Coefficient strengthening:A tool for formulating mixed integer programs

Providing a good formulation is an important part of solving a mixed integer program.We suggest to measure the quality of a formulation by whether it is possible tostrengthen the coefficients of the formulation. Sequentially strengthening coefficientscan then be used as a tool for improving formulations. We believe this method couldbe useful for analyzing and producing tight formulations of problems that arise inpractice. We illustrate the use of the approach on a problem in production scheduling.We also prove that coefficient strengthening leads to formulations with a desirableproperty: if no coefficient can be strengthened, then no constraint can be replaced byan inequality that dominates it. The effect of coefficient strengthening is tested on anumber of problems in a computational experiment. The strengthened formulationsare compared to reformulations obtained by the preprocessor of a commercialsoftware package. For several test problems, the formulations obtained by coefficientstrengthening are substantially stronger than the formulations obtained by thepreprocessor. In particular, we use coefficient strengthening to solve two difficultproblems to optimality that have only recently been solved....