On Equitable Resource Allocation Problems: A Lexicographic Minimax Approach

TLDR

Resource allocation problems arise in areas such as distribution planning, production scheduling, and emergency services location. The paper reviews resource allocation problems that require equitable distribution of limited resources and outlines efficient algorithms for solving them. Equitable solutions are defined via performance functions and lexicographic minimax, and the paper discusses efficient algorithms for large‑scale, multi‑knapsack, multi‑period, and substitutable‑resource allocation problems. The authors anticipate that the review will aid practitioners in formulating and solving diverse allocation problems and inspire researchers to develop new models and algorithms.

Abstract

In this expository paper, we review a variety of resource allocation problems in which it is desirable to allocate limited resources equitably among competing activities. Applications for such problems are found in diverse areas, including distribution planning, production planning and scheduling, and emergency services location. Each activity is associated with a performance function, representing, for example, the weighted shortfall of the selected activity level from a specified target. A resource allocation solution is called equitable if no performance function value can be improved without either violating a constraint or degrading an already equal or worse-off (i.e., larger) performance function value that is associated with a different activity. A lexicographic minimax solution determines this equitable solution; that is, it determines the lexicographically smallest vector whose elements, the performance function values, are sorted in nonincreasing order. The problems reviewed include large-scale allocation problems with multiple knapsack resource constraints, multiperiod allocation problems for storable resources, and problems with substitutable resources. The solution of large-scale problems necessitates the design of efficient algorithms that take advantage of special mathematical structures. Indeed, efficient algorithms for many models will be described. We expect that this paper will help practitioners to formulate and solve diverse resource allocation problems, and motivate researchers to explore new models and algorithmic approaches.