Polyhedral and complexity studies in integer optimization, with applications to maintenance planning and location-routing problems

This thesis develops integer linear programming models for and studies the complexity of problems in the areas of maintenance optimization and location–routing. We study how well the polyhedra defined by the linear programming relaxation of the models approximate the convex hull of the integer feasible solutions. Four of the papers consider a series of maintenance decision problems whereas the fifth paper considers a location–routing problem. In Paper I, we present the opportunistic replacement problem (ORP) which is to find a minimum cost replacement schedule for a multi-component system given a maximum replacement interval for each component. The maintenance cost consists of a fixed/set-up cost and component replacement costs. We show that the problem is NP-hard for time dependent costs, introduce an integer linear programming model for it and investigate the linear programming relaxation polyhedron. Numerical tests on random instances as well as instances from aircraft applications are performed. The stochastic opportunistic replacement problem (SORP) extends the ORP to allow for uncertain component lives/maximum replacement intervals. In Paper II, a first step towards a stochastic programmingmodel for the SORP is taken by allowing for non-identical lives for component individuals. This problem is shown to be NP-hard also for time independent costs. A new integer linear programming model for this problem is introduced which reduces the computational time substantially compared to an earlier model. In Paper III, we study the SORP and present a two-stage stochastic programming solution approach, which aims at — given the failure of one component — deciding on additional component replacements. We present a deterministic equivalent model and a decomposition method; both of which are based on the model developed in Paper II. Numerical tests on instances from the aviation and wind power industries and on two test instances show that the stochastic programming approach performs better than or equivalently good as simpler maintenance policies.