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On optimization in probabilistic design Tutek, Mehmet N. 1975

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ON OPTIMIZATION IN PROBABILISTIC DESIGN by MEHMET N. TUTEK B.S. (Mechanical), Robert C o l l e g e , I s t a n b u l , Turkey, 1970 M.B.A. (Operations Research), Syracuse U n i v e r s i t y , Syracuse, N.Y., U.S.A., 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of Mechanical Engineering We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1975 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th i s thes is for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is for f i nanc ia l gain sha l l not be allowed without my writ ten permission. Department of kA.er-W\r>. rn'ral n ^ ' i n e - e r l n ^ The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T' 1W5 Date { 5 , Q. ABSTRACT In c l a s s i c a l design the inputs to the design process, namely the r e l e -vant materia l property M and the load L are taken as de te rm in i s t i c values. The p r o b a b i l i s t i c design approach recognizes the va r i a t i on in design inputs , and the consequent random behavior of these va r i ab le s . The material property var ies from one l o t of production to another and wi th in the same production l o t . The app l ied load to a p a r t i c u l a r specimen var ies widely wi th in some range. The design output, a dimensional para-meter A, a l so var ies wi th in the given range of t o le rances , and there -fore i s randomly d i s t r i b u t e d among specimens. Fa i l u re occurs at the f i r s t instance the load value L i s l a rger than the r e s i s t i n g strength of the m a t e r i a l , S. Among n load app l i ca t i ons the f i r s t exceedance of S by L re su l t s in f a i l u r e . The re levant load va r i ab le i s therefore the extreme value of a number n of loads that correspond to a given design mission time. Within th i s framework, design r e l i a b i l i t y and mission time emerge as the appropr iate design inputs . In p r o b a b i l i s t i c design the designer has a wide range of choice f o r both input parameters, r e l i a b i l i t y R and mission time n. In th i s Thes i s , the optimal combination of n and R i s determined in a l o g i c a l way. A c o s t - b e n e f i t ana lys i s i s made, r e s u l t i n g in an optimal combination using the bene f i t and cost to the dec i s ion maker. Since benefit i s r e l a t i v e , the information reauired to determine the benefit function i s acquired from the decision maker through several questions and a "reference gamble",. Cost i s analyzed and a cost function i s constructed. Using the benefit and cost function?, i n d i f f -erence and constant cost functions are derived, r e s u l t i n g in suboptimal combinations of R and n. The optimal comb-ination i s chosen among the suboptimal combinations by minimizing the cost-benefit r a t i o . An example i s presented to i l l u s t r a t e t h i s -decision model. TABLE OF CONTENTS I. INTRODUCTION ££SL_1 " A. I n t r o d u c t i o n 1 1. General 1 2. Load L, and M a t e r i a l Propertv M 2 3. P r o b a b i l i s t i c Design 7 B. L i t e r a t u r e Survey 11 C. Present Research Problem 14 I I . BENEFIT A. D e f i n i t i o n of B e n e f i t 15 1. D e f i n i t i o n 15 2. C o n s i d e r a t i o n s i n Assessing B e n e f i t 16 3. Preference Functions 18 B. B e n e f i t of M i s s i o n Time at a Given Value of R e l i a b i l i t y 20 1. General 20 2. Non-decreasing Preference Functions 22 3. Non-increasing Slope of Preference Function 25 Page C. B e n e f i t of R e l i a b i l i t y a t a Given Value of M i s s i o n Time. 30 1. General 3 0 2. Non-decreasing Preference Function 32 3. Non-increasing slope of Preference Function 35 D. B e n e f i t of Both R e l i a b i l i t y and M i s s i o n Time. 41 1. Interdependence of preference Functions.. 41 2. B e n e f i t of R e l i a b i l i t y and M i s s i o n Time, I n d i f f e r e n c e Functions 42 E. D e r i v a t i o n of Preference Functions 44 1. Information Required t o Construct Preference Functions 44 2. Questions t o the D e c i s i o n Maker 45 3. Reference Gamble 52 4. C o n s t r u c t i o n of Preference Functions 57 5. C o n s t r u c t i o n of I n d i f f e r e n c e Functions... 60 I I I . COST 63 A. M a t e r i a l Cost 65 1. Dependence on Parameters of M a t e r i a l p.d.f 65 2. Dependence on S i z e Parameter.... 69 v i Page B. Cost of Operation 71 1. Dependence on R e l i a b i l i t y 71 2. Dependence on M i s s i o n Time 7 6 C. Cost Function 79 1. Dependence of Cost on R e l i a b i l i t v and M i s s i o n Time 79 2. Constant Cost Curves 81 IV. DECISION PROCESS A. I n t e r s e c t i o n of I n d i f f e r e n c e Curves 85 B. The Optimal Combination of R and/1 86 V. CONCLUSION AND RECOMMENDATIONS 88 BIBLIOGRAPHY 9 0 APPENDIX I . An Example.... 9 4 APPENDIX I I . A n a l y s i s of Preference F u n c t i o n 1 1 2 APPENDIX I I I . E s t i m a t i o n of W e i b u l l Extreme by Log-Normal Model 121 APPENDIX IV. A n a l y s i s of I n d i f f e r e n c e Function 125 V / i LIST OF FIGURES Figure Page 1.1 A T y p i c a l P r o b a b i l i t y D i s t r i b u t i o n Function f o r M a t e r i a l Property M 4 1.2 A T y p i c a l P r o b a b i l i t y D i s t r i b u t i o n Function f o r Load L 6 1.3 P.d.f. of Load, Extreme Value of Load, and M a t e r i a l Strength 9 2.1 The Preference Function f o r M i s s i o n Time at Constant R e l i a b i l i t y RJuCnjR') 29 2.2 The Preference Function f o r R e l i a b i l i t y at Constant M i s s i o n Time n ;y(RJn') 40 2.3 I n d i f f e r e n c e Functions f o r R e l i a b i l i t y and M i s s i o n Time n(Rfn=C) 43 2.4 L o c a t i o n of I n i t i a l and End P o i n t s of Preference Function f o r R e l i a b i l i t y a t Minimum M i s s i o n Time 47 2.5 L o c a t i o n of I n i t i a l P o i n t s of Preference Functions f o r R e l i a b i l i t y a t 50 2.6 L o c a t i o n of I n i t i a l and End P o i n t s of Preference Function f o r R e l i a b i l i t y at M i s s i o n Time Values, 51 2.7 L o c a t i o n of these P o i n t s f o r Each Preference Function f o r R e l i a b i l i t y at n. 55 2.8 Preference Function f o r D i f f e r e n t Values of R[ 59 2.9 Preference Functions u(RJi.) f o r R e l i a b i l i t y at Constant M i s s i o n Time 62 2.10 I n d i f f e r e n c e Functions n(P.Ju=C) 62 v i i i F i g u r e Page 3.1 U n i t - M a t e r i a l Cost as a Function of u£ 6 7 3.2 U n i t - M a t e r i a l Cost as a Function of u 2 68 3.3 Cost of F a i l u r e as Function of R f o r Specimens w i t h C a t a s t r o p h i c F a i l u r e 72 3.4 B e n e f i t - C o s t A n a l y s i s f o r M i s s i o n Time 7 7 3.5 Cost of M i s s i o n Time at Constant R e l i a b i l i t y C (R,nR=n) 8 2 3.6 Constant Cost Functions 84 4.1 Locus of Sub-optimal P o i n t s 87 A . l Preference F u n c t i o n s , u(R|n.), f o r R at constant n^..,. ^ 104 A.2 I n d i f f e r e n c e Functions, n(R|u) 104 A.3 Constant M a t e r i a l Cost Functions 107 A. 4 Locus of Suboptimal P o i n t s 110 ACKNOWLEDGEMENTS The author would l i k e to express his sincere gratitude to Dr. Karl V. Bury, who devoted considerable time and gave invaluable advice and guidance throughout a l l stages of the present work. The f i n a n c i a l support of Karadeni z Teknik Un?versitesi, Trabzon, Turkey, i s g r a t e f u l l y acknowledged. X NOMENCLATURE A : Design Parameter C : Cost : Cost of F a i l u r e C M : Material Cost Cp . : Operation Cost f" : P r o b a b i l i t y D i s t r i b u t i o n Function ^ I . M A A : Extreme Value Asymptote L- : Load M : Material Property ii : Number of Load Applications, Mission Time / 7 C " R / G ) ; Constant Cost Function fCR//*} : Indifference Function P : Price PCF-) : P r o b a b i l i t y of F a i l u r e r : Net Resources at Completion R : R e l i a b i l i t y p.' . : Certainty Equivalent R e l i a b i l i t y S : Material Strength •£ : Thickness 7" : Mission Length IX : U t i l i t y vV : Net Resources at Start $> : Exact Extreme Value D i s t r i b u t i o n r j - " * ' : Variance : Benefit /A, : Expected Value, Mean /*z ' Variance 2) : Safety Factor X /' L I S T OF DEFINITIONS B e n e f i t . : B e n e f i t i s t h e amount o f s a t i s f a c t i o n d e r i v e d by a d e c i s i o n m a k e r , r e s u l t i n g f r o m a s t a t e o f v a l u e s o f a v a r i a b l e (o r v a r i a b l e s ) , r e l a t i v e t o some s t a n d a r d amount o f s a t i s f a c t i o n . C o n s t a n t C o s t F u n c t i o n : A C o n s t a n t C o s t F u n c t i o n i s a t r a d e - o f f f u n c t i o n b e t w e e n r e l i a b i l i t y and m i s s i o n t i m e i n t e r m s o f c o s t . I t i s a l o c u s o f a l l c o m b i n a t i o n s o f R and n t h a t have t h e same g i v e n c o n s t a n t c o s t . I n d i f f e r e n c e F u n c t i o n : An I n d i f f e r e n c e F u n c t i o n i s a t r a d e -o f f f u n c t i o n between r e l i a b i l i t y and m i s s i o n t i m e i n t e r m s o f b e n e f i t . I t i s a l o c u s o f a l l c o m b i n a t i o n s o f R and n t h a t have t h e same g i v e n c o n s t a n t b e n e f i t . X / 1 1 M i s s i o n Time : M i s s i o n Time, n, o f t h e d e s i g n s p e c i m e n i s t h e t i m e l e n g t h , d u r i n g w h i c h s p e c i m e n i s s u b j e c t e d t o a p p l i e d l o a d s . n may be e x p r e s s e d i n t e r m s o f t i m e u n i t s on number o f a p p l i e d l o a d s . R e l i a b i l i t y : R e l i a b i l i t y i s t h e p r o b a b i l i t y o f s u r v i v a l o f d e s i g n s p e c i m e n s by t h e end o f t h e m i s s i o n t i m e . I t i s c o m p l i m e n t a r y t o t h e p r o b a b i l i t y o f f a i l u r e . 1 1A. INTRODUCTION 1. General In a design process, the physical d e t a i l s of a specimen are determined such that a given function i s performed by the specimen under given conditions. The inputs to a design process are a) the relevant material property of the material, b) the load which i s applied to design specimen and c) the chosen f a i l u r e c r i t e r i o n . Under the conventional design approach the inputs to the design process are taken as fixed values. A single representative value L 1 i s chosen for the load. The handbook value of the material property, M', i s used to represent the material property. A safety factor v i s introduced to take into account unforeseen contingencies. The specimen designed i s meant to sustain a l l future loads, although even designs with a very high safety factor v do f a i l o ccasionally. F a i l u r e of the design means that the load value L was larger than the relevant strength value on the occasion the f a i l u r e . The conventional design procedure of a specimen i s based on the r e l a t i o n . S = vL (1.1) Where S = strength of the material L = load value, v = safety factor. 2 S i s a function of the material property M, and relevant design dimension, A . A p a r t i c u l a r value of v i s selected on the basis of engineering experience, judgement and knowledge of s i m i l a r designs. By choosing a s u f f i c i e n t l y high safety factor, the p r o b a b i l i t y of f a i l u r e i s assumed to be eliminated. However, i n practice even the most conservative designs do f a i l . Furthermore, the safety factor does not predict the performance of the design specimen; that i s , the safety factor does not provide any information as to the l i k e l i h o o d of f a i l u r e and the l i k e l y time length the design would operate p r i o r to f a i l u r e . 2. Load L, and Material Property M If several s i m i l a r specimens of a p a r t i c u l a r material are tested for a s p e c i f i c material property, i t i s l i k e l y that d i f f e r e n t property values are obtained. Material properties of a given material vary from one production l o t to another, and within a p a r t i c u l a r production l o t . The handbook value i s usually a central value that represents the whole range of values. Material properties of a sample of specimens are randomly d i s t r i b u t e d . That i s material property i s a random variable among specimens. If measurements are taken of the load on s i m i l a r design specimens under given operating conditions, the r e s u l t s are l i k e l y to f a l l i n a wide range of values. A histogram of applied loads on a p a r t i c u l a r specimen often shows that load values occur randomlv, with larger load 3 values o c c u r i n g l e s s f r e q u e n t l y than smaller load v a l u e s . But these l a r g e load values are the ones t h a t cause f a i l u r e and are t h e r e f o r e of importance. As both the m a t e r i a l property M and load L should be recognized as random v a r i a b l e s , due t o inherent v a r i a b -i l i t i e s , these v a r i a b l e s are represented by some mathematical f u n c t i o n , c a l l e d the p r o b a b i l i t y d e n s i t y f u n c t i o n , ( p . d . f . ) . ^ The p r o b a b i l i t y d e n s i t y f u n c t i o n , f ( x ; 0 ) represents the magnitude of x i n terms of i t s frequency of occurence, where (2) G i s the parameter value of the s t a t i s t i c a l model Each property of each s p e c i f i c m a t e r i a l gives r i s e to a s p e c i f i c f a m i l y of d e n s i t y f u n c t i o n s . The c o n t r o l over the production process, ( c o n t r o l of temperature, homogenity, and percentage of a l l o y i n g elements,) etc . ) determines the observed v a r i a t i o n i n the m a t e r i a l property and thus d e t e r -mines the f u n c t i o n f and i t s paramet er v a l u e , 9. For load v a l u e s , the nature of the p.d.f. and parameter value 0 are determined by the o p e r a t i n g c o n d i t i o n s imposed on the specimen. The handbook value of a m a t e r i a l property M, i s u s u a l l y the mean value of the a s s o c i a t e d p.d.f. Data on m a t e r i a l property show th a t the p.d.f. of M i s u s u a l l y skewed to the r i g h t , s i n c e q u a l i t y c o n t r o l r e s u l t s i n e l i m i n a t i o n of variance i n low values of M, w h i l e high value of M o f t e n go unchecked. The lognormal, Gamma, and W e i b u l l d i s t r i b u t i o n s ( 3) are commonly used to represent the p.d.f. of M. A t y p i c a l p.d.f. f o r M i s shown i n F i g . 