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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. ( M e c h a n i c a l ) , Robert C o l l e g e , I s t a n b u l , Turkey, 1970 M.B.A. ( O p e r a t i o n s R e s e a r c h ) , Syracuse U n i v e r s i t y , S y r a c u s e , 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 t h e Department of Mechanical Engineering We accept t h i s t h e s i s as conforming t o t h e required  standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1975  In p r e s e n t i n g t h i s  thesis  in p a r t i a l  fulfilment of  an advanced degree at the U n i v e r s i t y of B r i t i s h the I  Library shall  f u r t h e r agree  for  freely available  that permission  for  Columbia,  I agree  r e f e r e n c e and  for e x t e n s i v e copying o f t h i s  for  that  study. thesis  s c h o l a r l y purposes may be granted by the Head of my Department or  by h i s of  make i t  the requirements  representatives.  this  thesis  written  for financial  kA.er-W\r>.  U n i v e r s i t y of B r i t i s h  2075 Wesbrook P l a c e Vancouver, Canada  V6T' 1W5  Date  {  i s understood that copying or p u b l i c a t i o n gain s h a l l  not be allowed without my  permission.  Department of The  It  5 , Q.  rn'ral Columbia  n^'ine-erln^  ABSTRACT  In c l a s s i c a l  d e s i g n the i n p u t s  vant m a t e r i a l  to the d e s i g n p r o c e s s , namely the  p r o p e r t y M and the load L are taken as  The p r o b a b i l i s t i c  design approach r e c o g n i z e s  deterministic  the v a r i a t i o n  in  property varies  values.  design  i n p u t s , and the consequent random b e h a v i o r o f these v a r i a b l e s . material  rele-  The  from one l o t o f p r o d u c t i o n to another and w i t h i n  the same p r o d u c t i o n l o t .  The a p p l i e d l o a d to a p a r t i c u l a r  v a r i e s w i d e l y w i t h i n some range.  specimen  The d e s i g n o u t p u t , a dimensional  para-  meter A, a l s o v a r i e s w i t h i n the given range o f t o l e r a n c e s , and t h e r e fore is first  randomly d i s t r i b u t e d among specimens.  i n s t a n c e the l o a d v a l u e L i s  o f the m a t e r i a l , S. S by L r e s u l t s  larger  mission  In  time.  Within this  time emerge as  probabilistic  the f i r s t  The r e l e v a n t load v a r i a b l e  the extreme v a l u e o f a number n o f loads design mission  than the r e s i s t i n g  Among n l o a d a p p l i c a t i o n s  in f a i l u r e .  F a i l u r e occurs at  therefore given  framework, d e s i g n r e l i a b i l i t y  the a p p r o p r i a t e d e s i g n  and  inputs.  d e s i g n the d e s i g n e r has a wide range o f c h o i c e f o r  combination o f n and R i s  A cost-benefit analysis using  strength  exceedance o f  t h a t c o r r e s p o n d to a  both i n p u t parameters, r e l i a b i l i t y R and m i s s i o n the o p t i m a l  is  the  is  time n.  In  determined i n a l o g i c a l  made, r e s u l t i n g  i n an optimal  the b e n e f i t and c o s t to the d e c i s i o n maker.  this  Thesis,  way.  combination  Since b e n e f i t i s r e l a t i v e , the i n f o r m a t i o n to determine the b e n e f i t f u n c t i o n i s a c q u i r e d d e c i s i o n maker through s e v e r a l q u e s t i o n s gamble",.  Cost i s analyzed  constructed.  and a  reauired  from the "reference  and a c o s t f u n c t i o n i s  Using the b e n e f i t and c o s t f u n c t i o n ? ,  erence and constant  indiff-  cost functions are derived, r e s u l t i n g  i n suboptimal combinations o f R and n.  The o p t i m a l  comb-  i n a t i o n i s chosen among the suboptimal combinations by m i n i m i z i n g the c o s t - b e n e f i t r a t i o . t o i l l u s t r a t e t h i s - d e c i s i o n model.  An example i s p r e s e n t e d  TABLE OF CONTENTS  I.  ££SL_1  INTRODUCTION " A.  Introduction  1  1.  General  1  2.  Load L, and M a t e r i a l P r o p e r t v  3.  P r o b a b i l i s t i c Design  B.  L i t e r a t u r e Survey  C.  Present Research  M  2 7 11  Problem  14  I I . BENEFIT A.  B.  D e f i n i t i o n of Benefit  15  1.  Definition  15  2.  Considerations i n Assessing Benefit  16  3.  Preference Functions  18  B e n e f i t o f M i s s i o n Time a t a G i v e n V a l u e o f Reliability  20  1.  General  20  2.  Non-decreasing  3.  Non-increasing Slope of Preference Function  Preference Functions  22  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  3.  Non-increasing slope of Preference  Preference Function  Function D.  35  B e n e f i t o f Both R e l i a b i l i t y and M i s s i o n Time. 4 1 1.  Interdependence o f p r e f e r e n c e F u n c t i o n s . . 4 1  2.  B e n e f i t o f R e l i a b i l i t y and M i s s i o n Time, Indifference Functions  E.  D e r i v a t i o n of Preference Functions 1.  III.  42 44  Information Required t o Construct Preference Functions  44  2.  Questions  t o t h e D e c i s i o n Maker  45  3.  Reference  Gamble  52  4.  Construction of Preference Functions  57  5.  Construction of I n d i f f e r e n c e Functions...  60  COST  A.  32  63  M a t e r i a l Cost 1.  2.  65  Dependence on Parameters o f M a t e r i a l p.d.f  65  Dependence on S i z e P a r a m e t e r . . . .  69  vi Page B.  C.  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  76  Cost F u n c t i o n 1.  2. IV.  V.  79  Dependence o f C o s t on R e l i a b i l i t v and M i s s i o n Time  79  C o n s t a n t C o s t Curves  81  DECISION PROCESS A.  I n t e r s e c t i o n o f I n d i f f e r e n c e Curves  85  B.  The O p t i m a l C o m b i n a t i o n o f R and/1  86  CONCLUSION AND RECOMMENDATIONS  88  BIBLIOGRAPHY  9 0  APPENDIX I . An Example.... APPENDIX I I . A n a l y s i s o f P r e f e r e n c e F u n c t i o n  94 1  1  2  APPENDIX I I I . E s t i m a t i o n o f W e i b u l l Extreme by L o g Normal Model APPENDIX IV. A n a l y s i s o f I n d i f f e r e n c e F u n c t i o n  121 125  V/i  LIST OF FIGURES  Figure 1.1  Page A Typical P r o b a b i l i t y D i s t r i b u t i o n Function for M a t e r i a l Property M  4  A Typical 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. o f Load, Extreme V a l u e o f Load, and M a t e r i a l Strength  9  2.1  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 a t Constant R e l i a b i l i t y RJuCnjR')  29  The P r e f e r e n c e F u n c t i o n f o r R e l i a b i l i t y a t C o n s t a n t 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 F u n c t i o n s 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 o f I n i t i a l and End P o i n t s o f 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  Location of I n i t i a l Points 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 o f I n i t i a l and End P o i n t s o f 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 V a l u e s ,  51  1.2  2.2  2.7  L o c a t i o n o f t h e s e P o i n t s f o r Each P r e f e r e n c e F u n c t i o n f o r R e l i a b i l i t y a t n. 55  2.8  Preference Function f o r D i f f e r e n t Values of  2.9  2.10  R[  59  P r e f e r e n c e F u n c t i o n s u(RJi.) f o r R e l i a b i l i t y a t C o n s t a n t M i s s i o n Time  62  I n d i f f e r e n c e F u n c t i o n s n(P.Ju=C)  62  viii Figure  Page  3.1  U n i t - M a t e r i a l C o s t as a F u n c t i o n o f u£  6 7  3.2  U n i t - M a t e r i a l C o s t as a F u n c t i o n o f u  68  3.3  C o s t o f F a i l u r e as F u n c t i o n o f 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 o f M i s s i o n Time a t C o n s t a n t  2  Reliability  C (R,nR=n)  8 2  3.6  Constant Cost Functions  84  4.1 A.l  Locus o f S u b - o p t i m a l P o i n t s Preference F u n c t i o n s , u(R|n.), f o r R a t  87  c o n s t a n t n^..,.  ^  104  A.2  I n d i f f e r e n c e F u n c t i o n s , n(R|u)  104  A.3  Constant M a t e r i a l Cost F u n c t i o n s  107  A. 4  Locus o f S u b o p t i m a l P o i n t s  110  ACKNOWLEDGEMENTS  The author would l i k e t o express h i s s i n c e r e  gratitude  to Dr. K a r l V. Bury, who devoted c o n s i d e r a b l e time and gave i n v a l u a b l e a d v i c e and guidance throughout a l l stages of the p r e s e n t work. The  f i n a n c i a l support of Karadeni z Teknik U n ? v e r s i t e s i ,  Trabzon, Turkey, i s g r a t e f u l l y  acknowledged.  X NOMENCLATURE  A  :  Design  C  :  Cost  :  Cost of F a i l u r e  :  M a t e r i a l Cost  Cp.  :  O p e r a t i o n 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.MAA  :  Extreme Value Asymptote  L-  :  Load  M  :  M a t e r i a l Property  ii  :  Number o f Load A p p l i c a t i o n s , M i s s i o n Time  /7C"R/G)  ;  Constant Cost F u n c t i o n  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  R  :  Reliability  C  M  p.'  .:  Parameter  a t Completion  Certainty Equivalent R e l i a b i l i t y  S  :  M a t e r i a l Strength  •£  :  Thickness  7"  :  M i s s i o n Length  IX  :  Utility  vV  :  Net Resources  $>  :  Exact Extreme Value  rj-"*'  :  Variance  at Start Distribution  :  Benefit  /A,  :  Expected Value,  /*z  '  Variance  2)  :  Safety  Factor  Mean  X /'  LIST  Benefit.  OF  :  DEFINITIONS  Benefit  i s the  derived  by  from (or  a  Cost  Function  :  A  amount  trade-off and is  Cost  locus n  maker,resulting  values  of  relative of  satisfaction  a variable to  some  satisfaction.  Function  is  a  f u n c t i o n between  mission a  and  of  variables),  Constant  of  decision  state  standard  Constant  a  amount  that  time of  i n terms  reliability of  cost.  a l l combinations  have  the  same  of  given  It R  constant  cost.  Indifference  Function  :  An  Indifference Function  off  f u n c t i o n between  mission  time  It  locus  R  is a and  n  constant  that  in  terms  of have  benefit.  is a  trade-  reliability of  benefit.  a l l combinations the  and  same  given  of  X  Mission  Time  :  Mission specimen which  terms  applied  Reliability  :  n,  i s the  specimen  applied in  Time,  the  time  1  design  length,  i s subjected  loads. of  of  /1  time  n may  be  units  during to  expressed  on  number  loads.  Reliability  i s the  probability  survival  design  specimens  the is of  end  of  of of  the mission  complimentary failure.  to  the  time.  of  by It  probability  1  1A. 1.  INTRODUCTION General In a d e s i g n p r o c e s s , the p h y s i c a l d e t a i l s o f a specimen  are determined  such t h a t a g i v e n f u n c t i o n i s performed by  the specimen under g i v e n c o n d i t i o n s . The i n p u t s t o a d e s i g n process a r e material, and  a)  b)  the r e l e v a n t m a t e r i a l p r o p e r t y o f the the l o a d which i s a p p l i e d t o d e s i g n  c) the chosen f a i l u r e  specimen  criterion.  Under the c o n v e n t i o n a l d e s i g n approach the i n p u t s t o the d e s i g n process are taken as f i x e d v a l u e s . A s i n g l e representative value  L  1  i s chosen f o r the l o a d . The  handbook v a l u e o f the m a t e r i a l p r o p e r t y , M', i s used t o r e p r e s e n t the m a t e r i a l p r o p e r t y . A s a f e t y f a c t o r v i n t r o d u c e d t o take i n t o account The  specimen designed  unforeseen  is  contingencies.  i s meant t o s u s t a i n a l l f u t u r e l o a d s ,  although even d e s i g n s w i t h a very h i g h s a f e t y f a c t o r do f a i l o c c a s i o n a l l y . l o a d value L  F a i l u r e o f the d e s i g n means t h a t the  was l a r g e r than the r e l e v a n t s t r e n g t h v a l u e  on the o c c a s i o n the f a i l u r e procedure  v  . The c o n v e n t i o n a l d e s i g n  o f a specimen i s based on the r e l a t i o n .  S  =  vL  Where S  =  s t r e n g t h o f the m a t e r i a l  L  =  load value,  v = safety factor.  (1.1)  2  S i s a f u n c t i o n of the m a t e r i a l p r o p e r t y dimension, A .  design  A p a r t i c u l a r value  on the b a s i s of e n g i n e e r i n g  and  relevant  of v i s s e l e c t e d  e x p e r i e n c e , judgement  knowledge of s i m i l a r designs. high  M,  By choosing a  and  sufficiently  s a f e t y f a c t o r , the p r o b a b i l i t y of f a i l u r e i s assumed  t o be e l i m i n a t e d . conservative  However, i n p r a c t i c e even the most  designs do  fail.  Furthermore, the  safety  f a c t o r does not p r e d i c t the performance o f the specimen;  t h a t i s , the  information  Load If  s a f e t y f a c t o r does not p r o v i d e  as t o the l i k e l i h o o d of f a i l u r e and  time l e n g t h the d e s i g n 2.  design  L, and  would operate p r i o r to  M a t e r i a l Property  the  any  likely  failure.  M  s e v e r a l s i m i l a r specimens o f a p a r t i c u l a r m a t e r i a l  are t e s t e d f o r a s p e c i f i c m a t e r i a l p r o p e r t y ,  i t is likely  t h a t d i f f e r e n t property  Material  values  are o b t a i n e d .  p r o p e r t i e s o f a g i v e n m a t e r i a l vary to another, and handbook value  from one  within a p a r t i c u l a r production i s u s u a l l y a c e n t r a l value  the whole range of v a l u e s .  If design  i s a random v a r i a b l e  specimens under g i v e n o p e r a t i n g l i k e l y to f a l l  l o t . The  that  represents  That i s m a t e r i a l  among specimens.  measurements are taken of the  r e s u l t s are  lot  M a t e r i a l p r o p e r t i e s of a sample  of specimens are randomly d i s t r i b u t e d . property  production  l o a d on s i m i l a r conditions,  the  i n a wide range of v a l u e s .  A  histogram of a p p l i e d loads on a p a r t i c u l a r specimen o f t e n shows t h a t l o a d v a l u e s  occur randomlv, w i t h l a r g e r l o a d  3  v a l u e s o c c u r i n g l e s s f r e q u e n t l y than s m a l l e r l o a d v a l u e s . But t h e s e l a r g e l o a d v a l u e s are the ones t h a t cause and a r e t h e r e f o r e o f  failure  importance.  As b o t h the m a t e r i a l p r o p e r t y  M  be r e c o g n i z e d as random v a r i a b l e s , due  and  load L  should  to inherent variab-  i l i t i e s , t h e s e v a r i a b l e s are r e p r e s e n t e d by some mathematical function, The  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 (p.d.f.).  ^  probability density function,  represents  f(x;0)  magnitude of x i n terms o f i t s f r e q u e n c y  of occurence,  the  where  (2)  G i s the parameter v a l u e o f t h e s t a t i s t i c a l  model  Each p r o p e r t y o f each s p e c i f i c m a t e r i a l g i v e s r i s e t o 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 . production process,  The  ( c o n t r o l of temperature,  c o n t r o l over  the  homogenity,  and p e r c e n t a g e o f a l l o y i n g elements,) e t c . ) determines the observed  v a r i a t i o n i n the m a t e r i a l p r o p e r t y and thus d e t e r -  mines t h e 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 , t h e n a t u r e o f t h e p . d . f . and parameter v a l u e 0 d e t e r m i n e d by t h e o p e r a t i n g c o n d i t i o n s imposed on  are  the  specimen. The handbook v a l u e o f a m a t e r i a l p r o p e r t y u s u a l l y t h e mean v a l u e o f t h e a s s o c i a t e d p . d . f .  M,  is  Data on  m a t e r i a l p r o p e r t y show t h a t the p.d.f. o f M i s u s u a l l y skewed t o 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 o f v a r i a n c e i n low v a l u e s o f M, w h i l e h i g h v a l u e o f M o f t e n go unchecked. The  l o g n o r m a l , 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 t o r e p r e s e n t the p . d . f . of M. t y p i c a l p.d.f.  f o r M i s shown i n F i g .  1.1.  A  f(M)  Fig.  1.1.  A Typical  Probability  D e n s i t y .Function f o r M a t e r i a l  P r o p e r t y M.  5 Most l o a d values f o r conventional occur l e s s  design  frequently.  