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OPTIMIZATION WITH RANDOM  ERROR  by  ROBIN B.A.Sc.,  A  University  THESIS THE  GEOFFREY  SUBMITTED  in  accept  required  Columbia,  1964  I N P A R T I A L F U L F I L M E N T OF FOR  THE DEGREE  OF  OF A P P L I E D S C I E N C E t h e Department  CHEMICAL  We  of British  REQUIREMENTS MASTER  BOOTH  this  of  ENGINEERING  thesis  as conforming  to the  standard  T H E U N I V E R S I T Y OF :  June,  BRITISH 1968  COLUMBIA  In p r e s e n t i n g  for  this  thesis  in partial  an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h  that  the L i b r a r y  Study.  thesis  shall  I further  make i t f r e e 1 y  agree that  f o r s c h o l a r l y p u r p o s e s may  publication  of this  w i t h o u t my w r i t t e n  thesis  Date  ^fu~*>  O ^ j  Columbia  ( 9 ^ %  Columbia,  I agree  for reference  for extensive  and  copying of  this  be g r a n t e d by t h e Head o f my  It i s understood  for financial  permission.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  avai1ab1e  permission  D e p a r t m e n t o r b y h.i)s r e p r e s e n t a t i v e s .  or  f u l f i l m e n t of the requirements  gain  shall  that  n o t be  copying  allowed  i i  ABSTRACT  A  new  method to,  i s presented.  i s that  stricted The  evolutionary  in  method  these  trial  array  design  The method  search  i s adhered  variables  a previous  regular  type  f o r t h e optimum.  when  are r e -  best  method  o f those  when t h e l a s t  and  trial. of the  Himsworth  arrays  The  (and o n l y  i r r e g u l a r array  formed  complicial  which  Spendley,-Hext  the simplicial  i n a  complicial  i n that when)  an  the  last  previously  i s such trial  that  a  i s proven  tested. regular not to  so f a r .  complicial  method  variety  and p r e s e n c e  complicial  by  t o be t h e b e s t  f o r a wide  absence  criterion,  from  employ  i s formed  of this  can be  the best  proposed  from  i s proven  array be  type  the  i s essentially a modification  authors  differs  irregular  main  step-size  method  method  sequential  The  small  complicial  which  The  called  changes i n the independent  to a  simplicial  operation  method  effectiveness  surfaces  Although  an  of response  o f random  i s much m o r e  i s very  response  i s compared  large)  f o r almost  i n both  I t i s found  effective  i n t h e amount  simplicial  surfaces  error.  involving a small  increase  to the  that  (i.e. the  the the  relative  a l lt h e t e s t  number  of  o f random  variables. error  decreases  the  effectiveness  of both  generally  remains  variables  i s increased  to  decrease  of  the basic  simplicity  Therefore^ surfaces  design  the r e l a t i v e  However,  the r e l a t i v e effectiveness This  i s explained  of the regular  the complicial  that where  method  the complicial  and  by  sacrifices  method be  involving  a  i s best small  used  is. found  arrays.  some o f t h e  method,  applied  variables.  i ti s  only  i s very  f o r a l l types  number, o f  of  considerations  irregular  the r e l a t i v e effectiveness  t h i s . method  effectiveness  a s t h e number  characteristic of the simplicial  recommended situations  unchanged.  markedly.  Because  methods,  of  i n  large. response  iv  ACKNOWLEDGEMENTS  The Dr.  author  would  D. A . R a t f c o w s k y  sistance, given  guidance  constantly  like  t o extend  h i sthanks  t o whom h e i s i n d e b t e d and encouragement  throughout  which  the course  to  f o r aswere  of this  project.  V  TABLE  OF  CONTENTS Page „ '  INTRODUCTION.  o ....... .  1  P R E V I O U S WORK IN. T H E F I E L D OF  EVOLUTIONARY  GENERAL  CRITERIA  SIMPLICIAL  OPERATION. . . . .  ..„..«..  0  .3  COMMON F O R BOTH T H E  AND C O M P L I C I A L METHODS  . ...  8  B A S I C D E S I G N OF T H E A R R A Y S . ............o.o 0-arrays. . 1  C t l T 37  S  # e  e  .....  e o  o  e  .....10  o  o  «  «  e  o  .......10 *  »  o  «  a  O  «  o  o  «  «  «  e  THE  SIMPLICIAL  THE  COMPLICIAL SEARCH TECHNIQUE. . . .  o  «  «  V  9  0  *  a  O  «  e  «  S I M U L A T I O N OF T H E COMPARISON  Format  Random  Error  Moving  Optimum.  SEARCHES  *  «  ^  *  ^  16  28  0  ....<.. o  ........28  <,<><,.<, . . . . . . .  29  c  o f Effectiveness  - Deterministic.....  Measure  of Effectiveness  - Non-deterministic  Program......  9  ..........28  ... o « ... o  COMPUTER PROGRAM  *  20  0  .«,  Effectiveness  *  .  Measure  Main  «  OF  C O M P L I C I A L AND S I M P L I C I A L  Relative  *  SEARCH TECHNIQUE  0  General  »  c  0  30 .30  ...o................31 33 33  Insert  A  33  Insert  B.  33  vi  Page Insert  C...  Plotter RESPONSE  ..........  Output  .  SURFACES USED  C O M P A R I S O N OF Normal Two  37  THE  IN  THE  METHODS  Multivariate  .. 4 1  ,  .47  Distribution  47  Variables.......'  Greater  than  Two  47  Variables  54  RESULTS  o  Brooks  R l , R2,  Multivariate Bivariate  Normal  Normal  Curvilinear Moving  R3,and  R4  57  Response  Response  Surfaces  Surfaces  84 -  86  Ridges  or Drifting  D I S C U S S I O N OF  57  98 Response  Surfaces  RESULTS  I l l 117  General The  H7  Number  Ridge  of Variables.  119  Sharpness  Curvilinear Moving  120  Ridges.......  Optimum  120  Surfaces.  CONCLUSIONS  121 o  122  NOMENCLATURE  124  LITERATURE  126  CITED.....  APPENDIX  I.....  APPENDIX  II  «...  *  1  -  1  I I - l  vii  Page APPENDIX  III  APPENDIX  IV  ....  Nomenclature  ....  f o r Computer  Computer  Listing.  Computer  Output.  III-l  P r o g r a m . . . . . . . . . . . . . . . . . IV-1 c  IV-2 .....................  APPENDIX V  IV-18 V - l  o  v i i i  LIST  OF  FIGURES  Figure  Geometric  Page  illustrations  1.  O-a^rays  2.  Rule  1  (simplicial)  17  3.  Rule  3  (simplicial)...  19  4.  Rule  4  (complicial)  22  5.  Rule  5  (complicial)  23  6.  Rule  6  (complicial)  26  Flow  charts  7.  Main  Program  8.  Insert  A  35  9.  Insert  B  . 36  10.  Insert  C - Initialization  11.  Insert  C - Search  12.  MOVE  40  13.  RETEST. . .  42  14.  MEMTRA  43  15.  LOGIC  44  16.  RESULT  45  Contour  and 1 - a r r a y s  a  o  . . .  section  <,.ol2  34  section  .....38 39  maps  17.  Brooks  R]  49  18.  Brooks  R2..;  51  ix  Figure  Page  19.  Brooks  R3  .52  20.  Brooks  R4  53  21.  Rosenbrock's  Number  surface  of t r i a l s  various  (modified)  required  percentages of  to  55  reach  optimum  22.  .Run  31  58  23.  Run  34....  24.  Run  35..  60  25.  Run  37.  61  26.  Run  39  .62  27.  Run  41  63  28.  _Run  29.  Run  44  30.  Run  45  31.  Run 46  ...59  43.  0  65  c  .  ...o...  Comparison searches  . . .. 64  of complicial  with  various  and  66  67  simplicial  d e g r e e s o f random  error  32.  Run  31  68  33.  Run  34.  69  34.  Run  35  70  35.  Run  37..  71  36.  Run  39  72  X  Figure  Page  37.  Run 41  '.  73  38.  Ran 43.......'  39.  Run 44  75  40.  Run 45  76  41.  Run 46  42.  Average R l , R2, R3 and R4  79  43.  O v e r a l l average  80  44.  R e l a t i v e e f f e c t i v e n e s s v e r s u s t h e amount  .  .  . . ..  77  o f random e r r o r 45.  Runs 31 t o 46  82  R e l a t i v e s c a t t e r v e r s u s random e r r o r ..' for  47.  81  R e l a t i v e s c a t t e r v e r s u s random e r r o r for  46.  Runs 40, 41 and 43  Number o f t r i a l s r e q u i r e d t o r e a c h  83 various  p e r c e n t a g e s o f optimum (Runs 61 t o 64) 48.  74  ..85  Number o f t r i a l s r e q u i r e d t o r e a c h 68.4% o f optimum v e r s u s t h e number o f v a r i a b l e s  87  Comparison o f c o m p l i c i a l and s i m p l i c i a l s e a r c h e s w i t h v a r i o u s degrees o f random e r r o r 49.  Run 61  88  50.  Run 62  89  51.  Run 63  90  52.  Run 64  91  xi  Figure 53.  Page Relative number  effectiveness  versus  the  of variables  Contour  S2  maps  54.  Run  47..  93  55.  Run  48  94  56.  Run  49  95  57.  Run  50  96  58.  Run  51  97  Comparison searches  of complicial  with  various  and  degrees  simplicial o f random  error  59.  Run  47  60.  Run  48.  100  61  Run  49  101  62.  Run  50.  102  63.  Run  51  103  64.  Relative  0  the  99  effectiveness  sharpness  Comparison searches  versus  of ridge...  of complicial  with  various  ...104  and  degrees „  simplicial o f random  error  65.  Run  55  105  66 „  Run  56  106  67.  Run  57.  107  xii  Figure  Page Number  of t r i a l s  various  required  percentages  of  to  reach  optimum  68.  Run  55...  *  69.  Run  56......  70.  Run  57  71.  Run  52.  112  72.  Run  54  113  109 ,  Comparison searches  108  110  o f c o m p l i c i a l and  with  various  degrees  simplicial o f random  error  73.  Run  52.  114  74.  Run  54  115  75.  "Relative for  76.  effectiveness versus  moving optimum  Possible  random  cases  explanation  for inferior  ...116 runs  118  1  INTRODUCTION  Optimization fection  o r near  throughout doing him  quest.  T h i s development  and d e s i g n ,  and t e c h n i q u e s  t oa i d  of optimization has  day i n t h e form  of "getting the  undertaken  with  methods  engineers  a concern  technique  new t r i a l s  pendent  i n the f i e l d  forindustrial  economics.  existing  commercial  method b y W i l d e  maximum  to  find This  to as "evolutionary operation" by small  technique  (24),r e f e r r e d  and B e i g h t l e r  with  profit.  changes  i n the de-  so a s t o i n s u r e t h e w o r k a b i l i t y One s u c h  and Himsworth  itself  plants —  yield  concerned  of optimization  concerns  chemical  which  are governed  unit.  study  The r e -  primarily  o f knowledge  i s referred  variables  has been  by  are motivated  and u n d e r s t a n d i n g .  the present  to optimize  motivated  i n research  i n the project  operating variables  general  time".  f o r knowledge  Specifically,  Hext  and i n  i n industry are mainly  engineers  per-  perfection  While  the broadening  where  forthis  to obtain  o f a r t and s c i e n c e  possible  search  those  strives  t o the present  the yearning  but  I n <A11 f i e l d s  i n the shortest  economics  v/ith  man  procedures  s o h a s c r e a t e d many p r o c e d u r e s  transcended  by  perfection.  t h e ages  i n this  best  describes these  of the  contributed by  t o as t h e  Spendley,  "simplicial"  (25),has been p o p u l a r l y  received where  f o r i t s s i m p l i c i t y and a p p l i c a b i l i t y  the effects  technique plicial order will  over out  proposed  method  be  i n this  referred  a much  range error.  greater  plicated  are important focuses  around  effectiveness.  t o as the " c o m p l i c i a l  method  s i m p l i c i a l method  random  thesis  the overall  complicial  a wide  error  (9).  The  the sim-  b u t s a c r i f i c e s some o f i t s s i m p l i c i t y i n  to increase  The the  o f random  i n situations  will  o f unimodal This  i s  response  method  would  can be  involved.  method  method".  compared  as to the degree  effectiveness  technique  be  This  of  d i r e c t l y with  effectiveness  surfaces  justify achieved  with  itself  and  with  only i f  a s a more  com-  3  P R E V I O U S WORK I N THE OF  The s t e e p e s t  EVOLUTIONARY  FIELD  OPERATION  a s c e n t method, f i r s t  proposed b y Cauchy  (10), was a p p l i e d t o problems i n i n d u s t r i a l s t a t i s t i c s b y .Box and W i l s o n ( 6 ) .  T h i s p r o c e d u r e i n v o l v e s the  to the region o f higher  response by s u c c e s s i v e  the g r a d i e n t d i r e c t i o n i n each s e t o f t r i a l s .  progression  inferences t o Researchers  developed t h i s t e c h n i q u e i n accordance w i t h p a r t i c u l a r c r i teria.  One such c r i t e r i o n chosen was t h e assumption t h a t  the system was d e t e r m i n i s t i c ( i . e . no random e r r o r ) . u s u a l l y i n v o l v e d a second c r i t e r i o n t h a t s t e p - s i z e  This  changes  o f t h e independent v a r i a b l e s need not be r e s t r i c t e d . A l though t h e r e s e a r c h  i n this particular f i e l d of optimization  (23), (8) and (14), s e r v e s t o e s t a b l i s h fundamental  (13),  p r i n c i p l e s i n the general  f i e l d of optimization, a f u l l  d i s c u s s i o n i s o u t s i d e t h e scope o f t h i s t h e s i s . Another c r i t e r i o n , t h a t o f assuming a degree o f random e r r o r , i s i m p o r t a n t i n the o p t i m i z a t i o n o f i n d u s t r i a l cesses.  F o r p i l o t p l a n t work and i n i t i a l  start-up  pro-  conditions  on l a r g e s c a l e u n i t s , s t e p - s i z e changes o f t h e o p e r a t i n g i a b l e s are e s s e n t i a l l y u n r e s t r i c t e d . plants i n production,  var-  Hov/ever, f o r e x i s t i n g  the c r i t e r i o n o f small step-size  changes  4  i s necessary substandard  t o ensure t h a t the product  does not become  i n q u a l i t y and a l s o t o prevent  i n equipment and o p e r a t i n g time.  disastrous losses  Methods o f e v o l u t i o n a r y  o p e r a t i o n g e n e r a l l y have been developed f o r t h i s  The  application.  p o p u l a r i z a t i o n o f the term " e v o l u t i o n a r y o p e r a t i o n "  i s mainly a t t r i b u t e d to the work o f G.E.P. Box ( 3 ) . He des c r i b e s e v o l u t i o n a r y o p e r a t i o n as a method o f p r o c e s s which not o n l y generates product  but also supplies  i n f o r m a t i o n t o enable the p l a n t p e r s o n n e l product.  operation  statistical  t o improve t h e  Box draws an analogy o f t h e Darwinian p r o c e s s by  which l i v i n g t h i n g s advance by g e n e t i c v a r i a b i l i t y and n a t u r a l s e l e c t i o n t o t h e e v o l u t i o n a r y i n d u s t r i a l o p e r a t i o n by which p r o f i t a b i l i t y i n c r e a s e s by small changes o f t h e p r o c e s s i a b l e s and s e l e c t i o n o f the most f a v o u r a b l e v a r i a n t s .  