Resource constrained assignment problems

In many applications it is necessary to find a minimum weight assignment that satisfies one or several additional resource constraints. For example, consider the problem of assigning persons to jobs where each assignment utilizes at least two scarce resources and the resource utilization is dependent on the person and the type of task. A practical situation where the above might occur is a slaughter house where the "cutters" are assigned to different cut patterns. In this case the resources are the time, the cost and the productivity measured in terms of quality and amount of the end products. In this paper we study the resource constrained assignment problem and derive several classes of valid inequalities based on the properties of the knapsack and assignment polytopes. We also present an algorithm that uses both the linear programming and the Lagrangean relaxation of the original problem in order to solve the separation problem. Some computational experiments are presented.