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October  2018, 14(4): 1501-1519. doi: 10.3934/jimo.2018018

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

Harish Garg

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

Received  February 2017 Revised  July 2017 Published  October 2018 Early access  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.

Keywords: Correlation coefficient, intuitionistic multiplicative set, pattern recognition, medical diagnosis, multi criteria decision making.
Mathematics Subject Classification: Primary: 68T35, 90B50, 62A86, 03E72.
Citation: Harish Garg. Novel correlation coefficients under the intuitionistic multiplicative environment and their applications to decision-making process. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1501-1519. 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

S. M. Chen, Similarity measures between vague sets and between elements, IEEE Transactions of System Man and Cybernetics, 27 (1997), 153-158.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

T. L. Saaty, Axiomatic foundation of the analytic hierarchy process, Management Science, 32 (1986), 841-855.  doi: 10.1287/mnsc.32.7.841.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. Xia, Z. Xu and H. Liao, Preference relations based on intuitionistic multiplicative information, IEEE Transactions on Fuzzy Systems, 21 (2013), 113-132.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[37]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

S. M. Chen, Similarity measures between vague sets and between elements, IEEE Transactions of System Man and Cybernetics, 27 (1997), 153-158.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

T. L. Saaty, Axiomatic foundation of the analytic hierarchy process, Management Science, 32 (1986), 841-855.  doi: 10.1287/mnsc.32.7.841.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. Xia, Z. Xu and H. Liao, Preference relations based on intuitionistic multiplicative information, IEEE Transactions on Fuzzy Systems, 21 (2013), 113-132.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[37]

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

Table 1.  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
Table Options

Download as excel

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
Table 2.  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 $C_2\succ C_1\succ C_3$
Table Options

Download as excel

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 $C_2\succ C_1\succ C_3$
Table 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 $Q_2\succ Q_1\succ Q_5 \succ Q_3 \succ Q_4$
Table Options

Download as excel

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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