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UBC Theses and Dissertations

A dynamic programming - Markov chain algorithm for determining optimal component replacement policies Young, G. Glen 1970

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A DYNAMIC PROGRAMMING - MARKOV CHAIN ALGORITHM FOR DETERMINING OPTIMAL COMPONENT REPLACEMENT POLICIES by .6. GLEN YOUNG B . A . S c . ( F o r e s t E n g . ) , U . B . C , 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS OF THE DEGREE OF M a s t e r o f A p p l i e d S c i e n c e ( i n F o r e s t E n g i n e e r i n g ) i n the F a c u l t y of FORESTRY We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA AUGUST, 1970 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is under-stood that publication, in part or in whole, or the copying of this thesis for financial gain shall not be allowed without my written permission. 6. GLEN YOUNG Department of Forestry Engineering The University of British Columbia Vancouver 8, B.C., Canada Date Ork °) j \91o A B S T R A C T An a l g o r i t h m i s d e v e l o p e d t o d e t e r m i n e t h e o p t i m a l c o m p o n e n t r e p l a c e m e n t r u l e s t o f o l l o w i n m a n a g i n g a p a r t i c -u l a r c l a s s o f e q u i p m e n t . T h e w o r k f o l l o w s b a s i c a l l y t h e m o d e l s d e v e l o p e d p r e v i o u s l y b y S . E . D r e y f u s a n d R.A. H o w a r d . H o w e v e r , a d i f f e r e n t M a r k o v s t a t e d e s c r i p t i o n h a s b e e n u s e d t o e x t e n d t h e a p p l i c a t i o n o f t h e s e m o d e l s t o s y s t e m s o f m o r e t h a n o n e c o m p o n e n t s u b j e c t t o s t o c h a s t i c f a i l u r e a n d f o r w h i c h t h e f a i l u r e o f a n y c o m p o n e n t r e n d e r s t h e e n t i r e s y s t e m i n o p e r a t i v e . T h e m o d e l , i n e f f e c t , s e l e c t s o p t i m a l r e p l a c e m e n t a l t e r n a t i v e s a s i n d i v i d u a l c o m p o n e n t s f a i l , u n d e r t h e u n c e r t a i n t y o f f u r t h e r f a i l u r e s o c c u r r i n g i n t h e same t r a n s i t i o n i n t e r v a l . T h e m o d e l was p r o g r a m m e d f o r an I.B.M. 3 6 0 / 6 7 c o m p u t e r a n d t h e r e s u l t s f o r a h y p o t h e t i c a l p r o b l e m w e r e c h e c k e d t h r o u g h r e n e w a l t h e o r y . ACKNOWLEDGEMENTS I w i s h t o acknowledge the encouragement and h e l p f u l c r i t i c i s m o f D r . C. W. B o y d , my m a j o r a d v i s o r i n t h i s t h e s i s . A l s o , t h e a s s i s t a n c e of D r . J . V. Z i d e k r e g a r d i n g the i n v e r s i o n of the moment g e n e r a t i n g f u n c t i o n i n C h a p t e r I I I i s g r e a t l y a p p r e c i a t e d . TABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION AND LITERATURE SURVEY 1 INTRODUCTION . . . . . . . . . 1 LITERATURE SURVEY 3 G e n e r a l 3 Dynamic Programming - Markov C h a i n A l g o r i t h m s and Equipment M a i n t e n a n c e . . . 5 I I . A MAINTENANCE MODEL 11 MODEL PURPOSE AND STRUCTURE . . . . . . . . . 11 System D e s c r i p t i o n . . . . . 11 M a t h e m a t i c a l Model o f the System . . . . . 12 The R e p l a c e m e n t C o s t s . . . . . . . . . . 13 MODEL DEVELOPMENT . . . . . . . . . . . . . . . 14 A s s u m p t i o n s of the Model 1.4 D e f i n i t i o n of S p e c i a l Terms Used . . . . . 15 The Rep lacement D e c i s i o n s . . . . . . . . 17 T r a n s i t i o n P r o b a b i l i t i e s . . . . . . . . . 18 A p p r o x i m a t i n g the f a i l u r e p d f ' s . . . . 19 C o n d i t i o n a l p d f ' s of component f a i l u r e . 19 C o n d i t i o n a l p r o b a b i l i t i e s o f t r a n s i t i o n . . . . . . . . . . . . . . 21 The O b j e c t i v e F u n c t i o n . . . . . . . . . . 27 The r e t u r n f u n c t i o n . 27 The o p t i m i z a t i o n 30 R e p l a c e m e n t b e f o r e f a i l u r e . 31 V CHAPTER PAGE I I I . A HYPOTHETICAL MAINTENANCE PROBLEM: VERIFICATION OF MODEL RESULTS . . . . 33 A D e s c r i p t i o n of the M a i n t e n a n c e P r o b l e m . 33 A d d i t i o n s to the Model 39 I m p r o v i n g the e s t i m a t e o f e x p e c t e d r e p l a c e m e n t c o s t . . . . . 39 A d j u s t i n g t h e s y s t e m age v e c t o r . . . . 41 Program D e s c r i p t i o n . . . . . . . . . . . . . . 41 V e r i f i c a t i o n o f Model R e s u l t s . . . . . . 43 IV . DISCUSSION AND CONCLUSION . 54 DISCUSSION . . . . . . . . . . 54 CONCLUSION . . . . . . . 57 BIBLIOGRAPHY . . . . . . . . . . 59 APPENDIX A - G l o s s a r y of Program V a r i a b l e s and Program L i s t i n g . • 60 APPENDIX B - Sample Program O u t p u t . . . . . 71 APPENDIX C - R e s u l t s of N u m e r i c a l I n t e g r a t i o n s o f E q u a t i o n ( 3 . 9 ) . . . . . . . . . . . . . . 74 APPENDIX D - G a i n / S t a g e f o r V a r i o u s Rep lacement C o s t S t r u c t u r e s . 75 APPENDIX E - R e p l a c e m e n t P o l i c i e s f o r F i r s t F a i l u r e s 76 LIST OF TABLES TABLE PAGE I. Alpha and Beta Values for the Piece-wise Linear Approximations to the Failure Pdf's 36 k II. Coefficients of At for Equations of Transition Probabilities Resulting from Piece-wise Linear Approximations to Failure Pdf's 3 7 " 3 8 III. Replacement Cost Data of the Hypothetical Maintenance Problem Used to Verify Model Results . . . . . . . . . . 4 3 IV. A Comparison of the Expected Replacement Cost Produced by the Maintenance Model with that Derived from Renewal Theory 48 LIST OF FIGURES FIGURE PAGE 1. Event t r e e showing a l l p o s s i b l e component f a i l u r e sequences t h a t can o c c u r d u r i n g i n t e r v a l ( t , , t ) f o r a t h r e e - c o m p o n e n t s y s t e m . . 7 16 2 . F i g u r e showing method o f a p p r o x i m a t i n g the p r o b a b i l i t y d e n s i t y of t i m e t o f a i l u r e o f the ] t n s y s t e m component . . . . . . . . . . 20 3 . F i g u r e showing the a p p r o x i m a t i n g f u n c t i o n s used t o c a l c u l a t e t r a n s i t i o n p r o b a b i l i t i e s when components 1 , 2 , and 3 e n t e r the r t h t r a n s i t i o n i n t e r v a l a t ages a - s i . a i _ i » and a j < _ 1 r e s p e c t i v e l y . . . . " . . . . . . . 22 4 ( a ) . G r a p h i c a l r e p r e s e n t a t i o n o f t h e e v e n t t h a t component 1 f a i l s f i r s t d u r i n g t r a n s i t i o n i n t e r v a l ( t „ , , t „ . ) . . . . . . . . . . . . . . 24 r - l r 4 ( b ) . G r a p h i c a l r e p r e s e n t a t i o n of the e v e n t t h a t component 1 f a i l s f i r s t and component m f a i l s second d u r i n g t r a n s i t i o n i n t e r v a l (t t ) 24 4 ( c ) . G r a p h i c a l r e p r e s e n t a t i o n of the e v e n t t h a t t h e r e be a f a i l u r e sequence 1, m, n i n t r a n s i t i o n i n t e r v a l ( t r _ i s t^ ) 24 5 . F a i l u r e d e n s i t y f u n c t i o n s f o r t h r e e m a j o r components of a h y p o t h e t i c a l s y s t e m . . . . . . 34 6. A c o n c e p t u a l f l o w c h a r t o f the r e p l a c e m e n t a l g o r i t h m showing the h i e r a r c h y of the o p t i m i z a t i o n p r o c e s s . . . . 42 7 . Graph of r e p l a c e m e n t c o s t per t r a n s i t i o n i n t e r v a l as a f u n c t i o n of number o f i n t e r v a l s t o end o f p r o c e s s , f o r a h y p o -t h e t i c a l m a i n t e n a n c e p r o b l e m , f o r two d i f f e r e n t r e p l a c e m e n t c o s t s t r u c t u r e s . . . . . 50 8. Graph showing s t e a d y s t a t e r e p l a c e m e n t c o s t per t r a n s i t i o n i n t e r v a l as a f u n c t i o n of c o s t r e d u c t i o n f o r m u l t i p l e r e p l a c e m e n t s . . . . . . 53 CHAPTER I INTRODUCTION AND LITERATURE SURVEY I. INTRODUCTION The r a t i o n a l management o f e q u i p m e n t i s o f paramount i m p o r t a n c e t o i n d u s t r i e s t h a t have l a r g e i n v e s t m e n t s i n e q u i p m e n t . For t h e s e i n d u s t r i e s the a b i l i t y t o m a i n t a i n r e a s o n a b l e l e v e l s of p r o f i t depends t o a l a r g e e x t e n t on the e f f i c i e n c y o f p r e s e n t and f u t u r e o p e r a t i o n of the equ ipment t h r o u g h w h i c h t h e i r income i s d e r i v e d . The p r o b l e m of d e c i d i n g when t o r e p l a c e a p i e c e o f e q u i p m e n t because i t i s i n a d e q u a t e , o b s o l e t e , or r e q u i r e s e x c e s s i v e r e p a i r s i s o f t e n r e f e r r e d t o as the r e p l a c e m e n t  p r o b l e m , and t h e p r o b l e m of m a i n t a i n i n g the equ ipment between r e p l a c e m e n t s i s r e f e r r e d t o as the m a i n t e n a n c e p r o b l e m . H o w e v e r , i n the b r o a d e s t sense r e p l a c e m e n t o f an e n t i r e s y s t e m can be c o n s i d e r e d t o be a m a i n t e n a n c e a c t i o n . T h i s c o n c e p t p r o v i d e s a u n i f i e d a p p r o a c h to m a i n t e n a n c e and r e -p l a c e m e n t and w i l l be used t h r o u g h o u t t h i s t h e s i s . The s e r i e s o f m a i n t e n a n c e a c t i o n s p e r f o r m e d on a s y s t e m d u r i n g i t s u s e f u l l i f e c o n s t i t u t e s a m a i n t e n a n c e p o l i c y . The t a s k o f m a i n t e n a n c e t h e o r y i s t o d e t e r m i n e the o p t i m a l m a i n t e n a n c e p o l i c y t o f o l l o w f o r a g i v e n s y s t e m , or p i e c e o f e q u i p m e n t . An o p t i m a l m a i n t e n a n c e p o l i c y i s ' t h a t p o l i c y t h a t o p t i m i z e s some measure of s y s t e m e f f i c i e n c y ; 2 u s u a l l y maximum p r o f i t d e r i v e d f r o m t h e s y s t e m o r m i n i m u m c o s t o f o w n i n g a n d o p e r a t i n g i t . T h i s t h e s i s i s c o n c e r n e d w i t h d e v e l o p i n g a m o d e l t h a t w i l l d e t e r m i n e t h e o p t i m a l m a i n t e n a n c e p o l i c i e s ( i n t h e f o r m o f c o m p o n e n t r e p l a c e m e n t ) f o r a c l a s s o f e q u i p m e n t f r e q u e n t l y f o u n d i n i n d u s t r y . One a p p r o a c h t o o b t a i n i n g o p t i m a l m a i n t e n a n c e p o l i c i e s i s t o d e s c r i b e t h e f a i l u r e b e h a v i o r o f t h e s y s t e m w i t h a M a r k o v m o d e l a n d u s e d y n a m i c p r o g r a m m i n g t o p e r f o r m t h e o p t i m i z a t i o n . T h i s i s t h e a p p r o a c h u s e d i n t h i s t h e s i s . C o n s i d e r a p i e c e o f e q u i p m e n t t h a t c o n s i s t s o f m o r e t h a n o n e c o m p o n e n t a n d t h a t p e r f o r m s a c o n t i n u o u s f u n c t i o n . T h e f a i l u r e o f a n y c o m p o n e n t l e a v e s t h e e n t i r e s y s t e m i n -o p e r a t i v e , a n d n o f u r t h e r f a i l u r e s c a n o c c u r u n t i l t h e f a i l e d c o m p o n e n t i s r e p l a c e d a n d t h e s y s t e m r e s t a r t e d . T h e d e s c r i p -t i o n o f t h i s s y s t e m w i t h a d i s c r e t e - p a r a m e t e r M a r k o v m o d e l r e q u i r e s t h a t d i s c r e t e t i m e i n t e r v a l s be a r t i f i c a l l y c r e a t e d d u r i n g w h i c h t h e s t a t e t r a n s i t i o n s a n d t h e p r o b a b i l i t i e s o f t h e s e t r a n s i t i o n s c a n be d e f i n e d . D r e y f u s [5] d e s c r i b e d t h e s t a t e s o f a d i s c r e t e s y s t e m a s t h e c o m b i n a t i o n o f f a i l u r e s t h a t a c c u m u l a t e d u r i n g o n e t r a n s i t i o n . M a i n t e n a n c e a l t e r n -a t i v e s w e r e t h e n d e r i v e d f o r a n y c o m b i n a t i o n o f f a i l u r e s t h a t m i g h t o c c u r . T h i s f o r m u l a t i o n i s n o t m e a n i n g f u l h o w e v e r , f o r t h e s y s t e m d e s c r i b e d a b o v e , s i n c e o n c e a f a i l u r e h a s o c c u r r e d no f u r t h e r f a i l u r e s c a n o c c u r u n t i l a m a i n t e n a n c e a c t i o n i s t a k e n . T h e i m p o r t a n t d i f f e r e n c e h e r e i s t h a t t h e 3 m a i n t e n a n c e a c t i o n s m u s t be t a k e n a s e a c h c o m p o n e n t f a i l s , u n d e r t h e u n c e r t a i n t y o f h a v i n g f u r t h e r f a i l u r e s a t l a t e r e p o c h s w i t h i n t h e same t r a n s i t i o n i n t e r v a l . T h e s t a t e s m u s t be d e f i n e d t o a c c o m m o d a t e t h i s d i f f e r e n c e a n d t h e m o d e l d e v e l o p e d i n t h i s t h e s i s d o e s s o b y u s i n g t h e o r d e r i n w h i c h t h e c o m p o n e n t s c a n f a i l as t h e b a s i c s t a t e d e s c r i p t i o n . I I . L I T E R A T U R E SURVEY G e n e r a l A s u m m a r y o f m o d e l s d e v e l o p e d up t o 1 9 6 1 c a n b e f o u n d i n [ 4 ] . F o r an e x p o s i t i o n o f t h e t h e o r y o f m a i n t e n a n c e a n d r e p l a c e m e n t a n a l y s i s t h e r e a d e r i s r e f e r r e d t o [ l ] , [ 7 ] , a n d [ 8 ] . F o r t h e p u r p o s e o f m a i n t e n a n c e a n d r e p l a c e m e n t a n a l y s i s e q u i p m e n t h a s g e n e r a l l y b e e n d i v i d e d i n t o t w o m a i n c a t e g o r i e s : e q u i p m e n t t h a t d e t e r i o r a t e s a n d e q u i p m e n t t h a t f a i l s . S y s t e m d e t e r i o r a t i o n r e f e r s t o a g r a d u a l r e d u c t i o n i n e f f i c i e n c y d u e t o i n c r e a s i n g c o s t s o f u s i n g t h e s y s t e m c a u s e d b y i n c r e a s i n g s y s t e m m a i n t e n a n c e , i n c r e a s i n g l o s s o f p r o d u c t i o n d u e t o d o w n t i m e a n d r e d u c t i o n i n p r o d u c t q u a l i t y , a n d o b s o l e s c e n c e . F a i l u r e , on t h e o t h e r h a n d , r e f e r s t o an a b r u p t c h a n g e i n t h e s y s t e m t h a t r e n d e r s i t , o r o n e o f i t s c o m p o n e n t s , u s e l e s s f o r i t s o r i g i n a l p u r p o s e . T h e 4 model as d e v e l o p e d i n t h i s t h e s i s i s f o r sys tems t h a t f a i l , a l t h o u g h i t c o u l d be e x t e n d e d t o i n c l u d e sys tems t h a t are s u b j e c t t o both d e t e r i o r a t i o n and f a i l u r e . The p r o b l e m of d e t e r m i n i n g optimum m a i n t e n a n c e p o l i c i e s f o r sys tems w i t h components t h a t f a i l d i r e c t s i t s e l f t o t h r e e main c h a r a c t e r i s t i c s o f such s y s t e m s : 1 . The components f a i l a c c o r d i n g t o some p r o b a b i l i t y f u n c t i o n and thus the f a i l u r e t i m e s c a n n o t be p r e d i c t e d a c c u r a t e l y . 2 . The c o s t o f r e p l a c i n g components a f t e r f a i l u r e i s u s u a l l y g r e a t e r t h a n r e p l a c i n g b e f o r e f a i l u r e . T h i s i s due t o t h e c o s t a s s o c i a t e d w i t h u n s c h e d u l e d i n t e r r u p t i o n s i n s y s t e m o p e r a t i o n c a u s e d by i n -s e r v i c e f a i l u r e s . 