The opportunistic replacement problem: theoretical analyses and numerical tests

We consider a model for determining optimal opportunistic maintenance schedules w.r.t. a maximum replacement interval. This problem generalizes that of Dickman et al. (J Oper Res Soc India 28:165–175, <CitationRef CitationID="CR12">1991</CitationRef>) and is a natural starting point for modelling replacement schedules of more complex systems. We show that this basic opportunistic replacement problem is NP-hard, that the convex hull of the set of feasible replacement schedules is full-dimensional, that all the inequalities of the model are facet-inducing, and present a new class of facets obtained through a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${\{0, \frac{1}{2}\}}$$</EquationSource> </InlineEquation> -Chvátal–Gomory rounding. For costs monotone with time, a class of elimination constraints is introduced to reduce the computation time; it allows maintenance only when the replacement of at least one component is necessary. For costs decreasing with time, these constraints eliminate non-optimal solutions. When maintenance occasions are fixed, the remaining problem is stated as a linear program and solved by a greedy procedure. Results from a case study on aircraft engine maintenance illustrate the advantage of the optimization model over simpler policies. We include the new class of facets in a branch-and-cut framework and note a decrease in the number of branch-and-bound nodes and simplex iterations for most instance classes with time dependent costs. For instance classes with time independent costs and few components the elimination constraints are used favorably. For fixed maintenance occasions the greedy procedure reduces the computation time as compared with linear programming techniques for all instances tested. Copyright Springer-Verlag 2012