Abstract
The study of unclear phenomena has been facilitated by fuzzy sets. Fuzzy set extensions have allowed for a more detailed investigation of these kinds of research. Finding quantitative measures for ambiguity and other characteristics of these occurrences thus becomes a challenge. As a fuzzy set extension, several researchers proposed intuitionistic fuzzy (IF) sets and used them in many contexts since they were first described by Atanassov. One such use is to solve multi-criteria decision-making issues. This study measure the amount of knowledge linked with an IF-set. An IF-knowledge measure is proposed. Using numerical examples, its utility and validity are examined. Besides this, the IF-accuracy measure, IF-information measure, similarity measure, and dissimilarity measure, are the four new measures that are derived from the proposed IF-knowledge measure. All these measures are checked for their validation and their properties are discussed. Pattern detection is taken as an application of the proposed accuracy measure. Finally, a modified VIKOR approach depending upon the proposed similarity and dissimilarity measure is proposed to deal with an MCDM issue in an intuitionistic fuzzy environment. The efficiency of the proposed approach is demonstrated by using a numerical example. A comparative study is also provided to assess the feasibility of the proposed approach.
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Notes
Vlsekriterijumska Optimizacija I Kompromisno Resenje.
If we use the proposed dissimilarity measure then \(b_{ij}=\zeta _m({\mathcal {B}}_j,m_{ij}), w_{ij}=\zeta _m({\mathcal {W}}_j,m_{ij})\).
References
Arya V, Kumar S (2021) Knowledge measure and entropy: a complementary concept in fuzzy theory. Gran Comput 6(3):631–643. https://doi.org/10.1007/s41066-020-00221-7
Atanassov KT (1986) Intutionistic fuzzy sets. Fuzzy Sets Syst 20:87–96. https://doi.org/10.1007/978-3-7908-1870-3_1
Bajaj RK, Kumar T, Gupta N (2012) R-norm intuitionistic fuzzy information measures and its computational applications. Proceedings of International Conference on Eco-friendly Computing and Communication System-2012 (ICECCS-12), 372–380. https://doi.org/10.1007/978-3-642-32112-2_43
Boekee DE, Vander Lubbe JCA (1980) The R-norm information measure. Inf Control 45(2):136–155. https://doi.org/10.1016/S0019-9958(80)90292-2
Boran FE, Akay D (2014) A biparametric similarity measure on intuitionistic fuzzy sets with applications to pattern recognition. Inf Sci 255:45–57. https://doi.org/10.1016/j.ins.2013.08.013
Burillo P, Bustince H (1996) Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy Sets Syst 78(3):305–316. https://doi.org/10.1016/0165-0114(96)84611-2
Bustince A et al (2015) A historical account of types of fuzzy sets and their relationships. IEEE Trans Fuzzy Syst 24(1):179–194. https://doi.org/10.1109/TFUZZ.2015.2451692
Bustince H (2000) Indicator of inclusion grade for interval-valued fuzzy sets. Application to approximate reasoning based on interval-valued fuzzy sets. Int J Approxi Reason 23(3):137–209. https://doi.org/10.1016/S0888-613X(99)00045-6
Chang TH (2014) Fuzzy VIKOR method: a case study of the hospital service evaluation in Taiwan. Inf Sci 271:196–212. https://doi.org/10.1016/j.ins.2014.02.118
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. https://doi.org/10.1016/j.ins.2014.07.033
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. https://doi.org/10.1016/j.ins.2016.02.049