3 . I t i s o f t e n c h e a p e r t o r e p l a c e components i n g roups than t o r e p l a c e them i n d i v i d u a l l y . T h i s i s p a r t i c u l a r l y so when the c o s t o f p r e p a r i n g t h e e q u i p m e n t f o r component r e p l a c e m e n t i s h i g h r e l a t i v e t o the c o s t o f a c t u a l l y r e p l a c i n g the c o m p o n e n t s . P o i n t s 2 and 3 wou ld s u g g e s t t h a t c o s t s a v i n g s c o u l d r e s u l t f rom r e p l a c i n g components b e f o r e t h e y f a i l . H o w e v e r , because component f a i l u r e s c a n n o t be p r e d i c t e d a c c u r a t e l y some p o l i c y o f r e p l a c i n g them b e f o r e they have aged t o o g r e a t l y must be a d o p t e d . On t h e o t h e r h a n d , t o o f r e q u e n t r e p l a c e m e n t o f f u n c t i o n a l components i n d u c e s e x c e s s i v e r e -5 p l a c e m e n t c o s t s and t h e r e i n l i e s the m a i n t e n a n c e p r o b l e m : the c o s t s a v i n g s r e s u l t i n g f rom f e w e r u n s c h e d u l e d sys tem i n t e r r u p t i o n s and economies of s c a l e r e s u l t i n g f rom group r e p l a c e m e n t of components must be we ighed a g a i n s t t h e c o s t of w a s t i n g e x p e c t e d r e s i d u a l l i f e o f w o r k i n g c o m p o n e n t s . Dynamic Programming - Markov C h a i n A l g o r i t h m s  and Equipment M a i n t e n a n c e E a r l y models o f sys tems s u b j e c t to s t o c h a s t i c f a i l u r e were f a i r l y r e s t r i c t e d i n scope s i n c e t e c h n i q u e s f o r s o l v i n g complex models were not a v a i l a b l e . The a p p r o a c h u s u a l l y t a k e n was to assume t h a t t h e o p t i m a l p o l i c y wou ld be o f a s p e c i f i c fo rm t h u s r e d u c i n g g r e a t l y the s e t o f c o m p e t i n g p o l i c i e s . T h i s r e s u l t e d i n many c a s e s i n s u b o p t i m a l s o l u -t i o n s s i n c e ' t h e o p t i m a l ' s o l u t i o n was n e v e r c o n s i d e r e d . The i n v e n t i o n of dynamic programming by R . E . B e l l m a n , i n 1 9 5 4 , p r o v i d e d an o p t i m i z a t i o n t e c h n i q u e t h a t c o u l d be used to s o l v e m a i n t e n a n c e p r o b l e m s w h i c h had no a p r i o r i a s s u m p t i o n s on the fo rm t h a t the o p t i m a l p o l i c y w o u l d t a k e . In 1 9 5 5 , B e l l m a n [ 2 ] p u b l i s h e d a dynamic programming model f o r d e t e r m i n i n g t h e o p t i m a l r e p l a c e m e n t t i m e f o r a h y p o -t h e t i c a l s y s t e m s u b j e c t to c o n t i n u o u s d e t e r i o r a t i o n . D r e y f u s [ 5 ] , i n 1 9 5 7 , p u b l i s h e d the f i r s t Dynamic P r o g r a m m i n g - M a r k o v C h a i n a l g o r i t h m f o r s o l v i n g an equ ipment m a i n t e n a n c e p r o b l e m . 6 I n 1 9 6 0 , H o w a r d [ 6 ] f o r m a l i z e d t h e t h e o r y o f d y n a m i c p r o g r a m m i n g a n d M a r k o v p r o c e s s . I n t h i s p u b l i c a t i o n he d e m o n s t r a t e s t w o m e t h o d s o f s o l v i n g f o r o p t i m a l p o l i c i e s i n a M a r k o v p r o c e s s t h e s e b e i n g v a l u e i t e r a t i o n a n d p o l i c y i t e r a t i o n . He a p p l i e d t h e p o l i c y i t e r a t i o n m e t h o d t o d e t e r m i n i n g t h e o p t i m a l r e p l a c e m e n t p o l i c y f o r an a u t o m o b i l e s u b j e c t t o f a i l u r e a n d c o n t i n u o u s d e t e r i o r a t i o n . T h e m o d e l d e v e l o p e d i n t h i s t h e s i s e m p l o y s t h e v a l u e i t e r a t i o n a p p r o a c h . T h i s m e t h o d was c h o s e n b e c a u s e i t d o e s n o t d e p e n d o n e r g o d i c i t y o f t h e m o d e l a n d c a n be u s e d i n s y s t e m s o f s h o r t d u r a t i o n w h e r e e n d e f f e c t s i n f l u e n c e t h e o p t i m a l p o l i c i e s . A b r i e f d i s c u s s i o n , f o l l o w i n g H o w a r d , i s g i v e n b e l o w o f t h e t h e o r y n e c e s s a r y t o d e v e l o p t h e m o d e l i n t h e f o l l o w i n g c h a p t e r . C o n s i d e r a d i s c r e t e - p a r a m e t e r , N - s t a t e , M a r k o v p r o c e s s i n w h i c h t h e s t a t e s a r e n u m b e r e d f r o m 1 t o N. L e t P = [ P - j - j ] d e n o t e t h e m a t r i x o f t r a n s i t i o n p r o b a b i l i t i e s f o r t h i s p r o c e s s , w h e r e p . . d e n o t e s t h e c o n d i t i o n a l p r o b a b i l i t y o f t r a n s i t i o n i n o n e s t e p t o s t a t e j g i v e n t h a t t h e s y s t e m i s i n s t a t e i . O b v i o u s l y , f o r a l l i N E p . . = 1 a n d o < p . . < 1 . j = l 1 J 1 J A l s o l e t R = [)".{,•] d e n o t e a r e t u r n m a t r i x f o r t h i s p r o c e s s , w h e r e r . . d e n o t e s t h e r e t u r n ( p o s i t i v e o r n e g a t i v e ) ' J t h a t i s o b t a i n e d f r o m t h e p r o c e s s w h e n e v e r a t r a n s i t i o n f r o m s t a t e i t o s t a t e j o c c u r s . I n a m a i n t e n a n c e p r o b l e m t h e r . . b e c o m e t h e c o s t s o f r e p l a c i n g c o m p o n e n t s . Now d e f i n e v n ( i ) a s t h e t o t a l e x p e c t e d r e t u r n f r o m t h e p r o c e s s a f t e r n t r a n s i t i o n s when t h e s y s t e m s t a r t s i n s t a t e i . T h i s r e t u r n c a n be e x p r e s s e d a s t h e sum o f t h e r e t u r n r e s u l t i n g f r o m t h e f i r s t t r a n s i t i o n f r o m i t o j , a n d t h e e x p e c t e d r e t u r n f r o m t h e r e m a i n i n g n-1 t r a n s i t i o n s s t a r t i n g t h e n i n s t a t e j o r N V n ( i ) = ^ / i j ^ i j + V l ( j ) ) . . . . . (1.1) w h i c h c a n be r e w r i t t e n a s V n ( i ) = q i + , L P i j V l ( j ) ..... (1.2) w h e r e N q . = I p . . r . . i s r e f e r r e d t o a s t h e i m m e d i a t e r e t u r n ( e x p e c t e d r e t u r n f r o m t h e f i r s t t r a n s i t i o n ) . I f a M a r k o v p r o c e s s i s l e f t u n d i s t u r b e d , t r a n s i t i o n s w i l l o c c u r r a n d o m l y w i t h p r o b a b i l i t i e s d e f i n e d by t h e t r a n - . s i t i o n m a t r i x P. H o w e v e r , i n some p r o c e s s e s c e r t a i n c o u r s e s o f a c t i o n ( a l t e r n a t i v e s ) a r e a v a i l a b l e t h a t w i l l c h a n g e t h e 8 t r a n s i t i o n p r o b a b i l i t i e s . I n t h e e q u i p m e n t m a i n t e n a n c e p r o b l e m , f o r i n s t a n c e , t h e r e p l a c e m e n t o f a g e d c o m p o n e n t s b y new o n e s w i l l r e d u c e t h e p r o b a b i l i t y o f t h e s y s t e m g o i n g i n t o a f a i l e d s t a t e , a n d i n c r e a s e t h e p r o b a b i l i t y o f i t r e m a i n i n g i n a w o r k i n g s t a t e . B u t s i n c e t h e r e i s a c o s t a s s o c i a t e d w i t h an a l t e r n a t i v e t h e r e t u r n m a t r i x i s a l s o c h a n g e d . L e t P k = [ p k . ] a n d R k = L > k . ] d e n o t e r e s p e c t i v e l y , 1 J i J t h e t r a n s i t i o n a n d r e w a r d m a t r i x t h a t r e s u l t s w h en t h e k t n a l t e r n a t i v e i s u s e d . T h e n q k = Z p k r k 1 / = 1 P i / i J i s t h e i m m e d i a t e e x p e c t e d r e t u r n . T h e p r o b l e m now b e c o m e s o n e o f d e t e r m i n i n g t h e k s o t h a t t h e o p t i m u m e x p e c t e d r e t u r n i s o b t a i n e d f r o m t h e M a r k o v p r o c e s s , a n d t h i s i s t h e r o l e o f d y n a m i c p r o g r a m m i n g . L e t f m ( i ) , m _< n , d e n o t e t h e o p t i m u m e x p e c t e d r e t u r n f r o m t h e r e m a i n i n g m t r a n s i t i o n s o f a n n - t r a n s i t i o n M a r k o v p r o c e s s when t h e p r o c e s s i s i n s t a t e i a n d a n o p t i m a l s e q u e n c e o f a l t e r n a t i v e s i s f o l l o w e d f o r t h e r e m a i n d e r o f t h e p r o c e s s . A l s o l e t k d e n o t e an a l t e r n a t i v e t h a t c a n be u s e d a t t h i s m p o i n t i n t h e p r o c e s s . T h e m i s r e f e r r e d t o a s t h e s t a g e n u m b e r o f t h e d y n a m i c p r o g r a m . W i t h t h i s l a b e l l i n g s y s t e m t h e ( n - m + l ) t n t r a n s i t i o n o c c u r s a t s t a g e m a n d t h e n t h 0 r l a s t t r a n s i t i o n i s made a t s t a g e 1. 9 By d e f i n i t i o n f m ( D = o p t [ q ^ m + Z pk.m. v ( j ) ] . . . ( 1 . 3 ) V m-1'-' , K1 J _ 1 D e c o m p o s i n g t h e o p t i m i z a t i o n y i e l d s f m ( i ) = o p t [q^ + I o p t v ( j ) ] . . ( 1 . 4 ) m m-1 1 o r fm<D • opt [q"m + j p J ; V l ( J ) ] . . . . ( i . s ) m J w h i c h i s t h e f u n c t i o n a l e q u a t i o n r e q u i r e d t o d e t e r m i n e t h e o p t i m a l p o l i c y f o r t h e ( n - m + l ) t n t r a n s i t i o n , a n d t h e o p t i m a l c o s t f o r t h e r e m a i n i n g m t r a n s i t i o n s o f t h e M a r k o v p r o c e s s . E q u a t i o n ( 1 . 5 ) i s s o l v e d r e c u r s i v e l y . T o b e g i n t h e e n d c o n d i t i o n , f Q ( i ) , m u s t b e s p e c i f i e d f o r i = 1,2,..,N. In e q u i p m e n t m a i n t e n a n c e p r o b l e m s t h e s e a r e t h e r e s a l e v a l u e s o f t h e e q u i p m e n t a t t h e e n d o f t h e p r o c e s s . T h e s o l u t i o n f o r an n - t r a n s i t i o n p r o c e s s i s t h e n o b t a i n e d b y s o l v i n g ( 1 . 5 ) f o r m = 1 ( s t a g e 1, c o r r e s p o n d i n g t o t h e l a s t t r a n s i t i o n ) w h i c h y i e l d s f - L ( i ) a n d t h e p o l i c y r e q u i r e d t o p r o d u c e f 1 ( i ) f o r a l l i . T h e n t h e f 1 ( i ) a r e u s e d t o o b t a i n t h e f 2 ( i ) , t h e o p t i m a l r e t u r n f o r a 2 - t r a n s i t i o n p r o c e s s . T h i s p r o c e s s i s i t e r a t e d f o r m = 3 , 4 , n a t w h i c h p o i n t t h e o p t i m a l s o l u t i o n f o r a n n - t r a n s i t i o n p r o c e s s i s o b t a i n e d . 10 T h i s r e c u r s i v e s o l u t i o n i s e q u i v a l e n t t o s t a r t i n g a t t h e e n d o f t h e p r o c e s s a n d w o r k i n g b a c k w a r d t h r o u g h t i m e u n t i l t h e p r e s e n t i s r e a c h e d . CHAPTER II A MAINTENANCE MODEL I. MODEL PURPOSE AND STRUCTURE T h i s c h a p t e r i s d e v o t e d t o the d e v e l o p m e n t o f an a l g o r i t h m t h a t can be used t o d e t e r m i n e the o p t i m a l component r e p l a c e m e n t p o l i c y t o use f o r a s y s t e m c o m p r i s e d of s e v e r a l m a j o r c o m p o n e n t s , t h e s e components b e i n g s u b j e c t t o f a i l u r e at random t i m e s w h i l e i n u s e . System D e s c r i p t i o n As used i n t h i s t h e s i s , the word s y s t e m means any a s s e m b l a g e of m e c h a n i c a l p a r t s t h a t p e r f o r m s some n e c e s s a r y f u n c t i o n and has t h e f o l l o w i n g c h a r a c t e r i s t i c s : 1 . The t o t a l s y s t e m can be r e d u c e d t o s e v e r a l ma jo r c o m p o n e n t s , a m a j o r component b e i n g d e f i n e d as a c o n t i g u o u s group o f p a r t s t h a t forms a u n i t w i t h i n t h e sys tem as a w h o l e . H e r e i n a f t e r a m a j o r com-ponent w i l l be r e f e r r e d t o s i m p l y as a ' c o m p o n e n t ' . 2. The components a r e s u b j e c t t o u n p r e d i c t a b l e f a i l u r e s d u r i n g u s e , where component f a i l u r e means t h a t a component has s u f f e r e d s u f f i c i e n t damage t o w a r r a n t i t s r e p l a c e m e n t . 3^ T h e f a i l u r e o f o n e c o m p o n e n t r e n d e r s t h e e n t i r e s y s t e m i n o p e r a t i v e . 4 . No f u r t h e r f a i l u r e s c a n o c c u r u n t i l t h e f a i l e d c o m p o n e n t ( p l u s p o s s i b l y o t h e r s ) i s r e p l a c e d a n d t h e s y s t e m r e s t a r t e d . 5. E a c h c o m p o n e n t m u s t be r e p l a c e d a s a u n i t . A s an e x a m p l e o f s u c h a s y s t e m t h e e n g i n e , t r a n s -m i s s i o n , a n d d i f f e r e n t i a l c o u l d be c o n s i d e r e d t o be t h r e e m a j o r c o m p o n e n t s i n t h e p o w e r t r a i n s y s t e m o f a v e h i c l e . M a t h e m a t i c a l M o d e l o f t h e S y s t e m T h e f a i l u r e b e h a v i o r o f t h e s y s t e m i s d e s c r i b e d b y m o d i f i e d , d i s c r e t e p a r a m e t e r , n o n - s t a t i o n a r y , f i n i t e s t a t e , M a r k o v p r o c e s s . T h e M a r k o v s t a t e s a n d p r o b a b i l i t i e s o f s t a t e t r a n s i t i o n a r e b a s e d on t h e o r d e r i n w h i c h c o m p o n e n t f a i l u r e s c a n o c c u r d u r i n g a s p e c i f i e d p e r i o d o f s y s t e m u s e ( a t r a n s i t i o n i n t e r v a l i n t h e M a r k o v p r o c e s s o r a s t a g e i n t h e D y n a m i c p r o g r a m ) . T h i s f o r m u l a t i o n o f t h e p r o b l e m , i n e f f e c t , a l l o w s r e p l a c e m e n t d e c i s i o n s t o o c c u r a s e a c h c o m p o n e n t f a i l s w i t h i n t h e t r a n s i t i o n i n t e r v a l . T h e p r o b a b i l i t i e s o f p o s s i b l e f a i l u r e o r d e r s a r e d e r i v e d f r o m t h e c o n d i t i o n a l p r o b a b i l i t y d e n s i t y f u n c t i o n s ( p d f ' s ) o f t i m e t o f a i l u r e o f t h e i n d i v i d u a l c o m p o n e n t s . T h i s i s d o n e b y c o n s t r u c t i n g t h e c o m p o u n d e v e n t a s s o c i a t e d w i t h e a c h f a i l u r e o r d e r . 13 T h e R e p l a c e m e n t ' C o s t s S i n c e e m p h a s i s was p l a c e d on t h e d e v e l o p m e n t a n d s t r u c t u r e o f t h e b a s i c r e p l a c e m e n t a l g o r i t h m , o n l y t h e d i r e c t c o s t s a s s o c i a t e d w i t h c o m p o n e n t r e p l a c e m e n t w e r e c o n s i d e r e d . F o r t h i s r e a s o n t h e f o l l o w i n g a s s o c i a t e d c o s t s w e r e n o t i n c l u d e d a t t h i s s t a g e : 1. T h e c o s t o f m o n e y o v e r t i m e , 2. C o r p o r a t e i n c o m e t a x d e d u c t i o n s on e x p e n d i t u r e s f o r e q u i p m e n t p u r c h a s e a n d e q u i p m e n t m a i n t e n a n c e , 3. T h e c o s t s o f l o s t p r o d u c t i o n d u e t o s y s t e m down t i m e , 4. T h e c o s t s o f o b s o l e s c e n c e d u e t o i m p r o v e d t e c h n o l o g y o f new e q u i p m e n t a v a i l a b l e , 5. T h e i n c r e a s i n g c o s t o f o p e r a t i o n d u e t o d e c r e a s i n g s y s t e m e f f i c i e n c y w i t h a g e , 6. T h e a s s o c i a t e d c o s t s o f s p a r e p a r t s i n v e n t o r i e s r e l a t e d t o r e p l a c e m e n t p o l i c i e s . I t i s t r u e t h a t a l l t h e s e c o s t s m u s t be c o n s i d e r e d i n a n y m a i n t e n a n c e p r o b l e m i f t h e p o l i c i e s d e r i v e d a r e t o be t r u l y o p t i m a l . H o w e v e r , t h e a d d i t i o n o f m o s t o f t h e s e c o s t s i s a s t r a i g h t f o r w a r d e x t e n s i o n o f t h e m o d e l d e v e l o p e d b e l o w . I I . MODEL D E V E L O P M E N T F o r c o n c r e t e n e s s t h e m o d e l was d e v e l o p e d f o r a t h r e e -c o m p o n e n t s y s t e m a n d c o m p o n e n t a g e was t a k e n t o mean r e a l t i m e s i n c e t h e c o m p o n e n t b e g a n o p e r a t i o n . T h e c o m p o n e n t s w i l l be l a b e l l e d c o m p o n e n t s 1 , 2, a n d 3. A s s u m p t i o n s o f t h e M o d e l I n t h e d e v e l o p m e n t o f t h e m o d e l i t was a s s u m e d t h a t : 1. T h e f a i l u r e o f e a c h c o m p o n e n t i s s t o c h a s t i c a l l y i n d e p e n d e n t f r o m f a i l u r e s o f t h e o t h e r c o m p o n e n t s . 