Some new similarity measures for intuitionistic fuzzy values and their application in group decision making

Abstract

We first propose a series of similarity measures for intuitionistic fuzzy values (IFVs) based on the intuitionistic fuzzy operators (Atanassov 1995). The parameters in the proposed similarity measures can control the degree of membership and the degree of non-membership of an IFV, which can reflect the decision maker’s risk preference. Moreover, we can obtain some known similarity measures when some fixed values are assigned to the parameters. Furthermore, we apply the similarity measures to aggregate IFVs and develop some aggregation operators, such as the intuitionistic fuzzy dependent averaging operator and the intuitionistic fuzzy dependent geometric operator, whose prominent characteristic is that the associated weights only depend on the aggregated intuitionistic fuzzy arguments and can relieve the influence of unfair arguments on the aggregated results. Based on these aggregation operators, we develop some group decision making methods, and finally extend our results to interval-valued intuitionistic fuzzy environment.

Keywords

Intuitionistic fuzzy value / interval-valued intuitionistic fuzzy value / similarity measure / aggregation operator

References

[1] Atanassov K.. Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 1986, 20: 87-96.

[2] Atanassov K., Gargov G.. Interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems, 1989, 31: 343-349.

[3] Atanassov K.. Remark on the intuitionistic fuzzy sets — III. Fuzzy Sets and Systems, 1995, 75: 401-402.

[4] Bustince H., Burillo P.. Vague sets are intuitionistic fuzzy sets. Fuzzy Sets and Systems, 1996, 79: 403-405.

[5] Burillo P., Bustince H.. Construction theorems of intuitionistic fuzzy sets. Fuzzy Sets and Systems, 1996, 84: 271-281.

[6] Chen S.M., Tan J.M.. Handling multicriteria fuzzy decision making problems based on vague set theory. Fuzzy Sets and Systems, 1994, 67: 163-172.

[7] Chen T.Y.. An outcome-oriented approach to multicriteria decision analysis with intuitionistic fuzzy optimistic / pessimistic operators. Expert Systems with Applications, 2010, 37: 7762-7774.

[8] Filev D.P., Yager R.R.. On the issue of obtaining OWA operator weights. Fuzzy Sets and Systems, 1998, 94: 157-169.

[9] Fischer R., Chalmers A.. Is optimism universal? A meta-analytical investigation of optimism levels across 22 nations. Personality and Individual Differences, 2008, 45: 378-382.

[10] Gau W.L., Buehrer D.L.. Vague sets. IEEE Transactions on Systems, Man, and Cybernetics, 1993, 23: 610-614.

[11] Grzegorzewski P.. Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Sets and Systems, 2004, 148: 319-328.

[12] Herrera F., Herrera-Viedma E.. Aggregation operators for linguistic weighted information. IEEE Transactions on Systems, Man, and Cybernetics, 1997, 27: 646-656.

[13] Hong D.H., Choi C.H.. Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems, 2000, 114: 103-113.

[14] Hung W.L., Yang M.S.. Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance. Pattern Recognition Letters, 2004, 25: 1603-1611.

[15] Hung W.L., Yang M.S.. Similarity measures of intuitionistic fuzzy sets based on Lp metric. International Journal of Approximate Reasoning, 2007, 46: 120-136.

[16] Hung W.L., Yang M.S.. On the J-divergence of intuitionistic fuzzy sets with its application to pattern recognition. Information Sciences, 2008, 178: 1641-1650.

[17] Hung W.L., Yang M.S.. On similarity measures between intuitionistic fuzzy sets. International Journal of Intelligent Systems, 2008, 23: 364-383.

[18] Li D.F., Cheng C.T.. New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recognition Letters, 2002, 23: 221-225.

[19] Liang Z.Z., Shi P.F.. Similarity measures on intuitionistic fuzzy sets. Pattern Recognition Letters, 2003, 24: 2687-2693.

[20] Li Y.H., Olson D.L., Qin Z.. Similarity measures between intuitionistic fuzzy (vague) sets: a comparative analysis. Pattern Recognition Letters, 2007, 28: 278-285.

[21] Liu H.W., Wang G.J.. Multi-criteria decision making methods based on intuitionistic fuzzy sets. European Journal of Operational Research, 2007, 197: 220-233.

