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

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A DYNAMIC PROGRAMMING FOR DETERMINING  MARKOV CHAIN OPTIMAL  REPLACEMENT  ALGORITHM  COMPONENT  POLICIES  by .6. B.A.Sc.  A THESIS THE  GLEN YOUNG  (Forest  SUBMITTED  Eng.),  U.B.C,  IN PARTIAL  REQUIREMENTS  1965  FULFILMENT  OF  OF THE DEGREE OF  Master of A p p l i e d S c i e n c e (in  Forest  Engineering)  i n the  Faculty of  FORESTRY  We a c c e p t t h i s required  t h e s i s as c o n f o r m i n g t o  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA AUGUST, 1970  the  In presenting  this thesis in partial f u l f i l l m e n t of the  requirements for an advanced degree at the University of B r i t i s h 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 of my  be granted by the Head  Department or by his representatives.  It is under-  stood that publication, in part or in whole, or the copying of this thesis for f i n a n c i a l gain shall not allowed without my written  permission.  6. GLEN YOUNG Department of Forestry  The  University of B r i t i s h Columbia  Vancouver 8, B.C., Date  Engineering  Ork  °) j  Canada \91o  be  ABSTRACT  An a l g o r i t h m component replacement  is developed rules  ular class of equipment. models  developed  to determine  to follow  by S.E.  b a s i c a l l y the  D r e y f u s a n d R.A.  However, a d i f f e r e n t Markov s t a t e d e s c r i p t i o n to extend the a p p l i c a t i o n more t h a n one for which system  replacement  The model,  alternatives  the u n c e r t a i n t y  to systems  to stochastic  360/67 computer were checked  of  f a i l u r e and entire  in effect, selects  failures occurring  optimal  T h e m o d e l was  i n the  renewal  theory.  under  same  p r o g r a m m e d f o r an  and 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  through  used  as i n d i v i d u a l c o m p o n e n t s f a i l ,  of further  interval.  Howard.  has been  t h e f a i l u r e o f any component r e n d e r s t h e  inoperative.  transition  of these models  component s u b j e c t  optimal  i n managing a p a r t i c -  The work f o l l o w s  previously  the  I.B.M.  problem  ACKNOWLEDGEMENTS  I  wish  to acknowledge the encouragement  criticism  of  thesis.  A l s o , the a s s i s t a n c e of  the  inversion  III  is  Dr.  greatly  of  C.  W. B o y d , my m a j o r a d v i s o r Dr.  t h e moment g e n e r a t i n g  appreciated.  J.  V.  and  in  this  Zidek  function  helpful  in  regarding Chapter  TABLE  OF CONTENTS  CHAPTER I.  PAGE INTRODUCTION AND LITERATURE SURVEY INTRODUCTION  1  . . . . . . . . .  1  LITERATURE SURVEY  3  General  3  Dynamic P r o g r a m m i n g Algorithms II.  Markov  and E q u i p m e n t  Chain  Maintenance  . . .  5  A MAINTENANCE MODEL  11  MODEL PURPOSE AND STRUCTURE  . . . . . . . . .  11  . . . . .  11  . . . . .  12  . . . . . . . . . .  13  . . . . . . . . . . . . . . .  14  t h e Model  1.4  System D e s c r i p t i o n Mathematical The  Model  Assumptions Definition  of of  Special  Terms  Used  . . . . .  15  . . . . . . . .  17  . . . . . . . . .  18  Replacement D e c i s i o n s  Transition  Probabilities  Approximating  The  the System  Replacement Costs  MODEL DEVELOPMENT  The  of  the  Conditional failure Conditional  pdf's  transition  .  Objective  failure of  pdf's  . . . .  19  component .  probabilities .  .  .  Function  The  return  The  optimization  .  .  .  .  21  . . . . . . . . . .  27  function  Replacement before  19  of .  .  .  .  .  .  .  27 30  failure  .  31  V  CHAPTER III.  PAGE A HYPOTHETICAL MAINTENANCE PROBLEM: V E R I F I C A T I O N OF MODEL RESULTS A Description Problem Additions  of  the  .  t o t h e Model  Adjusting  the  expected . . . .  41  . . . . . . . . . . . . . .  41  o f Model  Results  . . . . . .  43 54  . . . . . . . . .  .  CONCLUSION BIBLIOGRAPHY APPENDIX A -  Glossary  of  Program V a r i a b l e s  57  . . . . .  59  and . •  APPENDIX  B -  Sample P r o g r a m O u t p u t  APPENDIX  C -  APPENDIX  D -  R e s u l t s of N u m e r i c a l Integrations of Equation ( 3 . 9 ) . . . . . . . . . . . Gain/Stage f o r Various Replacement Cost S t r u c t u r e s  E -  60  . . . . .  Replacement P o l i c i e s f o r Failures  54  . . . . . . . . . . . .  Program L i s t i n g  APPENDIX  39  DISCUSSION AND CONCLUSION . DISCUSSION  33 39  s y s t e m age v e c t o r  Program D e s c r i p t i o n Verification  33  Maintenance  Improving the e s t i m a t e of replacement cost . . . . .  IV.  . . . .  71 .  .  .  74 .  75  First 76  LIST OF TABLES TABLE I.  II.  III.  IV.  PAGE Alpha and Beta Values for the Piece-wise Linear Approximations to the Failure Pdf's  36  k  Coefficients of At for Equations of Transition P r o b a b i l i t i e s Resulting from Piece-wise Linear Approximations to Failure Pdf's Replacement Cost Data of the Hypothetical Maintenance Problem Used to Verify Model Results . . . . . . . . . . A Comparison of the Expected Replacement Cost Produced by the Maintenance Model with that Derived from Renewal Theory  3 7  "  3 8  4 3  48  L I S T OF FIGURES FIGURE 1.  2.  PAGE E v e n t t r e e s h o w i n g a l l p o s s i b l e component f a i l u r e s e q u e n c e s t h a t can o c c u r d u r i n g interval(t , , t ) f o r a three-component system . . 7  16  F i g u r e s h o w i n g 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 time to f a i l u r e of the ] s y s t e m component . . . . . . . . . .  20  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 c o m p o n e n t s 1 , 2 , and 3 e n t e r t h e 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 . i_i» and a j _ r e s p e c t i v e l y . . . .". . . . . . .  22  t n  3.  a  <  4(a).  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 event 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„.) . . . . . . . . . . . . r-l r 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 event 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  4(c).  1  t  .  .  )  24  24  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 s e q u e n c e 1, m, n i n transition interval ( t _ t^)  24  5.  F a i l u r e density functions for three components of a h y p o t h e t i c a l system  major . . . . . .  34  6.  A c o n c e p t u a l f l o w c h a r t of the replacement 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 process . . . .  42  Graph o f r e p l a c e m e n t c o s t p e r 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 o f 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 replacement cost s t r u c t u r e s . . . . .  50  Graph s h o w i n g 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 p e r 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 o f c o s t reduction for m u l t i p l e replacements . . . . . .  53  r  7.  8.  i s  CHAPTER  I  INTRODUCTION AND LITERATURE SURVEY I. The importance  rational to  For  reasonable  levels  efficiency  of  The  management o f e q u i p m e n t  industries  equipment.  through which  INTRODUCTION  these of  profit  income i s  problem of it  is  excessive  repairs  is  often  and t h e  replacements However,  in  to  the  broadest  sense  policy.  The  optimal piece policy  its  replace  to  replacement  as t h e  requires  equipment  of  an  entire  action.  to maintenance  constitutes  task of maintenance t h e o r y  optimizes  follow  An o p t i m a l  for  This  and  re-  on a  a maintenance  is  to determine  a given  system,  maintenance p o l i c y  some m e a s u r e o f  between  problem.  performed  to  of  or  replacement  approach  a piece  obsolete,  as t h e m a i n t e n a n c e  life  the  equipment  of maintenance a c t i o n s  equipment.  that  the  thesis.  maintenance p o l i c y  of  maintain  be u s e d t h r o u g h o u t t h i s  useful  in  e x t e n t on  of  t o be a m a i n t e n a n c e  a unified  series  system during  when t o  referred  referred  and w i l l  The  to  a large  operation  inadequate,  is  provides  placement  ability to  paramount  investments  problem of m a i n t a i n i n g the  s y s t e m can be c o n s i d e r e d concept  the  of  derived.  deciding  because  large  d ep en d s  and f u t u r e  equipment  problem,  have  industries  present their  that  is  system  the or  is'that  efficiency;  2  u s u a l l y maximum p r o f i t d e r i v e d c o s t o f owning and This will  from the  system or  minimum  operating i t .  thesis is concerned with  d e t e r m i n e the  optimal  developing  a model  maintenance p o l i c i e s ( i n the  of component replacement) f o r a c l a s s of equipment found in industry.  One  approach to o b t a i n i n g  maintenance p o l i c i e s i s to d e s c r i b e the  system with  to perform this  the  a M a r k o v model and optimization.  use  This  form  frequently  optimal  f a i l u r e behavior dynamic  i s the  of  programming  approach used  in  thesis. Consider  than The  the  that  one  a piece  c o m p o n e n t and  f a i l u r e o f any  operative,  and  no  that performs a continuous  component leaves  and  t i o n of t h i s system with  during these  u n t i l the f a i l e d  a discrete-parameter  be d e f i n e d .  The  descrip-  Markov  model  one  Dreyfus  a t i v e s were then  derived  that might occur.  This  system described  transition. f o r any  [5]  The  the  of f a i l u r e s altern-  of f a i l u r e s  i s not meaningful  however,  a b o v e , s i n c e o n c e a f a i l u r e has  no f u r t h e r f a i l u r e s c a n is taken.  described  Maintenance  combination  formulation  created  the p r o b a b i l i t i e s of  o f a d i s c r e t e s y s t e m as t h e c o m b i n a t i o n  that accumulate during  occurred  function.  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  t r a n s i t i o n s can  f o r the  occur  system restarted.  w h i c h t h e s t a t e t r a n s i t i o n s and  states  action  the  more  the e n t i r e system i n -  f u r t h e r f a i l u r e s can  component i s replaced  requires  of equipment that c o n s i s t s of  important  occur  u n t i l a maintenance  d i f f e r e n c e here is that  the  3  maintenance actions under the  uncertainty  epochs within m u s t be  m u s t be t a k e n  the  defined  of having  as e a c h c o m p o n e n t f u r t h e r f a i l u r e s at  same t r a n s i t i o n i n t e r v a l .  The  in which the  components can  fail  as t h e  basic  later  states  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 and  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 by u s i n g  fails,  the  the  order  state  description.  II.  LITERATURE  SURVEY  General 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 found in [4].  F o r an e x p o s i t i o n  and  replacement analysis  and  [8]. For the  analysis  fails.  production  its  of  to i n c r e a s i n g  increasing due  obsolescence.  maintenance  b e e n d i v i d e d i n t o two and  costs  F a i l u r e , on  of using  components, useless  that  reduction the  system loss  reduction  in product  the other  hand, r e f e r s  change i n the system t h a t renders  main  equipment  system maintenance, increasing  t o d o w n t i m e and  be  replacement  System d e t e r i o r a t i o n r e f e r s to a gradual  c a u s e d by  an a b r u p t  generally  can  i s r e f e r r e d to [ l ] , [ 7 ] ,  equipment that deteriorates  i n e f f i c i e n c y due  and  reader  theory  p u r p o s e o f m a i n t e n a n c e and  e q u i p m e n t has  categories:  the  of the  t o 1961  quality,  i t , o r one  for i t s o r i g i n a l purpose.  of  The  to of  4 model  as d e v e l o p e d  although subject  it  policies itself 1.  for  systems w i t h  to three The  The  cost  fail,  to i n c l u d e  systems  that  are  and t h u s  of  replacing  Points  often  cheaper  be  failure  is  failure.  associated with  cost  to  unscheduled  c a u s e d by  r e p l a c e components  so when t h e  cost  of  This  in-  of  actually  2 and 3 w o u l d  suggest  components  replacing  functional  replacing that  before  the cost  they  t h e y have  On t h e o t h e r components  the  high  relative  components. savings  could  However,  accurately aged  hand, too  induces  groups  is  fail.  c a n n o t be p r e d i c t e d  them b e f o r e  in  preparing  component r e p l a c e m e n t i s  must be a d o p t e d .  replacement of  after  them i n d i v i d u a l l y .  