2. A l l c o m p o n e n t s h a v e f i n i t e l i v e s a t w h i c h t h e c o n d i t i o n a l p r o b a b i l i t y o f f a i l u r e i s u n i t y . 3. T h e c o n d i t i o n a l p r o b a b i l i t y o f f a i l u r e o f a com-p o n e n t i s a g e - d e p e n d e n t o n l y . 4. R e p l a c e m e n t c o m p o n e n t s a r e i d e n t i c a l t o t h e com-p o n e n t s t h e y r e p l a c e w i t h r e s p e c t t o r e p l a c e m e n t c o s t a n d f a i l u r e c h a r a c t e r i s t i c s . 5. T h e c o s t o f r e p l a c i n g a n y c o m b i n a t i o n o f c o m p o n e n t s i s k nown w i t h c e r t a i n t y . 6. E a c h c o m p o n e n t c a n f a i l o n l y o n c e d u r i n g a t r a n s -i t i o n i n t e r v a l . 1 7. C o m p o n e n t s r e p l a c e d i n a n y t r a n s i t i o n i n t e r v a l b e g i n t h e n e x t i n t e r v a l a s new ( 0 a g e ) c o m p o n e n t s . ^An e m p i r i c a l m e t h o d o f r e d u c i n g t h e e r r o r s c a u s e d by t h i s a s s u m p t i o n i s d i s c u s s e d i n C h a p t e r I I I . 2 I b i d . 15 D e f i n i t i o n o f S p e c i a l T e r m s U s e d L e t t h e a x i s o f r e a l t i m e be d i v i d e d i n t o a s e q u e n c e o f i n t e r v a l s , A t i n l e n g t h , s u c h t h a t t h e b e g i n n i n g a n d e n d o f t h e r t n i n t e r v a l i s a t t i m e t r - 1 a n d t . r e s p e c t i v e l y . T h e s e i n t e r v a l s w i l l b e c o m e t h e t r a n s i t i o n i n t e r v a l s o f t h e M a r k o v m o d e l , as w e l l a s t h e s t a g e s o f t h e D y n a m i c p r o g r a m . T h e y w i l l b e r e f e r r e d t o i n t e r c h a n g e a b l y a s t r a n s i t i o n i n t e r v a l s , i n t e r v a l s , o r s t a g e s . A l l p o s s i b l e s e q u e n c e s o f c o m p o n e n t f a i l u r e s t h a t c a n o c c u r d u r i n g t h e r t n i n t e r v a l h a v e b e e n e l a b o r a t e d i n t h e e v e n t t r e e o f F i g u r e 1. E a c h n o d e i n t h e t r e e r e p r e s e n t s t h e e v e n t o f f a i l u r e o f t h e c o m p o n e n t n u m b e r c i r c l e d . A z e r o a t a n o d e r e p r e s e n t s t h e s p e c i a l e v e n t o f no f a i l u r e a t t h a t l e v e l . N o d e s a t L e v e l j , j = 1 , 2 , 3 , r e p r e s e n t t h e j t h f a i l u r e i n a s e q u e n c e w i t h i n t h e i n t e r v a l ( t r _ 1 » * r ) ' A f i n a l c o n d i t i on s t a t e w i l l b e d e f i n e d a s a s e q u e n c e o f f a i l u r e s t h a t h a s o c c u r r e d d u r i n g a c o m p l e t e i n t e r v a l ( t 1 S t r ) , w h i c h c o r r e s p o n d s t o a c o m p l e t e p a t h i n t h e t r e e o f F i g u r e 1. T h e p r o b a b i l i t y o f a n y s u c h c o m p l e t e s e q u e n c e o c c u r r i n g w i l l be c a l l e d a t r a n s i t i o n p r o b a b i l i t y . A n i n t e r m e d i a t e c o n d i t i o n s t a t e w i l l be d e f i n e d a s a n i n c o m p l e t e s e q u e n c e o f f a i l u r e s t h a t h a s o c c u r r e d i n ( t r _ 1 , t ) , w h i c h c o r r e s p o n d s t o a n i n c o m p l e t e p a t h i n t h e t r e e down t o L e v e l 1 o r 2. T h e p r o b a b i l i t y t h a t t h e s y s t e m w i l l p a s s t h r o u g h a p a r t i c u l a r i n t e r m e d i a t e c o n d i t i o n ' s t a t e Level 1 Level 2 Level 3 Final y 10 120 123 130 132 20 210 213 230 231 30 310 312 320 321 0 Py (A) = conditional probability of transition to condition state y \j - failure of component I y = condition state Figure 1* Event tree showing all possible component failure sequences that can occur during interval t r _ i , t r for a three-component system* w i l l b e c a l l e d t h e p r o b a b i 1 i t y o f p a r t i a l t r a n s i t i o n f o r t h a t p a r t i c u l a r s t a t e . T h e c o n d i t i o n s t a t e ( f i n a l o r i n t e r m e d i a t e ) t h a t t h e s y s t e m o c c u p i e s a t t i m e t , t r 1 < t <_ t ^ , w i l l b e d e n o t e d b y y • T h e a g e s t a t e o f t h e s y s t e m w i l l b e d e n o t e d b y t h e v e c t o r , A , w h e r e t h e i t n c o m p o n e n t o f A i s t h e a g e o f t h e i t n c o m p o n e n t i n t h e s y s t e m d u r i n g t h e r t h i n t e r v a l , m e a s u r e d i n a p p r o p r i a t e u n i t s . A c o m p l e t e s t a t e d e s c r i p t i o n o f t h e s y s t e m a t a n y t i m e t , t r _ 1 < t <_ t r , c a n t h u s b e g i v e n b y t h e p a i r ( Y . - A ) . T h e R e p l a c e m e n t D e c i s i o n s A r e p l a c e m e n t d e c i s i o n m u s t be made i m m e d i a t e l y u p o n f a i l u r e o f a n y c o m p o n e n t i n t h e s y s t e m , o r e q u i v a l e n t l y , a t e a c h n o n - z e r o n o d e i n t h e e v e n t t r e e o f F i g u r e 1. R e -p l a c e m e n t d e c i s i o n s made a t t i m e o f f i r s t f a i l u r e d u r i n g an i n t e r v a l w i l l b e c a l l e d L e v e l 1 d e c i s i o n s a n d d e n o t e d b y ' d ' . T h o s e made a t t i m e o f s e c o n d f a i l u r e w i l l b e L e v e l 2 d e c i s i o n s a n d d e n o t e d b y ' e ' . No s p e c i a l n o t a t i o n w i l l b e u s e d f o r L e v e l 3 d e c i s i o n s s i n c e no r e p l a c e m e n t a l t e r n a t i v e s e x i s t , o t h e r t h a n r e p l a c e m e n t o f t h e f a i l e d c o m p o n e n t . T h e r e p l a c e m e n t a l t e r n a t i v e s a r e : 1. L e v e l 1 d = 1; R e p l a c e f a i l e d c o m p o n e n t o n l y , d = 2; R e p l a c e f a i l e d c o m p o n e n t p l u s o n e o f t h e tw o o p e r a t i v e c o m p o n e n t s , d = 3 ; R e p l a c e f a i l e d c o m p o n e n t p l u s b o t h t h e o p e r a t i v e c o m p o n e n t s . 2. L e v e l 2 e = 1; R e p l a c e f a i l e d c o m p o n e n t o n l y , e = 2; R e p l a c e f a i l e d c o m p o n e n t p l u s r e m a i n i n g c o m p o n e n t n o t a l r e a d y r e p l a c e d . 3 . L e v e l 3 R e p l a c e f a i l e d c o m p o n e n t . T r a n s i t i o n P r o b a b i l i t i e s T h e n o t a t i o n f o r t h e c o n d i t i o n a l p r o b a b i l i t i e s o f p a r t i a l a n d c o m p l e t e t r a n s i t i o n i s s h o w n a d j a c e n t t o t h e a p p r o p r i a t e l i n k s i n t h e t r e e o f F i g u r e 1. T h e s e p r o b a b i l i t i e s a r e o f t h e f o r m P ^ ( A ) , a n d w i l l b e d e v e l o p e d b y i n t e g r a t i n g f u n c t i o n s o f t h e c o n d i t i o n a l p d f ' s o f t i m e t o f a i l u r e o f t h e i n d i v i d u a l c o m p o n e n t s . S i n c e many p d f ' s a r e n o t e a s i l y i n t e g r a t e d an a p p r o x i m a t i o n t o t h e a c t u a l 3 p d f ' s w i l l b e u s e d a n d i s i n t r o d u c e d b e l o w . I n p r a c t i c e t h i s a p p r o x i m a t i o n w o u l d n o t be made o n l y f o r r e a s o n s o f i n t e g r a b i 1 i t y . I n m o s t c a s e s e i t h e r i n s u f f i c i e n t d a t a a r e a v a i l a b l e t o d e t e r m i n e t h e t r u e f o r m o f t h e p d f o r n o n e o f t h e common t h e o r e t i c a l p d f ' s w i l l f i t t h e d a t a a d e q u a t e l y . A p p r o x i m a t i ng t h e f a i 1 u r e p d f ' s A p d f o f t i m e t o f a i l u r e , f - | ( t ) , f o r a g e n e r a l i z e d c o m p o n e n t i s s h o w n i n F i g u r e 2. A c o n v e n i e n t a p p r o x i m a t i o n t o f - j ( t ) c a n b e o b t a i n e d a s f o l l o w s : 1. S u b d i v i d e t h e r a n g e o f t h e r a n d o m v a r i a b l e t i n t o m e q u a l - l e n g t h a g e i n t e r v a l s a s s h o w n i n F i g u r e 2. F o r r e a s o n s t h a t w i l l b e c o m e a p p a r e n t l a t e r t h e l e n g t h o f t h e s e i n t e r v a l s m u s t be t h e same a s t h e i n t e r v a l s ( t l S t r ) , o r A t . 2. D e f i n e a f u n c t i o n , g ^ - C t ) , t h a t a p p r o x i m a t e s f - | ( t ) o v e r t h e i t h a g e i n t e r v a l s u c h t h a t l i ( t ) * f ^ t ) f o r a i_1 < t < a . i = 1,2,....m . . ( 2 . 1 ) 0 o t h e rw i s e a n d m E i = l a.-a 1 g n ( t ) d t = l i - l . • ( 2 . 2 ) a n d 9 l i - l ( a i ) = 9 l i ( a i > ( 2 . 3 ) C o n d i t i o n a l p d f ' s o f c o m p o n e n t f a i l u r e T h e p r o b a b i l i t y o f f a i l u r e o f a c o m p o n e n t d u r i n g an i n t e r v a l c a n be c o n d i t i o n e d on t h e e v e n t t h a t a t t h e b e g i n n i n g o f t h e i n t e r v a l t h e c o m p o n e n t h a s a l r e a d y r e a c h e d a g i v e n a g e w i t h o u t f a i l i n g . A s t r a i g h t f o r w a r d a p p l i c a t i o n o f t h e e q u a t i o n o f c o n d i t i o n a l p r o b a b i l i t y l e a d s t o t h e r e s u l t , Figure 2- Figure showing method of approximating the probability density of time to failure of the I th system component-21 P { c o m p o n e n t 1 f a i l s i n ( a ^ _ 1 , t ) , a . . ^ <_ t <_ a^ h a s l i v e d t o a g e a 1-_ 1> t i - l g ^ C s J d s / d - E a i - l k = 1 9 l k ( s ) d s ) . ( 2 . 4 ) l k - l T h e d e n o m i n a t o r o f ( 2 . 4 ) i s a c o n s t a n t f o r a n y i n t e r v a l a n d t h e e q u a t i o n may be w r i t t e n a s P {• } ^ ( s j d s l i - l w h e r e i -1 * n ( s ) = g n ( s ) / ( i - E 1 1 1 1 k = l 9 l k ( s ) d s ) l k - l (2.5) i s t h e c o n d i t i o n a l p d f o f f a i l u r e o f t h e I t h c o m p o n e n t f o r i t s i t h a g e i n t e r v a l . C o n d i t i o n a l p r o b a b i 1 i t i e s o f t r a n s i t i on F i g u r e 3 s h o w s t h e p a r t i c u l a r a p p r o x i m a t i o n s t o t h e f a i l u r e p d f ' s , g l i ( t ) , a n d t h u s t h e i r r e s p e c t i v e ^ ^ ( t ) t h a t i n f l u e n c e t h e P^,(A) when t h e s y s t e m b e g i n s t h e r t h t r a n s i t i o n i n t e r v a l w i t h a g e v e c t o r A. H e r e , t h e a g e v e c t o r c a n be t a k e n t o b e t h e t r i p l e ( i , j , k ) , w h e r e i t i s u n d e r s t o o d t h a t t h e i t h a g e i n t e r v a l o f a c o m p o n e n t b e g i n s a t a g e a ^ _ ^ a n d e n d s a t a g e a . . T h e e n d p o i n t s o f t h e Figure 3 Figure showing the approximating functions used to calculate transition probabilities when components 1, 2 and 3 enter the r th transition interval at ages a i - l » a j - l * a n d a k - l respectively-c o m p o n e n t a g e i n t e r v a l s c o i n c i d e w i t h t h e e n d p o i n t s o f t h e t r a n s i t i o n i n t e r v a l s b e c a u s e o f A s s u m p t i o n 7 a n d t h e f a c t t h a t b o t h a g e a n d t r a n s i t i o n i n t e r v a l s a r e A t i n l e n g t h . T h e P ^ ( A ) c a n be o b t a i n e d f r o m t h e ^ ^ - ( t ) by c o n -s t r u c t i n g t h e c o m p o u n d e v e n t a s s o c i a t e d w i t h t h e f a i l u r e s e q u e n c e y. T h e P ^ ( A ) a r e d e r i v e d b e l o w . ( a ) P r o b a b i l i t y o f p a r t i a l t r a n s i t i o n t o L e v e l 1 P . j # > ( A ) d e n o t e s t h e p r o b a b i l i t y t h a t c o m p o n e n t 1 , 1 = 1 , 2 , 3 , f a i l s f i r s t d u r i n g ( t l S t r ) . F i g u r e 4 ( a ) i s a g r a p h i c a l r e p r e s e n t a t i o n o f t h i s s a m e e v e n t . F o r a s p e c i f i c v a l u e o f t , t ^ < t < t t h e n t h e p r o b a b i l i t y o f 1 f a i l i n g f i r s t c a n be w r i t t e n a s 5 1 # . ( t , A ) d t = P {1 f a i l s i n t , t + d t | l l i v e d t o a i - _ 1 n m a n d n do n o t f a i l i n t r l , t|m a n d n h a v e l i v e d t o a ^ _ 1 a n d a ^ _ 1 r e s p e c t i v e l y }, . . ( 2 . 6 ) f o r 1, m a n d n = 1 , 2 , 3 ; m f 1; n f 1 o r m. N o t i n g t h a t a s t v a r i e s o v e r ( t r _ 1 , t r ) v a r i e s o v e r ( a i - l ' a.. ) ( s e e F i g u r e 3 ) , p e r f o r m i n g t h e t r a n s f o r m a t i o n o f v a r i a b l e z-j = x ^ - a i - _ 1 > a n d r e c a l l i n g t h a t t h e a s s u m p t i o n o f i n d e p e n d e n c e o f f a i l u r e s h a s b e e n m a d e , t h e e q u a t i o n f o r P i . . ( A ) b e c o m e s , 24 >no Emor E n Figure 4(a)- Graphical representation of the event that component I fails first during transition interval ( t r _ j ,t r )• • f c •m t r -r-1 no E, Figure 4(b)- Graphical representation of the event that component I fails first and component m fails second during transition interval (tr.j.t,.)-•m *r-l ± Y | Figure 4(c)- Graphical representation of the event that there be failure sequence l,m,n in transition interval ( t r _ j , t r )• P T . . ( A ) = t ( 1 S ( t , A ) d t r - l rZ A t * i i < Z l > 1 K . - ( z , J d z J mj m m 2 5 . . . ( 2 . 7 ) ( 1 - IJI1 . (z )dz ) d z , r n k v n n ' 1 where the ^ . ( t ) a re the t r a n s f o r m e d ^ ^ ( t ) (b) P r o b a b i l i t y o f p a r t i a l t r a n s i t i o n t o L e v e l 2 p l m . ( A ) d e n o t e s the p r o b a b i l i t y o f an i n t e r m e d i a t e s t a t e t r a n s i t i o n t o L e v e l 2 t h r o u g h any f a i l u r e sequence 1 , m (1 and m = 1 , 2 , 3 ; m f 1 ) when r e p l a c e m e n t a l t e r n a t i v e d i s used on f a i l u r e of component 1 . A g r a p h i c a l r e p r e s e n -t a t i o n of t h i s e v e n t i s shown i n F i g u r e 4 (b ) and by r e a s o n i n g as above i t s p r o b a b i l i t y can be shown t o b e ; P T m . ( A > = L * 1 1 ( Z 1 } " J , Ki^' ( l - j 0 ^ n k ( z n ) d z n ) d z m d z l 1 ( 2 . 8 ) when d = 1 ; p f m ( A ) = I m • A t A t * l 1 ( 2 l H , Kj{zJ° ^~ > n k ( z n ) d z n ) d z m d z l . . . ( 2 . 9 ) 26 w h e n d = 2 a n d n r e p l a c e d w i t h 1, a n d ; = o i f m r e p l a c e d w i t h 1 ; P ^ m ( A ) = o f o r d = 3 ( 2 . 1 0 ) P 1 q ( A ) d e n o t e s t h e p r o b a b i l i t y o f t h e s p e c i a l e v e n t o f no f u r t h e r f a i l u r e s a f t e r t h e f i r s t . T h e s e p r o b a b i l i t i e s a r e ; , _ r A t f A t r A t p j 0 ( A ) - j Q * j 1 - ( z 1 ) . ( i - j o K^Jd2J'^-\0 * n k < 2 n > d z n > d z i ( 2 . 1 1 ) P ^ 0 ( A ) - } o * i 1 ( « 1 ) - ( l - j o 1 * i J ( z m ) d z | n ) . ( l . j o * ; k ( z n ) d z n ) d z 1 . ( 2 . 1 2 ) w h e r e c o m p o n e n t m r e p l a c e d w i t h 1, a n d ; P ? 0 ( A ) = 1 ( 2 . 1 3 ) ( c ) P r o b a b i l i t y o f c o m p l e t e t r a n s i t i o n t o L e v e l 3 P ^ n ( A ) d e n o t e s t h e p r o b a b i l i t y o f a t r a n s i t i o n t o L e v e l 3 t h r o u g h a n y f a i l u r e s e q u e n c e 1, m, n w i t h r e p l a c e m e n t a l t e r n a t i v e d u s e d on f a i l u r e o f c o m p o n e n t 1 a n d a l t e r n a t i v e e u s e d o n f a i l u r e o f c o m p o n e n t m. F i g u r e 4 ( c ) s h o w s a g r a p h i c a l r e p r e s e n t a t i o n o f s u c h a s e q u e n c e . T h e p r o b a b i l -i t i e s c a n be s h o w n t o b e : 27 1 mn A t a n d 1 mn ' A t ty' . ( z ) ^ m j v ' A t 'm 0 f o r a l l o t h e r d a n d e . * n k ( z n ) d z n d z m d z l ; . . . . ( 2 . 1 4 ) . . . . ( 2 . 