n Tables 53 and 54. I t should be noted t h a t the conf i d e n c e i n t e r v a l was not c o n s t r u c t e d f o r the p r e d i c t e d ES v a l u e s , s i n c e the ES model was found t o have a s i g n i f i c a n t l a c k of f i t ( R e s u l t s and D i s c u s s i o n s s e c t i o n B.3.1.). V a r i a b i l i t y of the q u a l i t y data of the d u p l i c a t e f o r m u l a t i o n s was observed. As mentioned before ( R e s u l t s and D i s c u s s i o n s e c t i o n B.2.) the Inherent v a r i a b i l i t y of the meat i n g r e d i e n t s , the p r o c e s s i n g steps i n v o l v e d i n the p r e p a r a t i o n of the f o r m u l a t i o n s -308-Table 53. Experimental v e r i f i c a t i o n of the predicted q u a l i t y values of Formula.*-Experimental q u a l i t y Quality Predicted Confidence values 0 parameters* q u a l i t y i n t e r v a l values 1 2 Shrink 9. .19 8, .01 10. .37 8. .33 10. .49 Tmloss 38. ,75 34. ,17 43. ,33 49 . 66 46 . ,11 Twloss 22. .92 19 . 54 26, ,30 29. .19 27, .10 ES 0, ,49 -- --_E> 2 , .20 0. ,90 PH 5. .69 5, .44 5, .94 5, . 52 5, .53 Ex f l u i d 9 . ,82 6 . ,24 13. , 35 7 . ,78 6. ,78 Exwater 5, .42 4 . ,01 6, .83 4, .00 3. .51 Exfat 4. , 47 2, , 55 6, , 39 3 . ,78 3. ,27 Hardl 159. ,19 141, ,14 177. .24 67. ,56 66, ,94 Shear 4 . ,79 2. ,83 6. ,75 3. ,72 4 . ,95 Cohes 0. ,255 0. , 219 0. .291 0. ,302 0, .295 Gummy 41. , 58 25. , 67 57 . ,49 20. ,33 19 , 75 Chewy 122. .12 43, .99 200 , .25 76, .61 71, .61 *• X i = 0.228/ X2=0.176, X 3 = 0.596 B Nomenclature and d e f i n i t i o n of the qu a l i t y parameters are given in Appendix E ° of duplicate formulations D not calculated -309-Table 54. Experimental v e r i f i c a t i o n of the predicted q u a l i t y values of LPl.*-Experimental qu a l i t y Quality Predicted Confidence v a l u e s c parameters* q u a l i t y i n t e r v a l values 1 2 Shrink 8, .63 7, .69 9 , .57 10, .29 15, .06 Tmloss 35, . 48 33, ,01 37, .95 49 , .73 57, .32 Twloss 21, .28 19, ,45 23, .11 29, .90 34, .46 ES 0. ,40 -- -- 2, ,40 4, ,75 PH 5. .79 5. ,60 5, .98 5, .67 5, , 69 E x f l u i d 8. .92 6. ,22 11, ,62 8. ,28 8 . ,13 Exwater 4, .93 4. ,08 5. ,78 4, .18 2. ,92 Exfat 3, .96 2. , 53 5, , 39 4 , 10 5, ,21 Hardl 113, .69 70, .97 156, .41 84, .30 106, .57 Shear 4 , .81 3. , 34 6, .28 4 , .75 4 , 53 Cohes 0, .257 0, .230 0, .284 0, .285 0. .326 Gummy 29 , , 77 17. ,23 42. . 31 24 , .16 34. ,60 Chewy 111, .09 52. .65 169, .53 91, .41 138. .48 *• Xx = 0 . 1 8 7 , X 2 = 0 . 3 1 3 , X a = 0 .500 B Nomenclature and d e f i n i t i o n of the qu a l i t y parameters are given in Appendix E a of duplicate formulations D not calculated - 3 1 0 -and the d i f f e r e n t steps Involved in the evaluation of the q u a l i t y parameters a l l contribute to 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 f e l l within t h e i r respective confidence Intervals. In some cases the qua l i t y data of one duplicate f e l l within the confidence Interval and the other did not. Before making any conclusions on the adequacy of the models for predicting q u a l i t y , i t must be considered that the q u a l i t y data used for generating the qu a l i t y prediction models were obtained with meat ingredients stored for a period of less than 2 months at -30<>C. The v e r i f i c a t i o n of the predicted q u a l i t y parameter values of the two formulations was performed using meat ingredients that had been stored for 6 months at -30°C. It has been reported that extended frozen storage a f f e c t s the functional properties of the meat proteins, i. e . water- and f a t - binding capacity and gelation a b i l i t y , and th i s in turn a f f e c t s the qu a l i t y of the finished product. (Lanier, 1990; Whiting, 1988; Smith, 1988; Pearson and Tauber, 1984a; Morrissey et a l . , 1982; Powrie, 1973; Schut, 1976; S a f f l e , 1968). Although the e f f e c t of frozen storage on the functional properties of the meat ingredients was not assessed in t h i s study, i t may be reasonable to suggest that the length of time in frozen storage affected the f u n c t i o n a l i t y of the meat ingredients and thus the q u a l i t y of the finished products. Therefore, the adequacy of the models for predicting the q u a l i t y of the formulations could only have been evaluated using the meat ingredients stored at -30<>C for less than 2 months, since the -311-models d i d not account f o r the e f f e c t of extended f r o z e n storage on the q u a l i t y of the f o r m u l a t i o n s . -312-SUMMARY AND CONCLUSIONS A computer program f o r c o n s t r a i n e d o p t i m i z a t i o n based on the m o d i f i e d v e r s i o n of the Complex method of Box was w r i t t e n i n IBM BASIC. For the purpose of t h i s study the computer program was named "FORPLEX". The s u i t a b i l i t y of the FORPLEX f o r l o c a t i n g the optimum of l i n e a r 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 t h a t were l i n e a r l y c o n s t r a i n e d was i n i t i