component f a i l u r e s of  probability  times cannot  than r e p l a c i n g before  replace  from r e p l a c i n g  some p o l i c y  systems:  t o some  in system o p e r a t i o n  equipment f o r the  directs  failures.  particularly  to  such  components  due t o t h e c o s t  is  of  fail  accurately.  This  is  that  the f a i l u r e  greater  It  failure.  according  usually  than to  greatly  that  components  components f a i l  service  because  systems  main c h a r a c t e r i s t i c s  interruptions  result  for  and  is  p r o b l e m o f d e t e r m i n i n g optimum m a i n t e n a n c e  predicted  3.  thesis  deterioration  function  2.  this  c o u l d be e x t e n d e d  to both The  in  too  frequent  excessive  re-  5 placement the  cost  costs  and t h e r e i n  savings  resulting  interruptions replacement of w a s t i n g  lies  components  expected  Dynamic  of  scale  life  Programming -  restricted  form thus  policies. tions  This  since  available.  The in  In for  invention  solve  assumptions  reducing  determining  [5], in Chain  of dynamic  form t h a t  the  optimal  system s u b j e c t  1957, published algorithm  for  the  published  to the  solving  cost  Algorithms  policy the  set in  of  optimal  replacement  be o f  a  competing solu-  considered. by R . E . that  had no a policy  be  priori  would  programming time f o r  Bellman,  could  take. model  a hypo-  deterioration.  Dynamic  solving  suboptimal  technique which  failure  usually  would  was n e v e r  continuous  for  approach  a dynamic  first  stochastic  programming  maintenance problems  1955, Bellman [2]  thetical  greatly  solution  to  The  optimal  an o p t i m i z a t i o n  on t h e  the  components.  techniques  i n many c a s e s  optimal'  1954, provided  used to  the  resulted  'the  subject  i n scope s i n c e  t a k e n was t o assume t h a t  group  Maintenance  systems  c o m p l e x m o d e l s were n o t  specific  of working  system  from  against  Markov C h a i n  and E q u i p m e n t  were f a i r l y  resulting  must be w e i g h e d  residual  E a r l y models of  problem:  from fewer unscheduled  and e c o n o m i e s  of  the maintenance  Dreyfus  Programming-Markov  an e q u i p m e n t m a i n t e n a n c e  problem.  6 In 1960,  Howard [6] f o r m a l i z e d the t h e o r y of  p r o g r a m m i n g and demonstrates  Markov p r o c e s s .  two  these being value i t e r a t i o n  iteration.  He  determining  the optimal  policies and p o l i c y  a p p l i e d the p o l i c y i t e r a t i o n method  subject to f a i l u r e  value  I n t h i s p u b l i c a t i o n he  methods of s o l v i n g f o r optimal  in a Markov process  The  dynamic  and  replacement  p o l i c y f o r an  approach.  in t h i s t h e s i s employs T h i s m e t h o d was  i n systems  the optimal  because  policies.  in the f o l l o w i n g  to develop  i n which  the s t a t e s are numbered from  P = [P-j-j] d e n o t e  the matrix  of t r a n s i t i o n  t h i s p r o c e s s , where p.. denotes i n one  is in state i . N j=l  the  model  chapter. a discrete-parameter, N-state,  E  be  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 Howard,  Consider  of t r a n s i t i o n  can  o f s h o r t d u r a t i o n where end e f f e c t s i n f l u e n c e  is given below of the theory necessary  process  the  chosen  i t d o e s n o t d e p e n d on e r g o d i c i t y o f t h e m o d e l a n d used  automobile  continuous d e t e r i o r a t i o n .  model developed  iteration  to  Markov  1 t o N.  Let  probabilitiesfor  the c o n d i t i o n a l p r o b a b i l i t y  step to s t a t e j given t h a t the  system  Obviously, for a l l i  p.. = 1 1J  and  o < p.. < 1 . 1J  A l s o l e t R = [)".{,•] d e n o t e p r o c e s s , where r . . denotes ' J  a return matrix  the r e t u r n ( p o s i t i v e or  for  this  negative)  that  i s o b t a i n e d from the process whenever a t r a n s i t i o n  from  state i to state j occurs.  In a m a i n t e n a n c e  problem  the r . . become the c o s t s of r e p l a c i n g components.  Now  d e f i n e v ( i ) as t h e t o t a l  process  n  expected  return  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 This return resulting  c a n be e x p r e s s e d  from  expected  the f i r s t  return  starts in state i .  a s t h e sum  of the  t r a n s i t i o n from  from the remaining  from the  return  i to j , and  the  n-1 t r a n s i t i o n s s t a r t i n g  then i n state j or N V  which  n  ( i )  ^ / i j ^ i j  =  c a n be r e w r i t t e n  V  n  ( i )  = q  +  V l  j  )  )  .....  (1.1)  as  i ,L ijVl +  (  P  ( j )  .....  (1.2)  where  q . =  N I  p ..r . .  i s r e f e r r e d t o as t h e i m m e d i a t e r e t u r n  (expected return  from  the f i r s t t r a n s i t i o n ) . If a Markov process will  occur randomly  s i t i o n m a t r i x P. of a c t i o n  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 ons  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 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  ( a l t e r n a t i v e s ) are a v a i l a b l e that w i l l  tran-. courses  change  the  8  transition  probabilities.  problem,for  In the equipment  instance,the replacement  maintenance  o f aged  components by  new o n e s w i l l  reduce  the probability  into a failed  state,  and i n c r e a s e the p r o b a b i l i t y  remaining  i n a working  associated  state.  of the system  But since  w i t h an a l t e r n a t i v e  the  going of i t  there i s a  return  cost  matrix i s also  changed. Let  P  = [ p . ] and R  k  k  = L> .]  k  the t r a n s i t i o n and reward i s used.  one  is o b t a i n e d from dynamic  k  1  =  t n  k  k  P  1  return.  the k so that  The problem  now  becomes  t h e optimum expected  return  t h e Markov p r o c e s s , and t h i s i s t h e r o l e o f  programming. Let  f ( i ) , m _< n , d e n o t e  t h e optimum expected  m  from the remaining m t r a n s i t i o n s p r o c e s s when t h e p r o c e s s of a l t e r n a t i v e s  i s in state  an a l t e r n a t i v e  i n the process.  t n  that  With  return Markov  i a n d an o p t i m a l  sequence  of the process.  c a n be u s e d  The m i s r e f e r r e d  number o f t h e dynamic program. the (n-m+l)  o f an n - t r a n s i t i o n  i s followed f o r the remainder  A l s o l e t km d e n o t e point  r e s u l t s when t h e k  Z p r / i / i J =  expected  of determining  matrix that  Then q  is the immediate  respectively,  iJ  1J  alternative  denote  k  at this  t o as t h e s t a g e  this labelling  system  t r a n s i t i o n occurs a t stage m and the nth 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  0  1.  9 By  f (D  definition  =  m  opt V  m  m-1'-' 1  [q^  "  m  f <D • opt[q m  m  cost  p o l i c y f o r the  opt  I  pJ;  V  l  (  (1.5)  .  f o r an  n-transition  for m = 1 (stage  equation required (n-m+l)  t n  is solved  to determine  t r a n s i t i o n , and of the  the  Markov  recursively.  of the  .  (i.s)  the optimal  process.  T obegin  the  policy  Then the return  the  process.  the  to the  1  by  to  produce  process.  i s i t e r a t e d f o r m = 3,4,  n  optimal  solution  process is  n-transition  (1.5)  transition)  process  f o r an  solution  used to obtain  for a 2-transition  values  solving  last  required  f ( i ) are  resale  The  is then obtained  1, c o r r e s p o n d i n g  L  optimal  end  process  f - ( i ) and  for a l l i .  f 2 ( i ) , the  .  Q  equipment at the  1  .  f ( i ) , m u s t b e s p e c i f i e d f o r i = 1,2,..,N.  of the  f (i)  . . (1.4)  ) ]  J  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  which y i e l d s  (j)]  v  1  remaining m transitions  condition,  . . . (1.3)  J  Equation end  (j)]  1  yields  +j  functional  f o r the  k m  m-1  or  optimal  +  _  v  p. .  Z J  optimization  = opt m  which i s the  +  m  , K  Decomposing the  f (i)  [q^  at which point  the This the  obtained.  10 This  r e c u r s i v e solution i s equivalent to starting  end o f the p r o c e s s and w o r k i n g the present i s reached.  backward through  at  time  the until  CHAPTER  II  A MAINTENANCE MODEL I. This  MODEL PURPOSE AND STRUCTURE  chapter  algorithm that  is  devoted  to  the development  can be u s e d t o d e t e r m i n e  replacement p o l i c y  t o use f o r  the  optimal  a system c o m p r i s e d of  major 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 a t random t i m e s w h i l e  in  System As u s e d i n t h i s  thesis,  characteristics:  and has t h e f o l l o w i n g  failure  performs  some  any  necessary  s y s t e m can be r e d u c e d t o s e v e r a l  major  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  contiguous  within  the  The  parts  be r e f e r r e d  that  during  to  u s e , where  simply to  its  replacement.  a unit a major  as a  com-  'component'.  unpredictable  component f a i l u r e  a component has s u f f e r e d  to warrant  forms  Hereinafter  components are s u b j e c t  failures that  group of  s y s t e m as a w h o l e .  ponent w i l l 2.  several  t h e word s y s t e m means  function  total  component  Description  that  The  to  an  use.  assemblage of m e c h a n i c a l p a r t s  1.  of  sufficient  means  damage  3^  The f a i l u r e o f one component system  4.  renders  inoperative.  No f u r t h e r f a i l u r e s c a n o c c u r component the system  5.  until  (plus possibly others)  the  failed  is replaced  and  restarted.  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 As an e x a m p l e o f s u c h  mission,  the entire  and d i f f e r e n t i a l  as a u n i t .  a system the engine,  c o u l d be c o n s i d e r e d  trans-  t o be  three  m a j o r components i n the power t r a i n system o f a v e h i c l e .  Mathematical  Model o f the System  The f a i l u r e b e h a v i o r modified,  of the system i s described  discrete parameter, non-stationary,  Markov process.  can occur  during  i n which  the Dynamic program).  This formulation  replacement  component f a i l s w i t h i n  component  a s p e c i f i e d period of system use  (a t r a n s i t i o n i n t e r v a l i n the Markov process  e f f e c t , allows  or a stage  from  (pdf's)  of time  in  of the problem, i n  decisions to occur  as each  the t r a n s i t i o n i n t e r v a l .  The p r o b a b i l i t i e s of p o s s i b l e f a i l u r e orders derived  state,  The Markov s t a t e s and 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 failures  finite  by  the conditional p r o b a b i l i t y density  are  functions  to f a i l u r e of the i n d i v i d u a l components.  This  i s done by c o n s t r u c t i n g  with  each f a i l u r e order.  the compound event  associated  13 The Since  Replacement'Costs  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  structure of the basic replacement direct  costs associated with  considered.  For this  reason  were n o t i n c l u d e d a t t h i s  algorithm,  component replacement  were  the following associated  costs  stage:  1.  The c o s t o f money o v e r  2.  Corporate  time,  income t a x deductions  on  f o r equipment purchase and equipment 3.  only the  The costs o f l o s t production  expenditures maintenance,  due t o s y s t e m  down  time, 4.  The costs o f obsolescence o f new e q u i p m e n t  5.  with  related to replacement  parts  inventories  c o s t s must be c o n s i d e r e d  extension  i n any  d e r i v e d a r e t o be  However, t h e a d d i t i o n o f most o f these  a straightforward  decreasing  policies.  maintenance problem i f the p o l i c i e s optimal.  due t o  age,  The a s s o c i a t e d costs of spare  It i s true that a l l these  technology  available,  The i n c r e a s i n g cost of o p e r a t i o n system e f f i c i e n c y  6.  due t o improved  truly  costs i s  o f t h e model developed  below.  II.  MODEL  DEVELOPMENT  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 component  system and component  time s i n c e t h e component will  be l a b e l l e d  a g e was t a k e n t o m e a n  began o p e r a t i o n .  components  The  real  components  1 , 2, a n d 3.  Assumptions o f t h e Model In t h e d e v e l o p m e n t o f t h e model 1.  The f a i l u r e o f each component independent from f a i l u r e s  2.  A l l components conditional  3.  that:  is stochastically  of the other  have f i n i t e  components.  lives at which the  probability of failure  The c o n d i t i o n a l  i s unity.  p r o b a b i l i t y o f f a i l u r e o f a com-  ponent i s age-dependent 4.  i t was a s s u m e d  Replacement components  only. are identical  t o t h e com-  ponents they replace with respect to replacement c o s t and f a i l u r e 5.  characteristics.  The cost o f r e p l a c i n g i s known w i t h  6.  Each component ition  7.  certainty. can f a i l  i nterval.  Components  any c o m b i n a t i o n o f components  only once during a t r a n s -  1  r e p l a c e d i n any t r a n s i t i o n  begin the next interval  interval  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 . Ibid. 2  15  D e f i n i t i o n of Special Let the of i n t e r v a l s , At of the  r  t  a x i s o f r e a l t i m e be in length,  These i n t e r v a l s w i l l  become the  M a r k o v m o d e l , as w e l l They w i l l  be  All  possible  during  the  event tree of Figure  the  r  t  level.  A final  1.  A n i ntermedi ate  (t _ , r  1  will  as t r a n s i t i o n  the  state will  will  The  interval (  t  be d e f i n e d  occurred  r  1.  tree  in  represents A  j , j = 1,2,3, r e p r e s e n t  t ) , which corresponds  during  r  _ » * ) ' 1  as  a  the r  a  complete  to a complete path  p r o b a b i l i t y o f any  such  in  complete  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 .  condition  state will  be d e f i n e d  s e q u e n c e o f f a i l u r e s t h a t has  t) , which corresponds  t r e e down t o L e v e l  program.  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  c o n d i t i on  sequence occurring  Dynamic  the  component number c i r c l e d .  Nodes at Level  tree of Figure  incomplete  of the  i n t e r v a l s of  Each node i n the  s e q u e n c e o f f a i l u r e s t h a t has 1S  end  .respectively.  t r a n s i t i on  jth f a i l u r e in a sequence within  the  t  and  i n t e r v a l have been e l a b o r a t e d  n  at a node r e p r e s e n t s  (t  and  1  beginning  stages.  the event of f a i l u r e of the  interval  -  into a sequence  sequences of component f a i l u r e s that  occur  at that  r  r e f e r r e d to interchangeably  can  zero  divided  as t h e s t a g e s  i n t e r v a l s , i n t e r v a l s , or  Used  such t h a t the  i n t e r v a l i s at time t  n  Terms  1 o r 2.  t o an The  as  occurred  incomplete  path  an in in  p r o b a b i l i t y t h a t the  pass through a p a r t i c u l a r intermediate  the system  condition'state  Level 1  Level 2  Level 3  10  Final y  120 123 130 132  0  \j - failure of component I y = condition state  Figure 1*  20  210 213 230 231  30  310 312 320 321  Py (A) = conditional probability of transition to condition state y  Event tree showing all possible component failure sequences that can occur during interval t _ i , t r  r  for a three-component system*  will  be 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 on f o r  that p a r t i c u l a r state. The c o n d i t i o n the  system occupies  d e n o t e d by y •  state  (final  a t time t , t  t  n  component  A complete  t  n  that be  be d e n o t e d  component o f A i s t h e age  i n the system during  measured i n appropriate  the rth interval,  units.  s t a t e d e s c r i p t i o n o f t h e system a t any  t i m e t , t _ < t <_ t r  < t <_ t ^ , w i l l  The age s t a t e o f t h e s y s t e m w i l l  by t h e v e c t o r , A , w h e r e t h e i of the i  r 1  or intermediate)  1  r  , c a n t h u s be g i v e n  by t h e p a i r  ( Y . - A ) .  The Replacement  Decisions  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 failure  o f any component  i n the system, or equivalently,  at each non-zero node i n t h e e v e n t t r e e o f F i g u r e placement interval 'd'.  decisions will  made a t t i m e o f f i r s t  be c a l l e d L e v e l  1 decisions  a n d d e n o t e d by 'e'.  used f o r Level e x i s t , other  3 decisions  1.  an  and d e n o t e d by be L e v e l  No s p e c i a l n o t a t i o n  s i n c e no r e p l a c e m e n t  than replacement o f the f a i l e d  The 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 :  Re-  failure during  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  decisions  upon  will  2 be  alternatives  component.  1.  Level  1  d = 1; R e p l a c e  failed  component  only,  d = 2; R e p l a c e  failed  component  plus one o f t h e  two o p e r a t i v e d = 3; Replace  failed  operative 2.  Level  component  plus both t h e  components.  2  e = 1; R e p l a c e  failed  component  only,  e = 2; R e p l a c e  failed  component  plus  component 3.  components,  Level  not already  remaining  replaced.  3 Replace  failed  component.  Transition Probabilities The 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 partial  and c o m p l e t e t r a n s i t i o n i s shown a d j a c e n t  appropriate ities  l i n k s i n t h e t r e e o f F i g u r e 1.  a r e o f t h e form  integrating failure  P^(A), and w i l l  These  be d e v e l o p e d  functions of the c o n d i t i o n a l pdf's  o f the i n d i v i d u a l components.  probabil by  o f time  S i n c e many  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  to the  to  pdf's  to the actual 3  pdf's  will  be u s e d  and i s i n t r o d u c e d  below.  In 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 only f o r reasons of i n t e g r a b i 1 i t y . In most cases e i t h e r i n s u f f i c i e n t data a r e a v a i l a b l e t o determine t h e true form 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 the data adequately.  A p p r o x i m a t i ng t h e f a i 1 u r e A pdf of time component  i s shown  to f a i l u r e , f-|(t), f o r a generalized  i n F i g u r e 2.  t o f - j ( t ) c a n be o b t a i n e d 1.  Subdivide  For reasons  approximation  as f o l l o w s :  age i n t e r v a l s as shown  that will  become  apparent  in Figure  2.  later the  length of these  i n t e r v a l s m u s t be t h e same as t h e  intervals  t ) , or At.  Define over  li (t) * and  A convenient  t h e range o f t h e random v a r i a b l e t i n t o  m equal-length  2.  pdf's  (t  l S  r  a f u n c t i o n , g^-Ct), that approximates f-|(t)  the i t h g e i n t e r v a l such  that  a  for a _  f^t)  < t < a.  i 1  i = 1,2,....m . . ( 2 . 1 )  0 o t h e rw i s e  m E i=l  a.-  g (t)dt = l  1  . • (2.2)  n  ai-l  and 9 l i - l  Conditional  (  a  i  )  =  pdf's  9 l i  (  a  (2.3)  i >  o f component  failure  The p r o b a b i l i t y o f f a i l u r e o f a component d u r i n g interval beginning a given  c a n be c o n d i t i o n e d on t h e e v e n t  that at the  o f t h e i n t e r v a l the component has a l r e a d y age w i t h o u t  of the equation  failing.  an  A straightforward  reached  application  of conditional p r o b a b i l i t y leads to the 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 {component has  1 fails  i n ( a ^ _ , t ) , a . . ^ <_ t <_ a^ 1  l i v e d t o a g e a -_ > 1  t g^CsJds/da  1  i - l E  i - l  9  k = 1  The denominator  l  l k  (s)ds)  .  (2.4)  k-l  o f ( 2 . 4 ) i s a c o n s t a n t f o r any i n t e r v a l  a n d t h e e q u a t i o n may b e w r i t t e n a s  P {• }  (2.5)  ^ ( s j d s l  i - l  where  *n(s)  =  1 1  gn(s)/(i1 1  i -1 E k=l  9 l  l k  (s)ds)  k-l  is the c o n d i t i o n a l pdf o f f a i l u r e o f the I t h component f o r its  i t h age i n t e r v a l .  Conditional  p r o b a b i 1 i t i es o f t r a n s i t i on  F i g u r e 3 shows t h e p a r t i c u l a r failure  pdf's, g  l i  (t),  approximations  to the  and thus 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) w h e n t h e s y s t e m  begins the r t h  transition  Here, t h e age  interval  w i t h age v e c t o r A.  v e c t o r c a n be t a k e n t o be t h e t r i p l e understood at  that the i t h ge interval a  age a ^ _ ^ and ends a t age a . .  ( i , j , k ) , where i t i s o f a component  begins  The end p o i n t s o f the  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» j-l* a  a n d  a  k-l  respectively-  c o m p o n e n t age transition  i n t e r v a l s coincide with  age  The structing  and  the  can  be  obtained  compound event The  (a)  P^(A)  are  derived  1,2,3, f a i l s  a graphical  of 1 f a i l i n g  i  n  1, m a n d  Noting ( i-l' a  fact  length. con-  the f a i l u r e  f i r s t can  be w r i t t e n  m and  = P {1 n do  fails  not  n = 1,2,3;  m f  Figure  then  Figure  the  4(a) For  a  probability  as  in t  a^_  t h a t as t v a r i e s o v e r a.. ) ( s e e  r  in t , t + d t | l l i v e d to  fail  and  1  t ).  lS  1  component  o f t h i s same e v e n t .  ^ < t < t  r  , t|m  l  and  n  r e s p e c t i v e l y },  1  n f  1;  1 or  (t _ ,  t )  3), performing  the  r  1  . .  varies  r  over  transformation assumption  i n d e p e n d e n c e o f f a i l u r e s has  equation  Pi..(A)  becomes,  (2.6)  m.  v a r i a b l e z-j = x ^ - a - _ a n d r e c a l l i n g t h a t t h e i  the  below.  (t  of t , t  . ( t , A)dt 1  f i r s t during  have l i v e d to a^_  for  with  probability that  representation  s p e c i f i c value  a -_  the  P r o b a b i l i t y of p a r t i a l t r a n s i t i o n to Level # >  1 #  of  f r o m t h e ^ ^ - ( t ) by  associated  P.j (A) denotes the  5  points  t r a n s i t i o n i n t e r v a l s are At i n  P^(A)  s e q u e n c e y.  is  end  i n t e r v a l s b e c a u s e of A s s u m p t i o n 7 and  t h a t both  1,1=  the  1 >  been made, the  of of  for  24  >no E or E m  n  Figure 4(a)- Graphical representation of the event that component I fails first during transition interval ( t _ j , t )• r  r  •m  • f c  tr-  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 ±  Y  |  *r-l Figure 4(c)- Graphical representation of the event that there be failure sequence l,m,n in transition interval ( t _ j , t )• r  r  P T . . ( A )  =  t  2 5  At  S(t,A)dt  *ii< l> Z  r-l rZ  Kmj. - ( m, J m J  1  ( 1  z  IJI . v(z r 1  nk  ( 1 -  where  the ^ . ( t )  (b)  p  state  is  Probability denotes  m  transition  of t h i s  as above  P  Tm.( > A  its  =  m  1 , 2 , 3 ;  transition  2 through  o f component 1 .  ( Z  1  }  ( 2 . 7 )  2  o f an i n t e r m e d i a t e sequence 1 ,  any f a i l u r e  A graphical 4(b)  e v e n t i s shown i n F i g u r e  *11  to Level  1 ) when r e p l a c e m e n t a l t e r n a t i v e  probability  L  .  ^^(t)  the p r o b a b i l i t y  f  .  )dz, n' 1  of p a r t i a l  to Level  used on f a i l u r e  tation  )dz  .  z  are the transformed  l .(A)  m ( 1 and m =  n  d  d  represen-  and by  reasoning  can be shown t o b e ;  Ki^'  "J,  ( l  -j ^nk( n) z  0  d z  n)  d z  m  l  d z  1  ( 2 . 8 )  when  d = 1 ;  At  At p  I m (• )  f m  A  =  *l1  (  2  lH,  Kj J° {z  ^~  >nk  ( z  n)  d z  n) .  d z  .  m .  d z  l ( 2 . 9 )  26 when d = 2 a n d n r e p l a c e d = o if m replaced  P^ (A) = o  f o rd  m  P  1 q  w i t h 1, a n d ; with 1 ;  =  3  (2.10)  ( A ) denotes the probability  of no f u r t h e r  o f the special  failures after the first.  These  event  probabilities  are; ,  rAt  _  pj (A) -  j  0  Q  f  At  *j -(z ).(i-j 1  1  r  K^J J'^-\ d2  o  At  *nk< n> 2  0  d z  n>  d z  i  (2.11)  P^ (A) - } 0  *i (« )-(l-j  o  1  1  1 o  *i (z )dz J  m  | n  ).(l.j  o  *; (z )dz )dz k  n  n  1  . (2.12) where component m replaced P? (A) 0  (c)  =  1  P^ (A) Level  (2.13)  Probability n  w i t h 1, a n d ;  o f complete t r a n s i t i o n t o Level  denotes the p r o b a b i l i t y  graphical  replacement  d used on f a i l u r e o f component 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.  ities  ofa transition to  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  alternative  3  representation  Figure  o f such a sequence.  c a n be shown t o b e :  4 ( c ) shows a The probabil-  27  At  At  At  .(z ) ^mj '  ty'  1 mn  v  'm  *nk  ( z  n  ) d z  n  d z  m  d z  l  ;  ....  (2.14)  ....  (2.15)  and 1 mn  0  '  f o r a l l other  d and e.  The O b j e c t i v e The dynamic programming  Function optimization  t h i s s e c t i o n by f i r s t d e r i v i n g t h e r e t u r n Markov model with decomposition  how  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  of  N stages  of the program.  