Optimal replacement with non-monotone failure rates
type: Item
instance of: scholarly article
title: Optimal replacement with non-monotone failure rates
zbMATH Open document ID: 0534.60080
DOI: 10.1007/BF01721253
published in: OR Spektrum
publication date: 1984-01-01T00:00:00Z
review text: Consider a system that is subject to random failure, let \(\{x_ t\), \(t\geq 0\}\) be a right continuous strong Markov process and suppose the system fails when \(x_ t\) first exceeds S, an r.v. independent of \(\{x_ t\), \(t\geq 0\}\) with known d.f., after failure the system is replaced by a new one and the process repeats with \(x_ 0=0\). Let \(\zeta\) denote the time to system failure, that is \(\zeta =\inf(t| x_ t\geq s)\) and C be the class of stopping times \(\tau\leq \zeta\) with respect to \(x_ t\). A planned replacement at any stopping time \(\tau<\zeta\) incurs a cost of \(c>0\), a failure, \(\tau =\zeta\) adds a penalty cost of \(k>0\), the problem is to minimize the long run average cost per unit time \(K_{\tau} (K_{\tau}=(C+kp(\tau =\zeta))/E\tau)\) that is to find \(\tau^*\in C\) such that \(K_{\tau^*}=\inf_{\tau \in C}K_{\tau}\). This problem has been studied by some earlier work under the assumption that the failure rate is nondecreasing, in this paper, an explicit formula for the optimal replacement policy is obtained under the assumption that the failure rate first decreases and then increases, this case is interesting for practical purposes.
Mathematics Subject Classification ID: 60K10
Mathematics Subject Classification ID: 90B25
Mathematics Subject Classification ID: 62N05
zbMATH DE Number: 3848370
zbMATH Keywords: damage process
zbMATH Keywords: optimal replacement
zbMATH Keywords: failure rate
zbMATH Keywords: stopping time
author: Uwe Jensen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3848370