1.1. f(M) F i g . 1 . 1 . A Typ ica l P r o b a b i l i t y Density .Function f o r Material Property M. 5 Most load values occur around a nominal value L ' , which i s used f o r conventional design procedures whi le r e l a t i v e l y high load values occur less f requent ly . Hence the p.d. f . of load L, i s usua l ly skewed to the r i g h t . A t y p i c a l p.d. f . f o r L i s shown in f i g . 1.2. Fa i l u re w i l l occur when a load value exceeds the r e s i s t i n g strength of the design specimen. S ince the f a i l u r e occurs at the f i r s t exceedance o f S by L, the i n t e r e s t l i e s not in what port ion of load values exceeds S, but in the number of load app l i ca t i ons p r i o r to the f i r s t exceedance o f S. The p r o b a b i l i t y that the strength S w i l l not be exceeded in n consecut ive load app l i ca t i ons i s given by the exact extreme value p .d . f . of the i n i t i a l load d i s t r i b u t i o n , •{n) ( S ) = [ F ( L ) |L=s] n 0.2) To represent the p.d. f . of L, f ( L ) , Lognormal, Gamma and Weibull models are employed. The extreme value asymptote, F T • (L) i s con-s t ruc ted from the i n i t i a l model of L, f o r a given number of contemplated loads , n^K The extreme value p .d . f . F j (L ) i s shown in f i g . 1.3. The mission length , T, of the design specimen i s the time length , during which the specimen is subjected to app l ied loads. T i s expressed in time un i t s . For many design specimens, the number of loads app l ied per un i t time i s a constant. I f An i s the number of loads app l ied per un i t time, n = T« An (1-3) 7 Hence n i s the number O-F a p p l i e d loads to which the design i s subjected during i t s m i s s i o n time. In the r e s t of the d i s c u s s i o n , n i s used to represent the m i s s i o n time and i t may be e a s i l y converted to time u n i t s by d i v i d i n g by An. 3. P r o b a b i l i s t i c Design When the random behavior of a r e l e v a n t m a t e r i a l property M and of the load L ; are recognized, the con v e n t i o n a l approach of designing w i t h d e t e r m i n i s t i c values of M' and L' i s r eplaced by designing w i t h the p.d.f's of the random v a r i a b l e s M and L, The r e s i s t i n g s t r e n g t h of the m a t e r i a l , F i s a f u n c t i o n of M and a c r i t i c a l design dimension, A. In the case of a simple t e n s i o n l i n k f o r which the f a i l u r e c r i t e r i o n .is rupture . , Where A i s the area of the c r i t i c a l c r o s s e c t i o n . F a i l u r e occurs when S i s exceeded by the l o a d : L>S The p r o b a b i l i t y of f a i l u r e , d u r i n g the mis s i o n time n i s The r e l i a b i l i t y of the design specimen i s de f i n e d as the complimentary p r o b a b i l i t y . S=MA, (1.4) P(n) = P(L>S) (1.5) R(n) = 1-P(L>S) (1.6) 8 The a c t u a l dimensions of a design element vary among specimens.. Q u a l i t y c o n t r o l r e s u l t s i n v a r i a t i o n s being between given l i m i t s on the average. But as long as each element i s not inspected, the design dimension, A, i s an unknown and must be considered a random v a r i a b l e . K ' Hence S i s a product of two random v a r i a b l e s and i s t h e r e f o r e i t s e l f a random v a r i a b l e . The p.d.f. of the r.v.S i s shown i n f i g . 1.3. Under the p r o b a b i l i s t i c design approach, the r e l i a b i l i t y R and mis s i o n time n are chosen by the d e c i s i o n maker. Using ecj. (1.6)^ R(n) = 1-P(L>S), the dimension of the design element j s c a l c u l a t e d , see r e f (2), The s a f e t y f a c t o r v, v=S/L } (1.7) i s the q u o t i e n t of two random v a r i a b l e s , and i s t h e r e f o r e i t s e l f a random v a r i a b l e . The r e l i a b i l i t y , R(n), may be d e f i n e d i n terms of V; f a i l u r e occurs when v<l, and hence the p r o b a b i l i t v t h a t f a i l u r e occurs d u r i n g a mis s i o n time n i s : P(n) = P(v<l) , R(n) = 1-P(v<l). (1.8) In f i g . 1.3. a t y p i c a l p.d.f. of L, f ( L ) i s shown. A corresponding extreme value model f (L)=F(f(L),n j d e r i v e d from the i n i t i a l model f ( L ) i s a l s o i l l u s t r a t e d . A s t r e n g t h p.d.f. f(S) i s presented i n f i g . 1.3, and the F i g . 1.3. P.d.f. of Load, Extreme Value of Load^and Material Strength. 10 p r o b a b i l i t y of f a i l u r e i s i nd i ca ted q u a l i t a t i v e l y . An example of obta in ing p . d . f ! s f o r M and L and c a l c u l a t i n g R(n) from these p . d . f . ' s i s i l l u s t r a t e d in Appendix I , w i th in the framework of i l l u s t r a t i n g the dec i s ion ana l y s i s . 11 I. B. LITERATURE SURVEY A study publ ished by Freudenthal v ' in 1947 introduced the random behavior of material strength and load, and the concept of r e l i a b i l i t y to the design process. This work was fol lowed by numerous d i scuss ions and s tudies in the l i t e r a t u r e by Mittenbergs, ( 6 ) ' ( 7 ) > Weibull ^ , and many others . An extensive survey of the l i t e r a t u r e per ta in ing to random behavior of materia l s trength and the concept of p r o b a b i l i s t i c (a) design i s presented by Agrawal v ; in his t he s i s . On the opt imizat ion of the p r o b a b i l i s t i c design process, that i s on searching fo r optimal values of the design parameter, r e l i a b i l i t y . R and mission time n, however, there i s very l i t t l e a v a i l a b l e l i t e r a t u r e . F r e u n d ^ ^ introduces r i s k in to a programming model. U t i l t y i s expressed as where u ( R ) = l - e " a R R = net revenue of p ro jec t and a = constant cost f a c t o r . Freund shows that i f R is normally d i s t r i b u t e d , u t i l i t y i s o p t i -mized by maximizing E(R) - aa 2/2, where E(R) i s the expected value of R 2 and o i s i t s var iance. This approach may prove to be useful when u t i l i t y can be expressed as above. But the condition that net revenue, R, i s to be of normal d i s t r i b u t i o n makes the approach very unpractical, since i t i s very u n l i k e l y that R, which i s a function of non-normally d i s t r i b u t e d variables, w i l l be a normally d i s t r i b u t e d variable, Weisman & Holzman argue that the u t i l i t y function can be considered concave everywhere and using Freund's function, they define r= W +P-C, where r = net resources at completion W = net resources at s t a r t P = Price or Revenue C = Cost . Hence the u t i l i t y function becomes / \ i -ar y (r) = 1-e Considering p r i c e as deterministic t , . , -a(w+P) f ° V C f (c) 40 . • y(r) = 1-e •»-•• • They show that i f the cost model i s unimodal and symmetric, the u t i l i t y i s maximized by minimizing E(c) +aa2/2 When the cost function i s not unimodai they give an upper bound for the objective, function. (12) Singh and Kumar discuss system r e l i a b i l i t y with pay-offs. Pay-offs are introduced to weigh the r i s k . The formulation given takes into account systems of two or more component's. A loss matrix i s assumed and used, but no study of getting the loss matrix i s indicated. In the f i e l d of optimization there i s a number of (13) excellent studies i n the l i t e r a t u r e . F:ishburn v ' gives a general theory of subjective p r o b a b i l i t i e s and expected (14) u t i l i t i e s . Debrew* presents a representation of (15) preference ordering by a numerical function. Rader ^ discusses the existence of a u t i l i t y function to represent preferences. .Suppes 'has an excellent study on the r o l e of subjective p r o b a b i l i t y and u t i l i t y function i n decision making. U t i l i t y functions for multi-attributed consequences and independence of these functions i s presented i n a r t i c l e (17 18\ by Keeney V J-'' X A methodical approach of estimating (19) additive u t i l i t i e s i s presented bv F.ishburn The s t a t i s t i c a l d ecision models are presented (20 to 22) thoroughly by S c h l a i f f e r ' and by Pratt, Raifa, (23 24) and S c h l a i f f e r • '.' ;. Other works i n the area include (25) (26) n (27) those by Weiss , DeGroot ' •, and Myron 14 I. C. PRESENT RESEARCH PROBLEM In p r o b a b i l i s t i c d esign, the designer can choose the r e l i a b i l i t y R and mission time n. He has a wide range of choice f o r both input parameters,.and the values of R and n determine the value of the dimensional parameter A. The purpose of t h i s study i s to c o n s t r u c t a r a t i o n a l d e c i s i o n process t h a t provides the most optimal combination of R and n f o r the d e c i s i o n maker. A r a t i o n a l d e c i s i o n maker would want to maximize the b e n e f i t he d e r i v e s from the design, w h i l e minimizing the cost of the design. So the problem i s de f i n e d as Minimize the c o s t - b e n e f i t r a t i o of the design subject to Minimum r e l i a b i l i t y , Maximum r e l i a b i l i t y , Minimum m i s s i o n time, I t i s t h e r e f o r e seen t h a t i n the problem as s t a t e d , the design parameters, R and n, become the r e l e v a n t d e c i s i o n v a r i a b l e s . I I . BENEFIT I I A . DEFINITION OF BENEFIT. 1. D e f i n i t i o n B e n e f i t i s t h e amount o f s a t i s f a c t i o n d e r i v e d by a d e c i s i o n maker, r e s u l t i n g f r o m a s t a t e o f v a l u e s o f a v a r i a b l e ( o r v a r i a b l e s ) , r e l a t i v e t o some s t a n d a r d amount o f s a t i s f a c t i o n . S a t i s f a c t i o n r e s u l t s f r o m a number o f f a c t o r s a f f e c t e d by t h e v a r i a b l e c o n s i d e r e d ; t h e s e f a c t o r s a r e d i s c u s s e d i n s e c t i o n I I A 2 . B e n e f i t i s an o v e r a l l t e r m t h a t i n c o r p o r a t e s t h e s a t i s f a c t i o n o f o u t p u t v a l u e s w i t h o u t any r e f e r e n c e t o i n p u t v a l u e s . I n a d e s i g n p r o c e s s , b e n e f i t i s t h e s a t i s f a c t i o n d e r i v e d f r o m t h e s t a t e o f v a l u e s o f m i s s i o n t i m e and o f t h e d e s i g n w i t h o u t r e f e r e n c e t o t h e c o s t o f d e s i g n and o t h e r i n p u t s t o t h e d e s i g n p r o c e s s . C o s t f a c t o r s a r e c o n s i d e r e d s e p a r a t e l y i n c h a p t e r I I I . B e n e f i t i s a r e l a t i v e t e r m . Some f a c t o r s o f b e n e f i t c a n be e x p r e s s e d i n a b s o l u t e t e r m s . F o r example t o t a l s a l e s r e v e n u e c a n be e x p r e s s e d i n t e r m s . o f m o n e t a r y u n i t s . B e n e f i t as a w h o l e composed o f a l l t h e f a c t o r s d i s c u s s e d i n s e c t i o n I I A 2 , c a n n o t be e x p r e s s e d i n a b s o l u t e t e r m s . F e n c e , b e n e f i t i s r e l a t i v e t o some norm and e x p r e s s e d as a r a t i o . The norm, o r s t a n d a r d , i s u s u a l l y an, e a s i l y d e f i n e d s t a t e o f o u t p u t f a c t o r s . B e n e f i t d e r i v e d from a s t a t e i s s u b j e c t i v e . That i s , i t i s d i f f e r e n t f o r di.fferent d e c i s i o n makers and at d i f f e r e n t s t a t e s of out s i d e f a c t o r s . I t i s n o t n e c e s s a r i l y the same f o r a l l d e c i s i o n makers but general trends can be analyzed. Sections IIB and IIC analyse these general trends f o r b e n e f i t d e r i v e d from design r e l i a b i l i t y and design mi s s i o n time. 2. C o n s i d e r a t i o n s i n Assessing B e n e f i t A d e c i s i o n maker, i n ass e s s i n g the b e n e f i t d e r i v e d from a c e r t a i n s t a t e of an output, must take the ..factors a f f e c t e d by t h a t output i n t o c o n s i d e r a t i o n . The s a t i s f a c t i o n d e r i v e d from the output depends on the r e l a t i v e e f f e c t of the output on these f a c t o r s and r e l a t i v e s t a t e s and importance of these f a c t o r s . The c o n s i d e r a t i o n s i n ass e s s i n g the b e n e f i t d e r i v e d from a p r o b a b i l i s t i c design process i n which the outputs are r e l i a b i l i t y R and m i s s i o n time n can be l i s t e d as f o l l o w s : a) Sales revenue. Increased r e l i a b i l i t y and/or mis s i o n time may a l l o w an increase i n s a l e s p r i c e or s a l e s volume. The inc r e a s e i n s a l e s revenue r e s u l t i n g from the a b i l i t y t o ask higher p r i c e s or the a b i l i t y to s e l l a l a r g e r volume i s o b v i o u s l y of b e n e f i t to the d e c i s i o n maker and h i s o r g a n i z a t i o n . 17 b) Reputation The r e p u t a t i o n of the d e c i s i o n maker and h i s o r g a n i z a t i o n i s a f f e c t e d by the r e l i a b i l i t y and the mis s i o n time of the design. The higher these values are, the higher i s the r e p u t a t i o n f o r the d e c i s i o n maker. This i m p l i e s present and f u t u r e b e n e f i t s f o r the d e c i s i o n maker., such as stronger market p o s i t i o n , higher p r i c e s , and favourable consumer b i a s . c) O b j e c t i v e s The o b j e c t i v e s of the d e c i s i o n maker are a f f e c t e d by the output v a l u e s . A longer m i s s i o n time i s of more b e n e f i t to a d e c i s i o n maker who wants the design perform f o r a long time, w h i l e i t i s of l e s s b e n e f i t to a d e c i s i o n maker who wants the design r e -placed i n the near f u t u r e . d) "Marketing C o n s i d e r a t i o n s The marketing s t r a t e g y of the d e c i s i o n maker should be considered i n a s s e s s i n g the benefit-. -The market, i n whjch the d e c i s i o n maker d e s i r e s to operate ( f o r example , high q u a l i t y , intermediate q u a l i t y , or low q u a l i t y markets,) fhe d e s i r e d market share, s i d e product markets, the 18 m a i n t a n e n c e m a r k e t , a l l a f f e c t t h e m a r k e t i n g s t r a t e g y . The b e n e f i t d e p e n d s o n t h e m a r k e t i n g s t r a t e g y f o r m u l a t e d b y t h e d e c i s i o n m a ker e) O b s o l e s c e n c e D e v e l o p m e n t i n t e c h n o l o g y , c h a n g e s i n c u s t o m e r t a s t e s , a n d o t h e r o u t s i d e f a c t o r s may b r i n g t h e d e s i g n i n t o o b s o l e s c e n c e , t h e s e f a c t o r s s h o u l d be c o n s i d e r e d i n a s s e s s i n g t h e b e n e f i t . A d e s i g n d e s i g n a t e d t o o p e r a t e a f t e r i t i s o b s o l e t e , i s o v e r - d e s i g n e d . I n c o n s i d e r i n g t h e a b o v e f a c t o r s t o a s s e s s t h e r e l a t i v e b e n e f i t s , t h e d e c i s i o n m a ker makes u s e o f a l l p e r t i n e n t i n f o r m a t i o n a n d u t i l i z e s h i s e x p e r i e n c e . How t h e s e c o n s i d e r a t i o n s a r e u t i l i z e d when d e t e r m i n i n g a d e c i s i o n maker*s b e n e f i t i s i l l u s t r a t e d i n S e c t i o n s I I B and I I C . 