Optimal age-replacement time with minimal repair based on cumulative repair-cost limit for a system subject to shocks

Abstract

An operating system is subject to random shocks that arrive according to a nonhomogeneous Poisson process and cause the system failed. System failures experience to be divided into two categories: a type-I failure (minor), rectified by a minimal repair; or a type-II failure (catastrophic) that calls for a replacement. An age-replacement model is studied by considering both a cumulative repair-cost limit and a system's entire repair-cost history. Under such a policy, the system is replaced at age T , or at the k-th type-I failure at which the accumulated repair cost exceeds the pre-determined limit, or at any type-II failure, whichever occurs first. The object of this article is to study analytically the minimum-cost replacement policy for showing its existence, uniqueness, and the structural properties. The proposed model provides a general framework for analyzing the maintenance policies, and presents several numerical examples for illustration purposes.Ann Oper Res (2011) 186:317-329 S k Arrival instant of the kth shock for k = 1, 2, 3, . . . W i Minimal repair cost due to the ith type-I failure for i = 1, 2, 3, . . . G(w) cdf (cumulative distribution function) of the r.v. (random variable) W i c w Mean cost of W i ; c w = E[W i ] Z j Accumulated repair cost until the j th type-I failure; Z jcdf of the r.v. Z j ; the j -fold Stieltjes convolution of the distribution G with itself M Number of shocks proceeding the first type-II failurē P k , p k sf (survival function), pmf (probability mass function) of M; P k = P (M > k) = Pr{first k shocks are type-I failures}, where the domain ofP k is {0, 1, 2, . . .} and 1 =P 0 ≥P 1 ≥P 2 ≥ · · ·;Pr{a type-I failure when shock k arrives} =P k /P k−1 θ k Pr{a type-II failure when shock k arrives} = 1 − q k T Replacement age of an operating system L Total repair-cost limit B(T ; L, {P k }) s-expected cost rate for an infinite time spanCost of a planned replacement c 1Cost of an unplanned replacement Y Waiting time until the first unplanned replacement (kth type-I failure at which the accumulated repair cost exceeds the pre-determined limit L or first type-II failure) when T = ∞ h(t), H (t) pdf (probability density function), cdf of the r.v. Ȳ H (t) sf of Y , which is 1 − H (t) r H (t) Failure (hazard) rate of Y ; r H (t) = h(t)/H (t) U i Length of successive replacement cycle for i = 1, 2, . . . V iOperational cost over U i D(t) s-expected cost of the operating system over [0, t]

A Joint Inspection-Based Preventive Maintenance and Spare Ordering Optimization Policy Using a Three-Stage Failure Process

“…The corresponding results obtained from them are given and shown in Figures 14(a) and 14(b), from which we can observe that there is a slight difference between them. More specifically, the optimal solutions from the analytical model (see (16)) are * = 42 days and * = 3 with the minimal expected cost per unit time, 0.9949; meanwhile, the decision variables optimized by the simulation algorithm designed in Figure 13 are also * = 42 days and * = 3 but with a different minimal expected cost per unit time, 0.9937. Obviously, the proposed simulation algorithm can be applied to optimize the joint policy of inspection-based PM and spare ordering.…”

“…This section presents and models an integrated inspectionbased PM policy and spare ordering provisioning policy, in which an assumption that the regular order is placed when the system begins to operate, that is, the initial time "0", is considered for comparison with the presented model in (16). We can conclude from this assumption that the spare must have been definitely ordered when needed; thus an emergency order due to the absence of a regular order for replenishment is not mentioned in this model.…”

“…Some works have investigated and presented the joint optimization model of preventive maintenance policy and spare ordering policy for single-unit systems. Most of the past studies modeled the age-based replacement policy and spare provisioning policy jointly; we can refer to Osaki and Yamada [11], Sheu and Griffith [12], Jhang [13], Sheu and Chien [14], and Sheu et al [15], Sheu et al [16]. In the proposed models of these works, only one spare is ordered at time 0, that is, the initial time of the system operation, and is delivered after a fixed or random lead time.…”

Fleet replacement under technological shocks

“…Discrete-time replacement models assume that the unknown lifetime of assets is integer (Hartman 2000;Kusaka and Suzuki 1990;Mehrez et al 2000;Regnier et al 2004;Rogers and Hartman 2005;Sheu et al 2011). The continuous models (e.g., Bethuyne 1998;Grinyer 1973;Yatsenko and Hritonenko 2008) are not restricted to this requirement.…”

“…Several models (Hopp and Nair 1991;Mehrez et al 2000;Rajagopalan et al 1998;Sheu et al 2011) assume the stochastic appearance of such shocks (with random instants or sizes). The shocks can positive as the arrival of new better assets in Kusaka and Suzuki (1990), Hopp and Nair (1991), Mehrez et al (2000), Rogers and Hartman (2005) or negative as system failures in Sheu et al (2011). The deterministic replacement with periodic shocks on the background of continuous technological development is considered by Rogers and Hartman (2005).…”