TOPSIS for Multiple Objective Programming With Rough Decision Set

Abstract

Many optimization problems have competing objectives that require being optimized at the same time. These problems are called ”multiple objective programming problems (MOPPs)”. Real-world MOPPs may have some imprecision (roughness) in the decision set and/or the objective functions. These problems are known as ”rough MOPPs (RMOPPs)”. There is no unique method able to solve all RMOPPs. Accordingly, the decision maker (DM) should have more than one method for solving RMOPPs at his disposal so that he can select the most appropriate method. To contribute in this regard, we propose a new method for solving a specific class of RMOPPs in which all the objectives are precisely defined, but the decision set is roughly defined by its lower and upper approximations. Our proposed method is a modified version of the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). TOPSIS was chosen as the foundation for our method because it is one of the most widely applied methods for solving MOPPs. The basic concept underlying TOPSIS is that the compromise solution is closer to the ideal solution while also being farther away from the anti-ideal solution. The conventional TOPSIS can only solve MOPPs with precise (crisp) definitions of the two main parts of the problem. We extend TOPSIS to optimize multiple precise objectives over an imprecise decision set. The proposed approach is depicted in a flowchart. A numerical example is given to demonstrate the effectiveness of our proposed method to solve RMOPPs with a rough decision set at different values of objectives’ weights and using different Lp-metrics.