1 5 ) T h e O b j e c t i v e F u n c t i o n T h e d y n a m i c p r o g r a m m i n g o p t i m i z a t i o n i s d e v e l o p e d i n t h i s s e c t i o n b y f i r s t d e r i v i n g t h e r e t u r n f u n c t i o n f o r t h e M a r k o v m o d e l w i t h f i x e d a l t e r n a t i v e s , a n d t h e n s h o w i n g how d e c o m p o s i t i o n c a n be e f f e c t e d t o y i e l d t h e m i n i m i z a t i o n o f r e p l a c e m e n t c o s t o v e r N s t a g e s o f t h e p r o g r a m . T h e r e t u r n f u n c t i o n An e q u a t i o n f o r t h e e x p e c t e d r e t u r n f r o m N t r a n -s i t i o n s o f t h e M a r k o v p r o c e s s d e v e l o p e d i n t h i s c h a p t e r c a n be o b t a i n e d b y t h e same r e a s o n i n g t h a t l e d t o e q u a t i o n (1.2) T h e i m m e d i a t e r e t u r n , h o w e v e r , i s o b t a i n e d b y s u m m i n g t h e c o s t s o f s e r i e s o f p a r t i a l t r a n s i t i o n s r a t h e r t h a n o f s i n g l e c o m p l e t e t r a n s i t i o n s . L e t V N < A > = q A + V l ( 2 . 1 6 ) be t h e e x p e c t e d r e t u r n f r o m N t r a n s i t i o n s w hen t h e s y s t e m s t a r t s t h e f i r s t t r a n s i t i o n i n a g e s t a t e A , w h e r e V ^ _ 1 i s t h e e x p e c t e d r e t u r n f r o m t h e N-1 r e m a i n i n g t r a n s i t i o n s a f t e r t h e f i r s t , a n d q i s a s d e f i n e d i n e q u a t i o n (1 .2 ) . T h e c o n d i t i o n s t a t e , y » h a s b e e n o m i t t e d f r o m t h e i n i t i a l s t a t e d e s c r i p t i o n s i n c e t h e s y s t e m a l w a y s s t a r t s i n c o n d i t i o n s t a t e y = o . 1. I m m e d i a t e r e t u r n s T h e I m m e d i a t e r e t u r n , q ^ , c a n be o b t a i n e d a s f o l l o w s : ( a ) T h e e x p e c t e d c o s t o f t r a n s i t i o n f r o m L e v e l o , YQ = o , t o L e v e l 1, y^ = i s t h e p r o b a b i l i t y o f t r a n s -i t i o n f r o m Y0 t o YT_ t i m e s t h e r e p l a c e m e n t c o s t i n d u c e d b y t h i s t r a n s i t i o n , summed o v e r a l l p o s s i b l e o r , I P l e . ( A ) C ^ , 1=1 1 1 w h e r e C d i s t h e r e p l a c e m e n t c o s t i n c u r r e d w hen 1 f a i l s a n d r e p l a c e m e n t a l t e r n a t i v e d i s u s e d . ( b ) T h e e x p e c t e d c o s t o f t r a n s i t i o n f r o m L e v e l 1, y^ = I * * , t o L e v e l 2, y^ = l m * , i s t h e p r o b a b i l i t y o f t r a n s i t i o n Y0 t o Y£ t i m e s t h e r e p l a c e m e n t c o s t i n c u r r e d , summed o v e r a l l Y]_ a n d y^ o r , 3 3 i _ I I P ° ( A ) C ® 1=1 m-1 l m ' m ( c ) T h e e x p e c t e d c o s t o f t r a n s i t i o n s f r o m L e v e l 2 , Y 2 = l m * > t o L e v e l 3 , Y3 = l m n , i n a s i m i l a r m a n n e r c a n be s h o w n t o b e , 3 3 3 . _ 3 3 H p _ Z Z Z P 1 m n ( A ) C n = E z P l m n ( A > C i 1=1 m = l n = l l m n n 1=1 m = l l m n 1 m ^ l n ^ l ,m m ^ l n ^ l ,m ( d ) T h e t o t a l e x p e c t e d r e p l a c e m e n t c o s t f o r o n e c o m p l e t e t r a n s i t i o n i s sum o f t h e c o s t s o f t h e p a r t i a l t r a n s i t i o n s . T h u s , a d d i n g t h e t e r m s i n ( a ) t o ( c ) a n d com-b i n i n g t h e s u m m a t i o n s y i e l d s 1=1 m=l „ m m ^ l n ? M ' m 2 . R e t u r n s f r o m t h e N - 1 r e m a i n i n g t r a n s i t i o n s L e t B Q = ( i + l , j + 1 , k + l ) be t h e s y s t e m a g e s t a t e a f t e r t h e f i r s t t r a n s i t i o n w h e n t h e s y s t e m s t a r t s i n a g e s t a t e A a n d no f a i l u r e s o c c u r d u r i n g t h e i n t e r v a l . T h e n B d d e n o t e s t h e s y s t e m a g e s t a t e w h en c o m p o n e n t 1 o n l y f a i l s a n d a l t e r n a t i v e d i s e x e c u t e d f o r i t s r e p a i r . S i m i l a r l y , " d e E3lm i s t h e r e s u l t i n g a g e s t a t e w h en f a i l u r e s e q u e n c e 1, m o c c u r s a n d a l t e r n a t i v e s d a n d e a r e u s e d . O b v i o u s l y , - de B l m n = ( i ' 1 ' 1 ) f ° r a n v d a n d e- N o w » s i n c e vM-I^ B) 1 S T N E e x p e c t e d r e t u r n f r o m N-1 t r a n s i t i o n s s t a r t i n g i n t h e g e n e r a l a g e s t a t e B r e s u l t i n g f r o m t h e f i r s t t r a n s i t i o n , we h a v e V i - 2 t p ? 0 ( A)» H. x(5f) + I (p? : 0 (A )v H . 1 ( i ? ; ) 1=1 m =l "tl . . (2.18) w h e r e t h e P . 0 ( A ) a r e t h e p r o b a b i l i t i e s o f t h e s y s t e m e n d i n g t h e f i r s t t r a n s i t i o n i n t h e c o r r e s p o n d i n g s t a t e B. T h e o p t i m i z a t i o n L e t f ( A ) d e n o t e t h e m i n i m u m e x p e c t e d r e p l a c e m e n t c o s t o v e r N t r a n s i t i o n i n t e r v a l s ( s t a g e s o f t h e D . P . ) ^ w h e n t h e s y s t e m i n i t i a l a g e s t a t e i s A , a n d an o p t i m a l r e p l a c e -m e n t p o l i c y i s f o l l o w e d . T h u s , f N ( A ) = m i n [ q ^ + F N ± ] , . . . . ( 2 . 1 9 ) d N ' e N w h e r e ( d ^ , e ^ ) i s t h e s e t o f r e p l a c e m e n t a l t e r n a t i v e s made i n t h e k t h s t a g e , a n d F w , = m i n [V., , ] . N ~ 1 ( d e )-"(d e ) H~1  l Q N - l ' e N - l ; ^ai»V S u b s t i t u t i n g ( 2 . 1 7 ) a n d ( 2 . 1 8 ) i n t o ( 2 . 1 9 ) a n d r e -a r r a n g i n g y i e l d s t h e f i n a l r e s u l t , 4 D . P . d e n o t e s D y n a m i c p r o g r a m . 31 3 + _ m i n {P d 1 n r ( A ) C ® + P ' m d e Imo ( A ) f m =l e + P d e l m n ( A ) ( C _ + f N-1 (B ))}] + P o < A > f N - l < Bo> • ( 2 . 2 0 ) w h e r e t h e s u b s c r i p t N h a s b e e n o m i t t e d f r o m e ^ a n d d ^ t o s i m p l i f y t h e n o t a t i o n . T h e o p t i m i z a t i o n o f ( 2 . 2 0 ) i s p e r f o r m e d r e c u r s i v e l y , as s h o w n i n C h a p t e r 1, b y s t a r t i n g a t s t a g e 1, t h e N t r a n s i t i o n i n t e r v a l , a n d w o r k i n g b a c k w a r d i n t i m e t o s t a g e N , t h e f i r s t t r a n s i t i o n i n t e r v a l . R e p l a c e m e n t b e f o r e f a i l u r e I n some s y s t e m s f o r w h i c h t h e c o s t o f n o n - s c h e d u l e d i n t e r r u p t i o n s o f o p e r a t i o n i s s u b s t a n t i a l t h e o p t i m a l com-p o n e n t r e p l a c e m e n t p o l i c y may i n v o l v e r e p l a c e m e n t o f c o m p o n e n t s a t t h e b e g i n n i n g o f a s t a g e b e f o r e t h e y h a v e f a i l e d . T h i s t y p e o f p o l i c y c a n e a s i l y b e i n c l u d e d i n t h e o p t i m i z a t i o n o f ( 2 . 2 0 ) b y c h e c k i n g t h e i n e q u a t i o n f w ( A ' ) + C f < f M ( A ) , ( 2 . 2 1 ) a f t e r t h e f ^ ( A ) h a v e b e e n c a l c u l a t e d , w h e r e C i s t h e c o s t o f t h e r e p l a c e m e n t b e f o r e f a i l u r e a l t e r n a t i v e , f , a n d A 1 i s t h e new i n i t i a l a g e s t a t e c r e a t e d by a l t e r n a t i v e f . I f ( 2 . 2 1 ) h o l d s t r u e f o r a g i v e n f , t h e n t h a t a l t e r n a t i v e s h o u l d be u s e d i n t h e o p t i m a l p o l i c y . C H A P T E R I I I A H Y P O T H E T I C A L M A I N T E N A N C E P R O B L E M : V E R I F I C A T I O N OF MODEL R E S U L T S T h i s c h a p t e r i s d e v o t e d t o t e s t i n g t h e a c c u r a c y o f t h e p o l i c i e s a n d e x p e c t e d c o s t s p r o d u c e d b y t h e m o d e l d e v e l o p e d i n C h a p t e r I I , f o r a h y p o t h e t i c a l m a i n t e n a n c e p r o b l e m . A b r i e f d e s c r i p t i o n i s g i v e n o f t h e m a i n t e n a n c e p r o b l e m a n d o f t h e c o m p u t e r p r o g r a m w r i t t e n t o p e r f o r m t h e o p t i m i z a t i o n . Two p r a c t i c a l a d d i t i o n s t o t h e m o d e l t h a t i m p r o v e i t s a c c u r a c y a r e o u t l i n e d . T h e r e n e w a l f u n c t i o n , f r o m r e n e w a l t h e o r y , i s u s e d t o d e t e r m i n e t h e a c c u r a c y o f c o s t s p r o d u c e d b y t h e m o d e l f o r t h e c a s e o f e c o n o m i c i n d e p e n d e n c e o f r e p l a c e m e n t s . T h e r e a l i s m o f t h e p o l i c i e s p r o d u c e d i s a l s o d i s c u s s e d f o r t h i s c a s e . R e s u l t s f o r o t h e r r e p l a c e m e n t c o s t s t r u c t u r e s a r e b r i e f l y d i s c u s s e d . A D e s c r i p t i o n o f t h e M a i n t e n a n c e P r o b l e m T h e h y p o t h e t i c a l s y s t e m c o n s t r u c t e d t o v e r i f y m o d e l a c c u r a c y was c o m p r i s e d o f t h r e e c o m p o n e n t s . T h e f a i l u r e p d f ' s o f t h e s e c o m p o n e n t s a r e s h o w n i n F i g u r e 5. C o m p o n e n t 1 h a d a maximum u s e f u l l i f e o f 5 t i m e u n i t s , i t s f a i l u r e p d f was s k e w e d r i g h t a n d h a d a mean o f 2 . 6 0 . C o m p o n e n t 2 h a d a maximum l i f e o f 4 t i m e u n i t s . I t s f a i l u r e p d f was 34 Component Age at Failure - Q| Component 2 E(a 2)n-70 Component Age at Failure — ag Component Age at Fai lure - 0 3 5- Failure density functions for three major components of a hypothetical system-s k e w e d l e f t a n d h a d a mean o f 1 . 7 0 . T h e t h i r d c o m p o n e n t h a d a maximum l i f e o f 6 t i m e u n i t s , i t s f a i l u r e p d f r o u g h l y a p p r o x i m a t e d an e x p o n e n t i a l d i s t r i b u t i o n a n d h a d a mean o f 2 . 0 0 . T h e t h r e e d i s t r i b u t i o n s c h o s e n c o v e r t h e g e n e r a l r a n g e o f t y p e s o f f a i l u r e p d f ' s t h a t a r e f o u n d i n many m e c h a n i c a l s y s t e m s . T h e i n t e r e s t i n g c a s e o f c o m p o n e n t s t h a t e x h i b i t i n f a n t m o r t a l i t y was n o t i n c l u d e d , h o w e v e r , t h i s t y p e o f d i s t r i b u t i o n w o u l d n o t p r e s e n t a n y new d i f f i c u l t i e s . As c a n b e s e e n f r o m F i g u r e 5 t h e a c t u a l p d f ' s w e r e a p p r o x i m a t e d b y t h e p i e c e - w i s e l i n e a r f o r m g ^  ^ ( t ) = a ^ . + 3 - | . j ( t ) , f o r a-j.-j^ £ t <_ a.. . T h i s a p p r o x i m a t i o n was c h o s e n f i r s t b e c a u s e o f i t s s i m p l i c i t y o v e r n o n - l i n e a r f o r m s a n d s e c o n d l y b e c a u s e i n m o s t p r a c t i c a l m a i n t e n a n c e p r o b l e m s s u f f i c i e n t d a t a w o u l d n o t u s u a l l y be a v a i l a b l e t o j u s t i f y t h e u s e o f a m o r e c o m p l i c a t e d n o n - l i n e a r a p p r o x i m a t i o n . T a b l e I s h o w s t h e c o e f f i c i e n t s o f t h e l i n e a r a p p r o x i m a t i o n s t o t h e p d f ' s a n d T a b l e I I s h o w s t h e c o e f f i c i e n t s o f t h e p o l y n o m i a l s r e s u l t i n g f r o m t h e s o l u t i o n o f e q u a t i o n s ( 2 . 7 ) t o ( 2 . 1 5 ) when t h e a b o v e a p p r o x i m a t i n g f u n c t i o n s a r e u s e d . T A B L E I A L P H A AND B E T A V A L U E S FOR T H E P I E C E - W I S E L I N E A R A P P R O X I M A T I O N S TO THE F A I L U R E P D F ' S COMP. AGE I N T E R V A L NO. 1 2 3 4 5 6 a 3 a 3 a 3 a 3 a 3 a 3 1 .0 .2 .2 .0 .2 .2 .4 - .2 - .2 -.2 - -2 .0 .4 .4 .1 .5 -.4 .1 -.1 - - - -3 . 3 3 3 3 - . 0 5 5 5 . 2 7 7 8 - . 0 5 5 5 . 2 2 2 3 - . 0 5 5 5 . 1 6 6 8 - . 0 5 5 5 . 1 1 1 3 - . 0 5 5 5 . 0 5 5 5 - . 0 5 5 5 CO TABLE II COEFFICIENTS OF A t k FOR EQUATIONS OF TRANSITION PROBABILITIES RESULTING FROM PIECE-WISE LINEAR APPROXIMATION TO FAILURE PDF'S* k P}0(A) pf 0 (A)** 1 *** a l a l • a l 0 2 -V2 [a 1 (a m + a n ) -3 1 ] V 2 - a l ( a m + a n } 3 1 / 2 -a 1 (a m / 2 + a n ) a i a / 2 1 m 3 +3 1 (a m + a n )] 4 1 / 4hKV Bm a n > / 2 + B l ( a m V ^ m + e n ) / 2 ^ hVV e l ( fV 2 + e n» / 4 + 6 l ( a m V * m ) / 2 l 5 l / 1 0 [ . l B m 3 n / 2 +3-,(am3n/2+3 m a n ) ] l m n m n 6 ¥ / n / 2 4 W n / 1 6 -3.3 3 /24 l m n Po(A) not shown. It was calculated from n ( l - (a- ,At+3,At /2) ) . ** 1=1 1 Component m replaced with component 1. The age i n t e r v a l s u b s c r i p t s i , j , k have been o m i t t e d t o s i m p l i f y the n o t a t i o n . TABLE II ( C o n t ' d . ) <l 1 k P?n.(A)** 0*> 1 0 0 0 2 a i V 2 1 m 0 3 l/3Ca 1 B N + a n ( B r a l a m ) / 2 ] V m / 3 - a m ( a 1 a n - 3 1 / 3 ) / 2 a l a m a n / 6 4 l / 4 [ 3 n ( 3 1 - a 1 a m ) / 2 - a l ( e m V 3 + W 4 > + M V 4 - V n / 3 > / 2 l/4 [3 1 a m a n /6 V V n / 2 + V V 3 ^ 5 -e n ( a 1B m / 2 + 6 l a m ) / 1 5 " W r / 4 0 - + M V n / 2 % 6 n / 3 ) ^ 1 / 5 ^ 3 ^ / 3 +3 n(a 3 /3+S a /4)/2] 1 v nrn' m n •' 6 " W n / 4 8 " W n / 1 6 3iB 3 /48 1 m n' 39 A d d i t i o n s t o t h e M o d e l I m p r o v i n g t h e e s t i m a t e o f e x p e c t e d r e p l a c e m e n t c o s t As d e v e l o p e d i n C h a p t e r I I t h e m o d e l a c c o u n t s f o r t h e e x p e c t e d c o s t o f e x a c t l y o n e f a i l u r e o f e a c h c o m p o n e n t i n a t r a n s i t i o n i n t e r v a l , s i n c e t h e p r o b a b i l i t y o f a d d i t i o n a l f a i l u r e s o f a c o m p o n e n t was a s s u m e d t o be z e r o ( A s s u m p t i o n 6, C h a p t e r I I ) . F o r t h e p r e s e n t p r o b l e m , h o w e v e r , t h e p r o b a b i l -i t i e s o f a d d i t i o n a l f a i l u r e s a r e n o t a l w a y s s m a l l . F o r e x a m p l e , i f c o m p o n e n t 1 i s s t a r t i n g i t s i t h a g e i n t e r v a l a t t h e b e g i n -n i n g o f a t r a n s i t i o n i n t e r v a l , t h e p r o b a b i l i t y o f a t l e a s t t w o f a i l u r e s o f t h i s c o m p o n e n t i s P ( a t l e a s t t w o f a i l u r e s o f 1 d u r i n g t 1 5 t |1 n a s r e a c h e d a g e a . ^ ) f A t f A t f ^ C t ) - i K , ( s ) d s d t . . . . . ( 3 . 1 ) J o ' t F o r 1 = 3, i = 6, a n d A t = 1, ( 3 . 1 ) y i e l d s a v a l u e o f 0 . 1 9 7 w h i c h i s s u b s t a n t i a l l y g r e a t e r t h a n z e r o . T h u s , t o c o m p e n s a t e f o r t h e u n d e r e s t i m a t i o n o f r e p l a c e m e n t c o s t t h a t w o u l d r e s u l t f r o m A s s u m p t i o n 6, a m e t h o d o f a p p r o x i m a t i n g t h e e x p e c t e d c o s t o f a d d i t i o n a l f a i l u r e s was d e v i s e d as f o l l o w s : 1. A d d i t i o n a l c o s t o f L e v e l 1 d e c i s i o n s . T h e e x p e c t e d f a i l u r e t i m e o f e a c h c o m p o n e n t f o r i t s i t h a g e i n t e r v a l , a , . , c a n be c a l c u l a t e d f r o m 40 a l i A t sif>j. ( s ) d s / A t ^' ( s ) d s . . . . . ( 3 . 2 ) o I f i t i s a s s u m e d t h a t t h e f i r s t f a i l u r e , 1, a n d a l l c o m p o n e n t s r e p l a c e d w i t h 1 a r e r e p l a c e d a t a^ . , t h e n t h e e x p e c t e d c o s t o f a d d i t i o n a l f a i l u r e s o f a c o m p o n e n t r e p l a c e d a t L e v e l 1 c a n be c a l c u l a t e d f r o m A d d . C o s t = c • m c A t - a , . ' > m l ( s ) d s , . . . . ( 3 . 3 ) o w h e r e m = 1 , 2, o r 3 i s t h e r e p l a c e d c o m p o n e n t . 2. A d d i t i o n a l c o s t s o f L e v e l 2 a n d 3 d e c i s i o n s . I f i t i s a s s u m e d t h a t t h e s e c o n d f a i l u r e i n a n i n t e r v a l o c c u r s a t t h e m i d p o i n t o f ( a ^ . , A t ) , a n d t h a t t h e t h i r d f a i l u r e o c c u r s m i d way b e t w e e n t h e t i m e o f t h e s e c o n d f a i l u r e a n d A t , t h e n i n g e n e r a l ( i . e . c o m p o n e n t s f o r w h i c h aQ = o ) t h e a d d i t i o n a l c o s t o f f a i l u r e s o f c o m p o n e n t s r e p l a c e d a t L e v e l 2 a n d 3 b e c o m e s 1/4 a n d 1/16 o f ( 3 . 