# Polyhedral and complexity studies in integer optimization, with applications to maintenance planning and location–routing problems Doctoral thesis, 2012

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 themodels 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.<\p> 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.<\p> 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.<\p> 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 fromthe 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.<\p> In Paper IV, we study the preventive maintenance scheduling problem with interval costs which again considers a multi-component system with set-up costs. As for the ORP, an optimal schedule for the entire horizon is sought for. Here, themaximum replacement intervals are replaced by a cost on the replacement intervals. The problem is shown to be a generalization of the ORP as well as of the dynamic joint replenishment problem from inventory theory. We present a model for the problem originally introduced for the joint replenishment problem. The model is utilized in three case studies from the railway, aircraft and wind power industries.<\p> Finally, in Paper V we consider the Hamiltonian p-median problem which belongs to the class of location–routing problems. It consists of finding p disjoint minimum weight cycles which cover all vertices in a graph. We present several new and existing models and analyze these from a computational as well as a theoretical point of view. The conclusion is that threemodels are computationally superior, two of which are introduced in this paper.<\p> The main contribution of this thesis is to develop models for maintenance decisions and thus take an important step towards efficient and reliable maintenance decision support systems.<\p>

maintenance optimization

complexity theory

integer linear programming

stochastic programming

Hamiltonian p-median problem

polyhedral analysis

Sal Pascal, Institutionen för Matematiska vetenskaper vid Chalmers tekniska högskola och Göteborgs universitet
Opponent: Karen Aardal, Delft University of Technology

## Author

### Adam Wojciechowski

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

### The opportunistic replacement problem: theoretical analyses and numerical tests

Mathematical Methods of Operations Research,; Vol. 76(2012)p. 289-319

Journal article

### Driving Forces

Sustainable development

### Subject Categories

Computational Mathematics

Energy

Basic sciences

### ISBN

978-91-7385-736-9

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie

Sal Pascal, Institutionen för Matematiska vetenskaper vid Chalmers tekniska högskola och Göteborgs universitet

Opponent: Karen Aardal, Delft University of Technology

10/8/2017