varIn  each case an e v o l u t i o n a r y f o r c e i s a t work t o move t h e system towards optimum c o n d i t i o n s .  The  search  f o r optimum c o n d i t i o n s i n v o l v e s a c y c l i c  route.  First,  a m o d i f i c a t i o n i s made i n the o p e r a t i n g  ables.  Secondly, i n f o r m a t i o n i s compiled  response s u r f a c e .  vari-  concerning the  F u r t h e r m o d i f i c a t i o n s are made based on  t h i s information to y i e l d yet additional information. t h i s c y c l e i s repeated,  As  s u f f i c i e n t i n f o r m a t i o n i s garnered  t o enable the p l a n t personnel  t o " e x p l o r e " t h e response  5  surface (5), and  Box  as  the  EVOP a  chemical  responses  than  array the  small  t o be  factorial these  that  previous  (16),  and  array.  Box  reviews  array of  Each  Box  (4), Koehler  array  format (or  Box  (determined analysis  information to  change  (19),  represents  (1)  on  higher a  Modifi-  p u b l i s h e d by  Baasel  by  determine  productivity  been  Box  performed  operating variables.  Lowe  paper,  Essentially,  average  EVOP h a v e  (18),  later  a more  Statistical  an  i n the  of  a  research  trials  provide  yield  on  (22).  design).  should  presents  based  and  trials  s t e p - s i z e change  cations  (17) an  In  evolutionary operation  known),  testing  of  Hunter  to  industry  of  productivity.  with  approach  i t became  consists  new  greater  i n association  prearranged  the a  achieve  quantitative  EVOP in  and  and  Hunter Barnett  (2).  The  most  recent  factorial  design  developed  by  that  method cited  type  show t h a t , compares  and  response  of  fundamentally EVOP)  Hext  and  surfaces. method,  i s the  observations  i n the  many  to  are  presence Box  p r e v i o u s l y by Due  the  different  simplicial  Himsworth  favourably with  compared  simplicial  (Box  Spendley,  replications  authors  method  of  not  EVOP  Brooks  who  and (7)  propose  error, other on  four  simplicity  industries  have  the  method  beneficial.  random  relative  chemical  (24)  from  The  their methods test of  found  the the  6  method  attractive,  as  ( 9 ) , Lowe  (19)  Sweeney  Modifications posed The  by  step-size  stricted Also, with in  pends  trials cases  and,  choice of on  the  outcomes  a mixed  and  the  but  method  violate  arrays of  avoided  type  using  (25).  have  been  certain  Nelder  pro-  criteria. are  and  response  their  and  unreMead.  surfaces  technique  belongs  category.  of  evolutionary operation  search  strategy:  experiments  simultaneous  before  Beightler  operating variables  type  search  (1), Carpenter  any  trials  are  a  are  known.  simultaneous)  In  se-  based  (non-sequential) plan  results  and  or  either  de-  where  some  strategy  i s  (25) .  experiments the  are  s e a r c h methods  very  experimental t o be  carried  simultaneous  methods  (12),  Eaasel  therefore,  (sequential  Simultaneous  trials  (20)  future  or  chosen  Wilde  a particular  p l a n where  are  chosen  and  have  by  simplicial  expanding  error  largely  past  the  changes o f  deterministic  quential  and  Mead  authors  random  the  to  and  i n the  these  The  on  Nelder  reviewed  Friedman  and  lengthy  are  (e.g.  used  i n cases  in fields  of  where  agriculture)  "room"  i s large  enough  to  permit  out  the  time.  The  more  at  which  Savage  same  have  (15)  been  and  proposed  Cochran  and  the  are by Cox  many important DeBaun  (11).  7  However,  i n optimizing  for  r e s u l t s  for  one  i s r e l a t i v e l y  set of operating  conditions  (7). to  a  large  (degree of  block  The  with  time)  variables tions  upon  position, these  unit,  searches  choice  of  extent  a  at  upon  (number  depend  and  upon  or  of  upon  w i l l and  method.  random  d r i f t  the  and  discussed  t h e method  scale i n the  proposed  only  be  time..  between such  tt-sted  These  simultaneous  factors  own  search  size  upon  as  time  experience depends  the response  shape  optimum  operating  system,  changes,  r e s t r i c s t a r t i n g  factors).  Most  comparison i n this  surface  and  the  of the  o f the  step-size  delay  methods.  error,  effects  dimensionality  be  of  the properties  orientation effects,  complicial  choice  the properties  the magnitude of  method  type  one  the experimenter's  amount  also  f o r a  any  may  p a r t i c u l a r sequential  constraints, and  the time  the unit  variables  e f f e c t s or  properties  simplicial the  type  of modality,  ridges,  l o w and  considerations  sequential  trends,  industrial  q u a l i f y i t f o r sequential  Further and  an  of  of the  thesis,  8  GENERAL  CRITERIA  SIMPLICIAL  Before and both  proceeding  complicial methods  AND  to a  methods,  a  are founded  COMMON FOR COMPLICIAL  formal  summary  BOTH  THE  METHODS  discussion  of the  of the c r i t e r i a  i s listed  Both  searches  a r e o f an e v o l u t i o n a r y  2.  Both  searches  are sequential  3.  The  best  4.  change o f t h e o p e r a t i n g  to a  New  chosen  step-size  trials  design.  variables  distance  a r e computed tested  5.  The  response  6.  The  optimum  having  The  i n  from  The  i s restricted  the  previous  a  limited  optimum  surface  of  i s unimodal.  i s a maximum  (the best  trial  i s that  response).  surface  may  number  trials.  the highest  response  from  may  possess  (non-deterministic).  8.  type.  trial.  previously  7.  upon which  below.  1.  '  simplicial  drift  with  time.  random  error  9.  Both  searches  enable  10.  them  are essentially  to follow  A l lvariables  not terminated  a moving  are scaled  before  optimum.  the search  i s  t o a n y number  of  initiated.  11.  Both  searches  operating  can be  variables.  /  applied  to  10  BASIC  The a  regular  array. of  basic  N  or  Each  design  DESIGN  of  an  THE  array  0-(zero)-array, array  OF  consists  or, of  ARRAYS  can an  be  one  of  irregular  N+1  points  two  or  forms;  l-(one)-  occupying  a  space  a  0-array:  dimensions.  0-arrays The  1.  following  All  the  another an the  2.  properties  points by  array  a  are  regular  Certain  P  to  L, J  (N+1)  length the  characteristic  are R.  equidistant Examples  in  a  their  0-array relative  That  point the  (lowest =  min.  That LL, J  from this  Tr  point the  sponse  K  (A  type  (N=2)  1,  be  of  and  labelled  responses.  in  the  lowest  _ for  next  array response  K  =  1,  N+1)  array  lowest  lowest  N+1,  These  (i.e. Z  i n the  (next  =  ) may  point). Z  having  for  one  (N=3).  having  P  of  of  equilateral triangle  tetrahedron  points  according  are  K ^  re-  point).  L)  are:  11  3.  The  new  point  which  nas  i n the  array  A^  been  computed  from  the  0 points  4.  Algebraically, ed  a  i s denoted  O-array  (relative  in  matrix  of  unit  to a  From  N +  edge  chosen  form.  P  by  2 j •  can  origin  Spendley,  be and  set  Hext  and  (24) , t h i s m a t r i x i s :  A  /  0  0  0  0  0  p  q  q  q  q  p  q  q  q  q  P  q  •  •  q  q  q  p  1  P  (1)  3, J 4, J  P  /  N+l, J  where:  r (VN+I + N-1) NV2  P  (2) (3)  5.  The  elements  one  of  The  above  can be array  the  N+l  to  row  represent  the  coordinates  of  points.  matrix  used  (relative  establish  to  the  a  starting  points  point)  i n the  first  (A^).  Geometric  examples  in  1.  Figure  in a  of  O-arrays  f o r N=2  and  N=3  can be  found  0-array  1-array  N = 2 Points  P  Z, J  and P  3J  a r e common  to the joint  figure,  t  Points  P , Pand P are colinear ' 1, J 3, J 4,J space o f one d i m e n s i o n ) .  (occupy  a  2, J  5, J  1-array  N •= 3 Points  P_ , P 2,J, 3J B  Points  P  1,u  space  Figure  1.  , P  and P 4,J  a r e common  to the joint  , P and P are coplanar J , J 4,J 5,J  figure,  (occupy  o f two d i m e n s i o n s ) .  Geometric  examples  o f O-arrays  and 1 - a r r a y s  a  13  1-arrays The  1.  following  A l l the  properties  points  equidistant particular each  of  length  D  except  from  one  pair,  the  are  characteristic  for  one  another  although  remaining  (referred to  a  1-array:  particular pair by  they  points, as  of  a  length  are are  the  R  R.  are This  units  from  separated  by  a  characteristic  diagonal).  2.  The  1-array  is,  the  . having --•-. N + l  space 3.  From  "complement"  combined N  of  is a  points  the of  in  points  N-1  the  array  of  a  O-array  common, h a s i n the  the  the  combined  O-array. and  a  1-array,  property array  That  that  occupy  two  properties,  characteristic  diagonal  can  Appendix  be  to:  to  equal  be  the  length  derived  of  the  (see  R  4.  Certain  points  according  to  a  dimensions.  above  I)  of  in  their  1-array  (4)  (A  relative  1  ) may  be  responses.  labelled These  are:  P  j  D D  That  point  eristic been in  computed  the  That  D, J  the  diagonal  (oldest That  L, J  having but  i s not  acteristic  JJ / J  =  The  new  pointsS  point which in the  a  (relative  of  axes)  i n matrix  is  derived  of  the  of  the  to  represent  1-array.  a  the  the  (lowest  point) Z  K, J  D,  unit  (i.e.  for K  by  from  two  the  P  N+2, J edge  can  origin  first  be  and  following  The  =  DD)  computed  The  I.  array  diagonal  chosen  form.  i n Appendix  matrix  of  point).  char-  K ^  been  1-array  diagonal  on  a r r a y A^. i s d e n o t e d J  Algebraically, presented  has  the  response  min.  .1, N+1,  ) J— 1  lowest  non-diagonal Z  on  i n the  the  (A  point).  diagonal  point  had  points  array  point  characteristic  P  which  diagonal  other  charact-  from  previous  (newest  P  on  two  diagonal  re-  set  matrix rows points  A.  2q  r  r  P  q  q  q  q  p  q  q  q  q  p  q  3  1,  J  2,  J  Y, J 3  q  (5)  4,J N+l,  J  can  be  where  r = N-1  Geometric in  Figure  examples 1.  (  of  (N-2)q  +  1-arrays  p)  f o r N=2  a n d E-3  found  16  THE  A  sequential  "moves"; setting  each  SIMPLICIAL  search  move  usually  o f "moving"  TECHNIQUE  consists  corresponding  o f the independent  technique  SEARCH  series of  t o a nex^ p o i n t  variables).  consists  of a  The  (i.e. a  new  simplicial  of discarding  one p o i n t  i n a  0 previously newly The  determined  computed  point  simplicial  Spendley, search  technique.  should  situations  each  technique  1.  t h e average  systematic  form  o  only  a new  O-arrays  that  this  response  an o p t i m a l  surface  tend  the lowest remaining  (  p N +  2 j) i  point points  (P  T  T  s  )  + 1  ).  i n i t s i s  repeated  of  value.  changes  with  each  For time  and t h e p o i n t s i n  to follow  o f moving  (A^  a  (referred to by  of the points  the optimal  point.  i n the simplicial  by the following  T h e nex\7 p o i n t  O-array  technique  i s not terminated will  i t with  (24) a s " s i m p l e x e s " )  approach  method  i s governed  the  t  the response  the search  of  j)  2  Providing  successive ^array The  +  (Aj) and r e p l a c i n g  employs  eventually  where  ("drifts"),  N  and Himsworth  systematically, array  (P  method  Hext  O-array  search  set of rules.  formed through  i n the array  from  the  reflection  the centroid of (A?.).  Thus:  17  N+1  - 7  ' N+2, J  -  p  N  P L, J  (7)  K, J  K=l This  formula  and a l l s u b s e q u e n t  determination  o f a new  point  formulae  are found  f o r the  i n  Appendix  II. The  new  (A^)  by  array  replacing  new p o i n t  A  J+1  =  P  ( A ^ ,) J+l  (P  the lowest  P  Geometric N=2  P  L+1,J'  illustrations  a n d N=3  from  point  the o l d array  (P_  _) w i t h t h e  ).  N + 2 # J  1 , J ' 2 , J ' *•*' L - 1 , J ' P  i s formed  *•"  f o r Rule  P  N+2,J  (  8  )  1 a r e shown f o r  below.  Centroid of  Centroid of  ,J' T,J' U,J  S,J » T , J  P  P  P  N+2.J  N=  N = 2  Figure  2.  Geometric  illustrations  3  o f Rule  1  (simplicial)  18  2  If a  a  and  (high)  occurs  i s not eliminated  then  3.  point  that  I f t h e new (A^. in not  computed  that  array  point  from  the o l d array  with  the lowest  (A?), ( i . e . u Instead  formed  from  the reflection  ) through  LL, J  i n the array  response  T  1.  points  1,  , , = P , ) , do N+z,J—1 Ii,J  Rule  (P  arrays  by t h e a p p l i c a t i o n o f Rule  apply  point  successive  i s retested.  point  ) i s also t h e new  is  point  i n N+l  t h e new  point  o f the next  the centroid  (A^).  Thus:  p  _ p  ( P „ .„ ,) N+2, J lowest  o f the remaining  N+l p  N+2,J  = -  V  N  LJ  K,J  LL,J  v  (9) '  K=l K^LL  The  new /  (A^.)  J  with  A  J+1  =  P  1,J' 2  array  by replacing t h e new  P  the next  point  J ' **•' L L - 1 , J '  rule  Geometric  P  back  examples  shown o n t h e n e x t  lowest  LL+1,J'  i s necessary  oscillating  from  the o l d array  point  (P  T T  T  )  LL, J  (P ). CJ +/J, u  P  (  . The above from  (A^. ) i s f o r m e d J+0.  page.  P  to prevent  and f o r t h  o f Rule  * * ° ' N+2,J  between  3 f o r N=2  (  the two  a n d N=3  1  0  )  search points.  are  19  P  N  =2  Figure  Appendix searching  3  0  L,J N*2,J-I = P  N  Geometric  III gives various  illustrations  examples types  of  of the response  =3  of  Rule  simplicial surfaces.  3  (simplicial).  technique  20  THE  COMPLICIAL  SEARCH  TECHNIQUE i  The an  array  array  technique  systematically discarding  and r e p l a c i n g  i s applicable  simplicial only  of  search.  0-arrays,  Although  point  must  point  i s not v i o l a t e d ,  search  within  systematic  technique  1.  (A  2.  