3. P r e f e r e n c e f u n c t i o n s A d e c i s i o n m a k e r w i l l p r e f e r a s t a t e o f o u t p u t v a r i a b l e w i t h a h i g h e r b e n e f i t t o a s t a t e o f o u t p u t v a r i a b l e s w i t h a l o w e r b e n e f i t . I n t h e c a s e w h e r e b e n e f i t i n c r e a s e s a s t h e v a l u e o f a n o u t p u t v a r i a b l e i n c r e a s e s w i t h i n a n i n t e r v a l o f v a l u e s f o r t h e v a r i a b l e , i n t h a t i n t e r v a l t h e d e c i s i o n maker w i l l p r e f e r a h i g h e r v a l u e o f t h e v a r i a b l e t o a l o w e r v a l u e o f t h e v a r i a b l e . B e n e f i t i s a f u n c t i o n o f t h e o u t p u t v a r i a b l e and t h i s f u n c t i o n i s c a l l e d a preference f u n c t i o n . In p r o b a b i l i s t i c design, the output i s the r e l i a b i l i t y and mission time, and b e n e f i t i s a f u n c t i o n of both, (see S e c t i o n s I I . B . and II.C.) B e n e f i t as a f u n c t i o n of r e l i a b i l i t y , when mis s i o n time i s f i x e d , forms the preference f u n c t i o n f o r r e l i a b i l i t y at the given m i s s i o n time. B e n e f i t as a f u n c t i o n of m i s s i o n time at a f i x e d r e l i a b i l i t y v a l u e , forms the preference f u n c t i o n f o r m i s s i o n time at the given r e l i a b i l i t y v a l u e . 20 11. B. B e n e f i t of M i s s i o n Time a t a Given Value of R e l i a b i l i t y 1. General The p r e f e r e n c e f u n c t i o n f o r m i s s i o n time of a co n s t a n t r e l i a b i l i t y , y ( n | R ) , i s analyzed i n the i n t e r v a l between minimum and maximum m i s s i o n time v a l u e s . Minimum m i s s i o n time, n . , i s d e f i n e d as the lowest b e n e f i c i a l v a l u e ' mm assessed by the d e c i s i o n maker from e x p e r i e n c e and p e r t i n e n t i n f o r m a t i o n . Maximum m i s s i o n time, n i s d e f i n e d ~ a s t h a t v a l u e a f t e r which an i n c r e a s e i n m i s s i o n time does not r e s u l t i n an i n c r e a s e i n the b e n e f i t o r r e s u l t s i n a decrease i n the b e n e f i t . In the i n t e r v a l between minimum and maximum m i s s i o n times, ( n m i n ' n m a x ^ ' t ^ i e P r e f e r e n c e f u n c t i o n f o r m i s s i o n time a t g i v e n r e l i a b i l i t y , y (n |R) , i s a non-decreasing f u n c t i o n . B e n e f i t i n c r e a s e s as the m i s s i o n time i n c r e a s e s , and maximum m i s s i o n time, by d e f i n i t i o n , i s the p o i n t where b e n e f i t i s maximum. For d i s c u s s i o n o f the above statement, see S e c t i o n II.B;2. As m i s s i o n time i n c r e a s e s the ma r g i n a l i n c r e a s e i n the p r e f e r e n c e f u n c t i o n , y ( n ) , f o r u n i t i n c r e a s e i n m i s s i o n time d e c l i n e s . That i s , t h e m a r g i n a l b e n e f i t d e r i v e d by the d e c i s i o n maker d e c l i n e s as m i s s i o n time i n c r e a s e s . The p r e f e r e n c e f u n c t i o n has a n o n - i n c r e a s i n g s l o p e r e a c h i n g zero a t maximum m i s s i o n time. The above statements are discussed and i l l u s t r a t e d w i t h an example i n the follov/ing sections. The purpose of the d e c i s i o n process i s to minimize the c o s t - b e n e f i t r a t i o , and f o r t h i s reason mission time values l a r g e r than maximum mission time, n , which ' max' r e s u l t i n increased cost but decreased b e n e f i t are not considered i n the a n a l y s i s . 22 2. Non-decreasing Preference F u n c t i o n , As s t a t e d i n the previous s e c t i o n , the preference i f u n c t i o n f o r m i s s i o n time at given r e l i a b i l i t y , V ( n i s non-decreasing. This i s due to f o l l o w i n g factor;-; a) The longer i s the m i s s i o n time, the higher i s the p r i c e t h a t can be charged and>the l a r g e r i s the revenue from the design. In the case the d e c i s i o n maker i s the designer-user of the d e s i g n , longer usage of the design i m p l i e s higher revenue per specimen, si n c e t h e r e are fewer replacements of the design. In the case the decisionmaker i s the designer-s e l l e r , l a r g e r p r i c e s and/or more s a l e s r e s u l t i n higher revenue. Since an i n c r e a s e occurs i n revenue t h i s i s an increase i n the b e n e f i t . For example, i f the design i s a machine t o o l , longer usage or higher p r i c e s i n c r e a s e s the revenue of the d e c i s i o n maker. A machine t o o l used by the d e c i s i o n maker i s i n use a longer time i f the m i s s i o n time i s longer and one s o l d by the d e c i s i o n maker y i e l d s higher p r i c e s and/or l a r g e r s a l e s . 23 b) A design with a longer mission time has a higher reputat ion value f o r the dec i s ion maker. It enjoys the reputat ion of d u r a b i l i t y . This i s of bene f i t to the dec i s ion maker, because i t may prompt buyers to regard a l l designs by the dec i s ion maker as durab le, i nc lud ing the design in quest ion. For example, a dec i s ion maker who designs a machine tool with a longer mission time has the reputat ion of durable design and th i s w i l l be of bene f i t in present and i n fu ture s a l e s . New designs by that dec i s i on maker w i l l be ea s i e r to introduce to the market, and are l i k e l y able to command higher p r i c e s . c) Market share f o r the design may increase as i t i s l o n g e r - l a s t i n g . Among the designs f o r the same purpose, one with a longer miss ion time may be p re fe r red by the buyer and the market share may inc rease , g i v ing the dec i s i on maker the opportunity to expand and/or operate at more b e n e f i c i a l p r i c e l e v e l s . This i s an increase in the bene f i t f o r the dec i s ion maker. For example, a machine tool with a higher mission time w i l l be chosen more of ten by the buyers and i t s share in the market i s l i k e l y 24 to increase. This increase in market share w i l l enable the dec i s ion maker to cons ider expanding i t s production l i n e , and enable him to increase his volume to a more optimal value. These factors cause the bene f i t to increase as the miss ion time increases , but t h e i r e f f ec t s are not constant. That i s , b e n e f i t does not increase at a constant rate as mission time increases . This i s discussed in the next sec t i on . 25 3. Non-increasing' Slope of Preference F u n c t i o n . As s t a t e d i n s e c t i o n J I B I as the m i s s i o n time i n c r e a s e s the marginal increase i n the preference f u n c t i o n f o r m i s s i o n time at given r e l i a b i l i t y , jin/R) , f o r u n i t i n c r e a s e i n mission time ft d e c l i n e s and approaches zero at the maximum mission time n m a x • That i s , i t has a non-increasing slope. The marginal increase i n b e n e f i t d e c l i n e s because of the f a c t o r s l i s t e d below'. a) Higher mission time values b r i n g the design c l o s e r to p o t e n t i a l obsolescence near the end of i t s l i f e . T h a t i s , the p r o b a b i l i t y t h a t the design w i l l be o b s e l e t e p r i o r t o completion of i t s m i s s i o n time i n c r e a s e s . The advancement of technology and change i n consumer-tastes and industry-needs and other o u t s i d e f a c t o r s cause designs to become ob s e l e t e . VThen a design becomes o b s e l e t e , the remaining l i f e time i s of l i t t l e v a lue. Buyers do not want to commit themselves t o a design t h a t has a high p r o b a b i l i t y of becoming obse l e t e before the completion of i t s m i s s i o n time. This f a c t o r negates the second f a c t o r i n S e c t i o n I I . B . 2 , t h a t longer m i s s i o n time r e s u l t s i n higher r e p u t a t i o n s i n c e o b s e l e t e designs w i l l r e s u l t i n reducing the p o s i t i v e 26 a f f e c t that longer l i f e time has on r e p u t a t i o n . In the case a machine t o o l i s the design considered, advancement i n technology (better design concepts), change i n i n d u s t r v needs ( s h i f t i n demand), and other expected and unexpected o u t s i d e f a c t o r s (such as energy shortage) may cause a machine t o o l to become obselete w i t h i n a c e r t a i n l i f e span (product l i f e ) . A machine t o o l b u i l t to l a s t longer has l e s s marginal b e n e f i t as the mis s i o n time i n c r e a s e s . b) As the mis s i o n time i n c r e a s e s the t o t a l market f o r the design d u r i n g some extended time p e r i o d s h r i n k s and the demand f o r new items d e c l i n e s because the items a l r e a d y i n use l a s t longer. When designs have a high m i s s i o n time, f a i l u r e occurs l e s s f r e q u e n t l y and the n e c c e s i t y of replacement decreases. Even though, as s t a t e d i n Factor C i n s e c t i o n I I B2., the market share f o r the design i n c r e a s e s , the t o t a l market s h r i n k s and the a c t u a l market volume f o r the design does not increase as much. Hence, the b e n i f i c i a l a f f e c t of i n c r e a s i n g market share decreases as the m i s s i o n time i n c r e a s e s . In the machine t o o l i n d u s t r y , f o r example, suppose t h a t a manufacturer designs f o r a higher mission time than others do. The market share increases, but since the items i n use l a s t longer, the t o t a l demand i n the long run decreases. Hence, the t o t a l volume does not increase as much as the increase i n the market share. Longer mission times and associated longer revenue may tend to price the design out of i t s market. The buyer, even i f i t may be more b e n e f i c i a l in the long run, may not be w i l l i n g or able to give the higher price for the design with the higher mission time. Hence, as mission time.'increases and pri c e increases i n accordance, the buyer group of the design gets smaller. Although the revenue for each design increases, as les s items are sold, the t o t a l revenue w i l l not increase as much, and i f mission time i s very high t o t a l revenue may, i n f a c t , d e c l ine. This factor lessens the effect of the factor A i n Section IIB2, that the higher mission time r e s u l t s i n higher revenue for the decision maker. In the machine t o o l industry, for example, the designs with higher mission time may have smaller buyer groups. The decision maker designing for a higher mission time may not receive as much revenue as those designing for lower mission time. Even though the 28 u n i t revenue ( p r i c e per design specimen) i s higher, s a l e s volume may be lower, and may r e s u l t i n a lower t o t a l revenue. In c o n c l u s i o n , the preference f u n c t i o n f o r mis s i o n time at constant r e l i a b i l i t y , y (n |R) , i s a non-decreasing f u n c t i o n w i t h a non-increasing slope. When the slope reache zero, the corresponding m i s s i o n time i s defined as the maximum mis s i o n time, n . The f u n c t i o n y ( n | R ) i s as shown ' max * i n Fig.2.1 i n the i n t e r v a l between minimum and maximum mission time v a l u e s , ( n m j _ n ' n m a x ) • 29 Benefit /i(n,/R) y / / / / Mission Time, n Dmln. Dinew. F i g . 2.1. The Preference Function for Mission Time at Constant R e l i a b i l i t y R', u(n|p.') 30 I I . C. B e n e f i t of R e l i a b i l i t y at a Given Value of M i s s i o n Time 1. General The preference f u n c t i o n f o r r e l i a b i l i t y at a given value of mis s i o n time, y(R|n). /is a non-decreasing f u n c t i o n i n the range between minimum and maximum r e l i a b i l i t y , (R . R ). Minimum r e l i a b i l i t y , R . , i s assessed by the min/ max ' mm' •* d e c i s i o n maker as the lowest value acceptable i n the i n d u s t r y . A r e l i a b i l i t y value of one i s not a t t a i n a b l e i n p r a c t i c e . Maximum r e l i a b i l i t y i s de f i n e d as t h a t value ( l e s s than one) t h a t i s t e c h n o l o g i c a l l y and/or economically a t t a i n a b l e . As the'value of r e l i a b i l i t y i n c r e a s e s , the b e n e f i t t o the d e c i s i o n maker i n c r e a s e s . In the subsequent a n a l v s i s , the r e l i a b i l i t y values between minimum and maximum r e l i a -b i l i t y are considered o n l y , s i n c e the values below the former are not acceptable, and values above the l a t t e r are not a t t a i n a b l e . As the r e l i a b i l i t y i n c r e a s e s , the marginal i n c r e a s e i n the preference f u n c t i o n f o r r e l i a b i l i t y a t constant mission time, u(R|n) y f o r a u n i t i n c r e a s e i n r e l i a b i l i t y d e c l i n e s . Hence, u(R|n) f e a t u r e s a non-increasing slope. Since the maximum r e l i a b i l i t y R i s d e f i n e d as the l a r g e s t max r e l i a b i l i t y value which i s t e c h n o l o g i c a l l y and/or econom-i c a l l y f e a s i b l e , the slope of the preference f u n c t i o n f o r constant mission time, must be zero at R , sin c e no f u r t h e r ' max' input of t e c h n o l o g i c a l and other resources could p o s s i b l y 31 r e s u l t in a r e l i a b i l i t y increase. Therefore , the preference funct ion f o r r e l i a b i l i t y at a constant mission t ime, y(R|n) i s a concave funct ion in the range between minimum and maximum r e l i a b i l i t y . The reason f o r the above statements are presented in Sect ion II.C.2 and I I.C.3. 