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Title of Journal: Soft Comput

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Abbravation: Soft Computing

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Springer Berlin Heidelberg

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DOI

10.1002/tl.125

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1433-7479

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An analytical solution to the TOPSIS model with in

Authors: Xiuzhi Sang Xinwang Liu
Publish Date: 2015/01/24
Volume: 20, Issue: 3, Pages: 1213-1230
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Abstract

TOPSIS is a popular used model for multiple attribute decisionmaking problems Recently Chen and Lee Exp Syst Appl 3742790–2798 2010 extended TOPSIS method to interval type2 fuzzy sets IT2 FSs environment They first compute the ranking values of the elements in fuzzyweighted decision matrix and used the ranking values to compute the crisp relative closeness through traditional TOPSIS computing process Such ranking computation leads to the information loss of the weighted decision matrix In this paper we introduce an analytical solution to IT2 FSsbased TOPSIS model First we propose the fractional nonlinear programming NLP problems for fuzzy relative closeness Second based on Karnik–Mendel KM algorithm the switch points of the NLP models are identified and the analytical solution to IT2 FSsbased TOPSIS model can be obtained Compared with Chen and Lee’s method the proposed method operates the IT2 FSs directly and keeps the IT2 FSs formats in the whole process and the result of which is precise in analytical form In addition some properties of the proposed analytical method are discussed and the computing process is summarized as well To illustrate the analytical solution an example is given and the result is compared with that of Chen and Lee’s method Exp Syst Appl 3742790–2798 2010The authors are very grateful to the Associate Editor and the anonymous reviewers for their constructive comments and suggestions to help improve my paper This work was supported by the National Natural Science Foundation of China NSFC 71171048 and 71371049 and the Research Fund for the Doctoral Program of Higher Education of China 20120092110038From the conclusions of Eqs 55 56 it is deduced that when tildetildex lies in the left region tildexL and reaches its minimum maximum value tildefLtildexL reaches its maximum minimum value and widetildemathrmRCLtildexL reaches its minimum maximum value when tildetildex lies in the right region tildexR and reaches its minimum maximum value tildefRtildexR reaches its maximum minimum value and widetildemathrmRCRtildexR reaches its minimum maximum valueTherefore when i k mathrmLl the derivative of function g mathrmLltildec tilded to d i fracpartial g mathrmLltildec tildedpartial d i ge 0 On the other word g mathrmLltildec tilded increases when d i i le k mathrmLl increases Hence the computation of minimal g mathrmLltildec tilded must use the minimal d i i le k mathrmLl as stated in Eq 30

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