J  Y,  general  method  as w e l l search  as t h e  involves  both  the  criterion  that  a  0-  new  the previously  best  by  i n the complicial  the following  set of  o f t h e newest p o i n t  array  must  by  points  a 0-array  array  from  i n the  a 1-array  o f a l lt h e other  (A ) , t h e n e x t J  must  be  rules.  i s higher  o f t h e newest p o i n t  the responses  method  new  involves  R from  o f moving  i s governed  generate  a  from  however.  I f the response  To  search  a distance  the next  array  t o form  search  r e s p o n s e s cf a l l t h e o t h e r  than  3.  The  I f the response the  point  the simplicial  the complicial  and t h e 1 - a r r a y .  The  a new  to the complicial  array  be  i t by  a point  array  (h\  J +1  ).  i s not  higher  points  i n the  a 0-array  a 0-array  than  the  (A*? ) . J +1  following  i s used.  The  new  point  the  lowest  i s formed  point  through  from  the r e f l e c t i o n  the centroid  of  of the r e -  21  maining  points  identical  i n the array  to that  o f Rule  (A^.) . T h e f o r m u l c J  1 of the simplicial  N+1  v  The by  J+l  = p  4.  •  search.  • -  2  N+2,J  new  N  array  replacing  P  1 , J ' 2,J'-" y  i s  ^ K=l  K, J  (A*? ) i s f o r m e d J+l the lowest  P  illustrations  in  2.  method  a 1-array  with  the o l d array t h e new  P  L+1,J'  Geometric  To g e n e r a t e  point  from  P  L~1,J'  ml  Figure  L,J  N+2,J  f o r Rule  from  3 above  a 0-array  point.  K  (1?) '  a r e shown  the following  i s used.  The  new p o i n t y  of  the lowest  ( P „ _ J) i s f o r m e d N+2 , J point  (P  from  ) through  the  reflection  the centroid of  Lij the  remaining  point  (P  T T  points  excluding  _) o f t h e a r r a y  LL, J  the next  (A^) . J  lowest  Thus:  N+1  p  2  N+2, J  N K=l K,£L,LL  The by  new  array u  (A^.,,) i s f o r m e d J+l  replacing the lowest  point  from  the o l d array  ( P _ _) w i t h L / u  the  new  ( P  point  P  l , J  f  P  +  2  f  j ) .  2,J'  Geometric for  N  .N=2  P  L-1,J'  P  illustrations  L+1>'  c f Rule  4.  To  of  (  4 a r e shown  1  4  )  below  N=3  Geometric  generate  method  The  N+2,J  N=3.  and  N = 2  e  P  a O-array  point  the oldest  centroid  Thus:  from  o f Rule  4  (complicial).  a 1-array the  following  i s used.  new  newest  illustrations  (P„,~ J N+2, J  i s formed  diagonal point  (P  from  the  reflection  ) through the  o f the remaining points 'excluding the  diagonal point  (P-.-. ) T  of the array  (h\).  23  P  The  N+2,J  new  N+l Z K,J K=l K^D,DD  N-1  =  array  "  P  (A^  ) is- formed  P  D,J  from  (  1  5  )  the o l d array  j+x by  replacing  the  A  J+1  =  P  new  point  1,J' 2,J' P  Geometric below  Figure  5.  the oldest (P.  P  T l 0  D-1,J'  Geometric  and  point  (P  )  with  _ ) .  P  D+1,J'  illustrations  f o r N=2  diagonal  •**' N + 2 , J P  o f t h e above  rule  (  1  6  >  a r e shown  N=3.  illustrations  o f Rule  5  (complicial)  24  To  generate  method  Rule  a 1-array  from  a 1-array,  the following  i s used.  5 i s used  t o e s t a b l i s h a temporary  new  point  N+2,j *  V  ;  N+1  2  P*  I  N 1  N+2,J  K , J  P  " D,J P  K=l K/^D,DD  The  P  temporary  D,J ™  3 1,J'  l  t  h  P  array  V 2 , J  p  D-1,J'  3 i s used  point  (P*  + 2 # J  by replacing  '  p  2,J'  Rule  i s formed  p  p*.  D+1,J'  t o e s t a b l i s h a second  *)  by letting  n a\  N+2,J  ( P *  +  2  >  temporary  =  3  p L  f  J *  1  '  V  new  " .  N+1  'P*...„ = N+2,J*  2 ~  Y  N  £->  P  *  K, J * v  T  - P  *  (19)  L , J * T  T  K=l  The  second  replacing  P  1,J*'  The  P  temporary P  L  <  J  *  2,J*'  above  manner.  array  array  (-P*  P  + 2 # J  ( A ^ ^ ) i s formed  ) with  P*  + 2 f  L - 1 , J * ' L+1,J*'  can also  by  P  be w r i t t e n  P  i n a  S+2,J*  ( 2 0 )  different  25  fiP  —  J**  p  p  1,J'  Rule  P N  a  4 i s used  +2, J * *  n  d  p  2,J'  (  =  L,J  P  b  +  =  P  LL J**  L  *  E  T  T  I  N  3  (  fpi )  p* N+2,J*  D+1,J'  to establish the actual  N 2, J  P  p  D--,J'  P N  +  2  , J* =  new  K  1  point  P L  f  *  )  f  N+l P  " - , J— v N+2 **  = P * "N + 2 , J  V  2  =  ~=—  N-1  P  L  _ p  K,J**  L,J**  1  (77) '  K=l K ^ L , L L  The  new  array  (A"!" J  { P  A  J+1  =  P  N+2,J*  =  P  L,J**>  1,J' 2 J' P  P  #  Thus  +1  t h e new  ) i s formed  W  i  t  h  D-1,J'  array  P  ( P  by  N+2,J**  replacing  ^+2,^*  =  D+1,J'  N+2,J  P  (A"!"..) i s f o r m e d  from  (  2  3  )  theo l d  J+x  ~~ array (P_,  (A"!") b y r e p l a c i n g  _) w i t h  Combining Rule  t h e new  the oldest  point  (P. ,T  diagonal  _) .  thesa f o r m u l a e , t h e f o l l o w i n g  6 can be  derived  point  formula f o r  (see Appendix I I ) .  N+l 1  N+2,J  N(N-l) [  (  N  +  1  )  Z  P K  /  J  P  DD,J  L , j ] "  D,J  K=l I^L,D,DD  Geometric Figure  6.  illustrations  o f Rule  6 a r e shown i n  (24)  Figure 6.  Geometric illustrations of  Rule 6  (complicial)  27  7.  If a and  (high)  i s not then  Rule is  8.  point  3,  then  (A^  u —X  in  point  t o Rule  point  ) i s also  that  not apply  formed  from  3,  i n the array  method.  with  the lowest • = N+2, u — 1  Instead,  (A^.) .  rule  the o l d array  t h e new  of the  response  p  ), L u #  point i s  o f the next  the centroid  T T  This  from  (i.e. P  the r e f l e c t i o n  (P _) t h r o u g h LL>, J points  point  arrays  by the application o f  2 of the simplicial  (h°) , J  Rule  successive  i s retested.  computed  t h e nev; a r r a y  do  i n N+1  eliminated  that  identical  I f t h e new  occurs  lowest  point  remaining  Thus:  J N+1 P  =  N+2,J  V  —  .  P  A  N  —  K,J  - P  LL,J  v  (25) '  K=l K/LL  The  new  array  (A^  ) i s formed  from  the o l d array  0 +JL by  replacing  new  A  J+1  =  P  point  1,J' 2,J' P  This  rule  method. shown  Appendix  the next  ( P  P  N  +  /  j  }  P  LL+1,J'  i s identical Geometric  some  point  (P  T T  T  )  with  the  •  LL-1,J'  i n Figure  I I I gives  2  lowest  t o Rule  P  N+2,J  3 of the  illustrations  o f Rule  (  2  6  )  simplicial 8 are  3.  examples  of the complicial  technique.  28  S I M U L A T I O N OF COMPLICIAL  General  order  complicial  method  Each  rules  other  the rules  starting  that  1.000.  The s t e p - s i z e s were  to  yield  good  operating  Random  from  searched  h a d a maximum roughly  t h e optimum  comparisons  variables  was  response run.  response by  to the starting  n o t done  follow-  and t h e  estimated  o f t h e two  one  (IBM  technique.  a n d t h e same  i n a particular  large  searches.  surface  Each  response  equal  to  dividing  point by  200  Scaling the  f o r any o f t h e r u n s .  Error  To response in  was  searches  technique  a  computer  o f search;  o f the complicial  position  of the  method,  on t h e u n i v e r s i t y  f o r the s i m p l i c i a l  f o r both  distance  effectiveness  o f the s i m p l i c i a l  simulated  surface  the  .  SEARCHES  r u n c o n s i s t e d o f two t y p e s  following  used  the relative  to that  were  given  The same v/as  t o compare  o f runs  7044). ing  SIMPLICIAL  OF  Format  In  number  AND  T H E COMPARISON  simulate  a degree  of a point,  the following  o f random  an a d d i t i o n a l  manner.  error factor  i n determining the was i n c o r p o r a t e d  29  R*  =  R'  +  ( 2 7 )  z'» S*  where: R'  =  Response  without  R*  =  Response  with  z'  =  Normal  S*  =  Modified  The by  an  standard  following slope  of  the  to  =  S ( l-  S*  =  S  Error  on  page  randomly  generated  IV-8.  error  was  computed  from  approximate  the  relative  change  response  S*  eiror  d e v i a t e s were  standard  formula  error  Deviate  Standard  normal  modified  random  Standard  algorithm listed The  random  the in  surface.  R )  • R '  R*>  .5  ( 2 8 )  R'^  .5  ( 2 9 )  where: S  =  Standard  Five  values of  and  . 0 2 ) .  S were  error used  f o r each  run  ( 0 , . 0 0 5 ,. 0 1 , . 0 1 5  / Moving In with  Optimum order  to  time,  "drift  were u s e d .  These  variables  to  simulate  a  factors" drift  compute  the  response as  a  factors  s u r f a c e v/hich  function v/ere  response.  of  added After  the to  "drifts"  trial  the  number  independent  computation,  these  30  factors  were  original  then  s u b t r a c t e d to  c o o r d i n a t e system.  Drift For  factor  Trial  =  .001  Number  restore  For  point  to i t s  example:  - Trial  500,  the  Number  coordinates are  ( 2,  -  3)  Therefore:  Moving  The  moving  Measure  of  The based  that  was  of  error)  the  I  Measure In  of  a  as  seen  For  limited  =  100  of  the  number  was  certain  • (1  -  the  two  of  -  2.5)  runs.  searches  deterministic  chosen  determined  the  number  percentages  by  the  case of  was (no trials  of  optimum.  following  formula.  (.l) ^ ) 1  (30)  2  y  1,2,3,4 . deterministic i n Appendix  Effectivenessthe  criterion  s e a r c h was  in a  ( 2.5,  Deterministic.  reach  were  f o r the  comparing  different of  =  = Function of  effectiveness  to  where:  displayed  -  criterion  required  percentages  output  used  principles.  Percent  The  Response  Effectiveness  two  were  These  optimum  measure  on  random  Optimum  used.  calculated  i was  graphically  IV.  Non-deterministic  searches  was  case  by  involving The  measure  summing  random of  a l l the  error  a  effectiveness responses  of  1000  test  trials  sum  has  the  optimum  the  value  a maximum  plotted  that  t h e random  the  markedly  error.  Therefore,  numbers  were  results  of  these  which  and  point  lies  method  their  a r e shown  A  the  obtained  was  from  levels  at  results  set of rules  different  of  the random  about  influenced  degrees  different  r e s u l t was  value  the plotted  the average  which  (RAND)  f o r high  run five  average on  o f t h e number  generator  average  y  above t h e l i n e ,  = x)  level  output).  The  i s called  i s also  i t indicates  the  shown  on  searches. that  random  obtained ( a l l  f o r each  o f t h e two  e f f e c t i v e than  of  initializing  the r e p r o d u c i b i l i t y of the  the comparison  i s more  Relative  an  (of the form  faciliate  over  plus  number  f o r each  chosen  points  line  display  value  the choice  especially  i s a m e a s u r e x>£ A  to  error  cumulative  of the response  graphically  for five  This  29 ) .  found  results  (value  the complicial  set of rules  was  variables.  the corresponding  (see page  random  To  form  initializes  five  o f 1000  i s 1.000).  against  It  value  obtained  simplicial error  of the operating  the  of spread  scatter search. the  graphs  If.a complicial  t h e s i m p l i c i a l method.  Effectiveness  term  called  simplicial)  relative effectiveness  i s defined  (of  as t h e d i f f e r e n c e  complicial i n the  32  cumulative  sums  Relative  f o r any p a r t i c u l a r r u n and  Effectiveness  =  Cumulative -  This  quantity  effectiveness simplicial such and  i s used  of the complicial  method  a s t h e number the  to determine  sharpness  changes w i t h  ridge.  error.  Sum(complicial)  Cumulative  (31) < Sum(simplicial)  i f the differences  method  as  compared  respect  to  certain  of variables, of a  random  t h e amount  i n the  to the criteria  o f random  error  33  COMPUTER  Main  Program Few  computations  program. the  This  results.  program. run  subroutines  shows t h e  main  loops  the  searches  flow  are  random  KRAND  loops  five  degrees  KRUN  loops  the  two  chart  number of  types  the to  the  that  main execute  main for  each  ( 5 x 5 x 2 ) . initializers  random  of  of  shows  performed  five  error  search  techniques  A A  shown  used.in  in Figure  statements  establish  Insert  text of  many  loops  to  the  (e.g.  storage  8  contains  DIMENSION,  primarily  DATA).  l o c a t i o n s f o r the  Its  nonpurpose  v a r i a b l e s to  be  program.  B  Insert be  7  i n the  on  NBIG  executable  used  variables are  Figure  performed  relies  Inspection of  Insert  to  are  program  (NEWT) f i f t y  Insert  is  PROGRAM  read.  starting  B  (shown  f o r each (N) , t h e Any  point  i n Figure of  the  stepsize  9)  fifty (R)  s t o r e s the searches.  and  response  parameters  are  v i a the  read  the  Run  (e.g.  as  The No,  SIGMA,  subroutine  c a l l e d . . A d d i t i o n a l v a r i a b l e s such  input  NP1,  data  number (LRUN) RHO)  and  CHARB w h i c h NP2,  of  EN,'  the  is  EMI,  are  34  MAIN  N  PROGRAM  CALL PLOTS ->( DO N E W T = 1, N T E S T  >{  DO  NBIG=I,5  X X X = Y Y Y + FLOAT(NBIG) -*(  DO  v K R A N D = I,5  r DEV= FLOAT(KRAND-l)/200.  T -*{D0  K R U N = I,2  CONTINUE  JL C A L L PLOTI (YP, L R U N ) C A L L PL0T2(RP, LRUN) CONTINUE CALL  4  PLOTND STOP,  re 7.  Flow chart of main program.  DIMENSION Y ( I O . I O ) , Z(IOIO)  1 DIMENSION  C(IO)  R H O ( I O , 10),  SIGMA(IO)  DIMENSION YP(2,5),  XP(5)  RP(2,5,5)  1 DIMENSION I DATA  NSTOP(IO)  NSTOP/2,3,  5,  7, 9 , II, 13, 16, 18, 21 /  D I M E N S I "Of N "  ZP(5)  KDO  K = I, 5 I Z P ( K ) = I.SQRTdO^^-(l-K))  LOGICAL NSUC  (READ,  figure 8.  Insert A.  N T E S T )  C A L L CHARB ( S I G M A , RH.O, C)  THE FOLLOWING ARE COMMON VARIABLES  EN = N  N P I = N+1 N P 2 = N+2. »  NMI = N - 1  I EMI = E N - I .  Y Y Y =. 1 2 3 4 1 5 6 7 8  ^Figure 9 .  Insert B.  37  calculated of  from  the value  the  execution  t h e ptogram.  Insert  C  The ally  first  from  name g i v e n for  part  of Insert  an i n i t i a l i z a t i o n  generated  a  starting  i s established.  (from  their  The vital 1.  relative  second  part  search  3.  (CALL  which  o f move  The p o i n t s  of Insert  mechanism  pute  the response  I f Rule  2  applicable through  - This  of this  (NCYCLE >  a retest  from  routine  response  point  i n the  executed  arranged  i n Figure  i n  order  11  contains  below. o f LMOVE, t h e  the formulae  f o r each  i s given  (simplicial)  RESP ( t h e  to be  t o the value  set of rules presented chart  array i s  ORDER).  as t a b u l a t e d  the  RESP  f o r each  C given  point  CALL  POINTS).  are then  i s calculated  i s essenti-  the f i r s t  (LMOVE)  new  A. f l o w  10)  calculates the  r e s p o n s e s ) (CALL  C A L L MOVE - A c c o r d i n g  nique. 2.  i n which  i s called  The t y p e  next  (see F i g u r e  point  i n the array)  array.  C  section  t o the subroutine  a point  starting  the  of N to facilitate  new  o r Rule  search  i n Figure  i s next  called  tech-  12. to  com-  point. 7  (complicial) i s  NP1), t h e program  s e c t i o n where  given iit  i s shunted  the highest  point  38  RINTI = R A N D ( X X X ) NT= 0 •r  NCYCLE  I  = 0  C A L L POINTS ( R , C ) -*{DO CALL  J - I, N P I  E  R E S P ( J , SIGMA,  R H O , NT,  DEV, Z )  1  CALL  ORDER  (1, N P I ) T (To  Figure  10.  Insert C - I n i t i a l i z a t i o n  F i g u r e II)  Section  39 (From  Figure 10)  •»(D0 J = NP2,I000  I  CALL  MOVE  CALL  RESP(NP2,S!GMA,  RHO,  (LMOVE)  NT, DEV. Z )  NSUC= Y(NPI,NP2),GT.Y(NPI,NPI)  C A L L R E S P (NPI .SIGMA, RHO, NT, DEV. Z)  NCYCLE= N C Y C L E + I  NCYCLE=0 CALL  RETEST  (LMOVE, KRUN.NSUC)  NCYCLE = 0  I  C A L L M E M T R A ( L M O V E , NSUC) C A L L LOGIC ( L M O V E , KRUN,  NSUC)  I  CONTINUE CALL  I  R E S U L T ( Z P , Z , NBIG,  KRAND, KRUN,YP, RP) Figure II. Insert C - Search Section.  40  SUBROUTINE  MOVE  (LMOVE)  DIMENSION S U M Y (10)  GOTO  ( 1 , 2 , 3 , 4 , 5 , 6 ) , LMOVE  1,5,6  4  DOT = l,N  DO 2 = I,N  •  SUM (2,NPI)  V  D0  I = I,N  f  SUM (3,NPI)  SUM(3,N)  J L  i  Y(I,NP2) (2./EN)  Y(X,NP2)  Y(X,MP2)  (2./(EN-l.))  (.5/(ElSS4EN»l.))  SUMY(I)  SUMY(l) Y(I,I)  (EH ®smmxy  Y(I,I)  Y d , i) JL  RETURN  RETURN  RETURN  S U M Y ( I ) = 0.  DO  SUM(L,M)  I  K= L.M  J L  S U M Y ( I ) = S U M Y ( X ) -r Y(I,K)  figure  12.  Flow chart of  MOVE  41  is  retested.  checks  CALL  RETEST  t h e new r e s p o n s e  (shown  i n Figure  and r e n u m b e r s  13)  the points  accordingly. 4.  C A L L MEMTRA this the  5.  routine points  Figure 14).  CALL  LOGIC  from  t h e MOVE  - Before  formed  next  given  search  (i.e.  (0 o r 1) w h i c h  i n Rules  technique.  routine  C A L L MOVE) trials  After  must  1 and 2 o r t h e  (A f l o w  and t h e c y c l e  chart i s  t h e new s e a r c h  t o Step  i s repeated  the results  and s t o r e d  (see Figure  back  1  until  0  1000 t r i a l s ,  a r e computed  RESULT.  i s now l o o p e d  are completed  completing  search  and  subroutine  i n Figure 15).  The search  1000  manner  i scalculated  t h e LOGIC  of array  evaluated,  i s t o number  i n a logical  a new p o i n t  as given  has been  I t s purpose  subroutine,  the type  complicial  7.  i scalled.  (see  be  t h e new p o i n t  i n t h e new a r r a y  determines  6.  - After  16).  by the  This  i sinitiated  ends from  f o r that  subroutine that  search  t h e main  program.  Plotter The  Output results  f o rf i f t y  searches  which  have been  stored  42  SUBROUTINE  RETEST  (LMOVE, K R U N , N S U C ) T—:  HDO  K = I,NP2 +  LOGICAL NSUC  Y(NP!,NP2r Y(NPI.I)  10,+ Y(NPIJ)  Y(NP!,NP2)  1 CALL ( l  4 s  ORDER NPI)  JL  CONTIMUI  LMOVE=1  RETURN  V  RETURN  figure 13.  Flow chart of RETEST.  43  SUBROUTINE  MEMTRA  (LMOVE, NSUC)  LOGICAL NSUC  GO T O ( 1 , 2 , 3 , 4 , 5 , 6 ) , L M O V E ,5,6  L  3,4  C A L L ORDER (3, NP2)  DO I = l , N P i  DO I=I,~NPI  DO 1 = I,NPI  SHUNT(I)  JL  SHUNT (I)  CONTINUE 1  CONTINUE  ) SHUNT(2)  RETURN  CONTINUE  DO K=M,NPI SHUNT (M) M >  KPI = K + I  I  Y(I,K) = Y(I,KPI)  figure  14.  Flow chart of  MEMTRA.  C A L L ORDER ( I, NPI)  RETURN  44  SUBROUTINE  LOGIC  (LMOVE,NSUC,KRUN)  LOGICAL NSUC  LMOVE= 1  LMOVE=4  LMOVE=3  J  t -1  1t  f  RETURN  ^Figure 15.  RETURN  Flow  chart of  LOGIC.  Jc. RETURN  SUBROUTINE R E S U L T (ZR Z , NBIG, K R A N D , K R U N , YP, R P )  DIMENSION  YP(2,5),  Z (IOIO), Z P ( 5 ) , R P ( 2 , 5 , 5 )  I SUMZ= 0. K =2 ->( DO  1=1,  1000  F r  YP(KRUN,K)= * I V  K = K * 1  I  S U M Z = S U M Z 4- Z ( I )  I  R P ( K R U N , NBIG, KRANQ)-  RETURN Figure 16.  Flow chart of  RESULT  SUM?]  are  translated  each  r u n two  output),  The along  and  graphs  variables.  graphical  run of  computer  a table  data  are p l o t t e d  the next  entire  with  into  giving  PLOT2).  Afte  ( s e e A p p e n d i x IV f o r a  typic  fifty  listing  (PLOTl  and  searches  i s given  the meaning  i s begun.  i n Appendix  o f the main  IV  computer  47  RESPONSE  S U R F A C E S USED  C O M P A R I S O N OF 1 H E W O  A pare  wide  variety  the simplicial  Normal  Multivariate  The  response  contours  s u r f a c e s were  method  the complicial  with  consists  t o com-  methods.  (a  v  of elliptical  contours  (N=3) a n d h y p e r - e l l i p s o i d a l  The r e s p o n s e  CT=  used  Distribution  contours  (N>3).  METHODS  o f response  surface  (N=2), e l l i p s o i d a l  I N THE  s u r f a c e h a s two  parameters:  (32)  c r ,c r ,  and  Pl , l '  'l,2'  ^2,1'  ^ 2 , 2 ' **'  Pi,  ^,1'  ^N,2'  ^N,N  r  having  the property  P =  X  The The  elements  for  Pj I  \P j\<  1.0  matrix  P,  '  l  (33)  that;  = 1.0  Pi J  P  1=1,  for I = for I =  i s positive  1, 1,  N N;' f o r J =. N;  for J =  1, 1,  • N N  definite.  o f t h e i n v e r s e o f t h e above  matrix  are  denoted  48  The  general  equation  Z„  where:  -  =  is  ness the  the  other  however,  contours of  addition  according  Two  interpretations  response to  I,K,  N  of  to  the  CT  of to  the  these  surfaces  the  variables,  J  that  surface  and  II, K  f  °II  L  yO  and  note  rY  J  CT,I  1=1  orientation  In  for  (34)  Pl),  I  sufficient, of  below  N  11=1  It  given  exp(-0.5'A)  N  I  Geometrical  is  can  be  J  (35)  •  found  G~  affects  P  primarily  in  the  (26),  sharp-  affects  contours.  normal were  number  of  Rl  -  multivariate  chosen.  These  distribution, can  be  classified  variables.  Variables 1.  Brooks  (7)  elliptical are  contours  parallel  variables  Z  The.response  as  to  the  shown  ~ A-B-exp(2  v  whose m a j o r  and  axes  two  in  =  (. 5  +  .5 • Y  1,K,  J  of  the  Figure  - A  where: A  surface  -  B)  c o n s i s t s , o f nearly minor  axes  independent  17.  (36)  49  Figure  17.  Contour map of Brooks  Rl.  50  Brooks of  R2  (7) - T h e r e s p o n s e  rotating  37°  the previous  a s shown  Z„  i n Figure  surface  surface  i s the  result  approximately  18.  = A - B e x p (2 - A - B)  (37)  where:  A  " -  B  = <-  (  3  Brooks ridge in  " -  8  Z„  I,K.J  4  -  +  Y  8  -  +  y  3  2,K./ Y  2.K  (7) - T h e r e s p o n s e  with  Figure  Y  • ' 1.K.J  +  R3  6  large  areas  ( J  »  4  surface  i s a  o f low response  as  sharp shown  19.  _ = A-exp(l  - A  - 20.25  B)  (38)  where:  A  "  E  =  < Y  ( Y  1,K./  1,K,J "  Brooks  R4  linear  ridge  Y  2,K,j'  (7) - T h e r e s p o n s e a s shown  ~ A-exp(l  Z  -  i n Figure  .6 B  where:  2 B  ~  (  Y  1,K,J~  Y  surface  2,K,J^  - A)  i s a  curvi-  20.  (39)  51  Figure  18.  Contour map  of Brooks  R2.  Figure  19.  Contour map of Brooks  R3.  53  Figure 20.  Contour map of Brooks R4.  54  5,  Roseribrock bolic  (23) - T h e r e s p o n s e  ridge  = e X P  K,J  Z  a s shown  [  i n Figure  f-llOO A W~\  surface  i s a  para-  21.  + 33)1  , ,\ /r  (  4  0  )  where:  B  The  (  "  1  previous  original to  Greater 1.  1.0  Than  \ , K , f equation  so t h a t  has been  t h e optimum  modified  from t h e  i s a maximum  equal  .  Two V a r i a b l e s  Powell of  (21) - T h e r e s p o n s e  a quartic  surface  function with  [  i s i n t h e form  4 independent  variables.  _ 2,n 2^ , 22 . „ 2 . „ 2 -(A + B -f C -fr D 1 (  ~  "  T  o  o  J  (41)  where: A  =  B  = Y  C = Y  „ , + 10 - Y ' 1,K,J 2,K,J  3,K,J  ~ -  2,K,J  D = Y:  i,K,J  The  above  original to  1.0  .  Y  4,K,J  Y 3,K,J Y 4,K,J  equation.has so t h a t  been  t h e optimum  modified  from t h e  i s a maximum  equal  55  Figure 21.  Contour map for Rosenbrock's surface (modified).  56  Fletcher in  and  Powell  (13) - T h e  the form o f a h e l i c a l  response  ridge with  3  surface i s independent  variables. Z  = exp [ - ( A * |_  v  K, J  + 100.B 1000  2  2  + C)]  (42)  J  where:  •'  A  •  -  B  C  The  Y  v/^'l.K.j'  3 . K . J -  0  +  ( V  2TT^  <* ,  =  Arctan  ( Y  3.K,J  2  .  Y  2  |  ,K,/  2  ~  1  K  K  i  / Y  , y^,  1  K  K  ,  J  )  Y  l , K , J - °  -  <  ) 2  above e q u a t i o n  1.0  <  ^  27T(£= A r c t a n  original to  1  2  so t h a t  has been modified  t h e optimum  from  i s a maximum  the equal  0  57  RESULTS  The of  results  the test  Brooks  will  response  R l , R2, . R3 Runs  Brooks  31  R =  .01  for  each  20)  the four  .  Four  surface.  the  runs  showed  was  more  effective  With  (40,41 from the  a n d 43)  were  as  which with  curved  line  (40,41 contour  order radius  and  t o reach  i s very More  not  this  presented the r e A l lof  22  to  reach  to 31). of the  runs  increased ( i . e .  (45°) l i n e ) . uniquely  be  These  i n common. crosses  straight  o r concave will  and  e f f e c t i v e n e s s went  o f the search  large)  runs  the complicial  required  error  thing  19  opposite).  are r e l a t i v e l y  on  were  t o 41), three  the straight  t h e optimum  of search.  were  the relative  the path  18,  by  chosen  to duplicate  error  as random  cited  were  17,  (see F i g u r e s 32  do h a v e one  that  of curvature  direction  random  (see F i g u r e s  43)  tended  trials  that  runs  diagonally  o f optimum,  crossed  lines  Several  no  as l e s s  showed  nature  f o r a l l the  points  (see F i g u r e s  v/ere  surfaces  chosen  starting  symmetry  p o s i t i v e to negative  runs The  error  on t h e f o u r  different  that  to the  chosen.  step-size  points.  percentages  random  The  These  corner  according  R4  48).  (e.g. points  various  surfaces  and  (32 , 3 3 , 3 6 , 3 8 , 40,.42) sults  discussed  t o 46 w e r e b a s e d  (see page  was  be  i n  ( i . e . the  towards  discussed  the later.  58 Symbol  Typ* of Search  0  Simplicial  t  Complicial  I 68.4  Figure 2 2.  I  I  90 96.8 PERCENT OF OPTIMUM  1_ 99  Number of trials required to reach various percentages of optimum.  59 Symbol  Type of Search  0  Simplicial  •  Complicial  A b s c i s s a coordinates  are e x p r e s s e d b y :  X . = 100.* ( I. -  (.l)  l / 2  )  I = 0,1,2,3,4  400  320 -  T  T  RUN NO. 34 N «2  co  <2  240 -  u. o 160 -  80  1  684  Figure 23  I 1 90 96.8 99 PERCENT OF OPTIMUM  1  Number of trials required percentages of optimum.  to reach  various  60  Symbol  Type of Search  0  Simplicial  t  Complicial  Abscissa coordinates are expressed by X. = I00.# ( I. -  (.l)  i / 2  )  1 = 0,1,2,3,4  400  320  co  240 ~  Li. O  : i6o  80 -  684  Figure 24  90 96.8 99 PERCENT OF OPTIMUM  Number of trials required to percentages of optimum.  reach various  . 61 Symbol  Type of Search  0  Simplicioi  t  Complicial  Abscissa coordinates ore expressed Xj = 100.#  ( I. -  (.l)  l / 2  by:  )  1 = 0,1,2,3,4  4»00  320  CO  O  240 "  160 -  684  Figure 25.  90 96.8 99 PERCENT OF OPTIMUM  Number of trials required to reach various percentages of optimum.  Symbol  Type of Search  0  Simplicial  •  Complicial  Abscissa coordinates are expressed Xj * 100. #  ( I. - ( , l )  l / 2  by  )  • • 0,1,2,3,4  400  320  0  Figure  68.4  90 96.8 PERCENT OF OPTIMUM  Number of trial* required to reaeh various percentages pf optimum,  63  Symbol  Type of Search  0  Simplicial  •  Complicial  Abscissa coordinates are expreesed X. = 1 0 0 . #  ( I. - ( . l )  l / 2  by:  )  I = 0,1,2,3,4  400  684  Figure 27.  90 96.6 99 PERCENT OF OPTIMUM  Number of trials required to reach various percentages of optimum.  64  Figure 28.  Symbol  Type of Search  0  Simplicial  ©'  Complicial  Number of "trials required to percentages of optimum.  reach various  65  Symbol  Figure 29.  Type of Search  .0  Simplicial  t  Complicial  Number of trials required to reach various percentages of optimum.  Figure 30.  Symbol  Type of Search  0  Simplicial  t  Complicial  Number of trials required to reach various percentages of optimum.  6 7  Figure 31  Symbol  Type of Search  0  Simplicial  •  Complicial  Number of trials required to reach various percentages of optimum.  68 Symbol  Standard  0  Error  .020 .015  A  Large symbols are  .010  Average  .005  values  .000  Coordinates  refer to cumulative  sum of responses for 1000  trials.  Maximum response s 1 . 0 0 0 Straight  880  840 -  line  1  is locus of squivglent cumulative  1  1  sums.  1  RUN NO. 31 N=2  o oc  V  #800  A  e  /  o  -  2 760 O  o 720  680 640  f  U  \ /  680  1  '  1  720  760  800  SIMPLICIAL  Figure 32.  840  SEARCH  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random  error.  69  Symbol  Standard  Error  0  .020  ©  .015  Large symbols are  .010  Average  .005  a Coordinates  values  .000  refer to cumulative  sum of responses for 1000  trials.  Maximum response = 1 . 0 0 0 Straight  line  is locus of equivalent  900  800 640  Figure 33.  cumulative  sums.  T  680  720 SIMPLICIAL  760 8 00 SEARCH  840  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random error.  Symbol  Standard  0  Error  .020 .015 .010  A  Large symbols are Average  .005 .000 Coordinates  refer to cumulative  sum of responses for 1000  Maximum response = Straight  line  Is  values  trials.  1.000  locus of equivalent  cumulative  sums.  920  8 80x o  < UJ E  840-  o  £  800-  o o 760  720 640  Figure 34.  680  720  SIMPLICIAL  760  SEARCH  800  840  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random error.  