Chen SM, Cheng SH, Lan TC (2016) Multicriteria decision making based on the TOPSIS method and similarity measures between intuitionistic fuzzy values. Inf Sci 367:279–295. https://doi.org/10.1016/j.ins.2016.05.044
Chen SM, Cheng SH, Lan TC (2016) 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. https://doi.org/10.1016/j.ins.2016.01.040
Chen T, Li C (2010) Determining objective weights with intutionistic fuzzy entropy measures: a comparative analysis. Inf Sci 180(21):4207–4222. https://doi.org/10.1016/j.ins.2010.07.009
Chu ATW, Kalaba RE, Spingarn K (1979) A comparison of two methods for determining the weights of belonging to fuzzy sets. J Optim Theory Appl 27:531–538. https://doi.org/10.1007/BF00933438
Couso I, Bustince H (2018) From fuzzy sets to interval-valued and Atanassov intuitionistic fuzzy sets: A unified view of different axiomatic measures. IEEE Trans Fuzzy Syst 27(2):362–371. https://doi.org/10.1109/TFUZZ.2018.2855654
Das S, Dutta B, Guha D (2016) Weight computation of criteria in a decision-making problem by knowledge measure with intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set. Soft Comput 20(9):3421–3442. https://doi.org/10.1007/s00500-015-1813-3
Das S, Guha D, Mesiar R (2017) Information measures in the intuitionistic fuzzy framework and their relationships. IEEE Trans Fuzzy Syst 26(3):1626–1637. https://doi.org/10.1109/TFUZZ.2017.2738603
De SK, Biswas R, Roy AR (2000) Some operations on intuitionistic fuzzy sets. Fuzzy Sets Syst 114(3):477–484. https://doi.org/10.1016/S0165-0114(98)00191-2
Diakoulaki D, Mavrotas G, Papayannakis L (1995) Determining objective weights in multiple criteria problems: the critic method. Comput Oper Res 22(7):763–770. https://doi.org/10.1016/0305-0548(94)00059-H
Dutta P, Saikia B (2021) Arithmetic operations on normal semi elliptic intuitionistic fuzzy numbers and their application in decision-making. Gran Comput 6:163–179. https://doi.org/10.1007/s41066-019-00175-5
Fan J (2002) Some new fuzzy entropy formulas. Fuzzy Sets Syst 128(2):277–284. https://doi.org/10.1016/S0165-0114(01)00127-0
Fan L, Zhangyan X (2001) Similarity measures between vague sets. J Softw 12(6):922–927
Farhadinia B (2020) A cognitively inspired knowledge-based decision-making methodology employing intuitionistic fuzzy sets. Cogn Comput 12(3):667–678. https://doi.org/10.1007/s12559-019-09702-7
Garg H (2019) Generalized intuitionistic fuzzy entropy-based approach for solving multi-attribute decision-making problems with unknown attribute weights. Proc Natl Acad Sci India Sect A 89(1):129–139. https://doi.org/10.1007/s40010-017-0395-0
Garg H, Kaur J (2018) A novel (R, S)-norm entropy measure of intuitionistic fuzzy sets and its applications in multi-attribute decision-making. Mathematics 6(6):92. https://doi.org/10.3390/math6060092
Ginevičius R, Podvezko V (2005) Objective and Subjective approaches to determining the criterion weight in multicriteria models. Transp Telecommun 6(1):133–137
Gomes LFAM, Lima MMPP (1991) TODIM: Basic and application to multicriteria ranking of projects with environmental impacts. Found Comput Decis Sci 16(3):113–127
Grzegorzewski P (2004) Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Sets Syst 148(2):319–328. https://doi.org/10.1016/j.fss.2003.08.005
Guo K (2015) Knowledge measure for Atanassov’s intuitionistic fuzzy sets. IEEE Trans Fuzzy Syst 24(5):1072–1078. https://doi.org/10.1109/TFUZZ.2015.2501434
Gupta P, Mehlawat MK, Grover N (2016) Intuitionistic fuzzy multi-attribute group decision-making with an application to plant location selection based on a new extended VIKOR method. Inf Sci 370:184–203. https://doi.org/10.1016/j.ins.2016.07.058
