Opportunity-based age replacement with a one-cycle criterion

Abstract

In this paper age replacement (AR) and opportunity-based age replacement (OAR) for a unit are considered, based on a one-cycle criterion, both for a known and unknown lifetime distribution. In the literature, AR and OAR strategies are mostly based on a renewal criterion, but in particular when the lifetime distribution is not known and data of the process are used to update the lifetime distribution, the renewal criterion is less appropriate and the one-cycle criterion becomes an attractive alternative. Conditions are determined for the existence of an optimal replacement age T ^* in an AR model and optimal threshold age T _opp ^* in an OAR model, using a one-cycle criterion and a known lifetime distribution. In the optimal threshold age T _opp ^*, the corresponding minimal expected costs per unit time are equal to the expected costs per unit time in an AR model. It is also shown that for a lifetime distribution with increasing hazard rate, the optimal threshold age is smaller than the optimal replacement age. For unknown lifetime distribution, AR and OAR strategies are considered within a nonparametric predictive inferential (NPI) framework. The relationship between the NPI-based expected costs per unit time in an OAR model and those in an AR model is investigated. A small simulation study is presented to illustrate this NPI approach.

Appendix

Proof of Theorem 1

It follows from that E _X|Y [C _opp(X,Y,T)]=E _X [C(X,T+Y)] where E _X|Y [·] denotes the expectation calculated with respect to the conditional distribution of X given Y. Hence, E _X,Y [C _opp(X,Y,T)] can be written as

where f _Y (y) denotes the pdf of Y. Integrating (A.1) by parts gives

where S _Y (y) denotes the survival function of Y. The result follows as S _Y (u−T)=e^λTe^−λu. □

Proof of Theorem 2

Differentiating (6) with respect to T gives

and, consequently, for T _opp ^* to be a stationary point of E _X,Y [C _opp(X,Y,T)] it must satisfy (7). From (6) it follows that T _opp ^* is a stationary point if E _X,Y [C _opp(X,Y,T _opp ^*)]=E _X [C(X,T _opp ^*)]. Differentiating (A.2) with respect to T gives

Using (7), it follows that for T _opp ^* to be a minimum, the condition E _X ′[C(X,T _opp ^*)]<0 must be satisfied. Using (2), E _X ′[C(X,T _opp ^*)]<0 is equivalent to (8). □

Proof of Theorem 3

If h _X (T) is monotonically increasing, then the AR model has a unique minimum T ^* which can be obtained by solving T ^* h _X (T ^*)=c _p/(c _f−c _p). Then E _X ′[C(X,T ^*)]=0 and E _X ′[C(X,T)]>0 for T>T ^*. Hence, for T⩾T ^*,

so that from (6), E _X,Y [C _opp(X,Y,T)]>E _X [C(X,T)]. In an AR model when T↓0, that is, a unit will be replaced as soon as it is installed, E _X [C(X,T)] becomes infinite (see also (1)). However, in an OAR model when T↓0, that is, a unit will be replaced upon the first opportunity after T = 0,E _X,Y [C _opp(X,Y,0)] is finite (see also (5)). Hence, there is at least one solution to E _X,Y [C _opp(X,Y,T)]=E _X [C(X,T)] and this must occur in the interval (0,T ^*). If h _X (T) is monotonically increasing, then for T in this interval, Th _X (T)<c _p/(c _f−c _p) holds, so that from Theorem 2, any solution to E _X,Y [C _opp(X,Y,T)]=E _X [C(X,T)] must correspond to a local minimum and is thus unique. Consequently, this unique minimum T _opp ^* satisfies both (10) and (11). □

Proof of Theorem 4

From Theorem 1 and (2) it follows that

For T∈(0,x ₁), using (14) and (22), it follows that

From (14), (15) and (22), it follows that

Using (22), it follows that

and similarly,

Noting that

it follows that

Substituting (A.6), (A.7) and (A.9) into (A.5) and the result together with (A.4) into (A.3) gives (24).

For T∈[x _j ,x _j+1),j=1,…,n, it follows from (14) and (22) that

From (14) and (15), it follows that

so that with (A.8), it can be deduced that

Substituting (A.12) into (A.11), and substituting the result together with (A.10) into (A.3) yields (25). □