On parallel machine replacement problems with general replacement cost functions and stochastic deterioration

Abstract

Abstract:The parallel machine replacement problem consists of finding a minimum cost replacement policy for a finite population of economically interdependent machines. In this paper, we formulate a stochastic version of the problem and analyze the structure of optimal policies under general classes of replacement cost functions. We prove that for problems with arbitrary cost functions, there can be optimal policies where a machine is replaced only if all machines in worse states are replaced (Worse Cluster Replacement Rule). We then show that, for problems with replacement cost functions exhibiting nonincreasing marginal costs, there are optimal policies such that, in any stage, machines in the same state are either all kept or all replaced (No-Splitting Rule). We also present an example that shows that economies of scale in replacement costs do not guarantee optimal policies that satisfy the No-Splitting Rule. These results lead to the fundamental insight that replacement decisions are driven by marginal costs, and not by economies of scale as suggested in the literature. Finally, we describe how the optimal policy structure, i.e., the No-Splitting and Worse Cluster Replacement Rules, can be used to reduce the computational effort required to obtain optimal replacement policies.

Resource-constrained management of heterogeneous assets with stochastic deterioration

“…Relatively little research addresses the constrained management of equipment with stochastic costs. Childress and Durango-Cohen (2005) generalize the replacement rules of Jones et al (1991) for stochastically deteriorating assets in which economic interdependence is induced by a fixed charge for replacing one or more assets. Hartman (2004) considers joint utilization and replacement decisions across a collection of machines; utilization decisions create interdependence between assets and stochastic demand induces a state-dependent stochastic cost structure, where the state is defined as a combination of age and utilization.…”

A review of asset management literature on multi-asset systems

“…Hence, previous researchers have incorporated performance and resource [112] dependencies into problems in this DC. The equipment replacement problem with resource dependence is more complex in a portfolio than in a fleet because different cost and deterioration functions are associated [113], [114].…”

Optimal replacement in a system of n-machines with random horizon

“…Then, if C t (π, x) ≥ J t (x), in particular for t = 0 and of (3.3), J τ (π, x) ≥ J 0 (x) and for π = π * , J τ (π * , x) = J 0 (x). 2 …”

“…Also, the optimal replacement has been implemented based on future technological advances, in this case, a non-stationary process is considered and the optimal decision is characterized using a forecast horizon approach (see [6]). On the other hand, in [2] the problem is studied with n-machines considering two heuristics rules of replacement that make possible the search of optimal policies. These rules are as follow: the first suggests that a machine is replaced only if all older machines are replaced and the second one indicates that in any stage all machines of the same age are either kept or replaced.…”