Fairness criteria for allocating scarce resources
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Fairness criteria for allocating scarce resources Bismark Singh1 Received: 15 June 2019 / Accepted: 4 March 2020 © The Author(s) 2020
Abstract We develop an optimization model to provide a fair allocation of multiple resources to multiple users. All resources might not be suitable to all users. We develop a notion of fairness, and then provide a general class of functions achieving it. Next, we develop more restricted notions of fairness—special cases of which exist in literature. Finally, we distinguish between scarce and abundant resources, and show that if a resource is abundant, all users seeking it achieve the maximum possible coverage. Keywords Proportional fairness · Equity · Optimization · Welfare · Resource allocation · KKT conditions
Notation Sets and indices i ∈ I Types of users k ∈ K Types of resources i ∈ Ik Subset of users eligible to receive resource k k ∈ K i Subset of resources eligible for user i Data and parameters ni Population of user i, n i > 0 bk Amount of resource k, bk > 0 Original coverage of user i, 0 ≤ f i < 1 fi Weight of user i, wi > 0 wi Decision variables Amount of resource k allocated to user i xik yi Final coverage of user i
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Bismark Singh [email protected] Discrete Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany
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B. Singh
1 Background We study the problem of allocating different kinds of substitutable resources to different populations seeking them. This is related to a fundamental problem in economics—how should different resources be produced and made available to agents; see, e.g., [10]. Our aim is to allocate resources in a fair manner, a term we make precise below, and we develop a class of optimization models that achieve specific notions of fairness. This work finds applications in resource allocation problems that have been studied in specific settings; see, e.g., distribution of coal among power companies [2], several military and defense examples [11], multiperiod manufacturing of high-tech products [12], wireless networks [9], healthcare [6], education [3], and conservation of threatened species [8]. Special cases of this general problem include the so-called waterfilling algorithm; see, e.g., [13]. We present general results, based on the KKT optimality conditions, that provide essential conditions for fair allocation of resources, as well as special cases in which resources are abundant. For an introduction to resource allocation problems, see, e.g., [7]. An excellent introduction to different concepts in fairness is available in Bertsimas et al. [1]. Similar to our study, the authors consider a central decision planner seeking to allocate resources in a “fair” manner. The authors describe several concepts of fairness based on utilities of each user, and primarily distinguish between max-min fairness and proportional fairness. Our work differs from this existing body of literature in the following sense. We do not seek to maximize utilities, but rather to ensure equitable coverages (or, shortages) when allocatin
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