Linear Programming Part I: The Simplex Method

Abstract

Linear programming (LP) problems occur in a diverse range of real-life applications in economic analysis and planning, operations research, computer science, medicine, and engineering. In such problems, it is known that any minima occur at the vertices of the feasible region and can be determined through a ‘brute-force’ or exhaustive approach by evaluating the objective function at all the vertices of the feasible region. However, the number of variables involved in a practical LP problem is often very large and an exhaustive approach would entail a considerable amount of computation. In 1947, Dantzig developed a method for the solution of LP problems known as the simplex method [1][2]. Although in the worst case, the simplex method is known to require an exponential number of iterations, for typical standard-form problems the number of iterations required is just a small multiple of the problem dimension [3]. For this reason, the simplex method has been the primary method for solving LP problems since its introduction.