a l l y t e s t e d u s i n g c o n s t r a i n e d mathematical models. The FORPLEX was able to l o c a t e the optimum r e g a r d l e s s of whether l i n e a r or 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 s were used. However, the FORPLEX was found to be i n e f f i c i e n t i n s o l v i n g l i n e a r 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 c o n s t r a i n e d problems. For these types of problems the FORPLEX cannot r e p l a c e l i n e a r programming. On the other hand, the FORPLEX was found to be an e f f e c t i v e method f o r s o l v i n g 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 -l i n e a r l y c o n s t r a i n e d problems, making i t s u i t a b l e f o r use f o r formula o p t i m i z a t i o n purposes. L i n e a r programming cannot s o l v e these type of problems. To t e s t the s u i t a b i l i t y of the FORPLEX computer program f o r 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 , s t a t i s t i c a l l y s i g n i f i c a n t q u a l i t y p r e d i c t i o n equations were developed f o r a 3-i n g r e d l 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 . The three i n g r e d i e n t s v e r e : pork f a t , m e c h a n i c a l l y deboned p o u l t r y meat and beef meat. Generation of the i n g r e d i e n t - q u a l i t y equations vas performed through mixture experimentation u s i n g an extreme v e r t i c e s d e s i g n . The e i g h t e e n q u a l i t y parameters ev a l u a t e d vere c l a s s i f i e d i n t o 5 -313-groups: (a) 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 p r o c e s s , (b) s t a b i l i t y of the raw emulsions to thermal treatment, (c) 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 , (d) 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 f r a n k f u r t e r s , and (e) pH. S c h e f f e ' s c a n o n i c a l s p e c i a l c u b i c model f o r three components was the model f i t t e d t o the e xperimental data u s i n g 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 . Not a l l of the seven 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 were r e q u i r e d t o d e s c r i b e the e f f e c t 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 of the q u a l i t y parameters e v a l u a t e d . Reduction of the f u l l model was performed by a Student's t - t e s t on the i n d i v i d u a l parameters i n each model. The s i g n i f i c a n c e and adequacy of each q u a l i t y p r e d i c t i o n model was assessed by a n a l y s i s of v a r i a n c e , a d j u s t e d m u l t i p l e c o e f f i c i e n t of d e t e r m i n a t i o n , standard e r r o r of the e stimate and a n a l y s i s of r e s i d u a l s . Seventeen q u a l i t y p r e d i c t i o n models were developed. Examination of the a n a l y s i s of v a r i a n c e of the f i t t e d models i n d i c a t e d t h a t a l l the r e g r e s s i o n models were s i g n i f i c a n t a t a p r o b a b i l i t y l e v e l of 0.05. With the e x c e p t i o n of the ES model, a l l the models possessed no s i g n i f i c a n t l a c k of f i t (p>0.05). The a d j u s t e d m u l t i p l e c o e f f i c i e n t of d e t e r m i n a t i o n values of three models (Cook s h r i n k , Vacuum s h r i n k and Firm) were found to be l e s s than 0.60, and thus were not c o n s i d e r e d adequate f o r p r e d i c t i o n purposes. The a n a l y s i s of r e s i d u a l s of Shrink, Tmloss, Exwater, E x f a t , pH, Hard2, Shear, Gummy and Chewy models showed t h a t these models appeared to be adequate. Although the a n a l y s i s of r e s i d u a l s of Twloss, E x f l u i d , Hardl and Cohes models -314-suggested t h a t the assumptions about the r e s i d u a l s were v i o l a t e d , the models were c o n s i d e r e d adequate f o r p r e d i c t i o n purposes. The q u a l i t y p r e d i c t i o n models developed i n c l u d e d e i t h e r l i n e a r terms o n l y or both l i n e a r and n o n l i n e a r terms. Three d i f f e r e n t techniques were used to p rovide a b e t t e r understanding of the r e l a t i o n s h i p between i n g r e d i e n t p r o p o r t i o n s and the q u a l i t y parameters. Response s u r f a c e contour a n a l y s i s helped v i s u a l i z e the e f f e c t of changes i n the i n g r e d i e n t p r o p o r t i o n s on the q u a l i t y parameters. C o r r e l a t i o n a n a l y s i s i d e n t i f i e d s t a t i s t i c a l l y s i g n i f i c a n t l i n e a r r e l a t i o n s h i p s between proximate composition of the meat blocks and raw emulsions and the q u a l i t y parameters, and between the q u a l i t y parameters. 