Gupta R, Kumar S (2022) Intuitionistic fuzzy scale-invariant entropy with correlation coefficients-based VIKOR approach for multi-criteria decision-making. Gran Comput 7(1):77–93. https://doi.org/10.1007/s41066-020-00252-0
Havdra JH, Charvat F (1967) Quantification method classification process: concept of structural \(\alpha\)-entropy. Kybernetika 3:30–35
Hong DH, Kim C (1999) A note on similarity measures between vague sets and between elements. Inf Sci 115(1–4):83–96. https://doi.org/10.1016/S0020-0255(98)10083-X
Hooda DS (2004) On generalized measures of fuzzy entropy. Math Slovaca 54(3):315–325
Hu J, Zhang X, Yang Y, Liu Y, Chen X (2020) New doctors ranking system based on VIKOR method. Int Trans Oper Res 27(2):1236–1261. https://doi.org/10.1111/itor.12569
Hung WL, Yang MS (2004) Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance. Pattern Recogn Lett 25(14):1603–1611. https://doi.org/10.1016/j.patrec.2004.06.006
Hung WL, Yang MS (2006) Fuzzy entropy on intuitionistic fuzzy sets. Int J Intell Syst 21(4):443–451. https://doi.org/10.1002/int.20131
Hwang CL, Yoon KP (1981) Multiple attribute decision-making: methods and applications. Springer, Berlin. https://doi.org/10.1007/978-3-642-48318-9
Joshi R (2023) Multi-criteria decision making based on novel fuzzy knowledge measures. Gran Comput 8(2):253–270. https://doi.org/10.1007/s41066-022-00329-y
Joshi R, Kumar S (2018) An exponential Jensen fuzzy divergence measure with applications in multiple attribute decision-making. Math Probl Eng. https://doi.org/10.1155/2018/4342098
Kadian R, Kumar S (2021) A novel intuitionistic Renyi’s-Tsallis discriminant information measure and its applications in decision-making. Gran Comput 6(4):901–913. https://doi.org/10.1007/s41066-020-00237-z
Kumar S (2009) Some more results on R-norm information measure. Tamkang J Math 40(1):41–58. https://doi.org/10.5556/j.tkjm.40.2009.35
Kumar S, Choudhary A, Kumar R (2014) Some more results on a generalized parametric R-norm information measure of type \(\alpha\). J Appl Sci Eng 17(4):447–453. https://doi.org/10.6180/jase.2014.17.4.12
Li J, Deng G, Li H, Zeng W (2012) The relationship between similarity measure and entropy of intuitionistic fuzzy sets. Inf Sci 188:314–321. https://doi.org/10.1016/j.ins.2011.11.021
Li Y, Zhongxian C, Degin Y (2002) Similarity measures between vague sets and vague entropy. J Comput Sci 29(12):129–132
Liang Z, Shi P (2003) Similarity measures on intuitionistic fuzzy sets. Pattern Recogn Lett 24(15):2687–2693. https://doi.org/10.1016/S0167-8655(03)00111-9
Luca AD, Termini S (1972) A definition of a Nonprobabilistic entropy in the setting of fuzzy sets theory. Inf Control 20:301–312. https://doi.org/10.1016/S0019-9958(72)90199-4
Mareschal B, Brans P Vincke P (1984) Promethee: a new family of outranking methods in multicriteria analysis. Technical report, Universite Libre de Bruxelles
Mitchell HB (2003) On the Dengfeng–Chuntian similarity measure and its application to pattern recognition. Pattern Recogn Lett 24(16):3101–3104. https://doi.org/10.1016/S0167-8655(03)00169-7
Nguyen H (2015) A new knowledge-based measure for Intuitionistic fuzzy sets and its application in multiple attribute group decision making. Expert Syst Appl 42(22):8766–8774. https://doi.org/10.1016/j.eswa.2015.07.030
Odu GO (2019) Weighting methods for multi-criteria decision making technique. J Appl Sci Environ Manag 23(8):1449–1457. https://doi.org/10.4314/jasem.v23i8.7
