Minimax resource allocation problems with resource-substitutions represented by graphs

Resource allocation problems focus on the allocation of limited resources among competing activities. The authors examine such problems when certain substitutions among resources are possible. The substitutional relations can be represented by a graph comprised of multiple components. In each component, the nodes correspond to resources and the arcs correspond to feasible substitutions. The objective is of the minimax form, where each term is a continuous, strictly decreasing function of a single activity level. The objective is to minimize the largest term, subject to a limited supply of multiple resources. Potential applications to such problems are found, for example, in the manufacture of high tech products. The authors develop an efficient algorithm to solve such problems. At each iteration, a relaxed minimax problem is solved. A max-flow algorithm applied to determine whether the solution of the relaxed problem is feasible for the original problem. If the solution is infeasible, a tighter relaxed problem is formulated and resolved. The algorithm is also extended to find the lexicographic minimax solution. Computational results are presented.