3 ) , r e s p e c t i v e l y . T h e s e a d d i t i o n a l c o s t s w e r e a d d e d t o t h e m o d e l , h o w e v e r , 1/9 i n s t e a d o f 1/16 o f ( 3 . 3 ) was a r b i t r a r i l y u s e d f o r t h e a d d i t i o n a l c o s t o f L e v e l 3 r e p l a c e m e n t s t o c o m p e n s a t e f o r C o m p o n e n t 3 f o r w h i c h a ^ o . I t s h o u l d b e n o t e d h e r e t h a t no a t t e m p t was made t o s e a r c h f o r o p t i m a l r e p l a c e m e n t d e c i s i o n s f o r a d d i t i o n a l 1 41 f a i l u r e s . T h e m o d e l a s s u m e s r e p l a c e m e n t o f t h e f a i l e d c o m-p o n e n t o n l y . A d j u s t i n g t h e s y s t e m a g e v e c t o r A s s u m p t i o n 7, C h a p t e r I I , s t a t e s t h a t a l l c o m p o n e n t s r e p l a c e d i n a n i n t e r v a l b e g i n t h e n e x t i n t e r v a l a s new c o m p o n e n t s . T h i s a s s u m p t i o n w o u l d p r o d u c e a n u n d e r e s t i m a t i o n o f r e p l a c e m e n t c o s t s i n c e , i n e f f e c t , some ' f r e e u s e ' i s o b t a i n e d f r o m e a c h r e p l a c e m e n t b e t w e e n t h e a c t u a l t i m e o f r e p l a c e m e n t a n d t h e e n d o f t h e i n t e r v a l . T o r e d u c e t h i s e r r o r t h e a l g o r i t h m was p r o g r a m m e d s o t h a t t h e f i r s t f a i l u r e i n e a c h i n t e r v a l , a n d a l l c o m p o n e n t s r e p l a c e d w i t h i t , w o u l d b e g i n t h e n e x t t r a n s i t i o n i n t e r v a l a t a g e i n t e r v a l 2. P r o g r a m D e s c r i p t i o n T h e a l g o r i t h m was p r o g r a m m e d i n FORTRAN IV a n d r a n on an IBM 3 6 0 / 6 7 c o m p u t e r . A c o n c e p t u a l f l o w c h a r t o f t h e p r o g r a m s h o w i n g t h e h e i r a r c h y o f t h e o p t i m i z a t i o n i s s h o w n i n F i g u r e 6. A l i s t i n g o f t h e s o u r c e p r o g r a m a n d a g l o s s a r y o f t h e v a r i a b l e n a m e s u s e d a r e c o n t a i n e d i n A p p e n d i x A. I n p u t s t o t h e p r o g r a m i n c l u d e t h e p r o g r a m o u t p u t c o n t r o l p a r a m e t e r s t h a t s p e c i f y a t w h a t s t a g e a n d s y s t e m a g e v e c t o r o u t p u t s h o u l d s t a r t , t h e maximum u s e f u l l i f e o f e a c h c o m p o n e n t , t h e ex's a n d $'s o f t h e a p p r o x i m a t i o n s t o t h e f a i l u r e p d f ' s , g - j ^ ( t ) , a n d t h e r e p l a c e m e n t c o s t s f o r a l l p o s s i b l e c o m b i n a t i o n s o f c o m p o n e n t r e p l a c e m e n t s . A l l o t h e r i n f o r m a t i o n r e q u i r e d i s c o m p u t e d w i t h i n t h e p r o g r a m . F i g u r e 6. A c o n c e p t u a l f l o w m o d e l s h o w i n g t h e m i z a t i o n p r o c e s s . c h a r t o f t h e m a i n t e n a n c e h i e r a r c h y o f t h e o p t i -START Data Input1 1- Output Control Paranv 3- Coefs of Failure PDF's 2- Max- Life of Components 4- Replacement Costs Calculation and output of absolute and conditional probabilities of failure for each component-Calculation of expected cost of additional failures of each component for replacement on failure of any component-Calculation of conditional probabilies of state transition for all possible age vectors A , states y,and alternatives d,e-0 —*• k 0 - * f k (A); all A ^ Set next stage-, ^ k+1 —>k •I ^ Set next age vector-, A Set transition to Level 1 ; I • I • Set transition to Levels 2 a 3;m,n Set next policy-, d Set next policy; e P|m-(A)Ce +Pfje0(A) V j l f O + de P|mn<A><cn +WB ,m„)) " » a No a < £ ' a - > /3' Save e, m,n 43 O u t p u t s i n c l u d e a s u m m a r y o f t h e a b s o l u t e a n d c o n -d i t i o n a l p r o b a b i l i t y o f f a i l u r e b y a g e i n t e r v a l f o r e a c h c o m p o n e n t , a n d t h e o p t i m a l c o m p o n e n t r e p l a c e m e n t p o l i c i e s a n d e x p e c t e d r e p l a c e m e n t c o s t f o r e a c h i n i t i a l a g e v e c t o r w i t h i n e a c h s t a g e . An e x a m p l e o u t p u t i s s h o w n i n A p p e n d i x B. T h e p r o g r a m c o n t a i n s e x c e s s i v e s u b s c r i p t i n g o f some v a r i a b l e s , a t t h i s p o i n t , t o f a c i l i t a t e f u r t h e r i n v e s t i -g a t i o n o f m o d e l b e h a v i o r . R e p l a c e m e n t b e f o r e f a i l u r e was n o t c o n s i d e r e d . V e r i f i c a t i o n o f M o d e i R e s u l t s T h e m o d e l was r u n on t h e s i x s e t s o f r e p l a c e m e n t c o s t d a t a s h o w n i n T a b l e I I I . T A B L E I I I R E P L A C E M E N T COST DATA OF T H E H Y P O T H E T I C A L M A I N T E N A N C E PROBLEM USED TO V E R I F Y MODEL R E S U L T S R E P L A C E M E N T C O M B I N A T I O N Run C o s t R e d . (%) ( 1 ) ( 2 ) ( 3 ) (1 ,2) (1 ,3) ( 2 , 3 ) (1 , 2 ,3) 1 0 1 0 C 0 2 0 0 0 3 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 2 10 1 0 0 0 2 0 0 0 3 0 0 0 2 7 0 0 3 6 0 0 4 5 0 0 5 4 0 0 3 20 1 0 0 0 2 0 0 0 3 0 0 0 2 4 0 0 3 2 0 0 4 0 0 0 4 8 0 0 4 30 1 0 0 0 2 0 0 0 3 0 0 0 2 1 0 0 2 8 0 0 0 3 5 0 0 4 2 0 0 5 40 1 0 0 0 2 0 0 0 3 0 0 0 1 8 0 0 2 4 0 0 3 0 0 0 3 6 0 0 6 50 1 0 0 0 2 0 0 0 3 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0 T h e s e d a t a r a n g e d f r o m t h e c a s e o f c o m p l e t e e c o n o m i c I n d e -p e n d e n c e o f r e p l a c e m e n t s ( t h e c o s t o f r e p l a c i n g t w o o r t h r e e c o m p o n e n t s t o g e t h e r i s t h e sum o f t h e i n d i v i d u a l r e p l a c e m e n t c o s t s ) t o t h e e x t r e m e c a s e w h e r e a 50 p e r c e n t r e d u c t i o n i n t o t a l r e p l a c e m e n t c o s t r e s u l t s f r o m m u l t i p l e r e p l a c e m e n t s . A d i s c u s s i o n o f t h e r e s u l t s i s c o n t a i n e d b e l o w . 1. Run 1: C o m p l e t e e c o n o m i c i n d e p e n d e n c e ( a ) P o l i c i e s . When n o c o s t s a v i n g r e s u l t s f r o m m u l t i p l e r e p l a c e m e n t s , o p t i m a l r e p l a c e m e n t p o l i c i e s s h o u l d i n v o l v e f a i l u r e r e p l a c e m e n t o n l y . When r u n a s f i r s t p r o -g r a m m e d t h e m o d e l p r o d u c e d p o l i c i e s t h a t i n v o l v e d o p p o r t u n -i s t i c r e p l a c e m e n t o f t h e t h i r d c o m p o n e n t w i t h t h e s e c o n d f a i l u r e , i n some c a s e s , when t h e t h i r d c o m p o n e n t was i n i t s l a s t a g e i n t e r v a l . I n v e s t i g a t i o n o f t h e r e l a t i v e c o s t s o f t h e f a i l u r e ( e = l ) a n d o p p o r t u n i s t i c ( e = 2) r e p l a c e m e n t a l t e r n a t i v e s f o r t h e s e c a s e s r e v e a l e d t h a t i n m o s t i n s t a n c e s t h e y d i f f e r e d i n t h e t h i r d d e c i m a l p l a c e . E r r o r s o f t h i s m a g n i t u d e a r e i n s i g n i f i c a n t a n d a r e m o s t l i k e l y c a u s e d b y t h e a p p r o x i m a t i o n t o t h e c o s t o f a d d i t i o n a l . r e p l a c e m e n t s t h a t was u s e d . A w e i g h t i n g f a c t o r was i n t r o d u c e d t o o v e r -come t h i s d e p a r t u r e f r o m t h e t h e o r e t i c a l l y c o r r e c t p o l i c i e s . W i t h t h i s w e i g h t i n g t h e a l g o r i t h m w o u l d n o t s e l e c t an e = 2 a l t e r n a t i v e a s b e i n g o p t i m a l u n l e s s i t c r e a t e d a t l e a s t a o n e p e r c e n t i m p r o v e m e n t i n e x p e c t e d r e p l a c e m e n t c o s t o v e r t h e e = 1 a l t e r n a t i v e . T h e r e s u l t i n g p o l i c i e s w e r e t h e n v e r y r e a l i s t i c . ( b ) E x p e c t e d r e p l a c e m e n t c o s t : An i n d e p e n d e n t c h e c k t h r o u g h r e n e w a l t h e o r y . W i t h t h e a s s u m p t i o n o f i n d e p e n d e n c e o f c o m p o n e n t f a i l u r e s a n d a p o l i c y o f f a i l u r e r e p l a c e m e n t o n l y , t h e h y p o t h e t i c a l t h r e e - c o m p o n e n t s y s t e m c a n be v i e w e d as t h r e e i n d e p e n d e n t r e n e w a l p r o c e s s e s r u n n i n g s i m u l t a n e o u s l y . T h e e x p e c t e d r e p l a c e m e n t c o s t o v e r a p e r i o d o f t i m e f o r e a c h i n d i v i d u a l c o m p o n e n t c a n t h u s be o b t a i n e d t h r o u g h i t s r e n e w a l f u n c t i o n a n d t h e r e s u l t s f o r e a c h c o m p o n e n t c o u l d t h e n be a d d e d t o o b t a i n t h e t h e o r e t i c a l r e p l a c e m e n t c o s t f o r t h e w h o l e s y s t e m . T h i s r e a s o n i n g , w h i c h i s o u t l i n e d b e l o w , was u s e d t o o b t a i n a c h e c k on m o d e l a c c u r a c y . F o r a r e n e w a l p r o c e s s o f d u r a t i o n T t h e e x p e c t e d r e n e w a l c o s t , E ( C - ) , c a n be c a l c u l a t e d f r o m E ( C T ) = E ( N j ) • c . . . . ( 3 . 4 ) w h e r e c i s t h e c o s t p e r r e n e w a l a n d E ( N . T ) i s t h e r e n e w a l f u n c t i o n . T h e r e n e w a l c o s t s a r e k n o w n ( R u n 1, T a b l e I I I ) s o t h e p r o b l e m r e d u c e s t o f i n d i n g E ( N T ) . I f t.j d e n o t e s t h e i t n i n t e r - r e n e w a l t i m e a n d r S = I t , i s t h e r a n d o m v a r i a b l e o f t i m e t o t h e r t n r e -i = l n e w a l , t h e p r o b a b i l i t y t h a t r r e n e w a l s o c c u r i n T c a n be s h o w n t o be P ( N T = r ) = F r ( T ) - F r + 1 ( T ) , . . . . (3.5) w h e r e F ^ ( T ) i s t h e c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n o f ( s e e [ 3 ] ) . T h u s t h e e x p e c t e d n u m b e r o f r e n e w a l s i n T c a n be e x p r e s s e d as E ( N T ) = _ r ( F ( T ) - F r + 1 ( T ) ) . 1 a l l r r r + 1 T h e F ^ ( T ) c a n be f o u n d b y t a k i n g t h e i n v e r s e t r a n s f o r m o f t h e k t h p o w e r o f t h e c o m p l e x moment g e n e r a t i n g f u n c t i o n , M t ( z ) , o f t h e f a i l u r e p d f , f ( t ) , a s f o l l o w s . S t a r t i n g w i t h Mt(z) e z t f ( t ) d t , . . . . (3.6) w h e r e z i s t h e c o m p l e x v a r i a b l e c + i u , a n d s u b s t i t u t i n g t h e l i n e a r a p p r o x i m a t i o n s t o f ( t ) y i e l d s , m M t ( z ) = E . . (3.7) w h e r e m i s t h e n u m b e r o f a p p r o x i m a t i n g i n t e r v a l s f o r f ( t ) , a n d a . , a . a n d 3- a r e a s p r e v i o u s l y d e f i n e d w i t h t h e J J J c o m p o n e n t n u m b e r s u b s c r i p t s o m i t t e d t o s i m p l i f y n o t a t i o n . E x p a n d i n g a n d p e r f o r m i n g t h e i n t e g r a t i o n o f ( 3 . 7 ) y i e l d s Mt(z) m Z a . e z t 3 j m + Z j = l z t ( z t - 1 ) ( 3 . 8 ) T h e F r ( T ) c a n t h e n be d e t e r m i n e d b y t h e i n v e r s i o n e q u a t i o n l - F ^ T ) = ± ( 1 - s i g n ( c ) ) + ^ w h e r e s i g n ( c ) =. , 1 f o r c > 0 e x p [ - z T + r l n M . f z ) ] ^ , . . ( 3 . 9 ) 0 f o r c = 0 l - l f o r c < 0 , ( s e e [ 9 ] ) . S i n c e i t w o u l d be d i f f i c u l t t o o b t a i n a c l o s e d f o r m s o l u t i o n t o ( 3 . 9 ) , a c o m p u t e r s u b r o u t i n e b a s e d on S i m p s o n ' s R u l e was w r i t t e n t o p e r f o r m t h e i n t e g r a t i o n n u m e r i c a l l y . T h e i n t e g r a t i o n was p e r f o r m e d f o r a l l v a l u e s o f r f r o m rm^n [ T / m a x l i f e o f c o m p o n e n t ] , f o r w h i c h F ^ ( T ) = 1, t o r m a x t h e r f o r w h i c h a n e g l i g i b l e c h a n g e i n E ( N T ) was p r o d u c e d . F o r t h i s s t u d y t h e c a l c u l a t i o n o f E ( N T ) t o ±.001 was c o n s i d e r e d s u f f i c i e n t a n d t h u s r „ was t a k e n t o be t h e r f o r max w h i c h ( r - l ) • ( F r _ 1 ( T ) - F ( T ) ) _< . o o l . T h e e x p e c t e d n u m b e r o f f a i l u r e s i n T was t h e n c a l c u l a t e d f r o m r max E ( N T > = r m i n + r z . + i F r ( T > • • • • • ( 3 ' 1 0 > m m T h e r e s u l t s o f t h e n u m e r i c a l i n t e g r a t i o n s a r e s h o w n i n A p p e n d i x C a l o n g w i t h t h e c a l c u l a t i o n s o f E ( N - ) . T a b l e IV s h o w s a c o m p a r i s o n o f t h e r e n e w a l t h e o r y r e s u l t s a s o b t a i n e d f r o m ( 3 . 4 ) , f o r T = 1 0 , T A B L E IV A C OMPARISON OF T H E E X P E C T E D R E P L A C E M E N T COST PRODUCED BY T H E M A I N T E N A N C E MODEL WITH TH A T D E R I V E D THROUGH RENEWAL THEORY Comp. No. R e p l a c e m e n t C o s t R e n e w a l T h e o r y R e s u 1 t s M o d e l R e s u l t s E (N j ) E ( C T ) E x p e c t e d C o s t 1 1 0 0 0 3 . 4 3 3 3 4 3 3 -2 2 0 0 0 5 . 4 8 2 1 0 9 6 4 -3 3 0 0 0 4 . 7 5 0 1 4 2 5 0 -E a l l - - 2 8 6 4 7 2 8 3 5 3 s t a r t i n g w i t h a l l new c o m p o n e n t s a t t i m e z e r o , w i t h t h o s e p r o d u c e d b y t h e m o d e l f o r t h e same c o n d i t i o n s . T h e r e s u l t s i n d i c a t e t h a t t h e m o d e l u n d e r e s t i m a t e s t h e r e p l a c e m e n t c o s t 49 b y o n e p e r c e n t . I n m o s t m a i n t e n a n c e p r o b l e m s t h i s w o u l d be a n e g l i g i b l e e r r o r . A s i m p l e c h e c k on t h e s t e a d y s t a t e r e p l a c e m e n t c o s t r a t e , o r g a i n p e r s t a g e o f t h e D . P . , i s a l s o a v a i l a b l e t h r o u g h some b a s i c r e s u l t s o f r e n e w a l t h e o r y . An o r d i n a r y r e n e w a l p r o c e s s r e a c h e s a n e q u i l i b r i u m s t a t e when r u n f o r a s u f f i c i e n t l e n g t h o f t i m e . As e q u i l i b r i u m i s o b t a i n e d t h e r e n e w a l f u n c t i o n c a n be s h o w n t o c o n v e r g e t o E ( N t ) = 1 . . . . ( 3 . 1 1 ) w h e r e t i s t h e p r o c e s s d u r a t i o n , a n d u i s t h e e x p e c t e d l i f e o f e a c h r e n e w a l ( a v e r a g e i n t e r - r e n e w a l t i m e ) . T h e t h e o r e t i c a l e x p e c t e d r e n e w a l c o s t , E ( C t ) , a t e q u i l i b r i u m , f o r t h e h y p o -t h e t i c a l p r o b l e m c a n t h u s be e x p r e s s e d a s 3 c,t E ( C . ) = l — , . . . . ( 3 . 1 2 ) 1 1=1 a 1 w h e r e c-| i s t h e c o s t o f o n e r e n e w a l o f t h e l t n c o m p o n e n t a n d a-j i s i t s e x p e c t e d l i f e . E q u a t i o n ( 3 . 1 2 ) y i e l d s a s t e a d y s t a t e r e p l a c e m e n t c o s t r a t e o f 3061 when t = 1 , t h e a-j a r e as s h o w n i n F i g u r e 5, a n d t h e c-j a r e a s s h o w n i n T a b l e I I I , Run 1. A t s t a g e 15 t h e m o d e l c o n v e r g e d t o a c o n s t a n t g a i n p e r s t a g e o f 3 0 1 5 ( s e e F i g u r e 7 a n d d a t a o f A p p e n d i x D) i n d i c a t i n g t h a t t h e m o d e l p r o d u c e d a 1.5 p e r c e n t u n d e r - e s t i m a t i on' o f t h e t h e o r e t i c a l v a l u e . o L _ i I i i I I i I i I I I i I I i i ' ' ' 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 Transition Interval Figure 7- Graph of replacement cost per transition interval as a function of number of intervals to end of process,for a hypothetical maintenance problem,for two different replacement cost structures • CD o 51 2. R u n s 2 t h r o u g h 6: I n c r e a s i n g e c o n o m i c d e p e n d e n c e . R u n s 2 t h r o u g h 6 i n v o l v e d i n c r e a s i n g t h e c o s t r e d u c -t i o n s f o r m u l t i p l e r e p l a c e m e n t s f r o m 10 p e r c e n t t o 50 p e r c e n t , r e s p e c t i v e l y . T h e c o s t s t r u c t u r e s a r e s h o w n i n T a b l e I I I a n d t h e m o d e l b e h a v i o r a s a f u n c t i o n o f t h e m i s d i s c u s s e d b r i e f l y b e l o w . ( a ) P o l i c i e s . F i r s t f a i l u r e r e p l a c e m e n t p o l i c i e s a r e s h o w n i n A p p e n d i x E , f o r a l l s y s t e m a g e v e c t o r s i n w h i c h t h e a g e o f c o m p o n e n t 3 i s 5 o r 6. T h e p o l i c i e s b e h a v e d w i t h c h a n g i n g c o s t s t r u c t u r e g e n e r a l l y a s w o u l d be e x p e c t e d - - i n c r e a s i n g c o s t s a v i n g s o f m u l t i p l e r e p l a c e m e n t s i n d u c i n g o p p o r t u n i s t i c r e p l a c e m e n t s o f i n c r e a s i n g l y y o u n g e r c o m p o n e n t s . T h e p o l i c i e s d i d n o t , h o w e v e r , f o l l o w a c o n s i s t e n t p a t t e r n . T h e o p p o r t u n i s t i c r e p l a c e m e n t o f a c o m p o n e n t a p p e a r e d i n some c a s e s t o be a f u n c t i o n o f t h e a g e s o f t h e o t h e r t w o c o m p o n e n t s , a s w e l l a s i t s own a g e . A c a s e i n p o i n t o c c u r r e d f o r t h e 10 p e r c e n t c o s t r e d u c t i o n f o r w h i c h a s e g m e n t o f t h e p o l i c y w e n t a s f o l l o w s : ( a ) i f t h e a g e v e c t o r i s ( 5 , l , 1 o r 2) r e p l a c e 1 a l s o i f 2 o r 3 f a i l s ; ( b ) i f t h e a g e v e c t o r i s ( 5 , 1, 3 t o 6 ) r e p l a c e 1 a l s o i f 3 f a i l s ; ( c ) i f t h e a g e v e c t o r i s ( 5 , 2 t o 4, 1 t o 6 ) r e p l a c e 1 i f 2 o r 3 f a i l s . No t h e o r e t i c a l c h e c k was made on t h e v a l i d i t y o f t h i s p h e n o m e n o n a n d i t c o u l d h a v e b e e n c a u s e d b y t h e a p p r o x i m a t i o n s a n d a s s u m p t i o n s o f t h e mode 1 . ( b ) E x p e c t e d r e p l a c e m e n t c o s t s . T h e e x p e c t e d r e p l a c e m e n t c o s t s p e r t r a n s i t i o n i n -t e r v a l r e a c h e d a s t e a d y s t a t e v a l ue f o r e a c h o f t h e r e p l a c e -m e n t c o s t s t r u c t u r e s . As w o u l d be e x p e c t e d t h i s v a l u e d e c r e a s e d w i t h i n c r e a s i n g c o s t r e d u c t i o n . I n F i g u r e 8 t h e s t e a d y s t a t e v a l u e s ( a v e r a g e d o v e r t h e l a s t 5 s t a g e s ) a r e s h o w n a s a f u n c t i o n o f t h e c o s t r e d u c t i o n . F i g u r e 7 s h o w s t h e n a t u r e o f t h e c o n v e r g e n c e f o r t h e z e r o a n d 40 p e r c e n t c o s t s t r u c t u r e s . T h e g a i n p e r s t a g e h a d c o n v e r g e d t o w i t h i n .1.5 p e r c e n t o f t h e f i n a l v a l u e b y s t a g e 6 i n a l l c a s e s . 53 Figure 8- Graph showing steady state replacement cost per transition interval as a function of cost reduction for mult iple replacements-C H A P T E R IV D I S C U S S I O N AND C O N C L U S I O N I . D I S C U S S I O N T h e r e s u l t s o f t h e r e n e w a l t h e o r y c h e c k s on t h e e x p e c t e d c o s t p r o d u c e d by t h e m o d e l i n d i c a t e d t h a t t h e m o d e l c l o s e l y a p p r o x i m a t e d t h e a b s t r a c t e d f o r m o f t h e h y p o t h e t i c a l m a i n t e n a n c e p r o b l e m u s e d . T h e 1.5 p e r c e n t e r r o r i n t h e r e s u l t s w o u l d be much l e s s t h a n t h e e r r o r s c a u s e d by t h e a b s t r a c t i o n i t s e l f . I m p r o v e d a p p r o x i m a t i o n s t o a c t u a l f a i l u r e p d f ' s w o u l d be o b t a i n e d t h r o u g h t h e u s e o f s h o r t e r a g e i n t e r v a l s , w i t h r e s p e c t t o t h e maximum c o m p o n e n t l i f e , t h a n w e r e u s e d i n t h e t e s t p r o b l e m . H o w e v e r , t h i s i m p r o v e m e n t w o u l d n o t be w i t h o u t p e n a l t y s i n c e t h e n u m b e r o f s y s t e m a g e s t a t e s a n d t h u s t h e a m o u n t o f c o m p u t a t i o n - a n d c o m p u t e r m e m o r y r e -q u i r e d i n c r e a s e s e x p o n e n t i a l l y w i t h t h e n u m b e r o f a g e i n t e r v a l s i n t o w h i c h t h e p d f ' s a r e d i v i d e d . I n t h e s o l u t i o n o f a n a c t u a l m a i n t e n a n c e p r o b l e m a t r a d e o f f w o u l d h a v e t o be made b e t w e e n t h e i n c r e a s e i n r e a l i s m a t t a i n e d a n d t h e i n c r e a s e d c o m p u t i n g c o s t s i n d u c e d by i n c r e a s i n g t h e n u m b e r o f a g e i n t e r v a l s . O f c o u r s e t h e a m o u n t a n d q u a l i t y o f f a i l u r e a n d c o s t d a t a a v a i l a b l e m u s t a l s o be t a k e n i n t o c o n s i d e r a t i o n , a n d i n t h e f i n a l a n a l y s i s t h e a g e i n t e r v a l m u s t be c h o s e n s o as t o p r o v i d e a m e a n i n g f u l b a s i s f o r p o l i c y i m p l e m e n t a t i o n . F u r t h e r , a s t h e n u m b e r o f a g e i n t e r v a l s i n t o w h i c h t h e f a i l u r e p d f ' s a r e d i v i d e d i n c r e a s e s s o w o u l d t h e d i f f i -c u l t y o f d e t e r m i n i n g t h e g ^ ( t ) s u b j e c t t o c o n s t r a i n t s ( 2 . 2 ) a n d ( 2 . 3 ) . S i n c e t h e p r o b a b i l i t y o f a g i v e n f a i l u r e o r d e r i s d e p e n d e n t on t h e r e l a t i v e m a g n i t u d e s o f t h e s l o p e s , 3 ^ , o f t h e l i n e a r a p p r o x i m a t i o n s s u f f i c i e n t d a t a a n d a r e l i a b l e f i t t i n g t e c h n i q u e m u s t be a v a i l a b l e i n o r d e r t o o b t a i n g o o d e s t i m a t e s o f t h e 3 -| ^ - G i v e n t h e r e q u i r e d d a t a i t a p p e a r s a s t h o u g h s e v e r a l t e c h n i q u e s c o u l d be u s e d r a n g i n g f r o m t r i a l a n d e r r o r g r a p h i c a l m e t h o d s t o m o r e s o p h i s t i c a t e d m e t h o d s e m p l o y i n g d y n a m i c p r o g r a m m i n g a n d t h e t h e o r y o f l e a s t s q u a r e s . T h i s l a t t e r c o u l d be p e r f o r m e d by s o l v i n g m n . ? m i n L £ [ f ( t . ) - ( a + g . . t J ] S . . . . ( 4 . 1 ) i =1 j = l J 1 1 1 1 J s u b j e c t t o ( 2 . 2 ) a n d ( 2 . 3 ) , w h e r e t h e f ( t . ) a r e t h e J e s t i m a t e s o f t h e f a i l u r e p d f c a l c u l a t e d f r o m t h e d a t a , n^ i s t h e n u m b e r o f e s t i m a t e s i n t h e i t h a g e i n t e r v a l , i i s t h e a g e i n t e r v a l , a n d t h e a ^ . a n d 3 ^ a r e as p r e v i o u s l y d e f i n e d . D y n a m i c p r o g r a m m i n g c o u l d be u s e d t o s o l v e ( 4 . 1 ) by t a k i n g t h e a g e i n t e r v a l s a s t h e s t a g e s , t h e a r e a u n d e r t h e p d f f o r k a g e i n t e r v a l s , A, , a s t h e i n i t i a l s t a t e v e c t o r , A k - l " A k " rAt ( a u + e l k ( t ) d t ) 56 as t h e s t a g e c o u p l i n g f u n c t i o n , a n d as t h e d e c i s i o n v a r i a b l e , h o w e v e r t h i s h a s n o t y e t b e e n d o n e . F o r p r o b l e m s i n w h i c h t h e r e a r e n o t e n o u g h d a t a t o w a r r a n t u s e o f t h e a b o v e t e c h n i q u e a n o t h e r m e t h o d c o u l d be u s e d t h a t i n v o l v e s l e a s t s q u a r e s f i t t i n g o f q u a d r a t i c s t o t h e c d f ' s o f t h e f a i l u r e d a t a , f o r e a c h a g e i n t e r v a l , a n d t h e n d i f f e r e n t i a t i n g t h e s e t o o b t a i n t h e l i n e a r a p p r o x i -m a t i o n s t o t h e p d f ' s . T h e q u e s t i o n o f s a t i s f y i n g c o n s t r a i n t s ( 2 . 2 ) a n d ( 2 . 3 ) h a s n o t b e e n i n v e s t i g a t e d f o r t h i s a p p r o a c h . T h e g r a p h i c a l m e t h o d w o u l d be u s e f u l when t h e d a t a a n d / o r s o l u t i o n s r e q u i r e d d o n o t w a r r a n t u s e o f m o r e s o p h i s -t i c a t e d e s t i m a t e s o f . a n d $^.. I n c a s e s w h e r e v e r y l i m i t e d d a t a a r e a v a i l a b l e a l l t h e 3 ^ c o u l d be a s s u m e d e q u a l t o z e r o a n d e s t i m a t e s o f t h e o n l y u s e d . T h e a l g o r i t h m , w i t h p o s s i b l y a f e w m i n o r c h a n g e s i n t h e p r o g r a m -m i n g , w o u l d h a n d l e t h i s c a s e a l t h o u g h many r e d u n d a n t c a l c u l a -t i o n s w o u l d be p e r f o r m e d s i n c e i f a l l t h e 3-j^ a r e z e r o d i f f e r e n t p e r m u t a t i o n s o f a f a i l u r e s e q u e n c e h a v e t h e same p r o b a b i l i t y o f o c c u r r i n g . S i n c e t h e m o d e l was s h o w n t o be a c c u r a t e t o a b o u t 1.5 p e r c e n t , f o r t h e c a s e o f e c o n o m i c i n d e p e n d e n c e o f r e -p l a c e m e n t s a t l e a s t , t h e v a l i d i t y o f a l t e r n a t i v e s s e l e c t e d on t h e b a s i s o f c o s t d i f f e r e n c e s o f l e s s t h a n 1.5 p e r c e n t b e t w e e n c o m p e t i n g a l t e r n a t i v e s c o u l d be q u e s t i o n e d . I n some c a s e s a l t e r n a t i v e s w e r e s e l e c t e d b e c a u s e o f d i f f e r e n c e s much l e s s t h a n 1.5 p e r c e n t . W i t h some m o d i f i c a t i o n t h e m o d e l c o u l d b e made t o s e l e c t o n l y t h o s e a l t e r n a t i v e s t h a t p r o d u c e a p r e s p e c i f i e d i m p r o v e m e n t i n e x p e c t e d c o s t o v e r a s i m p l e r a l t e r n a t i v e , t h u s s u p p r e s s i n g a l t e r n a t i v e s t h a t p r o d u c e i n s i g n i f i c a n t c h a n g e i n e x p e c t e d c o s t a n d w h i c h c o u l d be i n c o r r e c t b e c a u s e o f m o d e l e r r o r s . I n an a c t u a l r e p l a c e m e n t p r o b l e m t h i s l o w e r b o u n d on c o s t i m p r o v e m e n t w o u l d be s e t a c c o r d i n g t o t h e e s t i m a t e d a c c u r a c y o f t h e d a t a . I I . C O N C L U S I O N 1. A m a i n t e n a n c e m o d e l h a s b e e n d e v e l o p e d t h a t i s c a p a b l e o f h a n d l i n g s y s t e m s i n w h i c h c o m p o n e n t f a i l u r e s a r e n o t s i m u l t a n e o u s . 2. O t h e r t h a n A s s u m p t i o n s 1 a n d 2, C h a p t e r I I , no a s s u m p t i o n s n e e d be made a b o u t t h e f o r m o f t h e f a i l u r e p d f ' s , c o m p o n e n t r e p l a c e m e n t c o s t s t r u c -t u r e s , o r r e p l a c e m e n t p o l i c i e s . 3. R e n e w a l t h e o r y c h e c k s i n d i c a t e d t h a t t h e m o d e l e s t i m a t e d a l l e x p e c t e d r e p l a c e m e n t c o s t s t o w i t h i n 1.5 p e r c e n t o f t h e t h e o r e t i c a l v a l u e s , f o r t h e c a s e o f e c o n o m i c i n d e p e n d e n c e o f r e -p l a c e m e n t s . P o l i c i e s p r o d u c e d b y t h e m o d e l w e r e r e a l i s t i c b u t f u r t h e r w o r k i s r e q u i r e d t o e n s u r e t h a t t h e y a r e t o t a l l y c o r r e c t . A s y s t e m i s r e q u i r e d f o r s u p p r e s s i n g o p p o r t u n i s t i c p o l i c i e s t h a t w o u l d be c h o s e n b e c a u s e o f c o s t i m p r o v e -m e n t s o f t h e same o r d e r o f m a g n i t u d e a s e r r o r s c r e a t e d b y a s s u m p t i o n s i n t h e m o d e l . S u c h a s y s t e m c o u l d be e a s i l y a d d e d . F u r t h e r w o r k i s r e q u i r e d on t e c h n i q u e s o f o b t a i n -i n g t h e a p p r o x i m a t i o n s t o t h e f a i l u r e p d f ' s . B I B L I O G R A P H Y B a r l o w , R . E . a n d F. P r o s c h a n . P l a n n e d R e p l a c e m e n t , i n K . J . A r r o w , e t a l . ( e d s . ) , S t u d i e s i n A p p l i e d P r o b a b i l i t y a n d M a n a g e m e n t S c i e n c e , S t a n f o r d U n i v e r -s i t y P r e s s , S t a n f o r d , C a l i f o r n i a , 1 9 6 2 , p p . 6 3 - 8 7 . B e l l m a n , R . E . E q u i p m e n t R e p l a c e m e n t P o l i c y . J . S o c . I n d u s t . A p p l . M a t h . , V o l . 3, No. 3, S e p t e m b e r , 1 9 5 5 , p p . 1 3 3 - 1 3 6 . C o x , D.R. R e n e w a l T h e o r y . S c i e n c e P a p e r b a c k s , f r o m M e t h u e n ' s M o n o g r a p h s on A p p l i e d P r o b a b i l i t y a n d S t a t i s t i c s , 1 9 6 7 , 142 p p . D e a n , B.V. R e p l a c e m e n t T h e o r y , i n R . L . A c k o f f ( e d . ) , P r o g r e s s i n O p e r a t i o n s R e s e a r c h , J o h n W i l e y , New Y o r k , 1 9 6 1 , p p . 3 2 8 - 3 6 2 . D r e y f u s , S . E . A N o t e on an I n d u s t r i a l R e p l a c e m e n t P r o c e s s . P - 1 9 4 5 , T h e RAND C o r p o r a t i o n , S a n t a M o n i c a , C a l i f o r n i a , 1 9 5 7 . H o w a r d , R.A. D y n a m i c P r o g r a m m i n g a n d M a r k o v P r o c e s s e s . J o h n W i l e y a n d S o n s , 1 9 6 0 , 136 p p . R a d n e r , R. a n d D.W. J o r g e n s o n . O p t i m a l R e p l a c e m e n t a n d I n s p e c t i o n o f S t o c h a s t i c a l l y F a i l i n g E q u i p m e n t , i n K . J . A r r o w , §_t a j _ . ( e d s . ) , S t u d i e s i n A p p l i e d P r o b a b i l i t y a n d M a n a g e m e n t S c i e n c e , S t a n f o r d , U n i v e r -s i t y P r e s s , S t a n f o r d , C a l i f o r n i a , 1 9 6 2 , p p . 1 8 4 - 2 0 6 . R i f a s , B . E . R e p l a c e m e n t M o d e l s , i n C h u r c h m a n , e t a l . ( e d s . ) , I n t r o d u c t i o n t o O p e r a t i o n s R e s e a r c h , J o h n W i l e y a n d S o n s , . N e w Y o r k , 1 9 5 7 , p p . 