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 provided an o v e r a l l view of the r e l a t i o n s h i p between i n g r e d i e n t p r o p o r t i o n s , proximate composition and q u a l i t y parameters. I t was found that d i s p l a y i n g the data i n a s e r i e s of s c a t t e r p l o t m a t r i c e s was most h e l p f u l i n e x p l a i n i n g the e f f e c t s of the d i f f e r e n t f a c t o r s on the q u a l i t y parameters. Not o n l y the i n g r e d i e n t s had an e f f e c t , but the moisture, p r o t e i n and f a t content of the meat b l o c k s and moisture content of the raw emulsions, though dependent on the composition of the f o r m u l a t i o n s , a l s o a f f e c t e d the q u a l i t y parameters. In a d d i t i o n , the pH of the raw emulsions and the weight l o s s a f t e r thermal treatment f u r t h e r a f f e c t e d other q u a l i t y parameters. Some of the most important e f f e c t s of the i n g r e d i e n t s were the f o l l o w i n g : -315-(A) h i g h p r o p o r t i o n s o£ beef meat caused high product weight l o s s , high 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 and high values f o r the t e x t u r a l parameters. (B) high p r o p o r t i o n s of pork f a t caused low product weight l o s s , high 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 , low val u e s f o r the t e x t u r a l parameters and low emulsion s t a b i l i t y of the raw emulsions to thermal treatment ( i . e . h i g h l e v e l s of f a t r e l e a s e d ) . (C) high p r o p o r t i o n s of m e c h a n i c a l l y deboned p o u l t r y meat (MDPM) caused low water l o s s per u n i t moisture content of the meat b l o c k , thus c o n t r i b u t i n g to the s t a b i l i t y of the emulsions t o thermal treatment. (D) both beef meat and MDPM a f f e c t e d the pH of the raw emulsions. The pH in c r e a s e d as the p r o p o r t i o n of MDPM in c r e a s e d and decreased as the p r o p o r t i o n of beef meat i n c r e a s e d . S e v e r a l h y p o t h e t i c a l f r a n k f u r t e r f o r m u l a t i o n o p t i m i z a t i o n t r i a l s were performed t o t e s t the s u i t a b i l i t y of the FORPLEX program f o r meat formula o p t i m i z a t i o n . In each t r i a l , d i f f e r e n t combinations of 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 and thus were i n c o r p o r a t e d i n t o the Fu n c t i o n s u b r o u t i n e . When q u a l i t y parameters vere c o n s i d e r e d c o n s t r a i n t s these were i n c o r p o r a t e d i n the C o n s t r a i n t s u b r o u t i n e . The o b j e c t i v e was t o f i n d optimal combinations of i n g r e d i e n t p r o p o r t i o n s which gave "best q u a l i t y " f o r m u l a t i o n s t h a t met 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 " f o r m u l a t i o n s vere d e f i n e d as those f o r m u l a t i o n s vhose p r e d i c t e d -316-quality was as close as possible to a predetermined target quality. Target quality values were either selected based on a target formulation or were individually selected. When the former procedure was used the FORPLEX was able to find formulations whose predicted quality was no different from their target quality. Using the latter procedure, the FORPLEX was able to find formulations whose predicted quality was as close as possible to the target quality. In both cases the optimum formulations met the constraints imposed on them. Differences between predicted and target quality values existed in a l l the computed optimum formulations using the latter procedure. Differences existed because i t was d i f f i c u l t for the formulations to meet a l l the target quality values. Care should be taken in the selection of the target quality values. Setting the target value for ES far from the other four target quality values resulted in an optimum formulation (Form4) that did not meet the ES target. When a target quality value is not met by a formulation, i t may be necessary to consider modifying the target values set on other quality parameters. Five optimum formulations found by FORPLEX were compared with seven least-cost formulations which were found by increasing the lower limit of the fat binding constraint. The predicted quality of each FORPLEX optimum formulation was very close to its respective target quality. The least-cost formulations showed, in general, considerable departure from the target quality values set in the FORPLEX formulations. The cost of the least-cost -317-formulations f e l l within the l i m i t s Imposed on the FORPLEX formulations. However t h e i r costs depended on the lower l i m i t set on the fat binding constraint. The adequacy of the models for predicting the q u a l i t y of frankfurter formulations could not be evaluated since the meat ingredients had been stored frozen for 6 months. The models did not account for the e f f e c t of extended frozen storage on the q u a l i t y of the formulations. Formula optimization based on the Complex method can replace li n e a r programing programs presently being used in the meat processing industry. The l a t t e r searches for least-cost formulations that meet predetermined product s p e c i f i c a t i o n s and qu a l i t y . 