American Institute of Mathematical Sciences

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September  2021, 17(5): 2279-2306. doi: 10.3934/jimo.2020069

Multi-criteria decision making method based on Bonferroni mean aggregation operators of complex intuitionistic fuzzy numbers

Harish Garg and  Dimple Rani

School of Mathematics, Thapar Institute of Engineering & Technology (Deemed University), Patiala-147004, Punjab, India

Received  August 2019 Revised  December 2019 Published  September 2021 Early access  March 2020

Complex intuitionistic fuzzy sets (CIFSs), characterized by complex-valued grades of membership and non-membership, are a generalization of standard intuitionistic fuzzy (IF) sets that better speak to time-periodic issues and handle two-dimensional data in a solitary set. Under this environment, in this article, various mean-type operators, namely complex IF Bonferroni means (CIFBM) and complex IF weighted Bonferroni mean (CIFWBM) are presented along with their properties and numerous particular cases of CIFBM are discussed. Further, using the presented operators a decision-making approach is developed and is illustrated with the help of a practical example. Also, the reliability of the developed methodology is investigated with the aid of validity test criteria and the example results are compared with prevailing methods based on operators.

Keywords: Complex intuitionistic fuzzy set, Bonferroni mean, geometric operators, multicriteria decision-making, aggregation operators.
Mathematics Subject Classification: Primary: 68T35; 90B50; 62A86; 03E72.
Citation: Harish Garg, Dimple Rani. Multi-criteria decision making method based on Bonferroni mean aggregation operators of complex intuitionistic fuzzy numbers. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2279-2306. doi: 10.3934/jimo.2020069
References:
[1]

M. Ali and F. Smarandache, Complex neutrosophic set, Neural Computing and Applications, 28 (2017), 1817-1834. 

[2]

A. Alkouri and A. Salleh, Complex intuitionistic fuzzy sets, chap. 2nd International Conference on Fundamental and Applied Sciences, 1482 (2012), 464-470. 

[3]

A. U. M. Alkouri and A. R. Salleh, Complex Atanassov's intuitionistic fuzzy relation, Abstract and Applied Analysis, 2013 (2013), Article ID 287382, 18 pages. doi: 10.1155/2013/287382.

[4]

R. Arora and H. Garg, Group decision-making method based on prioritized linguistic intuitionistic fuzzy aggregation operators and its fundamental properties, Computational and Applied Mathematics, 38 (2019), Art. 36, 32 pp. doi: 10.1007/s40314-019-0764-1.

[5]

K. Atanassov and G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 31 (1989), 343-349.  doi: 10.1016/0165-0114(89)90205-4.

[6]

K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.  doi: 10.1016/S0165-0114(86)80034-3.

[7]

C. Bonferroni, Sulle medie multiple di potenze, Bollettino Dell'Unione Matematica Italiana, 5 (1950), 267-270. 

[8]

S. M. Chen and C. H. Chang, Fuzzy multiattribute decision making based on transformation techniques of intuitionistic fuzzy values and intuitionistic fuzzy geometric averaging operators, Information Sciences, 352/353 (2016), 133-149.  doi: 10.1016/j.ins.2016.02.049.

[9]

S. Dick, R. R. Yager and O. Yazdanbakhsh, On Pythagorean and complex fuzzy set operations, IEEE Transactions on Fuzzy Systems, 24 (2016), 1009-1021. 

[10]

H. Garg, Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision making, Computers and Industrial Engineering, 101 (2016), 53-69.  doi: 10.1016/j.cie.2016.08.017.

[11]

H. Garg, Some series of intuitionistic fuzzy interactive averaging aggregation operators, SpringerPlus, 5 (2016), 999. doi: 10.1186/s40064-016-2591-9.

[12]

H. Garg, Novel intuitionistic fuzzy decision making method based on an improved operation laws and its application, Engineering Applications of Artificial Intelligence, 60 (2017), 164-174.  doi: 10.1016/j.engappai.2017.02.008.

[13]

H. Garg, Intuitionistic fuzzy hamacher aggregation operators with entropy weight and their applications to multi-criteria decision-making problems, Iranian Journal of Science and Technology, Transactions of Electrical Engineering, 43 (2019), 597-613.  doi: 10.1007/s40998-018-0167-0.

[14]

H. Garg and R. Arora, Bonferroni mean aggregation operators under intuitionistic fuzzy soft set environment and their applications to decision-making, Journal of the Operational Research Society, 69 (2018), 1711-1724.  doi: 10.1080/01605682.2017.1409159.

[15]

H. Garg and K. Kumar, Linguistic interval-valued atanassov intuitionistic fuzzy sets and their applications to group decision-making problems, IEEE Transactions on Fuzzy Systems, 27 (2019), 2302-2311.  doi: 10.1109/TFUZZ.2019.2897961.

[16]

H. Garg and D. Rani, Complex interval- valued intuitionistic fuzzy sets and their aggregation operators, Fundamenta Informaticae, 164 (2019), 61-101.  doi: 10.3233/FI-2019-1755.

[17]

H. Garg and D. Rani, Exponential, logarithmic and compensative generalized aggregation operators under complex intuitionistic fuzzy environment, Group Decision and Negotiation, 28 (2019), 991-1050.  doi: 10.1007/s10726-019-09631-8.