function  An e q u a t i o n  f o r the expected  o f the Markov process  return  of  from N t r a n -  developed i n this chapter  by t h e same r e a s o n i n g  that l e d to equation  The immediate r e t u r n , however, i s o b t a i n e d costs  f o r the  showing  The r e t u r n  be o b t a i n e d  function  f i x e d a l t e r n a t i v e s , and then  replacement cost over  sitions  i s developed in  can (1.2)  by summing t h e  series of p a r t i a l transitions rather  than  of  single complete t r a n s i t i o n s . Let  V  N< > A  =  q  A  +  Vl  (2.16)  be t h e e x p e c t e d r e t u r n  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 ^ _ i s  starts the f i r s t  1  the e x p e c t e d r e t u r n after the f i r s t , The  condition  f r o m N t r a n s i t i o n s when t h e s y s t e m  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  and q i s as d e f i n e d  s t a t e , y»  i n e q u a t i o n (1.2).  has been o m i t t e d from t h e i n i t i a l  state d e s c r i p t i o n since the system always s t a r t s i n c o n d i t i o n state y  =  1.  o. Immediate The  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 as  (a) Y  Q  from Y  1, y^  =  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  0  I 1=1 d  P  l e  .(A)C^  1  o,  i s the probability of trans-  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  where C  follows:  The expected cost o f t r a n s i t i o n from Level  = o, t o Level  ition  returns  i n d u c e d by t h i s  or,  ,  1  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 when 1 f a i l s  and  replacement a l t e r n a t i v e d i s used. (b) y^  The expected cost o f t r a n s i t i o n from Level  = I * * , to Level  transition Y  0  summed  t o Y£  o v e r a l l Y]_  2,  y^  = lm* , i s t h e p r o b a b i l i t y o f  times the replacement cost a n d y^ o r ,  incurred,  1,  3  3  1=1  (c) Y  l *>  =  shown t o  Z  Y3  .  Z  n=l n ^ l ,m  (d)  The  m  =  P  ( A ) C  lmn  i s sum  E  1=1  n  3  be  B  Q  _  H p  m=l l m n > i m ^ l n ^ l ,m z  P  ( A  C  lmn  of the costs  1  of the  f o r one partial  i n (a) t o (c) and  = ( i + l , j+1,  „ n ? M  remaining  m  '  m  transitions  k + l ) be t h e s y s t e m a g e  t r a n s i t i o n when t h e s y s t e m s t a r t s  state i n age  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 .  denotes  the system  and a l t e r n a t i v e  com-  yields  m=l m^l  a f t e r the f i r s t  d  n  R e t u r n s f r o m t h e N-1 Let  B  3 =  Thus, adding the terms  1=1  state  2,  from Level  lmn, i n a s i m i l a r manner can  _  1mn  the summations  2.  of t r a n s i t i o n s  total expected replacement cost  transition  transitions. bining  3,  3  m=l m^l  complete  '  The e x p e c t e d c o s t  3  Z  l m  _  (A)C®  be,  3 1=1  P°  m-1  to Level  m  2  i  I  I  age s t a t e  when c o m p o n e n t 1 o n l y  d is executed f o r i t s repair.  Then fails  Similarly,  "de 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 e n f a i l u r e occurs de  and a l t e r n a t i v e s  sequence  d and e a r e u s e d .  1, m  Obviously,  -  B  lmn  (i' ' )  =  1  1  f°  r  a  n  v d a  n  de  -  N o w  »  s  i  n  c  e  v  M-I^ ) B  1  S  T  N  E  e x p e c t e d r e t u r n from N-1 t r a n s i t i o n s s t a r t i n g i n the general 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  Vi  -  2 tp? (A)» . (5f) 0  H  x  I (p?: (A)v . (i?;) 0  +  1=1  H  1  m=l "tl  . . (2.18)  where the P. (A) are the p r o b a b i l i t i e s o f the system  ending  0  the  first  t r a n s i t i o n i n the corresponding  The  optimization Let f ( A ) denote  cost the  the minimum e x p e c t e d  over N t r a n s i t i o n intervals (stages system  initial  age s t a t e  ment p o l i c y i s f o l l o w e d .  f (A)  d  w h e r e ( d ^ , e^)  replace-  N ±  ] ,  ....  (2.19)  e  N  w  ~  ,=  1  (2.17)  y i e l d s the f i n a l D.P.  D.P.)^when  Thus,  and F  Substituting  4  o fthe  i s the set o freplacement a l t e r n a t i v e s  the kth stage,  arranging  replacement  i s A , and an o p t i m a l  = min[q^ + F N' N  N  in  s t a t e B.  min (d e N-l' N-l l Q  and  e  (2.18)  result,  denotes Dynamic  program.  ;  )-"(d  e )  ^ i»V a  i n t o (2.19)  made  [V., , ] . ~  H  and  1  re-  31  3  de _ m i n {P d ( A ) C ® + P Imo ( A ) f 1 n r m=l e ' m  +  de + P lmn  +  P  (A)(C_ + f  o < > N - l < o> A  f  B  (2.20) o m i t t e d from e^ and d^ t o  the notation. The  o p t i m i z a t i o n o f (2.20)  as s h o w n i n C h a p t e r transition  transition  Replacement  backward  i n time to stage  interval.  before  failure  In some s y s t e m s interruptions  i s performed r e c u r s i v e l y ,  1, by s t a r t i n g a t s t a g e 1, t h e N  i n t e r v a l , and w o r k i n g  N, t h e f i r s t  ponent  ))}]  •  where t h e s u b s c r i p t N has been simplify  (B  N-1  f o r which  the cost of  non-scheduled  of operation i s substantial the optimal  replacement  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  components at the beginning of a stage before they failed.  com-  of have  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 be 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 (2.20)  f (A') + C w  f  by c h e c k i n g t h e i n e q u a t i o n  < f (A), M  (2.21)  a f t e r the  f^(A)  have been c a l c u l a t e d ,  where C  of the  replacement before failure alternative,  is the  new  initial  (2.21) holds true should  be  age  state  for a given  used i n the  optimal  created  by  cost  f , and  A  alternative  f, then that policy.  i s the  f.  alternative  1  If  CHAPTER I I I A HYPOTHETICAL MAINTENANCE PROBLEM: OF MODEL R E S U L T S This chapter i s devoted to testing the  policies  VERIFICATION  the accuracy of  and e x p e c t e d c o s t s p r o d u c e d by t h e model  developed i n Chapter I I , f o r a h y p o t h e t i c a l maintenance problem. A brief description problem and of the computer optimization. improve  program  Two p r a c t i c a l  written to perform the  a d d i t i o n s t o t h e model  i t saccuracy are outlined.  from renewal  The renewal  that  function,  t h e o r y , i s used t o determine the accuracy o f  costs produced independence produced  i s given of the maintenance  by t h e model  f o r the case of economic  of replacements.  The r e a l i s m o f the p o l i c i e s  i s also discussed f o r this case.  Results f o r other  replacement cost structures are briefly discussed.  A D e s c r i p t i o n of the Maintenance The  Problem  h y p o t h e t i c a l system c o n s t r u c t e d to v e r i f y  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 . pdf's o f these components 1 h a d a maximum u s e f u l p d f was s k e w e d  Component  of 5 time u n i t s , i t s f a i l u r e  r i g h t a n d h a d a mean o f 2.60.  had a maximum l i f e  The f a i l u r e  a r e s h o w n i n F i g u r e 5.  life  model  of 4 time u n i t s .  Component 2  I t s f a i l u r e p d f was  34  Component Age at Failure -  Q|  Component  2  E(a )n-70 2  Component Age at Failure — a g  Component Age at F a i l u r e - 0 3 5-  Failure density functions for three major components of a hypothetical system-  skewed l e f t had  a n d h a d a mean o f 1.70.  a maximum l i f e  approximated  o f 6 time  The t h i r d  component  units, i t sfailure  an e x p o n e n t i a l d i s t r i b u t i o n  pdf roughly  and h a d a mean  of 2.00. The range  three d i s t r i b u t i o n s chosen  of types  mechanical exhibit  o f f a i l u r e pdf's that are found  systems.  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 ,  As c a n b e s e e n approximated + 3-|.j(t),  Figure 5  data would  I shows t h e c o e f f i c i e n t s  polynomials  and T a b l e resulting  was  maintenance  chosen and  problems  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  the u s e o f a more c o m p l i c a t e d  to the pdf's  were  g^ ^ ( t ) = a^.  over n o n - l i n e a r forms  i n most p r a c t i c a l  this  difficulties.  This approximation  of i t ssimplicity  that  the actual pdf's  l i n e a r form  f o r a-j.-j^ £ t <_ a.. .  because  sufficient  from  n o t p r e s e n t a n y new  by t h e p i e c e - w i s e  secondly because  Table  i n many  The i n t e r e s t i n g case o f components  type o f d i s t r i b u t i o n would  first  cover the general  non-linear  approximation.  of the linear  approximations  I I shows t h e c o e f f i c i e n t s from  of the  the s o l u t i o n of equations (2.7)  t o ( 2 . 1 5 ) when t h e above a p p r o x i m a t i n g  f u n c t i o n s are used.  TABLE 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 T H E F A I L U R E P D F ' S  AGE  COMP. NO.  3  2  1  a  a  3  INTERVAL  a  3  a  3  6  5  4  a  3  a  3  3  1  .0  .2  .2  .0  .2  .2  .4  - .2  - .2  -.2  -  -  2  .0  .4  .4  .1  .5  -.4  .1  -.1  -  -  -  -  3  .3333 -.0555  . 2778 -.0555  .2223 -.0555  .1668 -.0555  .1113 -.0555 .0555 - .0555  CO  TABLE II COEFFICIENTS OF A t FOR EQUATIONS OF TRANSITION PROBABILITIES RESULTING FROM PIECE-WISE LINEAR APPROXIMATION TO FAILURE PDF'S* k  k  0  0  ***  1 2  pf (A)**  P} (A)  a  l  a  -V2[a (a a )-3 ] 1  m+  n  1  V - l 2  a  l  • m  ( a  + a  a  n  0  l  3 /2-a (a /2 a ) 1  }  1  m  +  aia/2 1 m  n  3 +3 (a a )] 1  4  1 / 4  m+  n  hKV m n> B  a  hVV l V  / 2  e  +  5  l/10[.  l B m  B  l (  a m  V ^ m  +  e  n )  /  2  ( f  2 + e  n»  / 4  ^  +  6  l  (  a  m V * m  )  /  2  l  3 /2 n  +3-,(a )] l m n 3 /2+3 ma n m  6  ¥ / n  /  2  Wn  4  /  1  6  n  -3.3 3 /24 lmn  Po(A) not shown. It was c a l c u l a t e d from n (l-(a-,At+3,At / 2 ) ) . ** 1=1 Component m replaced with component 1. 1  The  age i n t e r v a l  subscripts  i , j , k have been o m i t t e d t o s i m p l i f y  the notation.  m  n  TABLE II ( C o n t ' d . ) <l  1  k  n  0  1 2 3  iV  a  l/3Ca B 1  + a  4  0*>  P? .(A)**  n  2  Vm  N  ( B  r l m a  a  1  n  1  m  -e (a B /2 n  m  1  "Wr/ 6  "Wn  /  4  8  +6l  4 0  1 m  0  a  / 3  m  - l a  +  5  0  1  a  a  / 6  1  n  mV  ( e  3 +  MV -Vn 4  W >  l/4[3 a a /6  >/  V V n  / 3  1  4  2  a )/15  -  l m n  -a (a a -3 /3)/2  ) / 2 ]  l/4[3 (3 -a a )/2  0  m  n  /  2  +  V V  3  ^  1/5^3^/3  m  +  MVn  "Wn  /  1  / 2  6  % n 6  / 3  )^  +3 (a 3 /3+S a /4)/2] 1 nrn' m n •' n  v  3iB 3 /48 1 m n'  39 A d d i t i o n s to the Improving  Model  the estimate of expected replacement 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 t h e e x p e c t e d c o s t o f e x a c t l y one f a i l u r e in a t r a n s i t i o n failures Chapter ities  of each  component  i n t e r v a l , s i n c e the p r o b a b i l i t y of  o f a c o m p o n e n t was II).  accounts f o r  assumed  t o be z e r o ( A s s u m p t i o n  For the p r e s e n t problem, however,  of additional  failures  i f component 1 i s s t a r t i n g  additional  are not always  i t s i t h age  the  small.  interval  two f a i l u r e s  component i s  age  failures  1  5  t |1  least  reached  n a s  a.^) f  At  Jo  f^Ct)-  f  o f 0.197  which  compensate would  At  't  F o r 1 = 3,  iK,(s)ds  g  e  a n d A t = 1,  ....  (3.1) y i e l d s  (3.1)  a value  Thus,  to  f o r the u n d e r e s t i m a t i o n of replacement c o s t t h a t  Additional The  a  .  i s s u b s t a n t i a l l y g r e a t e r than zero.  r e s u l t from Assumption  1.  dt  i = 6,  expected c o s t of a d d i t i o n a l  ith  of 1 during t  example,  at the begin-  i n t e r v a l , the p r o b a b i l i t y of at  P ( a t l e a s t two  probabilFor  ning of a t r a n s i t i o n of t h i s  6,  6, a m e t h o d o f a p p r o x i m a t i n g f a i l u r e s was  the  d e v i s e d as f o l l o w s :  cost of Level 1 d e c i s i o n s .  expected failure  time of each component f o r i t s  i n t e r v a l , a , . , c a n be c a l c u l a t e d  from  40 At a  li  sif>j. ( s ) d s /  If i t i s assumed replaced  o  with 1 are replaced failures  be c a l c u l a t e d  from  Cost  ^'  that the f i r s t  of a d d i t i o n a l  Add.  At  =  (s)ds .  failure,  Additional  of a component replaced  cAt-a,. c • '> m o  costs  m l  (s)ds  of Level  If i t i s assumed interval third  failure a  Q  o c c u r s m i d way  2 a n d 3 b e c o m e s 1/4  however,  1/9  2 and 3  f o r Component 3 f o r which  (3.3)  decisions. i n an  o f ( a ^ . , A t ) , and t h a t  between  a n d 1/16  the time of the  of  o f ( 3 . 3 ) was  the a d d i t i o n a l cost of Level  1 can  component.  the second  f o r which  of components  replaced  (3.3),respectively.  a d d i t i o n a l c o s t s were added  i n s t e a d o f 1/16  cost  ....  