Multiple objective linear programming

Abstract

Finally, we consider the problem of modifying the algorithm for listing efficient extreme points to identify the extreme points contained in each maximal efficient face. As each efficient extreme point- is considered, an algorithm of Ecker and Kouada can be used to identify the incident maximal efficient faces. We show that by finding a vector ?? > 0 corresponding to each such face (so that the face is exactly the set of optimal points for the program P??) we can then identify all the extreme points of the face. The information provided by the ??'s also helps to simplify the calculations involved in identifying the efficient faces incident to each point. The modification of the algorithm can be extended to handle the unbounded and the degenerate cases

The second modification deals with solving min {||M - Cx||∞ 1| x ε E}. Under certain conditions on M (e.g., {x| Cx ≥ M} = 0)this is equivalent to min {max (Mᵢ - Cᵢx) | x ε E}. We show that the latter problem can be solved by solving min {max (Mᵢ - Cᵢx) | Ax = b, x ≥ 0} to get a value W AND then finding the efficient points (using a known algorithm) of the program v-max {Cx| Ax = b, x ≥; 0, Cᵢx ≥ Mᵢ - w for all i}. Computational considerations are discussed which show that this second vector maximum problem is easier to solve than the original one. Alternatively, rather than solve this multiple objective program, we can transform it by eliminating objective functions which are constant over the new feasible region and then again find l∞-approximations to an ideal vector. Iteratively applying this scheme, under certain assumptions, we find a vector M < M such that there is an efficient x with Cx = M and if CᵢX > Mᵢ, then there is a j such that CⱼX < Mⱼ and Mⱼ - Mⱼ ≥ Mᵢ - Mᵢ.

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.)

Algorithms are available for solving the multiple objective linear program v-max {Cx| Ax = b, x ≥ 0} by finding the extreme points of the set E of efficient or nondominated solutions. Here we present three modifications of such algorithms. One enables the existing algorithms to handle degenerate feasible regions. The second applies the algorithm to a problem of choosing a subset of E. The third is an extension to identify maximal efficient faces.