[18]

H. Garg and D. Rani, New generalized Bonferroni mean aggregation operators of complex intuitionistic fuzzy information based on Archimedean t-norm and t-conorm, Journal of Experimental and Theoretical Artificial Intelligence, 32 (2020), 81-109. 

[19]

H. Garg and D. Rani, Robust Averaging - Geometric aggregation operators for Complex intuitionistic fuzzy sets and their applications to MCDM process, Arabian Journal for Science and Engineering, 45 (2020), 2017-2033.  doi: 10.1007/s13369-019-03925-4.

[20]

H. Garg and D. Rani, A robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making, Applied Intelligence, 49 (2019), 496-512.  doi: 10.1007/s10489-018-1290-3.

[21]

H. Garg and D. Rani, Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision-making process, Arabian Journal for Science and Engineering, 44 (2019), 2679-2698. 

[22]

H. Garg and D. Rani, Some results on information measures for complex intuitionistic fuzzy sets, International Journal of Intelligent Systems, 34 (2019), 2319-2363.  doi: 10.1002/int.22127.

[23]

M. Goyal, D. Yadav and A. Tripathi, Intuitionistic fuzzy genetic weighted averaging operator and its application for multiple attribute decision making in E-learning, Indian Journal of Science and Technology, 9 (2016), 1-15.  doi: 10.17485/ijst/2016/v9i1/76191.

[24]

Y. He, H. Chen, L. Zhau, J. Liu and Z. Tao, Intuitionistic fuzzy geometric interaction averaging operators and their application to multi-criteria decision making, Information Sciences, 259 (2014), 142-159.  doi: 10.1016/j.ins.2013.08.018.

[25]

J. Y. Huang, Intuitionistic fuzzy Hamacher aggregation operator and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems, 27 (2014), 505-513.  doi: 10.3233/IFS-131019.

[26]

G. Kaur and H. Garg, Cubic intuitionistic fuzzy aggregation operators, International Journal for Uncertainty Quantification, 8 (2018), 405-427.  doi: 10.1615/Int.J.UncertaintyQuantification.2018020471.

[27]

G. Kaur and H. Garg, Generalized cubic intuitionistic fuzzy aggregation operators using t-norm operations and their applications to group decision-making process, Arabian Journal for Science and Engineering, 44 (2019), 2775-2794.  doi: 10.1615/Int.J.UncertaintyQuantification.2018020471.

[28]

G. Kaur and H. Garg, Multi - attribute decision - making based on Bonferroni mean operators under cubic intuitionistic fuzzy set environment, Entropy, 20 (2018), Paper No. 65, 26 pp. doi: 10.3390/e20010065.

[29]

K. Kumar and H. Garg, Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making, Applied Intelligence, 48 (2018), 2112-2119.  doi: 10.1007/s10489-017-1067-0.

[30]

T. Kumar and R. K. Bajaj, On complex intuitionistic fuzzy soft sets with distance measures and entropies, Journal of Mathematics, 2014 (2014), Article ID 972198, 12 pages. doi: 10.1155/2014/972198.

[31]

D. Li, W. Zeng and J. Li, Geometric bonferroni mean operators, International Journal of Intelligent Systems, 31 (2016), 1181-1197.  doi: 10.1002/int.21822.

[32]

C. Maclaurin, A second letter to martin folkes, esq.; concerning the roots of equations, with demonstration of other rules of algebra, Philos Trans Roy Soc London Ser A, 36 (1729), 59-96. 

[33]

A. A. Quran and N. Hassan, The complex neutrosophic soft expert set and its application in decision making, Journal of Intelligent & Fuzzy Systems, 34 (2018), 569-582. 

[34]

D. Ramot, M. Friedman, G. Langholz and A. Kandel, Complex fuzzy logic, IEEE Transactions on Fuzzy Systems, 11 (2003), 450-461. 

[35]

D. Ramot, R. Milo, M. Fiedman and A. Kandel, Complex fuzzy sets, IEEE Transactions on Fuzzy Systems, 10 (2002), 171-186. 

[36]

D. Rani and H. Garg, Distance measures between the complex intuitionistic fuzzy sets and its applications to the decision - making process, International Journal for Uncertainty Quantification, 7 (2017), 423-439.  doi: 10.1615/Int.J.UncertaintyQuantification.2017020356.

[37]

D. Rani and H. Garg, Complex intuitionistic fuzzy power aggregation operators and their applications in multi-criteria decision-making, Expert Systems, 35 (2018), e12325.

[38]

R. Verma, Generalized bonferroni mean operator for fuzzy number intuitionistic fuzzy sets and its application to multiattribute decision making, International Journal of Intelligent Systems, 30 (2015), 499-519.  doi: 10.1002/int.21705.

[39]

W. Wang and X. Liu, Intuitionistic fuzzy information aggregation using Einstein operations, IEEE Transactions on Fuzzy Systems, 20 (2012), 923-938.  doi: 10.1109/TFUZZ.2012.2189405.

[40]

X. Wang and E. Triantaphyllou, Ranking irregularities when evaluating alternatives by using some ELECTRE methods, Omega - International Journal of Management Science, 36 (2008), 45-63.  doi: 10.1016/j.omega.2005.12.003.

[41]

Z. Xu, Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators, Knowledge-Based Systems, 24 (2011), 749-760.  doi: 10.1016/j.knosys.2011.01.011.

[42]

Z. Xu and R. R. Yager, Intuitionistic fuzzy bonferroni means, IEEE Transactions on Systems, Man and Cybernetics, Part B: Cybernetics, 41 (2011), 568-578. 

