Economic discrete replacement policy subject to increasing failure rate shock model

Abstract

This paper presents an economic discrete replacement model for a single-unit system subjected to external shocks. In this model, the system is affected by both external shocks and its ageing process. The external shocks are divided into two types, namely non-lethal and lethal, depending on their effect to the system. A non-lethal shock damages the system by increasing the failure rate of a certain degree, while a lethal shock causes the system into instantaneous failure. The failure rate also increases with the system's ageing process. The system is replaced at the instant of the nth non-lethal shock, or on failure, whichever occurs first. The research proposed a model where the long run expected cost per unit time is formulated by introducing relative costs and derived as a criterion of optimality. By minimizing the cost, an optimal number, n * , is then found. The optimal number, n * , is also verified to be finite and unique under certain conditions.Keywords Ageing process · Failure rate · Prevention maintenance model · Replacement policy · Shock · Single-unit system Notation:Intensity rate of a Poisson process {N(t)} t 0 pThe probability that an external shock occurs is a nonlethal shock r j (t) Failure rate of the system when the age of the system is t and the number of non-lethal shocks incurred over [0, t] P j (t) The probability that there is just j non-lethal shocks incurred over [0, t] r(t, pλ) Failure rate of the system without lethal shocks F p (t) Survival function of the system at age t without lethal shocks G(t)Survival function of the occurrence time of lethal shock Y 1The time to first type I failure of the system with survival function F p (t) without lethal shocks X 1The time to first type II failure of the system with survival function G(t) without non-lethal shocks ZThe time to first failure (type I or type II) of the system,The occurrence time of the nth non-lethal WThe time to replacement, W = min(T n , Z) E(R 1 ) The expected cost incurred during a replacement cycle time E(W 1 ) The expected replacement cycle time C p (n) The long-run expected cost per unit time