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 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 although also applicable in other areas. The building of such a model raises technical issues, as the combined objective is not only non-convex, but its epigraph is not MIP representable in its initial form. A method of making it MIP representable is given. We also show that the resultant integer programming model 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).