# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018018

## Novel correlation coefficients under the intuitionistic multiplicative environment and their applications to decision-making process

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

Received  February 2017 Revised  July 2017 Published  January 2018

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 work is to present novel correlation coefficients under the intuitionistic multiplicative preference relation (IMPR), for measuring the relationship between the two intuitionistic multiplicative sets, instead of intuitionistic fuzzy preference relation (IFPR). As IFPR deals under the conditions that the attribute values grades are symmetrical and uniformly distributed. But in our day-to-day life, these conditions do not fulfill the decision maker requirement and hence IFPR theory is not applicable in that domain. Thus, for handling this, an intuitionistic multiplicative set theory has been utilized where grades are distributed asymmetrical around 1. Further, under this environment, a decision making method based on the proposed novel correlation coefficients has been presented. Pairs of membership and non-membership degree are considered to be a vector representation during formulation. Three numerical examples have been taken to demonstrate the efficiency of the proposed approach.

Citation: Harish Garg. Novel correlation coefficients under the intuitionistic multiplicative environment and their applications to decision-making process. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018018
##### 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] A. Bhaumik, S. K. Roy and D. F. Li, Analysis of triangular intuitionistic fuzzy matrix games using robust ranking, Journal of Intelligent & Fuzzy Systems, 33 (2017), 327-336. doi: 10.3233/JIFS-161631. [4] S. M. Chen, Similarity measures between vague sets and between elements, IEEE Transactions of System Man and Cybernetics, 27 (1997), 153-158. [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, 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. [8] H. Garg, A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem, Journal of Intelligent & Fuzzy Systems, 31 (2016), 529-540. doi: 10.3233/IFS-162165. [9] H. Garg, A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making processes, International Journal of Intelligent Systems, 31 (2016), 1234-1252. doi: 10.1002/int.21827. [10] H. Garg, Distance and similarity measure for intuitionistic multiplicative preference relation and its application, International Journal for Uncertainty Quantification, 7 (2017), 117-133. doi: 10.1615/Int.J.UncertaintyQuantification.2017018981. [11] H. Garg, Generalized interaction aggregation operators in intuitionistic fuzzy multiplicative preference environment and their application to multicriteria decision-making, Applied Intelligence, (2017), 1-17. doi: 10.1007/s10489-017-1066-1. [12] H. Garg, A new improved score function of an interval-valued Pythagorean fuzzy set based TOPSIS method, International Journal for Uncertainty Quantification, 7 (2017), 463-474. doi: 10.1615/Int.J.UncertaintyQuantification.2017020197. [13] H. Garg, A novel improved accuracy function for interval valued Pythagorean fuzzy sets and its applications in decision making process, International Journal of Intelligent Systems, 31 (2017), 1247-1260. doi: 10.1002/int.21898. [14] 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. [15] H. Garg, A robust ranking method for intuitionistic multiplicative sets under crisp, interval environments and its applications, IEEE Transactions on Emerging Topics in Computational Intelligence, 1 (2017), 366-374. doi: 10.1109/TETCI.2017.2739129. [16] H. Garg, Some picture fuzzy aggregation operators and their applications to multicriteria decision-making, Arabian Journal for Science and Engineering, 42 (2017), 5275-5290. doi: 10.1007/s13369-017-2625-9. [17] T. Gerstenkorn and J. Manko, Correlation of intuitionistic fuzzy sets, Fuzzy sets and Systems, 44 (1991), 39-43. doi: 10.1016/0165-0114(91)90031-K. [18] D. H. Hong and C. Kim, A note on similarity measures between vague sets and between elements, Information Sciences, 115 (1999), 83-96. doi: 10.1016/S0020-0255(98)10083-X. [19] Y. Jiang and Z. Xu, Aggregating information and ranking alternatives in decision making with intuitionistic multiplicative preference relations, Applied Soft Computing, 22 (2014), 162-177. doi: 10.1016/j.asoc.2014.04.043. [20] Y. Jiang, Z. Xu and M. Gao, Methods for ranking intuitionistic multiplicative numbers by distance measures in decision making, Computer and Industrial Engineering, 88 (2015), 100-109. doi: 10.1016/j.cie.2015.06.015. [21] 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. [22] 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, (2017), 1-8. doi: 10.1007/s10489-017-1067-0. [23] Z. Liang and P. Shi, Similarity measures on intuitionistic fuzzy sets, Pattern Recognition Letters, 24 (2003), 2687-2693. doi: 10.1016/S0167-8655(03)00111-9. [24] T. L. Saaty, Axiomatic foundation of the analytic hierarchy process, Management Science, 32 (1986), 841-855. doi: 10.1287/mnsc.32.7.841. [25] E. Szmidt and J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems, 114 (2000), 505-518. doi: 10.1016/S0165-0114(98)00244-9. [26] G. W. Wei, H. J. Wang and R. Lin, Application of correlation coefficient to interval-valued intuitionistic fuzzy multiple attribute decision-making with incomplete weight information, Knowledge and Information Systems, 26 (2011), 337-349. doi: 10.1007/s10115-009-0276-1. [27] M. Xia, Z. Xu and H. Liao, Preference relations based on intuitionistic multiplicative information, IEEE Transactions on Fuzzy Systems, 21 (2013), 113-132. [28] M. M. Xia and Z. S. Xu, Group decision making based on intuitionistic multiplicative aggregation operators, Applied Mathematical Modelling, 37 (2013), 5120-5133. doi: 10.1016/j.apm.2012.10.029. [29] Z. S. Xu, On correlation measures of intuitionistic fuzzy sets, Lecture Notes in Computer Science, 4224 (2006), 16-24. doi: 10.1007/11875581_2. [30] Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions of Fuzzy Systems, 15 (2007), 1179-1187. [31] 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. [32] Z. S. Xu, J. Chen and J. J. Wu, Cluster algorithm for intuitionistic fuzzy sets, Information Sciences, 178 (2008), 3775-3790. doi: 10.1016/j.ins.2008.06.008. [33] J. Ye, Multicriteria fuzzy decision-making method based on a novel accuracy function under interval -valued intuitionistic fuzzy environment, Expert Systems with Applications, 36 (2009), 6809-6902. doi: 10.1016/j.eswa.2008.08.042. [34] J. Ye, Cosine similarity measures for intuitionistic fuzzy sets and their applications, Mathematical and Computer Modelling, 53 (2011), 91-97. doi: 10.1016/j.mcm.2010.07.022. [35] D. Yu, J. M. Merigo and L. Zhou, Interval-valued multiplicative intuitionistic fuzzy preference relations, International Journal of Fuzzy Systems, 15 (2013), 412-422. [36] S. Yu and Z. S. Xu, Aggregation and decision making using intuitionistic multiplicative triangular fuzzy information, Journal of Systems Science and Systems Engineering, 23 (2014), 20-38. doi: 10.1007/s11518-013-5237-2. [37] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X.