3 2 2. Non-decreasing Preference Function. The preference function for r e l i a b i l i t y at a given mission time, y(R|n) j derived by the decision maker i s a non-decreasing function. That i s , the higher the r e l i a -b i l i t y i s , larger i s the benefit the design provides to the decision maker. The following factors comprise the reasoning behind the above statement. a) The higher the p r o b a b i l i t y that a design does not f a i l before the completion of the mission time, the higher i s the revenue derived from the design. Suppose there e x i s t two samples of, 1 0 0 design items with r e l i a b i l i t y 1-F • and 1 - 2 F ' , respectively. Before the comp-l e t i o n of mission time, on the average F' items f a i l i n the former case while 2F 1 ' items f a i l i n the l a t t e r case. In the case of items with higher r e l i a b i l i t y , ( 1-F'), the designer-user has more items i n operation and needs fewer replacements, so that he derives higher revenue than with the items with lov/er r e l i a b i l i t y , ( 1 - 2 F 1 ) . A designer-s e l l e r i s able to ask higher prices (revenue per item) for the items with higher r e l i a -b i l i t y , due to obvious higher revenue for the 33 user, than he i s able to ask for the lower r e l i a b i l i t y items. For example, i f the design item i s a machine t o o l , the higher r e l i a b i l i t y items w i l l return higher revenues through higher prices or through a higher percentage completing the mission time r e s u l t i n g i n fewer replacements and/or re p a i r s . The revenue of the decision maker w i l l increase as the r e l i a b i l i t y of the item increases. b) An increase i n the r e l i a b i l i t y of the design r e s u l t s i n a higher market share for the design item. A buyer t y p i c a l l y prefers a more r e l -i a b l e item to one with lower r e l i a b i l i t y at the same mission time. A l l other things being equal, a higher r e l i a b i l i t y item s e l l s better and therefore obtains a higher share of the market. This trend may reverse a f t e r a c e r t a i n value of r e l i a b i l i t y . This point i s discussed i n Section II.C.3. For example, a machine t o o l with a higher r e l i a b i l i t y w i l l be chosen more often by buyers than a machine t o o l with a lower r e l i a b i l i t y . A customer w i l l be i n c l i n e d to prefer a machine tool that has a lower prob-a b i l i t y to faili[higher r e l i a b i l i t y ) i n i t s given mission time to one that has a higher p r o b a b i l i t y to f a i l (lower r e l i a b i l i t y ) . c) A design with higher r e l i a b i l i t y has a safer operation, and for c r i t i c a l designs, t h i s feature i s of considerable benefit, An increase in r e l i a b i l i t y decreases the p r o b a b i l i t y of an undesired shutdown, work accidents, and unplanned and/or unwanted delays. An increase i n r e l i a b i l i t y makes the operation of design items safer and more dependable, and so i n -creases the benefit to a decision maker. A safer operation r e s u l t s from a machine with high r e l i a b i l i t y , as an example. Since there i s a lower p r o b a b i l i t y of f a i l u r e during i t s mission time, the machine t o o l with higher r e l i a b i l i t y has lower expected loss i n revenue r e s u l t i n g from delays and shutdowns. The above statements demonstrate that an increase i n r e l i a b i l i t y causes the benefit of the design to decision maker to increase. Since there i s an increase i n benefit with an increase i n r e l i a b i l i t y , at a given mission time, a higher r e l i a b i l i t y design i s preferred to a lower re-l i a b i l i t y item by the decision maker. Hence, the preference function of r e l i a b i l i t y at a constant mission time i s a non-decreasing function. The increase i n r e l i a b i l i t y does not imply an increase in benefit at a constant rate. As the r e l i a b i l i t y approaches i t s maximum value, the increase i n benefit declines. This point i s discussed i n Section II %C.3. 35 3. Non- increas ing Slope of Preference Function With increas ing r e l i a b i l i t y va lues , the marginal increase in preference funct ion f o r r e l i a b i l i t y at a given mission t ime, y(R|n), per un i t increase in r e l i a b i l i t y R, decreases. That i s , y(R|n) has a non- increas ing s lope. The factors that cause a dec l i ne in the marginal increase of y(R|n) are discussed below. a) A higher r e l i a b i l i t y impl ies that more of the design items w i l l not have f a i l e d at the completion of the mission t ime. This r e su l t s in less frequent replacements and there fore a shrinkage f o r the demand of the design i tem. Although the design item with higher r e l i a b i l i t y may cause more buyers to p re fe r i t to a lower r e l i a b i l i t y item (causing the market share to increase f o r the h igher r e l i a b i l i t y i tem) , the to ta l market shr inks due to a decrease in demand caused by fewer f a i l u r e s . Hence the increase in revenue may not be as much as a n t i c i p a t e d otherwise. As the r e l i a b i l i t y gets c l o s e r to i t s maximum value the increase in market share i s o f f s e t to some extent by shrinkage in the market demand. 36 When a machine t o o l i s considered as an example, a higher r e l i a b i l i t y t o o l may capture a higher share of the market, but as the r e l i a b i l i t y i n c r e a s e s , the t o t a l market de-mand decreases, because there i s l e s s need to r e p l a c e the e x i s t i n g machine t o o l s . The decrease i n market demand causes the inc r e a s e i n revenue to d e c l i n e . b) A higher r e l i a b i l i t y v a l u e , w i t h higher p r i c e (revenue per item) may tend t o p r i c e the design item out of the market.• This may cause a d e c l i n e i n the market share, and an accompanying de-c l i n e i n the t o t a l revenue. I n c r e a s i n g r e l i a b i l i t y i n c r e a s e s the market share, but the p r i c e of the item a l s o i n c r e a s e s . When the p r i c e goes beyond a c e r t a i n l i m i t f o r each buyer, the buyer chooses a: lower r e -l i a b i l i t y item because the r e q u i r e d i n i t i a l c a p i t a l o u t l a y surpasses the advantages of higher r e l i a b i l i t y f o r the buyer. Hence, as the r e l i a b i l i t y becomes higher the r a t e of increase i n revenue d e c l i n e s . For. example, a buyer f o r a machine t o o l chooses a higher r e l i a b i l i t y t o o l , a l l other t h i n g s being equal. But, a f t e r the p r i c e of the design item goes beyond the amount the buyer can or i s w i l l i n g t o a l l o c a t e , he may 3 7 choose a lower r e l i a b i l i t y item.. 1'he l i m i t i n g amount d i f f e r s f o r each buyer, but the p r i c e increase^ as r e l i a b i l i t y becomes very h i g h j may cause the p r i c e to be beyond the amount which most buyers can a l l o c a t e . c) A d e c i s i o n maker may a l s o be i n t e r e s t e d i n spare p a r t s and maintanence markets f o r the design item. As the r e l i a b i l i t y gets higher, the d e c i s i o n makers volume i n these side markets w i l l decrease. Hence, as the r e l i a b i l i t y gets h i g h e r , the in c r e a s e i n revenue through more sa l e s i n the primary market i s negated by the decrease i n volume i n p a r t s and main-tanence market. For example, a machine t o o l designer i s able to o b t a i n l e s s revenue i n h i s spare p a r t s and maintanence e n t e r p r i s e as the r e l i a b i l i t y of the design item increases,. The l o s s i n b e n e f i t i n s i d e markets negates the in c r e a s e i n b e n e f i t by the i n c r e a s i n g r e l i a b i l i t y f o r the d e c i s i o n maker. d) Higher r e l i a b i l i t y t y p i c a l l y r e s u l t s i n b u l k i e r and heavier designs which i s not d e s i r a b l e f o r many designs.. Increase i n s i z e and weight mav go beyond the d e s i r a b l e l i m i t s f o r many d e c i s i o n makers a f t e r a c e r t a i n value of r e l i a b i l i t y . 38 e) Higher r e l i a b i l i t y impl ies that a l a r ge r number of design items complete the mission time without f a i l u r e . This increases the p r o b a b i l i t y that a design item w i l l become obsolete whi le s t i l l in use. For designs subject to rap id technolog ica l change, high r e l i a b i l i t y often impl ies overdesign. For such des igns, the marginal increase in bene f i t c l e a r l y dec l ines with increas ing r e l i a b i l i t y . As an example, a power tool ( e s p e c i a l l y a portab le one) with excess ive dimensions and/or weight may be impract ica l in usage and even though i t i s more r e l i a b l e may not be des i r ab le f o r many dec i s ion makers. In conc lus ion , the above fac tor s cause the marginal increase in benef i t per uni t increase in r e l i a b i l i t y to dec l i ne as the r e l i a b i l i t y increases . In Sect ion II. B.2., i t was seen that the preference funct ion for. r e l i a b i l i t y at a constant mission t ime, (RJn), i s a non-decreasing func t i on . In th i s sec t ion i t i s i nd i ca ted that the s lope of (RJn) i s non- increas ing . S ince maximum r e l i a b i l i t y & m x i s def ined as the l a r ges t r e l i a b i l i t y value which i s t e chno log i c a l l y and/or economical ly f e a s i b l e , the s lope of the preference funct ion f o r con-stant mission t ime, (RJn), must be zero at maximum r e l i a b i l i t y , s ince no fu r the r input of techno log ica l and other resource could poss ib ly r e su l t in an increase in r e l i a b i l i t y . 39 The f u n c t i o n u(R|n) i s t h e r e f o r e a non-decreasing f u n c t i o n , w i t h a non-increasing slope ,and the slope i s zero at the maximum value R In the i n t e r v a l between max * minimum and maximum r e l i a b i l i t y values (R . ,R ), the J mm' max ' preference f u n c t i o n f o r r e l i a b i l i t y a t a given m i s s i o n time, u(R|n), i s as shown i n Fig.2.2. 40 Benefit /i(n.(R) V / / / / / Rmin. Rmax. Reliability F i g . 2.2. The Preference Function for R e l i a b i l i t y at Constant Mission Time n', y(R|n') 41 I I . D. Benefit of Both R e l i a b i l i t y and Mission Time. 1. Interdependence of Preference Functions. As stated i n Sectionn.C., r e l i a b i l i t y i s a function of the mission time, R(n). In that section, a preference function i s developed for r e l i a b i l i t y at constant mission time, u(R|n) and i n Section II. B. a preference function for mission time at constant r e l i a b i l i t y ^ y ( n | R ) , i s developed. Since r e l i a b i l i t y and mission time are dependent on each other, (see Section llA.).the benefit of r e l i a b i l i t y i s only meaningful when i t i s defined at a given mission time and vice versa. The benefit r e s u l t i n g from r e l i a b i l i t y and mission time i s a three dimensional preference function, y(R,n). y(R,n)increases as r e l i a b i l i t y and/or mission time increases and the slope of y(R,n) decreases as re-l i a b i l i t y and/or mission time increases (see Sections II.B. and II.C.), thus forming a concave surface within the i n t e r v a l s (n . n ) and (R . R ). N mm? max min' max' In order to express benefit as a function of both r e l i a b i l i t y and mission time, indifference (constant benefit) functions are introduced i n Section I I . D. 2 . 42 I I . D. 2. B e n e f i t of R e l i a b i l i t y and M i s s i o n Time I n d i f f e r e n c e Function A d e c i s i o n maker i s expected to be i n d i f f e r e n t to two choices of r e l i a b i l i t y and mission time combinations which y i e l d the same b e n e f i t t o the d e c i s i o n maker..That i s , the d e c i s i o n maker i s i n d i f f e r e n t between (Rjn 1) and (Ryn") when y(R',n'-) i s equal toy(R','n"). s i n c e the concern of the d e c i s i o n maker i s to increase h i s b e n e f i t , two sets of the parameters r e l i a b i l i t y and mis s i o n time would be of same value t o him as long as the combined b e n e f i t de-r i v e d by the d e c i s i o n maker i s the same. An i n d i f f e r e n c e f u n c t i o n I(R,n) i s the:"locus of a l l s e t s (R,n) w i t h an equal value of b e n e f i t , y ( R , n ) . The i n -d i f f e r e n c e f u n c t i o n , I(R,n) i s t h e r e f o r e a t r a d e - o f f f u n c t i o n between the r e l i a b i l i t y and m i s s i o n time. When the preference f u n c t i o n f o r r e l i a b i l i t y a t constant mission time,y(R |n), and preference f u n c t i o n f o r mis s i o n time at constant r e l i a b i l i t y , y ( n |R), are both concave f u n c t i o n s (as i s the case i n t h i s a n a l y s i s , see Sections I I . B. and I I . C.), the i n d i f f e r e n c e f u n c t i o n s are convex f u n c t i o n s , as shown i n F i g . 2.3. The c o n s t r u c t i o n and a n a l y s i s of i n d i f f e r e n c e f u n c t i o n s are discussed i n S e c t i o n I I . E. 5. 43 Mission Time n mox. n min. Rmin. • Reliability Rmax. F i g . 2.3. Indifference Functions for R e l i a b i l i t y and Mission Time, I(R|y=cons.). 44 II. E. Der ivat ion of Preference Functions 1. Information Required to Construct Preference Functions In order to construct the preference func t i ons , ce r t a i n i n f o r -mation is required from the dec i s ion maker. The dec i s ion maker i s to assess the r e l a t i v e bene f i t he der ives from a ce r t a i n combination of r e l i a b i l i t y and mission time compared to c e r t a i n other combinations. This information is acquired by having the dec i s ion maker respond to several quest ions , and from a " re ference gamble," which is expla ined in fo l lowing sec t i ons . This informat ion i s u t i l i z e d to construct preference funct ions fo r r e l i a b i l i t y at several constant values of mission t ime. These func t i ons , then, are used to const ruct the i n d i f f e r e n c e func t i on s , introduced in II.D.3. 