Multiple objective linear programming

Abstract

It is well known that a point x is efficient if and only if there is a vector ?? > 0, with each ??ᵢ > 0, such that x is optimal for the program P??: max {??TCx| Ax = b, x ≥ 0}. In the degenerate case, if x is an extreme point which solves P??,then there is at least one tableau T representing x with corresponding cost coefficient matrix C such that - ??TC ≥ 0. If we apply the available algorithms allowing only those tableaux T for which {?? > 0| - ??TC ≥ 0} is nonempty and breaking ties in the choice of pivot row by some rule used to avoid cycling in the Simplex Method, the algorithm will converge. For the lexico-feasible tie breaking rule, we give a proof based on the results in the nondegenerate case. To justify the use of other tie breaking rules, we reprove for the degenerate case the result that E is connected: i.e., given a tableau T such that {?? > 0| - ??TC ≥ 0} is nonempty and an efficient extreme point x, there is a tableau T representing x such that a series of pivots connects T and T and each pivot is between two tableaux which both satisfy - μTC ᵢ ≥ 0 for some μ > 0. (Such a pivot corresponds to an efficient edge if it is nondegenerate.)