71 Symbol  Standard  0  Error  .020 .015  Coordinates  A  .010  A  .005  a  .000  refer to cumulative  Large symbols are Average  values  sum of responses for 1000  trials.  Maximum response = 1 . 0 0 0 Straight  line  is locus of equivglent  cumulative  T  870  850  sums.  -  o or < $  830  < o CL  2 O  810  o 790  -  770 744  Figure 3 5 .  752  760 SIMPLICIAL  768 SEARCH  776  784  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random error.  72 Symbol  Standard  0  Error  .020  •  .015  Large symbols are  .010  Average  .005  • Coordinates  values  .000  refer to cumulative  sum of responses for 1000  trials.  Maximum response = 1 . 0 0 0 Straight  line  930  is  locus of  1  880 -  equivalent cumulative  1  1  1  /**  RUN NO. 39 N=2  a  sums.  A  *  /  cc  $830  — '  _i •< o  o!780  _  *  /  _  Figure 36.  9  /  0 0  680 640  /  0 0  O O  e  A 690  /  i 740 SIMPLICIAL  i 790 SEARCH  840  890  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random  error.  73 Symbol  Standard  0 •  .020 .015  A  .010  A  .005 .000  a Coordinates  Error  refer to cumulative  840i  •  640 728  Figure 3 7 .  line  r  732  Is  Average  values  sum of responses for 1000  Maximum response Straight  Large symbols are  trials.  1.000  locus of equivalent  cumulative  sums.  f  7 36  SIMPLICIAL  740  SEARCH  74 4  748  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random error.  74 Standard  Symbol  0  Error  .020 .015  A  Large symbols are  .010  Average  .005  values  .000 Coordinates  refer to cumulative  sum of responses for 1000  trials.  Maximum responses 1.000 Straight  line  is locus of equivalent  cumulative  sums.  880  480 4 8 0  560  6 4 0  SIMPLICIAL  Figure 3 8 .  720  800  880  SEARCH  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random error.  75 Symbol  Standard  0  Coordinates  •  .020  A  .010  A  .005  •  .000  Error  .015  refer to cumulative  560  line  640  Is  720  39.  are  values  trials.  1.000  locus of equivalent  SIMPLICIAL  Figure  Average  sum of responses for 1000  Maximum response Straight  Large symbols  cumulative  800  880  sums.  960  SEARCH  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random error.  76 Symbol  Standard  0  Error  .020 .015  A  Large symbols are  .010  Average  .005  values  .000  Coordinates  refer to cumulative  sum of responses for 1000  Maximum response = Straight  line  Is  trials.  1.000  locus of equivalent  cumulative  sums.  880  800  -  o oc  %7Z0 _i < o a!640 2 O  o 560  480 480  Figure 4 0 .  560  640 720 SIMPLICIAL SEARCH  800  880  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random  error.  77 Standard  Symbol  Error  .020 .015  Coordinates  A  .010  A  .005  •  .000  r e f e r to cumulative Maximum  Straight  860  line  is  Average  sum of  response =  l o c u s of  Large symbols  responses for  are  values  1000  trials.  1.000  equivalent  cumulative  760  800  sums.  -  x o <r < #855 _J <  o £ 8 5 0 -  o o 845  840 680  640  Figure  41.  720  SIMPLICIAL  C o m p a r i s o n of with  SEARCH  C o m p l i c i a l and S i m p l i c i a l  v a r i o u s d e g r e e s of  random  error.  840  searches  78  The  rest  method  o f the runs,  i s more  presence greater the  four  42.  error  zero).  43  o f the four  the  t h e s i m p l i c i a l method  results  computed  are given  the overall response  from  f o r each  average  surfaces.  random  a s shown  i n Figure  44.  a l l the runs  involving  the Brooks  was of  compiled data  (see Appendix  points  about  V)  the complicial  method  than  The  data  40,  and  was  inferior  Runs  rest  is  As  increased  scatter and  higher  of the runs  scatter.  46.  41  for larger  significantly  no  error  random  was  surface  43  Figure  a l l the  f o r an i n c r e a s e  i n  surfaces,  whether the  data  scatter  s i g n i f i c a n t l y higher  (where t h e c o m p l i c i a l o f random  However,  i s increased  error  on  relative  error)  the data  Graphs  from  are presented  a  the  i n the  the scatter showing  search  showed  significant difference  proportionally.  versus  i s  f o r t h e s i m p l i c i a l method.  amounts  scatter.  showed  random  to test  the average  for  from  changed  The  i s not markedly  From  i n the  averaging  combining  effectiveness error,  complicial  c  The  shows  show t h a t  ( i e , relative effectiveness  starting points  Figure  results  e f f e c t i v e than  o f random than  however,  also  the relative  i n Figures  45  79 Symbol  Standard  0  Error  .020 .015 .010  Large symbols are Average  .005 .000 Coordinates  refer to cumulative  sum of responses for 1000  Maximum response Straight  line  values  trials.  1.000  is locus of equivalent  cumulative  sums.  900  850-  o cr <  UJ  Cf)  800-  < £  750-  O o 700-  650 600 1  Figure 42.  650  700 SIMPLICIAL  750 SEARCH  800  850  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random error.  80 Symbol  Coordinates  Standard  Error  0  .020  •  .015  A  .010  A  .005  •  .000  rotor to cumulative  600  line  650  values  1.000  is locus of equivalent  700 SIMPLICIAL  Figure 43  Average  sum of responses for 1 0 0 0 trials.  Maximum response = Straight  Large symbols are  cumulative  750  800  sums.  850  SEARCH  Comparison of Complicial and Simplicial s t a r c h e s with various degrees of random erVor.  Figure 4 4.  Relative effectiveness versus the amount of random error.  82  Symbol  Typs of  Ssarch  0  S i m p l i c i al  ©  Complicial  50  Average of Runs  40  31 to 46 excluding 40,41,43  <b  .005  .010  STANDARD  Figure  45  Amcynt of  .015 ERROR  s c a t t e r versos  (S)  random e r r o r  020  J  0  Symbol  Type o? SeGrch  0  Simplicial  ©  Complicial  .005  .010  STANDARD  Figure  46.  Amount of  .0i5 ERROR  s c a t t e r versus  :020  (S)  random  error.  84  Multivariate Runs response  61,  62,  surface  order  N.  CT  that these that  1.0  in  for  distance point  to  (C^,  I  =  was  O"  1,N.  be  the  chosen  i n each The  C^)  !  =  the  has  Thus  the  normal  =  2,  be  a  unit  was  a  vector  of  ( i . e . the  units.  ...,  case  of  (N  to  response  only  point 4  multivariate  dimensionality  function  to  C^,  2  test  case  optimal  fixed  C ,  each  is a  the  was  64  parameter  conditions point  and  Surfaces  for varying  The  =  Response  63  p a r a m e t e r yO  The of  Normal  any  origin). initial  property  5).  matrix  point  distance  the  3,'4,  such under  from This starting  that:  N  I  C  2  4  ,  (36)  1=1 If  =  =  coordinates  ... can  = be  C^,  given  C  The  step-size  Figure search  than  true  number  47  of  for  I  =  each  the  less  f  run  r  with  trials  values  required  o  was  simplicial  a l l the  trials  the  values  for  the  starting  by:  */ff  shows t h a t  required  optimum was  for  then  to  to  1  =  1  N  (  calculated  no  random  reach  method of  '  N  a  required.  68.4%  be  error  given  chosen.  reach  to  A of  .02  the  plot the  of  7  )  (4/200).  complicial  percentage This  3  of  condition the  optimum  Symbol  Type of Search  0  Simplicial  t  Complicial  Abecieea coordinates are expressed by: X  (  - IOO.# ( I. - ( . 1 ) " ' ) i = 0,1,2,3,4  0  68.4  90  PERCENT OF  Figure 47.  96.8  99  OPTIMUM  Number of trials required to reach various percentages of optimum.  86  against in  the  Figure  number  48  to  of  variables  illustrate  that  methods  i ^ decreased  Another  interesting observation  complicial  and.  increases. of  each  error) can  Figures  as  N  response varying  run not  these  that  the  52  one  for  (N)  will  increased. for  as  Plotting of  N  Figure  the  random 53),  effectiveness be  the  effectiveness  levels  (see  relative  i s  both  another  show t h e  five  method  for  curves  i s present.  This  each  variables  approach  error  the  of  and  (average  to  Response 51  surface  v/as  Figures  unit  51  i s increased.  values  contours  was  50,  for  effectiveness  i s that  dimensionality  noted  Normal  R u n s 47  each  number  shown  discussed  i t de-  more  later.  Bivariate  in  the  is  the  method  random  effectiveness  r e a d i l y be  fully  49,  when  against  creases  the  simplicial  search  relative  as  N  matrix. runs.  using  the  0  O"  changed 55,  to  simulate  from  56,  shown on  changed  made  with  (i e. multivariate  of  54,  are  were  Surfaces  57  and  these  throughout  The  circles  starting  58  bivariate  normal  ridges. to (the  runs  point  for The  thin  and  normal  N=2)  The had  (20,0) was  for  shape  ellipses  values  figures). the  Ridges  of  0"  of as  form  chosen  shown  for  parameter the  the  p of  a  for a l l  Symbol  Type of Search  0  Simplicial  ©  Complici al  NUMBER OF V A R I A B L E S  Figure 4 8.  Number of t r i a l s required to  (N)  reach 68.4% of optimum  versus the number of v a r i a b l e s .  Symbol  Coordinates  Standard  0  .020  •  .015  A  .010  A  .005  q  .000  refer to cumulative  Error  Large symbols Average  are  values  sum of responses for 1000  trials  Maximum response = 1 . 0 0 0 Straight  840  line  Is  1  I-  800 -  locus of equivalent  1  RUN NO. 61  N«2  z o oc  cumulative  sums.  ^  / /  $760  —  0  /  s  Ve  -J <  e  •  o a|720 O  o  /  0  r  /  7  680 0  640 520  Figure 49.  1 560  i 600 SIMPLICIAL  i/ 640 SEARCH  i 680  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random error.  89 Symbol  Standard  0  .020  •  .015  A  QIQ  A  0  P Coordinates  Error  Q  Large symbols are Average  5  values  .000  refer to cumulative  sum of responses for 1000  trials.  Maximum response = 1 . 0 0 0 Straight  80  line  180  is locus of equivalent  280 SIMPLICIAL  Figure 5 0 .  cumulative  380  480  sums.  580  SEARCH  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random  error.  90  Symbol  Standard  0  Error  .020 .015  A  Large symbols  .010  Average  .005  are  values  .000 Coordinates  refer to cumulative  sum of responses for 1000  Maximum response = Straight  line  trials.  1.000  is locus of equivalent  cumulative  sums.  500  400 o rr $300  <  o a! 2 0 0  2E O o  100-  160 SIMPLICIAL  Figure 51.  240 SEARCH  320  400  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random  error.  91 Symbol  Coordinates  Standard  0  .020  •  .015  A  .010  A  .005  q  .000  refer to cumulative  Error  200  line  Is  1  -  RUN NO.  Average  values  sum of responses for 1000  Maximum response = Straight  Large symbols are  trials  1.000  locus of equivalent  cumulative  1  1  1  1  1  1  sums.  64  N = 5  80  40  /  1 40  Figure  52.  80 SIMPLICIAL  120 SEARCH  160  200  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random  error.  LU 200  of  N U M B E R  Figure 5 3 .  O F  V A R I A B L E S  ( N )  Relative effectiveness versus the number of variables.  Figure  54.  Contour  mop for Run 47.  S4  o o o  d. CM  OJ  o o o  -  = 5.0  C^=  d_  2.0  o o o "3.  o o Q  Starting Point (20.,0.)  O  «—i. 1  Optimal Point o o a  ( 0., 0.)  Contours are 0.1(0.1)0.9  d CM. I  -2d.oao  -lo.oao  ~T -,oao  lo.aoa  I.K.J  Figure  5 5.  Contour  map for Run 4 8 .  20-000  >95  o o o  »  o.  CT = 5.0  o o o  0^= 1.0  o o  CD  O «-». I  D O  Starting Point  (20..0.)  Optimal Point  ( 0 , 0.)  Contours are 0.1(0.1)0.9  O  CD CJ.  .  I  -20.000  ,  1  --lo.oao'  -.ooa Y  Figure  5 6.  Contour  LO.ooa  I,M  map for Run 49.  2a.ooa  96  S_  1 I  Starting Point  (20..0.)  Optimal Point  ( 0., 0.)  Contours are 0.1(0.1)0.9 1 -,000  I  I  -20.000  -10.GOO Y  Figure  57.  Contour  1 10.000  I,K,J  map for Run 50.  1  20.GOO  97  o o o o  o o o Q  ~3  Oj = 5.0 »  cr  _  2 = 0.2  O o  o o o o «—f ,  I  o o o  Starting Point  (20.,0.)  Optimal Point  ( 0., 0.)  Contours are 0.1(0.1)0.9  CD I  -2D.000  -10.000  -.000 Y  Figure  5 8.  Contour  ~! 10.ooo  I,K,J  map for Run 51  20.Q0Q  98  In the  the presence  complicial  However,  This  as t h e r i d g e  Curvilinear The  superior  55,  56  and  and  and  67.  68,  69  The  64)  later  59  to  63)  f o r a l l the runs  when p l o t t e d showed  sharper  57) w e r e  random  cited tested  against  a marked  CT^/  than  CT^  made. the  de-  =  5 .  section.  The  0  error  deterministic enough  t o hamper  of  points  f a r away  a l l these  from  always  positive  produced  f o r a given  i s that  v  65,  66  Figures the  compli-  the s i m p l i c i a l f o r the amount  evidenced  the deterministic surfaces,  o f random  o f random by  the  value.  error  "bunching" However,  the r e l a t i v e e f f e c t i v e -  f o r the average  degree  i n  the  ridges.  i n Figures  a r e shown  small  searches  curvilinear  n e s s was  a  (13)  t o compare  curvilinear  a r e shown  e f f e c t i v e than  both  on  observation  cases b u t even  is  ( 2 3 ) , (21) a n d  i n order  results  general  i s more  i n  methods  deterministic  70  method  surfaces  complicial  results with  and  (see F i g u r e  i n a  (see F i g u r e s  Ridges  The  for  again  b e c a m e '••  discussed  response  simplicial  cial  was  of the ridge  w i l l b e  (Runs  method  error  the relative effectiveness  sharpness crease  o f random  of five  error.  points  99 Symbol  Coordinates  Standard  Error  o  .020  •  .015  A  .010  A  .005  •  .000  refer to cumulative  520  Figure 59.  line  560  Average  values  sum of responses for IOOO  Maximum response = Straight  Large symbols are  1.000  is locus of equivalent  600 SIMPLICIAL  trials.  cumulative  640 SEARCH  680  sums.  720  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random  error.  100 Symbol  Coordinates  Standard  Error  0  .020  •  .015  A  .010  A  .005  •  .000  refer to cumulative  Large symbols are Average  values  sum of responses for 1000  trials.  Maximum response = 1 . 0 0 0 Straight  480  Figure  line  530  60.  Is  locus of equivalent  580 SIMPLICIAL  cumulative  630 SEARCH  680  sums.  730  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random error.  