45 2 . Questions to D e c i s i o n Maker. The d e c i s i o n maker i s to respond to the f o l l o w i n g questions, i n order to assess h i s preference f u n c t i o n f o r r e l i a b i l i t y a t a given m i s s i o n time. The purpose of the questions i n t h i s s e c t i o n i s to l o c a t e the i n i t i a l and end po i n t s of the preference f u n c t i o n f o r r e l i a b i l i t y a t given mission time, y (R|n), f o r s e v e r a l d i f f e r e n t m ission time values n^ using a common s c a l e . That i s , the i n f o r m a t i o n obtained here i s u t i l i z e d t o e s t a b l i s h the r e l a t i v e b e n e f i t s d e r i v e d by the d e c i s i o n maker f o r the design at minimum r e l i a b i l i t y , R . . and maximum r e l i a b i l i t y , R . at mission 1 ' mm' J ' max f time values n. i " As s t a t e d i n S e c t i o n II.B..& I I . C . , values f o r the minimum mis s i o n time n . i minimum r e l i a b i l i t y R„ • > m m ' -1 m m v and maximum r e l i a b i l i t y R ,are de f i n e d and t h e r e f o r e •* max,3 taken as given. a j The f i r s t q uestion r e l a t e s the b e n e f i t d e r i v e d from minimum and maximum r e l i a b i l i t y values at minimum mis s i o n time. The i n f o r m a t i o n i s used t o l o c a t e the i n i t i a l and end p o i n t s of the preference f u n c t i o n f o r r e l i a b i l i t y at minimum m i s s i o n time, y(R|n . ). These p o i n t s are a l s o min used as the b a s i s from which to determine the i n i t i a l and end p o i n t s of a l l other preference f u n c t i o n , y (R ln.i), by using i n f o r m a t i o n obtained_ from question 2 , i n t h i s s e c t i o n . Question 1. Given the minimum mis s i o n time n=n . , min' what would be the r a t i o of the b e n e f i t de-r i v e d a t maximum r e l i a b i l i t y R , to the J max' b e n e f i t d e r i v e d at minimum r e l i a b i l i t y R . ; •* min m.=y(R ,n . )/y(R . ,n . ) 1 K V max' min ' mm' mm' I f we l e t y(R . ,n . ) = y and since our concern i s K X mm' mm' Ko i n the r e l a t i v e b e n e f i t , l e t us assume some constant value f o r y Q. Therefore; y(R ,n . ) = m,y (2.1) H x max' min 1 o A value f o r y(R ,n . ), ( b e n e f i t a t maximum r e l i a -H v max' min'' b i l i t y and minimum mis s i o n time) i s acquired u t i l i z i n g the in f o r m a t i o n obtained from question 1., and assuming a constant f o r y . The i n i t i a l and end p o i n t s f o r the b e n e f i t r e l i a b i l i t y f u n c t i o n a t minimum mis s i o n time are l o c a t e d using the above i n f o r m a t i o n as shown i n Fig.2.4 (marked by c i r c l e s ) . b) The second question r e l a t e s the b e n e f i t s d e r i v e d at d i f f e r e n t values of mis s i o n time, n^, given the minimum r e l i a b i l i t y , R . . The r a t i o of b e n e f i t d e r i v e d a t these 2 ' min values of mis s i o n time, n^, given R m ^ n / t o t n e b e n e f i t d e r i v e d at the minimum r e l i a b i l i t y and minimum mis s i o n time, y(R . ,n . ) =y , l o c a t e s the i n i t i a l p o i n t s of the p r e f -H v min' min' p o ' c erence f u n c t i o n f o r r e l i a b i l i t y a t these values of mis s i o n time. 47 Benefit Po Reliability Rmax. F i g . 2.4. Location of I n i t i a l and End Points of Preference Function for R e l i a b i l i t y at Minimum Mission Time. (Note that the preference function i t s e l f -dashed l i n e - has not been obtained at t h i s point.) 48 The maximum mis s i o n time i s defined i n Se c t i o n I I . B . l as t h a t value of mis s i o n time, a f t e r which f u r t h e r i n c r e a s e s i n m i ssion time do hot r e s u l t i n f u r t h e r increases i n b e n e f i t . The minimum mission time i s defined i n the same s e c t i o n as the lowest u s e f u l value. In order to o b t a i n s e v e r a l values of mis s i o n time n^, denote the maximum mission time given by n 1 and d i v i d e (n 1 -n . ), ' J max max mm ' i . e . the d i f f e r e n c e between estimated maximum mi s s i o n time and given minimum mis s i o n time, by an i n t e g e r N to o b t a i n a valu e , say n , d e f i n e n. such t h a t ; n 1 n.=n . - f i - n ; •••.N i mm n ' / ' t ' Since an increase of n. from n, t o n ,, i n mi s s i o n 1 N /v.+ l time does not r e s u l t i n an inc r e a s e i n b e n e f i t , n^ i s by d e f i n i t i o n maximum mi s s i o n time, n . max Question 2. Given t h a t the r e l i a b i l i t y i s equal to the minimum r e l i a b i l i t y , R . , what would be the J ' mm' r a t i o of b e n e f i t d e r i v e d at n^ to b e n e f i t d e r i v e d at n . : mm u (R . ,n. . i = l , 2 , • • • ,N H v mm' x ' Ho ' ' ' ' Let the above r a t i o be donated by P^. Then i n i t i a l p o i n t s of preference f u n c t i o n s f o r r e l i a b i l i t y a t mission time n^ are y(R . ,n.) = P.u (2.2) K mm' i ' i K o o 49 Since a value was assumed fo r y Q these i n i t i a l points are now loca ted , using the information gained in Question 2, as shown in F i g . 2.5. (marked by squares). c) The t h i r d question seeks the r a t i o of the bene f i t der ived at d i f -ferent values o f the mission time n^, given the maximum r e l i a b i l i t y R „ , to the bene f i t der ived at the minimum mission time n . , and the max min maximum r e l i a b i l i t y R . The benef i t der ived at R , „ and n . , as max max min shown in Sect ion 11. E. 2b above, i s ^ (R m a x > n m - j n ) = m]VQ-The information thus obtained i s u t i l i z e d to loca te the end points of preference funct ions f o r r e l i a b i l i t y at n^. Question 3. Given that the r e l i a b i l i t y i s equal to the maximum r e l i a b i l i t y R m a „ , what i s the r a t i o of bene f i t der ived max at n.j to bene f i t der ived at n .. : Let the above r a t i o be denoted by Q^. Then end points o f the preference funct ions f o r r e l i a b i l i t y at mission time n^, are •^WV= Qi miv Since these points are a l so expressed in terms of y Q they are located on the same sca le as the i n i t i a l po in t s . U t i l i z i n g the information obtained in Question 3, the end points are located as shown in F i g . 2.6. (marked by c ros ses ) . 50 Benefit -Reliability Rmin. Rmax. F i g . 2.5. Location of I n i t i a l Points of Preference Function for R e l i a b i l i t y at n,. 51 t Q a ^ o Q4m| ^ J o (Q2m\Po Rmin. Rmax. Reliability F i g . 2.6. Location of I n i t i a l and End Points of Preference Function for R e l i a b i l i t y at Mission Time Values n^. (Note that the preference functions -dashed l i n e s - have not been obtained at t h i s point.) 3 . Reference Gamble In the previous section the information obtained en-abled the location of the i n i t i a l and end points for the preference functions for r e l i a b i l i t y at mission time values n^: u(R,n^). A reference gamble i s employed to locate a t h i r d point between i n i t i a l and end points. Having three points, a suitable concave curve i s f i t t e d using the general c r i t e r i a established i n Section II.C, see Section I I . E. 4. This section explains what a reference gamble i s as applied to the case i n hand. Suppose that at a given mission time n, the r e l i a b i l i t y of the design i s uncertain, Suppose also that there i s a pr o b a b i l i t y , q, that the r e l i a b i l i t y i s equal to tiie maximum value R , and that there i s a p r o b a b i l i t y (l-<3) , that the max' * J ' r e l i a b i l i t y i s equal to the minimum value R^. The decision maker derives a c e r t a i n benefit from the given mission time and minimum r e l i a b i l i t y , or from the given mission time and maximum r e l i a b i l i t y . Now, there i s not the ce r t a i n t y of minimum and maximum r e l i a b i l i t y , but a p r o b a b i l i t y 1-q or q of getting either one or the other. The benefit derived from t h i s uncertain s i t u a t i o n by the decision maker i s l i k e l y to • be higher than the benefit derived i f R . pertained for ^ mm c c e r t a i n and lower than the benefit derived i f R pertained max v for c e r t a i n . That i s ; y(R . ,n) <u(n,q)<y(R ,n), ' M V mm' ' H V '^' H X max' > where y(n,q) i s the benefit derived at n with q and (1-q) p r o b a b i l i t i e s of R and R . .respectively. r max mm' r J Since y(R,n) i s a non-decreasing function, there i s a value of R such that y(RJn) = y(n,q). R' i s c a l l e d the certainty equivalent of the uncertain state of q and (1-q) p r o b a b i l i t i e s of R and R . R1 i s a M r max mm . t function of q and depends on the given mission time n:R'=R'(q| n). The value of R 1(q) i s found as follows. The decision maker establishes what c e r t a i n benefit he derives from the state at which there are q and (1-q) p r o b a b i l i t i e s of R and R . , r e s p e c t i v e l y , and decides at which value of R1, he max mm' * J ' • ' i s i n d i f f e r e n t between the above state and the prospect of obtaining R1 for c e r t a i n . For c l a r i t y and convenience the p r o b a b i l i t y value q=l-q=.5 i s chosen. It i s thus supposed that there i s a f i f t y percent chance that the r e l i a b i l i t y i s R m a x a n c ^ f i f t y percent chance that the r e l i a b i l i t y i s R . . Using the c J mm reference gamble, a certainty equivalent R'(.5) i s obtained. At R=R'(.5), the decision maker derives the same benefit as having a f i f t y percent chance of each R m a x and R m^ n . Therefore, the benefit derived at R'(.5) i s i t s ex-pected value E{u(R' (.50) |n)}= °- 5-Vi(R m a x|n)+ 0 • 5 ' ^  ( R m i n I n> = 0.5 . { y(R In) + y (R • In) } L M N max 1 ' M v mm 1 ' This b e n e f i t value at the r e l i a b i l i t y value RJ provides a t h i r d p o i n t f o r the preference f u n c t i o n f o r r e l i a b i l i t y a t a given mission time n. The r e q u i r e d i n f o r m a t i o n , then , i s the c e r t a i n t y e q u i v a l e n t R 1(.5|n) at which the d e c i s i o n maker would be i n d i f f e r e n t to a 50-50 gamble on R . and R . This i n f o r -3 mm max mation i s obtained from the d e c i s i o n maker by the f o l l o w i n g question. Question 4. (Reference Gamble) G iv e n t h a t the mis s i o n time i s n^ , P r o p o s i t i o n 1 i s t h a t there i s a 50% chance of o b t a i n i n g R . and a 50% chance of 3 mm ;\ r o b t a i n i n g R . P r o p o s i t i o n 2 i s t h a t a ^ max ^ r e l i a b i l i t y value R| can be obtained f o r c e r t a i n , At which value of Rj would you be i n d i f f e r e n t between the two p r o p o s i t i o n s . where n . =n . , n-, ,n~ • • •, n._ l mm' 1' 2 ' N The r e s u l t i n g intermediate p o i n t s _ f o r the preference f u n c t i o n f o r r e l i a b i l i t y are y{R!(.50)|n.} = {y ( R ^ | n.) + y ( R m i n | n.) }/2 . = (Q. mx y Q + P. y Q ) / 2 . = (y Q/2.) (Q i m1 + P i) (2.3) Since these p o i n t s are a l s o expressed i n terms of y Q , they are l o c a t e d on the same s c a l e as the i n i t i a l and end p o i n t s . The l o c a t i o n of these p o i n t s i s shown i n Fig.2.7 (marked by t r i a n g l e s ) . 55 Benefit Q2m,P2 JJ0 2 Q, m, P, 2 Mo P,./l. /Jo / -/ A ' I i i i i • i i ! Rmii Ro R'iR*2 y. Q 2 m , p 0 >• Q ,m | A i 0 () m i/Jo — » - Reliability Rmax. R F i g . 2.7. Location of three Points for Each Preference Function for R e l i a b i l t y at n.. (Note that the l preference functions -dashed l i n e s - have not been obtained at t h i s point.) 56 In S e c t i o n I I . C . , i t i s observed t h a t y (R j n ^ i s a concave f u n c t i o n , and i n t h i s s e c t i o n we showed t h a t y(R!|n.) = 0.5{y(R In.) + y(R . In.)} (2.4) l 1 1 H max1 i ' M V mm 1 i ' For a concave f u n c t i o n F ( x ) , . i f F ( x 2 ) = O . S l F ^ ) + E ( x 3 ) } then x 2 = 0.5(x 1 + x 3 ) . Hence, R] < 0.5( Rmax + R ^ ). ' i = min' 0 57 4. Construction of Preference Functions. In Sections II. E. 2 and I I . E.3. an i n i t i a l , an intermediate^ and an end point were found for the preference function for. r e l i a b i l i t y at a given mission time, y (R). A suitable curve i s to be f i t t e d to these data that meets the c r i t e r i a established i n Section I I . C. As stated i n Section I I . C. the preference function for r e l i a b i l i t y ; a. i s a non-decreasing function, i . e . y '(R,n^)>o b. has a non-increasing slope, i . e . y"(R,n^)<o c. has a zero slope at R=R . i . e . U W R ,n.)=o ^ max' M v max'. I In Sections I I . E.2. & I I . E. 3. we established three points for each mission time value n^. Let a« ^ R t a i n , ' n i ) = y l i b. y(R!,n.) = y 2 . c. y (R ,n. ) = p., K x max ' i ' K 3 i A function y(R,n^) i s now suggested that f i t s the above three points suitably and eonf.irms the r e s t r i c t i o n stated above: y(R,n.) = A-B ( (C-R) e" ( C" R ) } D (2.5) Where A = y^^ C = R max D = l n ( y 3 i - y l i } - l n ( y 3 i - y 2 i ) R . -R!+ln(R -R . )-ln (R -R! ) min I \ max mm max I 1 _ ^ 3 i - y l i } {(R -R . ) e.xp (R . - R ) }° max mm'- ^ L min max 58 This funct ion i s i l l u s t r a t e d in F i g . 2.8 f o r d i f f e r e n t values of R i at given values R^ n and R m a x . A value of zero i s taken for/U^. and a value of one f o r y ' ^ - Values of .900 and .999 are used f o r R m i and R r e s p e c t i v e l y . The value of R. i s var ied from .905 to .949. max r J i 60 5. Construct ion of Ind i f ference Funct ions. In Sect ion II.C. we developed preference funct ion fo r r e l i a b i l i t y at constant mission time. In Sect ion II.E.4 a funct ion was suggested fo r y(R|n). This funct ion i s u t i l i z e d in th i s sec t ion to const ruct the i n d i f f e r e n c e funct ion f o r r e l i a b i l i t y and mission time. An i n d i f f e r e n c e f u n c t i o n , introduced in Sect ion II.D.2 i s a t r ade -o f f funct ion between r e l i a b i l i t y and mission time in terms of b e n e f i t . It i s a locus of a l l combinations of r e l i a b i l i t y and mission time that have the same given constant b e n e f i t . I f we l e t y(R.n.j) = u-where y. i s a constant, and so lve f o r R at a l l n. f o r which y. i s in J * J the range of y ( R , n . ) » we obtain several combinations of (Rin!) which r e s u l t in the bene f i t y.. S ince a l l these combinations (Rin'.) are of equal benef i t to the dec i s ion maker, he would be i n d i f f e r e n t among these combinations. In F i g . 2.9. curves of y(R np are presented at equal i n t e r v a l s of n.j. As discussed in Sect ion I I . C , these funct ions are concave func t ions , s ince the increase in benef i t at constant r e l i a b i l i t y , per un i t increase in mission t ime, decreases as n increases (see Sect ion I I .B.3). The d i f f e rence between y(R|n^ +^) and y(R|n^) decreases as the mission time n increases . Hence, the d i f f e r e n c e 61 between the intersections of y (R|n^ +D and u(R.|n^)with a v e r t i c a l (R=constant) l i n e decreases as n increases. This i s i l l u s t r a t e d in F i g . 2.9. In F i g . 2.10. indifference functions n(R|y=C) are i l l u s t r a t e d for the preference functions y(R|n^) shown in F i g . 2.9. Indifference functions derived from the preference functions for r e l i a b i l i t y and mission time, y(R| n) and y(n| R), are convex functions with negative slope and p o s i t i v e second d e r i v a t i v e . The reasonincr for the shape of indifference functions i s given i n Appendix 4. Through the points (n|,R') obtained by l e t t i n g y(R|n) = y^ a constant a smooth convex function i s f i t t e d . This i s i l l u s t r a t e d within the framework of the example given in Appendix I . A. F i g . 2.9. Preference Functions u(R|n^), f o r R e l i a b i l i t y Constant M i s s i o n Time. F i g . 2.10 I n d i f f e r e n c e Functions n(R|u) 63 III. COST The cost of a p r o b a b i l i s t i c design i s a funct ion of r e l i a b i l i t y and the mission time of the design. The cost of a design may be ana-lyzed in three ca tegor ie s : a) Materia l Cost, b) Operation Cost, c) Production Cost. a) Mater ia l Cost i s , among other th ings , a funct ion of the parameters of the p.d. f . of the materia l p r o p e r t i e s , see Sect ion III.A. I f a choice of material i s s p e c i f i e d , t h i s cost w i l l not vary and i s omitted from the ana l y s i s . However, when a choice of mater ia l (eg. aluminum, s t e e l , or d i f f e r e n t grades of s tee l ) i s in ques t ion , th i s cost w i l l be of importance. The ana lys i s presented in th i s report would then be repeated f o r d i f f e r e n t ma te r i a l s , and the resu l t s compared in order to choose the optimum mate r i a l . Mater ia l Cost i s a l so a funct ion of the s i z e o f the specimen. The s i z e of the specimen, in terms of some c r i t i c a l dimension A, de te r -mines the amount of materia l used f o r the production of the specimen. R e l i a b i l i t y and mission time are both funct ions of the parameters of the p.d. f . of mater ia l property and the s i z e of the specimen, see Sect ion I.e. Hence, the cost of mater ia l can be expressed as a funct ion of r e l i a b i l i t y and the mission time. Most r e l i a b l e and longer l a s t i n g designs usua l ly c a l l f o r more exo t i c mate r i a l s , r e s u l t i n g in increased mater ia l cos t s . See Sect ion III.A. 64 b) Total operat ion cost decreases as the r e l i a b i l i t y of the design increases due to a smal ler p r o b a b i l i t y of f a i l u r e and fewer shutdowns, e t c . , see Sect ion 11 I.B.I. Total operat ion cost increases as the mission time increases because of longer useful l i f e of the specimen. (Note that benef i t too, increases as mission time increases . ) This e f f e c t i s analyzed in Sect ion III.B.2. c) Production Cost fo r the specimen depends on the production method chosen fo r the design. Production cost i s a major f a c t o r of the cost of the design. Production methods are a f fec ted by the propert ies o f the ma te r i a l . Even though i t may be less co s t l y to use a given mater ia l compared to a second materia l on the basis of i t s mater ia l p r o p e r t i e s , i f the production cost assoc iated with the f i r s t mater ia l i s high compared to the second m a t e r i a l , i t o f f se t s the cost advantage of using the f i r s t ma te r i a l . The second mater ia l becomes the bet ter cho ice . Therefore production cost should always be considered wi th in the context of the cost of the mater ia l which the type of production i s assoc iated w i th . The changes in cost .by changes in mater ia l propert ies i s d iscussed in Sect ion I I I .A. l . I I I . A. MATERIAL COST 1 . Dependence on Parameters of M a t e r i a l p.d.f. M a t e r i a l cost per u n i t weight incre a s e s as the ex-pected value and the va r i a n c e of the r e l e v a n t m a t e r i a l property become more favourable; t h a t i s as y^ incre a s e s and/or decreases. y-[ and y 2 are p r o p e r t i e s of the m a t e r i a l p.d.f. and they are f u n c t i o n s of the parameters of the p.d.f. a) The cost of m a t e r i a l i n c r e a s e s as the expected v a l u e , y£ i n c r e a s e s , since a higher y£ i m p l i e s more e x o t i c raw m a t e r i a l s and more expensive production methods f o r the m a t e r i a l . I f C i s d e f i n e d as the cost of m a t e r i a l per u n i t weight; m,y ^ 3 then C m,y + as y£ + Hence, ^ C n ^ / a / i , ' 70. ( 3 . 1 ) and since a d d i t i o n a l incremental i n c r e a s e s i n y ^ become more c o s t l y as y ^ i n c r e a s e s , Hence, / a(///;* y o. (3.2) This analysis determines the general shape of the function C (u-! ) as shown i n F i g . 3.1. The exact shape m, u 1 d i f f e r s for each material. b) The cost of material increases as the variance u 2 decreases because a decrease i n y 2 implies t i g h t e r prod uction controls. Hence, ?> c-mt/+/Z/<(x ( 3 > 3 ) Additional incremental decreases i n the value of u 2 w i l l be increasingly expensive as u 2 decreases, hence, Hence, (3.4) The above analysis gives the general shape of the function C m ^ ( l - ^ a s s n o w n i - n Fi-9J« 3.2. 68 69 -III. A. 2. Dependence on S ize Parameter The design con f i gura t ion i s a r e s u l t of design engineering ana ly s i s . It i s not taken as a va r i ab le in th i s study. The ana lys i s presented i s app l i cab le to designing f o r a given con f i gu ra t i on . The s i z e parameter of the design a f f ec t s both the r e l i a b i l i t y and the mission time as expla ined i n Sect ion I.A.3. The cost i s a l so a funct ion of the s i z e parameter. The materia l cost increases as the amount of materia l used f o r the design increases . The volume of materia l used f o r the design i s a d i r e c t funct ion of the s i ze parameter. I f the s i z e parameter i s a c ros sec t iona l a rea , f o r example, the volume increases d i r e c t l y with the area. The mater ia l cost i s therefore a d i r e c t funct ion of the s i z e parameter. I f the choice of mater ia l i s des i red to be taken as a v a r i ab l e in the ana l y s i s , the cost o f mater ia l per un i t weight, C m , v a r i e s . S ince the s p e c i f i c weight of the mater ia l i s constant, the cost o f the mater ia l per uni t volume, C , i s a d i r e c t funct ion of C m . r v m,y Hence, the cost o f materia l per design specimen is where V i s the volume of materia l used. Since the volume is a d i r e c t funct ion of the s i z e parameter A, (3.5) 70 where K = V/A i s a c o n s t a n t r e l a t i n g the s i z e p a r a m e t e r and volume. I I I . B. COST OF OPERATION 1. Dependence on R e l i a b i l i t y The p r o b a b i l i t y of f a i l u r e of the design i s r e l a t e d to the r e l i a b i l i t y of the design* (Section 1) P(n) = 1 - R(n). That i s , t h e p r o b a b i l i t y t h a t f a i l u r e w i l l occur d u r i n g i t s intended m i s s i o n time n, i s the complementary p r o b a b i l i t y of r e l i a b i l i t y . For c r i t i c a l designs, f o r which f a i l u r e i s c a t o s t r o p h i c , the c o s t of f a i l u r e maybe e x c e s s i v e l y h i g h . For such designs, the minimum r e l i a b i l i t y w i l l be h i g h , and i n the range between minimum and maximum r e l i a b i l i t y , c o st w i l l decrease as the r e l i a b i l i t y i n c r e a s e s . For non-repairable and f o r non-replaceable specimens the ex-pected r e l i a b i l i t y - d e p e n d e n t o p e r a t i o n c o s t i s C R=(1-R(n))C p where P(F) = (l-R(n))-- p r o b a b i l i t y of f a i l u r e i s the com-plementary p r o b a b i l i t y of r e l i a b i l i t y and C p i s the assessed cost of f a i l u r e . See Fig.3.3. For r e p l a c e a b l e and/or r e p a i r a b l e design specimens, f o r which f a i l u r e i s not c a t o s t r o p h i c , the r e l i a b i l i t y -dependent co s t of f a i l u r e decreases as r e l i a b i l i t y i n c r e a s e s In the h y p o t h e t i c a l case when r e l i a b i l i t y i s equal to one, C CcF, S 73 i s equal to zero. As r e l i a b i l i t y inc reases , the p r o b a b i l i t y of f a i l u r e decreases; hence among K specimens ( large K), the expected number of specimens f a i l i n g during the mission t ime, n, decreases. Among K specimens the expected number of f a i l u r e s i s equal to P(F) -K=(l-R(n)).K. I f we assume a f a i l i n g specimen is replaced or repa i red immediately, . the new specimen i s to serve mission time n 1 , the remainder of the mission time n of the f a i l e d specimen. Hence, n' < n. Since the new specimen has a smal ler mission t ime, whi le the design parameter i s the same, R(n') >R(n). Hence the p r o b a b i l i t y of f a i l u r e f o r the replaced or repa i red specimen is lower than the p r o b a b i l i t y of f a i l u r e of the o r i g i n a l specimens, and i t i s less l i k e l y to f a i l during the mission time n. The r e l a t i o n between R(n) and R (n ' ) , g (S ,L ,A) , depends on the nature of the p.d. f . o f mater ia l s t reng th , S, load L, and the design parameter, A: R(n')=g(S,L,A) -R(n). Since R(n')>R(n) the r e l a t i n g f u n c t i o n g becomes l a r g e r as R(n) i n c r e a s e s . I f we assume K (large K) specimens were o r i g i n a l l y put i n t o o p e r a t i o n , ( l - R ( n ) ) K specimens w i l l l i k e l y f a i l and be replaced and/or r e p a i r e d . Among these (l-R(n))K new specimens, P(F) d-R(n) ) .K=(l-R(n- ) ) (l-R(n) )K w i l l l i k e l y f a i l and be replaced and/or r e p a i r e d . The new replacement, ( t h i r d g e n e r a t i o n ) , has a small m i s s i o n time, n", l e f t t o complete s i n c e n" << n, we can assume R(n") = 1.0. That is^R(n") i s very c l o s e to one and f a i l u r e during n" i s very u n l i k e l y and t h e r e f o r e i t s e f f e c t may be neglected. Since, s t a r t i n g w i t h K, and replacements of two generations, to have K specimens complete the mi s s i o n time^ K' specimens are needed where So (K'-K) r e p a i r s and/or replacements are needed I f we l e t C„ be the cost of f a i l u r e , i . e . the c o s t of F r e p a i r i n g and/or r e p l a c i n g t h e , f a i l e d specimens, the co s t f o r K specimen i s the cost per specimen i s CB.= RCn)) [| + tjCs yU,A) ( , _ R ( l n ) ) ^ C F 76 2. Dependence on M i s s i o n Time The o p e r a t i o n of a specimen i n v o l v e s a c o s t . T h i s c o s t i n c r e a s e s as the specimen ages, due to more fre q u e n t shutdowns and f a i l u r e s . The c o s t s of f a i l u r e and shutdowns are taken i n t o c o n s i d e r a t i o n i n p r e v i o u s s e c t i o n s i n c o n n e c t i o n w i t h r e l i a b i l i t y , because t h e i r frequency i s dependent on the r e l i a b i l i t y of the specimen. Hence, the f i x e d p a r t of the o p e r a t i o n c o s t i s taken i n t o c o n s i d e r a t i o n . The f i x e d c o s t i s d i r e c t l y p r o p o r t i o n a l to the l e n g t h of the m i s s i o n time. L e t C = Cost of o p e r a t i o n per specimen o, s CQfR = Cost of o p e r a t i o n per time u n i t per specimen. c o , s = n c o , n i n monetary u n i t s / specimens. Here, as the m i s s i o n time i s i n c r e a s e d , Co,s i n c r e a s e s , and i t may seem as a p e n a l t y c o s t . That i s , the l o n g e r m i s s i o n time appears to be l e s s d e s i r a b l e . But when the b e n e f i t of the m i s s i o n time was c o n s i d e r e d , the b e n e f i t was seen to be i n c r e a s i n g w i t h n. T h e r e f o r e an i n c r e a s e i n the m i s s i o n time i s a s s o c i a t e d both w i t h an i n c r e a s e i n c o s t and i n b e n e f i t . I f the c o s t and b e n e f i t of the m i s s i o n time were the o n l y v a r i a b l e s c o n s i d e r e d i n d e c i s i o n making at a c o n s t a n t r e l i a b i l i t y , R, then the r e l a t i o n graphed i n F i g 3.4 would have r e s u l t e d . 7 7 78 As seen from F i g . 3.4, i f only the benefit of the mission time (omitting r e l i a b i l i t y ) , i s considered, the smaller of the maximum mission time or the mission time value n 1 (at which the benefit i s equal to the cost) would be chosen, since that would be the value which provides the maximum benefit. In case n .,. i s the maximum mission max(l) time, n ,,« would be chosen; i f n i s the maximum ' max(l) ' max (2) mission time, n" would be chosen since n' < n /-,%. ' max(2) 79 III. C. COST FUNCTION 1. Dependence of Cost on R e l i a b i l i t y and Mission Time In Sect ion I, i t was shown that the r e l i a b i l i t y of a design may be expressed as (1.8) R(n) = 1-P(v<l) 3.10 where v = £ 3.11 see eg. (1.7) The r e s i s t i n g strength of the m a t e r i a l , S, i s a funct ion of the mater ia l property, M, and the s i z e parameter, A. Hence the r e l i a b i l i t y can be expressed as R(n) = 1 - P (MxA < L|n) 3.12 From th i s r e l a t i o n the s i z e parameter A, can be expressed in terms of r e l i a b i l i t y R and mission time n. When the maximum-load d i s t r i -bution and strength d i s t r i b u t i o n are approximated as log-normal models (see App. I l l ) : R(n) = F N ( ^ ; 0 , 1 ) 3.13 where from A. l . l l and A. 1.12 in App. I, yv = y s " yv a . _ f 2 . 2x0.5 3 , 1 4 and o y - ( a s + a y ) But )j = y m + In A. s m Hence In A = y y - y m + u v . 