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Influence of various s a l t s and water solubl compounds on the water and fat exudation and gel strength of meat batters. J . Food S c i . 52:1130. Whiting, R.C. 1988. Ingredients and processing factors that control muscle protein f u n c t i o n a l i t y . Food Technol. 42(4):104. -331-Wierbicki, E., Kunkle, L.E., and Deatherage, F.E. 1957. changes in the water-holding capacity and c a t i o n i c s h i f t s during the heating and freezing and thawing of meat as revealed by a single c e n t r i f u g a l method for measuring shrinkage. Food Technol. 11(2):69. Wilkinson, L. 1988a. "SYSTAT: The System for S t a t i s t i c s . " Evanston, IL: SYSTAT, Inc. Wilkinson, L. 1988b. SYGRAPH. Evanston, IL: SYSTAT, Inc. Wolfe, P.M., and Koelling, C P . 1983. Linear programming. Ch. 9. In "Basic Engineering and S c i e n t i f i c Programs for the IBM P C " p.169. Prentice-Hall Publishing and Communications Co., Bowie, MD Zar, J.H. 1984. " B i o s t a t i s t i c a l Analysis." 2nd ed. Prentice-Hall, Inc. Englewood C l i f f s , NJ. Ziegler, G.R., and Acton, J . C 1984. Mechanisms of gel formation by proteins of muscle tissue. Food Technol. 38(5):77. Ziegler, G.R., R i z v i , S.S.H., and Acton, J.C. 1987. Relationship of water content to textural c h a r a c t e r i s t i c s , water a c t i v i t y , and thermal conductivity of some commercial sausages. J . Food S c i . 52:901. -332-APPENDIX A. EXPLANATION OF HOW TO READ THE INGREDIENT PROPORTIONS IN TRIANGULAR GRAPHS Figure Al describes g r a p h i c a l l y the general mixture problem for 3 Ingredients (pork f a t , FAT; mechanically deboned poultry meat, MDPM; and beef meat, BEEF) where the ingredient proportions are to s a t i s f y the constraint FAT + MDPM + BEEF = 1.0. Each ingredient proportion can take values from zero to unity. In thi s figure, a corner of the t r i a n g l e corresponds to a proportion of 1.0 for one ingredient and proportions of 0.0 for the two other ingredients. For a point on or inside the t r i a n g l e , the proportion of a pa r t i c u l a r ingredient i s represented by the perpendicular distance from the point to the side opposite i t s respective corner (Traver et a l . 1981). This method should be used to read the proportions of the ingredients in subsequent figures. In t h i s study the ingredient proportions were r e s t r i c t e d by lower and upper l i m i t s (Table 1). This reduced the feasible mixture space. The excised section in Figure A2 shows the ingredient proportions within the experimental feasible mixture space. Figure A3 shows the ingredient proportions within the optimization feasible mixture space. The black areas within the excised section are not feasible areas and correspond to the lower and upper l i m i t s of the i m p l i c i t constraints of proximate composition and cost (equations 31 to 34). -333-MDPM 1.0 FAT 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 . - 3 3 4 -MDPM Figure A2. Experimental f e a s i b l e mixture space f o r three i n g r e d i e n t s . MDPM Figure A 3 . Optimization feasible mixture space for three ingredients. APPENDIX B. LOTUS 1-2-3 TEMPLATE FOR TEXTURE PROFILE ANALYSIS The following i s the description of the Lotus 1-2-3 template used for c a l c u l a t i n g TPA parameters. For the proper use of t h i s template the user should have a basic understanding of Lotus 1-2-3 and of Lotus macro commands. The Lotus 1-2-3 template for TPA i s shown in Figure B l . A description of each of the macros of the template i s given below. Before performing the analysis, the Instron raw data has to be Imported ( i . e . loaded) into the template. ASCII f i l e s can be e a s i l y imported into the template using the Lotus 1-2-3 "import" command. The Instron raw data consists of time (sec) and m i l l i v o l t s (raV) readings. The time scale i s converted to distance traveled by the crosshead using the sampling rate of the a c q u i s i t i o n system and the Instron crosshead speed. In t h i s study the sampling rate of the a c q u i s i t i o n system was set at 0.2 sec/reading and the crosshead speed at 100 mm/min (1.67 mm/sec). The conversion factor used was calculated as follows: 0.2 sec x 1.67 mm = 0.334 mm reading sec reading The values shown in column A labeled "Distance" in the template are the distance (mm) traveled by the crosshead, calculated using the above conversion factor. The raw mV values must be placed in column B, labeled "mV exp" in the template. Macros A, B and C must be run in t h i s order and must be run before c a l l i n g the other macros. Macro A The purpose of t h i s Macro i s to correct the raw mV values to r e a l mV values, where rea l mV values are defined as: r e a l mV = raw mV - baseline mV The template assumes that baseline mV i s the mV value of the f i r s t reading, that i s , when the distance t r a v e l l e d by the crosshead equals zero. The re a l or corrected mV values are placed by the Macro In column C, labeled "realmV" 'in the template. MaCKQ B This Macro transforms the r e a l mV values to kg-force (kgf) by multiplying every c e l l of column C by a c a l i b r a t i o n factor (kgf/mV) (conrav). This c a l i b r a t i o n factor must be entered in c e l l J4 before c a l l i n g the Macro. The kgf values are placed by the Macro in column D, labeled "kgf" in the template. -337-Figure B l . Lotus 1-2-3 template for texture p r o f i l e analysis. A B C D E F G 1 2 3 Distance mV exp realmV Kgf Nw Area 4 0.000 Macro A 5 0.334 CG0TOC4 6 0.668 7 1.002 +B4-$B$4~ 8 1.336 /C~{D0WN} 9 1.670 .