Anuj Mehrotra

A Column Generation Approach for Graph Coloring

We present a method for solving the independent set formulation of the graph coloring problem (where there is one variable for each independent set in the graph). We use a column generation method for implicit optimization of the linear program at each node of the branch-and-bound tree. This approach, while requiring the solution of a difficult subproblem as well as needing sophisticated branching rules, solves small to moderate size problems quickly. We have also implemented an exact graph coloring algorithm based on DSATUR for comparison. Implementation details and computational experience are presented.

Min-cut clustering

We describe a decomposition framework and a column generation scheme for solving a min-cut clustering problem. The subproblem to generate additional columns is itself an NP-hard mixed integer programming problem. We discuss strong valid inequalities for the subproblem and describe some efficient solution strategies. Computational results on compiler construction problems are reported.

Planning Merchandising Decisions to Account for Regional and Product Assortment Differences

The last decade has fundamentally changed the face of retailing. The genesis has been increased customer fragmentation, enabling technologies such as the internet and increased competition. In this era of “hypercompetition,” retailers need to have a better understanding of the performance of individual stores so they can more accurately plan their merchandise assortments and set more realistic merchandising goals. In this paper, we determine the performance of retail outlets relative to the “best practice” set of outlets and demonstrate the importance of accommodating both regional and assortment differences. We empirically assess the performance of stores from a major Fortune 500 multinational retailing chain. Multiple inputs and outputs from 59 stores in three regions were used to determine sales goals for two different product categories. The results of three alternative models suggest that incorporating both assortment and regional differences significantly affects both performance and predicted sales volume estimates. Implications and avenues for future research are discussed.

Cliques and clustering: A combinatorial approach

We use column generation and a specialized branching technique for solving constrained clustering problems. We also develop and implement an innovative combinatorial method for solving the pricing subproblems. Computational experiments comparing the resulting branch-and-price method to competing methodologies in the literature are presented and suggest that our technique yields a significant improvement on the hard instances of this problem.

Optimal shift scheduling: A branch-and-price approach

We present a branch-and-price technique for optimal staff scheduling with multiple rest breaks, meal break, and break windows. We devise and implement specialized branching rules suitable for solving the set covering type formulation implicitly, using column generation. Our methodology is more widely applicable and computationally superior to the alternative methods in the literature. We tested our methodology on 365 test problems involving between 1728 and 86400 shift variations, and 20 demand patterns. In a direct comparison with an alternative method, our approach yields significant improvements both in cpu time and in the number of problem instances solved to optimality. The improvements were particularly marked for problems involving larger numbers of feasible shifts.

An integrated simulation modeling and operations research approach to spatial management decision making

Abstract Worldwide declines in fishery yields emphasize the need for innovations in fishery management methods and models that encompass the complex dynamics of fish stocks and a broader range of human impacts. To this end, we have developed a generalized analytical framework that builds upon and extends principles of traditional fishery management by utilizing methodologies from the fields of computer simulation modeling and operations research. To address the spatial dynamics of fishery resources and human uses, we built a new spatial population dynamics computer simulation model. Several operations research methods were also developed and integrated with the simulation model to provide a robust, quantitative framework for addressing a wide range of spatial management decisions in fisheries. We illustrate the power and applicability of our framework with three diverse case studies. The tirst case study explores how the use of varying numbers of marine reserves impacts coral reef fish productivity by integrating the population dynamics simulation model, a recursive clustering algorithm, and an integer program. The second case study integrates simulation model results into an analytical hierarchy process model to produce recommendations concerning contigurations of alternative spatial management plans for the Dry Tortugas, Florida, USA. Finally, in the third case study the spatial model is adapted to simulate spatial dredging activities to assess the uses of alternate “environmental windows” that meet operational goals and protect tishcry resources for use in dredging projects in United States coastal waterways.

A Branch-And-Price Approach for Graph Multi-Coloring

We present a branch-and-price framework for solving the graph multi-coloring problem. We propose column generation to implicitly optimize the linear programming relaxation of an independent set formulation (where there is one variable for each independent set in the graph) for graph multi-coloring. This approach, while requiring the solution of a difficult subproblem, is a promising method to obtain good solutions for small to moderate size problems quickly. Some implementation details and initial computational experience are presented.

Cardinality constrained Boolean quadratic polytope

We study the polyhedral structure of an integer programming formulation of the cardinality constrained Boolean quadratic problem. We give many facet-defining inequalities. The separation problems for these inequalities appear to be difficult, which explains, in part, the difficulty encountered in solving these problems via a branch-and-cut methodology. As a special case of these inequalities, we obtain some previously known inequalities for the equipartition problem.

Extending the Horizons: Advances in Computing, Optimization, and Decision Technologies

This book represents the results of cross-fertilization between OR/MS and CS/AI. It is this interface of OR/CS that makes possible advances that could not have been achieved in isolation. Taken collectively, these articles are indicative of the state-of-the-art in the interface between OR/MS and CS/AI and of the high caliber of research being conducted by members of the INFORMS Computing Society.

Consolidating Maintenance Spares

The inventory of spare parts that a firm holds depends on the number of working parts and age of the equipment to be serviced, the expected failure rate associated with each working part, and the acceptable level of service. We model the problem of consolidation of spare parts to reduce overall inventory as an integer program with a nonlinear objective function. A linear reformulation of this model is obtained that helps solve some practical instances. A more compact implicit formulation is developed and solved using a specialized branch-and-price technique. We also demonstrate how this specialized branch-and-price technique is modified to devise a very effective heuristic procedure with a prespecifiable guarantee of quality of solution produced. This provides a practical and efficient methodology for maintenance spare consolidation.