3.15 80 Hence, i s c a l c u l a t e d from r e l a t i o n 3.13 and t h e r e f o r e i s a f u n c t i o n of the r e l i a b i l i t y R and y i s a f u n c t i o n Li of m i s s i o n time n. (see S e c t i o n I ) . Hence we have ex-pressed the s i z e parameter A as a f u n c t i o n of n, and R. See Appendix I , f o r an example. In S e c t i o n I I I .A. 2., (3.5), i t was shown th a t the m a t e r i a l c o s t i s a d i r e c t f u n c t i o n of the s i z e parameter A(R,n). Hence, the cost of m a t e r i a l i s C = C -K • A (R, n) 3.16 m v ' " As shown i n S e c t i o n I I I . B. 2, the co s t of o p e r a t i o n per specimen as a f u n c t i o n of mi s s i o n time i s C. = n C 3.17 o,s o,n Hence, the t o t a l c o s t f u n c t i o n i s the sum of three c o s t f u n c t i o n s , namely the cost of m a t e r i a l C r o(R,n), the cost of o p e r a t i o n (dependent on mi s s i o n time) C Q s ( n ) , and the cos t of o p e r a t i o n (dependent of r e l i a b i l i t y ) C (R): C(R,n) = C.m(R,n) +C 0 f S(n) + C R(R) 3.19 When the m a t e r i a l choice i s a l s o a v a r i a b l e i n the d e c i s i o n process, the co s t of m a t e r i a l C m(R,n) i s found s e p e r a t e l y f o r a l l choices of m a t e r i a l . The co s t f u n c t i o n f o r each m a t e r i a l i s , hence, given by 3.19 using appropriate C r n(R,n). 81 IV. C. 2. Constant Cost Curves Using r e l a t i o n 3.12, R(n) = 1-P(S<L), several values of n and R can be found fo r several f i xed values of A. Since A is a d i r e c t f u n c t i o n of cost of m a t e r i a l , using th i s i n -formation a funct ion i s developed of the cost of materia l as a funct ion of mission time at constant r e l i a b i l i t y : Cm(R,n|R) This i s i l l u s t r a t e d by the dashed l i n e in F i g . 3.5. Since the cost o f operat ion dependent on mission time is a d i r e c t funct ion of n, th i s funct ion C (n) ( i l l u s t r a t e d in F i g . 3.5 by the dotted l i n e ) a**el i s added to Cm(R,n|R). Since the value of cost of r e l i a b i l i t y at constant r e l i a b i l i t y i s a constant, C R ( R , n | R) ( i l l u s t r a t e d in F i g . 3.5 by dash-dot-dash l i ne ) i s a hor izonta l l i n e . Adding a l so C^(R|R=C), we get the t o t a l cost funct ion f o r mission time at constant r e l i a b i l i t y , C(R,n|R=C). See F i g . 3.5. (See Appendix I f o r numerical example.) This procedure is repeated f o r N constant r e l i a b i l i t y va lues , where N i s the number of constant mission time values chosen in Sect ion 2. Using these cost funct ions C(R,n|R=C), and taking f i x e d cost va lues, we get a set of values of (R,n) at the same f i xed cos t . Using these sets of (R,n), with the 82 C • ' 1 nmin nmax F i g . 3.5. Cost of Mission Time at Constant R e l i a b i l i t y , C(R,n|R=c.). method employed to construct indifference functions i n Section 2, constant cost functions are constructed, see Fi g . 3.6. Since t o t a l cost increases as mission time increases, and cost of material increases as r e l i a b i l i t y increases, the constant cost functions are concave functions, see Appendix IV. F i g . 3.6. Constant Cost Functions 85 IV. DECISION PROCESS A. In ter sect ion of Ind i f ference and Constant Cost Funct ions. In Sect ion I I .E.5. i t i s shown that i n d i f f e r e n c e funct ions are convex func t i ons , and are expressed in terms of R and n. In Sect ion IV.C.2., i t i s shown that constant cost functions are concave func t i on s , and a l so are expressed in terms of R and n. Hence some of the i n -d i f f e rence funct ions and some of the constant cost funct ions i n t e r -sect at two po in t s , and one or both of these points may be in the acceptable range ' ( R m l - n » R m a v ) and (n . , n m a v ) . An i n d i f f e r e n c e funct ion min max min INOA w i l l be tangent to a constant cost f u n c t i o n , i f many such funct ions are developed. At the point (Rjn 1 ) where an i n d i f f e r e n c e funct ion with bene f i t u' i s tangent to a constant cost funct ion with cost C ' , y ' i s r e a l i z e d with lowest cost and C is r e a l i z e d with highest b e n e f i t . Hence (R in 1 ) i s c a l l e d a suboptimal po in t , s ince the c o s t - b e n e f i t r a t i o i s lowest when cost C and bene f i t y ' are cons idered. In the actual dec i s ion process , when most of the i n d i f f e r e n c e funct ions and constant cost funct ions chosen are l i k e l y not to be tangent, the locus of l i k e l y tangency points i s approximated g raph i -c a l l y . F ig . 4.1. shows the r e s u l t . 86 J o i n i n g t h e s e p o i n t s g i v e s t h e 1 o c u s o f s u b o p t i m a l p o i n t s . B. The O p t i m a l C o m b i n a t i o n o f R and n. Once t h e s u b o p t i m a l p o i n t s a r e f o u n d , e a c h p o i n t i s a s s o c i a t e d w i t h a b e n e f i t v a l u e y, and c o s t v a l u e C„ The r a t i o o f c o s t t o b e n e f i t , X = C/y, i s f o u n d and a smooth f u n c t i o n i s f i t t o X v e r s u s R. The minimum p o i n t o f X(R) g i v e s t h e v a l u e o f o p t i m a l r e -l i a b i l i t y R * , f o r w h i c h t h e c o s t - b e n e f i t r a t i o i s a m i n -imum. The c o r r e s p o n d i n g o p t i m a l v a l u e f o r t h e m i s s i o n t i m e n* i s l o c a t e d f r o m t h e s u b o p t i m a l i t y l i n e o f F i g . 4 . 1 . a t R*. An e x t e n s i v e example o f t h e d e c i s i o n p r o c e s s i s p r e s e n t e d i n A p p e n d i x I . 87 F i g . 4.1. Locus of.Suboptimal Points. 88 V. CONCLUSION AND RECOMMENDATIONS In the c l a s s i c a l design approach, a sa fety f ac to r is chosen based on past engineering experience. This f ac to r does not contain i n f o r -mation on the mission length of the design nor i t s r e l i a b i l i t y . Even the most conservat ive designs do f a i l o cca s i ona l l y . A p r o b a b i l i s t i c design approach conveys the information on expected mission time to f a i l u r e , and the f r a c t i o n of designs expected to f u l f i l l t h i s mission t ime, by the values of the dec i s ion parameters n and R. In the present work, a method i s developed to obtain R and n values by a r a t i ona l dec i s ion ana l y s i s . The dec i s ion c r i t e r i o n used in the ana l y s i s , i s to minimize the c o s t - b e n e f i t r a t i o f o r the dec i s ion maker. Preference funct ions f o r the dec i s ion maker, f o r both r e l i a b i l i t y and mission t ime, are developed. Ind i f ference funct ions are der ived from these preference funct ions . Ind i f ference funct ions are t r a d e - o f f func -t ions between r e l i a b i l i t y and mission time in terms of b e n e f i t . Con-stant cost func t i ons , developed from cost funct ions f o r R and n, are t r a d e - o f f funct ions between r e l i a b i l i t y and mission time in terms of cos t . Ind i f ference and constant cost funct ions are u t i l i z e d to obtain the sub-optimal l i n e which defines the locus of points (n,R) that have the highest benef i t at given cost and lowest cost at given b e n e f i t . Among these po in t s , the one with the minimum c o s t - b e n e f i t r a t i o i s chosen. This combination of R* and n* gives the values of R and n at which the dec i s ion maker minimizes his c o s t - b e n e f i t r a t i o . The 89 p r o b a b i l i s t i c design parameter A* is obtained d i r e c t l y from R* and n*. This dec i s ion model may be adopted f o r use when weight is the c r i t i c a l dec i s ion va r i ab le rather than cos t . Constant weight funct ions may be developed using the same method as f o r constant cost func t ions . The f i n a l ana lys i s may be u t i l i z e d to f i n d minimum weight -benef i t r a t i o in the same manner. It i s recommended that the present work be extended by making material choice a dec i s ion parameter. In that case, the constant cost funct ions are developed for each mater ia l cons idered. A set of constant cost func t ions , der ived fo r each m a t e r i a l , and i nd i f f e rence funct ions re su l t in a sub-optimal l i n e f o r each m a t e r i a l . The minimum cos t -benef i t points on each sub-optimal l i n e are compared, and the one with the lowest cost bene f i t r a t i o gives the optimal r e l i a b i l i t y R*, optimal mission time n*, optimal design parameter A*, and optimal mater ia l choice M*. 90 BIBLIOGRAPHY (A) L i s t , of Journals Surveyed: 1. AIIE Conference Proceedings, 1959-1974. 2. AIIE T r a n s a c t i o n s , 1969-1974. 3. ASCE J o u r n a l of S t r u c t u r a l D i v i s i o n , 1958-1974. 4. 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Davidson, D., and Suppes, P., "A F i n i s t i c A x i o m a t i z a t i o n s of S u b j e c t i v e P r o b a b i l i t y and U t i l i t y , " Econemetrica, V o l . 24, 1956, pp. 264-75. ' 30. F i s h b u r n , Peter C., " U t i l i t y Theory," Management Science, B a ltimore, V o l . 14A, No. 5, January 1968, pp. 335-78. 31. Freudenthal, A.M., "Safety, R e l i a b i l i t y and S t r u c t u r a l Design," ASCE S t r u c t u r a l D i v i s i o n J o u r n a l , V o l . 87, St. 3, March 1961, pp. 1-6. 32. K e c e c i o g l u , D., And Haugen, E.B., "A U n i f i e d Look at Design Safety F a c t o r s , Safety Margins and Measures of R e l i a b i l i t y , " SAE Annals on R e l i a b i l i t y , 1968, pp. 520-30. 33. Luce, R.D., "A P r o b a b i l i s t i c Theory of U t i l i t y , " Econometrica, V o l . 26, 1958, pp. 193-224. 34. Majumdar, T., The Measurement of U t i l i t y , MacMillan and Company, London, 1958. 35. P f a n z a g l , J . , "A General Theory of Measurement: A p p l i c a t i o n s to U t i l i t y , " Naval Research L o g i s t i c  Q u a r t e r l y , V o l . 16, 1959, pp. 283-94. 36. Pope, J.A., F o s t e r , B.K., and Bloomer, N.T., "Li m i t e d L i f e Designs.- A Survey of the Problem," Engineering, London, No. 112, August, 1957, pp. 236-417 37. Shinojuka, M., and Yang, J . , "Optimum S t r u c t u r a l Design Based on R e l i a b i l i t y and Proof-Load Test," SAE Annals of R e l i a b i l i t y , 1969, pp. 375-91. 38. Spear, R.C., "Monte C a r l o Method.for Component S i z i n g , " J o u r n a l of Spacecraft and Rockets, V o l . 7,.No. 9, September, 19T0. pp. -1127-9. 39. Weir, T.W., "Decision Theory and Cost M o d e l l i n g , " SAE Annals on R e l i a b i l i t y , 1972, pp. 319-28. 94 APPENDIX I. AN EXAMPLE I. The Problem The d e c i s i o n process to f i n d an o p t i m a l combination of m i s s i o n time n. and r e l i a b i l i t y R, i s i l l u s t r a t e d w i t h an example of a shear panel f o r an a i r p l a n e . Problem: To f i n d the optimum combination of r e -l i a b i l i t y R* and mi s s i o n time, n*, such that the r a t i o of c o s t to b e n e f i t i s miminized, f o r a shear panel of an a i r p l a n e , probab-i l i s t i c a l l y designed, and to f i n d the c o r r e s -ponding design s p e c i f i c a t i o n , t * . i n s , t h i c k n e s s of the panel. Given Information: 1. Preference f u n c t i o n f o r r e l i a b i l i t y a t constant m i s s i o n time assessed ,by the d e c i s i o n maker. 2. D i s t r i b u t i o n of shear lo a d , F ( L ) : W e i b u l l d i s t r i b u t i o n w i t h mean l o a d , L =1 K i p , and me *• shape parameter, X=1.5. 3. Choice of m a t e r i a l : Aluminum 24S w i t h handbook values of Young's modulus, _ g E = 10.6 x 10 p s i and Poisson's r a t i o v = .33. Material d i s t r i b u t i o n i s assumed to be a log-normal d i s t r i b u t i o n , F T (E) 3 ' LN with ]}\ (E) = E, and Y(E) = 0.1. 4. a. Minimum mission time, n . =100,000 ' mxn ' applications. b. Minimum r e l i a b i l i t y , R . =0.90. mxn c. Maximum r e l i a b i l i t y , R =0.99. 1 ' max 5. Configuration: a l l edges -damped, rec-tangular side panel, with dimensions; a=6ft and b=3ft. II. The F a i l u r e Mode The mode i s assumed to be y i e l d i n g i n shear. From Handbook of Engineering Mechanics by Fl'ugge, the c r i t i c a l y i e l d stress i n shear, r t R t i s related to design para-meters as bxt/7T*k * /O.W (A.1.1) where W. = E / ia c \ Hence; r = /<?-3V n*B iz J is. (I-^J^x ( A ± 2^ The shear strength i s (A.1.3) 96 I I I . Probabi1istic Design Fa i l u re occurs at the f i r s t occurrence of the l oad , L, which i s greater than the strength S: L 7 5 L - - (A.1.4.) A. P.d.f. of Material Property E. P.d.f. of E i s assumed as a log-normal d i s t r i b u t i o n , with X LB) =• E y if) = f o r a log-normal d i s t r i b u t i o n , y LE)* Ce-trta^)-/)0'* (A.1.5.) Hence, using given E and Y(B) values^ ME= 14.849, (T£ = 0.0324 . Since from eg^ . (A.1,3)^ 5 . C ^ V P M 3 / / . ^ F , (A.1.6.) S i s log-normally d i s t r i b u t e d with ^ = <*i. (a.1.7.) B. P.d.f. of Load L. Load i s assumed Weibull d i s t r i b u t e d with 1 . 5 . 97 Since f o r a W e i b u l l d i s t r i b u t i o n ( T ^ - V.2.8 . The extreme value d i s t r i b u t i o n of the We i b u l l model i s estimated by a log-normal model, t h i s i s shown i n Appendix I I I . The log-mormal model th a t estimates the extreme value d i s t r i b u t i o n of the W e i b u l l model i s Where, from A. 3.15 and A.3.16 i n Appendix I I I , ' R E - . 3 3 (i.s Znn-iJJa£^-) ML. = An LO. S73 iZn n) (A.1.8.) G~u= V.CS " * * T O V ^ ^ " * , A 3 3 (A.1.9.) Here, as shown i n Appendix I I I , the values f o r X and o w w are s u b s t i t u t e d and the only independent v a r i a b l e i s n. C. R e l i a b i l i t y F a i l u r e occurs when 1L ^  S. Define a v a r i a b l e V such t h a t V = S/L, (A.1.10) Hence f a i l u r e occurs when V < 1. The v a r i a b l e V i s log-normally d i s t r i b u t e d w i t h i o .S • rr ~. t t r *-_i.rv. ~ a rv ± <rv - l*S-+xCY'* ( A . l . i i . ) 98 The p r o b a b i l i t y of f a i l u r e i s and the r e l i a b i l i t y i s S.