Combining Equity and Utilitarianism in a Mathematical Programming Model

We 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 maximizing 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 not only to health provision but to other areas as well. Building a mixed-integer/linear programming (MILP) formulation of the problem raises technical issues, because 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 healthcare planning problem and show that instances of realistic size are easily solved by standard MILP software.