and A t , t h e n i n g e n e r a l ( i . e . components  These  for  ,  = 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  at Level  components  at Level  that the second f a i l u r e  o c c u r s at the mid p o i n t  failure  1, a n d a l l  (3.2)  a t a^ . , t h e n t h e e x p e c t e d  w h e r e m = 1 , 2, o r 3 i s t h e r e p l a c e d 2.  ....  to the  model,  arbitrarily  3 replacements to  used  compensate  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 search f o r optimal replacement decisions  made t o  for additional  1  41 failures. ponent  The model  assumes replacement of the f a i l e d  com-  only.  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 Assumption replaced  7, C h a p t e r I I , s t a t e s t h a t a l l  i n an i n t e r v a l  components.  begin the next interval  This assumption  would  produce  components  as  new  an 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 from each r e p l a c e m e n t between  the actual  r e p l a c e m e n t and t h e end o f t h e i n t e r v a l . e r r o r t h e a l g o r i t h m was in each  time  To r e d u c e  this  programmed so t h a t the f i r s t  i n t e r v a l , and a l l components  begin the next t r a n s i t i o n  interval  of  failure  r e p l a c e d with i t , would a t age  interval  2.  Program D e s c r i p t i o n The  a l g o r i t h m was  p r o g r a m m e d i n F O R T R A N IV a n d  on an IBM 3 6 0 / 6 7 c o m p u t e r . program  showing  i n F i g u r e 6.  A conceptual flow chart of the  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 shown  A listing  of the source program  and a g l o s s a r y  o f t h e v a r i a b l e names 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 Inputs to the program control  parameters  i n c l u d e the program  A.  output  t h a t s p e c i f y a t what s t a g e and system  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 c o m p o n e n t , t h e ex's a n d $'s failure  ran  life  of  age each  of the approximations to the  p d f ' s , g - j ^ ( t ) , and 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 combinations of component replacements. i n f o r m a t i o n r e q u i r e d i s computed  w i t h i n the  A l l other  program.  F i g u r e 6.  A conceptual flow chart of the maintenance model showing the h i e r a r c h y o f the o p t i mization process .  START  Data Input  1  1- Output Control Paranv 2- Max- Life of Components  3- Coefs of Failure PDF's 4- Replacement Costs  Calculation and output of absolute and conditional probabilities of failure for each componentCalculation of expected cost of additional failures of each component for replacement on failure of any componentCalculation of conditional probabilies of state transition for all possible age vectors A , states y , a n d alternatives d,e0 —*• k 0 - * f (A); all A k  ^ ^  Set next stage-, k+1 —>k  ^  Set next age vector-, A  •I  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 +Pfje (A) 0  VjlfO +  de P|mn< >< n + W B , m „ ) ) " » a A  No  c  a < £'  a - > /3' Save e, m,n  43 Outputs ditional  i n c l u d e a summary o f t h e a b s o l u t e and c o n -  probability  o f f a i l u r e by age i n t e r v a l  f o r each  component, and t h e o p t i m a l component r e p l a c e m e n t and 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 w i t h i n each stage. The  age v e c t o r  An e x a m p l e o u t p u t i s shown i n A p p e n d i x  program  contains excessive subscripting  some v a r i a b l e s , a t t h i s  point, to f a c i l i t a t e  g a t i o n o f model b e h a v i o r . not  policies  Replacement  further  B.  of investi-  before failure  was  considered.  Verification  o f Modei R e s u l t s  The 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 shown i n T a b l e I I I .  TABLE I I I R E P L A C E M E N T C O S T D A T A 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 P R O B L E M U S E D TO V E R I F Y MODEL R E S U L T S REPLACEMENT Cost R u n R e d . (%) 1 2 3 4 5 6  0 10 20 30 40 50  COMBINATION  (1)  (2)  (3)  (1 ,2)  10C0 1000 1000 1000 1000 1000  2000 2000 2000 2000 2000 2000  3000 3000 3000 3000 3000 3000  3000 2700 2400 2100 1800 1500  (1 ,3) 4000 3600 3200 28000 2400 2000  (2,3) 5000 4500 4000 3500 3000 2500  (1 , 2 , 3 ) 6000 5400 4800 4200 3600 3000  These  data ranged from the case o f complete  pendence o f replacements  economic  (the cost of replacing  Inde-  two 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 replacement c o s t s ) t o the extreme reduction  i n total  replacements.  c a s e w h e r e a 50 p e r c e n t  replacement c o s t r e s u l t s from m u l t i p l e  A discussion of the results  i s contained  below. 1.  R u n 1:  Complete  (a) P o l i c i e s .  economic  independence  When n o c o s t s a v i n g r e s u l t s  multiple replacements, optimal replacement policies involve failure  replacement only.  grammed t h e model p r o d u c e d istic  When r u n a s f i r s t  policies  should pro-  that involved opportun-  replacement o f the t h i r d component with t h e second  failure,  i n some c a s e s , when t h e t h i r d  l a s t age i n t e r v a l . the f a i l u r e  of the relative costs of  f o r these cases r e v e a l e d t h a t i n most i n s t a n c e s  they d i f f e r e d magnitude  Investigation  c o m p o n e n t was i n i t s  (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  alternatives  in the third  are insignificant  decimal place.  t h a t was u s e d . come t h i s  Errors of this  and a r e most l i k e l y  the approximation t o t h e cost o f a d d i t i o n a l  With  from  c a u s e d by  .replacements  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 -  departure from t h e t h e o r e t i c a l l y c o r r e c t  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 would  policies.  n o t s e l e c t an e = 2  alternative  as 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  one 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 the e = 1 a l t e r n a t i v e . very  The  resulting  p o l i c i e s were  then  realistic. (b)  Expected replacement cost:  check through renewal With failures  the assumption  three-component  independent renewal  independent  theory. of independence  and a p o l i c y o f f a i l u r e  hypothetical  An  of  component  replacement o n l y , the  s y s t e m c a n be v i e w e d  as t h r e e  processes running simultaneously.  The  e x p e c t e d r e p l a c e m e n t c o s t over a p e r i o d of time f o r each individual renewal  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  f u n c t i o n and the r e s u l t s f o r each component 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 f o r the whole system. b e l o w , was  T h i s r e a s o n i n g , which  u s e d t o o b t a i n a c h e c k on m o d e l  For a renewal renewal  E(C ) T  =  so the problem  renewal  accuracy. expected  from  E (Nj) • c  where c i s the c o s t per renewal The  is outlined  process of d u r a t i o n T the  c o s t , E ( C - ) , c a n be c a l c u l a t e d  function.  replacement cost  ....  and E(N. ) i s t h e T  renewal  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 )  reduces to finding  E(N ). T  (3.4)  I f t.j d e n o t e s S  =  r It, i=l  the i  i n t e r - r e n e w a l time  t n  i s the random v a r i a b l e of time t o the r  newal , the p r o b a b i l i t y shown t o  that r renewals  t  n  o c c u r i n T can  rebe  be P(N  = r) = F (T)  T  r  - F  r + 1  (T)  ,  ....  where F^(T)  i s the cumulative d i s t r i b u t i o n  (see [3]).  Thus the expected  be e x p r e s s e d  E(N ) T  The  and  F^(T)  1  (3.5)  f u n c t i o n of  number of renewals  in T  can  as  =  _ r ( F (T) - F all r r  r + 1 r + 1  (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  of  t h e k t h power 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 (z), t  o f t h e f a i l u r e p d f , f ( t ) , as f o l l o w s . Starting  M (z) t  with  e  z t  f(t)dt  ,  ....  (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 , and s u b s t i t u t i n g the l i n e a r approximations m M (z)= E t  to f ( t ) y i e l d s ,  .  .  (3.7)  where m i s the number 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 ) , and a., a . J  a n d 3-  J  a r e as p r e v i o u s l y d e f i n e d w i t h  the  J  component number s u b s c r i p t s o m i t t e d to s i m p l i f y n o t a t i o n . Expanding  and p e r f o r m i n g the i n t e g r a t i o n  of  (3.7)  yields  m  M (z)  a  zt .e  3  j  Z  t  zt  m +  (3.8)  (zt-1)  Z  j = l  The  F ( T ) c a n t h e n be d e t e r m i n e d b y t h e  inversion  r  e q u a t i on  l - F ^ T ) = ± (1-sign  (c)) +  ^  exp  [-zT  + rln M.fz)]^ , . . (3.9)  where sign  (c)  , 1  for c > 0  =. 0  for c = 0  l-l  f o r c < 0,  Since i t would  (see [9]) .  be d i f f i c u l t  to o b t a i n a c l o s e d form  solution  to ( 3 . 9 ) , a computer  R u l e was  w r i t t e n to perform the i n t e g r a t i o n n u m e r i c a l l y .  The  i n t e g r a t i o n was  performed  s u b r o u t i n e b a s e d on  of component],  r f o r which  a n e g l i g i b l e change  f o r which  T  T  max  „ was  F ^ ( T ) = 1,  in E(N )  t h i s study the c a l c u l a t i o n of E(N ) and t h u s r  r^  f o r a l l values of r from  [T/max l i f e  sufficient  Simpson's  was  t o ±.001  to r  m  a  produced. was  m n  x  the For  considered  t a k e n t o be t h e r f o r  which  ( r - l ) • ( F _ ( T ) - F ( T ) ) _< . o o l . T h e e x p e c t e d r  of f a i l u r e s  1  i n T was t h e n c a l c u l a t e d  number  from  r  E ( N  T>  The in Appendix  =  r  min  results  max r . i r z  F  ( T  +  mm  >  •  of the numerical  • • • •  of the renewal  ( 3  '  1 0  >  i n t e g r a t i o n s a r e shown  C along with the calculations  shows a c o m p a r i s o n from  +  of E(N-).  T a b l e IV  t h e o r y r e s u l t s as o b t a i n e d  ( 3 . 4 ) , f o r T = 10,  T A B L E IV A C O M P A R I S O N OF 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 P R O D U C E D BY T H E M A I N T E N A N C E MODEL WITH T H A T D E R I V E D T H R O U G H RENEWAL T H E O R Y  Comp. No.  E  Replacement Cost  Model Results  Renewal Theory Resu1ts E (N j )  E(C )  Expected Cost  T  1  1000  3.433  3433  2  2000  5.482  10964  3  3000  4.750  14250  all  -  -  28647  28353  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  those  produced  results  by 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 .  i n d i c a t e t h a t t h e model u n d e r e s t i m a t e s  The  the replacement  cost  49  by o n e  per  cent.  negligible  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  error.  A simple r a t e , or gain  c h e c k on  per  stage  t h r o u g h some b a s i c renewal  process  a sufficient the  renewal  the  steady  of the  D.P.,  length  can  E(N ) t  process  renewal  thetical  p r o b l e m can  w h e r e c-| i s t h e and  c o s t o f one  as s h o w n i n F i g u r e  gain  A p p e n d i x D) cent  and  1.  —  c,t a  per stage  expected  time).  The  theoretical  15  o f 3015  hypo-  as  ,  ....  of the  Equation  5, a n d  l  t  n  (3.12)  component  (3.12) y i e l d s a when t =1,  t h e c-j a r e  as s h o w n  the in  the model c o n v e r g e d to a (see  Figure  7  and  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  u n d e r - e s t i m a t i on' o f t h e  life  1  renewal  At s t a g e  to  u i s the  r e p l a c e m e n t c o s t r a t e o f 3061  I I I , Run  obtained  at e q u i l i b r i u m , f o r the  life.  a-j a r e  for  . . . . (3.11)  3 l 1=1  =  state  constant  1  t h u s be e x p r e s s e d  steady  Table  =  t  a-j i s i t s e x p e c t e d  is  shown to c o n v e r g e  cost, E(C ),  1  ordinary  s t a t e when run  be  inter-renewal  E(C.)  An  equilibrium  duration,  (average  expected  As  cost  available  theory.  an e q u i l i b r i u m  of time.  function  of each renewal  is also  r e s u l t s of renewal  reaches  where t i s the  state replacement  theoretical  value.  data per  of  a  oL_i 0 Figure 7-  I  I  i  i  I  I  i  2  3  4  5  6  7  I  i  I  I  I  8 9 10 II 12 Transition Interval  i  I  I  i  i  '  13  14  15  16  17  18  '  '  19 20  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:  Increasing  Runs 2 t h r o u g h 6 i n v o l v e d tions f o rmultiple  The cost s t r u c t u r e s  and t h e model b e h a v i o r  briefly  increasing  dependence.  