The stochastic opportunistic replacement problem, part I: models incorporating individual component lives

Journal article, 2015

We consider an extension of the opportunistic replacement problem, which has been studied by Dickman et al. (The Journal of the Operational Research Society of India, 28:165–175, 1991), Andréasson (Optimization of opportunistic replacement activities in deterministic and stochastic multi-component systems, Licentiate thesis, Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Göteborg, Sweden, 2004), and Almgren et al. (The opportunistic replacement problem: analysis and case studies, Preprint, Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Göteborg, Sweden, 2011), that allows the individuals of the same component to have non-identical lives. Formulating and solving this problem constitute a first step towards solving the opportunistic replacement problem with uncertain component lives. We show that the problem is NP-hard even with time independent costs, and present two 0–1 integer programming models for the problem. We show that in model I the integrality requirement on a majority of the variables can be relaxed; this is in contrast to model II and the model from Andréasson (Optimization of opportunistic replacement activities in deterministic and stochastic multi-component systems, Licentiate thesis, Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Göteborg, Sweden, 2004). We remove all superfluous variables and constraints in model I and show that the remaining constraints are facet inducing. We also utilize a linear transformation of model I to obtain a stronger version of model II, i.e., model II+, which inherits the polyhedral properties of model I. Numerical experiments show that the solution time of model I is significantly lower than those of both model II and Andréasson’s model. It is also slightly lower than the solution time of model II+.