A comparative study of two key algorithms in multiple objective linear programming

Abstract

Multiple objective linear programming problems are solved with a variety of algorithms. While these algorithms vary in philosophy and outlook, most of them fall into two broad categories: those that are decision space-based and those that are objective space-based. This paper reports the outcome of a computational investigation of two key representative algorithms, one of each category, namely the parametric simplex algorithm which is a prominent representative of the former and the primal variant of Bensons Outer-approximation algorithm which is a prominent representative of the latter. The paper includes a procedure to compute the most preferred nondominated point which is an important feature in the implementation of these algorithms and their comparison. Computational and comparative results on problem instances ranging from small to medium and large are provided.

A Comparison of Benson’s Outer Approximation Algorithm with an Extended Version of Multiobjective Simplex Algorithm

“…Y N � Cx: x ∈ X E and Y WN � Cx: x ∈ { X WE } are the nondominated and weakly nondominated sets in the objective space of (2), respectively. e nondominated faces in the objective space of the problem constitute the nondominated frontier and the efficient faces in the decision space of the problem constitute the efficient frontier [1].…”

“…In [1], we presented the results of a more detailed computational investigation of the MSA in [30] and BOA [8] using existing small, medium, and realistic MOLP instances to evaluate the robustness and quality of the most preferred nondominated point (MPNP) returned by these two algorithms which was not discussed or considered in [30]. Also presented in [1] was a formal procedure for the computation of a MPNP of the problem.…”

“…In [1], we presented the results of a more detailed computational investigation of the MSA in [30] and BOA [8] using existing small, medium, and realistic MOLP instances to evaluate the robustness and quality of the most preferred nondominated point (MPNP) returned by these two algorithms which was not discussed or considered in [30]. Also presented in [1] was a formal procedure for the computation of a MPNP of the problem. Numerical results on the robustness, efficiency, and quality of a MPNP show that BOA outperforms PSA in terms of the quality of a MPNP it returns and robustness, as well as confirming what was reported in [30] that BOA is computationally more efficient than PSA on highly degenerate problems, while PSA is superior to BOA computationally on nondegenerate MOLP problems.…”

“…Prominent among them is Benson's outer approximation algorithm [8]. As was noted in [1], the author who presented an account of decision space based methods proposed an algorithm for computing the nondominated set in the objective space of the problem. According to him, his method is the first of its kind.…”

“…We noted in [1] that "in practice, MOLP is typically solved by the Decision-Maker (DM) in conjunction with the analyst who looks for a most preferred solution in the feasible region X. is is because optimizing all the objective functions at the same time is not possible due to their conflicting nature. Consequently, the concept of optimality is replaced with that of efficiency.…”