A specific minimax programming with the max-plus inequalities constraints

Abstract

Recently, the max-plus inequalities have been introduced for describing the time constraints of the workpiece transmission in a factory workshop. Each solution of the max-plus inequalities system represents actually a feasible time scheme in the factory workshop. For searching all the feasible time schemes, we attempt to solve the corresponding max-plus inequalities system. An effective approach, based on the so-called row-nonnegative matrix, is proposed to find out all the solutions to the max-plus inequalities system. Besides, considering the practical application background, the system manager usually hopes to postpone the processing finishing times. To coincide with such an objective, we establish a relevant minimax programming problem, in which the objective function is to minimize the maximum of all variables, while the constraint is the above-mention system of max-plus inequalities. A detailed minimum-discriminant-matrix-based algorithm is provided for searching an optimal solution of our studied minimax programming. Moreover, we further develop a novel approach to compute all its optimal solution(s), i.e., the whole optimal solution set. Our proposed resolution approaches are illustrated by some numerical examples.