The elastic generalized assignment problem

The generalized assignment problem (GAP) has been studied by numerous researchers over the past 30 years or so. Simply stated, one must find a minimum-cost assignment of tasks to agents such that each task is assigned to exactly one agent and such that each agent's resource capacity is honoured. The problem is known to be NP-hard. In this paper, we study the elastic generalized assignment problem (EGAP). The elastic version of GAP allows agent resource capacity to be violated at additional cost. Another version allows undertime costs to be assessed as well if an agent's resource capacity is not used to its full extent. The EGAP is also NP-hard. We describe a special-purpose branch-and-bound algorithm that utilizes linear programming cuts, feasible solution generators, Lagrangean relaxation and subgradient optimization. We present computational results on a large collection of randomly generated 'hard' problems with up to 4000 binary variables.