{LEFT}{END}{DOWN} 10 2.004 {RIGHT}~ 11 2.338 12 2.672 Macro B 13 3.006 {goto}d4~ 14 3.340 (C4*$conmv)~ 15 3.674 /c~{dovn}. 16 4.008 {left}{end}{down} 17 4.342 {right}~ 18 4.676 19 5.010 Macro C 20 5.344 {goto}e4~ 21 5.678 (d4*9.81)~ 22 6.012 /c~{down}. 23 6.346 {left}{end}{down} 24 6.680 {rlght}~ 25 7.014 26 7.348 Macro D 27 7.682 {home} 28 8.016 /gtxx{?}~ 29 8.350 a{?}~ 30 8.684 ofglqqv 31 9.018 32 9.352 Macro E 33 9.686 /grg~ 34 10.020 {escape 3} 35 10.354 36 10.688 Macro F 37 11.022 {HOME} 38 11.356 /DRX{?}~ 39 11.690 Y{?}~ 40 12.024 0{GOTO}J8~ 41 12.358 G 42 12.692 {goto}j7~ 43 13.026 44 13.360 Macro G 45 13.694 ({LEFT}{UP}+ 46 14.028 {LEFT})~ 47 14.362 {EDIT}*0.167~ 48 14.696 /C~{DOWN}~ 49 15.030 {DOWN} 50 15.364 {EDIT}+{UP}~ 51 15.698 /C~{D0WN}. -338-Macro C The kgf values obtained by Macro B are multiplied by 9.81 in order to convert them to Nevtons. The Newtons values are placed by the Macro In column E, labeled "Nw" In the template. Macro D This Macro i s used to generate an X vs Y graph. After c a l l i n g the Macro the user highlights the column containing the x-values ( i . e . x-data range) and the corresponding column of the y-values ( i . e . A-data range) a f t e r which the graph appears on the screen. By se l e c t i n g column A (distance t r a v e l l e d by the crosshead) as the x-data range and column E (Newtons) as the A-data range, the t y p i c a l TPA curve w i l l be drawn. The user may save the graph for p l o t t i n g using the standard Lotus 1-2-3 commands. Macro E This Macro i s used to cancel the graph currently defined. Macro F This macro i s used to calculate firmness, which i s defined as the slope of the linear region of the f i r s t compression cycle. The Macro uses the b u i l t - i n linear regression program of Lotus 1-2-3. The operation of t h i s Macro is s i m i l a r to that of Macro D. After c a l l i n g the Macro the user highlights the appropriate section of column A (x-data range) and the corresponding section of column E (y-range). Using these data ranges the Macro then performs linear regression. The regression r e s u l t s , which include the slope, are placed s t a r t i n g i n c e l l J8. The user should check that the value of the c o e f f i c i e n t of determination (r«) is close to 1. It i s also recommended that Macro D be used f i r s t to estimate the linear region of the f i r s t compression cycle v i s u a l l y . Macro G Cohesiveness i s defined as the r a t i o of the positive force area under the second and f i r s t compression curves. This Macro is used to calculated these areas. The c a l c u l a t i o n of the areas i s performed using the trapezoid method. The p r i n c i p l e of t h i s method i s to divide the t o t a l area under the curve into a large number of small areas, each resembling a trapezoid. Each small area i s formed by two consecutive readings. The t o t a l area under the curve i s found by adding the areas of a l l trapezoids. The area of a trapezoid i s given by: Area = H(a + b)h where a and b are the length of the two sides of the trapezoid ( i . e . the Nw values of two consecutive readings) and h i s the distance between them ( i . e . the distance between two readings, h=0.334). The s t a r t i n g row i s selected based on the s t a r t of the compression curve. The cursor i s placed in column F one row below -339-the s t a r t i n g row before c a l l i n g the Macro. Remaining in column F, the user highlights up to the point of maximum compression found in column E. The t o t a l area under the curve defined by the highlighted section i s found in the l a s t c e l l of the section. This procedure is repeated for the second compression curve. After the c a l c u l a t i o n of the areas, cohesiveness can be e a s i l y calculated. This Macro can also be used to calculate adhesiveness. -340-APPENDIX C. LISTING OP THE FPOINT COMPUTER PROGRAM 1 CLEAR:CLS:DIM X(10,300),Y(300),L(10),U(10),M(11,10),S(10), B<10):INPUT "Maxialzation?