[22] Mitchell H.B.. On the Dengfeng-Chuntian similarity measure and its application to pattern recognition. Pattern Recognition Letters, 2003, 24: 3101-3104.

[23] Ngwenyama O., Bryson N.. Eliciting and mapping qualitative preferences to numeric rankings in group decision making. European Journal of Operational Research, 1999, 116: 487-497.

[24] O’Hagan, M. (1998). Aggregating template rule antecedents in real-time expert systems with fuzzy set logic. In: Proceedings of the 22nd Annnual IEEE Asilomar Conference on Signals, Systems and Computers, 681–689, Pacific Grove, CA

[25] Szmidt E., Kacprzyk J.. Distances between intuitionistic fuzzy sets. Fuzzy Sets and Systems, 2000, 114: 505-518.

[26] Szmidt, E. & Kacprzyk, J. (2006). Distances between intuitionistic fuzzy sets: straightforward approaches may not work. In: 3rd Int. IEEE Conf. Intelligent Systems, 716–721

[27] Xu Z.S., Da Q.L.. The ordered weighted geometric averaging operators. International Journal of Intelligent Systems, 2002, 17: 709-716.

[28] Xu Z.S.. Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment. Information Sciences, 2004, 168: 171-184.

[29] Xu Z.S.. An overview of methods for determining OWA weights. International Journal of Intelligent Systems, 2005, 20: 843-865.

[30] Xu Z.S., Yager R.R.. Some geometric aggregation operators based on intuitionistic fuzzy sets. International Journal of General Systems, 2006, 35: 417-433.

[31] Xu Z.S.. Dependent OWA operators. Lecture Notes in Artificial Intelligence, 2006, 3885: 172-178.

[32] Xu Z.S.. Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making. Fuzzy Optimization and Decision Making, 2007, 6: 109-121.

[33] Xu Z.S.. Intuitionistic fuzzy aggregation operators. IEEE Transactions on Fuzzy Systems, 2007, 15: 1179-1187.

[34] Xu Z.S.. On similarity measures of interval-valued intuitionistic fuzzy set and their application to pattern recognitions. Journal of Southeast University (English Edition), 2007, 23: 139-143.

[35] Xu Z.S.. Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making. Control and Decision, 2007, 22: 215-219.

[36] Xu Z.S., Chen J.. On geometric aggregation over interval-valued intuitionistic fuzzy information. The 4th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD’07), 2007, 2: 466-471.

[37] Xu Z.S., Chen J.. An overview of distance and similarity measures of intuitionistic fuzzy sets. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2008, 16: 529-555.

[38] Xu Z.S.. Dependent uncertain ordered weighted aggregation operators. Information Fusion, 2008, 9: 310-316.

[39] Yager R.R.. Induced aggregation operators. Fuzzy Sets and Systems, 2003, 137: 59-69.

[40] Yager R.R.. Generalized OWA aggregation operators. Fuzzy Optimization and Decision Making, 2004, 3: 93-107.

[41] Yager R.R., Kacprzyk J.. The Ordered Weighted Averaging Operators. Theory and Applications, 1997, Norwell, MA: Kluwer

[42] Yager R.R.. Choquet aggregation using order inducing variables. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2004, 12: 69-88.

[43] Yager R.R.. OWA aggregation over a continuous interval argument with applications to decision making. IEEE Transactions on Systems, Man, and Cybernetics, 2004, 34: 1952-1963.

[44] Yager R.R., Xu Z.S.. The continuous ordered weighted geometric operator and its application to decision making. Fuzzy Sets and Systems, 2006, 157: 1393-1402.

[45] Yager R.R., Filev D.P.. Induced ordered weighted averaging operators. IEEE Transactions on Systems, Man, and Cybernetics, 1999, 29: 141-150.

[46] Yager R.R.. Families of OWA operators. Fuzzy Sets and Systems, 1993, 59: 125-148.

[47] Zhao H., Xu Z.S., Ni M.F., Liu S.S.. Generalized aggregation operators for intuitionistic fuzzy sets. International Journal of Intelligent Systems, 2010, 25: 1-30.