Fourier's Method of Linear Programming and its Dual

Abstract

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, 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 (' ) 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: