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.
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Optimizaiton with random error Booth, Robin Geoffrey 1968
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Title | Optimizaiton with random error |
Creator |
Booth, Robin Geoffrey |
Publisher | University of British Columbia |
Date Issued | 1968 |
Description | A new evolutionary operation called the complicial method is presented. The main criterion, which is adhered to, is that changes in the independent variables are restricted to a small step-size from a previous best trial. The complicial method is essentially a modification of the simplicial method proposed by Spendley, Hext and Himsworth in which these authors employ regular type arrays in a sequential type search for the optimum. The complicial method differs from the simplicial method in that an irregular array is formed when (and only when) the last trial is proven to be the best of those previously tested. The design of this irregular array is such that a regular array can be formed when the last trial is proven not to be the best so far. The complicial method is compared to the simplicial method for a wide variety of response surfaces in both the absence and presence of random error. It is found that the complicial method is much more effective (i.e. the relative effectiveness is very large) for almost all the test response surfaces involving a small number of variables. Although an increase in the amount of random error decreases the effectiveness of both methods, the relative effectiveness generally remains unchanged. However, as the number of variables is increased the relative effectiveness is found to decrease markedly. This is explained by considerations of the basic design of the regular and irregular arrays. Because the complicial method sacrifices some of the simplicity characteristic of the simplicial method, it is recommended that the complicial method be applied only in situations where the relative effectiveness is very large. Therefore, this method is best used for all types of response surfaces involving a small number, of variables. |
Subject |
Mathematical optimizaiton |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-07-15 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0059095 |
URI | http://hdl.handle.net/2429/36023 |
Degree |
Master of Applied Science - MASc |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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