[43]

Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems, 15 (2007), 1179-1187. 

[44]

Z. S. Xu and R. R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General Systems, 35 (2006), 417-433.  doi: 10.1080/03081070600574353.

[45]

R. R. Yager, On generalized bonferroni mean operators for multi-criteria aggregation, International Journal of Approximate Reasoning, 50 (2009), 1279-1286.  doi: 10.1016/j.ijar.2009.06.004.

[46]

R. R. Yager and A. M. Abbasov, Pythagorean membeship grades, complex numbers and decision making, International Journal of Intelligent Systems, 28 (2013), 436-452. 

[47]

O. Yazdanbakhsh and S. Dick, A systematic review of complex fuzzy sets and logic, Fuzzy Sets and Systems, 338 (2018), 1-22.  doi: 10.1016/j.fss.2017.01.010.

[48]

J. Ye, Intuitionistic fuzzy hybrid arithmetic and geometric aggregation operators for the decision-making of mechanical design schemes, Applied Intelligence, 47 (2017), 743-751.  doi: 10.1007/s10489-017-0930-3.

[49]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.

[50]

W. Zhou and Z. Xu, Extreme intuitionistic fuzzy weighted aggregation operators and their applications in optimism and pessimism decision-making processes, Journal of Intelligent and Fuzzy Systems, 32 (2017), 1129-1138. 

show all references

References:
[1]

M. Ali and F. Smarandache, Complex neutrosophic set, Neural Computing and Applications, 28 (2017), 1817-1834. 

[2]

A. Alkouri and A. Salleh, Complex intuitionistic fuzzy sets, chap. 2nd International Conference on Fundamental and Applied Sciences, 1482 (2012), 464-470. 

[3]

A. U. M. Alkouri and A. R. Salleh, Complex Atanassov's intuitionistic fuzzy relation, Abstract and Applied Analysis, 2013 (2013), Article ID 287382, 18 pages. doi: 10.1155/2013/287382.

[4]

R. Arora and H. Garg, Group decision-making method based on prioritized linguistic intuitionistic fuzzy aggregation operators and its fundamental properties, Computational and Applied Mathematics, 38 (2019), Art. 36, 32 pp. doi: 10.1007/s40314-019-0764-1.

[5]

K. Atanassov and G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 31 (1989), 343-349.  doi: 10.1016/0165-0114(89)90205-4.

[6]

K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.  doi: 10.1016/S0165-0114(86)80034-3.

[7]

C. Bonferroni, Sulle medie multiple di potenze, Bollettino Dell'Unione Matematica Italiana, 5 (1950), 267-270. 

[8]

S. M. Chen and C. H. Chang, Fuzzy multiattribute decision making based on transformation techniques of intuitionistic fuzzy values and intuitionistic fuzzy geometric averaging operators, Information Sciences, 352/353 (2016), 133-149.  doi: 10.1016/j.ins.2016.02.049.

[9]

S. Dick, R. R. Yager and O. Yazdanbakhsh, On Pythagorean and complex fuzzy set operations, IEEE Transactions on Fuzzy Systems, 24 (2016), 1009-1021. 

[10]

H. Garg, Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision making, Computers and Industrial Engineering, 101 (2016), 53-69.  doi: 10.1016/j.cie.2016.08.017.

[11]

H. Garg, Some series of intuitionistic fuzzy interactive averaging aggregation operators, SpringerPlus, 5 (2016), 999. doi: 10.1186/s40064-016-2591-9.

[12]

H. Garg, Novel intuitionistic fuzzy decision making method based on an improved operation laws and its application, Engineering Applications of Artificial Intelligence, 60 (2017), 164-174.  doi: 10.1016/j.engappai.2017.02.008.

[13]

H. Garg, Intuitionistic fuzzy hamacher aggregation operators with entropy weight and their applications to multi-criteria decision-making problems, Iranian Journal of Science and Technology, Transactions of Electrical Engineering, 43 (2019), 597-613.  doi: 10.1007/s40998-018-0167-0.

[14]

H. Garg and R. Arora, Bonferroni mean aggregation operators under intuitionistic fuzzy soft set environment and their applications to decision-making, Journal of the Operational Research Society, 69 (2018), 1711-1724.  doi: 10.1080/01605682.2017.1409159.

[15]

H. Garg and K. Kumar, Linguistic interval-valued atanassov intuitionistic fuzzy sets and their applications to group decision-making problems, IEEE Transactions on Fuzzy Systems, 27 (2019), 2302-2311.  doi: 10.1109/TFUZZ.2019.2897961.

[16]

H. Garg and D. Rani, Complex interval- valued intuitionistic fuzzy sets and their aggregation operators, Fundamenta Informaticae, 164 (2019), 61-101.  doi: 10.3233/FI-2019-1755.

[17]

H. Garg and D. Rani, Exponential, logarithmic and compensative generalized aggregation operators under complex intuitionistic fuzzy environment, Group Decision and Negotiation, 28 (2019), 991-1050.  doi: 10.1007/s10726-019-09631-8.

[18]

H. Garg and D. Rani, New generalized Bonferroni mean aggregation operators of complex intuitionistic fuzzy information based on Archimedean t-norm and t-conorm, Journal of Experimental and Theoretical Artificial Intelligence, 32 (2020), 81-109. 

