# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020069

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

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

Received  August 2019 Revised  December 2019 Published  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.

Citation: Harish Garg, Dimple Rani. Multi-criteria decision making method based on Bonferroni mean aggregation operators of complex intuitionistic fuzzy numbers. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020069
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##### References:
Variation in score values with parameter $p$ by fixing $q$
Score values of alternatives $\mathcal{H}_u$ for different values of $p$, $q$
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 $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
 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 $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Input information in the form of the complex intuitionistic fuzzy decision-matrix
 $\mathcal{C}_1$ $\mathcal{C}_2$ $\mathcal{C}_3$ $\mathcal{C}_4$ $\mathcal{H}_1$ $\big ( (0.7, 0.9), (0.1, 0.1) \big )$ $\big ( (0.8, 0.5), (0.1, 0.4) \big )$ $\big ( (0.6, 0.6), (0.3, 0.2) \big )$ $\big ( (0.7, 0.7), (0.3, 0.2) \big )$ $\mathcal{H}_2$ $\big ( (0.7, 0.6), (0.3, 0.3) \big )$ $\big ( (0.4, 0.9), (0.2, 0.1) \big )$ $\big ( (0.7, 0.7), (0.2, 0.3) \big )$ $\big ( (0.4, 0.6), (0.3, 0.1) \big )$ $\mathcal{H}_3$ $\big ( (0.3, 0.4), (0.6, 0.4) \big )$ $\big ( (0.6, 0.6), (0.3, 0.4) \big )$ $\big ( (0.3, 0.4), (0.5, 0.6) \big )$ $\big ( (0.7, 0.7), (0.1, 0.1) \big )$ $\mathcal{H}_4$ $\big ( (0.4, 0.8), (0.5, 0.1) \big )$ $\big ( (0.7, 0.3), (0.3, 0.3) \big )$ $\big ( (0.6, 0.5), (0.1, 0.3) \big )$ $\big ( (0.5, 0.5), (0.3, 0.4) \big )$ $\mathcal{H}_5$ $\big ( (0.9, 0.7), (0.1, 0.2) \big )$ $\big ( (0.7, 0.7), (0.2, 0.1) \big )$ $\big ( (0.7, 0.6), (0.2, 0.2) \big )$ $\big ( (0.8, 0.8), (0.1, 0.1) \big )$
 $\mathcal{C}_1$ $\mathcal{C}_2$ $\mathcal{C}_3$ $\mathcal{C}_4$ $\mathcal{H}_1$ $\big ( (0.7, 0.9), (0.1, 0.1) \big )$ $\big ( (0.8, 0.5), (0.1, 0.4) \big )$ $\big ( (0.6, 0.6), (0.3, 0.2) \big )$ $\big ( (0.7, 0.7), (0.3, 0.2) \big )$ $\mathcal{H}_2$ $\big ( (0.7, 0.6), (0.3, 0.3) \big )$ $\big ( (0.4, 0.9), (0.2, 0.1) \big )$ $\big ( (0.7, 0.7), (0.2, 0.3) \big )$ $\big ( (0.4, 0.6), (0.3, 0.1) \big )$ $\mathcal{H}_3$ $\big ( (0.3, 0.4), (0.6, 0.4) \big )$ $\big ( (0.6, 0.6), (0.3, 0.4) \big )$ $\big ( (0.3, 0.4), (0.5, 0.6) \big )$ $\big ( (0.7, 0.7), (0.1, 0.1) \big )$ $\mathcal{H}_4$ $\big ( (0.4, 0.8), (0.5, 0.1) \big )$ $\big ( (0.7, 0.3), (0.3, 0.3) \big )$ $\big ( (0.6, 0.5), (0.1, 0.3) \big )$ $\big ( (0.5, 0.5), (0.3, 0.4) \big )$ $\mathcal{H}_5$ $\big ( (0.9, 0.7), (0.1, 0.2) \big )$ $\big ( (0.7, 0.7), (0.2, 0.1) \big )$ $\big ( (0.7, 0.6), (0.2, 0.2) \big )$ $\big ( (0.8, 0.8), (0.1, 0.1) \big )$
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 $\mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3$
 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 $\mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3$
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) $2.93$ $H(\mathcal{H}_1)=1.8191$, $H(\mathcal{H}_5)=1.8656$ $\mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3$ $\mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3$ $\mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3$
 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) $2.93$ $H(\mathcal{H}_1)=1.8191$, $H(\mathcal{H}_5)=1.8656$ $\mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3$ $\mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3$ $\mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3$
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]
 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]
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.
 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.
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 $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
 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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