Minimization of Makespan in Generalized Assignment Problem

A quantitative combinatorial search problem consisting of m sources supplying in bulk to n(> m) destinations is considered in this paper. Each destination receives its full quota of a homogeneous product from a single source but a source can supply to many destinations subject to its capacity restrictions. If tij, which without loss of generality can be assumed to be a positive integer, is the time of transportation for shipping the goods from the source i to the destination j, then the total time used by the ith source is sum of the transportation time to the destinations that are served by this source. The time of transportation, called makespan, for a feasible solution of this generalized assignment problem (also called bulk transportation problem) is the maximum total time used by a source. The objective is to find a feasible solution that minimizes the makespan. A lexicographic search algorithm is developed which avoids scanning through all the possible feasible solutions though the search is exhaustive enough to obtain an optimal feasible solution at an early stage of the search. A starting upper bound on the value of objective function computed heauristically is used as a supplementary aid in reducing the search. Some results are established which, though of computational nature, help to reduce both the memory requirements and computational time.