[19]

H. Garg and D. Rani, Robust Averaging - Geometric aggregation operators for Complex intuitionistic fuzzy sets and their applications to MCDM process, Arabian Journal for Science and Engineering, 45 (2020), 2017-2033.  doi: 10.1007/s13369-019-03925-4.

[20]

H. Garg and D. Rani, A robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making, Applied Intelligence, 49 (2019), 496-512.  doi: 10.1007/s10489-018-1290-3.

[21]

H. Garg and D. Rani, Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision-making process, Arabian Journal for Science and Engineering, 44 (2019), 2679-2698. 

[22]

H. Garg and D. Rani, Some results on information measures for complex intuitionistic fuzzy sets, International Journal of Intelligent Systems, 34 (2019), 2319-2363.  doi: 10.1002/int.22127.

[23]

M. Goyal, D. Yadav and A. Tripathi, Intuitionistic fuzzy genetic weighted averaging operator and its application for multiple attribute decision making in E-learning, Indian Journal of Science and Technology, 9 (2016), 1-15.  doi: 10.17485/ijst/2016/v9i1/76191.

[24]

Y. He, H. Chen, L. Zhau, J. Liu and Z. Tao, Intuitionistic fuzzy geometric interaction averaging operators and their application to multi-criteria decision making, Information Sciences, 259 (2014), 142-159.  doi: 10.1016/j.ins.2013.08.018.

[25]

J. Y. Huang, Intuitionistic fuzzy Hamacher aggregation operator and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems, 27 (2014), 505-513.  doi: 10.3233/IFS-131019.

[26]

G. Kaur and H. Garg, Cubic intuitionistic fuzzy aggregation operators, International Journal for Uncertainty Quantification, 8 (2018), 405-427.  doi: 10.1615/Int.J.UncertaintyQuantification.2018020471.

[27]

G. Kaur and H. Garg, Generalized cubic intuitionistic fuzzy aggregation operators using t-norm operations and their applications to group decision-making process, Arabian Journal for Science and Engineering, 44 (2019), 2775-2794.  doi: 10.1615/Int.J.UncertaintyQuantification.2018020471.

[28]

G. Kaur and H. Garg, Multi - attribute decision - making based on Bonferroni mean operators under cubic intuitionistic fuzzy set environment, Entropy, 20 (2018), Paper No. 65, 26 pp. doi: 10.3390/e20010065.

[29]

K. Kumar and H. Garg, Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making, Applied Intelligence, 48 (2018), 2112-2119.  doi: 10.1007/s10489-017-1067-0.

[30]

T. Kumar and R. K. Bajaj, On complex intuitionistic fuzzy soft sets with distance measures and entropies, Journal of Mathematics, 2014 (2014), Article ID 972198, 12 pages. doi: 10.1155/2014/972198.

[31]

D. Li, W. Zeng and J. Li, Geometric bonferroni mean operators, International Journal of Intelligent Systems, 31 (2016), 1181-1197.  doi: 10.1002/int.21822.

[32]

C. Maclaurin, A second letter to martin folkes, esq.; concerning the roots of equations, with demonstration of other rules of algebra, Philos Trans Roy Soc London Ser A, 36 (1729), 59-96. 

[33]

A. A. Quran and N. Hassan, The complex neutrosophic soft expert set and its application in decision making, Journal of Intelligent & Fuzzy Systems, 34 (2018), 569-582. 

[34]

D. Ramot, M. Friedman, G. Langholz and A. Kandel, Complex fuzzy logic, IEEE Transactions on Fuzzy Systems, 11 (2003), 450-461. 

[35]

D. Ramot, R. Milo, M. Fiedman and A. Kandel, Complex fuzzy sets, IEEE Transactions on Fuzzy Systems, 10 (2002), 171-186. 

[36]

D. Rani and H. Garg, Distance measures between the complex intuitionistic fuzzy sets and its applications to the decision - making process, International Journal for Uncertainty Quantification, 7 (2017), 423-439.  doi: 10.1615/Int.J.UncertaintyQuantification.2017020356.

[37]

D. Rani and H. Garg, Complex intuitionistic fuzzy power aggregation operators and their applications in multi-criteria decision-making, Expert Systems, 35 (2018), e12325.

[38]

R. Verma, Generalized bonferroni mean operator for fuzzy number intuitionistic fuzzy sets and its application to multiattribute decision making, International Journal of Intelligent Systems, 30 (2015), 499-519.  doi: 10.1002/int.21705.

[39]

W. Wang and X. Liu, Intuitionistic fuzzy information aggregation using Einstein operations, IEEE Transactions on Fuzzy Systems, 20 (2012), 923-938.  doi: 10.1109/TFUZZ.2012.2189405.

[40]

X. Wang and E. Triantaphyllou, Ranking irregularities when evaluating alternatives by using some ELECTRE methods, Omega - International Journal of Management Science, 36 (2008), 45-63.  doi: 10.1016/j.omega.2005.12.003.

[41]

Z. Xu, Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators, Knowledge-Based Systems, 24 (2011), 749-760.  doi: 10.1016/j.knosys.2011.01.011.

[42]

Z. Xu and R. R. Yager, Intuitionistic fuzzy bonferroni means, IEEE Transactions on Systems, Man and Cybernetics, Part B: Cybernetics, 41 (2011), 568-578. 

[43]

Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems, 15 (2007), 1179-1187. 