the cost  r e p l a c e m e n t s f r o m 10 p e r c e n t  cent, r e s p e c t i v e l y . III  economic  t o 50 p e r  a r e shown i n T a b l e  as a f u n c t i o n  o f them i s d i s c u s s e d  below. (a) P o l i c i e s .  First  failure  replacement policies are  shown i n A p p e n d i x E, f o r a l l s y s t e m age v e c t o r s age  i n which the  o f c o m p o n e n t 3 i s 5 o r 6. The  generally multiple  p o l i c i e s behaved with  changing cost  as w o u l d be e x p e c t e d - - i n c r e a s i n g replacements inducing  however, follow a consistent  function  o f t h e ages o f t h e other  own a g e .  cost  reduction  follows:  A case  i n point  The  t o be a  two c o m p o n e n t s , as w e l l  as  f o r t h e 10 p e r c e n t  i s ( 5 , l , 1 o r 2)  (b) i f t h e age v e c t o r  1 also i f 3 f a i l s ;  ( 5 , 2 t o 4, 1 t o 6) r e p l a c e  replace  i s ( 5 , 1, 3  ( c ) i f t h e age v e c t o r i s  1 i f 2 or 3 fails.  c h e c k was made on t h e v a l i d i t y  of this  No  theoretical  phenomenon and i t  have been c a u s e d by t h e a p p r o x i m a t i o n s  o f t h e mode 1 .  opportunistic  f o r w h i c h a segment o f t h e p o l i c y went as  1 also i f 2 or 3 f a i l s ;  could  replacements  i n some c a s e s  occurred  ( a ) i f t h e age v e c t o r  to 6) r e p l a c e  of  The p o l i c i e s d i d n o t ,  pattern.  replacement o f a component appeared  structure  cost savings  opportunistic  of i n c r e a s i n g l y younger components.  its  reduc-  and assumptions  (b) The  Expected expected  replacement replacement  costs. costs per transition 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 ment 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  decreased with increasing cost reduction. steady state values (averaged  this  value  In F i g u r e 8 the  over the l a s t 5 stages) are  shown as 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 . the nature of the convergence  of the replace-  F i g u r e 7 shows  f o r t h e z e r o a n d 40 p e r c e n t  cost structures. The  g a i n p e r stage had converged  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 by 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 multiple replacements-  CHAPTER  IV  D I S C U S S I O N AND I. The expected  CONCLUSION  DISCUSSION  r e s u l t s of the renewal  c o s t p r o d u c e d by t h e m o d e l  closely  approximated the  The  abstraction  obtained  respect  in the  and  quired  penalty  cent  the  t h e e r r o r s c a u s e d by  the  to actual use  failure  since  i n t o which the  of s h o r t e r  pdf's  age  intervals,  than  were  t h e n u m b e r o f s y s t e m age  with  pdf's  increase  computing costs  intervals. and  failure  used  However, t h i s improvement would  exponentially  be m a d e b e t w e e n t h e  o f age  hypothetical  error in  not  states  the number of  are d i v i d e d .  actual maintenance problem a trade  increased  model  amount of c o m p u t a t i o n - a n d c o m p u t e r memory  increases  intervals  per  t o t h e maximum c o m p o n e n t l i f e ,  thus the  o f an  through the  test problem.  be w i t h o u t  the  itself.  Improved approximations w o u l d be  form of the  1.5  r e s u l t s w o u l d be m u c h l e s s t h a n  c h e c k s on  i n d i c a t e d that the  abstracted  maintenance problem used.  with  theory  consideration,  and  induced  Of c o u r s e  cost data  in realism  the  by  final  solution  o f f would have attained  increasing  amount and  analysis  age  In t h e  and the  quality  a v a i l a b l e m u s t a l s o be t a k e n  in the  re-  the  age  to  the number of into  interval  m u s t be c h o s e n s o as t o p r o v i d e policy  basis  for  implementation. Further,  as t h e n u m b e r o f age  the f a i l u r e pdf's  and  i n t e r v a l s into which  are d i v i d e d i n c r e a s e s  c u l t y of determining (2.2)  a meaningful  the g ^ ( t )  so w o u l d the  subject  to  diffi-  constraints  (2.3).  Since  the  p r o b a b i l i t y of a given  f a i l u r e order  is  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 ^ , the  l i n e a r approximations  fitting  technique  estimates  of the  and  squares.  This  3 -| ^ -  m  n.  i =1  j=l  £  subject estimates  techniques  the  could  required be  used  d y n a m i c p r o g r a m m i n g and  [f(t.)-(a J  to (2.2)  of the  good  i t appears  ranging  i n t e r v a l , and  the  from  11  and  11  the a^.  ....  J  ( 2 . 3 ) , where the c a l c u l a t e d from  in the and  f ( t . ) are J  the  the d a t a ,  n^  interval, i is  3^  as  a  are  the  f o r k age  (4.1)  i t h ge  by  age  of l e a s t  solving  ?  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  taking  theory  + g . . t J ] S  defined.  the pdf  to o b t a i n data  be p e r f o r m e d by  f a i l u r e pdf  the number of e s t i m a t e s age  Given  latter could  L  min  the  a reliable  e r r o r g r a p h i c a l methods to more s o p h i s t i c a t e d  methods employing  is  and  m u s t be a v a i l a b l e i n o r d e r  as t h o u g h s e v e r a l trial  s u f f i c i e n t data  of  i n t e r v a l s as t h e s t a g e s , i n t e r v a l s , A, , a s t h e  previously to solve  the area  initial  (4.1) under  state  vector,  56  rAt  k - l "  A  as t h e  stage  (a  k "  A  coupling  +  u  e (t)dt) l k  f u n c t i o n , and  v a r i a b l e , h o w e v e r t h i s has  as t h e  not y e t been  For problems in which there w a r r a n t use  of the  used that  involves  the  of the  cdf's  then  above technique  d i f f e r e n t i a t i n g these  mations to the (2.2)  and  pdf's.  (2.3)  The  has  l i m i t e d data equal  to zero  algorithm,  with  . and  estimates  possibly  of  the  i n t e r v a l , and  linear  approxi-  of s a t i s f y i n g  constraints approach.  $^..  In c a s e s 3^  a few  of more where  could  of the  only  be  sophis-  very  assumed  used.  The  minor changes in the although  data  program-  many r e d u n d a n t  i f a l l the  3-j^ a r e  calcula-  zero  Since  t h e m o d e l was  cent,  f o r the  basis  same  occurring.  case  p l a c e m e n t s at l e a s t , the on  the  to  p e r m u t a t i o n s of a f a i l u r e sequence have the  probability  per  of quadratics  do n o t w a r r a n t u s e  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  1.5  be  been i n v e s t i g a t e d f o r t h i s  ming, would handle t h i s case  different  another method could  f o r e a c h age  a v a i l a b l e a l l the  and  to  m e t h o d w o u l d be u s e f u l w h e n t h e  of  are  enough data  question  required  ticated estimates  are not  to obtain  The  not  graphical  and/or solutions  done.  least squares f i t t i n g  f a i l u r e data,  decision  s h o w n t o be  accurate  to  about  of economic independence of  validity  of cost d i f f e r e n c e s  of a l t e r n a t i v e s of l e s s than  1.5  re-  selected per  cent  between  competing  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 .  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 m u c h l e s s t h a n 1.5 p e r c e n t . model  because  of differences  W i t h some m o d i f i c a t i o n t h e  c o u l d be 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  produce  a prespecified  improvement  produce  insignificant  change  c o u l d be i n c o r r e c t b e c a u s e replacement problem  this  that  i n expected cost over a  simpler a l t e r n a t i v e , thus s u p p r e s s i n g a l t e r n a t i v e s  would  In  that  i n e x p e c t e d c o s t and which  o f model  errors.  lower bound  In an  on c o s t  actual  improvement  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  data.  II. 1.  A maintenance  model  CONCLUSION has been  capable of handling systems failures 2.  assumptions  3.  i n which  component  are not simultaneous.  Other than Assumptions  failure  developed that i s  1 a n d 2, C h a p t e r I I , no  n e e d b e made a b o u t t h e f o r m o f t h e  p d f ' s , component replacement c o s t s t r u c -  tures, or replacement  policies.  Renewal  i n d i c a t e d t h a t t h e model  theory checks  estimated a l lexpected replacement costs to within  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  for the case of economic placements.  independence  values, of re-  Policies  p r o d u c e d by t h e m o d e l were r e a l i s t i c  but  f u r t h e r work i s r e q u i r e d  are  totally  f o r suppressing  could  ing  by a s s u m p t i o n s  be e a s i l y  Further  opportunistic  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  m e n t s o f t h e same o r d e r created  they  correct.  A system i s required policies  to ensure that  of magnitude i n the model.  as  improve-  errors  Such  a system  added.  work i s r e q u i r e d  on t e c h n i q u e s  the approximations to the f a i l u r e  of  obtain-  pdf's.  BIBLIOGRAPHY B a r l o w , R.E. and K.J. Arrow, Probability sity Press,  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 et a l . (eds.), Studies in Applied and Management S c i e n c e , S t a n f o r d U n i v e r S t a n f o r d , C a l i f o r n i a , 1962, pp. 63-87.  B e l l m a n , R.E. Equipment Indust. A p p l . Math., pp. 133-136.  Replacement V o l . 3 , No.  Policy. J . Soc. 3, S e p t e m b e r , 1955,  C o x , D.R. Renewal T h e o r y . S c i e n c e P a p e r b a c k s , from 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 , 1961, pp. 328-362. 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 Process. 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 , 1957. 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 and M a r k o v J o h n W i l e y a n d S o n s , 1 9 6 0 , 136 p p .  Processes.  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 Inspection of S t o c h a s t i c a l l y F a i l i n g Equipment, in 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 and Management 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 , 1962, pp. 184-206. R i f a s , B.E. Replacement Models, i n Churchman, et al . ( e d s . ) , I n t r o d u c t i o n to O p e r a t i o n s Research, John W i l e y and S o n s , . N e w Y o r k , 1 957 , pp. 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 R a n d o m 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 , Stanford University, Stanford, C a l i f o r n i a , August, 1965.  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  A(L)  - c u r r e n t age i n t e r v a l  o f L t h component  AF(L)  - age i n t e r v a l of stage  APR0B(NC0MP,INT)  - absolute probability of failure of e n t NCOMP i n a g e i n t e r v a l INT  B(NCOMP,INT)  - Beta c o e f f i c i e n t  of Lth component at beginning  of linear  compon-  approximation  to f a i l u r e pdf C(«)  - policy  output  variable  CHAR-(')  - output  D(»)  - policy  output  variable  E(«)  - policy  output  variable  ETF(NCOMP , INT)  - 1 minus component expected  literal  i n age i n t e r v a l  INT  F(«)  - policy  FN(I,0,K)  - optimal expected NSTAGE  output  time to f a i l u r e  variable replacement cost f o r  stages starting  w i t h age v e c t o r  FNMl(I,d,K)  - same as F N ( - ) b u t f o r ( N S T A G E - l ) s t a g e s  G(«)  - policy  I  - age i n t e r v a l  IMAX  - maximum age o f c o m p o n e n t L  IN  - program output  IND  - composite policy  J  - age i n t e r v a l  JMAX  - maximum age o f c o m p o n e n t M  output  variable of component L control  parameter  index  of component M  (I,J,K)  JN  - program  K  - age i n t e r v a l  KMAX  - maximum age o f c o m p o n e n t N  KN  - program  control  L  - number stage  of first  M  control  parameter  o f component N  parameter component  to fail  in a  - number 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 in a s t a g e : second 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  MARK  - program  N  - number o f t h i r d component t o be c o n s i d e r e d in a stage: t h i r d f a i l u r e , o r replacement with f i r s t o r second f a i l u r e  NCMPO(• )  - number o f component n o t r e p l a c e d w i t h fai1ure  NCMPR(• )  - number o f component r e p l a c e d w i t h fai1ure  NCOMP  - component number when f a i l u r e implicit  NFL  - number  NPOL  - number o f c o m p o s i t e poli cy  optimal  NREP  - number  replaced with  NSPOL(-)  - value of the e replacement  NSTAGE  - s t a g e number  PER  - policy  PNOFL(NCOMP , INT)  - 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 I N