Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

Adapting the interior point method for the solution of LPs on serial, coarse grain parallel and massively parallel computers

In this paper we describe a unified scheme for implementing an interior point algorithm (IPM) over a range of computer architectures. In the inner iteration of the IPM a search direction is computed using Newton's method. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system, and the design of data structures to take advantage of serial, coarse grain parallel and massively parallel computer architectures, are considered in detail. We put forward arguments as to why integration of the system within a sparse simplex solver is important and outline how the system is designed to achieve this integration

Mathematical models for planning support

In this paper we describe how computer systems can provide planners with active planning support, when these planners are carrying out their daily planning activities. This means that computer systems actively participate in the planning process by automatically generating plans or partial plans. Active planning support by computer systems requires the application of mathematical models and solution techniques. In this paper we describe the modeling process in general terms, as well as several modeling and solution techniques. We also present some background information on computational complexity theory, since most practical planning problems are hard to solve. We also describe how several objective functions can be handled, since it is rare that solutions can be evaluated by just one single objective. Furthermore, we give an introduction into the use of mathematical modeling systems, which are useful tools in a modeling context, especially during the development phases of a mathematical model. We finish the paper with a real life example related to the planning process of the rolling stock circulation of a railway operator.optimization;mathematical models;modeling process;planning support;Planning

Approximation schemes for parallel machine scheduling with non-renewable resources

In this paper the approximability of parallel machine scheduling problems with resource consuming jobs is studied. In these problems, in addition to a parallel machine environment, there are non-renewable resources, like raw materials, energy, or money, consumed by the jobs. Each resource has an initial stock, and some additional supplies at a-priori known moments in time and in known quantities. The schedules must respect the resource constraints as well. The optimization objective is either the makespan, or the maximum lateness. Polynomial time approximation schemes are provided under various assumptions, and it is shown that the makespan minimization problem is APX-complete if the number of machines is part of the input even if there are only two resources. © 2016 Elsevier B.V