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Entropy of interval-valued intuitionistic hesitant fuzzy set and its application to group decision making problems

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Abstract

Very often, there exist some situations with a high degree of uncertainty and fuzziness when a group of decision makers are not agreeing on common membership and non-membership values. In such cases, interval-valued intuitionistic hesitant fuzzy set (IVIHFS) is useful in multi-criteria group decision making (MCGDM) problem to represent the assessments of decision criteria. Entropy has been an important and useful tool for MCGDM methods in measuring fuzziness in information and to quantify the uncertain information. In the present study, we have defined entropy measures for IVIHFS along with a novel approach to aggregate interval valued hesitant fuzzy (IVHF) information into interval valued intuitionistic hesitant fuzzy (IVIHF) information. To do this we have defined satisfactory and dissatisfactory intervals in the present study. Proposed aggregation method is used in expansion of a MCGDM in which criteria weights are determined using proposed IVIHF entropy measure. A descriptive example of candidate selection problem is taken in the study to explain methodology of proposed MCGDM method. A comparison analysis is also performed with validity test to validate the performance of proposed IVIHF entropy based MCGDM method.

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References

  • Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96

    Article  Google Scholar 

  • Atanassov KT, Gargov G (1989) Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31:343–349

    Article  MathSciNet  Google Scholar 

  • Burillo P, Bustince H (1996) Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy sets Syst 78(3):305–316

    Article  MathSciNet  Google Scholar 

  • Chatterjee K, Kar S (2017) Unified granular-number-based AHP-VIKOR multi-criteria decision framework. Granular Comput 2:199–221

    Article  Google Scholar 

  • Chen SM (1998) Aggregating fuzzy opinions in the group decision-making environment. Cybern Syst 29(4):363–376

    Article  Google Scholar 

  • Chen SW, Cai LN (2013) Interval-valued hesitant fuzzy sets. Fuzzy Syst Math 6:007

    Google Scholar 

  • Chen SM, Chang TH (2001) Finding multiple possible critical paths using fuzzy PERT. IEEE Trans Syst Man Cybern Part B (Cybern) 31(6):930–937

    Article  Google Scholar 

  • Chen SM, Chang CH (2015) A novel similarity measure between Atanassov’s intuitionistic fuzzy sets based on transformation techniques with applications to pattern recognition. Inf Sci 291:96–114

    Article  Google Scholar 

  • Chen SM, Chang CH (2016) Fuzzy multiattribute decision making based on transformation techniques of intuitionistic fuzzy values and intuitionistic fuzzy geometric averaging operators. Inf Sci 352:133–149

    Article  Google Scholar 

  • Chen SM, Chiou CH (2015) Multiattribute decision making based on interval-valued intuitionistic fuzzy sets, PSO techniques, and evidential reasoning methodology. IEEE Trans Fuzzy Syst 23(6):1905–1916

    Article  Google Scholar 

  • Chen SM, Chung NY (2006) Forecasting enrollments of students using fuzzy time series and genetic algorithms. Int J Inf Manag Sci 17(3):1–17

    MATH  Google Scholar 

  • Chen SM, Hong JA (2014) Multicriteria linguistic decision making based on hesitant fuzzy linguistic term sets and the aggregation of fuzzy sets. Inf Sci 286:63–74

    Article  Google Scholar 

  • Chen SJ, Hwang CL (1992) Fuzzy multiple attribute decision making methods. In: Beckmann MJ, Krelle W (eds) Fuzzy multiple attribute decision making: methods and applications. Springer, Berlin, Heidelberg, pp 289–486

    Chapter  Google Scholar 

  • Chen SM, Wang NY, Pan JS (2009) Forecasting enrollments using automatic clustering techniques and fuzzy logical relationships. Expert Syst Appl 36(8):11070–11076

    Article  Google Scholar 

  • Chen SM, Yang MW, Yang SW, Sheu TW, Liau CJ (2012a) Multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets. Expert Syst Appl 39(15):12085–12091

    Article  Google Scholar 

  • Chen SM, Lee LW, Liu HC, Yang SW (2012b) Multiattribute decision making based on interval-valued intuitionistic fuzzy values. Expert Syst Appl 39(12):10343–10351

    Article  Google Scholar 

  • Chen SM, Lin TE, Lee LW (2014) Group decision making using incomplete fuzzy preference relations based on the additive consistency and the order consistency. Inf Sci 259:1–15

    Article  MathSciNet  Google Scholar 

  • Chen SM, Cheng SH, Lan TC (2016a) Multicriteria decision making based on the TOPSIS method and similarity measures between intuitionistic fuzzy values. Inf Sci 367:279–295

    Article  Google Scholar 

  • Chen SM, Cheng SH, Lan TC (2016b) A novel similarity measure between intuitionistic fuzzy sets based on the centroid points of transformed fuzzy numbers with applications to pattern recognition. Inf Sci 343:15–40

    Article  MathSciNet  Google Scholar 

  • De Luca A, Termini SA (1972) definition of a non probabilistic entropy in the setting of fuzzy sets theory. Inf Control 20(4):301–312

