Combining Equity and Utilitarianism in a Mathematical Programming Model

Abstract

Revised 18 November 2011 AbstractWe discuss the problem of combining the conflicting objectives of equity and utilitarianism, for social policy making, in a single mathematical programming model. The definition of equity we use is the Rawlsian one of maximising the minimum utility over individuals or classes of individuals. However, when the disparity of utility becomes too great, the objective becomes progressively utilitarian. Such a model is particularly applicable to health provision but to other areas as well. Building a mixed integer/linear programming (MILP) formulation of the problem raises technical issues, as the objective function is nonconvex and the hypograph is not MILP representable in its initial form. We present a succinct formulation and show that it is "sharp" in the sense that its linear programming relaxation describes the convex hull of the feasible set (before extra resource allocation or policy constraints are added). We apply the formulation to a health care planning problem and show that instances of realistic size are easily solved by standard MILP software.

Inequity averse optimization in operational research

“…Applications include bandwidth or channel allocation (Tomaszewski, 2005;Lee et al, 2004;Lee & Cho, 2007;Luss, 2008;Salles & Barria, 2008;Ogryczak et al, 2008;Luss, 2010;Luss, 2012a;Jeong et al, 2005;Chang et al, 2006;Zukerman et al, 2008;Morell et al, 2008;Zhang & Ansari, 2010;Bonald et al, 2006;Heikkinen, 2004;Ogryczak et al, 2005;Kunqi et al, 2007), water rights allocation (Udías et al, 2012), health care planning (Earnshaw et al, 2007;Demirci et al, 2012;Hooker & Williams, 2012;Bertsimas et al, 2013), WIP (Kanban) allocation in production systems (Ryan & Vorasayan, 2005), fixed cost allocation (Li et al, 2013;Butler & Williams, 2006), and public resource allocation such as allocating voting machines to election precincts . There are also studies that consider general resource allocation settings such as Bertsimas et al (2011), Hooker (2010, Nace and Orlin (2007), Medernach and Sanlaville (2012) and Bertsimas et al (2014).…”

“…Bertsimas et al (2014) propose a modeling framework for general dynamic resource allocation Table 2 Classical problems in OR re-considered with equity concerns. Tomaszewski (2005), Lee, Moon, and Cho (2004), Lee and Cho (2007), Luss (2008), Salles and Barria (2008), Ogryczak, Wierzbicki, and Milewski (2008), Luss (2010), Luss (2012a), Jeong, Kim, and Lee (2005), Chang, Lee, and Kim (2006), Zukerman, Mammadov, Tan, Ouveysi, and Andrew (2008), Morell, Seco-Granados, and Vázquez-Castro (2008), Zhang and Ansari (2010), Bonald, Massoulié, Proutière, andVirtamo (2006), Heikkinen (2004), Ogryczak, Pioro, and Tomaszewski (2005), Udías, Ríos Insua, Cano, and Fellag (2012), Earnshaw, Hicks, Richter, and Honeycutt (2007), Demirci, Schaefer, Romeijn, and Roberts (2012), Hooker and Williams (2012), Bertsimas, Farias, and Trichakis (2013), Ryan and Vorasayan (2005), Li, Yang, Chen, Dai, and Liang (2013), Butler and Williams (2006), Yang, Allen, Fry, and Kelton (2013), Trichakis (2011), Bertsimas, Farias, andTrichakis (2012), Hooker (2010), Nace and Orlin (2007), Medernach and Sanlaville (2012), Bertsimas, Gupta, and Lulli (2014), Karsu and Morton (2014), Johnson, Turcotte, and Sullivan (2010), Kozanidis (2009), Eiselt and Marianov (2008), Vossen and Ball (2006), …”

“…For example, Hooker and Williams (2012) consider allocation of utilities to individuals (or classes of individuals) and propose a weakly Schur-concave aggregation function to be maximized. The function is based on the idea of combining objectives of equity -they use a Rawlsian approach-and efficiency.…”

“…The specific functional forms used are context dependent and different forms are adopted in different studies. Two common properties of these functions are that they are increasing or nondecreasing (in a maximization problem) and inequity-averse in the sense that they satisfy PD, though sometimes in a weak way as in Hooker and Williams (2012). Considering the aggregation function approach from a multicriteria perspective, one can relate such functions to the DM's preferences and specify a set of properties that an equity-averse DM's preference model should satisfy.…”

Multi-objective decision-making for dietary assessment and advice

“…We refer to this problem as a general 0-1 fractional programming problem with conditional fractional terms, because the fractional termsrequires weighting again. The problem is how to justify and interpret weights (Korhonen et al, 2013, Hooker andWilliams, 2012). Considering the key importance of preference structure (as implemented via achievement functions and weights) for the quality of decision-making in MCDM, it is crucial to keep an open mind for novel approaches.…”