to the r i g h t . will  o c c u r around a nominal  Hence the p . d . f . o f l o a d L, i s  A typical  specimen.  p.d.f.  for L is  consecutive load a p p l i c a t i o n s load  skewed  Failure  strength  of  the  p r i o r to the f i r s t  is  exceeds exceedance  not be exceeded i n n  given by the e x a c t extreme v a l u e  p.d.f.  distribution,  •{n)  (  S  )  =  To r e p r e s e n t the p . d . f . models a r e employed.  [  F(L)  |L=s]  0.2)  n  o f L, f ( L ) ,  Lognormal, Gamma and W e i b u l l  The extreme v a l u e asymptote,  s t r u c t e d from the i n i t i a l  F  T  •  (L)  model o f L, f o r a g i v e n number o f  is  con-  contemplated  n^K  The extreme v a l u e p . d . f . The m i s s i o n  i n time u n i t s . per u n i t time i s  Fj(L)  is  shown i n f i g .  l e n g t h , T , o f the d e s i g n  d u r i n g which the specimen i s  unit  1.2.  not i n what p o r t i o n o f l o a d values  The p r o b a b i l i t y t h a t the s t r e n g t h S w i l l  loads,  values  usually  shown i n f i g .  but i n the number o f l o a d a p p l i c a t i o n s  o f the i n i t i a l  load  used  S i n c e the f a i l u r e o c c u r s a t 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  o f S.  which i s  procedures w h i l e r e l a t i v e l y high  o c c u r when a l o a d v a l u e exceeds the r e s i s t i n g  design  S,  value L ' ,  the time  s u b j e c t e d to a p p l i e d l o a d s .  For many design a constant.  specimen i s  1.3.  If  T is  specimens, the number o f l o a d s An i s  the number o f loads  length, expressed applied  a p p l i e d per  time,  n = T« An  (1-3)  7  Hence n i s t h e number O-F a p p l i e d l o a d s t o w h i c h t h e design i s subjected during i t s m i s s i o n time.  In the r e s t  of t h e d i s c u s s i o n , n i s used t o r e p r e s e n t t h e m i s s i o n and  i t may be e a s i l y c o n v e r t e d 3.  Probabilistic  time  t o time u n i t s by d i v i d i n g by An.  Design  When t h e random b e h a v i o r o f a r e l e v a n t m a t e r i a l p r o p e r t y M  and o f t h e l o a d  L  ;  are recognized, the conventional  approach o f d e s i g n i n g w i t h d e t e r m i n i s t i c v a l u e s o f M' and L' i s r e p l a c e d by d e s i g n i n g w i t h t h e p . d . f ' s  o f t h e 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 o f t h e 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 d e s i g n d i m e n s i o n ,  A.  I n t h e case o f a  s i m p l e t e n s i o n l i n k f o r w h i c h t h e f a i l u r e c r i t e r i o n .is rupture . , (1.4)  S=MA, Where A i s t h e a r e a o f t h e 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 o c c u r s when S i s exceeded by t h e l o a d : L>S The p r o b a b i l i t y o f f a i l u r e , d u r i n g t h e m i s s i o n  time  n is P(n) The r e l i a b i l i t y  = P(L>S)  (1.5)  o f t h e d e s i g n specimen i s d e f i n e d as  the c o m p l i m e n t a r y p r o b a b i l i t y . R(n)  = 1-P(L>S)  (1.6)  8  The a c t u a l dimensions specimens..  o f a d e s i g n element v a r y among  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 g i v e n l i m i t s on t h e average.  But as l o n g as each  element i s n o t i n s p e c t e d , t h e d e s i g n d i m e n s i o n , A, i s an unknown and must be c o n s i d e r e d a random v a r i a b l e .  K  '  Hence S i s a p r o d u c t o f two random v a r i a b l e s and is therefore i t s e l f r.v.S  a random v a r i a b l e .  The p . d . f . o f t h e  i s shown i n f i g . 1.3. Under t h e p r o b a b i l i s t i c d e s i g n approach,  R  and m i s s i o n t i m e  n  the r e l i a b i l i t y  a r e chosen by t h e d e c i s i o n maker.  U s i n g ecj. (1.6)^ R(n) = 1-P(L>S), the d i m e n s i o n The  o f t h e d e s i g n element j s c a l c u l a t e d , see r e f (2), safety factor v=S/L  v, (1.7)  }  i s t h e q u o t i e n t o f 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 o f V;  f a i l u r e o c c u r s when v < l , and hence t h e p r o b a b i l i t v t h a t f a i l u r e o c c u r s d u r i n g a m i s 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 . o f L, f ( L ) i s shown. A c o r r e s p o n d i n g extreme v a l u e model d e r i v e d from t h e i n i t i a l model  f (L)=F(f(L),n j  f(L) i s also illustrated.  A s t r e n g t h p . d . f . f ( S ) i s p r e s e n t e d i n f i g . 1.3, and t h e  F i g . 1.3. P.d.f. of Load, Extreme Value of Load^and M a t e r i a l Strength.  10  p r o b a b i l i t y of f a i l u r e is  indicated qualitatively.  An example o f o b t a i n i n g from these p . d . f . ' s of i l l u s t r a t i n g  is  p.d.f!sfor  M and L and c a l c u l a t i n g  R(n)  i l l u s t r a t e d i n Appendix I , w i t h i n the framework  the d e c i s i o n  analysis.  11 I.  B.  LITERATURE SURVEY  A study p u b l i s h e d by Freudenthal behavior of material  strength  to the d e s i g n p r o c e s s . and s t u d i e s  v  '  i n 1947 i n t r o d u c e d the random  and l o a d , and the concept o f  T h i s work was  f o l l o w e d by numerous  i n the l i t e r a t u r e by M i t t e n b e r g s ,  and many o t h e r s .  reliability discussions  ( ) ' ( ) > Weibull 6  7  ^ ,  An e x t e n s i v e survey o f the l i t e r a t u r e p e r t a i n i n g  random b e h a v i o r o f m a t e r i a l  strength  and the concept o f  to  probabilistic  (a)  design is  p r e s e n t e d by Agrawal  v  in his  ;  thesis.  On the o p t i m i z a t i o n o f the p r o b a b i l i s t i c on s e a r c h i n g  f o r optimal  R and m i s s i o n  time  Freund^^ e x p r e s s e d as  values  o f the d e s i g n parameter,  n, however, t h e r e i s  introduces  design process, that  risk  u(R)=l-e"  reliability.  very l i t t l e a v a i l a b l e  i n t o a programming model.  is  literature.  Utilty  is  a R  where R = net revenue o f p r o j e c t and  a = constant cost  Freund shows t h a t i f  R is  factor.  normally d i s t r i b u t e d , u t i l i t y  mized by maximizing  E(R) - aa /2, 2  where  E(R)  is  the expected v a l u e o f R  2 and o  is  its  variance.  T h i s approach may prove to be u s e f u l when u t i l i t y can be  is  opti-  expressed as above.  But the c o n d i t i o n  t h a t net revenue,  R, i s to be o f normal d i s t r i b u t i o n makes the approach v e r y u n p r a c t i c a l , s i n c e i t i s very u n l i k e l y t h a t R, which i s a f u n c t i o n of non-normally d i s t r i b u t e d v a r i a b l e s , w i l l be a normally d i s t r i b u t e d v a r i a b l e , Weisman & Holzman can be considered f u n c t i o n , they  concave everywhere and u s i n g  function  Freund's  define  r= W where  argue t h a t the u t i l i t y  +P-C,  r = net r e s o u r c e s a t completion W = net r e s o u r c e s a t s t a r t P = P r i c e or Revenue C = Cost .  Hence the u t i l i t y  f u n c t i o n becomes  y /(r)\ = i1-e -ar Considering  p r i c e as d e t e r m i n i s t i c  , . , -a(w+P) f ° V y ( r ) = 1-e •»-••  C  t  f (c) •40 . •  They show t h a t i f the c o s t model i s unimodal and symmetric, the u t i l i t y  i s maximized by  minimizing E(c) +aa /2 2  When the c o s t f u n c t i o n i s not unimodai they g i v e upper bound  an  f o r the o b j e c t i v e , f u n c t i o n .  (12) Singh and Kumar pay-offs. formulation component's.  Pay-offs given  discuss are i n t r o d u c e d  system r e l i a b i l i t y t o weigh the r i s k .  with The  takes i n t o account systems of two or more  A l o s s matrix i s assumed and used, but no  study of g e t t i n g the l o s s m a t r i x i s i n d i c a t e d .  In the f i e l d  of o p t i m i z a t i o n t h e r e i s a number of (13) e x c e l l e n t s t u d i e s i n the l i t e r a t u r e . F:ishburn ' gives v  a g e n e r a l theory of s u b j e c t i v e p r o b a b i l i t i e s and  expected  (14)  utilities.  Debrew*  presents  a r e p r e s e n t a t i o n of (15)  p r e f e r e n c e o r d e r i n g by a numerical  function.  d i s c u s s e s the e x i s t e n c e of a u t i l i t y preferences.  .Suppes  'has  Rader  function to  represent  an e x c e l l e n t study on  r o l e 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  ^  the  function i n  d e c i s i o n making. Utility and  f u n c t i o n s f o r m u l t i - a t t r i b u t e d consequences  independence of these  by Keeney  (17 18\ -''  VJ  X  functions i s presented  A methodical  in article  approach of e s t i m a t i n g (19)  a d d i t i v e 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 e c i s i o n models are presented (20 to 22) thoroughly by S c h l a i f f e r ' and by P r a t t , R a i f a , (23 24) and S c h l a i f f e r • '.' ;. Other works i n the area i n c l u d e (25) (26) (27) those by Weiss , DeGroot ' •, and Myron n  14 I . C.  PRESENT RESEARCH PROBLEM  In p r o b a b i l i s t i c d e s i g n , t h e d e s i g n e r can choose t h e reliability  R  and m i s s i o n time  n.  He has a wide range  o f c h o i c e f o r b o t h i n p u t parameters,.and t h e v a l u e s o f R and n determine  t h e v a l u e o f t h e d i m e n s i o n a l parameter  A.  The purpose o f t h i s study i s t o 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 p r o c e s s t h a t p r o v i d e s t h e most o p t i m a l o f R and n f o r t h e d e c i s i o n maker.  combination  A rational decision  maker would want t o maximize t h e b e n e f i t he d e r i v e s from the design, w h i l e m i n i m i z i n g the cost of the design.  So  the problem i s d e 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 t i m e , I t i s t h e r e f o r e seen t h a t i n t h e problem as s t a t e d , t h e design parameters, variables.  R and n, become t h e r e l e v a n t d e c i s i o n  II.  BENEFIT IIA.  DEFINITION  1.  Definition  Benefit  OF  BENEFIT.  i s t h e amount  decision  maker,  variable  (or v a r i a b l e s ) ,  of  are  discussed  values.  considered  can  be  as  c a n be  a whole  IIA2, is  relative  or  standard,  factors.  that  be  standard  from  any  a  a amount  a number these  of factors  of mission to the cost Cost  the  reference  benefit i s the  process.  to  input  satisfaction time  and  of design  factors  and  are  i n chapter I I I . term.  i n absolute  expressed  Some  terms.  i n terms.of  norm  expressed  i s u s u a l l y an, e a s i l y  of benefit total  monetary u n i t s .  i n absolute and  factors  F o r example  of a l lthe f a c t o r s  expressed  t o some  of  incorporates  without  reference  relative  composed  cannot  values  to the design  expressed  revenue  results  term  process,  without  i s a  t o some  by  IIA2.  overall  separately  Benefit  state of values  the state of values  the design inputs  a  relative  i n section  of output  from  from  derived  the variable considered;  In a d e s i g n  derived  other  by  i s an  satisfaction  satisfaction  Satisfaction  affected  Benefit  of  resulting  satisfaction.  factors  of  discussed terms. as a  defined  sales Benefit  i n section  Fence, ratio.  state of  benefit The  norm,  output  Benefit derived  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 d i . f f e r e n t d e c i s i o n makers and a t d i f f e r e n t states of outside  factors.  It is  a l l d e c i s i o n makers b u t g e n e r a l Sections  I I B and I I C a n a l y s e  benefit derived  n  o  t n e c e s s a r i l y t h e same f o r  t r e n d s can be a n a l y z e d .  these general  trends f o r  from d e s i g n r e l i a b i l i t y and d e s i g n  mission  time.  2.  Considerations  i n Assessing  A d e c i s i o n maker, i n a s s e s s i n g  Benefit  the benefit  derived  from a c e r t a i n s t a t e o f an o u t p u t , must t a k e the ..factors a f f e c t e d by t h a t o u t p u t i n t o c o n s i d e r a t i o n . derived  The  satisfaction  from t h e o u t p u t depends on t h e r e l a t i v e e f f e c t o f t h e  o u t p u t on t h e s e f a c t o r s and r e l a t i v e s t a t e s and i m p o r t a n c e o f 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 assessing  the b e n e f i t  derived  from a p r o b a b i l i s t i c d e s i g n p r o c e s s i n w h i c h t h e o u t p u t s 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  follows : a)  Sales  revenue.  I n c r e a s e d r e l i a b i l i t y and/or m i s s i o n a l l o w an i n c r e a s e volume.  ability  i n s a l e s p r i c e or s a l e s  The i n c r e a s e  from t h e a b i l i t y  t i m e may  i n s a l e s revenue r e s u l t i n g  t o ask h i g h e r  p r i c e s or the  t o s e l l a l a r g e r volume i s o b v i o u s l y  o f b e n e f i t t o t h e d e c i s i o n maker and h i s organization.  17 b)  Reputation The r e p u t a t i o n o f t h e 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 t h e r e l i a b i l i t y and t h e m i s s i o n time o f t h e d e s i g n .  The h i g h e r  these values a r e , the higher i s the r e p u t a t i o n for  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 t h e d e c i s i o n maker., such as s t r o n g e r market p o s i t i o n , h i g h e r p r i c e s , and f a v o u r a b l e consumer b i a s . c)  Objectives The o b j e c t i v e s o f t h e d e c i s i o n maker a r e a f f e c t e d by t h e o u t p u t v a l u e s .  A longer  m i s s i o n time i s o f more b e n e f i t t o a d e c i s i o n maker who wants t h e d e s i g n p e r f o r m  f o r a long  time, while i t i s of l e s s b e n e f i t t o a d e c i s i o n maker who wants t h e d e s i g n  re-  p l a c e d i n t h e near f u t u r e . d) " M a r k e t i n g C o n s i d e r a t i o n s The m a r k e t i n g  s t r a t e g y o f t h e d e c i s i o n maker  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 - . The market, i n whjch t h e d e c i s i o n maker d e s i r e s t o operate  ( f o r example , h i g h q u a l i t y ,  i n t e r m e d i a t e q u a l i t y , o r low q u a l i t y m a r k e t s , ) d e s i r e d market s h a r e , s i d e p r o d u c t m a r k e t s , t h e  fhe  18 maintanence market, strategy. strategy  e)  The  the  marketing  by t h e d e c i s i o n maker  i n technology,  and o t h e r  design  should  into  outside  designated  obsolete,  i s  changes i n customer factors  obsolescence,  be c o n s i d e r e d  A design  may  these  i n assessing to operate  bring factors  the  after  benefit. i ti s  over-designed.  c o n s i d e r i n g t h e above f a c t o r s  to assess  the  relative  t h e d e c i s i o n maker makes u s e o f a l l p e r t i n e n t  information How  marketing  Obsolescence  tastes,  benefits,  the  b e n e f i t depends on t h e  formulated  Development  In  a l l affect  and u t i l i z e s  these  h i s experience.  considerations are utilized  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  when  i n  determining  Sections IIB  and I I C . 3.  Preference  functions  A d e c i s i o n maker w i l l with  a higher  b e n e f i t to a state of output  lower  benefit.  value  o f an output  values will of  prefer a state of output  In the case variable  f o r the variable,  prefer a higher  the variable.  variables with  a  where b e n e f i t i n c r e a s e s as t h e i n c r e a s e s w i t h i n an  i n that  value  variable  interval  of the variable  interval  of  t h e d e c i s i o n maker t o a lower  Benefit i s a function of the  output  value  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 p r e f e r e n c e f u n c t i o n . In p r o b a b i l i s t i c d e s i g n , the o u t p u t  i s the r e l i a b i l i t y  m i s s i o n t i m e , and b e n e f i t i s a f u n c t i o n o f b o t h , II.B.  and  and  (see S e c t i o n s  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 i s f i x e d , forms t h e p r e f e r e n c e the g i v e n m i s s i o n time.  , when m i s s i o n  function for r e l i a b i l i t y  time at  B e n e f i t as a f u n c t i o n o f m i s s i o n  time  a t a f i x e d r e l i a b i l i t y v a l u e , forms t h 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 at the g i v e n r e l i a b i l i t y  value.  20 11. B.  B e n e f i t o f M i s s i o n Time a t a G i v e n  Value o f  Reliability  1.  General  The  preference  function f o rmission  y(n|R),  reliability,  i s analyzed  minimum and maximum m i s s i o n time, n . , ' mm assessed  time  values.  beneficial  Maximum m i s s i o n  time,  n  constant between  Minimum  by t h e d e c i s i o n maker f r o m e x p e r i e n c e  that value  in  of a  i n the i n t e r v a l  i s d e f i n e d as the lowest  information.  result  time  mission value  and p e r t i n e n t  i s defined~as  a f t e r w h i c h an i n c r e a s e i n m i s s i o n  time  i n an i n c r e a s e i n t h e b e n e f i t o r r e s u l t s  does n o t  in a  decrease  the b e n e f i t . In t h e i n t e r v a l  times, time  (  n m  i '  at given  function. and  time  a  x  ^'  t  ^  P  i e  reliability,  r  e  f  e  r  e  n  c  time,  y (n |R) ,  isa  non-decreasing  by d e f i n i t i o n ,  time  increases,  i s t h e p o i n t where  F o r d i s c u s s i o n o f t h e above  statement,  II.B;2.  mission  preference declines.  time  increases the marginal  function, y(n), That  f o runit  is,the marginal  d e c i s i o n maker d e c l i n e s a s m i s s i o n preference  mission  function f o r mission  e  B e n e f i t i n c r e a s e s as t h e m i s s i o n  i s maximum.  Section As  the  m  maximum m i s s i o n  benefit see  n  n  b e t w e e n minimum and maximum  time.  increase i n mission  b e n e f i t d e r i v e d by the  time  f u n c t i o n has a n o n - i n c r e a s i n g  z e r o a t maximum m i s s i o n  increase i n  increases.  The  slope reaching  The above s t a t e m e n t s a r e d i s c u s s e d 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 o f the d e c i s i o n p r o c e s s i s t o m i n i m i z e 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 r e a s o n m i s s i o n time  v a l u e s l a r g e r t h a n maximum m i s s i o n t i m e , n , ' max'  which  r e s u l t i n i n c r e a s e d c o s t but d e c r e a s e d b e n e f i t a r e not considered i n the a n a l y s i s .  22  2.  Non-decreasing  Preference Function,  As s t a t e d i n the p r e v i o u s 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 a t g i v e n r e l i a b i l i t y , i s non-decreasing. a)  The  T h i s i s due  V(  n  t o f o l l o w i n g factor;-;  l o n g e r i s the m i s s i o n t i m e , the h i g h e r  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 d e s i g n .  In  the case the d e c i s i o n maker i s the d e s i g n e r u s e r o f the d e s i g n , l o n g e r usage of the d e s i g n i m p l i e s h i g h e r revenue per specimen, s i n c e t h e r e a r e fewer r e p l a c e m e n t s  of the d e s i g n .  In t h e case the decisionmaker  i s the d e s i g n e r -  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 i n h i g h e r revenue.  result  S i n c e an i n c r e a s e o c c u r s i n  revenue t h i s i s an i n c r e a s e i n the b e n e f i t . F o r example, i f t h e d e s i g n i s a machine t o o l , l o n g e r usage o r h i g h e r p r i c e s i n c r e a s e s t h e revenue o f 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 l o n g e r time i f the m i s s i o n time i s l o n g e r and  one  s o l d by t h e d e c i s i o n maker y i e l d s h i g h e r p r i c e s and/or l a r g e r  sales.  23 b)  A design with a longer mission  time has a  h i g h e r r e p u t a t i o n v a l u e f o r the d e c i s i o n  maker.  It  enjoys  This  is  o f b e n e f i t to the d e c i s i o n maker,  the r e p u t a t i o n o f d u r a b i l i t y .  i t may prompt buyers  to r e g a r d a l l  by the d e c i s i o n maker as d u r a b l e , the d e s i g n i n  because  designs including  question.  For example, a d e c i s i o n maker who d e s i g n s a machine t o o l w i t h a l o n g e r m i s s i o n  time has  the r e p u t a t i o n o f d u r a b l e d e s i g n and  this  will  future  be o f b e n e f i t i n p r e s e n t and i n  sales. will  New d e s i g n s by t h a t d e c i s i o n maker  be e a s i e r to i n t r o d u c e to the m a r k e t ,  and are l i k e l y a b l e to command h i g h e r c)  prices.  Market s h a r e f o r the d e s i g n may i n c r e a s e  as  it  for  is  longer-lasting.  Among the d e s i g n s  the same p u r p o s e , one w i t h a l o n g e r  mission  time may be p r e f e r r e d by the buyer and the market s h a r e may i n c r e a s e , g i v i n g the  decision  maker the o p p o r t u n i t y to expand a n d / o r  operate  a t more b e n e f i c i a l  is  price levels.  i n c r e a s e i n the b e n e f i t f o r the  This  an  decision  maker. For example, a machine t o o l w i t h a h i g h e r mission buyers  time w i l l and i t s  be chosen more o f t e n by the  share  i n the market i s  likely  24  to i n c r e a s e . will  This  i n c r e a s e i n market  enable the d e c i s i o n maker to  expanding  its  production l i n e ,  him to i n c r e a s e h i s  share  consider  and enable  volume to a more optimal  value.  These f a c t o r s  cause the b e n e f i t to i n c r e a s e as the m i s s i o n  i n c r e a s e s , but t h e i r e f f e c t s a r e not c o n s t a n t . not i n c r e a s e a t a c o n s t a n t discussed  That i s ,  b e n e f i t does  r a t e as m i s s i o n time i n c r e a s e s .  i n the next s e c t i o n .  time  This  is  25  3.  N o n - i n c r e a s i n g ' Slope of P r e f e r e n c e F u n c t i o n .  As s t a t e d i n s e c t i o n  as the m i s s i o n time i n c r e a s e s  J I B I  the m a 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 f o r m i s s i o n time  at given r e l i a b i l i t y ,  m i s s i o n time  ft  jin/R) ,  for unit increase i n  d e c l i n e s and approaches z e r o a t t h e  maximum m i s s i o n time  n m  a  x  •  That i s , i t has a n o n - i n c r e a s i n g  slope. The m a r g i n a l i n c r e a s e i n b e n e f i t d e c l i n e s because o f the f a c t o r s l i s t e d below'.  a)  H i g h e r m i s s i o n time v a l u e s b r i n g the d e s i g n c l o s e r to p o t e n t i a l obsolescence end o f i t s l i f e . T h a t i s , the  near the  probability  t h a t t h e d e s i g n w i l l be o b s e l e t e p r i o r t o c o m p l e t i o n o f i t s m i s s i o n time i n c r e a s e s . The  advancement of t e c h n o l o g y and change i n  consumer-tastes  and  i n d u s t r y - n e e d s and  other  o u t s i d e f a c t o r s cause d e s i g n s t o become obselete.  VThen a d e s i g n becomes o b s e l e t e ,  the r e m a i n i n g  l i f e time i s o f l i t t l e  value.  Buyers do not want t o commit themselves  to a  d e s i g n t h a t has a h i g h p r o b a b i l i t y o f becoming o b s e l e t e b e f o r e the c o m p l e t i o n o f i t s m i s s i o n time.  T h i s f a c t o r n e g a t e s the second  in Section II.B.2,that longer mission  factor time  r e s u l t s i n higher reputation since obselete d e s i g n s w i l l r e s u l t i n r e d u c i n g the  positive  26 a f f e c t t h a t l o n g e r l i f e time has on r e p u t a t i o n . In t h e case a machine t o o l i s t h e d e s i g n c o n s i d e r e d , advancement i n t e c h n o l o g y  (better  d e s i g n c o n c e p t s ) , change i n i n d u s t r v needs ( s h i f t i n demand), and o t h e r e x p e c t e d and unexpected o u t s i d e f a c t o r s  (such as energy  s h o r t a g e ) may cause a machine t o o l t o become obselete w i t h i n a certain l i f e life).  span  A machine t o o l b u i l t t o l a s t  (product longer  has l e s s m a r g i n a l b e n e f i t as t h e m i s s i o n time i n c r e a s e s . b)  As t h e m i s s i o n time i n c r e a s e s t h e t o t a l market f o r t h e d e s i g n d u r i n g some extended t i m e p e r i o d s h r i n k s and t h e demand f o r new items d e c l i n e s because t h e i t e m s a l r e a d y i n use l a s t l o n g e r .  When d e s i g n s have a h i g h  mission time, f a i l u r e occurs less f r e q u e n t l y and t h e n e c c e s i t y o f r e p l a c e m e n t Even  decreases.  though, as s t a t e d i n F a c t o r C i n s e c t i o n  I I B2., t h e market share f o r t h e d e s i g n i n c r e a s e s , t h e t o t a l market s h r i n k s and t h e a c t u a l market volume f o r t h e d e s i g n does n o t i n c r e a s e as much.  Hence, t h e b e n i f i c i a l  a f f e c t o f i n c r e a s i n g market share  decreases  as t h e m i s s i o n time i n c r e a s e s . In t h e machine t o o l i n d u s t r y , f o r example, suppose t h a t a m a n u f a c t u r e r  designs f o r a  h i g h e r m i s s i o n time than o t h e r s do.  The market  share  i n c r e a s e s , but s i n c e the items  i n use  l a s t l o n g e r , the t o t a l demand i n the long run decreases.  Hence, the t o t a l volume does not  i n c r e a s e as much as the i n c r e a s e i n the market share.  Longer m i s s i o n times and a s s o c i a t e d l o n g e r revenue may tend t o p r i c e the d e s i g n out o f i t s market. beneficial  The buyer, even i f i t may be more i n the long r u n , may not be w i l l i n g  or a b l e t o g i v e the h i g h e r p r i c e f o r the d e s i g n w i t h the h i g h e r m i s s i o n time.  Hence, as  m i s s i o n time.'increases and p r i c e i n c r e a s e s i n accordance,  the buyer group o f the d e s i g n  gets s m a l l e r .  Although  the revenue f o r each  d e s i g n i n c r e a s e s , as l e s s items a r e s o l d , the t o t a l revenue w i l l not i n c r e a s e as much, and i f m i s s i o n time  i s very h i g h t o t a l  revenue  may, i n f a c t , d e c l i n e . T h i s f a c t o r l e s s e n s the e f f e c t o f the factor A  i n S e c t i o n I I B 2 , t h a t the h i g h e r m i s s i o n  time  r e s u l t s i n h i g h e r revenue f o r the d e c i s i o n maker. In the machine t o o l i n d u s t r y , f o r example, the designs w i t h h i g h e r m i s s i o n time may have s m a l l e r buyer groups.  The d e c i s i o n maker  d e s i g n i n g f o r a h i g h e r m i s s i o n time may not r e c e i v e as much revenue as those f o r lower m i s s i o n time.  designing  Even though the  28  u n i t revenue ( p r i c e p e r d e s i g n specimen) i s h i g h e r , s a l e s volume may be l o w e r , 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 , t h e p r e f e r e n c e time a t c o n s t a n t r e l i a b i l i t y ,  function f o r mission  y (n |R) , i s a  function with a non-increasing slope.  non-decreasing  When t h e s l o p e r e a c h e  zero, t h e c o r r e s p o n d i n g m i s s i o n time i s d e f i n e d as t h e maximum m i s s i o n t i m e , n . ' max *  The f u n c t i o n  y ( n | R ) i s as shown  i n F i g . 2 . 1 i n t h e 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 ,  (  n m  j_ ' n  n m a x  )•  29  Benefit  /i(n,/R)  /  Dmln.  Fig.  /  /  /  y  Mission Time, n Dinew.  2.1. 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 a t 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 a t a Given Value o f M i s s i o n Time  1.  General  The p r e f e r e n c e 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 mission time, y(R|n). is a non-decreasing /  i n t h e range between minimum and maximum  function  reliability,  (R min/ . Rmax ) . Minimum r e l i a b i l i t y ,' Rmm' . , i s a s s e s s e d by •* t h e d e c i s i o n maker as t h e l o w e s t v a l u e a c c e p t a b l e i n t h e i n d u s t r y . A reliability  v a l u e o f one i s n o t 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 d e f i n e d as t h a t v a l u e  ( l e s s t h a n one)  t h a t i s t e c h n o l o g i c a l l y and/or e c o n o m i c a l l y a t t a i n a b l e . As t h e ' v a l u e o f r e l i a b i l i t y the d e c i s i o n maker i n c r e a s e s . the r e l i a b i l i t y  increases, the benefit t o  I n t h e subsequent a n a l v s i s ,  v a l u e s between minimum and maximum  relia-  b i l i t y a r e c o n s i d e r e d o n l y , s i n c e t h e v a l u e s below t h e former a r e n o t a c c e p t a b l e , and v a l u e s above t h e l a t t e r a r e not  attainable. As t h e r e l i a b i l i t y  increases, the marginal increase  in the preference function f o r r e l i a b i l i t y m i s s i o n t i m e , u(R|n) declines.  y  a t constant  for a unit increase i n r e l i a b i l i t y  Hence, u(R|n) f e a t u r e s a n o n - i n c r e a s i n g s l o p e .  S i n c e t h e maximum r e l i a b i l i t y  R max  i s d e f i n e d as t h e l a r g e s t  r e l i a b i l i t y v a l u e w h i c h i s t e c h n o l o g i c a l l y and/or economically  f e a s i b l e , the slope of the preference f u n c t i o n f o r  c o n s t a n t m i s s i o n t i m e , must be z e r o a t R , s i n c e no f u r t h e r ' max' i n p u t o f t e c h n o l o g i c a l and o t h e r r e s o u r c e s c o u l d p o s s i b l y  31  result in a r e l i a b i l i t y  increase.  T h e r e f o r e , the p r e f e r e n c e 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 c o n s t a n t mission  t i m e , y(R|n)  i s a concave 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 . presented i n Section  The reason f o r t h e above statements a r e  II.C.2  and I I . C . 3 .  32  2.  Non-decreasing  The  preference function for r e l i a b i l i t y  m i s s i o n time,  y(R|n) j  non-decreasing bility  Preference  Function.  at a given  d e r i v e d by the d e c i s i o n maker i s a  function.  That  i s , the h i g h e r the  relia-  i s , l a r g e r i s the b e n e f i t the d e s i g n p r o v i d e s to the  d e c i s i o n maker. The the above  f o l l o w i n g f a c t o r s comprise the r e a s o n i n g  behind  statement.  a)  The h i g h e r the p r o b a b i l i t y t h a t a d e s i g n does not f a i l b e f o r e the completion  of the m i s s i o n  time, the h i g h e r i s the revenue d e r i v e d the d e s i g n .  Suppose t h e r e e x i s t two  of, 1 0 0 d e s i g n items w i t h r e l i a b i l i t y and  1-2F',  respectively.  from  samples 1-F •  Before the comp-  l e t i o n of m i s s i o n time, on the average F' items  fail  i n the former  items  fail  i n the l a t t e r case.  of  the d e s i g n e r - u s e r has more items and needs fewer replacements,  i n operation  so t h a t he  (1-2F ). 1  i s a b l e to ask h i g h e r p r i c e s  due  to obvious  items  A designer-  per item) f o r the items w i t h h i g h e r bility,  case  (1-F'),  d e r i v e s higher revenue than w i t h the  seller  1  In the  items w i t h h i g h e r r e l i a b i l i t y ,  w i t h lov/er r e l i a b i l i t y ,  2F '  case w h i l e  (revenue relia-  h i g h e r revenue f o r the  33  user, than he i s a b l e t o ask f o r the lower reliability  items.  For example, i f the d e s i g n item i s a machine t o o l , the h i g h e r r e l i a b i l i t y h i g h e r revenues through a h i g h e r percentage time r e s u l t i n g repairs. will  items w i l l  return  h i g h e r p r i c e s o r through  completing  the m i s s i o n  i n fewer replacements  and/or  The revenue o f the d e c i s i o n maker  i n c r e a s e as the r e l i a b i l i t y  o f the item  increases. b)  An i n c r e a s e i n the r e l i a b i l i t y  o f the d e s i g n  r e s u l t s i n a h i g h e r market share item.  f o r the d e s i g n  A buyer t y p i c a l l y p r e f e r s a more r e l -  i a b l e item t o one w i t h lower the same m i s s i o n time.  r e l i a b i l i t y at  A l l other things being  e q u a l , a higher r e l i a b i l i t y  item s e l l s b e t t e r  and t h e r e f o r e o b t a i n s a h i g h e r share o f the market.  