    Article  Google Scholar 

  • Dyer JS, Fishburn PC, Steuer RE, Wallenius J, Zionts S (1992) Multiple criteria decision making, multiattribute utility theory: the next ten years. Manag Sci 38(5):645–654

    Article  Google Scholar 

  • Farhadinia B (2017) A multiple criteria decision making model with entropy weight in an interval-transformed hesitant fuzzy environment. Cognit Comput 9:513–525

    Article  Google Scholar 

  • Gangwar SS, Kumar S (2014) Probabilistic and intuitionistic fuzzy sets-based method for fuzzy time series forecasting. Cybern Syst 45(4):349–361

    Article  Google Scholar 

  • Gorzałczany MB (1987) A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst 21(1):1–17

    Article  MathSciNet  Google Scholar 

  • Grattan GI (1976) Fuzzy membership mapped onto intervals and many valued quantities. Math Logic Q 22(1):149–160

    Article  MathSciNet  Google Scholar 

  • Horng YJ, Chen SM, Chang YC, Lee CH (2005) A new method for fuzzy information retrieval based on fuzzy hierarchical clustering and fuzzy inference techniques. IEEE Trans Fuzzy Syst 13(2):216–228

    Article  Google Scholar 

  • Jiang Y, Xu Z, Shu Y (2017) Interval-valued intuitionistic multiplicative aggregation in group decision making. Granul Comput 2:387–407

    Article  Google Scholar 

  • Joshi D, Kumar S (2014) Intuitionistic fuzzy entropy and distance measure based TOPSIS method for multi-criteria decision making. Egypt Inform J 15(2):97–104

    Article  Google Scholar 

  • Joshi D, Kumar S (2016) Interval-valued intuitionistic hesitant fuzzy Choquet integral based TOPSIS method for multi-criteria group decision making. Eur J Oper Res 248(1):183–191

    Article  MathSciNet  Google Scholar 

  • Joshi DK, Kumar S (2017) Trapezium cloud TOPSIS method with interval-valued intuitionistic hesitant fuzzy linguistic information. Granul Comput. https://doi.org/10.1007/s41066-017-0062-5

    Article  Google Scholar 

  • Kosko B (1986) Fuzzy entropy and conditioning. Inf Sci 40(2):165–174

    Article  MathSciNet  Google Scholar 

  • Lee LW, Chen SM (2015a) Fuzzy decision making based on likelihood-based comparison relations of hesitant fuzzy linguistic term sets and hesitant fuzzy linguistic operators. Inf Sci 294:513–529

    Article  MathSciNet  Google Scholar 

  • Lee LW, Chen SM (2015b) Fuzzy decision making and fuzzy group decision making based on likelihood-based comparison relations of hesitant fuzzy linguistic term sets1. J Intell Fuzzy Syst 29(3):1119–1137

    Article  MathSciNet  Google Scholar 

  • Liao H, Xu Z (2017) Novel correlation and entropy measures of hesitant fuzzy sets. In: Hesitant fuzzy decision making methodologies and applications, Springer, Singapore, pp 37–72

    Google Scholar 

  • Liu P (2017) Multiple attribute group decision making method based on interval-valued intuitionistic fuzzy power Heronian aggregation operators. Comput Ind Eng 108:199–212

    Article  Google Scholar 

  • Meng F, Chen X (2016a) Entropy and similarity measure of Atanassov’s intuitionistic fuzzy sets and their application to pattern recognition based on fuzzy measures. Pattern Anal Appl 19(1):11–20

    Article  MathSciNet  Google Scholar 

  • Meng F, Chen X (2016b) Entropy and similarity measure for Atannasov’s interval-valued intuitionistic fuzzy sets and their application. Fuzzy Optim Decis Mak 15(1):75–101

    Article  MathSciNet  Google Scholar 

  • Mongin P (1997) Expected utility theory. In: Davis J, Hands W, Maki U (eds) Handbook of economic methodology. Edward Elgar, London, pp 342–350

  • Park JH, Kwark HE, Kwun YC (2017) Entropy and cross-entropy for generalized hesitant fuzzy information and their use in multiple attribute decision making. Int J Intell Syst 32(3):266–290

    Article  Google Scholar 

  • Pedrycz W, Chen SM (eds) (2011) Granular computing and intelligent systems: design with information granules of higher order and higher type, vol 13. Springer Science & Business Media, Berlin

    Google Scholar 

  • Pedrycz W, Chen SM (eds) (2014) Information granularity, big data, and computational intelligence, vol 8. Springer, Heidelberg

    Google Scholar 

  • Pedrycz W, Chen SM (eds) (2015) Granular computing and decision-making: interactive and iterative approaches, vol 10. Springer, Heidelberg

    Google Scholar 

  • Qin J, Liu X, Pedrycz W (2015) Hesitant fuzzy Maclaurin symmetric mean operators and its application to multiple-attribute decision making. Int J Fuzzy Syst 17(4):509–520

