American Institute of Mathematical Sciences

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January  2018, 14(1): 283-308. doi: 10.3934/jimo.2017047

Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision-making process

Harish Garg

School of Mathematics, Thapar University Patiala-147004, Punjab, India

Received  July 2016 Revised  October 2016 Published  January 2018 Early access  June 2017

Fund Project: The author would like to thank the Editor-in-Chief and referees for providing very helpful comments and suggestions.

The objective of this manuscript is to present some new interactive geometric aggregation operators for the interval-valued intuitionistic fuzzy numbers (IVIFNs). In order to achieve it, firstly the shortcomings of the existing operators have been highlighted and then resolved it by defining new operational laws based on the pairs of hesitation degree between the membership functions. By using these improved laws, some geometric aggregation operators, namely interval-valued intuitionistic fuzzy Hamacher interactive weighted and hybrid geometric labeled as IIFHIWG and IIFHIHWG operators, respectively have been proposed. Furthermore, desirable properties corresponding to these operators have been stated. Finally, a decision-making method based on the proposed operator has been illustrated to demonstrate the approach. A computed result is compared with the existing results.

Keywords: MCDM, intuitionistic fuzzy set, aggregation operator, Hamacher operations.
Mathematics Subject Classification: Primary: 90B50, 62A86, 03E72, 68T35.
Citation: 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
References:
[1]

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.

[2]

K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. 

[3]

W. -K. Chen and Y. -T. Chen, Fuzzy optimization in decision making of air quality management, Springer International Publishing, Cham, 2015,341-363,

[4]

H. Garg, Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process, Computational and Mathematical Organization Theory, (2017), 1-26.  doi: 10.1007/s10588-017-9242-8.

[5]

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

[6]

H. Garg, Generalized intuitionistic fuzzy multiplicative interactive geometric operators and their application to multiple criteria decision making, International Journal of Machine Learning and Cybernetics, 7 (2016), 1075-1092.  doi: 10.1007/s13042-015-0432-8.

[7]

H. Garg, Generalized pythagorean fuzzy geometric aggregation operators using einstein t-norm and t-conorm for multicriteria decision-making process, International Journal of Intelligent Systems, 32 (2017), 597-630.  doi: 10.1002/int.21860.

[8]

H. Garg, A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems, Applied Soft Computing, 38 (2016), 988-999.  doi: 10.1016/j.asoc.2015.10.040.

[9]

H. Garg, A new generalized Pythagorean fuzzy information aggregation using einstein operations and its application to decision making, International Journal of Intelligent Systems, 31 (2016), 886-920.  doi: 10.1002/int.21809.

[10]

H. Garg, A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem, Journal of Intelligent and Fuzzy Systems, 31 (2016), 529-540.  doi: 10.3233/IFS-162165.

[11]

H. Garg, A novel approach for analyzing the reliability of series-parallel system using credibility theory and different types of intuitionistic fuzzy numbers, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38 (2016), 1021-1035.  doi: 10.1007/s40430-014-0284-2.

[12]

H. Garg, A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making processes, International Journal of Intelligent Systems, 31 (2016), 1234-1253.  doi: 10.1002/int.21827.

[13]

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

[14]

H. Garg, N. Agarwal and A. Tripathi, Entropy based multi-criteria decision making method under fuzzy environment and unknown attribute weights, Global Journal of Technology and Optimization, 6 (2015), 13-20. 

[15]

M. Gupta, Group Decision Making in Fuzzy Environment -An Iterative Procedure Based on Group Dynamics, Springer International Publishing, Cham, 2015.

[16]

H. Hamacher, Uber logistic verknunpfungenn unssharfer aussagen und deren zugenhoringe bewertungsfunktione, Progress in Cybernatics and Systems Research, 3 (1978), 276-288. 

[17]

Y. He, H. Chen, L. Zhou, B. Han, Q. Zhao and J. Liu, Generalized intuitionistic fuzzy geometric interaction operators and their application to decision making, Expert Systems with Applications, 41 (2014), 2484-2495.  doi: 10.1016/j.eswa.2013.09.048.

[18]

K. Kumar and H. Garg, TOPSIS method based on the connection number of set pair analysis under interval-valued intuitionistic fuzzy set environment, Computational and Applied Mathematics, (2016), 1-11, doi: 10.1007/s40314-016-0402-0.

[19]

W. Li and C. Zhang, Decision Making-Interactive and Interactive Approaches, Springer International Publishing, Cham, 2015,

[20]

P. Liu, Some hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making, IEEE Transactions on Fuzzy Systems, 22 (2013), 83-97.  doi: 10.1109/TFUZZ.2013.2248736.

[21]

Nancy and H. Garg, An improved score function for ranking neutrosophic sets and its application to decision-making process, International Journal for Uncertainty Quantification, 6 (2016), 377-385. 

