Abstract
The multiple objective simplex algorithm and its variants work in the decision variable space to find the set of all efficient extreme points of multiple objective linear programming (MOLP). Other approaches to the problem find either the entire set of all efficient solutions or a subset of them and also return the corresponding objective values (nondominated points). This paper presents an extension of the multiobjective simplex algorithm (MSA) to generate the set of all nondominated points and no redundant ones. This extended version is compared to Benson’s outer approximation (BOA) algorithm that also computes the set of all nondominated points of the problem. Numerical results on nontrivial MOLP problems show that the total number of nondominated points returned by the extended MSA is the same as that returned by BOA for most of the problems considered.
Highlights
Multiobjective linear programming seeks to optimize two or more linear objective functions subject to a set of linear constraints with a view of obtaining either all the efficient solutions or nondominated points or a subset of them, or a most preferred solution depending on the approach adopted
Like other simplex-type algorithms, the extended multiobjective simplex algorithm (MSA) or EMSA is not an exception. e extended MSA may exhibit one or more redundancies whenever the problem is degenerate. e first is that EMSA may find the same efficient solutions in more than one iteration or find different efficient solutions that leads to the same nondominated point. is can be seen clearly in Table 1, as the number of efficient solutions returned by EMSA is the same as that returned by the original MSA, though this is corrected in the computation of the corresponding nondominated points as the Origin n m q
We recorded the number of efficient solutions (NES) returned by MSA, the number of nondominated points (NNP) returned by EMSA, and the NNP returned by Benson’s outer approximation (BOA) for each problem
Summary
Multiobjective linear programming seeks to optimize two or more linear objective functions subject to a set of linear constraints with a view of obtaining either all the efficient solutions or nondominated points or a subset of them, or a most preferred solution depending on the approach adopted. Is algorithm works in the decision variable space to find the entire set of all efficient solutions. It was noted in [6] that finding the nondominated points instead of the efficient set is more important for the DM. Advances in Operations Research compare this extended version with the original one and with the primal variant of Benson’s outer approximation (BOA) algorithm [8] which is an objective space based method that computes the set of all nondominated points of the problem.
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