Exact Algorithms

Contents

In this chapter two general methods for solving the Minmax Regret $\mathcal{P}$ problem are presented. The first one, discussed in Section 3.1, is based on a mixed integer programming formulation (shortly MIP) which is a standard tool for solving discrete optimization problems. We show that under some additional and general assumptions it is possible to construct a MIP model with linear constraints and linear objective function for Minmax Regret $\mathcal{P}$. This model is compact in the sense that it involves a polynomial number of variables and constraints. A considerable number of excellent software packages are available to solve the constructed MIP model. One that is widely regarded is CPLEX, which will be used to test the constructed MIP formulations for particular problems. After constructing a MIP model we can apply the results obtained in the previous chapter to speed up calculations. That is, we can remove from the model all variables that correspond to nonpossibly optimal elements and fix some variables that correspond to the necessarily optimal ones. After relaxing the integrality constraints the MIP formulation may also be used to compute the lower bound on the value of the maximal regret of the optimal robust solution.

The second method, discussed in Section 3.2, is based on a branch and bound technique, which is commonly used to solve hard optimization problems. We present a general framework of the branch and bound algorithm and a general method of calculating the lower bound on the maximal regret of the optimal robust solution. The application of the branch and bound algorithm to particular problems will be shown in the next chapters of this monograph.