Optimal replacement with non-monotone failure rates
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.
scientific article; zbMATH DE number 3848370
scientific article