Ohlan A (2022) Novel entropy and distance measures for interval-valued intuitionistic fuzzy sets with application in multi-criteria group decision-making. Int J Gen Syst 51(4):413–440. https://doi.org/10.1080/03081079.2022.2036138
Opricovic S (1998) Multicriteria optimization of civil engineering systems. Faculty of civil engineering, Belgrade 2(1):5–21
Opricovic S, Tzeng GH (2004) Decision aiding compromise solution by MCDM methods: a comparative analysis of VIKOR and TOPSIS. Eur J Oper Res 156(2):445–455. https://doi.org/10.1016/S0377-2217(03)00020-1
Pal A et al (2013) Uncertainties with Atanassov’s intuitionistic fuzzy sets: fuzziness and lack of knowledge. Inf Sci 228:61–74. https://doi.org/10.1016/j.ins.2012.11.016
Papakostas GA, Hatzimichailidis AG, Kaburlasos VG (2013) Distance and similarity measures between intuitionistic fuzzy sets: A comparative analysis from a pattern recognition point of view. Pattern Recogn Lett 34(14):1609–1622. https://doi.org/10.1016/j.patrec.2013.05.015
Ravichandran KS et al (2020) A novel extension to VIKOR method under intuitionistic fuzzy context for solving personnel selection problem. Soft Comput 24:1063–1081. https://doi.org/10.1007/s00500-019-03943-2
Rényi A (1961) On measures of entropy and information. In: Proceedings of the 4th Berkeley symposium on mathematical statistics and probability 1:547–561
Rostamzadeh R, Govindan K, Esmaeili A, Sabaghi M (2015) Application of fuzzy VIKOR for evaluation of green supply chain management practices. Ecol Ind 49:188–203. https://doi.org/10.1016/j.ecolind.2014.09.045
Sanayei A, Mousavi SF, Yazdankhah A (2010) Group decision making process for supplier selection with VIKOR under fuzzy environment. Expert Syst Appl 37(1):24–30. https://doi.org/10.1016/j.eswa.2009.04.063
Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27(3):379–423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
Shemshadi A, Shirazi H, Toreihi M, Tarokh MJ (2011) A fuzzy VIKOR method for supplier selection based on entropy measure for objective weighting. Expert Syst Appl 38(10):12160–12167. https://doi.org/10.1016/j.eswa.2011.03.027
Singh S et al (2020) On some knowledge measures of intuitionistic fuzzy sets of type two with application to MCDM. Cybern Inf Technol 20(1):3–20. https://doi.org/10.2478/cait-2020-0001
Singh A, Kumar S (2023) Novel fuzzy knowledge and accuracy measures with its applications in multi-criteria decision-making. Gran Comput. https://doi.org/10.1007/s41066-023-00374-1
Singh S, Lalotra S, Sharma S (2019) Dual concepts in fuzzy theory: entropy and Knowledge measure. Int J Intell Syst 34(5):1034–1059. https://doi.org/10.1002/int.22085
Song A et al (2017) Uncertainty measure for Atanassov’s intuitionistic fuzzy sets. Appl Intell 46(4):757–774. https://doi.org/10.1007/s10489-016-0863-2
Szmidt E, Kacprzyk J (1998) A fuzzy set corresponding to an intuitionistic fuzzy set. Int J Uncertain Fuzz Knowl-Based Syst 6(05):427–435. https://doi.org/10.1142/S0218488598000343
Szmidt E, Kacprzyk J (2001) Entropy for intuitionistic fuzzy sets. Fuzzy Sets Syst 118(3):467–477. https://doi.org/10.1016/S0165-0114(98)00402-3
Szmidt E, Kacprzyk J, Bujnowski P (2014) How to measure the amount of knowledge conveyed by Atanassov’s intuitionistic fuzzy sets. Inf Sci 257:276–285. https://doi.org/10.1016/j.ins.2012.12.046
Szmidt E, Kacprzyk J, Bujnowski P (2014) How to measure the amount of knowledge conveyed by Atanassov’s intuitionistic fuzzy sets. Inf Sci 257:276–285. https://doi.org/10.1016/j.ins.2012.12.046