T h i s t r e n d may r e v e r s e a f t e r a c e r t a i n  v a l u e 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 w i t h a h i g h e r reliability  will  be chosen more o f t e n by  buyers than a machine t o o l w i t h a lower reliability.  A customer w i l l  be i n c l i n e d t o  p r e f e r a machine t o o l t h a t 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 g i v e n m i s s i o n time t o one t h a t has a h i g h e r p r o b a b i l i t y to f a i l  (lower  reliability).  c)  A design with higher r e l i a b i l i t y o p e r a t i o n , and  has a s a f e r  f o r c r i t i c a l designs,  this  f e a t u r e i s of c o n s i d e r a b l e b e n e f i t , An in r e l i a b i l i t y an u n d e s i r e d  decreases  shutdown,  the p r o b a b i l i t y of work a c c i d e n t s , and  unplanned and/or unwanted d e l a y s . in r e l i a b i l i t y items  increase  An  increase  makes the o p e r a t i o n of d e s i g n  s a f e r and more dependable, and  so i n -  c r e a s e s the b e n e f i t to a d e c i s i o n maker. A s a f e r o p e r a t i o n r e s u l t s from a machine w i t h high r e l i a b i l i t y ,  as an example.  Since t h e r e  i s a lower 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 i t s m i s s i o n time, the machine t o o l w i t h reliability  has  lower  expected  r e s u l t i n g from d e l a y s and The  above statements  reliability  shutdowns.  demonstrate t h a t an i n c r e a s e i n  Since t h e r e i s an i n c r e a s e i n b e n e f i t  w i t h an i n c r e a s e i n r e l i a b i l i t y , a higher r e l i a b i l i t y  at a given mission  d e s i g n i s p r e f e r r e d t o a lower  item by the d e c i s i o n maker.  f u n c t i o n of r e l i a b i l i t y non-decreasing The  l o s s i n revenue  causes the b e n e f i t of the d e s i g n to d e c i s i o n  maker to i n c r e a s e .  liability  higher  Hence, the  a t a c o n s t a n t m i s s i o n time  time, re-  preference is a  function.  increase i n r e l i a b i l i t y  in b e n e f i t at a constant r a t e .  does not imply an i n c r e a s e As the r e l i a b i l i t y  approaches  i t s maximum v a l u e , the i n c r e a s e i n b e n e f i t d e c l i n e s . point i s discussed i n Section  II C.3. %  This  35  3.  Non-increasing  With i n c r e a s i n g  Slope of Preference Function  r e l i a b i l i t y values,  the marginal  increase  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 mission per 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 R, d e c r e a s e s . non-increasing  slope.  The f a c t o r s  i n c r e a s e o f y(R|n) a r e d i s c u s s e d  a)  That i s ,  y(R|n) has a  below.  d e s i g n items w i l l  t h a t more o f the  not have f a i l e d a t  c o m p l e t i o n o f the m i s s i o n  time.  This  the results  f r e q u e n t replacements and t h e r e f o r e  a shrinkage Although  y(R|n),  t h a t cause a d e c l i n e i n the m a r g i n a l  A higher r e l i a b i l i t y implies  in less  time,  in  f o r the demand o f the d e s i g n i t e m .  the d e s i g n item w i t h h i g h e r  may cause more buyers to p r e f e r i t reliability  item ( c a u s i n g  reliability  to a lower  the market share  i n c r e a s e f o r the h i g h e r r e l i a b i l i t y i t e m ) , total  market s h r i n k s  Hence the  i n revenue may not be as much as  its is  increase  anticipated  As the r e l i a b i l i t y gets c l o s e r  maximum v a l u e the i n c r e a s e i n market o f f s e t t o some e x t e n t by s h r i n k a g e  market demand.  the  due to a d e c r e a s e i n demand  caused by fewer f a i l u r e s .  otherwise.  to  to  share  i n the  36  When a machine t o o l i s c o n s i d e r e d as an example, a h i g h e r r e l i a b i l i t y  t o o l may c a p t u r e  a h i g h e r s h a r e o f t h e market, b u t as t h e reliability  i n c r e a s e s , t h e t o t a l market de-  mand d e c r e a s e s , because t h e r e i s l e s s need to  r e p l a c e t h e e x i s t i n g machine t o o l s .  The  d e c r e a s e i n market demand causes t h e i n c r e a s e i n revenue t o d e c l i n e . b)  A higher r e l i a b i l i t y  value, with higher p r i c e  (revenue p e r item) may t e n d t o p r i c e t h e d e s i g n i t e m o u t o f t h e market.• T h i s may cause a d e c l i n e i n t h e market s h a r e , and an accompanying dec l i n e i n t h e t o t a l revenue. reliability  Increasing  i n c r e a s e s t h e market s h a r e , b u t  the p r i c e o f t h e 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 b u y e r , t h e buyer chooses a: l o w e r r e l i a b i l i t y i t e m because t h e 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 s u r p a s s e s t h e advantages o f higher r e l i a b i l i t y  f o r the buyer.  Hence,  as t h e r e l i a b i l i t y  becomes h i g h e r t h e r a t e  o f i n c r e a s e i n revenue d e c l i n e s . For. example, a buyer f o r a machine chooses a h i g h e r r e l i a b i l i t y things being equal.  tool  t o o l , a l l other  But, a f t e r the p r i c e of  the d e s i g n i t e m goes beyond t h e amount t h e buyer can o r i s w i l l i n g  t o a l l o c a t e , he may  37  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 b u y e r , b u t t h e p r i c e increase^  as r e l i a b i l i t y  becomes v e r y  highj  may cause t h e p r i c e t o be beyond t h e amount which most b u y e r s c a n 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 t h e d e s i g n item.  As t h e 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 t h e s e s i d e markets w i l l decrease. gets higher,  Hence, as t h e r e l i a b i l i t y  the increase  i n revenue  through  more s a l e s i n t h e p r i m a r y market i s negated by t h e d e c r e a s e i n volume i n p a r t s and maintanence market. For example, a machine t o o l d e s i g n e r i s a b l e t o 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 t h e r e l i a b i l i t y of t h e d e s i g n i t e m increases,. The  loss i n  b e n e f i t i n s i d e markets n e g a t e s t h e i n c r e a s e i n b e n e f i t by t h e 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 b u l k i e r and h e a v i e r  typically results i n designs which i s not  d e s i r a b l e f o r many d e s i g n s . .  Increase i n s i z e  and  w e i g h t mav go beyond t h e d e s i r a b l e  for  many  limits  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 implies  o f d e s i g n items  complete the m i s s i o n  without f a i l u r e .  This  increases  t h a t a d e s i g n item w i l l still  i n use.  technological implies marginal  that a l a r g e r time  the  probability  become o b s o l e t e w h i l e  For designs s u b j e c t t o  with i n c r e a s i n g  reliability.  (especially  e x c e s s i v e dimensions impractical  often  For such d e s i g n s ,  increase in benefit c l e a r l y  a power t o o l  rapid  change, high r e l i a b i l i t y  overdesign.  number  the  declines  As an example,  a p o r t a b l e one)  with  a n d / o r weight may be  i n usage and even though  it  is  more r e l i a b l e may not be d e s i r a b l e f o r many decision  In  makers.  c o n c l u s i o n , the above f a c t o r s  cause the marginal  increase  b e n e f i t per 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 to d e c l i n e as the increases.  In S e c t i o n  II.  B.2.,  of as  (RJn)  function. is  In  this  non-increasing.  the l a r g e s t  reliability  i t was seen t h a t the p r e f e r e n c e  f u n c t i o n for. r e l i a b i l i t y a t a c o n s t a n t m i s s i o n decreasing  in  section  it  is  time,  (RJn),  i n d i c a t e d t h a t the  S i n c e maximum r e l i a b i l i t y &  r e l i a b i l i t y v a l u e which i s  is  m x  technologically  a nonslope  is  defined  and/or  e c o n o m i c a l l y f e a s i b l e , the s l o p e o f the p r e f e r e n c e f u n c t i o n f o r c o n stant mission  time,  ( R J n ) , must be zero a t maximum r e l i a b i l i t y ,  no f u r t h e r i n p u t o f t e c h n o l o g i c a l result  i n an i n c r e a s e i n  and o t h e r r e s o u r c e c o u l d  reliability.  since  possibly  39  The f u n c t i o n  u(R|n)  i s therefore  a  non-decreasing  f u n c t i o n , w i t h a n o n - i n c r e a s i n g s l o p e ,and t h e s l o p e i s z e r o a t t h e maximum v a l u e  R I n t h e i n t e r v a l between max * minimum and maximum r e l i a b i l i t y v a l u e s (R . ,R ), the mm' max ' J  preference function f o r r e l i a b i l i t y a t a given mission time, u(R|n),  i s as shown i n F i g . 2 . 2 .  40  Benefit  V  /i(n.(R)  /  /  /  /  /  Reliability Rmax.  Rmin.  Fig.  2.2. The P r e f e r e n c e F u n c t i o n 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(R|n')  41  II.  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.  1.  Interdependence of P r e f e r e n c e  As s t a t e d i n S e c t i o n n . C . , of  the m i s s i o n time, R(n).  f u n c t i o n i s developed time, u(R|n) and for  Functions.  reliability  i s a function  In t h a t s e c t i o n , a p r e f e r e n c e  for r e l i a b i l i t y  at constant  mission  i n S e c t i o n I I . B. a p r e f e r e n c e f u n c t i o n  m i s s i o n time a t c o n s t a n t r e l i a b i l i t y ^ y ( n | R ) , Since r e l i a b i l i t y  is  developed.  and m i s s i o n time are dependent on  each o t h e r , (see S e c t i o n llA.).the b e n e f i t of r e l i a b i l i t y o n l y meaningful and v i c e v e r s a . m i s s i o n time y(R,n). time  when i t i s d e f i n e d a t a g i v e n m i s s i o n  (n N  i s a three dimensional  y ( R , n ) i n c r e a s e s as r e l i a b i l i t y  and/or m i s s i o n  increases  as r e -  (see S e c t i o n s II.B.  I I . C . ) , thus forming a concave s u r f a c e w i t h i n the n  max  ) and  (R  intervals  . R ). min' max'  In order to express b e n e f i t as a f u n c t i o n of reliability  and  preference function,  i n c r e a s e s and the s l o p e of y(R,n) decreases  . mm?  time  The b e n e f i t r e s u l t i n g from r e l i a b i l i t y  l i a b i l i t y and/or m i s s i o n time and  is  and m i s s i o n time,  indifference  f u n c t i o n s are i n t r o d u c e d i n S e c t i o n I I . D.  both  (constant b e n e f i t ) 2.  42 I I . D. 2.  B e n e f i t o f R e l i a b i l i t y and M i s s i o n Time Indifference  Function  A d e c i s i o n maker i s e x p e c t e d t o be i n d i f f e r e n t t o two  choices  which y i e l d  o f r e l i a b i l i t y and m i s s i o n  time combinations  t h e same b e n e f i t t o t h e d e c i s i o n maker..That i s ,  t h e d e c i s i o n maker i s i n d i f f e r e n t between ( R j n ) and 1  (Ryn") when y(R',n'-) i s e q u a l toy(R','n").  since the concern  of t h e d e c i s i o n maker i s t o i n c r e a s e h i s b e n e f i t , two s e t s of t h e parameters r e l i a b i l i t y and m i s s i o n  t i m e would be  o f same v a l u e t o him as l o n g as t h e combined b e n e f i t der i v e d by t h e d e c i s i o n maker i s t h e 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 (R,n)  w i t h an e q u a l v a l u e  of benefit,y(R,n).  of a l l sets  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 t h e r e l i a b i l i t y and m i s s i o n  time.  When t h e p r e f e r e n c e f u n c t i o n f o r r e l i a b i l i t y a t c o n s t a n t mission  t i m e , y ( R |n), and 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 c o n s t a n t r e l i a b i l i t y , y ( n | R ) , a r e b o t h concave functions  (as i s t h e case i n t h i s a n a l y s i s , see S e c t i o n s  I I . B. and I I . C . ) , t h e i n d i f f e r e n c e f u n c t i o n s a r e 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 o f i n d i f f e r e n c e are discussed  i n Section  I I . E. 5.  functions  43  Mission Time  n mox.  n min. • Reliability Rmax.  Rmin.  Fig.  2.3. I n d i f f e r e n c e F u n c t i o n s f o r R e l i a b i l i t y and M i s s i o n Time,  I(R|y=cons.).  44 II.  E.  Derivation of Preference Functions  1.  I n f o r m a t i o n Required to C o n s t r u c t  In  Preference  Functions  o r d e r to c o n s t r u c t the p r e f e r e n c e f u n c t i o n s , c e r t a i n  mation i s  r e q u i r e d from the d e c i s i o n maker.  to a s s e s s the r e l a t i v e b e n e f i t he d e r i v e s  The d e c i s i o n maker  from a c e r t a i n  information  several  is  questions,  in following This  a c q u i r e d by having  which i s  explained  sections.  information  is  u t i l i z e d to construct preference functions constant  values  of mission time.  t h e n , are used to c o n s t r u c t the i n d i f f e r e n c e  introduced in  combinations.  the d e c i s i o n maker respond to  and from a " r e f e r e n c e gamble,"  for r e l i a b i l i t y at several functions,  is  combination  o f r e l i a b i l i t y and m i s s i o n time compared to c e r t a i n o t h e r This  infor-  II.D.3.  These  functions,  45  2.  Questions  t o D e c i s i o n Maker.  The d e c i s i o n maker i s t o respond t o t h e f o l l o w i n g questions, i n order t o assess h i s preference reliability questions  a t a given mission time.  functionf o r  The purpose o f t h e  i n t h i s s e c t i o n i s t o l o c a t e t h e i n i t i a l and end  points of the preference  function f o rr e l i a b i l i t y  at 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 mission v a l u e s n^ u s i n g a common s c a l e .  time  That i s , t h e 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 t h e d e c i s i o n maker f o r t h e d e s i g n a t minimum r e l i a b i l i t y ,' Rmm' . . and maximum r e l i a b i l i t y ', 1  time v a l u e s  J  Rmax f. a t m i s s i o n  n. i"  As s t a t e d i n S e c t i o n II.B..& I I . C . , v a l u e s f o r t h e minimum m i s s i o n time  n .  i  and maximum r e l i a b i l i t y •*  R„ • >  minimum r e l i a b i l i t y  m m '  -  1  R  m m v  ,are d e f i n e d and t h e r e f o r e max,  3  t a k e n as g i v e n . a  j  The f i r s t q u e s t i o n r e l a t e s t h e 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 time.  v a l u e s a t minimum m i s s i o n  The i n f o r m a t i o n i s used t o l o c a t e t h e i n i t i a l and  end p o i n t s o f t h e p r e f e r e n c e  function forr e l i a b i l i t y at  minimum m i s s i o n t i m e , y(R|n . min  ).  These p o i n t s a r e a l s o  used as t h e b a s i s from w h i c h t o d e t e r m i n e t h e i n i t i a l and end p o i n t s o f a l l o t h e r p r e f e r e n c e  f u n c t i o n , y ( R l n . i ) , by  u s i n g i n f o r m a t i o n obtained_ from q u e s t i o n 2 , i n t h i s s e c t i o n .  Question  1.  G i v e n t h e minimum m i s s i o n time  n=n . , min'  what would be t h e r a t i o o f t h e b e n e f i t der i v e d a t maximum r e l i a b i l i t y  R  , to the max' b e n e f i t d e r i v e d a t minimum r e l i a b i l i t y R . ; •* min m.