(Y/N) H,X$:PRINT:INPUT "No. of factors? ",NN:INPUT "Maximum ver t i c e s to compute? ",MV:INPUT "Terminating difference value? ",TERM 2 INPUT "How many vertic e s without prohibit-trespassing? ",ZR: PRINT:FOR 1=1 TO NN:PRINT "Factor No. ";I:INPUT " Enter lower then upper l i m i t s H,L(I),U(I):PRINT:NEXT I 3 LPRINT "Terminating difference value=";USING " i l l . I I I I";TERM: LPRINT:LPRINT "Lower and upper limits":LPRINT " LL:";:FOR J=l TO NN:LPRINT USING " l l l l l l . I I I " ; L ( J ) ; : N E X T J:LPRINT:LPRINT " UL:";:FOR K=l TO NN 4 LPRINT USING " l l l l l l . I I I " ; U ( K ) ; : N E X T K:LPRINT:LPRINT:LPRINT: P=(1/(NN*SQR(2)))*(NN-1+SQR(NN+1)): Q=(1/(NN*SQR(2)))*(SQR(NN+1)-1):FOR J=l TO NN:M(1,J)=L(J): NEXT J:FOR 1=2 TO NN+l:FOR J=l TO NN 5 IF I-1=J THEN M(I,J)=L(J)+P*(U(J)-L(J)) ELSE M(I,J)=L(J)+Q*(U(J)-L(J)) 6 NEXT J:NEXT I:FOR K=l TO NN+l:FOR J=l TO NN:S(J)=M(K,J): NEXT J:GOSUB 46:B(K)=R:FOR L=l TO NN:M(K,L)=S(L):NEXT L:NEXT K :FOR XX=1 TO NN+l:FOR 1=1 TO NN:X(I,XX)=M(XX,I):NEXT I:NEXT XX :FOR Y=l TO NN+1:Y(Y)=B(Y):NEXT Y 7 LPRINT TABOO) " I n i t i a l Simplex":LPRINT TAB(16) "X1";:LPRINT TAB(25) "X2";:LPRINT TAB(60) "RESPONSE":XX=XX-1:Y=Y-1 8 FOR J=l TO XX:LPRINT "Vertex ";USING "III";J;:LPRINT USING "!####.###";X(1,J);X(2,J);:LPRINT TAB(56) USING " l i l i l . l i l " ; Y(J):NEXT J:LPRINT:Q0=0:WHILE Q0=0:WORST=B(1):WL=1:FOR 1=2 TO NN+1 9 IF(X$="Y")=0 THEN 11 ELSE IF B(I)<WORST THEN WORST=B(I):WL=I 10 GOTO 12 11 IF B(I)>WORST THEN WORST=B(I):WL=I 12 NEXT I:BEST=B(1):BL=1:FOR J=2 TO NN+1:IF(X$="Y")=0 THEN 14 ELSE IF B(J)>BEST THEN BEST=B(J):BL=J 13 GOTO 15 14 IF B(J)<BEST THEN BEST=B(J):BL=J 15 NEXT J:T=0:FOR 1=1 TO NN+1:T=T+B(I):NEXT I:NXT=(T-WORST-BEST) /(NN-l):FOR K=l TO NN:S=0:FOR L=l TO NN+1:S=S+M(L,K):NEXT L: S=S-M(WL,K):N(K)=S/NN:NEXT K:C=1!:C$="(Reflection)":GOSUB 41: FOR M=l TO NN:R(M)=S(M):NEXT M:REFL=R 16 IF(X$="Y")=0 THEN 27 ELSE IF(REFL>BEST)=0 THEN 19 ELSE C=2t: C$="(Expansion)":GOSUB 41:IF(R>REFL)=0 THEN 17 ELSE FOR N=l TO NN:Q(N)=S(N):NEXT N:GOSUB 45:GOTO 18 17 FOR 1=1 TO NN:Q(I)=R(I):NEXT I:R=REFL:GOSUB 45 18 GOTO 26 19 IF(REFL>NXT)=0 THEN 20 ELSE FOR J=l TO NN:Q(J)=R(J):NEXT J : R=REFL:GOSUB 45:GOTO 26 20 IF(REFL>WORST)=0 THEN 24 ELSE C=.5:C$="(Contraction-R)": GOSUB 41:IF(R>REFL)=0 THEN 21 ELSE FOR 1=1 TO NN:Q(I)=S(I): NEXT I:GOSUB 45:GOTO 23 -341-21 C=.25:C$=M(Massive contraction-R)":GOSUB 41:IF(R<REFL)=0 THEN 22 ELSE FOR K = l TO NN:Q(K)=R(K):NEXT K:R=REFL:GOSUB 45:GOTO 23 22 FOR L = l TO NN:Q(L)=S(L):NEXT L:GOSUB 45 23 GOTO 26 24 C=-.5:C$="(Contraction-W)":GOSUB 41:IF(R>WORST)=0 THEN 25 ELSE FOR J = l TO NN:Q(J)=S(J):NEXT J:GOSUB 45:GOTO 26 25 C=-.25:C$="(Masslve contraction-W)":GOSUB 41:FOR K=l TO NN: Q(K)=S(K):NEXT K:GOSUB 45 26 GOTO 37 27 IF(REFL<BEST)=0 THEN 30 ELSE C=2!:C$="(Expansion)":GOSUB 41: IF(R<REFL)=0 THEN 28 ELSE FOR N=l TO NN:Q(N)=S(N):NEXT N: GOSUB 45.'GOTO 29 28 FOR 1=1 TO NN:Q(I)=R(I):NEXT I:R=REFL:GOSUB 45 29 GOTO 37 30 IF(REFL<NXT)=0 THEN 31 ELSE FOR J=l TO NN:Q(J)=R(J):NEXT J : R=REFL:GOSUB 45:GOTO 37 31 IF(REFL<WORST)=0 THEN 35 ELSE C=.5:C$="(Contraction-R) M: GOSUB 41:IF(R<REFL)=0 THEN 32 ELSE FOR 1=1 TO NN:Q(I)=S(I): NEXT I:GOSUB 45:GOTO 34 32 C=.25:C$="(Masslve contraction-R)" : G O S U B 41:IF(R>REFL)=0 THEN 33 ELSE FOR K=l TO NN:Q(K)=R(K):NEXT K:R=REFL:GOSUB 45: GOTO 34 33 FOR L=l TO NN:Q(L)=S(L):NEXT L:GOSUB 45 3 4 GOTO 37 35 C=-.5:C$="(Contraction-W)":GOSUB 41:IF(R<WORST)=0 THEN 36 ELSE FOR J=l TO NN:Q(J)=S(J):NEXT J:GOSUB 45:GOTO 37 36 C=-.25:C$="(Massive contraction-W) H:GOSUB 41:FOR K=l TO NN: Q(K)=S(K):NEXT K:GOSUB 45 37 IF(XX>MV)=0 THEN 38 ELSE GOTO 40 38 IF R=0 THEN T$="Y" ELSE T$="N" 39 Q0=T$="Y":WEND:LPRINT:LPRINT "0 RESPONSE VALUE FOUND":FOR J=l TO NN:LPRINT TAB(ll) USING "#t#.lt#t";X(J,XX);:NEXT J:LPRINT TAB(56) USING "######.######";Y(Y) 40 PRINT "END":END 41 FOR 1=1 TO NN:S(I)=N(I)+C*(N(I)-M(WL,I)):NEXT I:IF(XX>ZR)=0 THEN 43 ELSE FOR J = l TO NN:IF S(J)<L(J) THEN S(J)=L(J) ELSE IF S(J)>U(J) THEN S(J)=U(J) 42 NEXT J 43 GOSUB 46:Y=Y+l:XX=XX+l:Y(Y)=R:FOR J=l TO NN:X(J,XX)=S(J):NEXT J:PRINT "Vertex ";USING "#l# ";XX;-.PRINT C$:PRINT " ";:FOR 1=1 TO NN:PRINT USING "»»#*#».»»#*«#"; X(I XX)*:NEXT I 44 PRINT:PRINT TAB(60) "Response";USING "######.######";Y(Y): RETURN 45 B(WL)=R:FOR 1=1 TO NN:M(WL,I)=Q(I):NEXT I:RETURN 46 FUNCT=0 50 CONl=l-S(l)-S(2) 60 IF C0N1>1 THEN 70 ELSE 80 70 FUNCT=FUNCT+CONl-l 80 IF CONK. 5 THEN 90 ELSE 100 90 FUNCT=FUNCT+.5-CON1 100 CON2=4.96*S(l)+15.51*S(2)+22.49*CONl -342-110 IF CON2>21 THEN 120 ELSE 130 120 FUNCT=FUNCT+CON2-21 130 IF CON2<16 THEN 140 ELSE 150 140 FUNCT=FUNCT+16-CON2 150 CON3=80.37*S(1)+18.67*S(2)+3.72*CON1 160 IF CON3>28 THEN 170 ELSE 180 170 FUNCT=FUNCT+CON3-28 180 IF CON3<8 THEN 190 ELSE 200 190 FUNCT=FUNCT+8-CON3 200 CON4=14.58*S(l)+65.69*S(2)+73.64*CONl 210 IF CON4>89 THEN 220 ELSE 230 220 FUNCT=FUNCT+CON4-89 230 IF CON4<54.33 THEN 240 ELSE 250 240 FUNCT=FUNCT+54.33-CON4 250 CON5=.2*S(l)+.8*S(2)+3.2*CONl 260 IF CON5>2.8 THEN 270 ELSE 280 270 FUNCT=FUNCT+CON5-2.8 280 IF CON5<1.9 THEN 290 ELSE 300 290 FUNCT=FUNCT+1.9-CON5 300 R=FUNCT 310 RETURN -343-APPENDIX D. LISTING OP THE FORPLEX COMPUTER PROGRAM 10 REM **MAIN LINE PROGRAM FOR COMPLEX ALGORITHM OF BOX 20 DIM X(50,50), R(50,50), F(50), G(50), H<50), XC(50) 30 CLS:INPUT "NUMBER OF FACTORS"; N 40 INPUT "NUMBER OF EXPLICIT AND IMPLICIT CONSTRAINTS"; M 50 INPUT "COMPLEX SIZE K=2N"; K 60 INPUT "MAXIMUM NUMBER OF ITERATIONS"; ITMAX 70 INPUT "NUMBER OF IMPLICIT CONTRAINTS"; IC 80 INPUT "PRINT 1 IF YOU WANT ALL ITERATIONS PRINTED, 0 FOR ONLY FINAL