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##### 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] A. Bhaumik, S. K. Roy and D. F. Li, Analysis of triangular intuitionistic fuzzy matrix games using robust ranking, Journal of Intelligent & Fuzzy Systems, 33 (2017), 327-336. doi: 10.3233/JIFS-161631. [4] S. M. Chen, Similarity measures between vague sets and between elements, IEEE Transactions of System Man and Cybernetics, 27 (1997), 153-158. [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, 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. [8] H. Garg, A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem, Journal of Intelligent & Fuzzy Systems, 31 (2016), 529-540. doi: 10.3233/IFS-162165. [9] H. Garg, A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making processes, International Journal of Intelligent Systems, 31 (2016), 1234-1252. doi: 10.1002/int.21827. [10] H. Garg, Distance and similarity measure for intuitionistic multiplicative preference relation and its application, International Journal for Uncertainty Quantification, 7 (2017), 117-133. doi: 10.1615/Int.J.UncertaintyQuantification.2017018981. [11] H. Garg, Generalized interaction aggregation operators in intuitionistic fuzzy multiplicative preference environment and their application to multicriteria decision-making, Applied Intelligence, (2017), 1-17. doi: 10.1007/s10489-017-1066-1. [12] H. Garg, A new improved score function of an interval-valued Pythagorean fuzzy set based TOPSIS method, International Journal for Uncertainty Quantification, 7 (2017), 463-474. doi: 10.1615/Int.J.UncertaintyQuantification.2017020197. [13] H. Garg, A novel improved accuracy function for interval valued Pythagorean fuzzy sets and its applications in decision making process, International Journal of Intelligent Systems, 31 (2017), 1247-1260. doi: 10.1002/int.21898. [14] 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. [15] H. Garg, A robust ranking method for intuitionistic multiplicative sets under crisp, interval environments and its applications, IEEE Transactions on Emerging Topics in Computational Intelligence, 1 (2017), 366-374. doi: 10.1109/TETCI.2017.2739129. [16] H. Garg, Some picture fuzzy aggregation operators and their applications to multicriteria decision-making, Arabian Journal for Science and Engineering, 42 (2017), 5275-5290. doi: 10.1007/s13369-017-2625-9. [17] T. Gerstenkorn and J. Manko, Correlation of intuitionistic fuzzy sets, Fuzzy sets and Systems, 44 (1991), 39-43. doi: 10.1016/0165-0114(91)90031-K. [18] D. H. Hong and C. Kim, A note on similarity measures between vague sets and between elements, Information Sciences, 115 (1999), 83-96. doi: 10.1016/S0020-0255(98)10083-X. [19] Y. Jiang and Z. Xu, Aggregating information and ranking alternatives in decision making with intuitionistic multiplicative preference relations, Applied Soft Computing, 22 (2014), 162-177. doi: 10.1016/j.asoc.2014.04.043. [20] Y. Jiang, Z. Xu and M. Gao, Methods for ranking intuitionistic multiplicative numbers by distance measures in decision making, Computer and Industrial Engineering, 88 (2015), 100-109. doi: 10.1016/j.cie.2015.06.015. [21] 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. [22] 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, (2017), 