[22]

Nancy and H. Garg, Novel single-valued neutrosophic decision making operators under frank norm operations and its application, International Journal for Uncertainty Quantification, 6 (2016), 361-375. 

[23]

S. Singh and H. Garg, Distance measures between type-2 intuitionistic fuzzy sets and their application to multicriteria decision-making process, Applied Intelligence, 46 (2017), 788-799.  doi: 10.1007/s10489-016-0869-9.

[24]

W. Wang and X. Liu, Some interval-valued intuitionistic fuzzy geometric aggregation operators based on einstein operations, in 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery, 2012,604-608. doi: 10.1109/FSKD.2012.6234364.

[25]

W. Wang and X. Liu, The multi-attribute decision making method based on interval-valued intuitionistic fuzzy einstein hybrid weighted geometric operator, Computers and Mathematics with Applications, 66 (2013), 1845-1856.  doi: 10.1016/j.camwa.2013.07.020.

[26]

G. Wei and X. Wang, Some geometric aggregation operators based on interval -valued intuitionistic fuzzy sets and their application to group decision making, in Proceedings of the IEEE international conference on computational intelligence and security, 2007,495-499. doi: 10.1109/CIS.2007.84.

[27]

Z. Xu and J. Chen, Approach to group decision making based on interval valued intuitionistic judgment matrices, Systems Engineering -Theory and Practice, 27 (2007), 126-133.  doi: 10.1016/S1874-8651(08)60026-5.

[28]

Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transaction of Fuzzy System, 15 (2007), 1179-1187. 

[29]

Z. S. Xu, Intuitionistic preference relations and their application in group decision making, Information Sciences, 177 (2007), 2363-2379.  doi: 10.1016/j.ins.2006.12.019.

[30]

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

[31]

Z. Xu and J. Chen, On geometric aggregation over interval-valued intuitionistic fuzzy information, Fuzzy Systems and Knowledge Discovery, 2007. FSKD 2007. Fourth International Conference on, 2 (2007), 466-471.  doi: 10.1109/FSKD.2007.427.

[32]

Z. Xu and X. Gou, An overview of interval-valued intuitionistic fuzzy information aggregations and applications, Granular Computing, 2 (2017), 13-39.  doi: 10.1007/s41066-016-0023-4.

[33]

Z. Xu and H. Wang, Managing multi-granularity linguistic information in qualitative group decision making: An overview, Granular Computing, 1 (2016), 21-35.  doi: 10.1007/s41066-015-0006-x.

[34]

X. Zhao and G. Wei, Some intuitionistic fuzzy einstein hybrd aggregation operators and their application to multiple attribute decision making, Knowledge Based Systems, 37 (2013), 472-479. 

show all references

References:
[1]

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.

[2]

K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. 

[3]

W. -K. Chen and Y. -T. Chen, Fuzzy optimization in decision making of air quality management, Springer International Publishing, Cham, 2015,341-363,

[4]

H. Garg, Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process, Computational and Mathematical Organization Theory, (2017), 1-26.  doi: 10.1007/s10588-017-9242-8.

[5]

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

[6]

H. Garg, Generalized intuitionistic fuzzy multiplicative interactive geometric operators and their application to multiple criteria decision making, International Journal of Machine Learning and Cybernetics, 7 (2016), 1075-1092.  doi: 10.1007/s13042-015-0432-8.

[7]

H. Garg, Generalized pythagorean fuzzy geometric aggregation operators using einstein t-norm and t-conorm for multicriteria decision-making process, International Journal of Intelligent Systems, 32 (2017), 597-630.  doi: 10.1002/int.21860.

[8]

H. Garg, A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems, Applied Soft Computing, 38 (2016), 988-999.  doi: 10.1016/j.asoc.2015.10.040.

[9]

H. Garg, A new generalized Pythagorean fuzzy information aggregation using einstein operations and its application to decision making, International Journal of Intelligent Systems, 31 (2016), 886-920.  doi: 10.1002/int.21809.

[10]

H. Garg, A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem, Journal of Intelligent and Fuzzy Systems, 31 (2016), 529-540.  doi: 10.3233/IFS-162165.

[11]

H. Garg, A novel approach for analyzing the reliability of series-parallel system using credibility theory and different types of intuitionistic fuzzy numbers, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38 (2016), 1021-1035.  doi: 10.1007/s40430-014-0284-2.

[12]

H. Garg, A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making processes, International Journal of Intelligent Systems, 31 (2016), 1234-1253.  doi: 10.1002/int.21827.

[13]

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

[14]

H. Garg, N. Agarwal and A. Tripathi, Entropy based multi-criteria decision making method under fuzzy environment and unknown attribute weights, Global Journal of Technology and Optimization, 6 (2015), 13-20. 