RESULTS";IPRINT 90 INPUT "ALPHA VALUE, RECOMMEND 1.3"; ALPHA 100 INPUT "BETA VALUE, CONVERGENCE CRITERIA"; BETA 110 INPUT "GAMMA VALUE, CONVERGENCE CRITERI"; GAMMA 120 DELTA=.0001 130 FOR 1=1 TO N 140 PRINT "FOR FACTOR NO. ";I 150 INPUT "ENTER LOWER THEN UPPER LIMITS ",G(I),H(I) 160 NEXT I 170 NN=N+1 180 FOR J=NN TO M 190 PRINT "FOR IMPLICIT CONSTRAINT NO. ";J-N 200 INPUT "ENTER LOWER THEN UPPER LIMITS ",G(J),H(J) 210 NEXT J 220 PRINT "GIVE THE ESTIMATES OF THE FIRST VERTEX, MUST BE WITHIN CONSTRAINT LIMITS" 230 FOR J=l TO N 240 PRINT J ; " FACTOR="; 250 INPUT X(1,J) 260 NEXT J 270 RANDOMIZE 3 280 FOR 11=2 TO K 290 FOR JJ=1 TO N 300 R(II,JJ)=RND 310 NEXT JJ 320 NEXT II 330 LPRINT TAB(25) "COMPUTATIONAL COMPLEX OPTIMIZATION":LPRINT 340 LPRINT TAB(35) "(CONSTRAINED SIMPLEX)" 350 LPRINT:LPRINT TAB(30) "PARAMETERS" 360 LPRINT TAB(IO) "N=";N; "M=";M;"K=";K;"ITMAX=";ITMAX;"IC=";IC 370 LPRINT TAB(IO) "ALPHA=";ALPHA;"BETA=";BETA;"GAMMA=";GAMMA; "DELTA=";DELTA 380 LPRINT:LPRINT TAB(5) "LOWER AND UPPER LIMITS ":LPRINT " LL:"; 390 FOR J=l TO N 400 LPRINT USING "####.••#!";G(J); 410 NEXT J 420 LPRINT:LPRINT " UL:"; 430 FOR 1=1 TO N 440 LPRINT USING "!###.###I";H(I); 450 NEXT I -344-460 L P R I N T : L P R I N T 470 L P R I N T " S T A R T I N G I N I T I A L P O I N T " : L P R I N T 480 FOR J=l TO N 490 LPRINT T A B ( I O ) " X ( " ; J ; " ) = " ; X ( 1 , J ) 500 NEXT J 510 L P R I N T : L P R I N T 520 I F IPRINT=1 THEN 530 E L S E 610 530 PRINT "RANDOM NUMBERS" 540 S=0 550 FOR 3=2 TO K 560 FOR 1=1 TO N 570 S=S+1 580 PRINT " R ( " ; S " ) = " ; R ( J / I ) ; 590 NEXT I 600 NEXT J 610 GOSUB 720 620 I F IT-ITMAX<=0 THEN 630 E L S E 700 630 LPRINT " F I N A L VALUE OF THE FUNCTION="; F ( I E V 2 ) 640 LPRINT " F I N A L X V A L U E S " 650 FOR J=l TO N 660 LPRINT " X ( " ; J " ) = " ; X ( I E V 2 / J ) 670 NEXT J 680 LPRINT "NUMER OF ITERATIONS N E E D E D " ; I T 690 GOTO 710 700 L P R I N T " T H E NUMBER OF ITERATIONS HAS EXCEEDED, PROGRAM TERMINATED" 710 LPRINT " B Y E " : E N D 720 REM **SUBROUTINE CONSX S T A R T S * * 730 REM * T H I S SUBROUTINE COORDINATES A L L OTHER SUBROUTINES* 740 REM IT= ITERATION INDEX 750 REM IEV1= INDEX OF POINT WITH MINIMUM FUNCTION VALUE (WORST) 760 REM IEV2= INDEX OF POINT WITH MAXIMUM FUNCTION VALUE (BEST) 770 REM 1= POINT INDEX (VERTEX) 780 REM KODE= CONTROL KEY USED TO DETERMINE I F I M P L I C I T (FAT) CONSTRAINTS ARE PROVIDED 790 REM Kl= DO LOOP L I M I T 800 IT=1 810 KODE=0 820 I F M-N>0 THEN 830 E L S E 840 830 KODE=l 840 FOR 11=2 TO K 850 FOR J=l TO N 860 X ( I I , J ) = 0 870 NEXT J 880 NEXT I I 890 REM * C A L C U L A T E I N I T I A L COMPLEX POINTS AND CHECK AGAINST CONSTRAINTS 900 REM *PROGRAM ANY MORE GOES TO GET UPPER AND LOWER LIMITS 910 TT=1 920 FOR 11=2 TO K 930 FOR J=l TO N - 3 4 5 -940 1=1I 950 X(II,J)=G(J)+R(II,J)*(H(J)-G(J)):PRINT X(II,J); 960 NEXT J 970 K1=II 980 GOSUB 2000 990 NEXT II 1000 REM *FINISHES CALCULATING INITIAL COMPLEX* 1010 REM *NOW GOES TO CALCULATE RESPONSES OF INITIAL COMPLEX* 1020 K1=K 1030 FOR 1=1 TO K 1040 GOSUB 2540 1050 NEXT I 1060 REM *FINISHES CALCULATING RESPONSE OF INITIAL COMPLEX* 1070 KOUNT=l 1080 IA=0 1090 IF IPRINT =0 THEN 1180 ELSE 1100 1100 LPRINT:LPRINT TAB(25) "INITIAL COMPLEX" 1110 LPRINT:LPRINT TAB(10) "XI";TAB(20) "X2";TAB(30) "X3"; TAB(60) "RESPONSE" 1120 FOR 1=1 TO K 1130 FOR J=l TO N 1140 LPRINT USING "######.####";X(I,3) ; 1150 NEXT 3 1160 LPRINT TAB(55) USING"######.####";F(I) 1170 NEXT I 1180 TT=0 1190 REM *FINDS WORST VALUE AND KEEPS IT IN IEV1 1200 LLY=0 1210 IEV1=1 1220 FOR ICM=2 TO K 1230 IF F(IEV1)-F(ICM)<=0 THEN 1250 ELSE 1240 1240 IEV1=ICM 1250 NEXT ICM 1260 REM *FINDS POINT WITH HIGHEST FUNCTION VALUE, KEEPS IT IEV2 1270 IEV2=1 1280 FOR ICM=2 TO K 1290 IF F(IEV2)-F(ICM)<=0 THEN 1300 ELSE 1310 1300 IEV2=ICM 1310 NEXT ICM 1320 REM *CHECK CONVERGENCE CRITERIA 1330 IF F(IEV2)-(F(IEV1)+BETA)>=0 THEN 1340 ELSE 1360 1340 KOUNT=l 1350 GOTO 1400 1360 KOUNT=KOUNT+l 1370 IF KOUNT-GAMMA>=0 THEN 1870 ELSE 1400 1380 REM *REPLACE POINT WITH LOWEST FUNCTION VALUE 1390 CC=0 1400 GOSUB 2340 1410 FOR JJ= 1 TO N 1420 X(IEV1,JJ)=(1+ALPHA)*(XC(JJ))-ALPHA*X(IEV1,JJ) 1430 NEXT JJ 1440 I=IEV1 -346-1450 REM *REFLECTION POINT, TAKES IT TO CHECK IF IT IS WITHIN THE CONSTRAINTS 1460 GOSUB 2000 1470 REM *REFLECTION POINT, TAKES IT TO FUNCTION TO CALCULATE RESPONSE 1480 GOSUB 2540 1490 REM *FIND IF REFLECTION OR CONTRACTION REPEATS AS LOWEST POINT IF IT DOES CONTRACT IT* 1500 WORST=l 1510 FOR ICM= 2 TO K 1520 IF F(WORST)-F(ICM)>0 THEN 1530 ELSE 1540 1530 WORST=ICM 1540 NEXT ICM 1550 IF WORST-IEV1=0 THEN 1560 ELSE 1710 1560 REM *CONTRACT THE REFLECTION OR CONTRACTION POINT* 1570 IF LLY>=5 THEN 1580 ELSE 1590 1580 CNT=CNTfl:GOTO 2450 1590 LLY=LLY+1 1600 FOR JJ=1 TO N 1610 X(IEV1,JJ)=(X(IEV1,JJ)+XC(JJ))/2:PRINT "1";X(IEVl,JJ) 1620 NEXT JJ 1630 I=IEV1 1640 CC=CC+1 1650 REM *TAKE NEW CONTRACTED POINT TO CHECK CONSTRAINTS 1660 GOSUB 2000 1670 REM *TAKE NEW CONTRACTED POINT TO CALCULATE RESPONSE 1680 GOSUB 2540 1690 GOTO 1500 1700 REM *WENT BACK AGAIN TO CHECK WHETHER IT REPEATS AS A LOW RESPONSE, OTHERWISE CONTINUES PRINTING PROCEDURE* 1710 IF IPRINT=0 THEN 1850 ELSE 1720 1720 LPRINT "ITERATION NUMBER"; IT 1730 IF CC=0 THEN 1740 ELSE 1750 1740 LPRINT "COORDINATES OF REFLECTION POINT":LPRINT CC: GOTO 1760 1750 LPRINT "COORDINATES OF CONTRACTION POINT":LPRINT CC 1760 FOR JC=1 TO N 1770 LPRINT USING "######.