R. Kipp Martin

Solving multi-item capacitated lot-sizing problems using variable redefinition

Mixed-integer programming models are typically not used to solve realistic-sized production scheduling problems because they require exorbitant solution times. We impose a useful taxonomy on production scheduling problems and develop alternative formulations for a wide variety of problems within the taxonomy. The linear programming relaxation of the new models is very effective in generating bounds. We show that these bounds are equal to those that could be generated using Lagrangian relaxation or column generation. The linear programming bounds increase in effectiveness as the problems become larger. Perhaps of greatest significance is that practitioners can obtain our results using only standard “off-the-shelf” codes such as LINDO or MPSX/370. We report computational experience in several computing environments (hardware and software) on problems with up to 200 products and 10 time periods (2000 0-1 variables).

OR Practice-A Scenario Approach to Capacity Planning

Production capacity has always been one of the most important strategic variables for the major automobile companies. Decisions by individual companies concerning the overall level of capacity, the type of facility e.g., the level of flexibility, and the location of that capacity e.g., in the United States or abroad are discussed in great detail in the popular business press. In this paper, we describe a model developed for General Motors to aid in making decisions about capacity for four of their auto lines. The model incorporates elements of scenario planning, integer programming, and risk analysis. All the input and output is done using Lotus 1-2-3. Although the presentation is motivated by the particular application in the auto industry, the model represents a general purpose approach that is applicable to a wide variety of decisions under risk. An example in this paper uses actual data, appropriately transformed to ensure confidentiality.

Using separation algorithms to generate mixed integer model reformulations

The linear relaxation of mixed integer programming models can be strengthened by introducing auxiliary variables. We develop a new method for generating auxiliary variable reformulations for problems where the separation algorithm for finding violated cuts can be formulated as a linear program. Our results have important consequences for integrality proofs and efficient formulations.

Generating alternative mixed-integer programming models using variable redefinition

Dropping the “complicating” constraints in a mixed-integer linear program often yields a “special structure subproblem” that can be reformulated using a different set of decision variables. Once the new variables have been identified, the entire problem can be reformulated in terms of the new variables. We develop a theory of variable redefinition based on relating the two sets of decision variables by a linear transformation, and describe methods for reformulating the special structure problem. The reformulated models have a more accurate linear relaxation than the problems from which they were derived, an important property within the context of linear programming-based branch-and-bound modeling approaches.

Polyhedral characterization of discrete dynamic programming

Many interesting combinatorial problems can be optimized efficiently using recursive computations often termed discrete dynamic programming. In this paper, we develop a paradigm for a general class of such optimizations that yields a polyhedral description for each model in the class. The elementary concept of dynamic programs as shortest path problems in acyclic graphs is generalized to one seeking a least cost solution in a directed hypergraph. Sufficient conditions are then provided for binary integrality of the associated hyperflow problem. Given a polynomially solvable dynamic program, the result is a linear program, in polynomially many variables and constraints, having the solution vectors of the dynamic program as its extreme-point optima. That is, the linear program provides a succinct characterization of the solutions to the underlying optimization problem expressed through an appropriate change of variables. We also discuss projecting this formulation to recover constraints on the original variables and illustrate how some important dynamic programming solvable models fit easily into our paradigm. A classic multiechelon lot sizing problem in production and a host of optimization problems on recursively defined classes of graphs are included.

Subset Coefficient Reduction Cuts for 0/1 Mixed-Integer Programming

We describe a method for generating cuts for mixed-integer 0/1 programs. These cuts are designed to tighten an integer program prior to applying linear programming based branch and bound algorithms. The method involves two basic ideas: subset selection and coefficient reduction. Coefficient reduction is a process of reducing the coefficients of the 0/1 variables. Subset selection is combined with coefficient reduction by applying the coefficient reduction process to the coefficients of a subset of variables from the constraints in the problem formulation. The paper exploits these two simple ideas to derive a broad class of cuts for integer programs with both 0/1 and continuous variables. It also reports on the use of this methodology in solving a variety of fixed charge problems.

Gainfree Leontief substitution flow problems

Leontief substitution systems have been studied by economists and operations researchers for many years. We show how such linear systems are naturally viewed asLeontief substitution flow problems on directed hypergraphs, and that important solution properties follow from structural characteristics of the hypergraphs. We give a strongly polynomial, non-simplex algorithm for Leontief substitution flow problems that satisfy againfree property leading to acyclic extreme solutions. Integrality conditions follow easily from this algorithm. Another structural property,support disjoint reachability, leads to necessary and sufficient conditions for extreme solutions to be binary. In a survey of applications, we show how the Leontief flow paradigm links polyhedral combinatorics, expert systems, mixed integer model formulation, and some problems in graph optimization.

On Parallelizing Dual Decomposition in Stochastic Integer Programming

For stochastic mixed-integer programs, we revisit the dual decomposition algorithm of Caroe and Schultz from a computational perspective with the aim of its parallelization. We address an important bottleneck of parallel execution by identifying a formulation that permits the parallel solution of the master program by using structure-exploiting interior-point solvers. Our results demonstrate the potential for parallel speedup and the importance of regularization (stabilization) in the dual optimization. Load imbalance is identified as a remaining barrier to parallel scalability.

OSiL: An instance language for optimization

Distributed computing technologies such as Web Services are growing rapidly in importance in today’s computing environment. In the area of mathematical optimization, it is common to separate modeling languages from optimization solvers. In a completely distributed environment, the modeling language software, solver software, and data used to generate a model instance might reside on different machines using different operating systems. Such a distributed environment makes it critical to have an open standard for exchanging model instances.In this paper we present OSiL (Optimization Services instance Language), an XML-based computer language for representing instances of large-scale optimization problems including linear programs, mixed-integer programs, quadratic programs, and very general nonlinear programs. OSiL has two key features that make it much superior to current standard forms for optimization problem instances. First, it uses the object-oriented features of XML schemas to efficiently represent nonlinear expressions. Second, its XML schema maps directly into a corresponding in-memory representation of a problem instance. The in-memory representation provides a robust application program interface for general nonlinear programming, facilitates reading and writing postfix, prefix, and infix formats to and from the nonlinear expression tree, and makes the expression tree readily available for function and derivative evaluations.

Optimization Services: A Framework for Distributed Optimization

We describe a research project to design a distributed optimization environment in which solvers, modeling languages, registries, analyzers, and simulation engines can be implemented as services and utilities under a unified framework. Our work, which we call optimization services or OS, defines standards for all activities necessary to support decentralized optimization on the Internet: representation of optimization instances, results, and solver options; communication between clients and solvers; and discovery and registration of optimization-related software using the concept of Web services. In this paper we place emphasis on issues in distributed computing that are posed by the special character of optimization. We also describe a reference implementation that is freely available as an open-source project of COIN-OR.