[15]

M. Gupta, Group Decision Making in Fuzzy Environment -An Iterative Procedure Based on Group Dynamics, Springer International Publishing, Cham, 2015.

[16]

H. Hamacher, Uber logistic verknunpfungenn unssharfer aussagen und deren zugenhoringe bewertungsfunktione, Progress in Cybernatics and Systems Research, 3 (1978), 276-288. 

[17]

Y. He, H. Chen, L. Zhou, B. Han, Q. Zhao and J. Liu, Generalized intuitionistic fuzzy geometric interaction operators and their application to decision making, Expert Systems with Applications, 41 (2014), 2484-2495.  doi: 10.1016/j.eswa.2013.09.048.

[18]

K. Kumar and H. Garg, TOPSIS method based on the connection number of set pair analysis under interval-valued intuitionistic fuzzy set environment, Computational and Applied Mathematics, (2016), 1-11, doi: 10.1007/s40314-016-0402-0.

[19]

W. Li and C. Zhang, Decision Making-Interactive and Interactive Approaches, Springer International Publishing, Cham, 2015,

[20]

P. Liu, Some hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making, IEEE Transactions on Fuzzy Systems, 22 (2013), 83-97.  doi: 10.1109/TFUZZ.2013.2248736.

[21]

Nancy and H. Garg, An improved score function for ranking neutrosophic sets and its application to decision-making process, International Journal for Uncertainty Quantification, 6 (2016), 377-385. 

[22]

Nancy and H. Garg, Novel single-valued neutrosophic decision making operators under frank norm operations and its application, International Journal for Uncertainty Quantification, 6 (2016), 361-375. 

[23]

S. Singh and H. Garg, Distance measures between type-2 intuitionistic fuzzy sets and their application to multicriteria decision-making process, Applied Intelligence, 46 (2017), 788-799.  doi: 10.1007/s10489-016-0869-9.

[24]

W. Wang and X. Liu, Some interval-valued intuitionistic fuzzy geometric aggregation operators based on einstein operations, in 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery, 2012,604-608. doi: 10.1109/FSKD.2012.6234364.

[25]

W. Wang and X. Liu, The multi-attribute decision making method based on interval-valued intuitionistic fuzzy einstein hybrid weighted geometric operator, Computers and Mathematics with Applications, 66 (2013), 1845-1856.  doi: 10.1016/j.camwa.2013.07.020.

[26]

G. Wei and X. Wang, Some geometric aggregation operators based on interval -valued intuitionistic fuzzy sets and their application to group decision making, in Proceedings of the IEEE international conference on computational intelligence and security, 2007,495-499. doi: 10.1109/CIS.2007.84.

[27]

Z. Xu and J. Chen, Approach to group decision making based on interval valued intuitionistic judgment matrices, Systems Engineering -Theory and Practice, 27 (2007), 126-133.  doi: 10.1016/S1874-8651(08)60026-5.

[28]

Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transaction of Fuzzy System, 15 (2007), 1179-1187. 

[29]

Z. S. Xu, Intuitionistic preference relations and their application in group decision making, Information Sciences, 177 (2007), 2363-2379.  doi: 10.1016/j.ins.2006.12.019.

[30]

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

[31]

Z. Xu and J. Chen, On geometric aggregation over interval-valued intuitionistic fuzzy information, Fuzzy Systems and Knowledge Discovery, 2007. FSKD 2007. Fourth International Conference on, 2 (2007), 466-471.  doi: 10.1109/FSKD.2007.427.

[32]

Z. Xu and X. Gou, An overview of interval-valued intuitionistic fuzzy information aggregations and applications, Granular Computing, 2 (2017), 13-39.  doi: 10.1007/s41066-016-0023-4.

[33]

Z. Xu and H. Wang, Managing multi-granularity linguistic information in qualitative group decision making: An overview, Granular Computing, 1 (2016), 21-35.  doi: 10.1007/s41066-015-0006-x.

[34]

X. Zhao and G. Wei, Some intuitionistic fuzzy einstein hybrd aggregation operators and their application to multiple attribute decision making, Knowledge Based Systems, 37 (2013), 472-479. 

