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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