Using an enhanced integer NSGA-II for solving the multiobjective Generalized Assignment Problem

The traditional Generalized Assignment Problem (GAP) problem consists of assigning n different tasks to m different agents, while minimizing a cost function. Additionally, it is necessary to ensure that each task is assigned to a single agent (indivisible task) and that the maximum resource capacity of the agents is honored. In this paper, the problem is extended to a bi-objective formulation, in which an equilibrium function is included in the problem statement. This formulation is motivated by situations in which it is important to distribute the tasks uniformly amongst the agents. An integer enhanced version of the Non-dominated Sorting Genetic Algorithm II (NSGA-II) algorithm is proposed for solving such a multiobjective problem. The results obtained using this algorithm show that it is possible to find solutions which are very close to the exact optimum of the single objective problem. The approach still allows to perform a trade-off analysis of the objectives, offering the possibility of choosing solutions with slightly higher cost and considerably better distribution of the tasks. Such a trade-off decision cannot be performed in the mono-objective approaches.