[44]

Z. S. Xu and R. R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General Systems, 35 (2006), 417-433.  doi: 10.1080/03081070600574353.

[45]

R. R. Yager, On generalized bonferroni mean operators for multi-criteria aggregation, International Journal of Approximate Reasoning, 50 (2009), 1279-1286.  doi: 10.1016/j.ijar.2009.06.004.

[46]

R. R. Yager and A. M. Abbasov, Pythagorean membeship grades, complex numbers and decision making, International Journal of Intelligent Systems, 28 (2013), 436-452. 

[47]

O. Yazdanbakhsh and S. Dick, A systematic review of complex fuzzy sets and logic, Fuzzy Sets and Systems, 338 (2018), 1-22.  doi: 10.1016/j.fss.2017.01.010.

[48]

J. Ye, Intuitionistic fuzzy hybrid arithmetic and geometric aggregation operators for the decision-making of mechanical design schemes, Applied Intelligence, 47 (2017), 743-751.  doi: 10.1007/s10489-017-0930-3.

[49]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.

[50]

W. Zhou and Z. Xu, Extreme intuitionistic fuzzy weighted aggregation operators and their applications in optimism and pessimism decision-making processes, Journal of Intelligent and Fuzzy Systems, 32 (2017), 1129-1138. 

Figure 1.  Variation in score values with parameter $ p $ by fixing $ q $
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Figure 2.  Score values of alternatives $ \mathcal{H}_u $ for different values of $ p $, $ q $
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Table 1.  Comparison of CIFS model with existing models in literature
Model Uncertainty Falsity Hesitation Periodicity Ability to represent
two-dimensional information
Fuzzy set $ \checkmark $ $ \times $ $ \times $ $ \times $ $ \times $
Interval-valued fuzzy set $ \checkmark $ $ \times $ $ \times $ $ \times $ $ \times $
Intuitionistic fuzzy set $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \times $ $ \times $
Interval-valued intuitionistic fuzzy set $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \times $ $ \times $
Complex fuzzy set $ \checkmark $ $ \times $ $ \times $ $ \checkmark $ $ \checkmark $
Interval-valued complex fuzzy set $ \checkmark $ $ \times $ $ \times $ $ \checkmark $ $ \checkmark $
Complex intuitionistic fuzzy set $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \checkmark $
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Model Uncertainty Falsity Hesitation Periodicity Ability to represent
two-dimensional information
Fuzzy set $ \checkmark $ $ \times $ $ \times $ $ \times $ $ \times $
Interval-valued fuzzy set $ \checkmark $ $ \times $ $ \times $ $ \times $ $ \times $
Intuitionistic fuzzy set $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \times $ $ \times $
Interval-valued intuitionistic fuzzy set $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \times $ $ \times $
Complex fuzzy set $ \checkmark $ $ \times $ $ \times $ $ \checkmark $ $ \checkmark $
Interval-valued complex fuzzy set $ \checkmark $ $ \times $ $ \times $ $ \checkmark $ $ \checkmark $
Complex intuitionistic fuzzy set $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \checkmark $
Table 2.  Input information in the form of the complex intuitionistic fuzzy decision-matrix
$ \mathcal{C}_1 $ $ \mathcal{C}_2 $ $ \mathcal{C}_3 $ $ \mathcal{C}_4 $
$ \mathcal{H}_1 $ $ \big ( (0.7, 0.9), (0.1, 0.1) \big ) $ $ \big ( (0.8, 0.5), (0.1, 0.4) \big ) $ $ \big ( (0.6, 0.6), (0.3, 0.2) \big ) $ $ \big ( (0.7, 0.7), (0.3, 0.2) \big ) $
$ \mathcal{H}_2 $ $ \big ( (0.7, 0.6), (0.3, 0.3) \big ) $ $ \big ( (0.4, 0.9), (0.2, 0.1) \big ) $ $ \big ( (0.7, 0.7), (0.2, 0.3) \big ) $ $ \big ( (0.4, 0.6), (0.3, 0.1) \big ) $
$ \mathcal{H}_3 $ $ \big ( (0.3, 0.4), (0.6, 0.4) \big ) $ $ \big ( (0.6, 0.6), (0.3, 0.4) \big ) $ $ \big ( (0.3, 0.4), (0.5, 0.6) \big ) $ $ \big ( (0.7, 0.7), (0.1, 0.1) \big ) $
$ \mathcal{H}_4 $ $ \big ( (0.4, 0.8), (0.5, 0.1) \big ) $ $ \big ( (0.7, 0.3), (0.3, 0.3) \big ) $ $ \big ( (0.6, 0.5), (0.1, 0.3) \big ) $ $ \big ( (0.5, 0.5), (0.3, 0.4) \big ) $
$ \mathcal{H}_5 $ $ \big ( (0.9, 0.7), (0.1, 0.2) \big ) $ $ \big ( (0.7, 0.7), (0.2, 0.1) \big ) $ $ \big ( (0.7, 0.6), (0.2, 0.2) \big ) $ $ \big ( (0.8, 0.8), (0.1, 0.1) \big ) $
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$ \mathcal{C}_1 $ $ \mathcal{C}_2 $ $ \mathcal{C}_3 $ $ \mathcal{C}_4 $
$ \mathcal{H}_1 $ $ \big ( (0.7, 0.9), (0.1, 0.1) \big ) $ $ \big ( (0.8, 0.5), (0.1, 0.4) \big ) $ $ \big ( (0.6, 0.6), (0.3, 0.2) \big ) $ $ \big ( (0.7, 0.7), (0.3, 0.2) \big ) $
$ \mathcal{H}_2 $ $ \big ( (0.7, 0.6), (0.3, 0.3) \big ) $ $ \big ( (0.4, 0.9), (0.2, 0.1) \big ) $ $ \big ( (0.7, 0.7), (0.2, 0.3) \big ) $ $ \big ( (0.4, 0.6), (0.3, 0.1) \big ) $
$ \mathcal{H}_3 $ $ \big ( (0.3, 0.4), (0.6, 0.4) \big ) $ $ \big ( (0.6, 0.6), (0.3, 0.4) \big ) $ $ \big ( (0.3, 0.4), (0.5, 0.6) \big ) $ $ \big ( (0.7, 0.7), (0.1, 0.1) \big ) $