=y(R ,n . )/y(R . ,n . ) 1 max' min ' mm' mm' I f we l e t y(R . ,n . ) = y and s i n c e our c o n c e r n i s mm' mm' o J  K V  K X  K  i n t h e r e l a t i v e b e n e f i t , l e t us assume some c o n s t a n t  value  for y . Q  Therefore; y(R ,n . ) = m,y max' min 1 o  (2.1)  H x  A v a l u e f o r y(R . ) , ( b e n e f i t a t maximum r e l i a max' ,n min'' H v  b i l i t y and minimum m i s s i o n time)  i s acquired u t i l i z i n g the  i n f o r m a t i o n o b t a i n e d from q u e s t i o n 1., and assuming a constant f o r y . reliability  The i n i t i a l and end p o i n t s f o r t h e b e n e f i t  f u n c t i o n a t minimum m i s s i o n time a r e l o c a t e d  u s i n g t h e above i n f o r m a t i o n as shown i n Fig.2.4  (marked by  circles). b)  The second q u e s t i o n r e l a t e s t h e b e n e f i t s d e r i v e d a t  d i f f e r e n t v a l u e s o f m i s s i o n t i m e , n^, g i v e n t h e minimum reliability, R . . ' min 2  The r a t i o o f b e n e f i t d e r i v e d a t t h e s e  v a l u e s o f m i s s i o n t i m e , n^, g i v e n  R m  ^ / n  t o  t  n  e  benefit  d e r i v e d a t t h e minimum r e l i a b i l i t y and minimum m i s s i o n  time,  y(R . ,n . ) =y , l o c a t e s t h e i n i t i a l p o i n t s o f t h e p r e f m i n ' min' o ' H v  erence time.  p  c  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 v a l u e s of m i s s i o n  47  Benefit  Po  Reliability Rmax.  Fig.  2.4. L o c a t i o n o f I n i t i a l  and End P o i n t s o f  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. function i t s e l f obtained at t h i s  (Note t h a t the p r e f e r e n c e  -dashed l i n e - has not been point.)  48  The maximum m i s s i o n time i s d e f i n e d i n S e c t i o n  II.B.l  as t h a t v a l u e o f m i s s i o n t i m e , a f t e r w h i c h f u r t h e r i n c r e a s e s i n m i s s i o n time do h o t r e s u l t i n f u r t h e r i n c r e a s e s i n benefit.  The minimum m i s s i o n t i m e i s d e f i n e d i n t h e same  s e c t i o n as t h e l o w e s t u s e f u l v a l u e . s e v e r a l v a l u e s o f m i s s i o n time m i s s i o n time g i v e n by n ' max  n^, denote t h e maximum  and d i v i d e ( n -n . ) , max mm '  1  1  J  i.e.  In order t o o b t a i n  t h e d i f f e r e n c e between e s t i m a t e d maximum m i s s i o n  time  and g i v e n minimum m i s s i o n t i m e , by an i n t e g e r N t o o b t a i n a v a l u e , say n , d e f i n e n. such t h a t ; n 1 n.=n . - f i - nn ; i mm '  /  '  t  •••.N '  S i n c e an i n c r e a s e o f n. from n, t o n ,, i n m i s s i o n 1 N /v.+ l time does n o t r e s u l t i n an i n c 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 m i s s i o n t i m e , n  .  max Question  2.  Given t h a t the r e l i a b i l i t y  i s equal t o the  minimum r e l i a b i l i t y , R . , what would be t h e ' mm' r a t i o o f b e n e f i t d e r i v e d a t n^ t o b e n e f i t d e r i v e d a t nmm. : J  u (R . ,n. . i = l , 2 , • • • ,N mm' x ' o ' ' ' ' H v  H  L e t t h e above r a t i o be donated by P^. of preference  Then i n i t i a l  points  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 m i s s i o n time  are y(R K  . ,n.) = P.u mm' i ' i o K  o  (2.2)  n^  49 S i n c e a v a l u e was assumed f o r y using by  c)  Q  these i n i t i a l  points  a r e now l o c a t e d ,  the i n f o r m a t i o n gained i n Q u e s t i o n 2, as shown i n F i g . 2 . 5 . (marked  squares).  The t h i r d q u e s t i o n seeks  f e r e n t v a l u e s o f the m i s s i o n  the r a t i o o f the b e n e f i t d e r i v e d a t time n^, given  the maximum r e l i a b i l i t y  R „ , to t h e b e n e f i t d e r i v e d a t the minimum m i s s i o n max maximum r e l i a b i l i t y R shown i n S e c t i o n The i n f o r m a t i o n  .  m a x  >  n m  -j )  =  n  max ]V -  , as  min  m  Q  thus o b t a i n e d i s u t i l i z e d to l o c a t e the end p o i n t s  of preference functions Q u e s t i o n 3.  time n . , and the min  The b e n e f i t d e r i v e d a t R , „ and n .  max 11. E. 2b above, i s ^ ( R  dif-  f o r r e l i a b i l i t y a t n^.  Given t h a t t h e r e l i a b i l i t y i s equal reliability  to the maximum  R „ , what i s the r a t i o o f b e n e f i t d e r i v e d max m a  a t n.j to b e n e f i t d e r i v e d a t n .. :  Let the above r a t i o be denoted by Q^. functions  f o r r e l i a b i l i t y at mission  Then end p o i n t s  o f the p r e f e r e n c e  time n^, a r e  ^•WV i iv =Q  S i n c e these p o i n t s  are also  expressed i n terms o f y  on the same s c a l e as the i n i t i a l  points.  o b t a i n e d i n Q u e s t i o n 3, the end p o i n t s 2.6.  (marked by c r o s s e s ) .  m  Q  they a r e l o c a t e d  U t i l i z i n g the i n f o r m a t i o n  are l o c a t e d as shown i n F i g .  50  Benefit  -Reliability Rmax.  Rmin.  Fig.  2.5. L o c a t i o n o f I n i t i a l P o i n t s o f P r e f e r e n c e Function  for Reliability  a t n,.  51  tQa^o Q4m|  ^Jo  2 \Po  (Q  Rmin.  Fig.  m  Rmax.  Reliability  2.6. L o c a t i o n o f I n i t i a l and End P o i n t s o f Preference  Function  M i s s i o n Time V a l u e s preference  f o r R e l i a b i l i t y at n^. (Note t h a t the  f u n c t i o n s -dashed l i n e s -  not been o b t a i n e d  at this  point.)  have  3.  Reference Gamble In t h e p r e v i o u s  s e c t i o n the i n f o r m a t i o n o b t a i n e d en-  a b l e d the l o c a t i o n o f the i n i t i a l preference  and end p o i n t s f o r the  functions f o r r e l i a b i l i t y  n^: u(R,n^).  a t m i s s i o n time  values  A r e f e r e n c e gamble i s employed t o l o c a t e a t h i r d  p o i n t between i n i t i a l  and end p o i n t s .  s u i t a b l e concave curve  i s fitted  Having t h r e e p o i n t s , a  using the general  e s t a b l i s h e d i n S e c t i o n I I . C , see S e c t i o n I I . E. 4.  criteria This section  e x p l a i n s what a r e f e r e n c e gamble i s as a p p l i e d t o the case i n hand. Suppose t h a t a t a g i v e n m i s s i o n time n, the r e l i a b i l i t y of t h e d e s i g n i s u n c e r t a i n ,  Suppose a l s o t h a t t h e r e i s a  p r o b a b i l i t y , q, t h a t the r e l i a b i l i t y value  i s equal  R , and t h a t t h e r e i s a p r o b a b i l i t y max' *  (l-<3) , t h a t the '  J  reliability  t o tiie maximum  i s equal t o the minimum value R ^ .  The d e c i s i o n  maker d e r i v e s a c e r t a i n b e n e f i t from t h e g i v e n m i s s i o n and minimum r e l i a b i l i t y , maximum r e l i a b i l i t y .  o r from t h e g i v e n m i s s i o n  time  time and  Now, there i s not t h e c e r t a i n t y o f  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 o r q  of g e t t i n g e i t h e r one o r t h e o t h e r .  The b e n e f i t d e r i v e d  t h i s u n c e r t a i n s i t u a t i o n by the d e c i s i o n maker i s l i k e l y be h i g h e r than the b e n e f i t d e r i v e d i f R . ^ mm  from to •  pertained f o r  c  c e r t a i n and lower than the b e n e f i t d e r i v e d i f R pertained max for c e r t a i n . v  That i s ; ' where  y(R . ,n) <u(n,q)<y(R ,n), mm' ' '^' max' >  M V  H  V  H  X  y(n,q) i s the b e n e f i t d e r i v e d a t n with q and (1-q)  p r o b a b i l i t i e s of R r  Since  and  max  R  . .respectively. mm' r  J  y(R,n) i s a non-decreasing f u n c t i o n , there  i s a value  of R such t h a t y(RJn) = R'  y(n,q).  i s c a l l e d the c e r t a i n t y e q u i v a l e n t  of q and  (1-q)  M  r  of the u n c e r t a i n  p r o b a b i l i t i e s of R max  f u n c t i o n of q and  and  R . mm  R  is a . t  depends on the g i v e n m i s s i o n  time n:R'=R'(q| n ) .  The  v a l u e of R (q)  The  d e c i s i o n maker e s t a b l i s h e s what c e r t a i n b e n e f i t he  1  i s found as  1  state  from the  s t a t e at which t h e r e  R  R  max  and  follows.  are q and  . , r e s p e c t i v e l y , and mm' * ' •  (1-q)  R  1  1  and  I t i s thus  supposed t h a t t h e r e  f i f t y p e r c e n t chance t h a t the r e l i a b i l i t y p e r c e n t chance t h a t the r e l i a b i l i t y c  J  obtained.  gamble, a c e r t a i n t y  is R  is  R  . . mm  equivalent  a n c m  a  x  R'(.5)  At R=R'(.5), the d e c i s i o n maker d e r i v e s  Therefore,  the b e n e f i t d e r i v e d  =  °- -Vi(R 5  max  |n)+ In)  0.5.{y(R LM N  max  1  '  fifty the  is the same m a x  and  a t R'(.5) i s i t s ex-  value  E{u(R' (.50) |n)}=  is a  ^  Using  b e n e f i t as having a f i f t y p e r c e n t chance of each R  pected  of  convenience the p r o b a b i l i t y v a l u e  i s chosen.  reference  the p r o s p e c t  for certain.  For c l a r i t y q=l-q=.5  p r o b a b i l i t i e s of  d e c i d e s at which v a l u e of R , he '  J  i s i n d i f f e r e n t between the above s t a t e and obtaining  derives  • '^  0  5  + M v  (  y (R  mm  R m i n  In)  • 1  '  I> n  }  R^ m  n  .  This b e n e f i t value a t the r e l i a b i l i t y a t h i r d point f o r the preference a given mission  time  v a l u e RJ  provides  function for r e l i a b i l i t y at  n.  The r e q u i r e d i n f o r m a t i o n , t h e n , i s t h e c e r t a i n t y e q u i v a l e n t R ( . 5 | n ) a t w h i c h t h e d e c i s i o n maker would be 1  i n d i f f e r e n t t o a 50-50 gamble on R . mm  and R  mation i s obtained  .  This  infor-  max  3  from t h e d e c i s i o n maker by t h e f o l l o w i n g  question. Q u e s t i o n 4. (Reference Gamble) G i v e n t h a t t h e m i s 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 t h e r e i s a 50% and a 50% chance of ;\ r Proposition 2 i s that a ^  chance o f o b t a i n i n g R . mm 3  obtaining R . ^ max  r e l i a b i l i t y v a l u e R| can be o b t a i n e d  for certain,  At w h i c h v a l u e o f R j would you be i n d i f f e r e n t between t h e 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 i n t e r m e d i a t e  p o i n t s _ f o r the  preference  function f o r r e l i a b i l i t y are y{R!(.50)|n.} = {y ( R ^ | n.) y ( R +  = (Q. m  x  = (y /2.) Q  y  Q  | n.) }/2 .  m i n  + P. y ) / 2 . Q  (Q m i  1  + P ) i  (2.3)  S i n c e t h e s e p o i n t s a r e a l s o e x p r e s s e d i n terms o f y , Q  t h e y a r e l o c a t e d on t h e same s c a l e as t h e i n i t i a l and end points.  The l o c a t i o n o f t h e s e p o i n t s i s shown i n F i g . 2 . 7  (marked by t r i a n g l e s ) .  55  Benefit  Q m,p  y.  2  >• Q , m i | A  Q m,P JJ 2  2  2  ()  0  Q, m, P, 2  Mo  /  -/  m  0  0  i/Jo  A ' I  i  i  P,./l.  i i • i  /Jo  i Rmii  Fig.  !  — » - Reliability  Ro R'iR*2  Rmax.  2.7. L o c a t i o n o f t h r e e P o i n t s f o r Each Function f o r R e l i a b i l t y  R  Preference  a t n.. (Note t h a t the l  preference  f u n c t i o n s -dashed l i n e s - have not  been o b t a i n e d a t t h i s  point.)  56 In S e c t i o n  I I . C . , i t i s o b s e r v e d 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.)} l 1 max i ' mm i ' 1  H  1  M V  For a concave f u n c t i o n F ( x ) , . F(x ) = O . S l F ^ )  i f  2  then  x 2  = 0.5(x  1  + E(x )} 3  + x ). 3  Hence, R] < 0.5( Rmax + R ^ ). ' i = min'  0  1  (2.4)  57  4.  C o n s t r u c t i o n of P r e f e r e n c e  Functions.  In S e c t i o n s I I . E. 2 and I I . E.3. an i n i t i a l , intermediate^ and an end p o i n t were found  f o r the p r e f e r e n c e  f u n c t i o n for. r e l i a b i l i t y a t a g i v e n m i s s i o n time, A s u i t a b l e curve criteria  i s to be f i t t e d  an  y (R).  t o these data t h a t meets the  e s t a b l i s h e d i n S e c t i o n I I . C.  As s t a t e d i n S e c t i o n I I . C. the p r e f e r e n c e  function for  reliability; a.  i s a non-decreasing  f u n c t i o n , i . e . y'(R,n^)>o  b.  has a n o n - i n c r e a s i n g  c.  has a zero s l o p e a t R=R .i.e. U W R ,n.)=o ^ max' max'. I  s l o p e , i . e . y"(R,n^)<o  M  v  In S e c t i o n s I I . E.2. & I I . E. 3. we e s t a b l i s h e d t h r e e p o i n t s f o r each m i s s i o n time v a l u e n^. L e t a  «  ^ tain,' i R  n  )  =  y  l i  b.  y(R!,n.) = y .  c.  y (R ,n. ) = p., max ' i ' 3i  2  K x  K  A f u n c t i o n y(R,n^) i s now suggested  t h a t f i t s the  above t h r e e p o i n t s s u i t a b l y and eonf.irms  the  restriction  s t a t e d above: y(R,n.) = A-B Where  ( (C-R) e " " ( C  }  R )  (2.5)  D  A = y^^ C = R max D  =  l  n  ( y  3i- li y  }  -  l  n  ( y  3i- 2i y  )  R . -R!+ln(R -R . )-ln(R -R! ) min I \ max mm max I 1  _  ^3i- li y  {(R  max  -R  }  . ) e.xp (R  mm'-  ^  L  . -  min  R  max  ) }°  58  This R  i  function is  a t g i v e n values  R^  i l l u s t r a t e d i n F i g . 2.8 f o r d i f f e r e n t values n  and R  and a value o f one f o r y ' ^ and R respectively. max r  J  m a x  .  Values  A value o f zero i s of  The value o f R. i  taken for/U^.  .900 and .999 are used f o r is  of  v a r i e d from .905 to  R  .949.  mi  60  5.  Construction of Indifference  In S e c t i o n  II.C.  at c o n s t a n t m i s s i o n f o r y(R|n).  This  Functions.  we developed p r e f e r e n c e f u n c t i o n f o r r e l i a b i l i t y  time.  In S e c t i o n  I I . E . 4 a f u n c t i o n was s u g g e s t e d  function is u t i l i z e d in this  s e c t i o n to c o n s t r u c t  the i n d i f f e r e n c e f u n c t i o n f o r r e l i a b i l i t y and m i s s 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 n t r o d u c e d i n S e c t i o n 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 It i s a locus  o f a l l combinations  have the same given  constant  where y . i s a c o n s t a n t ,  time i n terms o f b e n e f i t . time  that  = u-  and s o l v e f o r R a t a l l n. f o r which y . i s i n  *  the range o f y ( R , n . ) » we o b t a i n s e v e r a l r e s u l t i n the b e n e f i t y..  these  is a trade-  I f we l e t  J  o f equal  II.D.2  o f r e l i a b i l i t y and m i s s i o n  benefit.  y(R.n.j)  time.  J  combinations  of  S i n c e a l l these combinations  (Rin!)  which  (Rin'.) a r e  b e n e f i t t o the d e c i s i o n maker, he would be i n d i f f e r e n t among  combinations. In F i g . 2 . 9 . curves o f y(R n p  o f n.j.  As d i s c u s s e d  in Section  a r e p r e s e n t e d a t equal  II.C,  these f u n c t i o n s  f u n c t i o n s , since the increase i n b e n e f i t at constant unit increase i n mission  t i m e , decreases  a r e concave  r e l i a b i l i t y , per  as n i n c r e a s e s  (see S e c t i o n  II.B.3). The d i f f e r e n c e between y(R|n^ ^) and y(R|n^) +  as the m i s s i o n  time n i n c r e a s e s .  intervals  decreases  Hence, the d i f f e r e n c e  61 between the  i n t e r s e c t i o n s of y ( R | n ^ D  (R=constant) l i n e  vertical  is illustrated  in Fig.  In F i g . 2.10. illustrated Fig.  and  +  u(R.|n^)with a  decreases as n i n c r e a s e s .  This  2.9.  i n d i f f e r e n c e f u n c t i o n s n(R|y=C) are  f o r the p r e f e r e n c e  f u n c t i o n s y(R|n^) shown i n  2.9. Indifference functions derived  functions for r e l i a b i l i t y y(n| R),  and  from the  mission  time, y(R| n)  are convex f u n c t i o n s w i t h n e g a t i v e  p o s i t i v e second d e r i v a t i v e .  The  y(R|n) = y^ a  i n Appendix  (n|,R') o b t a i n e d  by  and  and shape  4.  letting  constant  a smooth convex f u n c t i o n i s f i t t e d . w i t h i n the  slope  reasonincr f o r the  of i n d i f f e r e n c e f u n c t i o n s i s g i v e n Through the p o i n t s  preference  This i s  framework of the example g i v e n i n Appendix I .  A.  illustrated  Fig.  2.9.  Preference Functions u(R|n^), f o r R e l i a b i l i t y C o n s t a n t M i s s i o n Time.  Fig.  2.10  Indifference Functions  n(R|u)  63  III.  COST  The c o s t o f a p r o b a b i l i s t i c and the m i s s i o n lyzed i n three a)  a)  Material  time o f t h e d e s i g n .  i s a function of r e l i a b i l i t y  The c o s t o f a d e s i g n  b) O p e r a t i o n C o s t ,  c) P r o d u c t i o n  Cost i s , among o t h e r t h i n g s ,  of the p . d . f .  may be ana-  categories:  Cost,  Material  design  o f the m a t e r i a l  choice o f material  cost w i l l  III.A.  of steel)  The a n a l y s i s  ( e g . aluminum,  is in question, this  presented i n t h i s  be r e p e a t e d f o r d i f f e r e n t m a t e r i a l s ,  If a  not vary and i s o m i t t e d  However, when a c h o i c e o f m a t e r i a l  s t e e l , o r d i f f e r e n t grades be o f importance.  