Table 1.  Information about each alternative in the form of the IVIFNs
$C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $C_6$
$X_1$ $\langle[0.2, 0.3], [0.4, 0.5]\rangle$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.4, 0.5], [0.2, 0.4]\rangle$ $\langle[0.7, 0.8], [0.1, 0.2]\rangle$ $\langle[0.1, 0.3], [0.5, 0.6]\rangle$ $\langle[0.5, 0.7], [0.2, 0.3]\rangle$
$X_2$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.5, 0.6], [0.1, 0.3]\rangle$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.6, 0.7], [0.1, 0.2]\rangle$ $\langle[0.3, 0.4], [0.5, 0.6]\rangle$ $\langle[0.4, 0.7], [0.1, 0.2]\rangle$
$X_3$ $\langle[0.4, 0.5], [0.3, 0.4]\rangle$ $\langle[0.7, 0.8], [0.1, 0.2]\rangle$ $\langle[0.5, 0.6], [0.3, 0.4]\rangle$ $\langle[0.6, 0.7], [0.1, 0.3]\rangle$ $\langle[0.4, 0.5], [0.3, 0.4]\rangle$ $\langle[0.3, 0.5], [0.1, 0.3]\rangle$
$X_4$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.5, 0.6], [0.1, 0.3]\rangle$ $\langle[0.7, 0.8], [0.1, 0.2]\rangle$ $\langle[0.3, 0.4], [0.1, 0.2]\rangle$ $\langle[0.5, 0.6], [0.1, 0.3]\rangle$ $\langle[0.7, 0.8], [0.1, 0.2]\rangle$
$X_5$ $\langle[0.5, 0.6], [0.3, 0.4]\rangle$ $\langle[0.3, 0.4], [0.3, 0.5]\rangle$ $\langle[0.6, 0.7], [0.1, 0.3]\rangle$ $\langle[0.6, 0.8], [0.1, 0.2]\rangle$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.5, 0.6], [0.2, 0.4]\rangle$
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$C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $C_6$
$X_1$ $\langle[0.2, 0.3], [0.4, 0.5]\rangle$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.4, 0.5], [0.2, 0.4]\rangle$ $\langle[0.7, 0.8], [0.1, 0.2]\rangle$ $\langle[0.1, 0.3], [0.5, 0.6]\rangle$ $\langle[0.5, 0.7], [0.2, 0.3]\rangle$
$X_2$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.5, 0.6], [0.1, 0.3]\rangle$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.6, 0.7], [0.1, 0.2]\rangle$ $\langle[0.3, 0.4], [0.5, 0.6]\rangle$ $\langle[0.4, 0.7], [0.1, 0.2]\rangle$
$X_3$ $\langle[0.4, 0.5], [0.3, 0.4]\rangle$ $\langle[0.7, 0.8], [0.1, 0.2]\rangle$ $\langle[0.5, 0.6], [0.3, 0.4]\rangle$ $\langle[0.6, 0.7], [0.1, 0.3]\rangle$ $\langle[0.4, 0.5], [0.3, 0.4]\rangle$ $\langle[0.3, 0.5], [0.1, 0.3]\rangle$
$X_4$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.5, 0.6], [0.1, 0.3]\rangle$ $\langle[0.7, 0.8], [0.1, 0.2]\rangle$ $\langle[0.3, 0.4], [0.1, 0.2]\rangle$ $\langle[0.5, 0.6], [0.1, 0.3]\rangle$ $\langle[0.7, 0.8], [0.1, 0.2]\rangle$
$X_5$ $\langle[0.5, 0.6], [0.3, 0.4]\rangle$ $\langle[0.3, 0.4], [0.3, 0.5]\rangle$ $\langle[0.6, 0.7], [0.1, 0.3]\rangle$ $\langle[0.6, 0.8], [0.1, 0.2]\rangle$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.5, 0.6], [0.2, 0.4]\rangle$
Table 2.  Effect of the parameter $\gamma$ on the ranking of the alternatives by IIFHIWG and the existing operators
$\gamma=1$ $\gamma=2$ $\gamma=3$
Wei and Wang [26] Proposed Wang and Liu [24] Proposed Liu [20] Proposed
Score value Score value Score value
$X_1$ 0.0548 0.1346 0.0727 0.1454 0.0822 0.1517
$X_2$ 0.2874 0.3174 0.2998 0.3310 0.3065 0.3388
$X_3$ 0.2139 0.2713 0.2205 0.2760 0.2245 0.2793
$X_4$ 0.4463 0.4997 0.4535 0.5013 0.4576 0.5024
$X_5$ 0.2985 0.3119 0.3047 0.3166 0.3083 0.3197