$ \mathcal{H}_4 $ $ \big ( (0.4, 0.8), (0.5, 0.1) \big ) $ $ \big ( (0.7, 0.3), (0.3, 0.3) \big ) $ $ \big ( (0.6, 0.5), (0.1, 0.3) \big ) $ $ \big ( (0.5, 0.5), (0.3, 0.4) \big ) $
$ \mathcal{H}_5 $ $ \big ( (0.9, 0.7), (0.1, 0.2) \big ) $ $ \big ( (0.7, 0.7), (0.2, 0.1) \big ) $ $ \big ( (0.7, 0.6), (0.2, 0.2) \big ) $ $ \big ( (0.8, 0.8), (0.1, 0.1) \big ) $
Table 3.  Ranking on changing values of $ p $ and $ q $
Values of $ p $ and $ q $ Score values Ranking
$ \mathcal{H}_1 $ $ \mathcal{H}_2 $ $ \mathcal{H}_3 $ $ \mathcal{H}_4 $ $ \mathcal{H}_5 $
$ p=1 $; $ q=1 $ -0.8301 -0.9395 -1.2343 -1.1190 -0.6852 $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=1 $; $ q=2 $ -0.7549 -0.9000 -1.1946 -1.0670 -0.6387 $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=2 $; $ q=2 $ -0.7566 -0.8957 -1.1800 -1.0779 -0.6358 $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=2 $; $ q=3 $ -0.7017 -0.8657 -1.1530 -1.0419 -0.6009 $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=3.5 $; $ q=0.1 $ -0.4369 -0.7490 -1.0863 -0.8261 -0.4493 $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=4 $; $ q=0.1 $ -0.3898 -0.7234 -1.0694 -0.7911 -0.4178 $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=5 $; $ q=0.5 $ -0.3927 -0.7177 -1.0684 -0.7962 -0.4108 $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=6 $; $ q=1 $ -0.3966 -0.7141 -1.0639 -0.8049 -0.4050 $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
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Values of $ p $ and $ q $ Score values Ranking
$ \mathcal{H}_1 $ $ \mathcal{H}_2 $ $ \mathcal{H}_3 $ $ \mathcal{H}_4 $ $ \mathcal{H}_5 $
$ p=1 $; $ q=1 $ -0.8301 -0.9395 -1.2343 -1.1190 -0.6852 $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=1 $; $ q=2 $ -0.7549 -0.9000 -1.1946 -1.0670 -0.6387 $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=2 $; $ q=2 $ -0.7566 -0.8957 -1.1800 -1.0779 -0.6358 $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=2 $; $ q=3 $ -0.7017 -0.8657 -1.1530 -1.0419 -0.6009 $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=3.5 $; $ q=0.1 $ -0.4369 -0.7490 -1.0863 -0.8261 -0.4493 $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=4 $; $ q=0.1 $ -0.3898 -0.7234 -1.0694 -0.7911 -0.4178 $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=5 $; $ q=0.5 $ -0.3927 -0.7177 -1.0684 -0.7962 -0.4108 $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=6 $; $ q=1 $ -0.3966 -0.7141 -1.0639 -0.8049 -0.4050 $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Table 4.  Analysis of the Figures 1(a), 1(b), 1(c) and 1(d)
Value of $ \wp $ Accuracy for $ p=\wp $ Ranking of the alternatives
When $ p<\wp $ When $ p=\wp $ When $ p<\wp $
Figure 1(a) $ 5.55 $ $ H(\mathcal{H}_1)=1.8380 $, $ H(\mathcal{H}_5)=1.8689 $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $ $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Figure 1(b) $ - $ $ - $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Figure 1(c) $ 1.593 $ $ H(\mathcal{H}_1)=1.8307 $, $ H(\mathcal{H}_5)=1.8686 $ $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Figure 1(d) $ 2.93 $ $ H(\mathcal{H}_1)=1.8191 $, $ H(\mathcal{H}_5)=1.8656 $ $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
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Value of $ \wp $ Accuracy for $ p=\wp $ Ranking of the alternatives
When $ p<\wp $ When $ p=\wp $ When $ p<\wp $
Figure 1(a) $ 5.55 $ $ H(\mathcal{H}_1)=1.8380 $, $ H(\mathcal{H}_5)=1.8689 $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $ $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Figure 1(b) $ - $ $ - $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Figure 1(c) $ 1.593 $ $ H(\mathcal{H}_1)=1.8307 $, $ H(\mathcal{H}_5)=1.8686 $ $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Figure 1(d) $ 2.93 $ $ H(\mathcal{H}_1)=1.8191 $, $ H(\mathcal{H}_5)=1.8656 $ $ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $ $ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Table 5.  Comparative Analysis results with CIFS studies
Method used Score values Ranking
$ \mathcal{H}_1 $ $ \mathcal{H}_2 $ $ \mathcal{H}_3 $ $ \mathcal{H}_4 $ $ \mathcal{H}_5 $
Method based on CIFWA operator [19] 1.1605 0.8812 0.3491 0.6484 1.2545 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on CIFWPA operator [37] 1.1449 0.8829 0.3540 0.6432 1.2504 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on Distance measure [3] - $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on Euclidean distance measure [36] - $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on Correlation coefficient [20] - $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Proposed method with $ (p=1;q=1) $ -0.8301 -0.9395 -1.2343 -1.1190 -0.6852 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Proposed method with $ (p=1;q=10) $ -0.2174 -0.6143 -0.9993 -0.6744 -0.2686 $ \mathcal{H}_1 \succ \mathcal{H}_5 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Proposed method with $ (p=1;q=0) $ -0.6921 -0.8863 -1.1942 -1.0262 -0.6041 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
