Nonparametric adaptive opportunity-based age replacement strategies

Abstract

We consider opportunity-based age replacement (OAR) using nonparametric predictive inference (NPI) for the time to failure of a future unit. Based on n observed failure times, NPI provides lower and upper bounds for the survival function for the time to failure X_n+1 of a future unit which lead to upper and lower cost functions, respectively, for OAR based on the renewal reward theorem. Optimal OAR strategies for unit n+1 follow by minimizing these cost functions. Following this strategy, unit n+1 is correctively replaced upon failure, or preventively replaced upon the first opportunity after the optimal OAR threshold. We study the effect of this replacement information for unit n+1 on the optimal OAR strategy for unit n+2. We illustrate our method with examples and a simulation study. Our method is fully adaptive to available data, providing an alternative to the classical approach where the probability distribution of a unit's time to failure is assumed to be known. We discuss the possible use of our method and compare it with the classical approach, where we conclude that in most situations our adaptive method performs very well, but that counter-intuitive results can occur.

Appendix A

Derivation of (13)

We derive an expression for E[L_op(T)] which is similar to Equation (9) of Dekker and Dijkstra,⁸ but in a form more suitable for this paper. This expression holds for exponentially distributed Y, with probability density function f _Y (.).

where the penultimate equality follows from the fact that S _Y (x)=E[Y]dF _Y (x)/dx, as Y has an Exponential distribution.

Proof of Lemma 1

• 

  Here,

  

  see, for example Dekker and Dijkstra,⁸ and

  

  Substituting (A.2) and (A.3) into (A.1) yields

  

  As the last expression is equal to , and both and are positive, we have, for T∈[x_(j), x_(j+1)) and j=0, …, n

  
  

  Obviously, we only prove this for T∈(x_(j), x_(j+1)) and j=0, …, n−1, as the second derivatives do not exist in the observed values x_(j). Differentiating C_op(T) of Equation (14) with respect to T, and using (A.2) and (A.3) with X_n+1 replaced by X, obtain

  

  where

  

  Hence, as E[L(T)] and E[L_op(T)] are both strictly positive, we have

  

  Now,

  

  where

  

  with C′(T) the first derivative of the age replacement cost function (11). Moreover, we have

  

  and using (A.2), with X_n+1 replaced by X,

  

  so that

  

  Substituting (A.9) into (A.8) yields

  

  Differentiating the age replacement cost function C(T) (11) with the respect to T yields

  

  and since E[L(T)]>0 for all T>0,

  

  so that (A.10) can be written as

  

  Using (A.11), we have

  

  Substituting (A.12) into (A.7) yields

  

  However, at the optimal OAR threshold T_op^* we have g(T_op^*)=0 according to (A.6). From Lemmas 2.2 and 3.2 of Coolen-Schrijner and Coolen⁶ we know that the first derivatives of our NPI based age replacement cost functions are all negative. Substituting the lower and upper survival functions in E[L(T)] and E[S _X (T+Y)] yields corresponding lower and upper bounds which are all positive. Then, from (A.13), it follows that are all positive.

Proof of Lemma 3

• • 1 T∈[x_(j), x_(j+1)) with j=0, …, k−1. In this case, Equation (32) can be obtained by noting that

    

  • 2 T∈[x_(k), x_(k+1)). In this case, Equation (33) can be obtained by noting that

    

  • 3 T∈[x_(j), x_(j+1) with j=k+1, …, n. In this case, Equation (34) can be obtained by noting that

    

  □

Proof of Lemma 6

• We only prove part (a)-1 of the lemma, the proofs of the other parts are similar. Suppose T∈(x_(j), x_(j+1)] with j=0, …, k−1, then

  

  As Y has an Exponential distribution with parameter λ, the result follows.