101 Symbol  Standard  Error  0  .020  •  .015 Large symbols are  A  A  0  • Coordinates  0  Average  5  values  .000  refer to cumulative  sum of responses for 1000 trials.  Maximum response = 1.000 Straight  520  line  560  is locus of equivolent cumulative  600  SIMPLICIAL  Figure  61.  640  680  sums.  720  SEARCH  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random  error.  102  Symbol  Standard  Error  .020 .013  Large symbols are  .010  Average  .005  values  .000  Coordinates  refer to cumulative  sum of responses for 1000  trials.  Maximum response = 1 . 0 0 0 Straight  line  is locus of equivalent  cumulative  sums.  800  760  x o or #720  _i < o CL  680  3E  O O  640  600 520  Figure  560  62.  600 SIMPLICIAL  640 SEARCH  680  720  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random error.  103 Standard  Symbol  Error  .020 .015 Large symbols  .010  Average  .005  are  values  .000 Coordinates  r e f e r to cumulative Maximum  Straight  line  Is  sum of  response =  l o c u s of  responses for  1000  trials.  1.000  equivalent  cumulative  sums.  800  720  -  o or #640  < o 5^560  O a 480  -  400 360  Figure  400  440  480  SIMPLICIAL  63.  C o m p a r i s o n of with  520  SEARCH  C o m p l i c i a l and S i m p l i c i a l  v a r i o u s d e g r e e s of  560  random  error.  searches  Figure 64.  Relative effectiveness versus the sharpness of ridge.  1 0 5  Symbol  Standard  0  Error  .020 .015  Large symbols are  .010  Average  .005 .000 Coordinates  refer to cumulative  sum of responses for 1000  Maximum response * Straight  line  Is  values  trials.  1.000  locus of equivalent  cumulative  sums.  960  560 600  Figure  680  65.  760 840 9 20 SIMPLICIAL SEARCH  1000  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random error.  106  Symbol  Standard  0  .020  •  .015  A  .010  A  .005  • Coordinates  Error  Average  values  .000  refer to cumulative  sum of responses for 1000  Maximum response = Straight  Large symbols are  line  Is  trials.  1.000  locus of equivalent  cumulative  sums.  1000  800 800  840  860 SIMPLICIAL  Figure 6 6  920  960  1000  SEARCH  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random  error.  107  Symbol  Coordinates  Standard  0  .020  ©  .015  A  .010  A  .005  •  .000  refer  to cumulative  Error  1000  is  l o c u s of  1  800 -  r e s p o n s e s for  values  1000  trials.  1.000  equivalent  1  ere  cumulative  1  i  sums.  /  RUN NO. 57 N = 3  o or £  line  Average  sum of  Maximum r e s p o n s e = Straight  Large, symbols  • —  600  < o CL 400  o o 200  / ,  0  200  ,  ,  400  600  SIMPLICIAL  Figure  67.  C o m p a r i s o n of with  1  800  000  SEARCH  C o m p l i c i a l and S i m p l i c i a l s e a r c h e s  v a r i o u s d e g r e e s of  random  error.  108  Symbol  Type of Search  0  Simplicial  t  Complicial  Abscissa coordinates are expressed by X, = 1 0 0 . * ( I. - ( . l )  )  , / 2  1 = 0,1,2,3,4  1000  1  800 _  1  1  1  RUN NO. 5 5 N • 2  co < 600  -  • • •  .1U.  O  400  ' 9 9 % of Optimum >. not reached after 1000 trials  -  •  0  #  200  0  —  1  0  68.4  90 PERCENT OF  Figure 68.  1  1  96.8  99  OPTIMUM  Number of trials required to percentages of optimum.  reach various  109  Symbol  Type of Search  0  Simplicial  •  Complicial  A b e c i e s a c o o r d i n a t e s are X. = 100.#  ( I. -  expressed (.l)  l / 2  by:  )  i = 0,1,2,3,4  700  560  ~  < 420 rr  ~  co  o  280  -  140  "  0  68.4  90 PERCENT  Figure  69.  96.8 OF  99  OPTIMUM  Number of trials required percentages of optimum.  to  reach  various  Symbol  Type of Search  0  Simplicial Complicial  A b s c i s s a coordinates X . = 100. *  are  expressed  (* I. — (.1 )  l / 2  by  )  i = 0, 1 , 2 , 3 , 4  1000  T  RUN NO.  800  9 9 % of Optimum not reached  57  'after lOOOtriaJs  N = co <  600  E 400  -  200  0  684  90 PERCENT  Figure 7 0  1  1  96.8 OF  99  OPTIMUM  Number of t r i a l s required p e r c e n t a g e s of optimum.  to  reach  various  Ill  Moving  or  D r i f t i n g Response  Brooks (2,0)  and  surface  R4  (Runs  (2,2)  to  from  (1,1)  units  per  trial  so  to  as  arrays that  the  plicial  (see  would  the  be  effective  although  the  the  away  74  error the  the  results (see  as  amount  remains  of  60°  This  75)  optimum  at  a  of  the  results,  as  the  rate  was  relatively  "drifted" of  so  optimum w h i l e  methods the  Figure  complicial  shows t h e  runs)  error  rate  response  .005 chosen  simplicial chosen the  sim-.  search  point  was  has  the  Although  there  i s present, constant  the  is  a  far more  percentage  optimum  path).  plotted  be  the  approaching would  Figures  effectiveness  been  was  to  a high  however,  of  shows t h a t  complicial  achieved  optimal  71  relative effectiveness  0  test  of  to.  same d e g r e e The  the  d i r e c t i o n was  The  the  as  The  s i m p l i c i a l search  from both Figure  20).  (eventually,  the  i s present.  small  ness  from  display  error a  value  optimum..  catch  also  both  starting point  drift and  optimum  72  starting points  chosen  arrays.  deterministic  Figure  v/as  orientation  could  correct  using  d i r e c t i o n of  the  unable  superior.  of  a  complicial  choice was  54),  moving  Figure  for  complicial  For above  the  a  along  account  and  and  respectively,  simulate  away  52  Surfaces  when  73 random  (averaging against  large  random  drop  when  relative effective-  thereafter.  112  Symbol  Type of Search  0  Simplicial Complicial  Abscitea coordinates are expressed X. = 1 0 0 . *  ( I. - ( . l )  l / 2  by:  )  i = 0, 1 , 2 , 3 , 4  IOOO  _  800  ^  —  These percentages were  RUN NO. 5 2 '  not  N « 2 »'  trials were completed.  /  m  -i < 600 rr  O  reached after 1 0 0 0  400  200  1 68.4  90 PERCENT OF  Figure  7i  96.8  99  OPTIMUM  Number of trials required to reach various percentages of optimum.  113  Symbol  Type of Search  0  Simplicial  ©  Complicial  Abscissa  c o o r d i n a t e s ar®  X . = 100. #  ( I. -  I = 0 , I,  300  sxprtssod (,l)  i / 2  by:  )  2,3,4  r—  68.4  90 PERCENT OF  Figure  72.  96.8  99  OPTIMUM  Number of trials required percentages of optimum.  to  reach  various  114 Symbol  Coordinates  Standard  0  .020  o  .015  A  .010  A  .005  O  .000  refer to cumulative  Error  line  Is  Average  values  sum of responses for 1000  Maximum response = Straight  Large symbols are  trials.  1.000  locus of equivalent  cumulative  sums.  480  120 SIMPLICIAL  Figure  73.  160 SEARCH  200  240  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random error.  115 Standard  Symbol  Error  .020 .015  Coordinates  A  .010  A  .005  •  .000  refer to cumulative  tine  Average  values  sum of reeponses for 1000  Maximum response Straight  Large symbols are  trials.  1.000  is locus of equivalent  cumulative  sums.  3 6 0  1 4 0  180  2 2 0 S I M P L I C I A L  Figure 74  2 6 0  3 0 0  3 4 0  S E A R C H  Comparison of Complicial and Simplicial s e a r c h e s with various degrees of random  error.  116  Figure 75.  R e l a t i v e e f f e c t i v e n e s s versus random error (moving optimum)  117  DISCUSSION  OF  RESULTS  General The  plots  against  that  plicial  method  of the cumulative  o f t h e s i m p l i c i a l method i s generally  more  method.  For the deterministic  positive  relative effectiveness.  error  ( i . e . f o r the average  initialize  t h e random  level  o f random  below  the equivalent  chance) of  40, 41  relative  would  e r r o r ) was  and  The  always  scatter  of  always  indicate  line  points  of  a  random was  numbers  to  irrespective of the  the occasional sum  showed  o f the runs  different  point  (due t o  fell  random  ( f o r any g i v e n  above t h e l i n e  at each  i s about  that  o f each  f o r a given  scatter  f o r most  com-  the s i m p l i c i a l  In the presence  of the successful  effectiveness  positive)  same  of the five  the  a l l the runs  generator),  cumulative  that  effective that  of five  Although  complicial  (except  level  f o r Runs  43).  reliability) the  error.  the average  random  number  showed  cases,  the relative effectiveness  positive  sums o f t h e  runs  level  t h e same  o f random  f o r each  the r e p r o d u c i b i l i t y search level  v/ith random  technique o f random  error  (where  i s t o be  the  average  error  i s  method.  This  (and h e n c e t h e  i s expected error.  The  expected  t o be variation  as t h e  118  search of  p a t h s become more  random e r r o r  The relative  three  be  The  presence  a to  to influence  runs  explained  from  show  move.  negative  o f random e r r o r  o f the response  o r concave  contours  leads  o f s u c c e s s f u l moves  f o r both  methods.  where t h e c o n t o u r s  the direction i s shorter.  shown b e l o w .  values  l a r g e r amounts  o f a next  and 43) w h i c h  considerations  of straight  sequence  situation  moves  ( 4 0 , 41  with  the choice  effectiveness f o r higher  may  longer  diversified  o f the search)  are quite  convex  t h e sequence  This' s i t u a t i o n  can be  surface.  to a For  (relative  of successful  illustrated  as  J  Optimum  Figure  76  c  Possible  explanation  for inferior  runs.  119  New  consider  established considered  by  the choice  the s i m p l i c i a l  conservative  point  for  t h e c o m p l i c i a l method  the  highest  of  i s formed  thus  (a) do  corrected  to  (which  thereby  fully  presence  of large  for  t h e c o m p l i c i a l method  ing  far adrift  The  Number The  of  can be  runs  be  path  explained error,  by  by  towards to  The  t o go  the  far adrift.  direction  43)  The as  i s scatter  compared Brooks  theory.  degree  to that  the search  random  contours  to adjust  this  a high  the  However,  the remaining  (as compared caused  and  can be  due  contours.  (40, 41 on  high  O-array)  In  of scatter  of the path  wander-  (a).  Variables  zero  considerations it  be  the  which  points.  o f an e r r o n e o u s  random  may  choice  erroneous.  search  as i n s i t u a t i o n  results  approaches  the  f o r the runs  more  method)  be  to the surface  the  simplicial  high  not r e a l l y  could  the choice  q u i c k l y due  can be  two  the complicial search  for the unsuccessful  surfaces  on  may  causing  the scatter data  This  the.direction i s biased  the direction  ( b ) , however,  data  i s based  not permit  direction, In  point  rules.  (from  as t h e c e n t r o i d through  new  error)?  of a direction  noted  show t h a t  the relative  a s t h e number  effectiveness  of variables increases.  of the properties of 0-arrays that  the length  and  From  1-arrays,  of the c h a r a c t e r i s t i c  diagonal  120  ( D = ••/rg ! "^ ) a p p r o a c h e s  the length  as  (N) b e c o m e s  -  t h e r.umber o f v a r i a b l e s  1-array Thus,  becomes  synonymous w i t h  i t i s natural  becomes  Ridge  zero  to expect  a s t h e two  certain  relative  infinite.  the 0-array  that  sides  ( R  (i.e. the  i n the  the relative  arrays.become  range  effectiveness  of ridge  made-sharp enough  limit).  effectiveness  t h e same  (relative  of the ridge  for  method  either  effectiveness methods  shape.  t o search  shows  that  point  has  those  searches with  indicates ridge.  this  condition  a. f a r l o w e r  that  Only  metric  explanation  Curvilinear The  cumulative  See  (principal  Run  sum  degree  51  o f random reference  of this  for a the  resolution)  Thus,  the  relative  sums f o r  (see F i g u r e  as t h e  63)  deterministic  f o r each  method  o f random  error.  error  ridge  step-size), the  are c y c l i n g around  the presence  to progress.  zero.  a higher  once  as t h e cumulative  i s reached  the arrays  search  to a given  effectively;,  zero  approach  constant  However,  i s too small  approaches also  remains  sharpness.  resolution  both  of the other  Sharpness The  is  -  a point  will  allow  (25,p.320) f o r a  than This on t h e the geo-  phenomenon.  Ridges  curvilinear ridge  surfaces  (Runs 57,  58  and  59)  )  121  again the  show t h a t  simplicial  comparison random for  points  tested  easily  Hence,  observed.  yields  ridges.  thus  cause  i s more  the complicial  method,  a  small  degree  subsequent  path  effective  o f random  Even  a high  The  i t s search  of the simplicial  search  In the absence  however,  curved  O-arrays. path  search.  can be  error,  these  the -complicial  thereby  error,  the  amount  of  o f erroneous  failures search  starts  than  of the  t o form  t o resemble  reducing  moves new  many  the  the  relative  effectiveness.  Moving  Optimum  The  results  emphasize when In  t h e optimum  to catch  causing  array  due  ( i n f a c t , new t o wander  error  over  results  effectiveness  f o r the  range  simplicial  However,  Although  o f random  52  -were  the  and  constant error  compli-  diagonal  t o cause both o f Runs  failures  the  i f t h e optimum  e f f e c t i v e n e s s remains  the entire  direction.  of the long  point.  i s sufficient  the average  the relative  length than  54)  t e s t e d would be  aimlessly).  the moving optimal  o f random  positive  points  slower  (52 a n d  i n one  i s moving too f a s t  (although  " l o s e ground",  show t h a t as  on  rapidly  to the additional  gradually  stationary)  to  up  the arrays  presence  runs  the difference i n the relative  52  search  gains  f o r t h e moving optimum  t h e n e c e s s i t y i s t o move  Run  cial  Surfaces  searches 54 as w e l l  simulated.  122  CONCLUSIONS  Based  on  conclusions  1.  the results previously  and  The  recommendations can be  c o m p l i c i a l method  than  the  cases  2.  