ranking $X_4 \succ X_5 \succ X_2 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_5 \succ X_2 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_5 \succ X_2 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$
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$\gamma=1$ $\gamma=2$ $\gamma=3$
Wei and Wang [26] Proposed Wang and Liu [24] Proposed Liu [20] Proposed
Score value Score value Score value
$X_1$ 0.0548 0.1346 0.0727 0.1454 0.0822 0.1517
$X_2$ 0.2874 0.3174 0.2998 0.3310 0.3065 0.3388
$X_3$ 0.2139 0.2713 0.2205 0.2760 0.2245 0.2793
$X_4$ 0.4463 0.4997 0.4535 0.5013 0.4576 0.5024
$X_5$ 0.2985 0.3119 0.3047 0.3166 0.3083 0.3197
ranking $X_4 \succ X_5 \succ X_2 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_5 \succ X_2 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_5 \succ X_2 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$
Table 3.  Effect of the parameter $\gamma$ on the ranking of the alternatives by using IIFHIHWG and the existing operators
$\gamma=1$ $\gamma=2$ $\gamma=3$
Wei and Wang [26] Proposed Wang and Liu [24] Proposed Liu [20] Proposed
Score value Score value Score value
$X_1$ 0.1221 0.2080 0.1434 0.2163 0.1558 0.2151
$X_2$ 0.3304 0.3674 0.3443 0.3734 0.3522 0.3795
$X_3$ 0.2535 0.3019 0.2630 0.3068 0.2692 0.3113
$X_4$ 0.3705 0.4853 0.3815 0.4815 0.3880 0.4828
$X_5$ 0.3141 0.3414 0.3164 0.3510 0.3203 0.3500
ranking $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$
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$\gamma=1$ $\gamma=2$ $\gamma=3$
Wei and Wang [26] Proposed Wang and Liu [24] Proposed Liu [20] Proposed
Score value Score value Score value
$X_1$ 0.1221 0.2080 0.1434 0.2163 0.1558 0.2151
$X_2$ 0.3304 0.3674 0.3443 0.3734 0.3522 0.3795
$X_3$ 0.2535 0.3019 0.2630 0.3068 0.2692 0.3113
$X_4$ 0.3705 0.4853 0.3815 0.4815 0.3880 0.4828
$X_5$ 0.3141 0.3414 0.3164 0.3510 0.3203 0.3500
ranking $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$
Table 4.  Ordering of the attributes for different $\gamma$
$\gamma$ By IIFHIWG By IIFHIHWG
Aggregated IVIFN Score values Aggregated IVIFN Score values
0.1 $X_1$ $\big\langle[0.3771, 0.5753], [0.2996, 0.4247]\big\rangle$ 0.1140 $\big\langle[0.4562, 0.6029], [0.2818, 0.3971]\big\rangle$ 0.1901
$X_2$ $\big\langle[0.5042, 0.6545], [0.2324, 0.3455]\big\rangle$ 0.2904 $\big\langle[0.5016, 0.6889], [0.1971, 0.3111]\big\rangle$ 0.3411
$X_3$ $\big\langle[0.4719, 0.6441], [0.2310, 0.3559]\big\rangle$ 0.2646 $\big\langle[0.4808, 0.6605], [0.2187, 0.3395]\big\rangle$ 0.2916
$X_4$ $\big\langle[0.6130, 0.7519], [0.1218, 0.2481]\big\rangle$ 0.4975 $\big\langle[0.5404, 0.7670], [0.1097, 0.2330]\big\rangle$ 0.4824
$X_5$ $\big\langle[0.5306, 0.6405], [0.2027, 0.3595]\big\rangle$ 0.3045 $\big\langle[0.5338, 0.6603], [0.1876, 0.3397]\big\rangle$ 0.3334
Ranking $X_4\succ X_5 \succ X_2 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
0.5 $X_1$ $\big\langle[0.3805, 0.5819], [0.2933, 0.4181]\big\rangle$ 0.1255 $\big\langle[0.4597 0.6086], [0.2764 0.3914]\big\rangle$ 0.2003
$X_2$ $\big\langle[0.5092, 0.6634], [0.2249, 0.3366]\big\rangle$ 0.3056 $\big\langle[0.5062 0.6977], [0.1897 0.3023]\big\rangle$ 0.3560
$X_3$ $\big\langle[0.4734, 0.6455], [0.2285, 0.3545]\big\rangle$ 0.2679 $\big\langle[0.4826 0.6633], [0.2157 0.3367]\big\rangle$ 0.2967