a f u n c t i o n o f t h e parameters  p r o p e r t i e s , see Section  is specified, this  from the a n a l y s i s .  Cost.  cost  will  r e p o r t would then  and the r e s u l t s  compared i n o r d e r  to choose the optimum m a t e r i a l . Material  Cost i s a l s o  a f u n c t i o n o f the s i z e o f the specimen.  The s i z e o f t h e specimen, i n terms o f some c r i t i c a l mines t h e amount o f m a t e r i a l Reliability the p . d . f . Section  and m i s s i o n of material  I.e.  used f o r the p r o d u c t i o n o f the specimen.  time a r e both f u n c t i o n s  o f r e l i a b i l i t y and the m i s s i o n usually  material  costs.  call  o f t h e parameters o f  p r o p e r t y and t h e s i z e o f t h e s p e c i m e n , see  Hence, t h e c o s t o f m a t e r i a l  designs  dimension A , d e t e r -  time.  can be expressed as a f u n c t i o n  Most r e l i a b l e and l o n g e r  f o r more e x o t i c m a t e r i a l s ,  See S e c t i o n  III.A.  resulting  lasting  i n increased  64 b)  T o t a l o p e r a t i o n c o s t decreases as the r e l i a b i l i t y o f the  increases  due to a s m a l l e r p r o b a b i l i t y o f f a i l u r e and fewer  e t c . , see S e c t i o n mission  11 I.B.I.  time i n c r e a s e s  Total operation cost increases  because o f l o n g e r u s e f u l  (Note t h a t b e n e f i t t o o , i n c r e a s e s effect is  c)  design  analyzed in Section  as m i s s i o n  shutdowns, as  the  l i f e o f the specimen.  time i n c r e a s e s . )  This  III.B.2.  P r o d u c t i o n Cost f o r the specimen depends on the p r o d u c t i o n method  chosen f o r the d e s i g n . o f the d e s i g n .  Production cost is  a major f a c t o r o f the  cost  P r o d u c t i o n methods are a f f e c t e d by the p r o p e r t i e s  of  the m a t e r i a l . Even though  i t may be l e s s c o s t l y  to a second m a t e r i a l  to use a g i v e n m a t e r i a l  on the b a s i s o f i t s  material  p r o d u c t i o n c o s t a s s o c i a t e d w i t h the f i r s t m a t e r i a l to the second m a t e r i a l , i t o f f s e t s material.  The second m a t e r i a l  compared  properties, i f is  the c o s t advantage  the  high compared of using  becomes the b e t t e r c h o i c e .  the  first  Therefore  p r o d u c t i o n c o s t s h o u l d always be c o n s i d e r e d w i t h i n the c o n t e x t o f the c o s t o f the m a t e r i a l which the type o f p r o d u c t i o n i s The changes in Section  i n c o s t .by changes  III.A.l.  in material  associated with.  properties is  discussed  I I I . A.  MATERIAL COST  1.  Dependence on Parameters o f M a t e r i a l p.d.f.  M a t e r i a l c o s t p e r u n i t w e i g h t i n c r e a s e s as t h e expected value  and t h e v a r i a n c e  of the relevant  m a t e r i a l p r o p e r t y become more f a v o u r a b l e ; t h a t i s as y^ i n c r e a s e s and/or  decreases.  y-[ and y  of t h e m a t e r i a l p.d.f.  and they a r e f u n c t i o n s o f t h e  2  are properties  parameters o f t h e p.d.f. a)  The c o s t o f m a t e r i a l i n c r e a s e s as t h e expected  value,  y£ i n c r e a s e s , s i n c e a h i g h e r 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 e x p e n s i v e  p r o d u c t i o n methods f o r t h e  material.  If C i s d e f i n e d as the c o s t o f m a t e r i a l per u n i t weight; m,y ^ 3  then C m,y  + as y £  +  Hence, ^Cn^/a/i,'  70.  (3.1)  and s i n c e a d d i t i o n a l i n c r e m e n t a l 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)  T h i s a n a l y s i s determines the g e n e r a l shape o f t h e function C (u-! ) as shown i n F i g . 3.1. m, u 1 differs  b)  The exact  shape  f o r each m a t e r i a l .  The c o s t o f m a t e r i a l  increases  decreases because a decrease i n y uction controls.  as the v a r i a n c e implies  2  t/  x  (  prod  3  >  3  )  i n c r e m e n t a l decreases i n the v a l u e o f u  w i l l be i n c r e a s i n g l y expensive as u  2  d e c r e a s e s , hence,  Hence,  (3.4)  The above a n a l y s i s g i v e s function C  tighter  2  Hence,  ?> c-m +/Z/<(  Additional  u  m  ^(l-^  a  s  s  n  o  w  n  t h e g e n e r a l shape o f t h e  i - i-9J« 3.2. n  F  2  68  69 -III.  A. 2.  Dependence on S i z e  The d e s i g n c o n f i g u r a t i o n analysis.  It  presented is  is  Parameter  is  a r e s u l t of design  not taken as a v a r i a b l e i n t h i s  a p p l i c a b l e to d e s i g n i n g  f o r a given  The s i z e parameter o f the d e s i g n a f f e c t s mission  time as e x p l a i n e d i n S e c t i o n  f u n c t i o n o f the s i z e The m a t e r i a l  size  The a n a l y s i s  configuration.  both the r e l i a b i l i t y and the  I.A.3.  cost  increases  as  The c o s t i s  also a  the amount o f m a t e r i a l  The volume o f m a t e r i a l  a d i r e c t f u n c t i o n o f the s i z e parameter.  the a r e a .  study.  parameter.  the d e s i g n i n c r e a s e s .  a crossectional  engineering  If  cost is  for  used f o r the d e s i g n the s i z e parameter  a r e a , f o r example, the volume i n c r e a s e s  The m a t e r i a l  used  is  is  d i r e c t l y with  therefore a direct function of  the  parameter. If  the c h o i c e o f m a t e r i a l  i n the a n a l y s i s ,  is  d e s i r e d t o be taken as a v a r i a b l e  the c o s t o f m a t e r i a l  S i n c e the s p e c i f i c weight  per u n i t w e i g h t ,  o f the m a t e r i a l  the m a t e r i a l  is  constant,  C  m  , varies.  the c o s t  of  per u n i t volume, C , i s a d i r e c t f u n c t i o n o f C . v m,y Hence, the c o s t o f m a t e r i a l per d e s i g n specimen i s m  r  where V i s  the volume o f m a t e r i a l  used.  S i n c e the volume i s  a direct  f u n c t i o n o f 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.  1.  The  COST OF OPERATION  Dependence on R e l i a b i l i t y  p r o b a b i l i t y of f a i l u r e of the design  r e l i a b i l i t y of the design* P(n)  i s r e l a t e d t o the  ( S e c t i o n 1) = 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 o c c u r intended m i s s i o n time of  during i t s  n, i s t h e complementary p r o b a b i l i t y  reliability.  For c r i t i c a l d e s i g n s ,  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 o f f a i l u r e maybe e x c e s s i v e l y h i g h . F o r such d e s i g n s , t h e minimum r e l i a b i l i t y w i l l be h i g h , and  i n t h e range between minimum and maximum  c o s t w i l l d e c r e a s e as t h e r e l i a b i l i t y non-repairable  and f o r n o n - r e p l a c e a b l e  reliability,  increases.  For  specimens t h e e x -  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 =(1-R(n))C R  p  where P(F) = ( l - R ( n ) ) - - p r o b a b i l i t y o f f a i l u r e i s t h e comp l e m e n t a r y p r o b a b i l i t y o f r e l i a b i l i t y and C cost of f a i l u r e .  p  i s the assessed  See F i g . 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 d e s i g n  specimens, f o r  which f a i l u r e i s n o t c a t o s t r o p h i c , t h e r e l i a b i l i t y dependent c o s t o f f a i l u r e d e c r e a s e s as r e l i a b i l i t y In t h e 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  increases  i s e q u a l t o one, C  CcF, S  73 is  equal  to z e r o .  decreases; specimens  As r e l i a b i l i t y  hence among K failing  Among K specimens  the new specimen i s  ( l a r g e K), t i m e , n,  specimen i s  replaced  to serve mission  n'  Hence the p r o b a b i l i t y  is  less  equal  to  or r e p a i r e d i m m e d i a t e l y , . the remainder o f  1  the  Hence,  t i m e , w h i l e the d e s i g n  >R(n).  o f f a i l u r e f o r the r e p l a c e d o r r e p a i r e d specimen  likely  to f a i l  The r e l a t i o n between R(n)  o f f a i l u r e o f the o r i g i n a l during  the m i s s i o n  specimens,  time n.  and R ( n ' ) , g ( S , L , A ) , depends  n a t u r e o f the p . d . f . o f m a t e r i a l parameter,  is  < n.  lower than the p r o b a b i l i t y  and i t  the expected number o f  same,  R(n')  is  failure  decreases.  time n ,  S i n c e the new specimen has a s m a l l e r m i s s i o n the  of  -K=(l-R(n)).K.  time n o f the f a i l e d specimen.  parameter i s  the p r o b a b i l i t y  the expected number o f f a i l u r e s  we assume a f a i l i n g  mission  specimens  d u r i n g the m i s s i o n  P(F)  If  increases,  strength,  A:  R(n')=g(S,L,A)  -R(n).  S,  on the  l o a d L, and the d e s i g 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 ( l a r g e 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  and be r e p l a c e d and/or r e p a i r e d . new  Among  fail  these (l-R(n))K  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 r e p l a c e d and/or r e p a i r e d .  The  r e p l a c e m e n t , ( t h i r d g e n e r a t i o n ) , has a s m a l l m i s s i o n  new  time,  n", l e f t t o c o m p l e t e s i n c e n" <<  n,  we can assume R(n") = 1.0. That i s ^ R ( n " ) i s v e r y c l o s e t o one and f a i l u r e d u r i n g i s v e r y 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  n"  neglected.  S i n c e , s t a r t i n g w i t h K, and r e p l a c e m e n t s o f two g e n e r a t i o n s , t o have K specimens complete t h e m i s s i o n  time^  K' specimens a r e needed where  So  (K'-K)  r e p a i r s and/or r e p l a c e m e n t s a r e needed  I f we l e t C„ be t h e c o s t o f f a i l u r e , i . e . t h e c o s t o f 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, t h e c o s t for  K specimen i s  t h e c o s t per specimen i s  C  B.=  RCn)) [| + t j C s U , A ) ( , _ y  R ( l n )  )^  C  F  76  2.  Dependence on  The cost  operation  increases  shutdowns and are  taken  as  of the  Mission  a specimen  failures.  The  the  Hence, t h e  fixed  part  to the  mission  C  length  o, s  H e r e , as and  the  fixed  operation  per  time u n i t per  c  the  o,n  i  mission  time  i s increased,  a penalty  time appears  t o be  mission  cost.  the  was  s e e n t o be  i n c r e a s i n g w i t h n.  in  the  time  in  cost  and  in benefit.  If  the  cost  and  reliability, w o u l d have  R,  b e n e f i t of  then the  resulted.  That  t i m e was  i s associated  variables considered  Co,s  proportional  specimen.  increases,  i s , the  less desirable.  b e n e f i t of  mission  taken  monetary u n i t s / specimens.  n  seem as  is  is directly  = Cost of n  is  time.  the  only  cost  specimen  =  in  specimen.  operation  cost  shutdowns  frequency  per  i t may  mission  the  frequent  sections  operation  Qf  o,s  the  of  This  and  = Cost of  C R c  of  failure  because t h e i r  of  The  of  a cost. t o more  i n previous  reliability  into consideration.  Let  costs  with r e l i a b i l i t y ,  d e p e n d e n t on  involves  s p e c i m e n a g e s , due  into consideration  connection  Time  But  when  considered, Therefore  mission  the  an  b o t h w i t h an  the  longer  increase increase  t i m e were  i n d e c i s i o n making a t a relation  graphed  benefit  in Fig  the  constant 3.4  77  78 As seen from F i g . 3 . 4 , i f only the b e n e f i t of the m i s s i o n time  (omitting r e l i a b i l i t y ) ,  i s c o n s i d e r e d , the s m a l l e r  of the maximum m i s s i o n time or t h e m i s s i o n time v a l u e n  1  (at which the b e n e f i t i s equal to the cost) would be chosen, s i n c e t h a t would be the v a l u e which  provides  the maximum b e n e f i t .  In case n .,. i s the maximum m i s s i o n max(l) time, n ,,« would be chosen; if n i s the maximum ' max(l) ' max (2) m i s s i o n time, n" would be chosen s i n c e n' < n /-,%. ' max(2)  79  III.  C.  COST FUNCTION  1.  Dependence o f Cost on R e l i a b i l i t y  In S e c t i o n  I,  be expressed as  and M i s s i o n Time  i t was shown t h a t the r e l i a b i l i t y o f a d e s i g n may  (1.8) R(n)  = 1-P(v<l)  3.10  where  3.11  v = £ see eg.  (1.7)  The r e s i s t i n g  strength  o f the m a t e r i a l , S,  p r o p e r t y , M, and the s i z e parameter, A. expressed  Hence the r e l i a b i l i t y can be  R and m i s s i o n  b u t i o n and s t r e n g t h  time n.  distribution  When the maximum-load  are approximated as  where from A . l . l l  and A. 1.12 y  .  But  distri-  log-normal  models  3.13  N  a  terms  Ill):  R(n) = F ( ^ ; 0 , 1 )  and  3.12  = 1 - P (MxA < L|n)  r e l a t i o n the s i z e parameter A, can be expressed i n  of r e l i a b i l i t y  (see App.  a f u n c t i o n o f the m a t e r i a l  as R(n)  From t h i s  is  v  o  =  y  i n App.  s  "  y  v  _ f 2 .  -  y  )j  s  (a  s  I,  2x0.5  +  3  ,  1  4  a ) y  = y + In A. m 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  i s a f u n c t i o n o f the r e l i a b i l i t y R and y  and t h e r e f o r e i s a function  Li  of m i s s i o n t i m e n. (see S e c t i o n I ) . Hence we have exp r e s s e d t h e s i z e parameter A as a f u n c t i o n o f 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 t h a t t h e 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 o f t h e s i z e parameter A ( R , n ) . Hence, t h e c o s t o f m a t e r i a l i s C  m  = C  v  -K • A (R, n) ' "  3.16  As shown i n S e c t i o n I I I . B. 2, t h e c o s t o f o p e r a t i o n p e r specimen as a f u n c t i o n o f m i s s i o n t i m e i s C. = n C o,s o,n  3.17  Hence, t h e t o t a l c o s t f u n c t i o n i s t h e sum o f t h r e e f u n c t i o n s , namely t h e c o s t o f m a t e r i a l of o p e r a t i o n  C (R,n), the cost ro  (dependent on m i s s i o n time) C  cost of operation  Q  s  (dependent o f r e l i a b i l i t y )  C(R,n) = C. (R,n) + C m  0 f S  cost  ( n ) , and t h e C (R):  ( n ) + C (R) R  3.19  When t h e m a t e r i a l c h o i c e i s a l s o a v a r i a b l e i n t h e d e c i s i o n p r o c e s s , t h e c o s t o f m a t e r i a l C ( R , n ) i s found s e p e r a t e l y m  for  a l l choices of m a t e r i a l .  The c o 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, g i v e n by 3.19 u s i n g a p p r o p r i a t e  C (R,n). rn  81  IV.  C. 2.  Using  Constant  relation  Cost  3.12, R(n)  several A.  values  Since A is  = 1-P(S<L),  o f n and R can be found f o r s e v e r a l  f i x e d values  a d i r e c t f u n c t i o n o f c o s t o f m a t e r i a l , using  formation a f u n c t i o n i s of mission  Curves  developed o f the c o s t o f m a t e r i a l  time a t c o n s t a n t This  is  reliability:  this  of in-  as a f u n c t i o n  C (R,n|R) m  i l l u s t r a t e d by the dashed l i n e i n F i g .  3.5.  S i n c e the c o s t o f o p e r a t i o n dependent on m i s s i o n  time i s  o f n, t h i s  3.5 by the d o t t e d  a**el i s  function C  (n)  added to C ( R , n | R ) . m  constant  r e l i a b i l i t y is  3.5 by d a s h - d o t - d a s h we get the t o t a l C(R,n|R=C). This where N i s  (illustrated  is  C ( R , n | R)  procedure i s  (illustrated  R  a horizontal  line.  cost function f o r mission  See F i g . 3.5.  f o r numerical  repeated f o r N c o n s t a n t  the number o f c o n s t a n t m i s s i o n  Adding a l s o  time a t c o n s t a n t  (See Appendix I  in  these c o s t f u n c t i o n s  v a l u e s , we get a s e t o f values Using  these s e t s  of  (R,n), with  (R,n)  reliability,  chosen i n  the  a t the same f i x e d  Fig.  values,  C(R,n|R=C), and t a k i n g f i x e d of  at  example.)  reliability  time v a l u e s  line)  C^(R|R=C),  2. Using  function  S i n c e the v a l u e o f c o s t o f r e l i a b i l i t y  a constant,  line)  in Fig.  a direct  cost  cost.  Section  82  C  • '  1  nmin  Fig.  nmax  3.5. Cost o f M i s s i o n Time a t Constant C(R,n|R=c.).  