3.  simplicial  i n random  change  the relative  method  compared  complicial  --with  relative  the  simplicial  its  simpler  t o be  more e f f e c t i v e  for practically  error  number  do  not  a l lthe  effectiveness  i s f a r wore With  method  method  be  are thus  would  have  utations  no  very  required.  chosen  variables and  because  which have  a  ( i . e . N=2  i n the complicial  simple  difficulty  of  towards zero  for industrial  of operating'variables  rules  surfaces  of  Thus t h e a p p l i c a t i o n  f o r a l lsurfaces  involved  complicial  method.  numbers  tends  should  set of rules.  of the  effective for large  effectiveness  formulae  appreciably  to the simplicial  the complicial  commended  The  made.  shown  method  few v a r i a b l e s .  the  of  was  certain  tested.  Increases  The  presented,  and  plant  i n doing  use  i s re-  small or  N=3).  set of  personnel  the l o g i c a l  comp-  123  4.  The  c o m p l i c i a l method  simplicial Until  the  reached, iveness  5.  In  the  method  presence method  to  of  the  ridged  change the  large  should  be  the  d i r e c t i o n of of  rectify to  Rule  this  the  1 of  of  the  response  the  com-  contours  a  1 would  to  modifi-  c o m p l i c i a l method Rule  effect-  relative  Perhaps  condition.  i s  ridge.  the  i f the  "-.he  tested.  relative  concave  search. the  ridge  error  avoided  e x p e c t e d to be s t r a i g h t o r  a  i n the  random  than  surfaces  sharpness  are  cation  effective  r e s o l u t i o n of  i s no  relative  plicial  for  principal there  i s more  be  could changed  read: If  the  is  higher  the by  other  than  a  the  points  some v a l u e  must be  of  £  newest  response  i n the , the  1-array  of a l l  array  next  point  (A  )  array  (A"!",.).  J+l Rule -  The as the  2  ( c o m p l i c i a l ) would  value a  of  function response  further  work  £  would of  the  be  changed  probably amount  surface„  It  should  done  be  of  be  accordingly.  best  random  expressed error  i s recommended in this  in  that  respect.  124  NOMENCLATURE  A  J  h\ J  -  The J  array  (0 t y p e )  =  The  array  (1 t y p e )  th C  =  The I  coordinate  C'  =  Centroid  D  =  Length  =  The s u b s c r i p t  of the starting  point  point  of the characteristic  diagonal  pertaining  t o t h e newest  diagonal  The s u b s c r i p t ' p e r t a i n i n g  to the oldest  diagonal  point DD  =  point I  =  The s u b s c r i p t dependent  pertaining  variable  II  =  An a d d i t i o n a l  subscript  J  =  The s u b s c r i p t  pertaining  of  to a particular i n -  points  o f t h e above  type  to a particular  array  point  r  K  =  The s u b s c r i p t  pertaining  to a particular  L  =  The s u b s c r i p t  pertaining  to the point  with the  to the point  with the  lowest  LL  =  The s u b s c r i p t next  N  =  response  lowest  T h e number  pertaining  response.  o f independent  variables  125  P  =  The  R  =  The  length  R'  =  The  response  without  R*  =  The  response  with  R.E.  =  Relative  effectiveness  S  =  Standard  error  S*  =  Modified  standard  Y  =  The  K, J  I, K, J  point  I  i n the  of a  array  side  of a 0-array  random  random  (step-size)  error  error  error  coordinate  of the K  point  i n the  J  array  Z  =  K, J  Z'  Greek  The  response  The  normal  of the K  standard  point  i n the J  deviate  Symbols  P  —  0~ -  Parameter  used  i n the m u l t i v a r i a t e  response  surface  Parameter  used  response  i n the multivariate  normal  normal  surface  A O"  =  Variable  indicating  t h e amount  of  scatter  array  126  LITERATURE  1.  Baasel,  W.D.,  optimum  conditions",  CITED  "Exploring  responses  Chemical  surfaces  Engineering,  to establish (0ct  o  25,  1965),  147-151.  2.  Barnett, Ind.  3.  B o x , G.E.P.,  6,  industrial  (1957),  Box,  for  G.E.P.,  "Some  6.  B13,  7.  500-503  Operation:  operation",  o  A method  considerations  ASME,  operation  for  i n -  Statistics,  (March,  I960),  i n  process  113-119.  "Condensed c a l c u a l t i o n s  programs",  Technometrics,  77-95.  a n d K.B. W i l s o n ,  o f optimum  (1951),  general  and J.S. Hunter,  1959),  B o x G.E.P., tainment  1960),  productivity", Applied  Trans,  evolutionary  (February  (June,  to evolutionary  81-101.  G.E.P.,  Box,  52,  Evolutionary  optimization",  5.  "Introduction  E n g . Chem.,  creasing  4.  E.H.,  "On  conditions",  the experimental J . Roy. S t a t .  at-  Soc , 0  1-38.  B r o o k s , S.H. , "A  comparison  o f maximum  Qper.  7,  430-457.  Research,  (1959),  seeking  methods",  127  8.  Brooks,  S.H.,  gradient  direction  Biometrics,  9.  'Simplex'  Chemical  10.  C a u c h y , A.,  Sci.  11.  12.  13.  and  (Wiley,  New  conditions",  Forsythe, optimum (1951),  "Block  pour  Cox,  improvement  117-126.  l a resolution Compt.  York,  effects  Experimental  rend.  des Acad.  Designs,  1957).  i n the determination  Biometrics,  12,  a n d M.J.D. P o w e l l ,  method  (1963),  1965),  simultanees",  G.M.  optimum  R.,  "Process  evolutionary operation",  5,  generale  experiments",  :  R.M.,  Fletcher,  (July  estimation of  (1847), 536-38.  DeBaun,  6,  14.  self-directing  W.G.,  Ed.,  descent  Sweeney,  d'equations  Cochran, 2nd.  H.C.  and  25,  ascent  48-56.  "Methode  Paris,  "Optimum  1961),  Engineering,  systemes  /  B.H.,  Mickey,  i n steepest  (March,  Carpenter, with  a n d M.R,  "A  f o r minimization",  (1956),  rapidly  Computer  of  20-22.  converging Journal,  163-168.  C.E.,  and  T.S.  g r a d i e n t method", 304-305.  Motzkin,  "Acceleration  Bull.  Math.  Amer.  of the  Soc.,  57,  128  15.  Friedman,  M.  Analysis,  16.  Hunter,  and L . J . Savage,  (McGraw-Hill,  J.S., "Optimize  evolutionary 19,  17.  1960),  Koehler,  55,  19.  20.  Proc. 2nd.  T.,  L o w e , C.W.,  Chemical  Stevens  1959),  "Some  Ch. 13.  process  with  Engineering  (Sept.  techniques  J . A . , a n d R. M e a d , Computer  Powell,  M.J.D.,  minimum  of a function  calculating  Symp.,  some  actual  (1958).  Prog.,  76-104.  Chem. E n g . , 4 2 ,  minimization",  21.  chemical  Statistical  " E v o l u t i o n a r y o p e r a t i o n " , Chem. E n g .  Inst.  Nelder,  1947),  "Evolutionary operation,  (October,  Trans.  your  operations",  T.,  Koehler,  York,  of  193-201.  examples",  18.  New  Techniques  of evoltionary (1964),  "An e f f i c i e n t  7,  method  (1965),  method  of several  derivatives",  T334-344.  "A s i m p l e x  Journal,  f o r function 308-313.  f o r finding the  variables  Computer  operation",  without  Journal,  7,  (1964),  155-162.  22.  Riordan,  R.S.,  "Problems  i n t h e a d m i n i s t r a t i o n o f evo-  lutionary  o p e r a t i o n " , Amer. S o c . f o r Q u a l i t y  (Buffalo,  New  York,  1958).  Control,  129  23.  24.  Rosenbrock,  H.H.,  "An  greatest  or  least  value  Journal,  3,  (October,  Spendley,  W.,  application evolutionary  25.  of  of  441-459.  Wilde,  D.J.,  Hext  simplex  and  C.  a  1960),  and  method  function",  F.R.  S.S.,  Press,  Princeton,  1943)  the  Computer  "Sequential  in optimization  T e c h nome t r i c s,  Beightler,  Mathematic  finding  Himsworth,  4,  Foundations  ( P r e n t i c e - H a l l , Englewood,  Wilks,  for  175-184.  designs  operation",  1962),  zation,  26.  G.R.  automatic  Statistics,  N.J.,  and  (November,  of  1967),  (Princeton  Optimi318-321.  University  I-l  APPENDIX  Problem: from  Derive  I  the c h a r a c t e r i s t i c  the f i r s t  two p r o p e r t i e s s t a t e d  1.  with  Start  the 0-array  /  0  length of a on page  given  o  0  1-array  13.  below:  o\ P  I . J  p  q  q  q  P  2, J  q  p  q  q.  P  3, J  q  q  p  q  q  q  q  (I-D  P, „. 4, J  N+1, J  where:  (JW+1 +  NV2  P  N-1)  (1-2)  1  (1-3)  NV2 2. _ A u g m e n t such  t h e above m a t r i x  with  a new  point  P  N+2, J  that: N+1 =  N+2 , J  y  _2_ N-1  £->  K,J  1,  (1-4)  K=3 Thus:  P  N+2,J  ~  2  q  '  r  '  r  '  * *''  r  (1-5)  where:  r = N- '— ( ( N - 2 ) q + p )  (1-6)  1-2  3.  "The  1-array  r/  is  Thus  4.  the  . q  q  q  q  p  q  •  •  •  •  •  q  q  q  p  P  i 2q  r  r  combined  N  points  The  combined  N-1  in  q \  •  array  common  array  dependent  of  a  the K  =  (from  P  one  P  another the  R  2  a  by  a  from  distant  =  K  property N+2;  J  N+2,  J  and  a  K  =  that  1-4)  1-array 2,  N+l  K f- 2)  Equation  (2q-q)  R  2  =  +  =  of  N+l).  of  the  are  and  occupy  a  1  as  •  "N+l  are  R  =  P  (r-p)  +  2  obtain  R  Point K  =  1-3 =  P N+2, 3,  J  N+l  is also  below:  (N-2)(r-q)  1-2,  from  considerations  for  shown  •  equidistant  1 by  O-array.  points  Equations  s i m p l i f y to  2,  distant  properties  Substitute and  for  K, J  equidistant by  N+l,  dimensions.  Points  of  (1-7)  for  K, J  i  5.  .  N+2 , J  2, J p 3, J  O-array  1,  P  P  i  r  (i.e. P  has  ( i . e . 'P__ _ f o r K, J  linearly  P. with I, J  p  has  points  replacing  q  ~  1  A  by  q  q  A J+1  formed  1.  and  1-5  (1-8)  2  into  the  above  1-3  6.  The  distance  between  characterisitic  D  2  =.  (2q-p)  Substitute and  diagonal  2  +  1-2,  to obtain  P. , „ T  the  1-3  T  i s therefore  the  N+2 , J  and h a s  (N-1)(r-q)  Equations  simplify  P_ _ a n d 2, u  a  length  D.  Thur>:  (1-9)  2  and  1-5  following  into  the  above  expression:  (1-10)  for  R =  1  .  II-l  APPENDIX I I  Problem: rules.  1.  Derive Show  t h e formulae  that  f o r the simplicial  t h e a r r a y s formed  Generally, i fP  centroid  pression  i sa reflection  P~ S,J  2.  3.  C o f some o t h e r p o i n t s ,  f o rthis.computed  Cc"- P  =  The formulae 3,  above  formula.  of  6  m  f o rRules  Rules  Rule  4,  ,) + C'  T,J  5 and 8  t h e form  2c'-  =  1 and 3  consists  o f Equation  of P T, J then  through  t h eex-  point i s : P , T,J m  are based  o f three  I I - l  (II-D  ( s i m p l i c i a l ) and  (complicial)  (complicial)  complicial  are correct.  S, u a  and  on t h e  reflections  These a r e :  N+1 .  P  S  +  2  .j  -  i?r  £  P  K.J  -  P  "  P  D.J  -  ( I I  2 )  K=l K/D,DD  N+1  2  p*  N+2,J* - N - 1 .  P  K,J*  L,J*  (  I  I  "  3  )  K=l  N+1 p  N+2,J  _  „?_ N-1  >  ^  p  K=l K^L,LL  K,J**  _ p*  L,J**  V  (II-4) '  II-2  Equation  II-2 can be w r i t t e n a s : N+l  N+2,J  P  X  N^T  =  K  P  #  J  * L,J|  +  ?  P  D,J  (H-5)  K=l K^L,D,DD Equation  P* N+2,J*  =  II-3 can be written as: N+l V LJ K=l  N  p *K,J  _ p  L,J*  l  (II-6) ° ;  K/D from  considerations  o f Equations  20 a n d 2 1 .  Thus: N+l P* ^N+2,J*  =  — N[  V L,  P  K,J  +P  L , J  +P  I DD,jJ  - P* N+2,J  K  (II-7) '  K  (II-8) '  K=l K/L,D,DD Equation  p  N+2,J  =  I I - 4 can be w r i t t e n a s :  —— N-1  N+l V L, K=l  P  - p* N+2,J*  K,J  K/D,L from -considerations and  23.  o f Equations  21  Thus: N+l  P  N+2  f  J =  £  P  K , J  K=l K^L,D,DD  +  P  DD,j]  - N+2,J* P  (  I  I  "  9  )  II-3  Combine the  Equations  formula  II-5,  f o r Rule  6  I I - 7 and  II-9 to  obtain  (complicial).  N+1 N+2,J  4.  N ( Nb  [  y  Rule  1  p  i<  +  P  ; J  DD.J  +  P  L.J]  - ' M  < I I  ~  : O )  K=l K^L,D,DD  ( s i m p l i c i a l ) and Rule  O-arrays (see  Z  <N+1!  from  O-arrays.  Appendix  3  (complicial)  Apply  I) l e t t i n g  the formula  P  = LJ ,  P  J  .  form t o A^.  The  new  x , J  point i s : P  KT-I.O  ,  =  S,  S,  S,  . . . ,  (11-11)  S  N+2,J where: s  = |  The is  ( (N-l)q  distance  from  + p)  P  (11-12)  „ , to N+2,J  P„ , f o r K = K,J  2,  N+1  .  given by:  R  2  =  ( -p) s  Substitute above  2  +  (N-D(s-q)  Equations  and o b t a i n  R =  Thus  a l l the points  from  one another  tion  applies  (complicial).  1-2,  *  2  1-3  and  into the  1.  i n t h e new  array  and i s a 0 - a r r a y .  f o r Rule  11-12  (11-13)  3  are The  equidistant same  ( s i m p l i c i a l ) and. R u l e  deriva8  11-4  5.  Rule  4  Apply  (complicial) the formula  P '•_  = P.  r  P  N+2  _  J  to A^  and  =  2  '  q  r  forms  P  '  r  a 1-array  (see A p p e n d i x = P  '  r  from  .  '  a  I)  O-array.  letting  T h e new p o i n t i s :  (11-14)  r  where:  r  = ~  ( (N-2)q  Appendix points  6.  Rule  P  5  Apply  I shows  + p)  that  P„,_ forms N+2, J  a 1-array  T  with .  f o r K = 2, N + l .  K, J  (complicial)  the formula  forms  t o A"!" J+1  P^ , , = P , and D,J+1 2,J T  (11-15)  a 1-array  from  (see A p p e n d i x  P ^ ,, = P„, _ . DD,J+1 N+2,J  0  T  T  a I)  O-array, letting  T h e new  point i s :  P  N+2  J+l  =  2  q  ~  -  3  P  ' ' t  '  t  (H-16)  fc  where:  t  The is  i l l  =  (  (  distance  N  from  ^  +  2  P)  (H-17)  P„ ^ , to N+2,J+l  P „ , f o r K = 3, N + l K,J  given by:  R  2  =  (q_p)  Substitute above  2  +  (t-p)  equations  and o b t a i n  2  +  (N-2)(t-q)  2  (11-18)  1-2, 1-3 a n d 1 1 - 1 7 i n t o t h e  R = 1.  II-5  The  R  distance  2  =  (2q-p -  Substitute above  R  2  to  =  Thus  Rule  T  2q)  +  2  (N-l)q  substitute  and  "derived  T  (N-l)(t-r)  11-16  hy:  and.11-17  1  (11-18)  2  into  the  obtain  from  (11-19)  2  Equations R =  a l l the points  6  +  2  Equations  equidistant  7.  P._,_ ., t o P „ , . i s given N+2,J+l N+2,J  obtain:  p  Further above  from  one  (complicial) i n the text  1  1-2  and  1-3  into  the  .  i n t h e new another  gives o f the  the  array  and  i s a  correct  rule.  A  0 J+2  are  0-array.  