$X_4$ $\big\langle[0.6133, 0.7526], [0.1214, 0.2474]\big\rangle$ 0.4986 $\big\langle[0.5406 0.7681], [0.1093 0.2319]\big\rangle$ 0.4838
$X_5$ $\big\langle[0.5318, 0.6429], [0.2009, 0.3571]\big\rangle$ 0.3084 $\big\langle[0.5352 0.6639], [0.1855 0.3361]\big\rangle$ 0.3388
Ranking $X_4\succ X_5 \succ X_2 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
1 $X_1$ $\big\langle[0.3834, 0.5868], [0.2878, 0.4132]\big\rangle$ 0.1346 $\big\langle[0.4626, 0.6125], [0.2717, 0.3875]\big\rangle$ 0.2080
$X_2$ $\big\langle[0.5134, 0.6699], [0.2185, 0.3301]\big\rangle$ 0.3174 $\big\langle[0.5100, 0.7041], [0.1835, 0.2959]\big\rangle$ 0.3674
$X_3$ $\big\langle[0.4750, 0.6467], [0.2260, 0.3533]\big\rangle$ 0.2713 $\big\langle[0.4845, 0.6659], [0.2126, 0.3341]\big\rangle$ 0.3019
$X_4$ $\big\langle[0.6136, 0.7533], [0.1210, 0.2467]\big\rangle$ 0.4997 $\big\langle[0.5409, 0.7693], [0.1089, 0.2307]\big\rangle$ 0.4853
$X_5$ $\big\langle[0.5330, 0.6449], [0.1991, 0.3551]\big\rangle$ 0.3119 $\big\langle[0.5141, 0.6751], [0.1813, 0.3249]\big\rangle$ 0.3414
Ranking $X_4\succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
2 $X_1$ $\big\langle[0.3872, 0.5923], [0.2809, 0.4077]\big\rangle$ 0.1454 $\big\langle[0.4663, 0.6161], [0.2659, 0.3839]\big\rangle$ 0.2163
$X_2$ $\big\langle[0.5186, 0.6770], [0.2106, 0.3230]\big\rangle$ 0.3310 $\big\langle[0.5268, 0.7023], [0.1847, 0.2977]\big\rangle$ 0.3734
$X_3$ $\big\langle[0.4774, 0.6484], [0.2221, 0.3516]\big\rangle$ 0.2760 $\big\langle[0.4890, 0.6646], [0.2047, 0.3354]\big\rangle$ 0.3068
$X_4$ $\big\langle[0.6141, 0.7543], [0.1202, 0.2457]\big\rangle$ 0.5013 $\big\langle[0.5556, 0.7623], [0.1172, 0.2377]\big\rangle$ 0.4815
$X_5$ $\big\langle[0.5348, 0.6473], [0.1963, 0.3527]\big\rangle$ 0.3166 $\big\langle[0.5166, 0.6813], [0.1772, 0.3187]\big\rangle$ 0.3510
Ranking $X_4\succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
5 $X_1$ $\big\langle[0.3922, 0.5987], [0.2716 0.4013]\big\rangle$ 0.1590 $\big\langle[0.4324, 0.6341], [0.2537, 0.3659]\big\rangle$ 0.2235
$X_2$ $\big\langle[0.5256, 0.6849], [0.1999 0.3151]\big\rangle$ 0.3478 $\big\langle[0.5195, 0.7130], [0.1776, 0.2870]\big\rangle$ 0.3840
$X_3$ $\big\langle[0.4815, 0.6506], [0.2153 0.3494]\big\rangle$ 0.2837 $\big\langle[0.4939, 0.6691], [0.1969, 0.3309]\big\rangle$ 0.3176
$X_4$ $\big\langle[0.6150, 0.7558], [0.1189 0.2442]\big\rangle$ 0.5039 $\big\langle[0.5565, 0.7642], [0.1158, 0.2358]\big\rangle$ 0.4846
$X_5$ $\big\langle[0.5379, 0.6504], [0.1917 0.3496]\big\rangle$ 0.3235 $\big\langle[0.5521, 0.6726], [0.1906, 0.3274]\big\rangle$ 0.3534
Ranking $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
10 $X_1$ $\big\langle[0.3952, 0.6020], [0.2659, 0.3980]\big\rangle$ 0.1667 $\big\langle[0.4360, 0.6387], [0.2474, 0.3613]\big\rangle$ 0.2330
$X_2$ $\big\langle[0.5300, 0.6889], [0.1932, 0.3111]\big\rangle$ 0.3573 $\big\langle[0.5220, 0.7368], [0.1572, 0.2632]\big\rangle$ 0.4192
$X_3$ $\big\langle[0.4847, 0.6519], [0.2102, 0.3481]\big\rangle$ 0.2891 $\big\langle[0.4767, 0.6968], [0.1744, 0.3032]\big\rangle$ 0.3479
$X_4$ $\big\langle[0.6158, 0.7568], [0.1178, 0.2432]\big\rangle$ 0.5058 $\big\langle[0.5573, 0.7657], [0.1147, 0.2343]\big\rangle$ 0.4870