1-8. doi: 10.1007/s10489-017-1067-0. [23] Z. Liang and P. Shi, Similarity measures on intuitionistic fuzzy sets, Pattern Recognition Letters, 24 (2003), 2687-2693. doi: 10.1016/S0167-8655(03)00111-9. [24] T. L. Saaty, Axiomatic foundation of the analytic hierarchy process, Management Science, 32 (1986), 841-855. doi: 10.1287/mnsc.32.7.841. [25] E. Szmidt and J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems, 114 (2000), 505-518. doi: 10.1016/S0165-0114(98)00244-9. [26] G. W. Wei, H. J. Wang and R. Lin, Application of correlation coefficient to interval-valued intuitionistic fuzzy multiple attribute decision-making with incomplete weight information, Knowledge and Information Systems, 26 (2011), 337-349. doi: 10.1007/s10115-009-0276-1. [27] M. Xia, Z. Xu and H. Liao, Preference relations based on intuitionistic multiplicative information, IEEE Transactions on Fuzzy Systems, 21 (2013), 113-132. [28] M. M. Xia and Z. S. Xu, Group decision making based on intuitionistic multiplicative aggregation operators, Applied Mathematical Modelling, 37 (2013), 5120-5133. doi: 10.1016/j.apm.2012.10.029. [29] Z. S. Xu, On correlation measures of intuitionistic fuzzy sets, Lecture Notes in Computer Science, 4224 (2006), 16-24. doi: 10.1007/11875581_2. [30] Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions of Fuzzy Systems, 15 (2007), 1179-1187. [31] 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. [32] Z. S. Xu, J. Chen and J. J. Wu, Cluster algorithm for intuitionistic fuzzy sets, Information Sciences, 178 (2008), 3775-3790. doi: 10.1016/j.ins.2008.06.008. [33] J. Ye, Multicriteria fuzzy decision-making method based on a novel accuracy function under interval -valued intuitionistic fuzzy environment, Expert Systems with Applications, 36 (2009), 6809-6902. doi: 10.1016/j.eswa.2008.08.042. [34] J. Ye, Cosine similarity measures for intuitionistic fuzzy sets and their applications, Mathematical and Computer Modelling, 53 (2011), 91-97. doi: 10.1016/j.mcm.2010.07.022. [35] D. Yu, J. M. Merigo and L. Zhou, Interval-valued multiplicative intuitionistic fuzzy preference relations, International Journal of Fuzzy Systems, 15 (2013), 412-422. [36] S. Yu and Z. S. Xu, Aggregation and decision making using intuitionistic multiplicative triangular fuzzy information, Journal of Systems Science and Systems Engineering, 23 (2014), 20-38. doi: 10.1007/s11518-013-5237-2. [37] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X.
Comparison between the 0.1 - 0.9 scale and the 1 - 9 scale
 1 - 9 scale 0.1 - 0.9 scale Meaning 1/9 0.1 Extremely not preferred 1/7 0.2 Very strongly not preferred 1/5 0.3 Strongly not preferred 1/3 0.4 Moderately not preferred 1 0.5 Equally preferred 3 0.6 Moderately preferred 5 0.7 Strongly preferred 7 0.8 Very strongly preferred 9 0.9 Extremely preferred other values between 1/9 and 9 other values between 0 and 1 Intermediate value used to present compromise
 1 - 9 scale 0.1 - 0.9 scale Meaning 1/9 0.1 Extremely not preferred 1/7 0.2 Very strongly not preferred 1/5 0.3 Strongly not preferred 1/3 0.4 Moderately not preferred 1 0.5 Equally preferred 3 0.6 Moderately preferred 5 0.7 Strongly preferred 7 0.8 Very strongly preferred 9 0.9 Extremely preferred other values between 1/9 and 9 other values between 0 and 1 Intermediate value used to present compromise
