On Equitable Resource Allocation Problems: A Lexicographic Minimax Approach

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.

The Price of Fairness

“…Max-min fairness was first implemented in networking and telecommunications applications and has also initiated a lot of research in this area (see Bertsekas and Gallager (1987), Bonald and Massoulié (2001), Luss (1999)). It has many applications in bandwidth allocation, routing and load balancing problems, as well as in general resource allocation or multiobjective optimization problems.…”

Combining Equity and Utilitarianism in a Mathematical Programming Model

“…These and others are discussed by Tempkin (1993). Maximin and lexicographic maximum functions aim to capture a Rawlsian fairness criterion and are discussed, for example, by Blackorby et al (2002), Daniels (1989), Dworkin (1977), Luss (1999), Roemer (1998), and Stein (2006). Structural properties of optimal solutions for utilitarian and lexmax objectives are derived in Hooker (2010).…”

A multi-criteria approach to fair and efficient bandwidth allocation☆

“…This paper deals with problems of bandwidth allocation within telecommunication networks. Therefore, we focus on the approaches that attempt to provide a fair (equal) treatment of all the activities (demands) [4,5] while allocating resources.…”