$X_5$ $\big\langle[0.5403, 0.6522], [0.1882, 0.3478]\big\rangle$ 0.3282 $\big\langle[0.5608, 0.6594], [0.1933, 0.3406]\big\rangle$ 0.3432
Ranking $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_3 \succ X_5 \succ X_1$
25 $X_1$ $\big\langle[0.3979, 0.6046], [0.2610, 0.3954]\big\rangle$ 0.1730 $\big\langle[0.4217, 0.6566], [0.2234, 0.3434]\big\rangle$ 0.2558
$X_2$ $\big\langle[0.5338, 0.6920], [0.1874, 0.3080]\big\rangle$ 0.3652 $\big\langle[0.5570, 0.7047], [0.1689, 0.2953]\big\rangle$ 0.3988
$X_3$ $\big\langle[0.4878, 0.6530], [0.2051, 0.3470]\big\rangle$ 0.2943 $\big\langle[0.5176, 0.6687], [0.1871, 0.3313]\big\rangle$ 0.3339
$X_4$ $\big\langle[0.6165, 0.7576], [0.1167, 0.2424]\big\rangle$ 0.5075 $\big\langle[0.6057, 0.7394], [0.1297, 0.2606]\big\rangle$ 0.4774
$X_5$ $\big\langle[0.5426, 0.6536], [0.1847, 0.3464]\big\rangle$ 0.3325 $\big\langle[0.5600, 0.6525], [0.1944, 0.3475]\big\rangle$ 0.3353
Ranking $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
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$\gamma$ By IIFHIWG By IIFHIHWG
Aggregated IVIFN Score values Aggregated IVIFN Score values
0.1 $X_1$ $\big\langle[0.3771, 0.5753], [0.2996, 0.4247]\big\rangle$ 0.1140 $\big\langle[0.4562, 0.6029], [0.2818, 0.3971]\big\rangle$ 0.1901
$X_2$ $\big\langle[0.5042, 0.6545], [0.2324, 0.3455]\big\rangle$ 0.2904 $\big\langle[0.5016, 0.6889], [0.1971, 0.3111]\big\rangle$ 0.3411
$X_3$ $\big\langle[0.4719, 0.6441], [0.2310, 0.3559]\big\rangle$ 0.2646 $\big\langle[0.4808, 0.6605], [0.2187, 0.3395]\big\rangle$ 0.2916
$X_4$ $\big\langle[0.6130, 0.7519], [0.1218, 0.2481]\big\rangle$ 0.4975 $\big\langle[0.5404, 0.7670], [0.1097, 0.2330]\big\rangle$ 0.4824
$X_5$ $\big\langle[0.5306, 0.6405], [0.2027, 0.3595]\big\rangle$ 0.3045 $\big\langle[0.5338, 0.6603], [0.1876, 0.3397]\big\rangle$ 0.3334
Ranking $X_4\succ X_5 \succ X_2 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
0.5 $X_1$ $\big\langle[0.3805, 0.5819], [0.2933, 0.4181]\big\rangle$ 0.1255 $\big\langle[0.4597 0.6086], [0.2764 0.3914]\big\rangle$ 0.2003
$X_2$ $\big\langle[0.5092, 0.6634], [0.2249, 0.3366]\big\rangle$ 0.3056 $\big\langle[0.5062 0.6977], [0.1897 0.3023]\big\rangle$ 0.3560
$X_3$ $\big\langle[0.4734, 0.6455], [0.2285, 0.3545]\big\rangle$ 0.2679 $\big\langle[0.4826 0.6633], [0.2157 0.3367]\big\rangle$ 0.2967
$X_4$ $\big\langle[0.6133, 0.7526], [0.1214, 0.2474]\big\rangle$ 0.4986 $\big\langle[0.5406 0.7681], [0.1093 0.2319]\big\rangle$ 0.4838
$X_5$ $\big\langle[0.5318, 0.6429], [0.2009, 0.3571]\big\rangle$ 0.3084 $\big\langle[0.5352 0.6639], [0.1855 0.3361]\big\rangle$ 0.3388
Ranking $X_4\succ X_5 \succ X_2 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
1 $X_1$ $\big\langle[0.3834, 0.5868], [0.2878, 0.4132]\big\rangle$ 0.1346 $\big\langle[0.4626, 0.6125], [0.2717, 0.3875]\big\rangle$ 0.2080
$X_2$ $\big\langle[0.5134, 0.6699], [0.2185, 0.3301]\big\rangle$ 0.3174 $\big\langle[0.5100, 0.7041], [0.1835, 0.2959]\big\rangle$ 0.3674
$X_3$ $\big\langle[0.4750, 0.6467], [0.2260, 0.3533]\big\rangle$ 0.2713 $\big\langle[0.4845, 0.6659], [0.2126, 0.3341]\big\rangle$ 0.3019
$X_4$ $\big\langle[0.6136, 0.7533], [0.1210, 0.2467]\big\rangle$ 0.4997 $\big\langle[0.5409, 0.7693], [0.1089, 0.2307]\big\rangle$ 0.4853
$X_5$ $\big\langle[0.5330, 0.6449], [0.1991, 0.3551]\big\rangle$ 0.3119 $\big\langle[0.5141, 0.6751], [0.1813, 0.3249]\big\rangle$ 0.3414
