Solving 0-1 Integer Programming Problems Arising from Large Scale Planning Models

We present methods that are useful in solving some large scale hierarchical planning models involving 0-1 variables. These 0-1 programming problems initially could not be solved with any standard techniques. We employed several approaches to take advantage of the hierarchical structure of variables (ordered by importance) and other structures present in the models. Critical, but not sufficient for success, was a strong linear programming formulation. We describe methods for strengthening the linear programs, as well as other techniques necessary for a commercial branch-and-bound code to be successful in solving these problems. T HIS STUDY was motivated by a desire to solve some large, important integer programming problems that arose in planning applications at General Motors. The problems were pure 0-1 programming problems from planning models involving project selection with fixed charges subject to various constraints. We describe the structure of these problems in more detail in Section l. Solving the larger problems was initially beyond the capability of current codes; finding solutions that were even feasible was often impossible. In the process of testing the code described in this paper, a set of benchmark problems were run on another commercial code and on an enumeration code (Ibaraki et al. [1972], Young et al. [1977]). For the smallest problem, all performed well, but on the medium-sized and large problems only the code described in this paper was able to find the optimum answer and develop close bounds, or to prove optimality. The methods used can be divided into three phases: logical methods applied before linear programming; constraint generation in conjunction Subject classification: 181 planning models. 625 integer programming algorithms. 804 Johnson et al. with linear programming; and specialized branch-and-bound for solving the preprocessed integer optimization problem .. Sections 2, 3, and 4 describe the three main parts of the logical preprocessing done before linear programming. Section 5 summarizes the constraint generation phase, and Sections 6 and 7 deal with the branch-and-bound strategy used. Section 8 presents the computational results on four benchmark problems, and Section 9 gives our conclusions.