4 8 1 - 5 1 8 . R u b i n , H. a n d J . Z i d e k . A p p r o x i m a t i o n s t o t h e D i s t r i -b u t i o n o f Sums o f I n d e p e n d e n t C h i Random V a r i a b l e s . T e c h n i c a l R e p o r t No. 1 0 6 , D e p a r t m e n t o f S t a t i s t i c s , S t a n f o r d U n i v e r s i t y , S t a n f o r d , C a l i f o r n i a , A u g u s t , 1 9 6 5 . A P P E N D I X A G L O S S A R Y OF PROGRAM V A R I A B L E S AND PROGRAM L I S T I N G A ( L ) - c u r r e n t a g e i n t e r v a l o f L t h c o m p o n e n t A F ( L ) - a g e i n t e r v a l o f L t h c o m p o n e n t a t b e g i n n i n g o f s t a g e A P R 0 B ( N C 0 M P , I N T ) - a b s o l u t e p r o b a b i l i t y o f f a i l u r e o f c o m p o n -e n t NCOMP i n a g e i n t e r v a l INT B ( N C O M P , I N T ) - B e t a c o e f f i c i e n t o f l i n e a r a p p r o x i m a t i o n t o f a i l u r e p d f C ( « ) - p o l i c y o u t p u t v a r i a b l e CHAR-(') - o u t p u t l i t e r a l D ( » ) - p o l i c y o u t p u t v a r i a b l e E ( « ) - p o l i c y o u t p u t v a r i a b l e E T F ( N C O M P , I N T ) - 1 m i n u s c o m p o n e n t e x p e c t e d t i m e t o f a i l u r e i n a g e i n t e r v a l INT F ( « ) - p o l i c y o u t p u t v a r i a b l e F N ( I , 0 , K ) - o p t i m a l e x p e c t e d r e p l a c e m e n t c o s t f o r N S T A G E s t a g e s s t a r t i n g w i t h a g e v e c t o r ( I , J , K ) F N M l ( I , d , K ) - same a s F N ( - ) b u t f o r ( N S T A G E - l ) s t a g e s G ( « ) - p o l i c y o u t p u t v a r i a b l e I - a g e i n t e r v a l o f c o m p o n e n t L IMAX - maximum a g e o f c o m p o n e n t L IN - p r o g r a m o u t p u t c o n t r o l p a r a m e t e r IND - c o m p o s i t e p o l i c y i n d e x J - a g e i n t e r v a l o f c o m p o n e n t M JMAX - maximum a g e o f c o m p o n e n t M J N K KMAX KN L M MARK N N CMPO(• ) N C M P R ( • ) NCOMP NF L NPOL NREP N S P O L ( - ) N S T A G E PER P N O F L ( N C O M P , I N T ) P P O L ( L ) P O ( I , J , K ) - p r o g r a m c o n t r o l p a r a m e t e r - a g e i n t e r v a l o f c o m p o n e n t N - maximum a g e o f c o m p o n e n t N - p r o g r a m c o n t r o l p a r a m e t e r - n u m b e r o f f i r s t c o m p o n e n t t o f a i l i n a s t a g e - n u m b e r o f s e c o n d c o m p o n e n t t o be c o n s i d e r e d i n a s t a g e : s e c o n d f a i l u r e o r r e p l a c e m e n t wi t h f i r s t f a i 1 u r e - p r o g r a m o u t p u t c o n t r o l p a r a m e t e r - n u m b e r o f t h i r d c o m p o n e n t t o be c o n s i d e r e d i n a s t a g e : t h i r d f a i l u r e , o r r e p l a c e m e n t w i t h f i r s t o r s e c o n d f a i l u r e - n u m b e r o f c o m p o n e n t n o t r e p l a c e d w i t h f i r s t f a i 1 u r e - n u m b e r o f c o m p o n e n t r e p l a c e d w i t h f i r s t f a i 1 u r e - c o m p o n e n t n u m b e r w h e n f a i l u r e o r d e r n o t i m p l i c i t - n u m b e r o f f a i l e d c o m p o n e n t - n u m b e r o f c o m p o s i t e o p t i m a l r e p l a c e m e n t p o l i c y - n u m b e r o f c o m p o n e n t r e p l a c e d w i t h f a i l u r e - v a l u e o f t h e e r e p l a c e m e n t a l t e r n a t i v e - s t a g e n u m b e r o f D y n a m i c P r o g r a m - p o l i c y c o r r e c t i o n f a c t o r - p r o b a b i l i t y o f no f a i l u r e o f c o m p o n e n t NCOMP b e f o r e a g e i n t e r v a l INT - n u m b e r o f c o m p o s i t e o p t i m a l r e p l a c e m e n t p o l i c y when c o m p o n e n t L f a i l s f i r s t P l ( L , I , J ,K) P 1 0 ( L , I , J , K ) P 1 R 0 ( L , M , I , J , K ) P 2 ( L , M , I , J ,K) P 2 0 ( L , M , I , J , K ) P 2 R ( L , M , N , I , J , K ) P 3 ( L , M , N , 1 , J , K ) SAVEM S A V E Y T TMIN U ( N C O M P , I N T ) Y Z $ C O S T ( I N D ) $ E C O S T ( L ) PT - .CA ) P T ( A ) P 2 Q ( A ) f o r M r e p l a c e d w i t h L P l m o < S > P 2 n ( A ) f o r M r e p l a c e d w i t h L s a v e s v a l u e s o f M s a v e s v a l u e s o f Y t i m e v a r i a b l e i n p r o b a b i l i t y e q u a t i o n s m i n i m u m o f $ T E M P 1 a n d $ T E M P 2 a l p h a c o e f f i c i e n t o f l i n e a r a p p r o x i m a t i o n t o f a i 1 u r e p d f a g e i n d e x o f c o m p o n e n t L u s e d a s s u b s c r i p t i n p r o b a b i l i t y c o m p u t a t i o n s same as X f o r c o m p o n e n t M same as X f o r c o m p o n e n t N e x p e c t e d c o s t o f c o m p o s i t e p o l i c y n u m b e r IND o p t i m a l e x p e c t e d r e p l a c e m e n t c o s t f o r a l l p o s s i b l e t r a n s i t i o n s t h a t b e g i n w i t h t h e f a i l u r e o f c o m p o n e n t L $ E T F ( N F L , N R E P , I N T ) - e x p e c t e d c o s t o f a d d i t i o n a l f a i l u r e s o f c o m p o n e n t NREP when i t i s r e p l a c e d w i t h c o m p o n e n t NFL t h a t was INT t i m e u n i t s o l d a t f a i l u r e $ T C O S T ( N S T A G E ) $ T E M P 1 $ T E M P 2 o p t i m a l e x p e c t e d c o s t f o r a l l p o s s i b l e f a i l u r e s e q u e n c e s e x p e c t e d c o s t o f a l t e r n a t i v e e = 1 / d = 1 e x p e c t e d c o s t o f a l t e r n a t i v e e = 2 / d = 1 C DYNAMIC PROGRAMMING-MARKOV CHAIN ALGORITHM F O R DETERMINING OPTIMAL JC C-O.ESJl!!iENT_^E_lLAC.Ejy.ERT PJ2LJLC_LES DIMENSION F N I5,4,6}, F N M l ( 6 , 5 , 7 » , A ( 3 1 , P 1 ( 3 , 5 , 4 , 6 3 , P 1 0 ( 3 , 5 , 4 , 6 ) , 1 P?{3,3,5.4,6),P20(3,3.5,4,6) , P 3 ( 3 , 3 , 3 , 5,4,6),P1R0( 3,3,5,4,6) ,  2 P2R{ 3 ,3,3, 5,4,6) , PO ( 5,4,6) , C l ( 3 ) ,C2 ?3,3 ) , C3 t 3 ,3 , 3 ) , $C£3ST (4) , 3 $ECOST{3),$TCOST(25),NSPOL13,3),MCMPR< 3 ) ,NCMPO( 3 ) , P P 0 L ( 4 ) , C ( 3 ) , 4. Di3L,.Ei3L,„E<3 3 (.3,„6>, 5 CO(6) ,APROB( 3,6) , C P R O B ( 3 , 6 ) , A F ( 3 ) , E T F ( 3 , 1 0 ) , $ETF( 3,3,10) INTEGER A, A F, X , Y ,Z,PPOL»SAVEM,SAVEY DATA CHAR/'1* ,'2' ,* 3' ,* 0*,*R 1/ C DATA INPUT C ** PROGRAM CONTROL PARAMETERS 96 FORMAT(412,F4. 2) C COEFFICIENTS OF LINEAR APPROXIMATIONS TO FA IL OR E PDF'S READ ( 5 , 200? I MAX,JMAX,KMAX 200 FORMAT(312) NC0MP=1. B_EADJJi^2111JJlJj^^ : . NC0MP=2 RE AD( 5*201) (U (NCOMP, INT ) , B( NCOMP, I NT ) , I NT = .1 , JMAX ) NC0MP=3 ; R E A O ( 5 , 2 0 1 ) ( U ( N C O M P , I N T ) , B C N C O M P ? I N T ) , I N T = 1 , K M A X ) 2 0 1 F O R M A T ( 8 F 1 0 . 0 ) C r n j ; T p _ T A - i »M,N A R F C O M P O N E N T N U M B E R S c * S I N G L E R E P L A C E M E N T S R E A D ( 5 , 2 0 1 ) < C 1 ( L ) , L = 1 , 3 ) C '* D O U B L E R E P L A C E M E N T S R E A D ( 5 , 2 0 1 ) ( ( C 2 ( L , M ) , L = 1 , 3 ) , M = 1 , 3 ) c * T R I P L E R E P L A C E M E N T S R E A D ( ^ , ?01 ) < ( L . C 3 ( L tJL,M L » 1 = 1 , 3 } , M = 1 , 3 ) , N - 1 , 3 ) C A L C U L A T E T H E A B S O L U T E A N O C O N D I T I O N A L P R 0 8 S OF F A I L U R E I N E A C H c A G E I N T E R V A L F O R E A C H C O M P O N E N T M A X ( 1 ) = I M A X MAX!" 2)-=JMAX M A X ( 3 ) = K M A X pn 4 0 NC.nMP = 1 , 3 P N O F L * N C O M P , 1 ) = 1 . 0 M A X 1 = M A X ( N C O M P ) A P R 0 8 I N C O M P , 1 ) = U ( N C O M P , 1 ) + . 5 ^ 8 ( N C O M P , 1 ) C P R O B ( N C O M P , 1 ) = A P R O B ( N C O M P , 1 ) D O 4 0 I N T = 2 , M A X 1 I M = T N T - 1 A P R O B ( N C O M P , .I N T ) = U ( N C O M P , I N T ) + . 5 * B ( N C O M P , I N T ) P N O F L ( N C O M P , I NT) = P N O F L ( N C O M P , IM ) - A P R O B ( N C O M P , I M ) C P R O B ( N C O M P , I N T ) = A P R O B ( N C O M P , I N T ) / P N O F L ( N C O M P , I N T ) 4 0 C O N T I N U E c C H A N G E A B S O L U T E C O E F F I C I E N T S TO C O N D I T I O N A L C O E F F I C I E N T S DO 4 3 NmMP=l , 3 M A X 1 = M A X ( N C O M P ) DO 4 3 I N T = 1 , M A X 1 U ( N C O M P . I N T 1 - I H NCOMP. INT) / P N O F I ( N C O M P , I NT) J B ( N C O M P , INT ) = B< N C O M P , INT ) / P N O F L (NCOMP , INT ) 4 3 C O N T I N U E c *#* L I S T A B S O L U T E P R O B A B I L I T I E S W R I T E R , 1 0 7 ) 1 0 7 F O R M A T ( T 1 0 , ' S U M M A R Y OF A B S O L U T E P R O B A B I L I T I E S OF F A I L U R E * » / 1 T 1 0 , ' D U R I N G G I V E N A G E I N T E R V A L 1 / / ) ) W R I T E ( 6 , 1 0 8 ) < 1 0 8 F O R M A T ( T 3 3 , ' A G E I N T E R V A L ' , / i T i n r « r n M P . N n . l ? 3 4 «s 6 » / ) DO 3 3 N C 0 M P = 1 , 3 MAX 1 = M A X ( N C O M P ) W R T T F ( 6 . 1 0 9 ) N C O M P , < A P R O B ( N C G M P , I N T ) * I N T = 1 . M A X 1 ) 1 0 9 F O R M A T ( 1 2 X , 1 1 » 4 X , 1 O F ? . 3 ) 3 3 C O N T I N U E W R I T E fh.1 I D ) 1 1 0 F O R M A T ! 1 H 0 ) c ** L I S T C O N D I T I O N A L P R O B A B I L I T I E S W R I T F ( 6 , 1 I 1 ) 1 1 1 F O R M A T I T 1 0 , ' S U M M A R Y O F C O N D I T I O N A L P R O B A B I L I T I E S OF F A I L U R E ' , / 1 T l O t ' D U R I N G G I V E N A G E I N T E R V A L S / / ) WR I TF ( fS » 1 C 8 ) DO 3 4 N C 0 M P = 1 » 3 M A X l = M A x t N C O M P ) W R I T E ( 6 , 1 0 9 ) N C O M P , ( C P R O B ( N C O M P , I N T ) , I N T = 1 , M A X 1 ) 3 4 C O N T I N U E C A L C U L A T E E X P E C T E D T I M E T O F A I L U R E FOR E A C H C O M P O N E N T HO 4 4 N f . n M P = l T 3 MAX 1 = M A X ( N C O M P ) DO 4 4 I N T = 1 , M A X 1 E T F ( N C O M P , I N T } = ! . - ( . 5 * U ( N C O M P , 1 NT )+ B ( N C O M P , I N T ) / 3 . ) / ( U ( N C O M P , I N T ) 1 + . 5 * B ( N C O M P , I N T ) ) 4 4 C O N T I N U E c CAI f.UI A T F THF F X P E C T F D C O S T OF A D D I T I O N A L F A I I l l R E S OF R E P I A C E O c C O M P O N E N T S WHEN R E P L A C E D W I T H ANY O T H E R C O M P O N E N T DO 4 5 N F L = 1 , 3 MAX 1= M A X ( N F L 5 DO 4 5 N R E P = 1 , 3 DO 4 5 I N T = 1 , M A X 1 T = F T F ( NFI . INT 1 $ E T F ( N F L , N R E P , I NT ) ='< . 5 * B< N R E P , I )*T+U ( N R E P , 1 ) ) * T * C l t N R E P ) 4 5 C O N T I N U E T = l . c B E G I N C A L C U L A T I O N OF T R A N S I T I O N P R O B A B I L I T I E S FOR A L L A G E V E C T O R S C * S E T L E V E L 1 F A I L U R E DO 2 0 1 = 1 f 3 c * S E T L E V E L 2 AND 3 F A I L U R E S GO TO { 2 1 , 2 2 , 2 3 ) , L 21 M=2 N=3 GO TO 2 4 ?? M=1 N = 3 GO TO 2 4 2 3 M = l N=2 2 4 C O N T I N U E DO 2 0 K = 1 , K M A X A ( 3 ) = K DO 20 J = 1 , J M A X A ( 2 ) = J J DO 2 0 I = 1 , 1 MAX A { 1 ) = I X=A(L.) Y = A C M ) Z=A(N) C »» CALCULATE PROB OF TRANS TO GANHA=Q  P 0 ( I, J,K)={ l . - C P R O B ( l , I ) 3 * ( l . - C P R O B ! 2 , J ) )*( 1 „ - C P R O S ( 3 , K ) ) C * * CALCULATE PRO 8 OF TRANS TO GAMMA = L . . C J Q ± J S 1 = J 3 J L 1 J J . . ^ ^ C O i 5 ) = - 1 * { U ( L , X ) * B ( M , Y ) * B ( N , Z ) / 2 . + B < L , X 3 * ( U ( M , Y ) * B ( N , Z ) + B ( M , Y 3 * 1 U ( N , Z > ) ) C O ( 4 ) = . 2 5 * ( U ( t , X ) * t U ( M , Y i * B ( N , Z 3 - + B f M , Y ) * U ( N , I 3 ) / 2 . - B ( L , X ) » ( ( 8 ( M , Y ) 1 + B ( N , Z ) ) / 2 . - U ( M , Y ) * U ( N , Z ) > ) C O O ) •=-. 3 3 3 * < U ( L , X ) * ( ( B ( M , Y ) + B < N , Z 1 ) / 2 .-U ( M , Y ) * I H N , Z ) ) + L _ M J ^ J L l * I J J L L l ! ^ ^ : C 0 ( 2 ) = - . 5 * ( U l L , X ) * < I M M , Y ) + U ( N , Z ) ) - B ( L , X ) ) C O * 1 ) = U ( L , X ) c * F O R M P O L Y P 0 L Y = 0 . D O 2 5 I N D = 1 , 6 N 0 = 7 - I N D 2 5 POLY=(POLY + COfNO.H*T P H L , I , J , K ) = P O L Y C * * C A L C U L A T E P R O B O F T R A N S T O G A M M A = L 0 F O R 0 = 1 C 0 ( 6 ) = 8 ( L , X 3 « B ( M , Y 3 * B ( N , Z ) / 8 . C 01 5 3 = . 2 5 * ( U ( L , X ) * 8 ( M , Y I • B < N , Z ) + 8 ( L , X ) * U ( M , Y ) * 8 ( N , Z ) ^JL + J J J L j i ) . * B J J ^ j L ^ L * U X N „ , Z J L L _ . _ _ • _ C0(4) = . 5 * < U ( L , X ? * U < M , Y ) * B < N , Z ) + U ( L , X ) * B ( M , Y ) * U ( N , Z ) 1 + B ( L , X ) * U ( M , Y ) * U ( N , Z ) - ( B C L t X ) * B < N , Z ) + B < L , X ) * B < M , Y ) ) / 2 . 3 C O { 3 ) = U { L , X 3 * U < M , Y ) * U ( N , 1 3 - { U ( L , X 3 * B ( N , Z ) + U ( L , X ) * B i M , Y ) 1 + B ( L , X ) * U ( N , Z 3 + B J L , X 3 * U ( M , Y ) ) / 2 . C O ( 2 3 = B ( L , X ) / 2 . - U ( L , X ) « U ! N , Z ) - U ( L , X ) * U C M , Y ) _ £ D J J L ) = U J J U - X J - . _ . . _ . . . C * F O R M P O L Y P O L Y = 0 . 0 0 3 2 I N 0 = 1 , 6 N0=7-IN0 3 2 P O L Y = ( P O L Y + C 0 ( N 0 ) 3 * T PTO ( 1 T T t . l , K -) = Pni Y D O 2 7 1 1 = 1 , 2 C C A L C U L A T E P R O B O F T R A N S T O G A M M A = L M . FOR 0 = 1 C O ( 6 ) = — B ( L , X ) * 8 ( M , Y ) * B ( N , Z 3 / 2 4 • C 0 ( 5 ) = - . l * ( U ( L , X ) * 8 ( M , Y ) « 8 < N , Z ) + B ( L , X ) * ( U ( M , Y ) * B ( N , Z ) / 2 . + B ( M , Y ) * 1 U ( N i Z ) J ) C O ( 4 ) = - . Z 5 * 1 J L L L L J X ) * ( 11 ( J 1 J _ Y _ L * B J N , Z 3 / 2 , ±Bi M , . Y 1 * U J J 5 U L L 1 + _ . 5±MLJ,JU * 1 ( U < M , Y ) « U ( N , Z ) - 8 ( M , Y ) 3 ) C O ( 3 3 = . 3 3 3 * { B ( L , X ) * U ( M , Y ) / 2 . - U ( L , X 3 * ( U < M , Y ) * U ( N , Z 5 - B ( M , Y ) ) 3 C 0 ( 2 ) = U ( M , Y ) « U ( L , X 3 / 2 . C 0 ( 1 .3=0. c * F O R M P O L Y P O I Y = 0 -D O 2 6 I N D = 1 » 6 M 0 = 7 - I N D 2 6 P 0 L Y = ( P 0 L Y + C 0 ( N 0 ) 3 * T P 2 ( L , M , I , J , K ) = P G L Y C * * CALCULATE PROB OF TRANS TO GAMMA=LMN FOR D AND E=l _..C.0J..6J_=J1.(.JU^^ „ _ . . C 0 ( 5 ) = . 2 * ( U ( L , X 3 *B ( M, Y 3 * 8 ( N, Z ) / 3 . + 8 ( L , X1 * U ( M, Y 3 * B ( N , Z 3 / 6 . 1 + B ( L , X 3 * 8 ( M , Y 3 « U ( N , Z ) / 8 . 3 m i 4 ) = . 2 5 » ( B f I . X ) * U f W , Y>*t ) f 7 ) /6.+\Ui . X 3 » l ) f M . Y ) » B ( N . 7 ) / 2 . ; J 1 + UIL»X)*B(M,YJ*U(N,Z)/3.) C0( 3) =UC UX)*U< MtY)*UIN fZ) /6. C * FORM POLY PGLY=0. OO 28 lND=l,4 28 H U - I — i 1111 POLY=(POLY+COtNO))*T < P3(L,M,N, I,J,K )=POLY r * * r .A i r .U IATF PROR OF TRANS TO GAMMA=t MO FOR D AMD E=l CO(6)=-B(L,X)*BCM,Y)*BiN,Z)/16. C0(5)=-<U(L,X)*B<M,Y)*B(N»ZI/6.+B(L,X)*BtM,Y)*U<N»Z)/8. 1 + Bd. .X)*U(M.Y1*8<N.7 1/12-) C0(^)=-(U(L,X)*B{M,Y)*U(N tZ)/3.+U(L,X)*U(M,Y)*-B(N,Z)/4. 1 + B(LtX)*U(M,YJ*U(N,Z)/6.-B(LtXJ*B<M,Y)/8.) -J_Ul(--U-=aLU-^^ *U C M ,X±Lh-.. CO<2)=U(L,X)*U{M,Y)/2. C * FORM POLY PQLY=0.  DO 29 IND=1,5 N0=7-IND 2.3. _m_y_^(_pj-jui- + r_a.{j-_3jj._ii : P20 (LtM, I, J tK ) = POLY C CALCULATE PROB OF TRANS TO GAMMA=LN FOR 0=2 CO(6)=-B(L,X)#B(M,Y)»B(N,Z)M8.  CO(5)=-(B(N,Z)*U< L,X1*B <M tY)/30. + B(L,XJ*U(M,YJ*B< N,Z)/15. 1 + B( L,X) *8<M, Y) *UIN,Z) MO. ) C__LL__J___3_LI^^ 1 U(L,Xl*B(M,Y)/2.)/12. C0<3)=(U(L tX)*B(N,Z)+(B(L,X)*U(N,Z)-U(L,X)*U(MtY)*U(NtZ))/2.)/3. CO (2 )=U ( L. X ) » 1 J { N, Z )/2  C * FORM POLY POLY=0. DO 30 TND = 1 .5 . . . . . _____ N0=7-IND 30 POLY=(POLY + CO<NO))*T P2R<L,M,N,I, J.K) = P0LY C ** CALCULATE PROB OF TRANS TO GAMMA=L0 FOR C=2 CCM6) = B{L»X»*B(M f Y)#B<N,Z) /lb. C.0.(5±=.U LL, XJJiB.LM,Y )2ilB {N ,Z >/12 .>.BT.L,.VJ_v,U_{.M.,.Yj.S;BJ.N..,.ZJ./_ . . 1 + B(L,X)*B(M,Y»*U<N,Z)/8. C0(4) = .25*<U(L,X)*U(M,Y)"B(N,Z}-8(L,X)*8(N,Z)-B(L,X)*B(M,Y)/2.) I + .3 33333»( BH . X) *U iM , Y ) *U ( N , 7) +U ( L . X ) B ( M , Y U ( N, Z ) / 2 . )  C0{3)=-.5#(U(L,X)*B<N,Z]-U{L,X)*U(M,Y)*U( N, Z) + B{L,X)*U(N » Z) ) 1 333333*(U(L,X.*B(M,Y)/2.+BlL,X)*U(M,Y) ) C£LL2J___LUI_,.XJ_J.LN^^ C0(1I= U(L,X) C * FORM POLY POL.Y=0 .  DO 31 IN0=1,6 N0=7-I ND _ - l pni y=(pni Y + cntNOi >»T : P1RO(L,M,I,J,K)=POLY C ** INTERCHANGE M AND N AND RETURN TO CALC PROBS FOR NEW GAMMAS S AVEM=M  M=N N = SA VEM SA.V_EY_=Y , ..__ Y = Z Z=SAVEY 21 HOMTTMHF ) tSi <L 2C < UJ z tt D LL OO CD D a. CL o < o o 2: Oi UJ cc o z o a X c -> co O X UJ UJ z a I - * • z t— < z ac O NO UJ < UJ X O 1 X 1 a C O _ ) c _ 3 •K-o U N O a: o CO IS> UJ o CL s: UJ o < Q i z OJ z o CL ST O O c • r- x c UJ UJ Qi UJ x _J (/! — >• < 00 LL o i a a o c r - » X LU I— X - J < > UJ LLI CC CD Z) < < < LL 00 c a Z CC a LU O M LU ^ Q LU s: X 00 LL! 00 C H < — s ZJ — r * ^ < LL r - r - i to «-CC H-I LL — - a. o •» O X LU —1 O r s j s C LU I — < 2: o z •Tr •ft-O O .—I O + + X X <r < I I I I x X i s: - 5 X X X s: s: s: i«c -5 t--11 11 00 CO CO 0 0 0 0 0 0 z* o z U-LU > o LU - 5 CD O LL O O 00 LU > u LU cc cc o LL C IP z CM • LU r o a II II < LL — H- CC *C 00 <0 LU < H 00 LU CD < s: LU 00 >-</) LU 00 LU • K Z < Qi f s . LU - 1 Z UJ c u. 0 c 00 < Qi LU Z LU CD ro c-a x < s: x < I I I I I I I I ~* ro 11 — > ro m H H N H - CM w l i t Q ~| U. O < C <. < O LU Oi Xi < LL ro LU > LU _ ! CM LU < 00 00 LU Oi a. ro o z <r CM _ l LU > LU 00 O r-CM O l l CD X OO r c r - j I I O l l z o s: o I I co 2: o a: LU LL X ! -Z _ r-i <3 (— (M Z > 11 C a z u f-1 r - l O 00 «"» UJ s ^ 01 r X - 5 < -LL _J o i O LU i - t x a. l— + ci — 3 — LL — . O — Z LL < O0 X> — 1 LU LL Oi X> LU _J « • r-l + <[ « ~ LL J I- r - l 00 O 01 ~ -LL — LL ~ 00 o LJ LU a r — I I r - l X l II S — s — ' _J o — u < < + <— 00 ro LU cc <5 X + <C — LL CM — O < Qi • CO + Q — Z rH <t < a — z r - l CM c j Z Di" < t -00 z I—I LU Oi a. Q z o LU 00 s: o o LL O LU Oi X < LL UJ 00 X < u 00 LU o - J o a > Oi <r. O z o a LU OO LU > o 00 00 Q O O I I s: cc w a o CM + LL < LL h-LU + s; w + r - l — n o r 1  0 # < >v <~ » H ^ H II • + LU -7 — LU H V > » <• r - X . ( - » < _ l + Z w 01 CM LU CL — H I! < _J r — I s— <t CL r ~ s a; >- LU Z 0C r - LL < z H LU + * — i i i r - l » • + • ro - < z s: + 1*. — . - J CM CO «sx CX. • + r - i >~l * f CL «~ LU <— r- < I I r - l r-l s o. Z S LL LU H - + r - l C ** TRY A L T E R N A T I V E E=2/0=1 $TEMP2 = P 2 ( L »M f I , J » K ) * ( C 2 ( M , N ) + ( S E T F { M , M , A F ( M ) )+ $ E T F I M , N , A F { M 3 3 ) / 4 . 