An optimal proportion of perfect repair

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Abstract

At each failure of a system, with probability p it is perfectly repaired, and with probability 1−p it is restored to its condition just prior to failure. It is shown that there exists a unique p minimizing the long-run repair cost, provided the failure time distribution of the system has strictly DMRL.

Introduction

Brown and Proschan [1] introduced a model for imperfect repair. When a system fails, with probability p it is completely repaired or, if necessary, replaced with a new system (perfect repair), and with probability 1−p it is returned to the functioning state but only restored to its condition just prior to failure (imperfect repair). Both repairs take negligible time. Thus, if F is the failure time distribution of the system following perfect repair, then the failure time distribution following imperfect repair done at age t is given byF̄(x|t)=F̄(x+t)F̄(t),x>0.

In this paper, it is assumed that a system is maintained for a long time under the above repair policy, and hence p is now considered as the long-run proportion to perfect repair. After assigning costs to both perfect and imperfect repairs, we show that there exists a unique p which minimizes the long-run average cost per unit time, provided F has strictly decreasing mean residual life (DMRL).

Section snippets

Optimal proportion

Let C1 be the cost of perfect repair and let C2 be the cost of imperfect repair. Notice that the sequence of points where the perfect repair is done forms an embedded renewal process. Thus, applying the renewal reward theorem [2, p. 133], we obtain the long-run average cost per unit time given byC(p)=pC1+(1−p)C2pμ(p)for0<p⩽1,where μ(p) is the mean time between perfect repairs. For p=0, we put C(0)=limp→0C(p) (possibly ∞).

Conditioning on the number of imperfect repairs between perfect repairs

Acknowledgements

We would like to thank the associate editor for several helpful comments.

References (2)

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Cited by (4)

Research has been supported by the Korea Science and Engineering Foundation (KOSEF), under Grant Number 951-0103-036-2.

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