$ Used: t(a)=-\log(a) $ for $ 0< a \leq 1 $ with $ t(0)=\infty $ in [19], $ \alpha_1=\beta_1=\sigma_1=\alpha_2=\beta_2=\sigma_2=\frac{1}{3} $ in [3]
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Method used Score values Ranking
$ \mathcal{H}_1 $ $ \mathcal{H}_2 $ $ \mathcal{H}_3 $ $ \mathcal{H}_4 $ $ \mathcal{H}_5 $
Method based on CIFWA operator [19] 1.1605 0.8812 0.3491 0.6484 1.2545 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on CIFWPA operator [37] 1.1449 0.8829 0.3540 0.6432 1.2504 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on Distance measure [3] - $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on Euclidean distance measure [36] - $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on Correlation coefficient [20] - $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Proposed method with $ (p=1;q=1) $ -0.8301 -0.9395 -1.2343 -1.1190 -0.6852 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Proposed method with $ (p=1;q=10) $ -0.2174 -0.6143 -0.9993 -0.6744 -0.2686 $ \mathcal{H}_1 \succ \mathcal{H}_5 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Proposed method with $ (p=1;q=0) $ -0.6921 -0.8863 -1.1942 -1.0262 -0.6041 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
$ Used: t(a)=-\log(a) $ for $ 0< a \leq 1 $ with $ t(0)=\infty $ in [19], $ \alpha_1=\beta_1=\sigma_1=\alpha_2=\beta_2=\sigma_2=\frac{1}{3} $ in [3]
Table 6.  Comparative Analysis results with IFS studies
Method used Score values Ranking
$ \mathcal{H}_1 $ $ \mathcal{H}_2 $ $ \mathcal{H}_3 $ $ \mathcal{H}_4 $ $ \mathcal{H}_5 $
Xu and Yager [42] method based on IFWBM operator $ (p=1;q=1) $ -0.3968 -0.5370 -0.6319 -0.5754 -0.3136 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Xu [41] method based on IFPWA operator 0.5653 0.3332 0.1484 0.2441 0.6839 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Wang and Liu [39] method based on IFEWA operator 0.5670 0.3276 0.1183 0.2181 0.6871 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Xu and Yager [44] based on IFWG operator 0.5314 0.2826 -0.0179 0.1466 0.6536 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Xu [43] method based on IFWA operator 0.5701 0.3351 0.1432 0.2301 0.6898 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Garg [10] method based on IFEWGIA operator 0.6563 0.4787 0.0142 0.2849 0.7193 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
He et al. [24] method based on IFGIA method 0.6484 0.4768 -0.0085 0.2707 0.7172 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Huang [25] method based on IFHWA operator 0.5658 0.3241 0.1064 0.2127 0.6860 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Garg [11] method 0.4307 0.1603 0.0710 0.0694 0.6375 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_3 \succ \mathcal{H}_4 $
Chen and Chang [8] method 0.4339 0.1804 0.1000 0.0845 0.6435 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_3 \succ \mathcal{H}_4 $
Goyal et al.[23] method 0.7982 0.6623 0.3109 0.4510 0.8604 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Garg [12] method 0.4316 0.1669 0.0809 0.0743 0.6392 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_3 \succ \mathcal{H}_4 $
Ye [48] method 0.5506 0.3084 0.0596 0.1876 0.6715 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Zhou and Xu [50] method 0.5868 0.3824 0.3288 0.3776 0.6979 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
The proposed CIFWBM operator $ (p=1;q=1) $ -1.3968 -1.5370 -1.6319 -1.5754 -1.3136 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
The proposed CIFWBM operator $ (p=1;q=0) $ -1.3485 -1.5108 -1.6119 -1.5700 -1.2525 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Abbreviations. IFWA: Intuitionistic fuzzy weighted averaging; IFWG: Intuitionistic fuzzy weighted geometric; IFEWA: intuitionistic fuzzy Einstein weighted averaging; IFPWA: intuitionistic fuzzy power weighted averaging; IFWBM: intuitionistic fuzzy weighted Bonferroni mean; IFGIA: intuitionistic fuzzy geometric interactive averaging; IFEWGIA: intuitionistic fuzzy Einstein weighted geometric interactive averaging; IFHWA: intuitionistic fuzzy Hamacher weighted averaging; CIFWBM: complex intuitionistic fuzzy weighted Bonferroni mean.
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Method used Score values Ranking
$ \mathcal{H}_1 $ $ \mathcal{H}_2 $ $ \mathcal{H}_3 $ $ \mathcal{H}_4 $ $ \mathcal{H}_5 $