1 + F N M 1 ( A ( 1) + 1 , A { 2 3 +1 »A ( 3 3 +1) ) C * CHOOSE BEST OF TWO SECONDARY P O L I C I E S TMIN=$TEMP1 N S P 0 L ( L , M ) = 1  I F ( S T E M P 2 . G E . T M I N * P E R ) G O TO 2 TMIN=$TEMP2 MSPnt f l . M ) = ? 2 C CONT INUE ADD COST N S P O L ( L , M ) TO $ C 0 S T 1 4C0ST I 1 3 = $C0ST( 13 +TMIN C EXCHANGE M FOR N <MAKE OTHER COMPONENT SECOND F A I L U R E ) AND RETURN A { M ) = A F ( « ) A { M ) = A F { N ) SAVEM=M M = N N=SAVFM 1 CONTINUE C TRY A L T E R N A T I V E D=2 nn 3 TNQ=?, 3 A ( M ) = l c *V -»* FORM EXPECTED COST OF P O L I C Y NO. IND $COST( IND3 = P l ( L , I , J , K 1 * ( C 2 ( L , M ) + $ E T F ( L , L , A F ( L ) ) + $ E T F ( L , M , A F ( L ) ) ) 1 + P 1 R 0 ( L , M , I , J , K ) * F N M 1 { A ( 1 ) + 1 , A < 2 J + 1 , A<3 ) + 1 ) A { N ) = 0 SCnST ( TND ) = .$CnST ( TND) +P?R (1. ,M ,N , I , J ,K ) *( C 1 ( N) + . tFTF( N t N , AF ( N) ) / 4 . 1 + F N M H A ( 1 3 + 1 , A { 2 ) + 1 . , A(3 ) + 1 ) 3 c SAVE NO. OF REPLACEMENT AND SECOND F A I L U R E NCMPR( IND)=M NCMPO( IND) =N c * EXCHANGE M AND N AND RETURN A ( M) = AF{M) A ( N)=AF < N3 SAVEM=M M = N N=SAVEM 3 CONTINUE c iA> •A. _ f TRY Al TFRNAT IVF 0=3 $ C 0 S T ( 4 ) = P 1 ( L f I , J , K ) * { C 3 ( L , M , N 3 + $ E T F ( L , L , A F U ) 3 + $ E T F ( L,M,A F I L ) ) 1 + $ E T F ( L , N , A F ( L 3 3 + F N M K 2 , 2 , 2 ) 3 C . SEARCH FOR LEAST COST P O L I C Y NPOL=0 $ E C O S T I L 3 = 1 0 . E 1 0 n n 4 TND=1.4 I F { $ C O S T ( I N D ) . G E . 4 E C 0 S T I L 3 ) G 0 TO 4 $ E C 0 S T t L )=$COST( ' IND) NPOL=IND 4 CONTINUE C IF STAGE . G E . MARK CONSTRUCT OUTPUT FOR NPOL ANO L I S T A<1 ) = A F { | ) I F ( I . N E . I N . O R . J . N E . J N . O R . K . N E . K N ) GO TO 12 c BRANCH TO A P P R O P R I A T E OUTPUT ACCORDING TO NPOL P P O L ( L ) = N P O L GO TO ( 5 » 6 , 6 t 7 3 , N P O L 5 CONTINUE r. aj; sj? P01 TOY OUTPUT T F D= 1 C ( L 3 = C,HAR( 1) C ( M ) = C H A R { 4 ) r . (N)=r.HflR (4) LU 11J > I—1 r-< CC LU < UJ ct _> LU > LU < I CJ II ct — _> Z r -— LU O Ct — Q (Ni Z • < o LU Z — — — 3C —-,• cvi -j- •> _> Ct Qt ct — _; _ < - < < _ X X O Oi o c_ a. on II II 00 Ct — — z. uu s: z > — — U_ LU O Q H O C I— X _> u C II I— I LU •-• 00 Q CJ s: n _ LU > Z oo < II II oo X Z H (M >t Ct Ct Ct < < <t X. X u o II X o II — J — z U J U J LU Ct < X u II LU (V LU Z — o SC z - < —- SC - J O UJ a <_> (•) < Z _J — a LL LU ct _ <_> LU Ct LL UJ > LU -—I ct O u _> sc Cu < t~ X _> o o LU —< 00 LL CJ z c — 2 (M rn < Ct ct <x <t X X CJ (_> II II SC z LL LL CJ _J z _c LL LL II II _ z O LD X ro CM XI X oc IN L-z a _ c — c u o • >-i --I X I <N O vO — a — \~ LL! <J (~ r- S 00 l - l Ct w _ c J 3. LL O a. _J • o O i Cv u < — LL CNJ -o H —i x •- CM o — — t-U J < t i— „ : >-! Ct Ct O 3: LL (NJ o X K> O I • < I—I •» <t < • • - <£ -*>. —. . « m ' •> o r—I r - l <r -• <j x — •4- LU — t~ (\J p—i Ct 3 < cn SC cr C LU Ct CD < LL U J > U J X LU a U- CNI z ct a _> i — UJ a ct o (Ni II D _> CL Ct LU H-J <r LLI Ct _> —1 — — <r LO <f LL Ct Ct (Ni < < X X - J U CJ LU II II > . LU _J —I Ct — o J J u-O O a. CL o — LJ z 3 O o CL __ (_> z —- •—• UJ CJ CJ oo ITi <N Ct < I CJ I! ctj < X CJ II 3 j a a CU r - l r-H •—i o CL ct Q -X. Ct CJ a. II sc ~* a j z Q a c t/) t-H P—' ct et 3 -CJ LU Z a __ X 00 <r (N! X •d-- < < m n a < < LL — - rn • o X - 1 _ cg *• (Ni o UJ — - v0 •—I ~> H- « -< LU O s: r - (-Ct _ i o a c a, ~ o ct _ <£ _  X >- Z l_ CJ || ^ J o o w" -a o r-LO > —• c ct •-< -X c CJ Q. Z (/ o CJ U -z a z X cn CN X C a CJ < i—• _• on o o r - l X »• •> (N vO vO — w |_ U J U J < K - M a CC CC o 3C S o W R I T E ( 6 , 1 0 3 ) A , A , A , A , A C * * * R E T U R N F O R N E X T F I R S T F A I L U R E 12 C O N T I N U E C * * * F O R M FN< I , J ,K ) 1 5 $ T C O S T ( N S T A G E ) = 0 . DO 1 4 L = l ,3 , 1 4 S T C O S T f N S T A G E ) = $ T C O S T ( N S T A G E ) + S E C O S T U . ) C * I F C O S T NO F A I L U R E E X I S T S ADD I N H E R E C * * * S £ J L _ E J ^ J ^ _ J ^ J l l _ E i L j ^ F N ( I t J , K ) = S T C O S T ( N S T A G E ) + P O < I , J , K ) *F N M 1 ( I + 1 , J + 1 , K + I ) I F U . N E . I N . O R . J . N E . J N . O R . K . N E . K N ) G O T O 1 3 W R I T E ( 6 . I 0 6 ) N S T A G E , A , F N ( 1 . J . K )  1 0 6 F O R M A T ( 2 X , ' N O . O F T I M E U N I T S S Y S T E M TO R U N = ' , 1 2 , / 1 2 X , ' P R E S E N T A G E V E C T O R O F S Y S T E M = ' , 3 1 2 , / 2 7 X . ' F X P F ' r . T F D C O S T OF F O L L O W I N G A R O V F P 0 1 I C Y = $ ' . F 8 . 0 . / / / ) C * * * R E T U R N F O R N E X T A G E V E C T O R 13 C O N T I N U E C B L O C K D A T A S H I F T F O R K + l - S T A G E P R O C E S S  DO 1 6 K = l , KMA X DO 1 6 J = 1 , J M A X nn 16 I = I . T M A X 1 6 F N M K I , J ,K) = F N ( I , J , K ) DO 1 6 1 K = 1 , K M A X DO 1 6 1 J = 1 , J M A X  1 6 1 F N M K I M X , J , K ) = F N M 1 ( 1 , J , K ) + C 1 ( 1 ) DO 1 6 2 K = 1 , K M A X 1.62 F N M K I , J M X » K ) = F NM K I , 1 , K ) + C K 2 ) DO 1 6 3 J = 1 , J M A X DO 1 6 3 1 = 1 , I M A X  1 6 3 F N M K I , J , K M X ) = F N M K I , J , 1 ) + C K 3 ) C * * * R E T U R N F O R N E X T S T A G E Al CMHIOUS- . _ . •_ 4 4 4 S T O P E N D A P P E N D I X B S AMPLE PROGRAM OUTPUT S L P M A R Y QF A B S O L U T E P R O B A B I L I T I E S CF F A I L U R E D U R I N G G I V E N AGE I N T E R V A L AGE I N T E R V A L C O M P . N O . 1 1 2 3 C . 1 0 0 C . 2 0 0 0 . 3 0 6 0 . 2 0 0 0 . 4 5 0 0 . 2 5 0 0 . 3 0 0 . 0 . 3 0 0 0 . 19 4 0 . 3 0 0 C . 0 5 C 0 . 1 3 9 0. 100 0 . C 8 3 0 . .0 2 8 S U M M A R Y OF C O N D I T I O N A L P R O B A B I L I T I E S OF F A I L U R E D U R I N G G I V E N A G E . I N T E R V A L AGE I N T E R V A L C O P P . N O . 1 2 3 4 5 6 1 0 . 1 0 0 . 0 . 2 2 2 0 . 4 2 9 0 . 7 5 0 . 1 . 0 0 0 _._ 2 C . 2 0 0 0 . 5 6 2 0 . 8 5 7 L O C O 3 0 . 3 0 6 0 . 3 6 0 C . 4 3 7 0 . 5 5 6 C . 7 5 0 I . 0 0 0 v. >-O P T I M A L C O M P O N E N T R F P L A C F M E N T P O L I C I E S _ * — # - ) J : >|c_ if — £ F I R S T F A I L U R E S E C O N D F A I L U R E . . . . TH I RO.. F A I L UR E. C C M P NC 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 F A I L / P O L 1 , 0 , 0 1 , 2 , 0 1 , 0 , 2 1 , 2 , 3 1 , 3, ? A G E V E C T 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 COMP NO I 2 3 1 2 3 1 2 3 1 2 3 1 2 3 F A I L / P O L 0 , 1 , 0 2 , 1 , 0 0 , 1 , 2 2 , 1 , 3 3 , 1 , 2 AGE V E C T 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 C C M P NO 1 2 3 1 2 2 1 2 3 1 2 3 1 3 F A I L / P O L 0 , 0 , 1 2 , 0 , 1 0 , 2 , 1 2 , 3 , 1 3 , 2 , 1 AGE V E C T 1 1 1 1 1 1 1 I .1 1 1 1 1 1 1 K G . OF T I M E U N I T S S Y S T E M TO RUN - 15 P R E S E N T AGE V E C T O R OF S Y S T E M = 1 1 1 E X P E C T E C COST O F F O L L O W I N G A B O V E P 0 L I C Y = $ 4 3 4 0 0 . COMP NO 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 F A I L / P O L 1 , 0 , 0 1 , 2 , C 1 , C 2 1 , 2 , 3 1 , 3 , 2 AGE V E C T 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 C C M P KO 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 F A I L / P O L 0 , 1 , 0 2 , 1 , 0 0 , 1 , 2 2 , 1 , 3 3 , 1 , 2 " AGE V E C T 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 COMP NO 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 F A I L / P O L 0 , 0 , 1 2 , 0 , 1 0 , 2 , 1 2 , 3 , 1 3 , 2 , 1 AGE V E C T 2 1 1 2 1 1 2 .1 1 ... 2 1 1 ... . 2 1. I N C . OF T I M E U N I T S S Y S T E M TO RUN = 1 5 P R E S E N T AGE V E C T O R CF S Y S T E M = 2 1 1 E X P E C T E D COST OF F O L L O W I N G A B O V E P O L I C Y = $ 4 3 6 2 0 . C C M P NG F A I L / P O L 1 2 3 1 , 0 , 0 1 2 3 I f 2 , 0 1 2 3 1 , 0 , 2 1 2 3 I t 2 , 3 1 2 3 I t ?t 2 AGE V E C T 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 C C M P NC 1 2 3 F A I L / P O L 0 , 1 , 0 A G E V E C T 3 1 1 1 2 3 2 , 1 , 0 3 1 1 1 2 3 C l , 2 3 1 1 1 2 3 2 , 1 , 3 3 1 1 1 2 3 3 , 1 , 2 3 1 1 C C M P NO FA I L / P O L A G E V E C T 1 2 3 0 , 0 , 1 3 1 1 1 2 3 2 , 0 , 1 3 1 1 1 2 3 0 , 2 , 1 3 1 1 I 2 3 2 , 3 , 1 3 1 1 1 2 3 3 , 2 , 1 3 1 1 N O . OF T I M E U N I T S S Y S T E M TO RUN P R E S E N T AGE V E C T O R CF S Y S T E M 1 5 3 1 1. A P P E N D I X C RESULTS OF NUMERICAL INTEGRATION OF EQUATION ( 3 . 9 ) AND CALCULATION OF E ( N T ) FOR COMPONENTS OF HYPOTHETICAL SYSTEM Component No. 1 Component No. 2 Component No. 3 R e a l ( l - F r ( T ) ) I m a g ( l - F r ( T ) ) R e a 1 ( 1 - F r ( T ) ) I m a g ( l - F r ( T ) ) R e a l ( 1 - F r ( T ) ) I m a g ( l - F r ( T ) ) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 .1053007-10 .1272445-10 C .5777496-10 C .8848199-10° .9796745-10 C .9974457-10 c .9997573-10° .9999821-10° ,9999993-10° -12 -.749771-10 -.396556-10 .137536-10 .712346-10' .111529-10' .138178-10' .707037-10 -.964214-10 -.981097-10' •17 •17 -15 •15 •14 •13 -13 •13 -12 .7303263 .1013889 .1969723 .1895046 .5365838 .8259798 .9554641 .9918122 .9988676 .9998780 .9999897 .9999996 10 10 10 10 10 c 10° 10 c 10° 10° 10° 10° 10° -14 -3 -1 o .712067 .344683 .345893 .140480 .230250 .236828 .391999 .330739 .110722 .154300 .138187 .412040 10 10 10 10 10 10 10 10 10 10 10 10 -18 -17 -18 •16 -16 -15 -15 -15 -14 •14 •13 -13 .2978908 .2057618 .6156081 .2378845 .4828113 .7049503 .8559559 .9387073 .9769055 .9921936 .9976077 .9993300 .9998279 .9999599 .9999923 10 10 10 10° 10° 10° 10° 10° 10° 10° 10° 10° 10° 10° 10° -8 -2 -1 .414999 .480331 .159347 .161989 .111888 .639291 .490498 .223913 .784574 .175167 .471971 .337551 .228850 .907031 .357435 10" 19 10" 17 10" 16 10" 16 10" 15 10" 15 10" 14 10" 13 10" 13 10" 12 10" 13 10" 11 10" 10 10" 10 10" 9 . - 2 1 + C -6.56667287 8.00000000 1. 3.43332713 -7.51787807 11.00000000 _2. 5.48212193 -10.24974444 14.00000000 _ L 4.75025556 75 A P P E N D I X D G A I N / S T A G E FOR V A R I O U S R E P L A C E M E N T COST S T R U C T U R E S S T A G E COST A D V A N T A G E 1 0 % 2 0 % 3 0 % 4 0 % 5 0 % 0 1 5 6 5 1 5 6 5 1 4 9 3 1 3 4 8 1 1 8 7 1 2561 2 5 6 1 2 4 5 7 2 2 4 6 1 9 9 2 . 2 3 0 3 4 2 9 5 5 2 7 5 5 2 4 9 4 2 2 4 3 3 3 0 8 4 2 9 5 4 2 7 7 3 2 5 4 9 2 2 6 7 4 3 0 1 4 2 9 7 7 2811 2 5 5 7 2 2 7 7 5 3 0 0 0 2 9 4 3 2 7 4 3 2 4 9 5 2 2 3 0 6 3 0 1 4 2 9 2 9 2 7 4 8 25 1 1 2 2 4 3 7 3 0 0 9 2 9 4 4 2 7 6 4 2 5 2 2 2 2 5 0 8 3 0 0 7 2 9 5 2 2 7 6 3 2 5 2 0 2 2 4 8 9 3 0 1 9 2 9 4 4 2761 2 5 1 6 2 2 4 6 10 3 0 1 5 2 9 6 1 2 7 6 1 2 5 1 7 2 2 4 7 11 3 0 0 9 2 9 5 2 2761 2 5 1 8 2 2 4 7 12 3 0 1 4 2 9 5 3 2 7 6 9 2 5 1 9 2 2 4 8 13 3 0 1 4 2 9 5 5 2 7 6 6 2 5 1 8 2 2 4 7 14 3 0 1 3 2 9 5 4 2 7 7 8 2 5 2 4 2 2 4 8 15 3 0 1 2 2 9 5 5 2 7 7 7 2 5 2 8 2 2 4 8 16 3 0 1 4 2 9 6 7 2781 2 5 2 2 2 2 6 4 17 3 0 1 3 2 9 5 6 2 7 8 8 2 5 2 4 2 2 5 0 18 3 0 1 3 2 9 7 5 2 7 8 2 2 5 2 4 2 2 5 3 19 3 0 1 3 2 9 6 7 2 7 8 4 2 5 2 4 2 2 5 3 20 3 0 1 3 2 9 7 0 2 7 8 4 2 5 2 4 2 2 5 3 21 3 0 1 3 2 9 7 3 2 7 8 5 2 5 2 5 2 2 5 3 22 3 0 1 4 2 9 7 2 2 7 8 6 2 5 2 4 2 2 5 4 23 3 0 1 3 2 9 7 7 2 7 8 6 2 5 2 5 2 2 5 4 24 3 0 1 3 2 9 7 3 2 7 8 6 2 5 2 4 2 2 5 4 76 A P P E N D I X E R E P L A C E M E N T P O L I C I E S FOR F I R S T F A I L U R E S A G E C 0 M P . 1 1 AGE COMPONENT No. 3 = 5 0% COST R E D U C T I O N 1 0 % COST R E D U C T I O N AGE COMP. 2 AGE COMP. 2 1 2 3 4 1 2 3 4 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1,0,0 0,1 ,0 0,0,1 2 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 3 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0 , 1 , 0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 4 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 R,0,1 1 ,0,0 0,1 ,0 R,0,1 5 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 R,0,1 1 ,0,0 R,1 ,0 R,0,1 1 ,0,0 R , l ,0 R,0,1 1 ,0,0 R J ,0 R,0,1 1 AGE COMPONENT No. 3 = 6 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 2 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 3 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0 ,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 4 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 R.0',1 1 ,0,0 0,1 ,0 R,0,1 5 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 R,0,1 1 ,0,0 R J ,0 R,0,1 1 ,0,0 R , l ,R R,0,1 1 ,0,0 R , l ,R R,R,1 77 APPENDIX E (continued) A G E C 0 M P 1 1 AGE COMPONENT No. 3 = 5 2 0 % COST R E D U C T I O N 3 0 % COST R E D U C T I O N AGE COMP. 2 AGE COMP. 2 1 2 3 4 1 2 3 4 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0 J ,0 0,0,1 1 ,0,0 0 J ,0 0,R J 1 ,0,0 0 J ,R R,0,1 1 ,0,0 0 J ,R R ,0 J 1 ,0,0 0 J ,R 0,R J 1 ,0,0 0 J ,R 0, R J 2 1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 R,0,1 1 ,0,0 0,1 ,0 R ,0 J 1 ,0,0 0 J ,0 0,R ,1 1 ,0,0 0 J ,0 R ,0 J 1 ,0,0 0 J ,R R ,0 J 1 ,0,0 0 J ,R 0 ,R J 1 ,0,0 0 J ,R 0,R J 3 1 ,0,0 0,1 ,0 R,0,1 1 ,0,0 0,1 ,0 R,0,1 1 ,0,0 0,1,0 R,0 J 1 ,0,0 0 J ,0 R,0 ,1 1 ,0,0 0 J ,0 R ,0 J 1 ,0,0 0 J ,0 R ,0 J 1 ,0,0 0,1 ,0 R ,0 J 1 ,0,0 0 J ,0 R ,0 J 4 1 ,0,0 0,1 ,0 R,0,1 1 ,0,0 R , l ,0 R,0,1 1 ,0,0 R J ,0 R ,0 J 1 ,0,0 R J ,0 R,0 ,1 1 ,0,0 0 J ,0 R,0 J 1 ,0,0 R J ,0 R,0 J 1 ,0,0 R J ,0 R ,0 J 1 ,0,0 R J ,0 R ,0 J 5 1 ,0,0 0,1 ,0 R,0,1 1 ,0,0 R J ,0 R,0,1 1 ,0,0 R J ,0 R ,0 J 1 ,0,0 R J ,0 R,R J 1 ,0,0 0 J ,0 R ,0 J 1 ,0,0 R J ,0 R ,0 J 1 ,0,0 R J ,R R ,0 J 1 ,0,0 R J ,R R,R J 1 AGE C0MP0NEN1 " No. 3 = 6 1 ,0,R 0,1 ,R 0,0,1 1 ,0,0 O J ,R 0,0,1 1 ,0,0 0 J ,R 0,0,1 1 ,0,0 0,1 ,R 0 ,R J 1 ,0,0 0 J ,R R,0 ,1 1 ,0,0 0 J ,R R ,0 J 1 ,0,0 0 J ,R 0,R ,1 1 ,0,0 0 J',R 0,R J 2 1 ,0,0 0,1 ,R 0,0,1 1 ,0,0 0 J ,R R,0,1 1 ,0,R 0,1 ,R R,0 ,1 1 ,0,R 0 J ,R 0,R ,1 1 ,0,R 0 J ,R R ,0 J 1 ,0,0 0,1 ,R R,0 J 1 ,0,0 O J ,R 0,R J 1 ,0,0 0 J ,R 0,R J 3 1 ,0,R 0,1 ,0 R.,0,1 1 ,0,0 0 J ,R R,0,1 1 ,0,0 0 J ,R R ,0 J 1 ,0,R 0 J ,R R, 0 J 1 ,0,R 0,1 ,R R ,0 J 1 ,0,0 0 J ,R R ,0 J 1 ,0,0 0,1 ,R R ,0 J 1 ,0,0 0 J ,R R ,0 J 4 1 ,0,0 0,1 ,0 R,0,1 1 ,0,0 0 J ,R R,0 J 1 ,0,R R J ,R R,0 J 1 ,R,R R J ,R R,0 ,1 1 ,0,R 0 J ,0 R ,0 J 1 ,0,0 R J ,R R ,0 J 1 ,0,0 R J ,R R ,0 J 1 ,R,R R J ,R. R,R J 5 1 ,0,0 0,1 , o R,0,1 1 ,0,0 R J ,R R,0 ,1 1 ,0,0 R J ,R R ,0 J 1 ,R,R R J ,R R,R J •1 ,0 ,R 0 J ,0 R ,0 J 1 ,0,0 R J ,R R ,0 J 1 ,0,0 R J ,R R,R J 1 ,R,R R J ,R R , R J 78 A P P E N D I X E ( c o n t i n u e d ) A G E C 0 M P 1 AGE COMPONEN T No. 3 = 5 40% COST REDUCTION 50% COST REDUCTION AGE COMP. 2 AGE COMP. 2 1 2 3 4 1 2 3 4 1 1,0,0 0,1,R R,0,1 1,0,0 0,1,R 0,R,1 1,0,0 O.l.R 0,R,1 1,0,0 O.l.R O.R.l 1,0,0 0,1 ,R R,0,1 1,0,0 0,1,R 0,R,1 1,0,0 O.l.R O.R.l 1,0,0 0,1,R O.R.l 2 1,0,0 0,1,R R,0,1 1,0,0 0,1,R R.0,1 1,0,0 0,1,R O.R.l 1.0,0 0,1, R O.R.l 1,0,0 O.l.R R.0,1 1,0,0 0,1,R R.0,1 1,0,0 0.1 .R O.R.l 1.0,0 0,1 .R O.R.l 3 1,0,0 0,1,R R,0,1 1,0,0 0,1,0 R.O.l 1.0,0 0,1,0 R.0,1 1,0,0 0,1,0 O.R.l 1,0,0 0,1 ,R R.0,1 1,0,0 0,1,R R.0,1 1,0,0 0,1,R O.R.l 1,0,0 O.l.R R.R.l 4 1,0,0 0,1,0 R.0,1 1.0,0 R,1,0 R.0,1 1,0,0 R.l.R R.0,1 1,0,0 R.l.R R.R.l 1,0,0 O.l.R R.O.l 1,0,0 R.l.R R.0,1 1,0,0 R.l.R R.R.l 1.0,0 R.l.R R.R.l 5 1,0,0 0,1,0 1,0,0 R.l.R R.0,1 1,0,0 R.l.R R.R.l 1,0,0 R.l.R R.R.l 1,0,0 0,1,0 R.0,1 1,0,0 R.l.R R.0,1 1.0,0 R.l.R R.R.l 1,0,0 R.l.R R.R.l 1 AGE COMPONENT No. 3 = 6 1,0,0 0,1,R R.0,1 1,0,0 0,1,R R.0,1 1,0,0 O.l.R O.R.l 1.0,0 0,1,R O.R.l 1,0, R 0,1 ,R R.0,1 1,0,0 O.l.R O.R.l 1,0,0 O.l.R O.R.l 1,0,0 0,1 .R O.R.l 2 l.O.R 0,1,R R.0,1 1,0,0 O.l.R R.0,1 1,0,0 0,1,R O.R.l l.R.R O.l.R O.R.l 1,0,R 0,1 ,R R.0,1 1,0,0 0,1 ,R R.0,1 1,0,0 O.l.R O.R.l l.R.R 0,1 ,R O.R.l 3 1,0,R 0,1,R R.0,1 1,0,0 O.l.R R.0,1 1,0,0 O.l.R R.0,1 1,0,0 R.l.R R.R.l 1,0 ,R 0,1,R R.0,1 1,0,0 O.l.R R.O.l 1.0,0 R.l.R R.R.l 1,0,0 R.l.R R.R.l 4 1,0,R 0,1,0 R.0,1 1,0,0 R.l.R R.0,1 1,0,0 R.l.R R.0,1 l.R.R R.l.R R.R.l 1,0, R 0,1,0 R.0,1 1,0,0 R.l.R R.0,1 1,0,0 R.l.R R.R.l l.R.R R,1,R R.R.l 5 1,0,0 0,1,0 R.0,1 1.0,0 R.l.R R.0,1 1,0,0 R.l.R R.R.l l.R.R R.l.R R.R.l 1,0,0 0,1,0 R.0,1 1,0,0 R.l.R R.0,1 1,0,0 R.l.R R.R.l l.R.R R,1,R R,R,1 

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