Comparative analysis of Example 2
 Method Calculate value of Ranking $C_1$ $C_2$ $C_3$ Aggregation Operator[30] 0.6693 0.9110 0.3078 $C_2\succ C_1\succ C_3$ Hamming distance measure[25] 0.3509 0.0886 0.5367 $C_2\succ C_1\succ C_3$ Novel accuracy function[33] 0.3386 0.8220 0.1127 $C_2\succ C_1\succ C_3$ Correlation coefficient[34] 0.7465 0.9576 0.5612 $C_2\succ C_1\succ C_3$ Similarity measure ($S_C$)[4] 0.6401 0.9134 0.6262 $C_2\succ C_1\succ C_3$ Similarity measure ($S_H$)[18] 0.6401 0.9134 0.6093 $C_2\succ C_1\succ C_3$ Cosine Similarity measure[34] 0.5383 0.6913 0.4453 $C_2\succ C_1\succ C_3$ Improved score function [7] 0.6280 0.9466 0.5545 $C_2\succ C_1\succ C_3$
 Method Calculate value of Ranking $C_1$ $C_2$ $C_3$ Aggregation Operator[30] 0.6693 0.9110 0.3078 $C_2\succ C_1\succ C_3$ Hamming distance measure[25] 0.3509 0.0886 0.5367 $C_2\succ C_1\succ C_3$ Novel accuracy function[33] 0.3386 0.8220 0.1127 $C_2\succ C_1\succ C_3$ Correlation coefficient[34] 0.7465 0.9576 0.5612 $C_2\succ C_1\succ C_3$ Similarity measure ($S_C$)[4] 0.6401 0.9134 0.6262 $C_2\succ C_1\succ C_3$ Similarity measure ($S_H$)[18] 0.6401 0.9134 0.6093 $C_2\succ C_1\succ C_3$ Cosine Similarity measure[34] 0.5383 0.6913 0.4453 $C_2\succ C_1\succ C_3$ Improved score function [7] 0.6280 0.9466 0.5545 $C_2\succ C_1\succ C_3$
Comparative analysis of Example 3
 Method Calculate value of Ranking $Q_1$ $Q_2$ $Q_3$ $Q_4$ $Q_5$ Aggregation Operator[30] 0.6685 0.8705 0.5791 0.1606 0.4731 $Q_2\succ Q_1\succ Q_3 \succ Q_5 \succ Q_4$ Hamming distance measure[25] 0.4059 0.3806 0.3406 0.4686 0.4281 $Q_3\succ Q_2\succ Q_1 \succ Q_5 \succ Q_4$ Novel accuracy function[33] 0.3375 0.7410 0.1582 0.2066 0.5198 $Q_2\succ Q_5\succ Q_1 \succ Q_4 \succ Q_3$ Correlation coefficient [34] 0.7208 0.7047 0.7207 0.6103 0.6957 $Q_1\succ Q_3\succ Q_2 \succ Q_5 \succ Q_4$ Similarity measure ($S_C$)[4] 0.6856 0.6767 0.7249 0.5701 0.6490 $Q_3\succ Q_1\succ Q_2 \succ Q_5 \succ Q_4$ Similarity measure ($S_H$) [18] 0.6742 0.6713 0.7249 0.5701 0.6490 $Q_3\succ Q_1\succ Q_2 \succ Q_5 \succ Q_4$ Cosine Similarity measure [34] 0.4982 0.5091 0.5091 0.4328 0.5076 $Q_2 = Q_3\succ Q_5 \succ Q_1 \succ Q_4$ Improved score function [7] 0.6894 0.7343 0.5248 0.4676 0.5846 $Q_2\succ Q_1\succ Q_5 \succ Q_3 \succ Q_4$
 Method Calculate value of Ranking $Q_1$ $Q_2$ $Q_3$ $Q_4$ $Q_5$ Aggregation Operator[30] 0.6685 0.8705 0.5791 0.1606 0.4731 $Q_2\succ Q_1\succ Q_3 \succ Q_5 \succ Q_4$ Hamming distance measure[25] 0.4059 0.3806 0.3406 0.4686 0.4281 $Q_3\succ Q_2\succ Q_1 \succ Q_5 \succ Q_4$ Novel accuracy function[33] 0.3375 0.7410 0.1582 0.2066 0.5198 $Q_2\succ Q_5\succ Q_1 \succ Q_4 \succ Q_3$ Correlation coefficient [34] 0.7208 0.7047 0.7207 0.6103 0.6957 $Q_1\succ Q_3\succ Q_2 \succ Q_5 \succ Q_4$ Similarity measure ($S_C$)[4] 0.6856 0.6767 0.7249 0.5701 0.6490 $Q_3\succ Q_1\succ Q_2 \succ Q_5 \succ Q_4$ Similarity measure ($S_H$) [18] 0.6742 0.6713 0.7249 0.5701 0.6490 $Q_3\succ Q_1\succ Q_2 \succ Q_5 \succ Q_4$ Cosine Similarity measure [34] 0.4982 0.5091 0.5091 0.4328 0.5076 $Q_2 = Q_3\succ Q_5 \succ Q_1 \succ Q_4$ Improved score function [7] 0.6894 0.7343 0.5248 0.4676 0.5846 $Q_2\succ Q_1\succ Q_5 \succ Q_3 \succ Q_4$

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