    Article  MathSciNet  Google Scholar 

  • Qin J, Liu X, Pedrycz W (2016) Multi-attribute group decision making based on Choquet integral under interval-valued intuitionistic fuzzy environment. Int J Comput Intell Syst 9(1):133–152

    Article  Google Scholar 

  • Quirós P, Alonso P, Bustince H, Díaz I, Montes S (2015) An entropy measure definition for finite interval-valued hesitant fuzzy sets. Knowl Based Syst 84:121–133

    Article  Google Scholar 

  • Selvachandran G, Maji PK, Faisal RQ, Salleh AR (2017) Distance and distance induced intuitionistic entropy of generalized intuitionistic fuzzy soft sets. Appl Intell 47:132–147

    Article  Google Scholar 

  • Shannon CE (1949) Communication theory of secrecy systems*. Bell Syst Tech J 28(4):656–715

    Article  MathSciNet  Google Scholar 

  • Szmidt E, Kacprzyk J (2001) Entropy for intuitionistic fuzzy sets. Fuzzy Sets Syst 118(3):467–477

    Article  MathSciNet  Google Scholar 

  • Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539

    MATH  Google Scholar 

  • Torra V, Narukawa Y (2009) On hesitant fuzzy sets and decision. In: The 18th IEEE international conference on fuzzy systems, Jeju Island, Korea, pp 1378–1382

  • Wang X, Triantaphyllou E (2008) Ranking irregularities when evaluating alternatives by using some ELECTRE methods. Omega 36(1):45–63

    Article  Google Scholar 

  • Wang JG, Lin CJ, Chen SM (2010) Applying fuzzy method to vision-based lane detection and departure warning system. Expert Syst Appl 37(1):113–126

    Article  Google Scholar 

  • Wei CP, Wang P, Zhang YZ (2011) Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications. Inf Sci 181(19):4273–4286

    Article  MathSciNet  Google Scholar 

  • Xu Z, Gou X (2017) An overview of interval-valued intuitionistic fuzzy information aggregations and applications. Granul Comput 2(1):13–39

    Article  Google Scholar 

  • Xu Z, Wang H (2016) Managing multi-granularity linguistic information in qualitative group decision making: an overview. Granul Comput 1(1):21–35

    Article  Google Scholar 

  • Xu Z, Xia M (2011) Distance and similarity measures for hesitant fuzzy sets. Inf Sci 181(11):2128–2138

    Article  MathSciNet  Google Scholar 

  • Xu Z, Xia M (2012) Hesitant fuzzy entropy and cross-entropy and their use in multiattribute decision-making. Int J Intell Syst 27(9):799–822

    Article  Google Scholar 

  • Ye J (2014) Interval-valued hesitant fuzzy prioritized weighted aggregation operators for multiple attribute decision making. J Algorithms Comput Technol 8(2):179–192

    Article  MathSciNet  Google Scholar 

  • Ye J (2016) Cross-entropy of dual hesitant fuzzy sets for multiple attribute decision-making. Int J Decis Support Syst Technol (IJDSST) 8(3):20–30

    Article  Google Scholar 

  • Yuan J, Li C, Xu F, Sun B, Li W (2016) A group decision making approach in interval-valued intuitionistic hesitant fuzzy environment with confidence levels. J Intell Fuzzy Syst 31(3):1909–1919

    Article  Google Scholar 

  • Yue Z (2011) An approach to aggregating interval numbers into interval-valued intuitionistic fuzzy information for group decision making. Expert Syst Appl 38(5):6333–6338

    Article  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–356

    Article  Google Scholar 

  • Zeleny M (1973) Compromise programming. In: Cochrane JL, Zeleny M (eds) Multiple criteria decision making. University of South Carolina Press, Columbia, pp 262–301

    Google Scholar 

  • Zhang Z (2013) Interval-valued intuitionistic hesitant fuzzy aggregation operators and their application in group decision-making. J Appl Math. https://doi.org/10.1155/2013/670285

    Article  Google Scholar 

  • Zhao N, Xu Z (2016) Entropy measures for interval-valued intuitionistic fuzzy information from a comparative perspective and their application to decision making. Informatica 27(1):203–229

    Article  Google Scholar 

  • Zhu YJ, Li DF (2016) A new definition and formula of entropy for intuitionistic fuzzy sets. J Intell Fuzzy Syst 30(6):3057–3066

    Article  Google Scholar 

  • Zhu B, Xu Z, Xia M (2012) Dual hesitant fuzzy sets. J Appl Math. https://doi.org/10.1155/2012/879629

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, G. B. Pant University of Agriculture and Technology, Pantnagar, Uttarakhand, 263145, India

    Dheeraj Kumar Joshi & Sanjay Kumar

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  1. Dheeraj Kumar Joshi
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  2. Sanjay Kumar
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Correspondence to Sanjay Kumar.

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Joshi, D.K., Kumar, S. Entropy of interval-valued intuitionistic hesitant fuzzy set and its application to group decision making problems. Granul. Comput. 3, 367–381 (2018). https://doi.org/10.1007/s41066-018-0077-6

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