On the Relationship Between the Value Function and the Efficient Frontier of a Mixed Integer Linear Optimization Problem

In this paper, we investigate the connection between the efficient frontier (EF) of a general multiobjective mixed integer linear optimization problem (MILP) and the so-called restricted value function (RVF) of a closely related single-objective MILP. We demonstrate that the EF of the multiobjective MILP is comprised of points on the boundary of the epigraph of the RVF so that any description of the EF suffices to describe the RVF and vice versa. In the first part of the paper, we describe the mathematical structure of the RVF, including characterizing the set of points at which it is differentiable, the gradients at such points, and the subdifferential at all nondifferentiable points. Because of the close relationship of the RVF to the EF, we observe that methods for constructing so-called value functions and methods for constructing the EF of a multiobjective optimization problem, each of which have been developed in separate communities, are effectively interchangeable. By exploiting this relationship, we propose a generalized cutting plane algorithm for constructing the EF of a multiobjective MILP based on a generalization of an existing algorithm for constructing the classical value function. We prove that the algorithm is finite under a standard boundedness assumption and comes with a performance guarantee if terminated early

The asymmetric traveling salesman problem with replenishment arcs

We consider a constrained asymmetric traveling salesman problem with knapsack-like constraints on subpaths of the tour. This problem arises in routing aircraft. We formulate the problem with an exponential number of variables that correspond to feasible subpaths. We study certain polyhedral aspects of the reformulation and present a branch-and-price-and-cut algorithm for solving it. We test the algorithm on both random instances and real instances that arise in the airline application

An optimisation approach to maintenance scheduling for capacity alignment in the Hunter Valley coal chain

The Hunter Valley Coal Chain (HVCC) consists of mining companies, rail operators, rail track owners and terminal operators, together forming the world’s largest coal exporting facility. In 2008, the throughput of HVCC was about 92 million tonnes, or more than ten per cent of the world’s total trade in coal for that year. The coal export operation generates around $15 billion in annual export income for Australia. As demand has increased significantly in recent years, and is expected to increase further in the future, efficient supply chain management is crucial. The Hunter Valley Coal Chain Coordinator Limited (HVCCC) was founded to enable integrated planning and coordination of the interests of all parties, thus improving the efficiency of the system as a whole. One of the many planning challenges faced by the HVCCC is that of annual maintenance planning. Different supply chain elements, such as trains, railway track, terminal equipment and load points, must undergo regular preventive and corrective maintenance, leading to significant reductions in system capacity (up to 15 per cent). However good alignment of the maintenance tasks can reduce their impact, and the HVCCC undertakes an annual process to ensure the impact of maintenance on the supply chain capacity is as small as possible. This is achieved in an iterative negotiation process between HVCCC and individual service providers. In the past, maintenance schedule optimisation was largely manual, which for the more than 1000 tasks involved is quite labour-intensive. In this paper the authors describe an approach developed at the University of Newcastle in partnership with the HVCCC to automate this schedule optimisation process. We will discuss our experience in applying exact (mixed integer programming) and heuristic techniques from mathematics and computer science to address the problem. This work is anticipated to lead to new decision-support tools for the HVCCC’s capacity planning team

Finding an optimal stationing policy for the US Army in Europe after the force drawdown

With the continuing reduction of forces in Europe, it is apparent that the base support structure cannot be maintained at current levels. The purpose of this effort is to develop a methodology to assign US Army units remaining in Europe to installations in an economical manner, and to make recommendations regarding which installations are candidates for deactivation and closure. An integer programming model has been formulated which minimizes annual costs subject to constraints on resources, implementation costs, unit proximity, and support requirements. The model can be used to provide decision makers with insights regarding resource utilization and shortfalls, and costs of implementing various stationing plan alternatives. Model development and data collection issues are discussed. Computational experience is given and techniques used to improve model performance are described