Ranking $X_4\succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
2 $X_1$ $\big\langle[0.3872, 0.5923], [0.2809, 0.4077]\big\rangle$ 0.1454 $\big\langle[0.4663, 0.6161], [0.2659, 0.3839]\big\rangle$ 0.2163
$X_2$ $\big\langle[0.5186, 0.6770], [0.2106, 0.3230]\big\rangle$ 0.3310 $\big\langle[0.5268, 0.7023], [0.1847, 0.2977]\big\rangle$ 0.3734
$X_3$ $\big\langle[0.4774, 0.6484], [0.2221, 0.3516]\big\rangle$ 0.2760 $\big\langle[0.4890, 0.6646], [0.2047, 0.3354]\big\rangle$ 0.3068
$X_4$ $\big\langle[0.6141, 0.7543], [0.1202, 0.2457]\big\rangle$ 0.5013 $\big\langle[0.5556, 0.7623], [0.1172, 0.2377]\big\rangle$ 0.4815
$X_5$ $\big\langle[0.5348, 0.6473], [0.1963, 0.3527]\big\rangle$ 0.3166 $\big\langle[0.5166, 0.6813], [0.1772, 0.3187]\big\rangle$ 0.3510
Ranking $X_4\succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
5 $X_1$ $\big\langle[0.3922, 0.5987], [0.2716 0.4013]\big\rangle$ 0.1590 $\big\langle[0.4324, 0.6341], [0.2537, 0.3659]\big\rangle$ 0.2235
$X_2$ $\big\langle[0.5256, 0.6849], [0.1999 0.3151]\big\rangle$ 0.3478 $\big\langle[0.5195, 0.7130], [0.1776, 0.2870]\big\rangle$ 0.3840
$X_3$ $\big\langle[0.4815, 0.6506], [0.2153 0.3494]\big\rangle$ 0.2837 $\big\langle[0.4939, 0.6691], [0.1969, 0.3309]\big\rangle$ 0.3176
$X_4$ $\big\langle[0.6150, 0.7558], [0.1189 0.2442]\big\rangle$ 0.5039 $\big\langle[0.5565, 0.7642], [0.1158, 0.2358]\big\rangle$ 0.4846
$X_5$ $\big\langle[0.5379, 0.6504], [0.1917 0.3496]\big\rangle$ 0.3235 $\big\langle[0.5521, 0.6726], [0.1906, 0.3274]\big\rangle$ 0.3534
Ranking $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
10 $X_1$ $\big\langle[0.3952, 0.6020], [0.2659, 0.3980]\big\rangle$ 0.1667 $\big\langle[0.4360, 0.6387], [0.2474, 0.3613]\big\rangle$ 0.2330
$X_2$ $\big\langle[0.5300, 0.6889], [0.1932, 0.3111]\big\rangle$ 0.3573 $\big\langle[0.5220, 0.7368], [0.1572, 0.2632]\big\rangle$ 0.4192
$X_3$ $\big\langle[0.4847, 0.6519], [0.2102, 0.3481]\big\rangle$ 0.2891 $\big\langle[0.4767, 0.6968], [0.1744, 0.3032]\big\rangle$ 0.3479
$X_4$ $\big\langle[0.6158, 0.7568], [0.1178, 0.2432]\big\rangle$ 0.5058 $\big\langle[0.5573, 0.7657], [0.1147, 0.2343]\big\rangle$ 0.4870
$X_5$ $\big\langle[0.5403, 0.6522], [0.1882, 0.3478]\big\rangle$ 0.3282 $\big\langle[0.5608, 0.6594], [0.1933, 0.3406]\big\rangle$ 0.3432
Ranking $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_3 \succ X_5 \succ X_1$
25 $X_1$ $\big\langle[0.3979, 0.6046], [0.2610, 0.3954]\big\rangle$ 0.1730 $\big\langle[0.4217, 0.6566], [0.2234, 0.3434]\big\rangle$ 0.2558
$X_2$ $\big\langle[0.5338, 0.6920], [0.1874, 0.3080]\big\rangle$ 0.3652 $\big\langle[0.5570, 0.7047], [0.1689, 0.2953]\big\rangle$ 0.3988
$X_3$ $\big\langle[0.4878, 0.6530], [0.2051, 0.3470]\big\rangle$ 0.2943 $\big\langle[0.5176, 0.6687], [0.1871, 0.3313]\big\rangle$ 0.3339
$X_4$ $\big\langle[0.6165, 0.7576], [0.1167, 0.2424]\big\rangle$ 0.5075 $\big\langle[0.6057, 0.7394], [0.1297, 0.2606]\big\rangle$ 0.4774
$X_5$ $\big\langle[0.5426, 0.6536], [0.1847, 0.3464]\big\rangle$ 0.3325 $\big\langle[0.5600, 0.6525], [0.1944, 0.3475]\big\rangle$ 0.3353
Ranking $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
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