Reliability,  method employed  to construct i n d i f f e r e n c e functions i n  S e c t i o n 2, c o n s t a n t c o s t f u n c t i o n s are c o n s t r u c t e d , Fig.  see  3.6.  Since t o t a l c o s t i n c r e a s e s as m i s s i o n time i n c r e a s e s , and c o s t of m a t e r i a l i n c r e a s e s as r e l i a b i l i t y  increases,  the c o n s t a n t c o s t f u n c t i o n s are concave f u n c t i o n s , see Appendix IV.  F i g . 3.6. Constant Cost F u n c t i o n s  85  IV.  DECISION PROCESS  A.  In  Intersection  Section  of  II.E.5.  Indifference  it  is  and Constant  shown t h a t  Cost  Functions.  indifference functions  convex f u n c t i o n s , and are e x p r e s s e d i n terms o f R and n. IV.C.2., i t and a l s o  is  shown t h a t c o n s t a n t  f u n c t i o n s are concave  a r e e x p r e s s e d i n terms o f R and n.  difference  functions  s e c t a t two p o i n t s ,  and some o f the c o n s t a n t  m l  - »R n  min be tangent  m a v  )  and (n  max  . ,n  cost  functions,  cost  ).  functions may be i n  interthe  An i n d i f f e r e n c e f u n c t i o n  INOA  min  to a c o n s t a n t  m a v  Section  Hence some o f the i n -  and one o r both o f these p o i n t s  a c c e p t a b l e range ' ( R  will  cost  In  are  f u n c t i o n , i f many such  functions  are d e v e l o p e d . At the p o i n t u'  is  tangent  w i t h lowest  c o s t and C 1  is In  cally.  is  is  the a c t u a l  Fig.  f u n c t i o n with cost C ' , y '  r e a l i z e d with highest  C  and b e n e f i t  realized  benefit.  p o i n t , s i n c e the y'  is  benefit  cost-benefit  are c o n s i d e r e d .  d e c i s i o n p r o c e s s , when most o f the i n d i f f e r e n c e  and c o n s t a n t the l o c u s  cost  c a l l e d a suboptimal  lowest when c o s t  functions tangent,  1  to a c o n s t a n t  Hence ( R i n ) ratio  ( R j n ) where an i n d i f f e r e n c e f u n c t i o n w i t h  cost  functions  o f l i k e l y tangency  4 . 1 . shows the  result.  chosen are l i k e l y not t o be points  is  approximated  graphi-  86 Joining  these  points  gives  the  1ocus  of  suboptimal  points.  B.  The  Once  the  associated The  Optimal  Combination  suboptimal  with  ratio  a of  points  benefit cost  to  found  minimum  and point  liability imum.  at  n*  smooth of  R*,for  The  time  a  X(R) which  y,  and  n.  found, and  each  cost  point  value  =  C„  C/y, X  gives  optimal  the  from  the  value  of  cost-benefit  optimal the  value  versus  ratio  R.  is a  f o r the  suboptimality line  extensive  presented  example  i n Appendix  I.  of  the  decision  The  remin-  mission of  Fig.4.1.  R*. An  is  benefit,  function i s f i t to  corresponding  i s located  R  are  value  X is  of  process  i s  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 s a f e t y  on p a s t e n g i n e e r i n g e x p e r i e n c e . mation on the m i s s i o n  This  d e s i g n approach conveys  In  analysis,  is  a method i s  decision analysis.  n and  time to  this  mission  R.  The d e c i s i o n c r i t e r i o n used i n the f o r the d e c i s i o n maker.  f o r the d e c i s i o n maker, f o r both r e l i a b i l i t y  t i m e , are d e v e l o p e d .  these p r e f e r e n c e f u n c t i o n s .  Indifference Indifference  functions functions  tions  between r e l i a b i l i t y and m i s s i o n  stant  c o s t f u n c t i o n s , d e v e l o p e d from c o s t f u n c t i o n s  t r a d e - o f f functions  Even  developed to o b t a i n R and n v a l u e s  to minimize the c o s t - b e n e f i t r a t i o  Preference functions  infor-  A probabilistic  the i n f o r m a t i o n on expected m i s s i o n  the p r e s e n t work,  by a r a t i o n a l  mission  occasionally.  o f the d e c i s i o n parameters  based  reliability.  and the f r a c t i o n o f d e s i g n s expected to f u l f i l l  t i m e , by the values  chosen  f a c t o r does not c o n t a i n  l e n g t h o f the d e s i g n nor i t s  the most c o n s e r v a t i v e d e s i g n s do f a i l  failure,  factor is  are d e r i v e d from are t r a d e - o f f  time i n terms o f b e n e f i t .  between r e l i a b i l i t y and m i s s i o n  and  funcCon-  f o r R and n , a r e time i n terms  of  cost. Indifference the s u b - o p t i m a l the h i g h e s t  and c o n s t a n t  l i n e which d e f i n e s  functions the locus  are u t i l i z e d to of points  (n,R)  b e n e f i t a t given c o s t and lowest c o s t at given  Among t h e s e p o i n t s , chosen.  cost  This  obtain that  benefit.  the one w i t h the minimum c o s t - b e n e f i t r a t i o  combination o f R* and n* g i v e s the v a l u e s  a t which the d e c i s i o n maker minimizes  his  have  is  o f R and n  cost-benefit ratio.  The  89 probabilistic  design parameter A* i s  o b t a i n e d d i r e c t l y from R*  and  n*. This critical  d e c i s i o n model may be adopted f o r use when weight d e c i s i o n v a r i a b l e r a t h e r than c o s t .  may be developed using The f i n a l  the same method as  is  Constant weight  f o r constant  cost  the functions  functions.  a n a l y s i s may be u t i l i z e d t o f i n d minimum w e i g h t - b e n e f i t  ratio  i n the same manner. It  is  material  c h o i c e a d e c i s i o n parameter.  functions cost  recommended t h a t the p r e s e n t work be extended by making In  are developed f o r each m a t e r i a l  that case, considered.  the c o n s t a n t A set of  f u n c t i o n s , d e r i v e d f o r each m a t e r i a l , and i n d i f f e r e n c e  result  in a sub-optimal  benefit points the lowest mission  on each s u b - o p t i m a l  The minimum  constant  functions cost-  l i n e are compared, and the one w i t h  c o s t b e n e f i t r a t i o g i v e s the optimal  time n * , optimal  c h o i c e M*.  l i n e f o r each m a t e r i a l .  cost  r e l i a b i l i t y R*,  d e s i g n parameter A * , and optimal  optimal  material  90  BIBLIOGRAPHY  (A)  L i s t , o f J o u r n a l s Surveyed:  1.  A I I E Conference P r o c e e d i n g s ,  2.  AIIE Transactions,  3.  ASCE J o u r n a l o f S t r u c t u r a l D i v i s i o n ,  4.  A p p l i e d Mechanics Review,  5.  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Debreu, G., " R e p r e s e n t a t i o n of a P r e f e r e n c e O r d e r i n g by a Numerical F u n c t i o n , " i n R.M. T h r a l l , C.H. Coombs,, and R.L. Davis (Eds.), D e c i s i o n P r o c e s s e s , John Wiley and Sons, New York, N.Y. 1954.  15.  Rader, J.T., "The E x i s t e n c e of a U t i l i t y F u n c t i o n to Represent P r e f e r e n c e s , " Q u a r t e r l y J o u r n a l of Economics, V o l . 75, 1961, pp. 229-32.  16.  Suppes, P., "The Role 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 i n D e c i s i o n Making," Proceedings of the T h i r d B e r k e l e y Symposium on Mathematical S t a t i s t i c s arid" P r o b a b i l i t y , 1954-1955, V o l . 5, 1956, pp. 61-73.  17.  Keeney, R.L., " U t i l i t y Independence and P r e f e r e n c e s f o r M i l t i a t t r i b u t e d Consequences," O p e r a t i o n a l Research, Operations Research s o c i e t y of America, V o l . 19, No. 4, J u l y - A u g u s t 1971, pp. 875-93.  18.  Keeney, R.L., " U t i l i t y Functions for M u l t i - a t t r i b u t e d Consequences," Management S c i e n c e , B a l t i m o r e , V 0 I . I 8 A , No. 5, January 19T2~7~p~p. 276-87. "  19.  F i s h b u r n , P.C., "Methods of E s t i m a t i n g A d d i t i v e Utilities," Management S c i e n c e ^ B a l t i m o r e , V o l . 13A, No. 7. March 1967, pp. 435-53. ~  20.  S c h l a i f f e r , R., P r o b a b i l i t y and S t a t i s t i c s f o r Business Decisions", McGraw-Hill, New York, N.Y. 19 59 .  21.  S c h l a i f f e r , R., I n t r o d u c t i o n to S t a t i s t i c s f o r Business D e c i s i o n s ^ McGraw-Hill, New York, N.Y.,  22  S c h l a i f f e r , R., McGraw-Hill, New  23.  P r a t t , J.W., R a i f f a , H. , and S c h l a i f f e r , R. , "The Foundations of D e c i s i o n Under U n c e r t a i n t y , An Elementary E x p o s i t i o n , " J o u r n a l of ASA^_ V o l . 59, pp. 3 53-75.  1961.  A n a l y s i s of D e c i s i o n s Under U n c e r t a i n t y , York, N.Y., 1969.  24.  P r a t t , J.W., R a i f f a , H., and S c h l a i f f e r , R., I n t r o d u c t i o n to S t a t i s t i c a l D e c i s i o n Theory, McGraw-Hill, New York, N.Y., 1965.  25.  Weiss, L., S t a t i s t i c a l D e c i s i o n Theory, New York, N.Y., 1961.  26.  DeGroot, M., Optimal S t a t i s t i c a l McGraw-Hill, New York, N.Y., 1970.  1964,  McGraw-Hill,  Decisions,  .  93  27.  Myron, T., R a t i o n a l D e s c r i p t i o n s , D e c i s i o n ? and D e s i g n s , Pergamon P r e s s , New York, N.Y., 1969.  L i s t of Related A r t i c l e s 28.  Perused:  B e c k e r , P.W., and J a r k l e r , B., "A S y s t e m a t i c Procedure f o r the G e n e r a t i o n o f C o s t M i n i z e d D e s i g n s , " IEEE T r a n s a c t i o n s on R e l i a b i l i t y , V o l . R-21, No. 1, F e b r u a r y 1972, pp. 41-5.  29. s  D a v i d s o n , 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 o f 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 , P e t e r C., " U t i l i t y Theory," Management S c i e n c e , B a l t i m o r e , V o l . 14A, No. 5, J a n u a r y 1968, pp. 335-78.  31.  F r e u d e n t h a l , A.M., " S a f e t y , R e l i a b i l i t y and S t r u c t u r a l D e s i g n , " 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, S t . 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 D e s i g n S a f e t y F a c t o r s , S a f e t y M a r g i n s and Measures of R e l i a b i l i t y , " SAE A n n a l s 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 o f U t i l i t y , " E c o n o m e t r i c a , V o l . 26, 1958, pp. 193-224.  34.  Majumdar, T., The Measurement o f U t i l i t y , M a c M i l l a n and Company, London, 1958.  35.  P f a n z a g l , J . , "A G e n e r a l Theory o f Measurement: Applications to U t i l i t y , " N a v a l 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., " L i m i t e d L i f e Designs.- A Survey o f t h e Problem," E n g i n e e r i n g , London, No. 112, August, 1957, pp. 236-417  37.  S h i n o j u k a , M., and Yang, J . , "Optimum S t r u c t u r a l D e s i g n Based on R e l i a b i l i t y and P r o o f - L o a d T e s t , " SAE A n n a l s o f 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 o f S p a c e c r a f t and R o c k e t s , V o l . 7,.No. 9, September, 19T0. pp. -1127-9.  39.  W e i r , T.W., " D e c i s i o n Theory and C o s t M o d e l l i n g , " SAE A n n a l s 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 p r o c e s s t o f i n d an o p t i m a l  of m i s s i o n t i m e  combination  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 o f a shear p a n e l f o r an a i r p l a n e .  Problem:  To f i n d t h e optimum c o m b i n a t i o n o f r e liability  R*  and m i s s i o n t i m e , n*, such  that the r a t i o of cost to b e n e f i t i s miminized, f o r a shear p a n e l o f an a i r p l a n e , probabi l i s t i c a l l y d e s i g n e d , and t o f i n d t h e c o r r e s ponding d e s i g n s p e c i f i c a t i o n ,  t*. ins,  thickness of the panel.  Given Information: 1.  Preference function f o r r e l i a b i l i t y a t c o n s t a n t m i s s i o n t i m e a s s e s s e d ,by t h e d e c i s i o n maker.  2.  D i s t r i b u t i o n o f shear l o a d , F ( L ) : W e i b u l l distribution  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 2 4 S w i t h  handbook v a l u e s o f Young's modulus,  _  g  E = 10.6 x 10  p s i and P o i s s o n ' s  v = .33.  Material distribution  to  3  be a log-normal  ratio i s assumed  d i s t r i b u t i o n , F (E) ' LN T  w i t h ]}\ (E) = E, and ( E ) = 0.1. Y  4.  a.  Minimum m i s s i o n time, n . =100,000 ' mxn ' applications.  b.  Minimum r e l i a b i l i t y ,  c.  Maximum r e l i a b i l i t y , 1  5.  Configuration:  '  R . =0.90. mxn R =0.99. max  a l l edges -damped, r e c -  t a n g u l a r s i d e p a n e l , w i t h dimensions; and II.  a=6ft  b=3ft.  The F a i l u r e Mode The mode i s assumed t o be y i e l d i n g i n shear.  From  Handbook o f E n g i n e e r i n g Mechanics by Fl'ugge, t h e c r i t i c a l yield  s t r e s s i n shear, r  t R t  i s related  to design para-  meters as b t/7T*k x  * /O.W  where  W. =  E  Hence;  r  = /<?-3V n*B i J is. (I-^J^  The  (A.1.1)  / ia c \  z  x ( A  ±  2  ^  shear s t r e n g t h i s  (A.1.3)  96  III.  Probabi1istic  F a i l u r e occurs  Design  a t the f i r s t  o c c u r r e n c e o f the l o a d , L, which  i s g r e a t e r than the s t r e n g t h S: 7 5  L  -  L  A.  -  P.d.f. o f M a t e r i a l P r o p e r t y  (A.1.4.)  E.  P.d.f. o f E i s assumed as a log-normal  distribution,  with  X  LB) =• E y if)  =  f o r a log-normal  y  LE)*  distribution,  0  Hence, u s i n g g i v e n E ME= (T  £  (A.1.5.)  Ce-trta^)-/) '* and  Y(B)  values^  14.849, = 0.0324 .  Since from eg^. (A.1,3)^ 5  .  C ^ V P M  3  S i s log-normally d i s t r i b u t e d  ^ B.  / / . ^  F,  (A.1.6.)  with  (a.1.7.)  = <*i.  P.d.f. o f Load  L.  Load i s assumed W e i b u l l d i s t r i b u t e d  1.5.  with  97 Since f o r a Weibull  (T^-  The  distribution  .  V.2.8  extreme v a l u e d i s t r i b u t i o n o f t h e W e i b u l l model  i s e s t i m a t e d by a l o g - n o r m a l model, t h i s i s shown i n Appendix I I I . The  log-mormal model t h a t e s t i m a t e s t h e extreme v a l u e  d i s t r i b u t i o n o f t h e 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  ML. = An  E  LO. S73 iZn  - . 3 3 (i.s  Znn-iJJa£^-)  n)  (A.1.8.) V.CS  G~u=  " * *  T  O  V ^ ^ " *  ,  A  3  (A.1.9.)  3  Here, as shown i n Appendix I I I , t h e v a l u e s f o r X  and o  w  w  a r e s u b s t i t u t e d and t h e o n l y independent v a r i a b l e i s n. C.  Reliability Failure  o c c u r s when  1L ^ S.  D e f i n e 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 o c c u r s when V < 1. The v a r i a b l e V i s l o g - n o r m a l l y d i s t r i b u t e d  a  rv  ±•  o.S  rr ~. l*S-+xCY'* t t r *-_i.rv. ~ <r i  v  with  (A.l.ii.)  98 The p r o b a b i l i t y o f f a i l u r e i s  and t h e r e l i a b i l i t y i s S.= where  i  s  F „ l m 0 , 1 ) ,  t h e s t a n d a r d i z e d normal d i s t r i b u t i o n  function.  See Chapter 13 i n r e f . (2)  Equation  (Al.12) i s a r e l a t i o n among t h e d e s i g n p a r a -  meter R and n, and t h e c o r r e s p o n d i n g thickness  IV.  (probabilistic)  design  t.  The P r e f e r e n c e 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 G i v e n M i s s i o n Time  A.  Answers t o Q u e s t i o n s : Q u e s t i o n 1;  (see S e c t i o n II.E.2)  G i v e n t h e minimum m i s s i o n time o f 100,0 00 a p p l i c a t i o n s ^ t h e r a t i o o f b e n e f i t d e r i v e d a t R = .99 t o b e n e f i t d e r i v e d a t R = .90 i s m  = 2.0  1  Q u e s t i o n 2; Estimate  (see S e c t i o n I I . E . 2 )  maximum m i s s i o n t i m e n max  = 200,000 applications ' (n' _ n . ) by N= 5 ma>C min' 1  divide  J  hence n = n  20,000 a p p l i c a t i o n s .  n.1 = n min . + i nn ; '  i = l>2 '"  *  »N • 1N  §9  Given t h a t the r e l i a b i l i t y  i s R . =.90 the r a t i o o f 3  m m  b e n e f i t d e r i v e d a t n. t o b e n e f i t d e r i v e d a t n . , x min=100,000 n  is  n  P. = u(n.IR . )/y. l l mm ' 0 1  n.  p  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 i n c r e a s e beyond 200,000, the maximum mission  time  n i s 200,000 a p p l i c a t i o n s , as was e s t max / * . i. i imated p r e v i o u s l y . Question 3;  (see S e c t i o n  II.E.2.C.)  Given t h a t t h e r e l i a b i l i t y  i3 R =.99. the r a t i o max ) of b e n e f i t d e r i v e d a t n. t o b e n e f i t d e r i v e d a t l x  n  min  n  =  1  0  0  i  '  0  0  0  i  s  i ^  ( n  il max R  ) A j  0  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  Q  n  = m  a  x  200,000 a p p l i c a t i o n s , as b e f o r e .  