Solving A Stochastic Generalized Assignment Problem with Branch and Price

In this chapter, we investigate the generalized assignment problem with the objective of finding a minimum-cost assignment of jobs to agents subject to capacity constraints. A complicating feature of the model is that the coefficients for resource consumption and capacity are random. The problem is formulated as a stochastic integer program with a penalty term associated with the violation of the resource constraints and is solved with a branch-and-price algorithm that combines column generation with branch and bound. To speed convergence, a stabilization procedure is included. The performance of the methodology was tested on four classes of randomly-generated instances. The principal results showed that the value of the stochastic solution (V SS), i.e., the gap between the stochastic solution and the expected value solution, was 35.5% on average. At the root node of the search tree, it was found that the linear programming relaxation of the master problem associated with column generation provided a much tighter lower bound than the relaxation of the original constraint-based formulation. In fact, two thirds of the test problems evidenced no gap between the optimal integer solution and the relaxed master problem solution. Additional testing showed that (1) initializing the master problem with a feasible solution outperforms the classical big-M approach; (2) SOS type 1 branching is superior to single-variable branching; and, (3) variable fixing based on reduced costs provides only a slight decrease in runtimes.