An LP-based heuristic procedure for the generalized assignment problem with special ordered sets

• Authors: Alan P. French;John M. Wilson
• Venue: Computers and Operations Research
• Year: 2007

The generalized assignment problem with special ordered sets (GAPS2), is the problem of allocating n tasks to m time-periods, where each task must be assigned to a time-period, or shared between two consecutive time-periods. For reasonably large values of m and n the NP-hard combinatorial problem GAPS2 becomes intractable for standard mathematical programming software, hence there is a need for heuristic algorithms to solve such problems. It will be shown how an LP-based heuristic developed previously for the well-established generalized assignment problem can be modified and extended to solve GAPS2. Encouraging results, in terms of speed and accuracy, in particular when compared to an existing heuristic for GAPS2, are described. Scope and purpose: This paper considers an extension of the hard combinatorial problem-the generalized assignment problem (GAP). GAP is the problem of assigning to best advantage a set of jobs to a set of machines where each job must be assigned to exactly one machine, but each machine, which has limited capacity, may have more than one job assigned to it. For practical cases the terms 'jobs' and 'machines' are just indicative of the sets of objects to be assigned and the sets of objects to which they are to be assigned. In the extension to GAP considered in this paper, the assignment is of tasks to time periods, and tasks are permitted to be split across two periods, if required, provided the time periods are consecutive. This problem is termed GAPS2. A heuristic is developed to solve GAPS2 after it is established that there is unlikely to be any exact method which will solve the problem. The heuristic turns out to be very effective and the results of computational testing indicate that near-optimal solutions can be derived rapidly to instances of GAPS2 of substantial size. Although the linear programming based heuristic is relatively straightforward, the contribution of this paper is to show that very good quality solutions can be obtained rapidly to difficult combinatorial problems of substantial size. The described heuristic considerably outperforms the only other published heuristic for GAPS2.