= F „ l m 0 , 1 ) , where i s the standardized normal d i s t r i b u t i o n f u n c t i o n . See Chapter 13 i n r e f . (2) Equation (Al.12) i s a r e l a t i o n among the design para-meter R and n, and the corresponding ( p r o b a b i l i s t i c ) design t h i c k n e s s t . IV. The Preference Function f o r R e l i a b i l i t y at a Given M i s s i o n Time A. Answers to Questions: Question 1; (see S e c t i o n II.E.2) Given the minimum m i s s i o n time of 100,0 00 a p p l i c a t i o n s ^ the r a t i o of b e n e f i t d e r i v e d at R = .99 to b e n e f i t d e r i v e d at R = .90 i s m1 = 2.0 Question 2; (see S e c t i o n II.E.2) Estimate maximum mission time n 1 = 200,000 a p p l i c a t i o n s max ' d i v i d e (n' _ n . ) by N= 5 ma>C min' J hence n n= 20,000 a p p l i c a t i o n s . n. = n . + i n ; i = l 2 " * N 1 min n ' > ' »1N • §9 Given that the r e l i a b i l i t y i s R . =.90 the r a t i o of m m 3 benefit derived at n. to benefit derived at n . , n n A x min=100,000 i s P. = u(n.IR . )/y. l l 1 mm ' p0 n. p. 120,000 1.7 140,000 2.2 160,000 2.6 180,000 2.9 200,000 3.0 220,000 3.0 Since p^ does not increase beyond 200,000, the maximum mission time n i s 200,000 applica t i o n s , as was est-max / * . i. i imated previously. Question 3; (see Section II.E.2.C.) Given that the r e l i a b i l i t y i 3 R =.99. the r a t i o x max ) of benefit derived at n. to benefit derived at l n m i n = 1 0 0 ' 0 0 0 i s Q i ^ ( n i l R m a x ) A j 0 n i Qi 120,000 1.30 140,000 1.50 160,000 1.65 180,000 1.75 200,000 1.80 220,000 1.80 Hence n m a x = 200,000 applications, as before. i6o B. Reference Gamble (see section I I . B ^ ) ; Given the mission time n. 1 Proposition 1: there i s a 50% chance of obtaining R . =.90 and 50% chance of obtaining R =.99. mm 3 max Proposition 2: A r e l i a b i l i t y value R| can be obtained for c e r t a i n . At what value of R^ would you be i n d i f f e r e n t between the two propositions r n. l R. • l 100,000 .930 120,000 .927 140,000 .924 160,000 .922 180,000 .921 200,000 .920 C. Preference Function for R e l i a b i l i t y at Constant Mission Time As explained i n Section II.E.4. the above i n f o r -mation i s u t i l i z e d to construct u(R|n.) i 2.' ' Let u ( b e n e f i t at minimum mission time n . and o v min minimum r e l i a b i l i t y R . ) be 1: 1 min' y 0 = i . o . 101 From Section III.E,2^ y (R In . ) =my max' min o J y(R • b•)=p•y K min 1 ^ i p o / y(R b•)=0•y max i ' ^ I ^ O From Section II.E,3^ y(R!_ (.50) /n i) = (y Q/2) (Qimi+pi) . The preference function for r e l i a b i l i t y at constant mission time y (RJru)^ suggested i n Section I I . E. 4^ has been f i t t e d to these points; y (F|i i)=A -B'<<C-R) i ( c " R ) > Q ( A . l The values of the constants for n. are as l f ollov . 'S : n. l A B C D 100,000 2.00 100.63 0.99 1.846 120,000 2.60 171.85 0.99 2.103 140,000 3.00 339.52 0.99 2.422 160,000 3.30 * 651.99 0.99 2.683 180,000 3.50 859.71 0.99 2.833 200,000 3.60 1068.90 0.99 2.997 These functions are i l l u s t r a t e d i n F i g . (A.l) 102 D. I n d i f f e r e n c e Functions For s e v e r a l Constant values of b e n e f i t , the preference f u n c t i o n s f o r r e l i a b i l i t y at constant m i s s i o n time are solved to get values f o r R and n^. That i s , f o r given values of n and y, the value of R i s obtained from equation (A.1.13.). The r e s u l t s are: y n. l R n. l R n. l R 1 .83 100,000 0. 972 120,000 0. 912 130,000 0. 900 2 .33 110,000 0. 990 120,000 0. 943 140,000 0. 906 2 .60 120,000 0. 990 140,000 0. 925 160,000 0. 900 2 .97 140,000 0. 980 160,000 0. 920 180,000 0. 903 3 .17 160,000 0. 940 180,000 0. 918 200,000 0. 908 3 .33 160,000 0. 970 180,000 0. 936 200.000 0. 925 A convex f u n c t i o n , n(R|y), i s f i t t e d to the p o i n t s (n^,R) corresponding to each f i x e d v a l u e , y : n(R|;y)= a 2R 2 + ajR + a Q (A. 1.14.) The r e s u l t i n g constants are: 103 (10 6) (10 6) 1.83 6.58 -13.42 6.94 2.83 3.96 - 7.76 3.90 2.60 5,43 -10.78 5.47 2.97 10.34 -21.14 10.95 3.17 30.44 -64.26 34.10 3.33 25.55 -52.69 27.33 These functions are graphed i n F i g . A.2. This concludes the analysis of the benefit to the decision maker of various combinations of n and R. 105 V Constant Cost Functions There are f a i l u r e cost, production cost, and material cost. Production cost i s fixed and proportional to panel thickness, hence t h i s cost component does not influence the location of the optimum cost-benefit r a t i o . F a i l u r e cost i s e s s e n t i a l l y part of the operational cost. Since r e a l i s t i c data for t h i s cost component are d i f f i c u l t to obtain^this component i s excluded i n the a n a l y s i s . This omission does not a f f e c t the nature of the analysis, but c e r t a i n l y a l t e r s the location of the true optimum cost-benefit r a t i o . The following cost analysis deals with material cost only and i s therefore only a f i r s t approx-imation to the true optimum cost-benefit r a t i o . Since cost of the material i s equal to cost per weight, C , times the weight W of the material, and since the specific weight v of the material i s constant: C=C • W, w The volume V i s V=a-b.t. Hence the cost i s proportional to t: C=Gt, where G = C a-b.v, i s a constant. w It follows that a constant value of t implies constant value of material cost . Previously, i t was shown that 0 , 1 ) 1 0 6 where y = y ^ s and cr^  - +<rJL)„ But ^ - ^ , j^, r,p.zw n3- t3/ /2{/-vJ-JJ>] i s a function of only t, since other values are constants, given the material choice. Furthermore, are functions of n only. Since p N '> °> 0 (A.1 . 1 4 ) gives R as a function of n and t, and since the cost of material i s a d i r e c t function of thickness t, we have a function r e l a t i n g the cost of material to the mission time and the r e l i a b i l i t y . The constant material cost functions (see Chap. I l l ) are, therefore, obtained from equation (A.1 . 1 4 ) by putting the thickness, t, equal to a series of constant values i n that equation. These functions of n i n terms of R are obtained for various constant thickness (constant cost) values. Figure A.3. shows the r e s u l t i n g curves. 107 ncooo) R F i g . A.3. Constant material cost functions; each value of t implies a constant value of cost. material 108 VI Decision Analysis The next step i s to obtain the locus of sub-optimal points of the cost-benefit r a t i o s . F i g . A.4. shows the super—position of F i g . A.2. (indifference functions) and F i g . A.3. (constant cost functions). The l i n e of tangency between these two sets of curves i s the required locus of suboptimal points, see F i g . A.4. Assuming that aluminium panels i n thickness-multiples_ of 0.010 in s . , the cost-benefit r a t i o of these thickness i s obtained from F i g . A.4. by i n t e r p o l a t i n g between i n -difference curves by the appropriate values of benefit, y. The following table shows the r e s u l t i n g suboptimal cost-benefit r a t i o s . Material cost Benefit Cost-benefit Ratio <e;t) (vo (et/y) .11C. 2.48 .0443<G .12G 3. 00 .04 00 G .13G 3. 24 .0402(G .14G 3. 38 .0415 6 .15G 3.47 .0422-G .16G 3.56 .0450 G From the above table, the optimum cost-benefit r a t i o i s seen to be 0.0400 , so that the optimum design thickness i s t*=0.120 ins. The corresponding optimum values of the decision parameters are R*=0.918 and n*=164,000, The 109 interp r e t a t i o n of t h i s optimum solution (based on material cost only) i s that the decision maker minimizes cost-benefit; furthermore, for a panel mission time corresponding to n=164,000 load applications, the proportion of surviving shear panels i s 0.918. ncooo) LEAF 111 OMITTED IN PAGE NUMBERING. does / v ^ t e . / l s t 112 APPENDIX IT  A n a l y s i s of Preference Function. In S e c t i o n III.E.4 a f u n c t i o n u(R,n), was suggested (A.2.1) where A = / ^ 3 t C = fcr*?** This f u n c t i o n . i s to meet the c o n d i t i o n s , e s t a b l i s h e d i n S e c t i o n I I I - C , t h a t a) yuYrZ\,->;> >sO b) yu." (R\rri) ±0 The purpose of t h i s appendix i s to show t h a t u(Rjn^) meets the c o n d i t i o n s l i s t e d above. (A.2.1) yu' /fctnL) = Bo e x p/Zc-*0 o] <L - *3 D ~ ' Lc-^i) y ' ( ? l n i ) ~ - & P [e-^f C c - ^ t l c c ^ ) 0 ' 2 : (A.2.2) (A.2.3) 1 1 3 Substitute R = R i n y 1(R|n.) max p v i ' Since C = R m a x % (C-R ) = O max' > M' (Km*.}\ m) - o. The t h i r d condition i s met. In order to analyze the f i r s t two conditions, f i r s t the constants are analyzed. The c h a r a c t e r i s t i c s of inputs of the function are a. O <c R m , n < */ ^ R-mc^ ± I.' b. from Section II.E.3 R-'i + 0.5 ( X.m^ + Z m ! n ) c. since 0 * R m . n <_ £c< ^ ^ I. O (A.2.4) (A.2.5) (A.2.6) (A. 2. 7) ^ £.'L - Rmlf> *> (A. 2. 8) 115 To analyze constant D, Let X = R.-R . 1 mm Y = R -R. max, 1 X+Y = R -R . max mm From eg. A.2.6. thru eg. A.2.8. o + X I (A.2.9) o ^ y < I (A.2.10) O < * + y + 1 (A.2.11) We have and X -1 Looking into above expansions of e and (1-X) Therefore (A.2.12) /-X since X>o C e X - / V A < 1/(!'*) (A.2.13) from eg. A.2.11. and since & <£ <j / 116 using eg. A.2.13 /< i - X y (A.2.14) * ^\ -t- y-R e s u b s t i t u t i n g values f o r x and y. e x f C R c - B m « „ ^ -1 ( ^ - ^ , ) / ^ ^ / (A.2.15) t a k i n g n a t u r a l l o g a r i t h m of both side s X A ^ CK-^^- R ^ i n ) - J U ^ e ^ ^ - f c ^ -»• (A. 2.16) This i s the denominator of constant D, t h e r e f o r e we have shown th a t denominator of D i s p o s i t i v e . The numerator of constant D i s From S e c t i o n II.E.3, /*3i - / I L I - ZC^l-yKtL) (A.2.17) Hence • Since both numerator and denominator of D are p o s i t i v e & 7 O. (A.2.18) Looking into constant B. ^31 y /in 3 ; 3 -7 O (A.2.19) Constant A. A-y* 3<- ?° (A.2.20) Constant C; Pm*<* 70 (A. 2. 21) Looking into condition (a) (R\nO ^ O eg. A.2.2. gives ^'(ti.\m)* BP e*'*>D ic-***"'Cc-n+o Since C - P - + \ o & 7 O So, condition!a)is s a t i s f i e d . 118 From A.2.8 fit 7-R, Adding expression Ln [ce^-em-,n) I to both sides  Sbn rce^ g^o) I ti. - m/n) (A.2.22) From eg. A.2.5 Adding (R A -R'. ) to both sides Comparing with eg. A.2.22. (A.2.23) From eg. A.2.16 Then d i v i d i n g both sides of A.2.24. by the above expression, 2 I Since l e f t hand side i s equal to D (A.2.1c)y O -7, \ . (A. 2 . 25) By d e f i n i t i o n of R , J max/ Squaring both sides C * W X - 1 ^ 0 Z ^ » (A.2.26) Since the expressions on l e f t sides of A.2.25. and A. 2.26. are both equal to or greater than one, t h e i r product i s also equal to or greater than one. Since C=R ' max D t C - l i T | ) ^ | 7, O (A.2.27) eg. A.2.3. gives From Eg. A.2.19. B>o From eg. A.2.18. D>o Since C=R max C-R>o by d e f i n i t i o n of R — •* max From eg A.2.27 [ D ( c - a + i ) x - i ] ^ o 120 The expression i n the large bracket on the r i g h t side of eg. A.2.28., i s a product of a l l p o s i t i v e expressions, hence i t i s equal to or larger than zero. The whole bracket i s m u l t i p l i e d by (-1) , therefore The constraint (b) i s , therefore, met. 121 APPENDIX XII  Estimation of Weibull Extreme by Log-normal In Appendix I the load, a random variable, i s modeled by a Weibull model. In t h i s Section i t w i l l be shown how the type I asymptote of a largest observation of a Weibull model i s estimated by a Log-normal model. The load i s modeled by a Weibull d i s t r i b u t i o n with parameters a and A . The density and d i s t r i b u t i o n function r w w 1 of load are F w ( L . - < T W ) \ - t r ^ ) ^ (A.3.2) The type I asymptote of a largest observation from a Weibull model has parameters A ; CT W ( JLo (A.3.3) - "7^ C J L ^ ° } ' (A.3.4) see reference (2). In order to estimate the type I asymptote by a log-normal model, "estimation by quantiles" "is employed. Since the r i g h t t a i l of the maximum-load distribution i s of high importance, two high-order quantiles, L Q and L Q are chosen for the estimation process. The quantile of orderL for an extreme value variable q i s L 1 c: yUe ~ (. l*n ^) CT^ (A. 3.5) 122 For a log-normal model the parameters u T and a T i n L Li terms of quantiles L and L are q x q 2 ^ v. - — ) *•"«*- ' . (A.3.6) yUu =• ^ h f i l , ' ^ ' (A.3.7) Substituting A.3.6. for L and L into the above q l q2 equations: cr - ^ t>& -LLnJLn ^;<TeJ - In C^t-(L^t^ 4r)<r,l u ' ^ 77" — '(A.3.8) • /^ /> /e - ^ i ^ ^ a - J - ^ ^ (A.3.9) Substituting A.3.4. and A.3.5. for u e and a i n the above equations: < r u = ^ ( " " w ^ ^ f ) (A. 3.10) ML. = 1 <rv*J*">>n> • (A.3.11) In equations A.3.11. and A.3.12., aT and u T . Li Li, tne parameters of log-normal approximation of the type I asymptote model of the o r i g i n a l Weibull model for load, are expressed i n terms of the parameters of the o r i g i n a l Weibull model for load, a , and Xw* Substituting q1=0.85 and q2=0.95 i n eg. A.3.11. and rearranging (A.3.12 ) u L > u > n + 1.8-2-J 123 Substituting q 1=0.85, q 2 = 0 i 9 5 , and A.3.13 a for a i n eg. A.3.12., and rearranging gives (A3.14) In Appendix I the parameter values for the o r i g i n a l Weibull model for load are and }\ «->o = V. S" Substituting these values i n equation A.3.13 and A. 3.14, we have 5 D /-3.00)' M u . L, J U.S Jinn + oo)'™J (A.3.15) (A.3.16) 124 APPENDIX IV Analysis Qf Indiffer@nG§ Function. In Section IT. £, 5 . , i n d i f f e r e n c e functions are developed. An indifference function i s a c o l l e c t i o n of points at which th© benefit has the same value. Thus, any spe c i f i e d l e v e l of benefit defines an i m p l i c i t function between the mission time and r e l i a b i l i t y . Since the benefit y i s a function of both R and n, the indifference function for benefit l e v e l y Q may be obtained from which i s an i m p l i c i t function i n two var i a b l e s . Hence there e x i s t s a function g such that t h i s i s the indifference function for benefit l e v e l y In f a c t , function g defines one indifference function for each given l e v e l of benefit, y. To investigate the slope of g, substitute for n, i n (A.4.1) from (A,4.2)j y Q=f(n,R), (A.4.1) n * g ( R ; H 0 ) (A.4.2) -(A.4.3) d i f f e r e n t i a t i n g with respect to R; o=f ng'+f R (A.4.4) where g1adg/dR f n -df/dn , f =*df/dft ; 125 and so -g'=fp/n (A.4.5) But from Sections II.B.2 and II.C.2 we know that the slopes of preference f u n c t i o n f o r r e l i a b i l i t y f and preference f u n c t i o n f o r mi s s i o n time f , are both p o s i t i v e . Hence -g'=f R/f n>o (A. 4. 6) This shows t h a t when one v a r i a b l e i s decreased, and b e n e f i t i s to remain constant, the other v a r i a b l e must be in c r e a s e d . D i f f e r e n t i a t i n g g' w i t h respect t o R, we get From Sections II.B.3 and II.C.3 we know t h a t the second d e r i v a t i v e s of y(R/n) w i t h respect to R, f . . . . - . , and y (n/RO) w i t h r e s p e c t to n are both negative. Since both f and f„ r 3 n R are p o s i t i v e , f i s a l s o p o s i t i v e . Hence the terms w i t h i n the bracket i n (A.4.7) are a l l negative terms, and so g" >o (A.4.8) Hence i n d i f f e r e n c e f u n c t i o n at a constant b e n e f i t value i s a convex f u n c t i o n w i t h negative slope and p o s i t i v e second d e r i v a t i v e . 

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