Xu and Yager [42] method based on IFWBM operator $ (p=1;q=1) $ -0.3968 -0.5370 -0.6319 -0.5754 -0.3136 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Xu [41] method based on IFPWA operator 0.5653 0.3332 0.1484 0.2441 0.6839 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Wang and Liu [39] method based on IFEWA operator 0.5670 0.3276 0.1183 0.2181 0.6871 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Xu and Yager [44] based on IFWG operator 0.5314 0.2826 -0.0179 0.1466 0.6536 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Xu [43] method based on IFWA operator 0.5701 0.3351 0.1432 0.2301 0.6898 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Garg [10] method based on IFEWGIA operator 0.6563 0.4787 0.0142 0.2849 0.7193 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
He et al. [24] method based on IFGIA method 0.6484 0.4768 -0.0085 0.2707 0.7172 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Huang [25] method based on IFHWA operator 0.5658 0.3241 0.1064 0.2127 0.6860 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Garg [11] method 0.4307 0.1603 0.0710 0.0694 0.6375 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_3 \succ \mathcal{H}_4 $
Chen and Chang [8] method 0.4339 0.1804 0.1000 0.0845 0.6435 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_3 \succ \mathcal{H}_4 $
Goyal et al.[23] method 0.7982 0.6623 0.3109 0.4510 0.8604 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Garg [12] method 0.4316 0.1669 0.0809 0.0743 0.6392 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_3 \succ \mathcal{H}_4 $
Ye [48] method 0.5506 0.3084 0.0596 0.1876 0.6715 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Zhou and Xu [50] method 0.5868 0.3824 0.3288 0.3776 0.6979 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
The proposed CIFWBM operator $ (p=1;q=1) $ -1.3968 -1.5370 -1.6319 -1.5754 -1.3136 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
The proposed CIFWBM operator $ (p=1;q=0) $ -1.3485 -1.5108 -1.6119 -1.5700 -1.2525 $ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Abbreviations. IFWA: Intuitionistic fuzzy weighted averaging; IFWG: Intuitionistic fuzzy weighted geometric; IFEWA: intuitionistic fuzzy Einstein weighted averaging; IFPWA: intuitionistic fuzzy power weighted averaging; IFWBM: intuitionistic fuzzy weighted Bonferroni mean; IFGIA: intuitionistic fuzzy geometric interactive averaging; IFEWGIA: intuitionistic fuzzy Einstein weighted geometric interactive averaging; IFHWA: intuitionistic fuzzy Hamacher weighted averaging; CIFWBM: complex intuitionistic fuzzy weighted Bonferroni mean.
Table 7.  The characteristic comparison of different approaches
Method Captures interrelationship among arguments Ability to capture information using complex numbers Ability to handle two-dimensional information Ability to integrate Information Flexible according to decision-maker's preferences
In [37] $ \times $ $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \times $
In [19] $ \times $ $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \times $
In [3] $ \times $ $ \checkmark $ $ \checkmark $ $ \times $ $ \checkmark $
In [36] $ \times $ $ \checkmark $ $ \checkmark $ $ \times $ $ \times $
In [42] $ \checkmark $ $ \times $ $ \times $ $ \checkmark $ $ \checkmark $
In [41] $ \times $ $ \times $ $ \times $ $ \checkmark $ $ \times $
In [39] $ \times $ $ \times $ $ \times $ $ \checkmark $ $ \times $
The proposed approach $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \checkmark $
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Method Captures interrelationship among arguments Ability to capture information using complex numbers Ability to handle two-dimensional information Ability to integrate Information Flexible according to decision-maker's preferences
In [37] $ \times $ $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \times $
In [19] $ \times $ $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \times $
In [3] $ \times $ $ \checkmark $ $ \checkmark $ $ \times $ $ \checkmark $
In [36] $ \times $ $ \checkmark $ $ \checkmark $ $ \times $ $ \times $
In [42] $ \checkmark $ $ \times $ $ \times $ $ \checkmark $ $ \checkmark $
In [41] $ \times $ $ \times $ $ \times $ $ \checkmark $ $ \times $
In [39] $ \times $ $ \times $ $ \times $ $ \checkmark $ $ \times $
The proposed approach $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \checkmark $ $ \checkmark $
[1]

Harish Garg. Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision-making process. Journal of Industrial and Management Optimization, 2018, 14 (1) : 283-308. doi: 10.3934/jimo.2017047
[2]

Muhammad Qiyas, Saleem Abdullah, Shahzaib Ashraf, Saifullah Khan, Aziz Khan. Triangular picture fuzzy linguistic induced ordered weighted aggregation operators and its application on decision making problems. Mathematical Foundations of Computing, 2019, 2 (3) : 183-201. doi: 10.3934/mfc.2019013
[3]

Harish Garg, Kamal Kumar. Group decision making approach based on possibility degree measure under linguistic interval-valued intuitionistic fuzzy set environment. Journal of Industrial and Management Optimization, 2020, 16 (1) : 445-467. doi: 10.3934/jimo.2018162
[4]

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