###»";X<IEVl,JC); 1780 NEXT JC 1790 LPRINT TAB(55) USING"######.####";F<I) 1800 LPRINT "COORDINATES OF THE CENTROID" 1810 FOR JC=1 TO N 1820 LPRINT USING"##||.I|##";XC(JC); 1830 NEXT JC 1840 LPRINT 1850 IT=IT+1:PRINT IT 1860 IF IT-ITMAX<=0 THEN 1190 ELSE 1870 1870 RETURN 1880 LPRINT "ITERATION NUMBER";IT 1890 LPRINT "COORDINATES OF POINT CONTRACTED 5 TIMES OVER THE BEST" 1900 FOR CPC=1 TO N -347-1910 LPRINT USING "######. # M i";X( IEV1,CPC) ; 1920 NEXT CPC 1930 LPRINT TAB(55) USING "######.####M;F(I) 1940 LPRINT "COORDINATES OF THE CENTROID" 1950 FOR JC=1 TO N 1960 LPRINT USING "####!».I#II";XC(JC); 1970 NEXT JC 1980 LPRINT "BYE, TRY AGAIN":END 1990 REM **END OF CONSX** 2000 REM **SUBROUTINE CHECK STARTS** 2010 KT=0 2020 REM *WENT TO PICK CONSTRAINTS 2030 REM *CHECK AGAINST EXPLICIT CONSTRAINTS* 2040 FOR J=l TO N 2050 IF X(I,J)-G(J)<=0 THEN 2060 ELSE 2080 2060 X(I,J)=G(J)+DELTA 2070 GOTO 2100 2080 IF H(J)-X(I,J)< = 0 THEN 2090 ELSE 2100 2090 X(I,J)=H(J)-DELTA 2100 NEXT J 2110 IF KODE >0 THEN 2120 ELSE 2320 2120 REM *KODE>0 IT CHECKS AGAINST IMPLICIT CONSTRAINTS* 2130 NN=N+1 2140 FOR J=NN TO M 2150 GOSUB 3000 2160 REM *GETS IMPLICIT CONSTRAINTS* 2170 IF X(I,J)-G(J)>=0 THEN 2180 ELSE 2190 2180 IF H(J)-X(I,J)<0 THEN 2190 ELSE 2290 2190 IEV1=I 2200 KT=1 2210 REM *GOES TO CENTR SUBROUTINE BECAUSE IMPLICIT CONSTRAINTS ARE VIOLATED AND CONSTRACTION TOWARDS THE CENTROID IS DONE 2220 IF TT=1 GOTO 2250 2230 CC=CC+1 2240 IF CC>=1 THEN 2260 2250 GOSUB 2340 2260 FOR JJ=1 TO N 2270 X(I,JJ)=(X(I,JJ)+XC(JJ))/2:PRINT "2";X(I,JJ); 2280 NEXT J J 2290 NEXT J 2300 IF KT<=0 THEN 2320 ELSE 2010 2310 REM *IT DOES GOES BACK TO CHECK EXPLICIT CONSTRAINTS BECAUSE POINT WAS MOVED TOWARDS THE CENTROID* 2320 RETURN 2330 REM **END OF SUBROUTINE CHECK 2340 REM "SUBROUTINE CENTR STARTS** 2350 FOR J=l TO N 2360 XC(J)=0 2370 FOR IL=1 TO Kl 2380 XC(J)=XC(J)+X(IL,J) 2390 NEXT IL 2400 RK=K1 -348-2410 XC(J)=(XC(J)-X(IEVl,J))/(RK-l) 2420 NEXT J 2430 RETURN 2440 REM *END OF SUBROUTINE CENTROID** 2450 REM **REPLACE CONTRACTION POINTS WITH A POINT REFLECTED OVER BEST** 2460 IF CNT>=2 THEN 2530 2470 CC=0 2480 FOR JJ = 1 TO N 2490 X(IEV1,JJ)=X(IEV2,JJ)+ALPHA*(X(IEV2,JJ)-XC(JJ)) 2500 NEXT JJ 2510 LLY=0 2520 GOTO 1440 2530 LLY=0:CNT=0:GOTO 1880 2540 REM **SUBROUTINE FUNCTION STARTS** 2550 TWLOSS=0*X(I,1)+29.747*(1-X(I,1)-X(I,2))+10.107*X(I,2) +29.111*X(I,1)*X(I,2)+16.481*X(I,1)*(1-X(I,1)-X(I,2)) 2560 FAT=15.013*X(I,1)+1.241*(1-X(I,1)-X(I,2))+1.718*X(I,2) 2570 WATER=0*X(I,1)+10.577*(1-X(I,1)-X(I,2))+18.151*X(I,2) -35.567*X(I,1)*X(I,2)-25.314*X(I,2)*(1-X(I,1)-X(I,2)) 2580 HARD1=0*X(I,1)+212.252*(1-X(I,1)-X(I,2))+795.988*X(I,2) -8308.693*X(I,1)*X(I,2)-1131.689*X(I,2)*(1-X(I,1)-X(I,2)) +14 410.514#*X(I,1)*X(I,2)*<1-X(I,1)-X<I,2)) 2590 COHES=.291*X(I,1)+.328*(1-X(I,1)-X(I,2))+.222*X(I,2) -.336*X(I,1)*(1-X(I,1)-X(I,2)) 2600 IF TWLOSS<=22.5 THEN 2610 ELSE 2710 2610 STW=l:GOTO 2730 2710 STW=ABS((TWLOSS-22.5)/.068) 2720 IF STW<=1 THEN STW=1 2730 SFT=ABS((FAT-4.5)/.034) 2740 IF SFT<=1 THEN SFT=1 2750 SWA=ABS((WATER-6)/4.600001E-02) 2760 IF SWA<=1 THEN SWA=1 2770 SH1=ABS((HARDl-160)/.803) 2780 IF SH1<=1 THEN SH1=1 2190 SCO=ABS((COHES-.255)/.0006) 2800 IF SC0<=1 THEN SCO=l 2910 F(I)=-(STW*SFT*SWA*SHl*SCO) 2920 PRINT IT;TWLOSS;WATER;FAT;HARDl;COHES;F(I) 2930 RETURN 2990 REM **END OF SUBROUTINE FUNCTION** 3000 REM **SUBROUTINE CONSTRAINT STARTS** 3010 X(I,3)=1-X(I,1)-X(I,2) 3020 X<I,4)=4.96*X(I/1)+15.51*X(I,2)+22.49*X(I,3) 3030 X(I,5)=80.37*X(I,1)+18.67*X(I,2)+3.72*X(I,3) 3040 X(I,6)=14.58*X(I /l)+65.69*X(I,2)+73.64*X(I /3) 3050 X(I,7)=.2*X(I,1)+.8*X(I,2)+3.2*X(I,3) 3060 RETURN 3070 REM **END OF SUBROUTINE CONSTRAINTS** -349-APPENDIX E. NOMENCLATURE AND DEFINITION OF THE QUALITY PARAMETERS Product weight loss parameters Shrink: Per cent weight loss a f t e r thermal processing Vacuum shrink: Per cent weight loss a f t e r 13 days under vacuum packaged storage Cook shrink: Per cent weight loss a f t e r the consumer cook test Effluxion s t a P l U t y parameters ES: Per cent fat released af t e r thermal treatment Tmloss: Per cent water loss per moisture content of the meat block aft e r thermal treatment Twloss: Per cent weight loss a f t e r thermal processing J u i c i n e s s parameters E x f l u i d : Per cent expressible f l u i d Exwater: Per cent expressible water Exfat: Per cent expressible fat T e x t u r a l parameters Hardl: Hardness at f i r s t compression, Newtons Hard2: Hardness at second compression, Newtons Firm: Firmness, Newtons/millimetre Cohes: Cohesiveness Spring: Springiness, millimetres Gummy: Gumminess, Newtons Chewy: Chewiness, Newtons millimetres Shear: Maximum shear force, Newtons -350-

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