A  i6o B.  Reference  Gamble  (see s e c t i o n I I . B ^ ) ;  Given the m i s s i o n time n. 1  P r o p o s i t i o n 1: R . =.90 mm  t h e r e i s a 50% chance o f o b t a i n i n g  and 50% chance o f o b t a i n i n g R =.99. max  P r o p o s i t i o n 2:  3  A reliability  v a l u e R| can be  obtained f o r c e r t a i n .  At what v a l u e o f R^ 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 r  n. l  C.  100,000  R. • l .930  120,000  .927  140,000  .924  160,000  .922  180,000  .921  200,000  .920  P r e f e r e n c e 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  As e x p l a i n e d i n S e c t i o n II.E.4. the above i n f o r mation i s u t i l i z e d  t o c o n s t r u c t u(R|n.) i  Let  2.' '  u ( b e n e f i t a t minimum m i s s i o n time n . o min v  minimum r e l i a b i l i t y 1  y  0  R . ) be 1: min' = i.o.  and  101 From S e c t i o n I I I . E , 2 ^ y (R In . ) =my max' m i n o J  • b•)=p•y  y(R  min 1  K  ^ i  y(R maxb•)=0•y i ' ^ I  p  o  /  ^ O  From S e c t i o n I I . E , 3 ^ y(R!_  (.50) n ) = ( y / 2 ) (Q m +p ) . /  i  Q  i  i  i  The p r e f e r e n c e f u n c t i o n f o r r e l i a b i l i t y  at constant  m i s s i o n time y (RJru)^ suggested i n S e c t i o n I I . E. 4^ has been f i t t e d t o these p o i n t s ; y (F|i )=A -B'<<C-R) i  i  (  c  "  R  )  >  (A.l  Q  The v a l u e s o f thec o n s t a n t s f o r n. a r e as f ollov.'S : l n. l 100,000  A  B  C  D  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 f u n c t i o n s a r e i l l u s t r a t e d  i n Fig.  (A.l)  102 D. I n d i f f e r e n c e F u n c t i o n s  For s e v e r a l C o n s t a n t v a l u e s o f b e n e f i t , t h e  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 c o n s t a n t m i s s i o n time a r e s o l v e d to g e t v a l u e s f o r R and n^. That i s , f o r g i v e n v a l u e s o f n and y, t h e v a l u e o f R i s o b t a i n e d from e q u a t i o n  (A.1.13.).  The r e s u l t s a r e :  1 .83  n. l 100,000  2 .33  0. 912  n. l 130,000  0. 900  120,000  0. 943  140,000  0. 906  0. 990  140,000  0. 925  160,000  0. 900  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  y  R 0. 972  n. l 120,000  110,000  0. 990  2 .60  120,000  2 .97  R  A convex f u n c t i o n , n ( R | y ) , i s f i t t e d (n^,R) c o r r e s p o n d i n g t o each f i x e d v a l u e , y  n(R| y)= a R ;  2  2  + ajR + a  The r e s u l t i n g c o n s t a n t s a r e :  Q  R  to the points :  (A. 1.14.)  103  (10 )  (10 )  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  6  These f u n c t i o n s This  6  a r e graphed  i n F i g . A.2.  concludes the a n a l y s i s o f the b e n e f i t t o the  d e c i s i o n maker of v a r i o u s  combinations o f n and R.  105 Constant Cost  V  Functions  There are f a i l u r e c o s t , p r o d u c t i o n cost.  Production  c o s t i s f i x e d and  c o s t , and  p r o p o r t i o n a l to panel  t h i c k n e s s , hence t h i s c o s t component does not the  material  influence  l o c a t i o n of the optimum c o s t - b e n e f i t r a t i o .  Failure  c o s t i s e s s e n t i a l l y p a r t of the o p e r a t i o n a l c o s t .  Since  r e a l i s t i c d a t a f o r t h i s c o s t component are d i f f i c u l t  to  o b t a i n ^ t h i s component i s excluded i n the a n a l y s i s . T h i s o m i s s i o n does not a f f e c t the nature of the a n a l y s i s , but c e r t a i n l y a l t e r s the benefit ratio.  The  l o c a t i o n of the t r u e optimum c o s t following cost analysis deals  m a t e r i a l c o s t o n l y and imation  to the t r u e optimum c o s t - b e n e f i t  Since C  i s therefore only a f i r s t  c o s t of the m a t e r i a l  , times the weight W  weight  v  C=C The  volume  is  approx-  ratio.  i s equal t o c o s t per  of the m a t e r i a l , and  of the m a t e r i a l  with  weight,  since the specific  constant:  • W,  w  V is V=a-b.t.  Hence the c o s t i s p r o p o r t i o n a l to t : C=Gt, where G = C  a-b.v, i s a  constant.  w It  f o l l o w s t h a t a constant  value  of m a t e r i a l c o s t  .  value  of t i m p l i e s  Previously, 0,1)  i t was  constant shown t h a t  106  where  y = y^s  cr^ -  and But  ^  +<rJ )„ L  - ^  , j^, r,p.zw n - t / /2{/-v -JJ>] 3  3  J  i s a f u n c t i o n of o n l y t , s i n c e other v a l u e s are  constants,  g i v e n the m a t e r i a l c h o i c e . Furthermore,  are f u n c t i o n s of n o n l y . Since p  N  '>  °>  0  (A.1.14)  g i v e s R as a f u n c t i o n of n and material i s a direct  t , and  s i n c e the c o s t of  f u n c t i o n of t h i c k n e s s t , we  have a  f u n c t i o n r e l a t i n g the c o s t of m a t e r i a l to the m i s s i o n and  the  The  time  reliability.  constant m a t e r i a l c o s t f u n c t i o n s  are, t h e r e f o r e , o b t a i n e d the t h i c k n e s s , t , equal t h a t equation.  from equation  (see Chap. I l l )  ( A . 1 . 1 4 ) by p u t t i n g  to a s e r i e s of c o n s t a n t  values  These f u n c t i o n s of n i n terms of R are  obtained  f o r v a r i o u s constant  values.  Figure A.3.  thickness  shows the r e s u l t i n g  (constant curves.  cost)  in  107  ncooo)  R  Fig.  A.3. Constant m a t e r i a l c o s t f u n c t i o n s ; value cost.  of t i m p l i e s a c o n s t a n t  each  value  of m a t e r i a l  108 Decision  VI  The  next step  Analysis  i s t o o b t a i n the locus o f sub-optimal  p o i n t s o f the c o s t - b e n e f i t r a t i o s . F i g . A.4. shows the super—position  of F i g . A.2. ( i n d i f f e r e n c e f u n c t i o n s ) and  F i g . A.3. (constant  cost functions).  The l i n e o f tangency  between these two s e t s of curves i s the r e q u i r e d  locus of  suboptimal p o i n t s , see F i g . A.4. Assuming t h a t aluminium panels i n t h i c k n e s s - m u l t i p l e s _ of 0.010 i n s . , the c o s t - b e n e f i t r a t i o o f these 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 -  d i f f e r e n c e curves by the a p p r o p r i a t e The  thickness  values  of b e n e f i t , y.  f o l l o w i n g t a b l e shows the r e s u l t i n g suboptimal c o s t -  benefit  ratios.  Material cost  <e;t)  Benefit  Cost-benefit  (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  Ratio  From the above t a b l e , the optimum c o s t - b e n e f i t i s seen t o be 0.0400 , so t h a t the optimum d e s i g n i s t*=0.120 i n s .  The c o r r e s p o n d i n g optimum v a l u e s  d e c i s i o n parameters are R*=0.918 and n*=164,000,  ratio  thickness o f the The  109 i n t e r p r e t a t i o n o f t h i s optimum s o l u t i o n  (based on m a t e r i a l  c o s t only) i s t h a t the d e c i s i o n maker minimizes  cost-benefit;  furthermore, f o r a panel m i s s i o n time c o r r e s p o n d i n g t o n=164,000 load a p p l i c a t i o n s , the p r o p o r t i o n shear panels i s 0.918.  of s u r v i v i n g  ncooo)  LEAF 111  OMITTED IN PAGE NUMBERING.  does  / v ^ t e./lst 112  APPENDIX Analysis of Preference  IT  Function.  I n S e c t i o n I I I . E . 4 a f u n c t i o n u ( R , n ) , was s u g g e s t e d (A.2.1) where  A =/^3t C  =  fcr*?**  T h i s f u n c t i o n . i s t o meet t h e c o n d i t i o n s , e s t a b l i s h e d in Section III-C, that a) yuYrZ\,->;> >sO b) yu." (R\rri)  ±0  The purpose o f t h i s a p p e n d i x i s t o show t h a t u ( R j n ^ ) meets t h e c o n d i t i o n s l i s t e d  above.  (A.2.1)  yu' / f c t n L ) = Bo  y'(?lni)  e x p/Zc-*0 o] < L - * 3 ~ ' D  Lc-^i)  ~ - & P [e-^f C c - ^ t l c c ^ ) ' : 0  2  (A.2.2)  (A.2.3)  1 1 3  Substitute R = R  in max  Since  C = R  m a x  y (R|n.) i ' 1  p  v  %  (C-R max' ) = O >  M' The In first  (Km*.}\ m)  -  o.  t h i r d c o n d i t i o n i s met. order to a n a l y z e the f i r s t  the c o n s t a n t s are a n a l y z e d .  two c o n d i t i o n s ,  The c h a r a c t e r i s t i c s o f i n p u t s o f the f u n c t i o n a r e a.  O  b.  from S e c t i o n  <c R  R-'i  m  ,  n  +  < */  0.5  ^  R- ^  ±  mc  I.'  (A.2.4)  II.E.3 ( X. ^ m  +  Z  m !  n)  (A.2.5) c.  since  0  *  R  m  .  n  <_ < £c  ^  ^  I.  (A.2.6) (A. 2. 7) O ^  £.'  L  -  R > mlf  *>  (A. 2. 8)  115 To analyze constant D, Let  X = R.-R 1  Y = R max, X+Y = R max  . mm -R. 1  -R . mm  From eg. A.2.6. t h r u eg. A.2.8. o  + X  I  (A.2.9)  o ^ y < I O <  *+y +  (A.2.10) (A.2.11)  1  We have  and  Looking i n t o above expansions  of e  X  and (1-X)  -1  Therefore  (A.2.12)  /-X since  X>o Ce -/VA X  from  <  eg. A.2.11.  and s i n c e  & <£ <j  /  1  /(!'*)  (A.2.13)  116  using  eg. A.2.13 i-X  /< *  ^\ -t-  y  (A.2.14)  y-  R e s u b s t i t u t i n g v a l u e s f o r x and y. exf  CRc-B «„^ m  -1 ( ^ - ^ , ) / ^ ^ /  taking natural logarithm  (A.2.15)  of both sides  X A ^ CK-^^- R ^ i n ) - J U ^ e ^ ^ - f c ^ -»• T h i s i s t h e denominator o f c o n s t a n t  (A. 2.16) D, t h e r e f o r e we  have shown t h a t denominator o f D i s p o s i t i v e .  The numerator o f c o n s t a n t D i s  From S e c t i o n  /*3i  II.E.3,  - / I L I - ZC^l-yKtL)  Hence  (A.2.17)  •  S i n c e b o t h numerator and denominator o f D a r e p o s i t i v e  &  7  O.  (A.2.18)  Looking i n t o c o n s t a n t B.  ^31  ;  y /in  3  3  -7 O  (A.2.19)  Constant A. A-y* <3  ?°  (A.2.20)  Constant C;  Pm*<* 70 Looking i n t o c o n d i t i o n  (A. 2. 21)  (a)  (R\nO ^ O eg. A.2.2. g i v e s  BP e*'*>  ^'(ti.\m)* Since C - P -  +  \  o  & 7O  So, c o n d i t i o n ! a ) i s  satisfied.  D  ic-***"'Cc-n+o  118 From  A.2.8 fit 7-R,  Adding e x p r e s s ioonn Sbn Ln [ce^-e -,o))I I tto o both rce^-g^ m  n  i. -  sides  m/n)  (A.2.22) From eg. A.2.5  Adding  (R  A  -R'. ) t o both  sides  (A.2.23) Comparing w i t h eg.  A.2.22.  From eg. A.2.16  Then d i v i d i n g both s i d e s of A.2.24. by the above e x p r e s s i o n ,  2 I  Since l e f t  hand s i d e i s equal t o D  (A.2.1c)y  O -7, \ . By d e f i n i t i o n o f R J  (A. 2 . 25)  , max/  Squaring both s i d e s C * W  X  - 1 ^ 0  Z  ^  »  Since t h e e x p r e s s i o n s on l e f t  (A.2.26)  s i d e s o f A.2.25. and  A. 2.26. are both equal t o o r g r e a t e r than one, t h e i r product i s a l s o equal t o or g r e a t e r than one.  Since  C=R  '  max  D t C - l i T | ) ^ | 7, O eg.  A.2.3. g i v e s  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) -i]^ o x  (A.2.27)  120 The e x p r e s s i o n i n the l a r g e b r a c k e t on the r i g h t s i d e of eg. A.2.28., i s a product of a l l p o s i t i v e hence i t i s equal t o or l a r g e r than z e r o . i s m u l t i p l i e d by  (-1) , t h e r e f o r e  The c o n s t r a i n t (b) i s , t h e r e f o r e , met.  expressions,  The whole b r a c k e t  121 APPENDIX X I I E s t i m a t i o n o f W e i b u l l Extreme by Log-normal  In Appendix I the l o a d , a random v a r i a b l e , i s modeled by a W e i b u l l model.  In t h i s S e c t i o n i t w i l l be shown how  the type I asymptote o f a l a r g e s t o b s e r v a t i o n o f a W e i b u l l model i s estimated by a Log-normal model. The l o a d i s modeled by a W e i b u l l d i s t r i b u t i o n parameters a and A . w w  with  The d e n s i t y and d i s t r i b u t i o n f u n c t i o n  r  1  of l o a d are  F (L.-<T w  \ -  W )  (A.3.2)  t r ^ ) ^  The type I asymptote o f a l a r g e s t o b s e r v a t i o n from a W e i b u l l model has parameters A  CT  ;  W  (A.3.3)  ( JLo  - "7^  C J L  ^ °  }  '  (A.3.4)  see r e f e r e n c e (2). In order t o estimate the type I asymptote by a log-normal  model, " e s t i m a t i o n by quantiles"  Since the r i g h t  t a i l o f the maximum-load  i s o f h i g h importance, LQ  and L Q  "is employed. distribution  two h i g h - o r d e r q u a n t i l e s ,  are chosen f o r the e s t i m a t i o n p r o c e s s .  The q u a n t i l e of o r d e r L  f o r an extreme value v a r i a b l e q  is  L  1  c: yU  e  ~ (. l*n  ^)  CT^  (A. 3.5)  122  For  a log-normal model the parameters u  T  and a i n T  L  Li  terms o f q u a n t i l e s L and L a r e q q x  ^ v. yU  — *•"«*-  =• ^  u  2  h  ) '  . (A.3.6)  '  (A.3.7)  f i l , ' ^  S u b s t i t u t i n g A.3.6. f o r L q  equations:  -LLnJLn ^;<TeJ ^ 77"  cr - ^ t > & u  '  •  /^/>/e  and L l  - In  q  i n t o the above 2  C^t-(L^t^ 4r)<r,l — '(A.3.8)  - ^ i ^ ^ a - J - ^ ^  (A.3.9)  S u b s t i t u t i n g A.3.4. and A.3.5. f o r u  e  and a  i n the  above e q u a t i o n s : < r  u=  ML.  ^ ( " " w ^ ^ f )  =  (A. 3.10)  1 <r * *">> > •  (A.3.11)  n  v  J  In equations A.3.11. and A.3.12., a  T  Li  and u  .  T  Li,  tne  parameters o f log-normal approximation o f t h e type I asymptote model o f the o r i g i n a l W e i b u l l model f o r l o a d , are expressed i n terms o f t h e parameters o f the o r i g i n a l W e i b u l l model f o r l o a d , a , and Xw* S u b s t i t u t i n g q =0.85 and q =0.95 i n eg. A.3.11. and 1  2  rearranging u  L > u >  n + 1.8-2-J  (A.3.12 )  123 S u b s t i t u t i n g q = 0 . 8 5 , q = 0 i 9 5 , and A.3.13 1  2  eg. A.3.12., and r e a r r a n g i n g  a  for a  in  gives  (A3.14) In Appendix I the parameter v a l u e s  f o r the o r i g i n a l  W e i b u l l model f o r l o a d are  and  }\  «->o =  V. S"  S u b s t i t u t i n g these v a l u e s  i n equation  A.3.13 and  A. 3.14, we have  M  u  .  L,  J  U.S 5 Jinn D /-3.00)' +3.oo)'™  J  (A.3.15)  (A.3.16)  124 APPENDIX A n a l y s i s Qf Indiffer@nG§  Function.  IT. £, 5 . , i n d i f f e r e n c e f u n c t i o n s a r e  In S e c t i o n developed.  IV  An i n d i f f e r e n c e f u n c t i o n i s a c o l l e c t i o n o f  p o i n t s a t which th© b e n e f i t has the same v a l u e . s p e c i f i e d l e v e l of b e n e f i t defines  an i m p l i c i t  Thus, any function  between the m i s s i o n time and r e l i a b i l i t y . Since the b e n e f i t the  y  i s a f u n c t i o n o f both R and n,  indifference function for benefit level y  Q  may be  o b t a i n e d from y =f(n,R),  (A.4.1)  Q  which i s an i m p l i c i t there  f u n c t i o n i n two v a r i a b l e s .  Hence  e x i s t s a f u n c t i o n g such t h a t (A.4.2) -  n*g(R H ) ;  0  t h i s i s the i n d i f f e r e n c e f u n c t i o n f o r b e n e f i t l e v e l y In f a c t , f u n c t i o n g d e f i n e s each given  one i n d i f f e r e n c e f u n c t i o n f o r  l e v e l o f b e n e f i t , y.  To i n v e s t i g a t e the s l o p e o f g, s u b s t i t u t e f o r n, i n (A.4.1)  from  (A,4.2)j (A.4.3)  d i f f e r e n t i a t i n g with respect o=f g'+f n  where  R  g adg/dR 1  f -df/dn , n  f =*df/dft ;  t o R; (A.4.4)  125  and so -g'=fp/n  (A.4.5)  But from S e c t i o n s II.B.2 and II.C.2 we know t h a t t h e s l o p e s 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 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 f , a r e b o t h  positive.  Hence -g'=f /f >o R  (A. 4. 6)  n  T h i s shows t h a t when one v a r i a b l e i s d e c r e a s e d ,  and b e n e f i t  i s t o remain c o n s t a n t , t h e o t h e r v a r i a b l e must be i n 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 r e s p e c t t o R, we g e t  From S e c t i o n s I I . B . 3 and I I . C . 3 we know t h a t t h e second d e r i v a t i v e s o f y(R/n) w i t h r e s p e c t t o R, w i t h r e s p e c t t o n are both negative. r  are p o s i t i v e , f  3  f....-.,  and y (n/RO)  Since both f n  and f„ R  i s also positive.  Hence t h e terms w i t h i n t h e b r a c k e t i n (A.4.7) a r e a l l n e g a t i v e 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 a t a c o n s t a n t b e n e f i t v a l u e i s a convex f u n c t i o n w i t h n e g a t i v e s l o p e and p o s i t i v e second derivative.  

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