Lagrangean/surrogate relaxation for generalized assignment problems

This work presents Lagrangean/surrogate relaxation to the problem of maximum pro®t assignment of n tasks to m agents (n > m), such that each task is assigned to only one agent subject to capacity constraints on the agents. The Lagrangean/surrogate relaxation combines usual Lagrangean and surrogate relaxations relaxing ®rst a set of constraints in the surrogate way. Then, the Lagrangean relaxation of the surrogate constraint is obtained and approximately optimized (one-dimensional dual). The Lagrangean/surrogate is compared with the usual Lagrangean relaxation on a computational study using a large set of instances. The dual bounds are the same for both relaxations, but the Lagrangean/surrogate can give improved local bounds at the application of a subgradient method, resulting in less computational times. Three relaxations are derived for the problem. The ®rst relaxation considers a vector of multipliers for the capacity constraints, the second for the assignment constraints and the other for the Lagrangean decomposition constraints. Relaxation multipliers are used with ecient constructive heuristics to ®nd good feasible solutions. The application of a Lagrangean/surrogate approach seems very promising for large scale problems. Ó 1999 Elsevier Science B.V. All rights reserved.