T  PPOL(L)  - number 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  PO(I,J,K)  output control  of failed  parameter  first  order not  component  o f component  o f Dynamic  correction  first  replacement failure  alternative  Program  factor  P l ( L , I , J ,K)  PT-.CA)  P10(L,I,J,K)  P  (A)  T  P1R0(L,M,I,J,K) P 2 ( L , M , I , J ,K)  P (A) f o r M replaced 2  Q  with  L  with  L  P20(L,M,I,J,K) P2R(L,M,N,I,J,K)  P  lmo< > S  P3(L,M,N,1,J,K)  P (A) f o r M replaced  SAVEM  saves values  of M  SAVEY  saves values  of Y  T  time v a r i a b l e  TMIN  minimum  U(NCOMP,INT)  alpha c o e f f i c i e n t of l i n e a r to f a i 1 u r e pdf  2  n  in probability  equations  o f $TEMP1 and $TEMP2 approximation  age i n d e x o f c o m p o n e n t L u s e d as in p r o b a b i l i t y computations  subscript  Y  same  as X f o r c o m p o n e n t  M  Z  same  as X f o r c o m p o n e n t N  $COST(IND)  expected  $ECOST(L)  optimal expected replacement cost f o ra l l possible t r a n s i t i o n s that begin with the f a i l u r e of component L  c o s t o f c o m p o s i t e p o l i c y number  $ETF(NFL,NREP , INT) - 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 N F L t h a t was I N T t i m e u n i t s o l d at f a i l u r e $TCOST(NSTAGE)  optimal expected cost f o r a l l possible f a i l u r e sequences  $TEMP1  expected  c o s t o f a l t e r n a t i v e e = 1/d = 1  $TEMP2  expected  c o s t o f a l t e r n a t i v e e = 2/d = 1  IND  C JC  DYNAMIC PROGRAMMING-MARKOV C H A I N C-O.ESJl!!iENT_^E_lLAC.Ejy.ERT PJ2LJLC_LES  ALGORITHM  F O R DETERMINING  OPTIMAL  DIMENSION F N I 5 , 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 ) 3),PP0L(4) ,C(3), 4. Di3L,.Ei3L,„E<3 3 (.3,„6>, 5 C O ( 6 ) ,APROB( 3,6) , C P R O B ( 3 , 6 ) , A F ( 3 ) , E T F ( 3 , 1 0 ) , $ E T F ( 3 , 3 , 1 0 ) I N T E G E R A, A F, X , Y DATA CHAR/'1* , ' 2 ' ,* 3' ,* 0 * , * R / DATA I N P U T  ,NCMPO(  ,Z,PPOL»SAVEM,SAVEY 1  C C  **  PROGRAM  CONTROL  PARAMETERS  96  FORMAT(412,F4. 2)  C  C O E F F I C I E N T S OF L I N E A R A P P R O X I M A T I O N S TO FA IL OR E P D F ' S READ ( 5 , 200? I MAX,JMAX,KMAX FORMAT(312) NC0MP=1. B_EADJJi^2111JJlJj^^ NC0MP=2 RE AD( 5 * 2 0 1 ) (U (NCOMP, INT ) , B( NCOMP, I NT ) , I NT = . 1 , JMAX ) 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 ) NC0MP=3 FORMAT(8F10.0)  200  201 C  r S nI j N ; GT L Ep _ T RA E P -L A iC E»M, MN E N T A S R F COMPONENT  c  *  C  '*  DOUBLE  c  *  T R I P L E  R E A D ( 5 , 2 0 1 )  <C1(L)  :  .  NUMBERS  , L = 1 , 3 )  REPLACEMENTS  R E A D ( 5 , 2 0 1 ) ( ( C 2 ( L , M ) , L = 1 , 3 ) , M = 1 , 3 )  c  REPLACEMENTS  R E A D ( ^ , ?01  )  CALCULATE  T H E ABSOLUTE  AGE  INTERVAL  < ( L . C 3 ( L tJL,M FOR EACH  L» 1 = 1 , 3 } , M= 1 , 3 ) , N A N O CONDITIONAL  1,3)  PR08S  OF  F A I L U R E  IN  EACH  COMPONENT  MAX(1)=IMAX  MAX!" 2)-=JMAX MAX ( 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 MAX 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(  NCOMP, 1)  C P R O B ( N C O M P , 1 ) = A P R O B ( N C O M P , 1 ) DO  4 0  INT=2,MAX1  IM=TNT-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 CPROB(NCOMP,I 40  CONTINUE  c  CHANGE  DO  ABSOLUTE  C O E F F I C I E N T S  ) - A P R O B ( N C O M P , I M )  NT)/PNOFL(NCOMP,INT)  TO  CONDITIONAL  C O E F F I C I E N T S  4 3 NmMP=l ,3  MAX 1= MAX DO  NT)=APROB(NCOMP,I  4 3  (NCOMP)  INT=1,MAX1  U(NCOMP.  I N T 1 - I H NCOMP. INT) / P N O F I  (NCOMP,I  NT)  J  B ( N C O M P , I N T ) = B< N C O M P , I N T ) / P N O F L ( N C O M P , I N T ) CONTINUE 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 WRITER,107) 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 * » / 107 1 T 1 0 , ' D U R I N G G I V E N AGE INTERVAL //) WRITE(6,108) FORMAT ( T 3 3 , ' A G E I N T E R V A L ' , / 108 i T i n « r n M P . Nn. 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 , <APROB(NCGMP, INT) * INT=1.MAX1 ) FORMAT(1 2 X , 1 1 » 4 X , 1 O F ? . 3 ) 109 33 CONTINUE W R I T E fh.1 I D ) FORMAT!1H0) 110 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 WRITF(6,1I 1 ) 111 F O R M A T I T 1 0 , ' 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 FAILURE',/ 1 T l O t ' D U R I N G G I V E N AGE INTERVALS//) WR I T F ( fS » 1 C 8 ) DO 3 4 N C 0 M P = 1 » 3 MAXl=MAxtNCOMP) WRITE ( 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 ) 34 CONTINUE 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 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 , INT} = ! . - ( . 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 NT) 1 + . 5 * B ( N C O M P , INT) ) CONTINUE 44 43  1  )  <  r  T  c c  45  c  C  c 21  ??  23 24  CAI f.UI A T F T H F 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 O M P O N E N T S WHEN R E P L A C E D W I T H A N Y 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 . I N T 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 ) CONTINUE T=l . 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 F O R A L L A G E V E C T O R S * SET L E V E L 1 F A I L U R E DO 2 0 1 = 1 3 * S E T L E V E L 2 AND 3 F A I L U R E S GO TO {21,22,23),L M=2 N=3 GO TO 2 4 f  M=1  N=3 GO TO 2 4 M= l N=2 CONTINUE DO 2 0 K = 1 , K M A X A(3)=K DO 2 0 J = 1 , J M A X A(2)=J  J  DO  I=  2 0  A { 1 ) =  MAX  1 , 1  I  X=A(L.) Y=  C  »»  C  **  A C M )  Z=A(N) C A L C U L A T E 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 A L C U L A T E PRO 8 OF TRANS TO GAMMA = L . . CJQ±JS1=J3JL1JJ..^^ COi5) = - 1 * { U ( L , X ) * B ( M , Y ) * B ( N , Z ) / 2 . + B < L , X 1  C O ( 4 ) = . 2 5*(U( 1  3 * ( U ( M , Y ) * B ( N , Z ) + B ( M , Y 3 *  U ( N , Z > ) )  + B ( N , Z ) COO)  t,X)*t U(M,Yi*B(N,Z3-+BfM,Y)*U(N,I3)/2.-B(L,X)»(  )/2.-U(M,Y)*U(N,Z  333*<U( L,X)*((B(M,Y) + B < N , Z 1  •=-.  (8(M,Y)  ) > ) ) / 2 .-U  (M , Y )*IH  N , Z)  ) +  L_MJ^JLl*IJJLLl!^^  :  C0(2)=-.5*(UlL,X)*<IMM,Y)+U(N,Z))-B(L,X))  c  CO*1) F O R M  *  = U ( L , X ) P O L Y  P0LY=0. DO  2 5  I N D = 1 , 6  N 0 = 7 - I N D  POLY=(POLY  2 5  +  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  TO  G A M M A = L 0  FOR  0 = 1  C 0 ( 6 ) = 8 ( L , X 3 « B ( M , Y 3 * B ( N , Z ) / 8 . C 01 ^JL  53=  . 2 5 * ( U ( L , X) * 8 ( M , YI • B < N , Z) + 8 ( L , X) * U ( M, Y) * 8 (  +JJJLji).*BJJ^jL^ L*UXN„, ZJLL C0(4) = .5*<U(L,X?*U<M,Y)*B<N,Z) + 1  + +  B ( L , X  C O ( 2 3  =  _£DJJL) C  *  Z ) __•  _  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 1  N,  _ . U ( L , X ) * B ( M , Y ) * U ( N , Z )  * U < M , Y ) * U ( N , 1 3 - { U ( L , X 3 * B ( N , Z ) + U ( L , X ) * B i M , Y )  ) * U ( N , Z 3 + B J L , X 3 * U ( M , Y ) ) / 2 .  B ( L , X ) / 2 . - U ( L , X ) « U ! N , Z ) - U ( L , X ) * U C M , Y )  = U J J U - X J  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 P O L Y = ( P O L Y  3 2  (1  PTO DO  2 7  +  CO ( 6 )  Y  t  1 1 = 1 , 2  C A L C U L A T E  C  C 0 ( N 0 ) 3 * T  T . l , K -) = Pni  T  P R O B  OF  =— B ( L , X ) * 8  T R A N S  ( M , Y ) *  TO  B ( N , Z  GAMMA=LM.  0=1  FOR  3 / 2 4•  C 0 ( 5 ) = - . l * ( U ( L , X ) * 8 ( M , Y ) « 8 < N , Z ) + B ( L , X ) * ( U ( M , Y  1 1  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 , ( 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  C 0 ( 2 ) = U ( M , Y ) « U ( L , X C 0 (  c  *  ) * B ( N , Z ) / 2 . + B ( M , Y ) *  U(NiZ)J)  3 / 2  ±Bi M , . Y 1  *UJJ5ULL1+_.  5±MLJ,JU *  3 * ( U < M , Y ) * U ( N , Z 5 - B ( M , Y ) ) 3  .  1.3=0.  F O R M  P O L Y  POIY=0-  DO  2 6  IND=1»6  M 0 = 7 - I N D P 0 L Y = ( P 0 L Y  2 6  P 2 ( L , M , I  C  **  +  C 0 ( N 0 ) 3 * T  , J , K ) = P G L Y  C A L C U L A T E PROB OF _..C.0J..6J_=J1.(.JU^^ C 0 ( 5 ) = . 2 * ( U ( L , X 1  +  m i  TRANS 3  *B  (  M, Y  TO 3* 8  GAMMA=LMN ( N, Z ) /  FOR  3 . + 8  D  AND E=l „ ( L , X1 * U ( M, Y  3  _ *B(N , Z  . 3 / 6  .  .  B ( L , X 3 * 8 ( M , Y 3 « U ( N , Z ) / 8 . 3 4)=.25»(Bf  I  . X )  *Uf  W,Y>*t)f  7)  /6.+\Ui  .X3»l)f M . Y  ) » B ( N . 7 ) / 2 .  ;  J  1 + UIL»X)*B(M,YJ*U(N,Z)/3.) C0( 3) =UC U X ) * U < M t Y ) * U I N Z ) /6. * FORM POLY PGLY=0. OO 28 lND=l,4 f  C  28  H U - I—  <  i 1111  POLY=(POLY+COtNO))*T P3(L,M,N, I,J,K )=POLY r * * r . A i r . U I A T F 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 Z)/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 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 K ) = 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 Y)/30. + B(L,XJ*U(M,YJ*B< N , Z ) / 1 5 . 1 + B( L,X) *8<M, Y) *UIN,Z) MO. ) t  ,X±Lh-..  :  t  t  C__LL__J___3_LI^^  1 U(L,Xl*B(M,Y)/2.)/12. C0<3)=(U(L X)*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 3 0 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 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) = . 2 5 * < U ( L , X ) * U ( M , Y ) " B ( N , Z } - 8 ( L , X ) * 8 ( N , Z ) - B ( L , X ) * B ( M , Y ) / 2 . ) t  .  _____  f  I +  C  _-l C  21  .3 3 3 3 3 3 » ( B H  . 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^^ C 0 ( 1 I = U(L,X) * FORM POLY POL.Y=0 . DO 31 IN0=1,6 N0=7-I ND pni y=(pni Y + c n t N O i > » T P1RO(L,M,I,J,K)=POLY ** 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 HOMTTMHF  :  ,  ..__ )  z*  o UJ Qi UJ  z  x  tSi  _J (/! — >•  U-  LL  LU >  00  <  <L  2C <  o i  a  a o c  IS> UJ  r - » X  I—  UJ  LU  o  D  LL  < -  UJ  CL  o <  o  < o  ci —  a  a  LU  O  LU ^  2:  X  Oi  LL! 00  ST O O  ZJ —  C H — s  CL  UJ  cc o z  u  00  o  LU  r  o  LL  * ^  <  LL r  C  r-i  -  CC  c ->  ac co O UJ O UJ z  a I-* • z t— < Oz X 1  _  X  c NO  UJ < UJ X  ) c  a C  1  _  3  a: o  —1 O  r  s  j  s C  UJ  LU  I—  -  + + X X <r < x  s: - 5  •Tr  •K-  11 11  Xi  s:  x < s:  LU  x <  < LL  LU  i  f  II  Qi . ro  s  -1  0 0 0 Z UJ c 0 0 0 u. 0 c  LU Z  LU CD  LL  ro  o  z  <r  LJ  a:  c-  a  II  II  ~* ro 11 — > ro m H H N H CM w l  <  i  t  Q  C  ~| U.  <. <  O  LU LL X !Z _  O  LU <  00  <3 II S — s — ' (— (M Z > _J o 11 C a  CM r c O l l II CD X z  r - j O  ll  o s:  o co  I  I  2:  — u  z u  < <  •ft-  o U N O  CO  O  O  .—I  00 O  OO  o  Q  X  Z  — z -  l  < LL CMUJ 00  Z  X <  Di"  u  z  O  s: +  >v <~ » H ^ H  +  r-i  >~l  *  00 Q r-  O  II  a  » X  <  <•  »  _ l  +  Z 01 LU H _J  CM CL — I! <  <t  CL  w  r—I  LU <— r- < I I  r-l  r-l  s  o. Z S LL LU  s— r  CO «sx CX. • CL «~  .  (-  s: cc w  1*. — .  - J CM  >  O  +  -<  —  II • + LU -7 — 00 LU H V  o  Oi  r-l  i  • ro l  s a;  f-1 r - l  +  cj  i  + -  11  LU  —  »•  0 # <  O  rH < t  r  i  nor  LL  < a  r-i  r-  + r  — LL CM — O  I I  *  w  s: o + <C o  r —  LL hLU  s;  l  ro LU cc <5 X  a —  l  LU >  +  LU  r-l X  LU OO  00  < Qi • CO  o  LU LU > > 00 LU LU _ ! CM O  II  ~  00  _ l  LU 00 LU • K Z < < Qi  LL —  a.  ro CM  *C 00 <0  II X  LU Oi  a  LU  <—- 00  r  00 O 01 ~ -  II 00 LL >— H- CC </)  00 CO CO II  o z  r  oa s: s: s: i«c -5 t- II <  <  2:  CM  • LU  X X X  LL  - a. •— o •» r- x O X LU  X  o  H-I  c  I-  00  LU CD IP <  z  t o «-  a  < H  cc  J  LL  +  o  z o  _J « • r-l + <[ « ~  LU Oi  LU  cc  <  H  z  Q  X> LU  00  LU  O  LU LL  s:  z  <r.  Oi  Q  OJ  Oi  a.  LU  z  >  1  00  LU >  LL <  -J  LU Oi  O  M  < z  o a  I—I  O0  X> —  +  o  z  3 —  LL — . O — Z LL <  Z CC  Q i  o  O  c  LU  00  l— +  LL O  CM  00  < t-  x a.  O  < < LL 00  s:  a.  o  oi O LU i - t  OO  CD D  - 5  LL _J  CD  UJ LLI CC CD Z) <  CL  ^ r  X  - 5  >  s  01  LU  X <  tt  UJ  o  - J  z  00 «"»  H ~  >- LU Z 0C r - LL  +  r - l  f  C  ** TRY ALTERNATIVE E=2/0=1 $TEMP2 = P 2 ( L »M 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) ) * CHOOSE B E S T OF TWO SECONDARY P O L I C I E S TMIN=$TEMP1 f  C  NSP0L(L,M)=1  I F ( S T E M P 2 . G E . T M I N * P E R ) G O TO 2 TMIN=$TEMP2 MSPnt f l . M ) = ? CONT INUE ADD COST N S P O L ( L , M ) TO $ C 0 S T 1 4 C 0 S T I 1 3 = $ C 0 S T ( 13 +TMIN EXCHANGE M FOR N <MAKE OTHER COMPONENT SECOND F A I L U R E ) AND RETURN A{M)=AF(«) A { M )=AF {N ) SAVEM=M M=N N=SAVFM CONTINUE TRY A L T E R N A T I V E D=2 nn 3 TNQ=?, 3 A(M)=l FORM E X P E C T E D COST OF P O L I C Y N O . IND $ C O S T ( IND3 = Pl(L,I,J,K1*(C2(L,M)+$ETF(L,L,AF(L))+$ETF(L,M,AF(L)))  2 C C  1 C  c  *V -»*  1 + P1R0(L,M,I,J,K)*FNM1{A(1)+1,A<2J+1,A<3)+1) A{ N) = 0 S C n S T ( TND ) = .$CnST ( TND) +P?R (1. ,M , N , I , J , K ) *( C 1 ( N) + . t F T F ( N N , AF ( N) ) / 4 . 1 + F N M H A ( 1 3 + 1 , A{2)+1.,A(3)+1) 3 S A V E N O . OF R E P L A C E M E N T AND SECOND F A I L U R E NCMPR(IND)=M NCMPO( I N D ) =N * EXCHANGE M AND N AND RETURN A ( M) = AF{M) A ( N ) = A F < N3 SAVEM=M M=N N=SAVEM CONTINUE f TRY Al TFRNAT IVF 0=3 t  c c  3  c_  iA> •A.  $C0ST(4)=P1(L I ,J,K)*{C3(L,M,N3+ $ E T F ( L , L , A F U ) 3 + $ETF(L,M,AFIL)) 1 + $ETF(L , N , A F ( L 3 3 + F N M K 2 , 2 , 2 ) 3 . S E A R C H FOR L E A S T COST P O L I C Y NPOL=0 $ECOSTIL 3 = 1 0 . E 1 0 n n 4 TND=1.4 IF{$COST(IND).GE.4EC0STIL3)G0 TO 4 $ E C 0 S T t L )=$COST('IND) NPOL=IND CONTINUE IF STAGE . G E . MARK CONSTRUCT OUTPUT FOR NPOL ANO L I S T f  C  4 C  A<1  c 5 r.  aj; sj?  )=AF{|  )  I F ( I . N E . I N . O R . J . N E . J N . O R . K . N E . K N ) GO TO 12 BRANCH TO A P P R O P R I A T E OUTPUT ACCORDING TO NPOL PPOL(L)=NPOL GO TO ( 5 » 6 , 6 t 7 3 , N P O L CONTINUE P01 TOY OUTPUT T F D= 1 C ( L 3 = C,HAR( 1) C(M)=CHAR{4) r.(N)=r.HflR(4)  LU  Z  a  X  __  X  00  ro CM  U-  z  XI  LU  a  z  11J  >  <r  I—1  (N!  r-  <  Ct  CC LU  LU HJ <r  X  <  oc  UJ ct  < I  _>  u II  _>  Z r— LU O Ct  LU > LU  — (Ni  Z • <  (V  Z  LU  o  LU  — — — 3C —-,• cvi -j-  _> Ct  Qt  •>  H  ct — _;  _<-<<_  I— X X X O Oi s: _> u o c_ a. on n C II II II 00 Ct I—I  uu _  — — z. s: z > LU  LU •-• — — U_ LU 00 Q O Q H O C  > Z oo < II II oo X Z  (M >t  Ct Ct Ct < < <t X X. X ou o  II II  —J UJ  — UJ  Z  - <  —- SC -J  UJ  a <_>  (•) < Z _J  z —  a  LL LU LU ct  Lz  LU -—I  a _ c — c  z  ct  O  c  u  —  2  (M rn < _> scCt ct _J z _c Cu < <x <t X X t~ X CJ (_> LL LL _> o II II o II II _ z LU —< SC z 00  LL  u o >-i  I  --I  • X  <N  O vO — a — \~ LL! <J (~ r - S 00 l - l Ct  O LD w _ c J 3. LL  LL LL  a. _J • o  O i Cv  u  _> —1  <  CJ  O  K>  O  •  — — <r  I  UJ  (Ni  UJ  II D  >  <  •» <t <  —  •  —i  <£ *>. —. .  LL  CNJ -  o  •  < -  « m' H •> o  x  r—I  •- CM  r-l  o — <r — t- • <j  CN  LO <f LL  ITi <N  Ct Ct (Ni  Ct ctj < <  < < X X U  II II >  <t  O O  u-  a. CL  a  U- CNI  o  CL  x — 3  o  J J  j  •—i o ct  3  CL CU  <  m n a  C  a  < <  CJ <  LL —  a r-l  Q-  Ct 3 X. CJ a.  __ O (_>  X  <  a r-H  - rn • o X 1  LO >  ct  _  cg *• (Ni o _U J <£ _> X — - v0 •—I ~> H- « >- Z  l_  < LU O CJ || s: r - (- ^ J ~* a z z t/) t-H P—'Ct _ i o o w" — —- •—• UJ j z ct et o a c CJ CJ oo Q a c a, ~ 3LJ  o  -  •d-  I! II  LU  _J —I Ct — o  _> CL  X  I X CJ CJ  -J CJ LU  .  X LU  z < cn ct a SC _> i— „ : •4- LU i— >-! Ct — t~ UJ a Ct O (\J p—i ct o Ct 3: LL  UJ  (NJ  CJ  Ct  I—I  _  <_>  LLI  LL  UJ >  — o SC z  O  X  LL  LU  Q  LU Ct CD  IN  < X  II ct  —  LU Ct  Ct  CJ  CJ  X  cn  II sc  i—•  —• c ct •- _• on < - o o X c r-l X CJ Q.  Z (/ o  »•  • > (N  vO vO — UJ  w |_ UJ  <  K-M a CC CC o  3C S  -a cr  C  CJ  o  o  r-  CJ  o  C  ***  WRITE(6,103)A,A,A,A,A RETURN FOR NEXT F I R S T  FAILURE  12 CONTINUE C * * * F O R M F N < I , J ,K ) 15 $TCOST(NSTAGE)=0. DO 1 4 L = l ,3 , 14 S T C O S T f N S T A G E )=$TCOST(NSTAGE) C * I F COST NO F A I L U R E E X I S T S ADD C  ***  106  C  ***  13 C  S£JL_EJ^J^_J^Jll_EiLj^ F N ( I t J , K ) = S T C O S T ( N S T A G E ) + P O < I ,J , K ) *FNM 1 ( I+ 1, J + 1 , IFU.NE.IN.OR.J.NE.JN.OR.K.NE.KN) G O T O 13 W R I T E ( 6 . I0 6 ) N S T A G E , A , F N ( 1 . J . K ) F O R M A T ( 2 X , ' N O . OF T I M E U N I T S S Y S T E M TO R U N 1 2 X , ' P R E SENT AGE VECTOR OF S Y S T E M 2 7 X . ' FXPF'r.TFD C O S T OF F O L L O W I N G AROVF P 0 1 I C Y RETURN  161  1.62  FOR  NEXT  CONTINUE BLOCK DATA S H I F T DO 1 6 K = l , KMA X DO 16 J = 1 , J M A X nn  16  + SECOSTU.) IN H E R E  16  AGE FOR  K+I)  = = =  ',12,/ ',312,/ $'.F8.0.///)  VECTOR K+l-STAGE  PROCESS  I = I . T M A X  F N M K I , J ,K) = F N ( I , J , K ) DO 161 K=1,KMAX DO 1 6 1 J = 1 , J M A X F N M K I M X , J , K ) = F N M 1 (1 , J , K ) + C 1 ( 1 ) DO 162 K=1,KMAX  163 C ***  F N M K I , J M X » K ) = F NM K I , 1 , K ) + C K 2 ) DO 163 J=1,JMAX DO 16 3 1 = 1 , IMAX F N M K I , J , K M X ) = F N M K I , J , 1 ) + C K 3 ) R E T U R N FOR NEXT STAGE  Al  CMHIOUS-  444  STOP END  .  _  .  •_  APPENDIX B  S A M P L E PROGRAM O U T P U T  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 D U R I N G G I V E N AGE I N T E R V A L  AGE COMP.  NO.  INTERVAL  0.200 0.450 0.250  0 .300. 0 . 300 0 . 19 4  0 .300 C.05C 0 . 139  SUMMARY 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 DURING G I V E N AGE. INTERVAL  AGE COPP. 1 2 3  v.  NO.  FAILURE  1 C. 100 C . 200 0.306  1 2 3  CF  1 0.100. C.200 0.306  2 0.222 0.562 0.360  INTERVAL 3 4  0.429 0.857 C.437  0 .750 LOCO 0.556  0. 1 0 0 0 . C83  OF  0 . .0 2 8  FAILURE  5 .1.000 C.750  6 _._ I.000  OPTIMAL  COMPONENT  _ *  >FIRST  —#  FAILURE  RFPLACFMENT  POLICIES  -)J:  SECOND  >|c_  FAILURE....  if —  £  TH I RO.. F A I L UR E.  CCMP NC FAIL/POL AGE VECT  1 2 3 1,0,0 1 I 1  1 2 3 1, 2,0 1 1 1  1 2 3 1,0,2 1 1 1  1 2 3 1,2,3 1 1 1  1 2 3 1 , 3, ? 1 1 1  COMP NO FAIL/POL AGE V E C T  I 2 3 0,1,0  1 1 1  1 2 3 2, 1 , 0 1 1 1  1 2 3 0,1,2 1 1 1  1 2 3 2, 1,3 1 1 I  1 2 3 3, 1 ,2 1 1 1  CCMP NO FAIL/POL AGE V E C T  1 2 3 0,0,1 1 1 1  1 2 2 2,0,1 1 1 1  1 2 3 0,2, 1 1 I .1  1 2 3 2, 3, 1 1 1 1  1 3 3, 2, 1 1 1 1  K G . OF T I M E U N I T S S Y S T E M T O R U N P R E S E N T A G E V E C T O R OF S Y S T E M E X P E C T E C COST OF F O L L O W I N G ABOVE  COMP NO F A IL/POL AGE V E C T  1 2  2  1 1  CCMP KO FAIL/POL AGE VECT  1  2 3  COMP NO FAIL/POL AGE V E C T  3 1 ,0,0  0,1,0  2 1 1 1 2 3 0,0, 1 2  1 1  - 15 = P0LICY=  1 $  1 1 43400.  1 2 3 1 ,2, C 2 1 1  1 2 3 1 , C 2 2 1 1  1 2 3 1,2,3 2 1 1  1 2 3 1, 3 , 2 2 1 1  1 2 3 2,1,0 2 1 1  1 2 3 0,1,2 2 1 1  1 2 3 2,1,3 2 1 1  1 2 3 3 , 1 ,2 " 2 1 1  1 2 3 2,0, 1 2 1 1  1 2 3 0,2, 1 2 .1 1 ...  1 2 3 1 2 3 2,3, 1 3, 2, 1 2 1 1 ... . 2 1. I  N C . O F T I M E U N I T S S Y S T E M TO R U N P R E S E N T AGE V E C T O R CF S Y S T E M E X P E C T E D C O S T OF F O L L O W I N G A B O V E  = 15 = 2 POLICY=  $  1 1 43620.  CCMP NG FAIL/POL AGE V E C T  1 2 3 1 ,0,0 3 1 1  1 2 3 If 2 , 0 3 1 1  1 2 3 1,0,2 3 1 1  1 2 3 It 2 , 3 3 1 1  1 2 3 I t ?t 2 3 1 1  CCMP NC FAIL/POL AGE VECT  1 2 3 0,1,0 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  CCMP NO FA I L / P O L AGE VECT  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 P R E S E N T AGE  U N I T S S Y S T E M TO R U N VECTOR CF S Y S T E M  15 3  1 1.  APPENDIX C RESULTS OF NUMERICAL INTEGRATION OF EQUATION ( 3 . 9 ) AND CALCULATION OF E ( N ) FOR COMPONENTS OF HYPOTHETICAL SYSTEM T  Component No. 1 Real(l-F (T)) r  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 .-21  + C  .1053007-10 -12 .1272445-10 .5777496-10 .8848199-10° .9796745-10 .9974457-10 .9997573-10° .9999821-10° ,9999993-10° C  C  C  c  -6.56667287 8.00000000 1.  3.43332713  Component No. 2  Imag(l-F (T)) r  -.749771-10 •17 -.396556-10 •17 .137536-10 -15 .712346-10' •15 .111529-10' •14 .138178-10' •13 .707037-10 -13 -.964214-10 •13 -.981097-10' -12  Rea1(1-F (T)) r  .7303263 .1013889 .1969723 .1895046 .5365838 .8259798 .9554641 .9918122 .9988676 .9998780 .9999897 .9999996  -14 10 10 -3 10 -1 10 o 10 10° 10 10° 10° 10° 10° 10°  -7.51787807 11.00000000  _2.  5.48212193  c  c  Component No. 3  Imag(l-F (T)) r  .712067 .344683 .345893 .140480 .230250 .236828 .391999 .330739 .110722 .154300 .138187 .412040  10 -18 10 -17 10 -18 10 •16 10 -16 10 -15 10 -15 10 -15 10 -14 10 •14 10 •13 10 -13  Real(1-F (T))  Imag(l-F (T))  .2978908 10 -8 .2057618 10 -2 .6156081 10 -1 .2378845 10° .4828113 10° .7049503 10° .8559559 10° .9387073 10° .9769055 10° .9921936 10° .9976077 10° .9993300 10° .9998279 10° .9999599 10° .9999923 10° -10.24974444 14.00000000 _L 4.75025556  .414999 .480331 .159347 .161989 .111888 .639291 .490498 .223913 .784574 .175167 .471971 .337551 .228850 .907031 .357435  r  r  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  75 APPENDIX 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 STRUCTURES  COST  ADVANTAGE  STAGE  10%  20%  30%  40%  50%  0 1 . 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  1565 2561 3034 3084 3014 3000 3014 3009 3007 3019 3015 3009 3014 3014 3013 3012 3014 3013 3013 3013 3013 3013 3014 3013 3013  1565 2561 2955 2954 2977 2943 2929 2944 2952 2944 2961 2952 2953 2955 2954 2955 2967 2956 2975 2967 2970 2973 2972 2977 2973  1493 2457 2755 2773 2811 2743 2748 2764 2763 2761 2761 2761 2769 2766 2778 2777 2781 2788 2782 2784 2784 2785 2786 2786 2786  1348 2246 2494 2549 2557 2495 2511 2522 2520 2516 2517 2518 2519 2518 2524 2528 2522 2524 2524 2524 2524 2525 2524 2525 2524  1187 1992 2243 2267 2277 2230 2243 2250 2248 2246 2247 2247 2248 2247 2248 2248 2264 2250 2253 2253 2253 2253 2254 2254 2254  76  APPENDIX 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 A G E C 0 M P. 1 1  AGE COMPONENT No. 0% C O S T  FAILURES  3 = 5  REDUCTION  10% COST  A G E COMP. 2  REDUCTION  A G E 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  AGE COMPONENT No. 3 = 6 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  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  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  1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 0,1 ,0 R,0,1  1 ,0,0 0,1 ,0 0,0,1 1 ,0,0 R J ,0 R,0,1  1 ,0,0 0,1 ,0 R.0',1 1 ,0,0 R , l ,R R,0,1  1 ,0,0 0,1 ,0 R,0,1 1 ,0,0 R , l ,R R,R,1  4 5  77 APPENDIX E  A G E C 0 M P 1  (continued)  AGE COMPONENT No. 3 = 5 20% COST  REDUCTION  30% COST  A G E COMP. 2 1  2  3  REDUCTION  A G E COMP. 2 4  1  2  3  4  1  1 ,0,0 1 ,0,0 1 ,0,0 1 ,0,0 0,1 ,0 0,1 ,0 0 J ,0 0 J ,0 0,0,1 0,0,1 0,0,1 0,R J  1 ,0,0 1 ,0,0 1 ,0,0 1 ,0,0 0 J ,R 0 J ,R 0 J ,R 0 J ,R R,0,1 R ,0 J 0,R J 0, R J  2  1 ,0,0 1 ,0,0 1 ,0,0 1 ,0,0 0,1 ,0 0,1 ,0 0,1 ,0 0 J ,0 0,0,1 R ,0 J 0,R ,1 R,0,1  1 ,0,0 1 ,0,0 1 ,0,0 1 ,0,0 0 J ,0 0 J ,R 0 J ,R 0 J ,R R ,0 J R ,0 J 0 ,R J 0,R J  3  1 ,0,0 1 ,0,0 1 ,0,0 1 ,0,0 0,1 ,0 0,1 ,0 0 , 1 , 0 0 J ,0 R,0,1 R,0,1 R,0 J R,0 ,1  1 ,0,0 1 ,0,0 1 ,0,0 1 ,0,0 0 J ,0 0 J ,0 0,1 ,0 0 J ,0 R ,0 J R ,0 J R ,0 J R ,0 J  4  1 ,0,0 1 ,0,0 1 ,0,0 1 ,0,0 0,1 ,0 R , l ,0 R J ,0 R J ,0 R,0,1 R,0,1 R ,0 J R,0 ,1  1 ,0,0 1 ,0,0 1 ,0,0 1 ,0,0 0 J ,0 R J ,0 R J ,0 R J ,0 R,0 J R,0 J R ,0 J R ,0 J  5  1 ,0,0 1 ,0,0 1 ,0,0 1 ,0,0 0,1 ,0 R J ,0 R J ,0 R J ,0 R,0,1 R ,0 J R,R J R,0,1  1 ,0,0 1 ,0,0 1 ,0,0 1 ,0,0 0 J ,0 R J ,0 R J ,R R J ,R R ,0 J R ,0 J R ,0 J R,R J  1  AGE 1 ,0,R 1 , 0 , 0 1 , 0 , 0 0,1 ,R O J ,R 0 J ,R 0,0,1 0,0,1 0,0,1  C 0 M P 0 N E N 1" 1 ,0,0 0,1 ,R 0 ,R J  No. 3 = 1 ,0,0 0 J ,R R,0 ,1  6 1 ,0,0 1 ,0,0 1 ,0,0 0 J ,R 0 J ,R 0 J ' , R R ,0 J 0,R ,1 0,R J  2  1 , 0 , 0 1 , 0 , 0 1 ,0,R 1 ,0,R 0,1 ,R 0 J ,R 0,1 ,R 0 J ,R 0,0,1 R,0 ,1 0,R ,1 R,0,1  1 ,0,R 1 , 0 , 0 1 , 0 , 0 1 , 0 , 0 0 J ,R 0,1 ,R O J ,R 0 J ,R R ,0 J R,0 J 0,R J 0,R J  3  1 ,0,R 1 , 0 , 0 1 , 0 , 0 1 ,0,R 0,1 ,0 0 J ,R 0 J ,R 0 J ,R R ,0 J R, 0 J R.,0,1 R,0,1  1 ,0,R 1 , 0 , 0 1 , 0 , 0 1 , 0 , 0 0,1 ,R 0 J ,R 0,1 ,R 0 J ,R R ,0 J R ,0 J R ,0 J R ,0 J  4  1 , 0 , 0 1 , 0 , 0 1 ,0,R 1 ,R,R 0,1 ,0 0 J ,R R J ,R R J ,R R,0,1 R,0 J R,0 J R,0 ,1  1 ,0,R 1 , 0 , 0 1 , 0 , 0 1 ,R,R 0 J ,0 R J ,R R J ,R R J ,R. R ,0 J R ,0 J R ,0 J R,R J  5  1 , 0 , 0 1 , 0 , 0 1 , 0 , 0 1 ,R,R 0,1 ,o R J ,R R J ,R R J ,R R,0,1 R,0 ,1 R ,0 J R,R J  •1 ,0 ,R 1 , 0 , 0 1 , 0 , 0 1 ,R,R 0 J ,0 R J ,R R J ,R R J ,R R ,0 J R ,0 J R,R J R , R J  78  APPENDIX E A G E C 0  A G E COMPONEN T N o . 3 = 5 50% COST REDUCTION 40% COST REDUCTION AGE COMP. 2  M  P 1  (continued)  1  1,0,0 1 0,1,R R,0,1 1,0,0 2 0,1,R R,0,1 1,0,0 3 0,1,R R,0,1 1,0,0 4 0,1,0 R.0,1 1,0,0 5 0,1,0  AGE COMP. 2  2  3  4  1  2  3  4  1,0,0 0,1,R 0,R,1 1,0,0 0,1,R R.0,1 1,0,0 0,1,0 R.O.l 1.0,0 R,1,0 R.0,1 1,0,0 R.l.R R.0,1  1,0,0 O.l.R 0,R,1 1,0,0 0,1,R O.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 O.l.R O.R.l 1.0,0 0,1, R O.R.l 1,0,0 0,1,0 O.R.l 1,0,0 R.l.R R.R.l 1,0,0 R.l.R R.R.l  1,0,0 0,1 ,R R,0,1 1,0,0 O.l.R R.0,1 1,0,0 0,1 ,R R.0,1 1,0,0 O.l.R R.O.l 1,0,0 0,1,0 R.0,1  1,0,0 0,1,R 0,R,1 1,0,0 0,1,R R.0,1 1,0,0 0,1,R R.0,1 1,0,0 R.l.R R.0,1 1,0,0 R.l.R R.0,1  1,0,0 O.l.R O.R.l 1,0,0 0.1 .R O.R.l 1,0,0 0,1,R O.R.l 1,0,0 R.l.R R.R.l 1.0,0 R.l.R R.R.l  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.R.l 1.0,0 R.l.R R.R.l 1,0,0 R.l.R R.R.l  1,0,0 O.l.R O.R.l 1,0,0 O.l.R O.R.l 1.0,0 R.l.R R.R.l 1,0,0 R.l.R R.R.l 1,0,0 R.l.R R.R.l  1,0,0 0,1 .R O.R.l  AGE COMPONENT No. 3 = 6 1,0,0 1 0,1,R R.0,1 l.O.R 2 0,1,R R.0,1 1,0,R 3 0,1,R R.0,1 1,0,R 4 0,1,0 R.0,1 1,0,0 5 0,1,0 R.0,1  1,0,0 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.0,1 1.0,0 R.l.R R.0,1  1,0,0 O.l.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 R.l.R R.0,1 1,0,0 R.l.R R.R.l  1.0,0 0,1,R O.R.l l.R.R O.l.R O.R.l 1,0,0 R.l.R R.R.l l.R.R R.l.R R.R.l l.R.R R.l.R R.R.l  1,0, R 0,1 ,R R.0,1 1,0,R 0,1 ,R R.0,1 1,0 ,R 0,1,R R.0,1 1,0, R 0,1,0 R.0,1 1,0,0 0,1,0 R.0,1  1,0,0 O.l.R O.R.l 1,0,0 0,1 ,R R.0,1 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.0,1  l.R.R 0,1 ,R O.R.l 1,0,0 R.l.R R.R.l l.R.R R,1,R R.R.l l.R.R R,1,R R,R,1  

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