A note on the TOPSIS method in MADM problems with linguistic evaluations

• The ranking ambiguity of the extended fuzzy TOPSIS method in MADM is revealed.
• Graphical interpretations are demonstrated to fully stress the ranking inadequacy.
• Two supplemental indexes are formulated to assist the extended fuzzy TOPSIS method.
• The sufficiency of the two indexes plus the original fuzzy TOPSIS method is proved.
• Impacts caused by the factors such as alternative quantity are discussed.

The TOPSIS method, commonly known as the technique for order preference by similarity to ideal solutions, is one of the most popular approaches used in multi-attribute decision making (MADM). The fundamental procedure of the traditional TOPSIS method is rather straightforward, the ranking position of an alternative depends on its relative closeness to the positive ideal solution and the negative ideal solution, respectively. In order to encompass uncertain and ambiguous decision elements, an extension of the original TOPSIS method has been coined. With the help of fuzzy sets based TOPSIS, an overwhelming trend of fuzzy decision making applications has been witnessed. In the present work, however, it is found that the extended fuzzy TOPSIS method is unable to distinguish all the different alternatives under linguistic environment. Moreover, the undistinguishable alternatives are countless in quantity, and they have formed specific patterns with respect to the parameters of TOPSIS methods. To dampen this ranking ambiguity, we designed a set of supplemental methods to construct a revised TOPSIS approach with linguistic evaluations. Correspondingly, the sufficiency of the revised TOPSIS method to guarantee total orders has been proven. Furthermore, a numerical example concerning the production line improvement of a manufacturing company is demonstrated to validate the feasibility and supremacy of the proposed method. Finally, a series of further discussions are performed to shed some lights on the impacts caused by the changes of the alternative quantity, the attribute quantity, and the type of linguistic term.

These figures demonstrate the areas where the extended fuzzy TOPSIS method cannot manage to differentiate the alternatives. The TOPSIS method is based on the ratio formula (11), various λ values can cause the shape to change dramatically. Figure optionsDownload as PowerPoint slide