Tsallis C (1988) Possible generalization of Boltzman–Gibbs statistics. J Stat Phys 52(1):479–487. https://doi.org/10.1007/BF01016429
Verma R, Sharma BD (2011) A measure of inaccuracy between two fuzzy sets. Cybern Inform Technol 11(2):13–23
Verma R, Sharma BD (2014) A new measure of inaccuracy with its application to multi-criteria decision making under intuitionistic fuzzy environment. J Intell Fuzzy Syst 27(4):1811–1824. https://doi.org/10.3233/IFS-141148
Wan SP, Wang QY, Dong JY (2013) The extended VIKOR method for multi-attribute group decision making with triangular intuitionistic fuzzy numbers. Knowl-Based Syst 52:65–77. https://doi.org/10.1016/j.knosys.2013.06.019
Wang TC, Chang TH (2005) Fuzzy VIKOR as a resolution for multicriteria group decision-making. In: The 11th International Conference on Industrial Engineering and Engineering Management, pp 352–356. https://doi.org/10.1109/ICSSSM.2006.320751
Wang W, Xin X (2005) Distance measure between intuitionistic fuzzy sets. Pattern Recogn Lett 26:2063–2069. https://doi.org/10.1016/j.patrec.2005.03.018
Xia M, Xu Z (2012) Entropy/cross entropy-based group decision making under intuitionistic fuzzy environment. Inf Fus 13(1):31–47. https://doi.org/10.1016/j.inffus.2010.12.001
Xiao F (2019) Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy. Inf Fus 46:23–32. https://doi.org/10.1016/j.inffus.2018.04.003
Yager RR (2020) Decision-making with measure modeled uncertain payoffs and multiple goals. Gran Comput 5(2):149–154. https://doi.org/10.1007/s41066-019-00203-4
Yang Y, Chiclana F (2012) Consistency of 2D and 3D distances of intuitionistic fuzzy sets. Expert Syst Appl 39(10):8665–8670. https://doi.org/10.1016/j.eswa.2012.01.199
Ye J (2010) Fuzzy dcision-making method based on the weighted correlation coefficient under intutionistic fuzzy enviornment. Eur J Oper Res 205(1):202–204. https://doi.org/10.1016/j.ejor.2010.01.019
Ye J (2010) Fuzzy decision-making method based on the weighted correlation coefficient under intuitionistic fuzzy environment. Eur J Oper Res 205(1):202–204. https://doi.org/10.1016/j.ejor.2010.01.019
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23(2):421–427. https://doi.org/10.1016/0022-247X(68)90078-4
Zeng S, Chen SM, Kuo LW (2019) Multiattribute decision making based on novel score function of intuitionistic fuzzy values and modified VIKOR method. Inf Sci 488:76–92. https://doi.org/10.1016/j.ins.2019.03.018
Zeng W, Li H (2006) Relationship between similarity measure and entropy of interval valued fuzzy sets. Fuzzy Sets Syst 157(11):1477–1484. https://doi.org/10.1016/j.fss.2005.11.020
Zhang H, Yu L (2013) New distance measures between intuitionistic fuzzy sets and interval-valued fuzzy sets. Inf Sci 245:181–196. https://doi.org/10.1016/j.ins.2013.04.040
Zhang QS, Jiang SY (2008) A note on information entropy measures for vague sets and its applications. Inf Sci 178(21):4184–4191. https://doi.org/10.1016/j.ins.2008.07.003
Zoraghi A et al (2013) A fuzzy MCDM model with objective and subjective weights for evaluating service quality in hotel industries. J Ind Eng Int 9(1):1–13. https://doi.org/10.1186/2251-712X-9-38
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Singh, A., Kumar, S. Intuitionistic fuzzy entropy-based knowledge and accuracy measure with its applications in extended VIKOR approach for solving multi-criteria decision-making. Granul. Comput. 8, 1609–1643 (2023). https://doi.org/10.1007/s41066-023-00386-x
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DOI: https://doi.org/10.1007/s41066-023-00386-x