1-array ^  as  III-l  'APPENDIX I I I  To  compare  complicial  i s a  search paths  methods,  shown b e l o w . method  the  two  In both  sample  cases  of the  searches  the f i r s t  O-array. —••  simplicial  simplicial complicial  f o r N=2  array  and are  f o r each  IV-1  A P P E N D I X .IV  Nomenclature  f o r Computer  Program  C  Starting  point  DEV  Standard  error  KRAND  Parameter  to loop  five  levels  KRUN  Parameter  to loop  each  o f the search  LMOVE  Variable  LRUN  Run  NBIG  Parameter  to indicate  the type  o f random  error  methods  o f move  number to loop  five  random number  initial-  izers N  Number  NCYCLE  Counter  NSUC  Logical variable new  of variables to record  point  i s a  number —  of successive  i t s value  i fthe  success  NT  Trial  NTEST  The number  R  Step-size  change  RHO  Parameter  for multivariate  RP  Cumulative  SIGMA  Parameter  Y  Value an  i s true  failures  counter o f runs  sum  per program  of the variables  f o r a given  search  for multivariate  normal  of the coordinate  array  normal  of a given  distribution  distribution point  in  IV- 2  YP  Number of  Z  Computer  The program  trials  to  reach  a  certain  percentage  optimum  Response  of  a  point  Listing  following that  simplicial the  of  was  used  method.  multivariate  pages to  The  normal  give  a  test  the  test  complete  listing  complicial  response  distribution.  surface  of  against  the the  shown i s  PLEASE RETURN  TO THE CHEMICAL ENGINEERING BUILDING JOB NUMBER  THE  >  ANALYSIS OF VARIANCE JOB START  i  $J08  16075  1  $FOR TRAN '  MAIN  1  !  '  17 20 21 22 23 24 25 26 27 30 31 32 33 34 35 36 37 40 41 42 A3 44 45 46 47 50 51 52 53 54 55  CATEGORY (MFAV  F  IV-3 USER'S NAME- ROBIN BOOTH  ) WRITE UP IS NOW  15HRS 28MIN 30.3SEC  USE  AVAILABLE AT THE FRONTDESK OFF  V9M011  i  ROBIN BOOTH  $TIME  1 2 3 4 5 6 7 10 11 12 13 14 15 16  16075  COMMON Y ( 1 0 » 1 0 ) , N» N P l , NP2 » EN, EMI, NM1, DIMENSION SIGMA(LO), RH0(10,10), C ( 1 0 ) DIMENSION Z ( 1 0 1 0 ) , X P ( 5 ) t Y P ( 2 » 5 ) t Z P ( 5 ) DIMENSION R P ( 2 , 5 , 5 ) , NSTOP(IO) DATA NSTOP/3,3,5,7,9,11,13,16,18,21 / DO 8619 K = 1,5 8619 ZP(K) - ( 1 0 0 . - SORT(10.**(5 —K ) ) )/100. LOGICAL NSUC CALL PLOTS READ 5021, NTEST 5021 F0RKATU5) DO 333 NEWT = 1, NTEST READ 5003, NRON, N , R, LRUN 5003 FORMAT12I5, F20.8 ,15) C ALL THESE VARIABLES WILL BE COMMON NPl = N + 1 NP2 = N + 2 EN = N EMI = EN - 1. NM 1 = N - 1 . NM2 = N - 2 CALL CHARB(SIGMA, RHO, C) YYY = .123415678 DO 9911 NBIG = 1,5 XXX = YYY * FLOAT(N31G ) + .1 DO 888 KRAND = 1,5 KKTY = KRAND - 1 DEV = FLOAT(KKTY)/200. DO 888 KRUN = 1, 2 RINTI = RAND(XXX) NT = 0 NCYCLE = 0 CALL POINTS (R, C) LMOVE = 0 DO 80 J = 1, N P l CALL RESPO( J , SIGMA RHO, NT, DEV, Z ) IF(NT.GE.IOOO) GO TO 854 80 CONTINUE LMOVE = 3 IF(KRUN.EQ.2) LMOVE = 6 NP = N CALL ORDER(NP) DO 300 J = NP2, NRUN CALL MOVE(LMOVE) CALL RESP0(NP"2, SIGMA, RHO , NT , DEV, Z ) I F(NT.GE. 1000) GO TO 854  NM2, VERT  _'_  5.6 57 60 61 62 63 64 .65 66  75  bl  70 71 72 73 74 75 76 77 100 101 102 103 104 105 106 107 110 111 112 113 114 115 116  300 854 888 9911 9090 909  333  NSUC = Y ( N P l f N P 2 ) . G T . Y ( N P l t N P l ) NCYCLE = NCYCLE + 1 IF(NSUC) NCYCLE = 0 IFINCYCLE.LT.NPI) GO TO 75 CALL R E S P 0 1 N P 1 , S I G M A , R H O , N T , D E V , Z ) I F I N T . G E . 1000) GO TO 854 CALL RETEST (LMOVE,KRUN,NSUC) NCYCLE = 0 . ._ CONTINUE CALL MEMTRAtLMOVE,NSUC) CALL LOGIC (LMOVE, KRUN,NSUC) CONTINUE CONTINUE CALL INVERT{RHO,N,10,DET,COND) CALL RESULT{ ZP , Z , N B I G , K R A N D , K R U N , Y P ,RP ) CONTINUE CONTINUE PRINT 9 0 9 0 , ( Z P ( K ) , Y P ( 1 , K ) , Y P ( 2 , K ) , K = 1,5) FORMAT(1X,F8.4,2F8.0/) PRINT 9 0 9 , ( ( ( R P ( I , K , L ) , 1 = 1 , 2 ) , L = l , 5 ) , K = 1 5 ) . FORMAT(1X,10F10.2/) CALL S Y M B 0 L ( 2 . 6 , 5 . 6 , . 1 4 , 3HN = , 0 . , 3) CALL N U M B E R ! 3 . 0 , 5 . 6 , . 1 4 , E N , 0 . , - 1 ) CALL PL0T1 ( Y P , L R U N ) CALL P L 0 T ( 2 . , 0 . , - 3 ) CALL S Y M B O L I O . 6 , 3 . 6 , . 14, 3HN = , 0 . , 3.) . . . . CALL NUMBER(1.0,3.6,.14,EN,0.,-l) CALL PL0T2 1RP,LRUN). CALL S K I P TO (1) CONTINUE CALL PLOTNO STOP END V  IV-4  117 120 121 122 123 124 125 126 127 130 131 132 133  5008  SUBROUTINE CHARB(SIGMA , RHO,. C) COMMON Y ( 1 0 , 1 0 ) , N , N P I , N P 2 , ENt E M I , NM1, NM2, VERT DIMENSION C M T ( 1 2 ) , R H M A T ( 1 2 ) , SMT(12) DIMENSION S I G M A ( I O ) , R H O ( 1 0 , 1 0 ) , C ( 1 0 ) READ 5 0 0 8 , (RHMAT(L) , L = 1, 12) READ RHMATl, ( ( RHO ( I , J ) , I =1, N) , J = 1, N) READ 5 0 0 8 , ( S M T ( L ) , L= 1, 12) READ SMT, ( S I G M A ( M ) , M= 1, N) READ 5 0 0 8 , ( C M T ( L ) , L= 1, 12) FORMAT(12A6) READ CMT , (C ( I ) , I = 1 , N) RETURN END  134 135 136 137 140 141 142 143 144 145 146 147 150 151  SUBROUTINE POINTStR, C) COMMON Y ( 1 0 , 1 0 ) , N» NPlt NP2 > EN, EMI, NM1, NM2, VERT DIMENSION C(10) 2 DO 40 'I = 1, N DO 40 J = 1, NPI IP1 = I + 1 Q = l./(EN*SQRT(2.))*(SORT(EN+1.) - 1.) ._, P = . l . / ( E N * S Q R T ( 2 . ) ' ) * ( EN - 1. + SQRT ( EN+1 . )_ ) IF(J.EQ.1) Q = 0. IF <J.EQ.IP1) Q = p Y ( I , J ) = C ( I ) + Q»R 40 CONTINUE RETURN END 1  152 153 154 155 156 157 160 161 162 163 164 165 166 167 170 171 172 173 ,174  SUBROUTINE RESPO ( J , S I G M A , RHO, NT, D E V , Z ) " COMMON Y ( 1 0 , 1 0 ) , N , N P I , N P 2 , E N , E M I , NM1, NM2, VERT DIMENSION S I G M A ( I O ) , R H O ( 1 0 , 1 0 ) , Z ( 1 0 1 0 ) IF(NT.NE.O) GO TO 650 PI = 3 , 1 4 1 5 9 CALL I N V E R T ( R H 0 , N , 1 0 , D E T , C 0 N D ) SUMP = 0 . DO 75 K = 1,N DO 75 L = 1,N SUMP = S U M P + ( R H O ( K , L ) ) # ( Y ( K , J ) / S I G M A ( K ) ) * { Y ( L » J ) / S I G M A ( L ) ) Y ( N P 1 , J) = E X P ( - . 5 * S U M P ) NT = NT + 1 I F ( D E V . E Q . 0 . ) GO TO 59 CALL STNDV (DEV,ERROR) Y ( N P 1 , J ) = Y ( N P 1 , J ) * ( 1 . + ERROR) IF{' Y I N P l , J ) . G T . 0 . 5 ) Y ( N P 1 , J ) = Y ( NPI , J ) * ( 1 . - E R R O R ) + ERROR Z(NT) = Y ( N P 1 , J ) RETURN END I  650 75  59  V  7  175 176 177 200 201 202 203 204 205 206  IV-8  SUBROUTINE STNDV (DEV,ERROR) rnMMriM M. MPI N P l., MP? N P 2 , E N , E M I , NM1, NM2, VERT COMMON vYn( 1n0. ,i1n0i) . N W=0. DO 20 1=1 ,10 V=RAND(0.) W=W+V =(iW/10.)-0.5)*10.9544511 SANDV ERROR = DEV * SANDV RETURN END f  20  207 210 211 212 213 214 215 216 217 220 221 222 223 224. 225  SUBROUTINE ORDER(NP) COMMON Y ( 1 0 , 1 0 ) , N, NPI, NP2, EN, EMI, NMl, NM2, VERT NP1P = NP + 1 DO 125 I = 1,NP IP1 = I + 1 DO 125 L = IP1, NP1P I F ( Y ( N P l , I ) . L E . Y ( N P 1 , L ) ) GO TO 125 DO 135 K = 1,NP1 T = Y(K,I) Y(K,I) = Y(K,L) Y { K , L) = T __ CONTINUE CONTINUE RETURN END 1  .  135 125  226 227 230 231 232 233 234 235 236 237 240 241 242 243__ 244 245 246 247 250 251 252 253 254 255 256 257 260 261 262 263 264 265 266 267 270  500  161 160 700  185 182 900  807 505 800  165 164  IV-10 SUBROUTINE MOVE (LMOVE) COMMON Y ( 1 0 , 1 0 ) , N , N P l , N P 2 , EN, E M I , N M l , NM2, VERT DIMENSION SUMY(IO) GO T 0 ( 5 0 0 , 7 0 0 , 8 0 0 , 9 0 0 , 5 0 0 , 5 0 0 ) , LMOVE CONTINUE DO 160 I = 1, N SUMY(I) = 0 . DO 161 K = 2 , N P l SUMY(I) = SUMY1I) + Y ( I , K ) Y(I,NP2)=(2./EN)*SUMY(I) - Y ( I »1> RETURN CONTINUE DO 182 I = 1, N SUMY(I) = 0 . DO 185 K = 3,NPl SUMY(I) = SUMY(I) + Y ( I , K ) Y(I,NP2) = (2./EMl)*SUMY(I) - Y(I ,1) RETURN CONTINUE DIV = - N * N M l / 2 DO 505 I = 1,NP1 SUMY (I") = 0 IF(N.EQ.2) GO TO 505 DO 807 K =. 3 , N SUMY(I) = SUMYII) + Y ( I » K ) Y(I,NP2) =4FL0AT(NP1)*SUMYU) +Y(I,NP1) +Y(I,2) J/OIV - Y ( I , 1 ) RETURN CONTINUE DO 164 I = 1, N SUMY(I) = 0 . DO 165 K = 3,NPl SUMY(I) = SUMY(I) + Y ( I i K ) Y(I,NP2) = (2./EMI)*SUMY(I) - Y (I , 1 ) RETURN END  271 272 273 274 275 276 211  300 _. 32 301 302 35 303 45 304 305 46 306 99 307 310 311 312 313 314 315  SUBROUTINE RETEST(LMOVE,KRUN,NSUC) LOGICAL NSUC COMMON Y ( l O t l O ) , N, N P I , N P 2 , E N , E M I , DO 32 K = 1,NP2 IF(K.EQ.NPl) GO TO 32 IF(Y(NP1,NP1).GT.Y(NP1,K) ) G O T O 32 GO TO 35 CONTINUE RETURN I F ( Y ( N P 1 , N P 2 ) - Y ( N P 1 , 1 ) ) 4 5 , 4 6 , 46 Y(NP1,NP2) = - 1 . 0 E 1 6 GO TO 99 Y ( N P 1 , 1 ) = - 1.0E16 CONTINUE NP = N + 1 CAUL ORDER(NP) LMOVE = 1 IF (IKRUN . EQ. 2 ) LMOVE = 5 NSUC = . T R U E . RETURN ' END  IV-11 NM1, NM2, VERT  316 317 320 321 322 323 324 325 326 327 330 331 332 333 334 335 336 337 340 341 342 343 344 345 346 347 350 351 352 353 354 355 356 357 360 361 362  IV-12 SUBROUTINE MEMT.RA ( LMOVE , NSUC ) COMMON Y ( 1 0 , 1 0 ) , N , N P l , N P 2 E N , E M I , N M l , NM2, VERT LOGICAL NSUC GO TO { 2 , 3 , 1 , 1 , 2 , 2 ) , LMOVE DO 198 I = 1,NP1 Y( I , 1) = Y(I ,2 ) IF(NSUC) GO TO 305 Y( I , 2 ) = Y d , N P 2 ) GO TO 198 DO 197 K = 2 , N P l KP1 = K + 1 Y { I , K ) = Y d ,KP1 ) CONTINUE RETURN DO 125 K = 3 , N P l KP1 = K + 1 DO 125 L = KP1,NP2 I F ( Y ( N P 1 , K ) . L E . Y ( N P 1 , L ) ) GO TO 125 DO 135 I = 1,NP1 T = Y{ I ,K ) Y { I »K ) = Y { I , L ) Y(I,L) = T CONTINUE CONTINUE DO 389 I = 1, N P l DO.389 K = 1,NP1 KP1 = K + 1 Y(ItK) = Y U t K P l ) RETURN DO 388 I - l . N P l DO 388 K = 1,NP1 KP1 = K + 1 Y(I ,K) = YdtKPl) NP = N CALL ORDER (NP) RETURN END f  1  305 197 198 2  135 125  389 3 388  363 364 365 366 367 370 371 372 373 374 375 376 377 400 401  92 91  SUBROUTINE LOG IC(LMOVE,KRUN,NSUC) COMMON Y ( 1 0 , 1 0 ) , N, N P l , N P 2 , ENt LOGICAL NSUC IFCKRUN.EQ.2) GO TO 91 IFCLM0VE.LE.2) GO TO 92 LMOVE = 2 IF(NSUC) LMOVE = 4 RETURN LMOVE = 1 IF(NSUC) LMOVE = 3 RETURN LMOVE = 5 IF(NSUC) LMOVE = 6 RETURN END  i  EMl,  NMl,  NM2,  VERT  402 403 404 405 406 407 410 411. 412 413 414 415 416 417 420 421  .  58  SUBROUTINE R E S U L T ( I P , Z , N B I G , K R A N D , K R U N , Y P , R P ) COMMON Y ( 1 0 , 1 0 ) , N ».NP 1, NP2 , EN , EMI , NM1, NM2 , VERT DIMENSION Y P ( 2 , 5 ) , Z ( 1 0 1 0 ) , Z P ( 5 ) , R P ( 2 , 5 , 5 ) SUMZ = 0 . YP 1 KRUN 11 ) = 0 . K = 2 DO 58 I = 1 , 1 0 0 0 I F ( K . E Q . 6 ) GO TO 58 IF(KRAND.GT.1) GO TO 58 I F i i i I ) . L T . Z P ( K ) )G0 TO 58 YP ( KRUN r K ) = I K = K + 1 SUMZ = SUMZ + Z ( I ) _RP(KRUN,NBIG,KRAND) = SUMZ RETURN END  IV-14  422 423 424 425 426 427 430 431 432 433 434 435 436 437 440 441 442 443 444  IV-15 SUBROUTINE PLOT1 ( Y P , N T R L ) DIMENSION Y P < 2 , 5 ) , X P ( 5 ) DATA X P / 2 . , 3 . , 4 . , 5 . , 6 . / MY = Y P ( 2 , 5 ) Q =MY/100 + 1 DY = 2 0 . *Q DX = 1.25 CALL A X I S ( 2 . , 2 . , 15H NO. OF TRIALS , + 1 5 , 5 . , 9 0 . , 0 . , D Y ) CALL A X I S ( 2 . , 2 . , 20H PERCENT OF OPTIMUM 20,5. ,0.,0.,DX) TRL = NTRL DO 19 K = 1,5 DO 19 I = 1,2 YP(I,K) = YP(I,K)/DY + 2. CALL SYMBOL ( XP(K) , YP I I , K ) , . 2 1 , I, 0 . - 1 ) CALL SYMBOL ( 2 . 2 , 6 . , . 1 4 , 8H RUN N O . , 0 . 8) CALL NUMBER ( 3 . 3 , 6 . , . 1 4 » T R L , 0 . , - 1 ) CALL P L 0 T ( 9 . 0 , 0 . 0 , - 3 ) RETURN END  445 446 447 450 451 452 453 454 455 456 457 460 461 462 463 464 465 466 467 470 471 472 473 474 475 476 477 5C0 501  21  -  22 23  IV-16 SUBROUTINE PLOT2 {RP » NTRL) COMMON Y ( 1 0 , 1 0 ) , . N , NPI» NP2 » E N , E M I , NML, NM2, VERT DIMENSION Y P ( 5 , 5 ) , X P ( 5 , 5 ) , X A V G { 5 ) , Y A V G ( 5 } DIMENSION R P ( 2 , 5 , 5 ) DO 21 I = 1,5 DO 21 K = 1,5 YP( I,K) = R P ( 1 , I , K ) XP( I , K ) R P ( 2 , I ,K) CALL SCALE(XP,25,5.,XMIN,DX,1) CALL S C A L E ( Y P , 2 5 , 5 . , Y M I N , D Y , 1 ) CALL AXIS < 0 . , 0 . , 19H S I M P L I C I A L SEARCH , - 1 9 , 5 . , 0 . , XMIN, DX ) CALL AXIS ( 0 . , 0 . , 19H COMPLICIAL SEARCH , + 1 9 , 5. , 9 0 . , YMIN, DY ) DO 23 K = l , 5 SUMX= 0 . SUMY= 0 . DO 22 1=1,5 SUMX = SUMX+ X P ( I , K ) SUMY = SUMY + Y P ( I , K ) CALL SYMBOL IXP( I , K ) , YP ( I , K) , . 07 , K , 0 . , -1 ) XAVG(K) = S U M X / 5 . YAVG(K) = S U M Y / 5 . CALL S Y M B O L ( X A V G ( K ) , Y A V G ( K ) , . 1 4 , K , 0 . , - 1 ) TRL = NTRL CALL SYMBOL ( 0 . 2 , 4 . , . 1 4 , 8 H RUN N O . , 0 . , 8) CALL NUMBER ( 1 . 3 , 4 . , . 1 4 , T R L , 0 . , - 1 ) CALL L 4 5 ( D X , D Y , X M I N , Y M I N ) CALL PLOT ( 7 . , 0 . , - 3 ) RETURN END  IV-17 502  SUBROUTINE  503  COMMON  504 505  DIMENSION YMX = Y M I N  Y(10,10),  506  IF ( Y M X . L T . O . ) XX( 1 )  =  YMX/DX  510  YY(1)  =  0.  511 512  XX(2) YY(2)  =  513  IF(YY{2).LT.0.) GO T O  515 516 517 520 521 522 523 524 525 .COMPILE OBJECT  990  222  N,  XX(2), XMIN  507  514  .$$  145(OX,DY,XMIN,YMIN) NPl,  NP2,  EN,  EMI,  NMl,  NM2,  VERT  YY{2)  GO T O 9 9 0  = 5 . 15.*DX  -  YMX)/DY RETURN  222  YMX = - YMX XX(l) = 0. YY(1) = YMX/DY XX (.2) =(5.*DY YY(2) = 5. IF(XX(2).LT.O.) CALL LINE(XX,YY  YMX)/DX RETURN ,2,1)  RETURN END TIME  PROG  0 0  TOTAL TIME DATA STORAGE  13633 0  AVAILABLE  0  CORE  00  MX550  IV-18  Computer  Output  The from  following  t h e computed  graphs are samples o f the output  program.  o o o o o.  o o o d. co  RUN NO. 25  A  A  O  «—"~>  x ° d.  (Jl  iO  T  an  ©  !—o L > O O O  o.  o o  CD  -.000  i 1.250  I 2.500  I 3.750  PERCENT OF OPTIMUM  I 5.000  I 6-250  I V-19  o o o  D  V-1  APPENDIX  The of  five  following cumulative  Complicial  table  shows  sums f o r  V  the  standard deviations  each l e v e l  o f random  error  Scatter Standard  Error  .005  .010  .015  .020  31  5.82  12.58  18.47  42.41  32  1.38  6.63  15.50  27.80  33  5.56  18.15  12.03  24.21  34  0.44  3.13  17.88  9.35  35  2.26  4.94  18.28  26.70  36  7.46  20.53  35.33  28.24  37  5.14  7.04  13.87  20.56  38  0.81  3.99  14.31  17.51  2.50  3.62  17.87  29.73  13.98  43.57  42.94  27.22  25.18  ^Run  No.  39  -  40  9.84  •  41  3.82  42  •8.33  17.91  38.42  41.19  43  5.57  7.38  3.57  17.15  44  5.01  13.29  21.00  26.72  45  16.06  50.21  33.95  55.62  46  0.50  6.14  2.71  6.13  6.41  10.29  24.79  28.42  4.71  12.94  19.97  27.40  " 9 . 5 2  Average for  40,  41,  43  Average excl. 41,  40,  43  A  V-2  Simplicial  Scatter Standard  Error  .005  .010  „015  .020  31  2.25  11.72  19.52  23.32  32  3.75  10.52  26.60  37.49  33  4.79  13.60  23.39  43.29  34  2.63  14.86  11.03  13.07  35  1.01  11.59  14.24  17.98  36  8.28  12.14  22.14  55.81  37  0.68  2.53  3.52  2.83  38  2.26  9.17  17.90  17.69  39  3.62  9.64  13.53  27.79  40  0.32  .1.05  2.73  1.35  41  0.33  0.82  0.57  2.06  42  9.94  12.22  14.57  33.45  43  0.98  1.11  5.34  14.76  7.06  14.33  46.10  19.12  45  9.20  26.10  39.63  19.32  46  1.85  20.27  13.56  19.51  0.54  0.99  2.88  6.05  4.41  12.98  18.74  25.44  Run  44  No .  ..  Average for  40,  41,  43  Average excl.40, 41,  43  These  results  a r e shown  i n Figures  45  and  46.