A labeling algorithm for the fuzzy assignment problem

This paper concentrates on the assignment problem where costs are not deterministic numbers but imprecise ones. Here, the elements of the cost matrix of the assignment problem are subnormal fuzzy intervals with increasing linear membership functions, whereas the membership function of the total cost is a fuzzy interval with decreasing linear membership function. By the max–min criterion suggested by Bellman and Zadeh, the fuzzy assignment problem can be treated as a mixed integer nonlinear programming problem. We show that this problem can usually be simplified into either a linear fractional programming problem or a bottleneck assignment problem. Here, we propose an efficient algorithm based on the labeling method for solving the linear fractional programming case. The algorithm begins with primal feasibility and proceeds to obtain dual feasibility while maintaining complementary slackness until the primal optimal solution is found. The computational results show that the proposed labeling algorithm offers an effective and efficient way for handling the fuzzy assignment problem.