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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.

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Introduction. There has been widespread popular interest in recent years in suggested improved methods for solving Linear Programming (LP) models. In 1977 Shor [13] described a new algorithm for LP. Khachian [7] modified this algorithm in order to prove that the number of computational steps was, in the worst case, bounded by a polynomial function of the size of the data. This method has become known as the Ellipsoid Method. It has in practice been disappointing in experimental computational performance. In 1984 Karmarker [6] produced another algorithm which was also "polynomially bounded" with spectacular practical computational claims. Controversy continues as to whether Karmarker's method will displace the Simplex Method. The Simplex Method was invented by Dantzig in 1948 and is well explained in Dantzig [1]. Although it is not polynomial in the worst case it has proved a remarkably powerful method in practice and its major extension, the Revised Simplex Method, is the method used in all commercial systems.The reason for the widespread popular interest (both Khachian and Karmarker's methods received headlines in the national press) is that LP models are among the most widely used type of Mathematical Model. Applications of LP arise in Manufacturing, Distribution, Finance, Agriculture, Health, Energy and general Resource Planning. A practical discussion of application areas is contained in Williams [16].In this article we show that, predating all these methods, a method discovered by Fourier in 1826 for manipulating linear inequalities can be adapted to Solving Linear Programming models. The theoretical insight given by this method is demonstrated as well as its clear geometrical interpretation. By considering the dual of a linear programming model it is shown how the method gives rise to a dual method. This dual method generates all extreme solutions (including the optimal solution) to a linear programme. Therefore if a polytope is defined in terms of its facets the dual of Fourier's method provides a method of obtaining all vertices.An LP model consists of variables (e.g., xI, x2,...,etc.) contained in a linear expression known as an objective function. Values are sought for the variables which maximise or minimise the objective function subject to constraints. These constraints are themselves linear expressions which must be either less-than-or-equal to ('<), greater-than-or-equal to (>) or equal to (=) some specified value. For example, the following is a small LP model.Find values for xl, x2,. . . among the real numbers so as to:Maximise -4x, + 5x2 + 3x3 P subject to -xl + X2 -X3 < 2 constraints x1 + ? X2 + 2x3 < 3? Xl>0, X 2 > 0,1 x3 >1It is usually the case that the var...