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January  2020, 16(1): 445-467. doi: 10.3934/jimo.2018162

Group decision making approach based on possibility degree measure under linguistic interval-valued intuitionistic fuzzy set environment

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

Received  October 2017 Revised  July 2018 Published  October 2018

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

In the present article, we extended the idea of the linguistic intuitionistic fuzzy set to linguistic interval-valued intuitionistic fuzzy (LIVIF) set to represent the data by the interval-valued linguistic terms of membership and non-membership degrees. Some of the desirable properties of the proposed set are studied. Also, we propose a new ranking method named as possibility degree measures to compare the two or more different LIVIF numbers. During the aggregation process, some LIVIF weighted and ordered weighted aggregation operators are proposed to aggregate the collections of the LIVIF numbers. Finally, based on these proposed operators and possibility degree measure, a new group decision making approach is presented to rank the different alternatives. A real-life case has been studied to manifest the practicability and feasibility of the proposed group decision making method.

Citation: Harish Garg, Kamal Kumar. Group decision making approach based on possibility degree measure under linguistic interval-valued intuitionistic fuzzy set environment. Journal of Industrial & Management Optimization, 2020, 16 (1) : 445-467. doi: 10.3934/jimo.2018162
References:
 [1] R. Arora and H. Garg, Prioritized averaging/geometric aggregation operators under the intuitionistic fuzzy soft set environment, Scientia Iranica, 25 (2018), 466-482.  doi: 10.24200/sci.2017.4410.  Google Scholar [2] R. Arora and H. Garg, A robust correlation coefficient measure of dual hesistant fuzzy soft sets and their application in decision making, Engineering Applications of Artificial Intelligence, 72 (2018), 80-92.  doi: 10.1016/j.engappai.2018.03.019.  Google Scholar [3] 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 [4] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.  doi: 10.1016/S0165-0114(86)80034-3.  Google Scholar [5] Z. Chen, P. Liu and Z. Pei, An approach to multiple attribute group decision making based on linguistic intuitionistic fuzzy numbers, Journal of Computational Intelligence Systems, 8 (2015), 747-760.   Google Scholar [6] F. Dammak, L. Baccour and A. M. Alimi, An exhaustive study of possibility measures of interval-valued intuitionistic fuzzy sets and application to multicriteria decision making, Advances in Fuzzy Systems, 2016 (2016), Article ID 9185706, 10 pages. doi: 10.1155/2016/9185706.  Google Scholar [7] F. Gao, Possibility degree and comprehensive priority of interval numbers, Systems Engineering Theory and Practice, 33 (2013), 2033-2040.   Google Scholar [8] H. Garg, Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision making, Computers and Industrial Engineering, 101 (2016), 53-69.  doi: 10.1016/j.cie.2016.08.017.  Google Scholar [9] 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 [10] 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 [11] H. Garg, Hesitant Pythagorean fuzzy sets and their aggregation operators in multiple attribute decision making, International Journal for Uncertainty Quantification, 8 (2018), 267-289.  doi: 10.1615/Int.J.UncertaintyQuantification.2018020979.  Google Scholar [12] H. Garg, Linguistic Pythagorean fuzzy sets and its applications in multiattribute decision-making process, International Journal of Intelligent Systems, 33 (2018), 1234-1263.  doi: 10.1002/int.21979.  Google Scholar [13] H. Garg, Novel correlation coefficients under the intuitionistic multiplicative environment and their applications to decision - making process, Journal of Industrial and Management Optimization, 14 (2018), 1501-1519.  doi: 10.3934/jimo.2018018.  Google Scholar [14] H. Garg, Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision -making process, Journal of Industrial & Management Optimization, 14 (2018), 283-308.  doi: 10.3934/jimo.2017047.  Google Scholar [15] H. Garg and R. Arora, Dual hesitant fuzzy soft aggregation operators and their application in decision making, Cognitive Computation, (2018), 1-21.  doi: 10.1007/s12559-018-9569-6.  Google Scholar [16] H. Garg and R. Arora, Generalized and group-based generalized intuitionistic fuzzy soft sets with applications in decision-making, Applied Intelligence, 48 (2018), 343-356.  doi: 10.1007/s10489-017-0981-5.  Google Scholar [17] H. Garg and R. Arora, Novel scaled prioritized intuitionistic fuzzy soft interaction averaging aggregation operators and their application to multi criteria decision making, Engineering Applications of Artificial Intelligence, 71 (2018), 100-112.  doi: 10.1016/j.engappai.2018.02.005.  Google Scholar [18] H. Garg and K. Kumar, An advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making, Soft Computing, 22(2018), 4959-4970. doi: 10.1007/s00500-018-3202-1.  Google Scholar [19] H. Garg and K. Kumar, Improved possibility degree method for ranking intuitionistic fuzzy numbers and their application in multiattribute decision-making, Granular Computing, (2018), 1-11.  doi: 10.1007/s41066-018-0092-7.  Google Scholar [20] H. Garg and K. Kumar, Some aggregation operators for linguistic intuitionistic fuzzy set and its application to group decision-making process using the set pair analysis, Arabian Journal for Science and Engineering, 43 (2018), 3213-3227.  doi: 10.1007/s13369-017-2986-0.  Google Scholar [21] Garg and Nancy, Linguistic single-valued neutrosophic prioritized aggregation operators and their applications to multiple-attribute group decision-making, Journal of Ambient Intelligence and Humanized Computing, (2018), 1-23.  doi: 10.1007/s12652-018-0723-5.  Google Scholar [22] H. Garg and D. Rani, Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision-making process, Arabian Journal for Science and Engineering, (2018), 1 - 20, doi: 10.1007/s13369-018-3413-x doi: 10.1007/s13369-018-3413-x.  Google Scholar [23] F. Herrera and L. Martinez, A 2- tuple fuzzy linguistic representation model for computing with words, IEEE Transactions on Fuzzy Systems, 8 (2000), 746-752.   Google Scholar [24] F. Herrera and L. Martínez, A model based on linguistic 2-tuples for dealing with multigranular hierarchical linguistic contexts in multi-expert decision-making, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 31 (2001), 227-234.  doi: 10.1109/3477.915345.  Google Scholar [25] G. Kaur and H. Garg, Cubic intuitionistic fuzzy aggregation operators, International Journal for Uncertainty Quantification, 8 (2018), 405-427.  doi: 10.1615/Int.J.UncertaintyQuantification.2018020471.  Google Scholar [26] G. Kaur and H. Garg, Multi - attribute decision - making based on bonferroni mean operators under cubic intuitionistic fuzzy set environment, Entropy, 20 (2018), Paper No. 65, 26 pp. doi: 10.3390/e20010065.  Google Scholar [27] K. Kumar and H. Garg, Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making, Applied Intelligence, 48 (2018), 2112-2119.  doi: 10.1007/s10489-017-1067-0.  Google Scholar [28] 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, 37 (2018), 1319-1329.  doi: 10.1007/s40314-016-0402-0.  Google Scholar [29] P. Liu and X. Qin, Power average operators of linguistic intuitionistic fuzzy numbers and their application to multiple - attribute decision making, Journal of Intelligent & Fuzzy Systems, 32 (2017), 1029-1043.   Google Scholar [30] P. Liu and P. Wang, Some improved linguistic intuitionistic fuzzy aggregation operators and their applications to multiple-attribute decision making, International Journal of Information Technology & Decision Making, 16 (2017), 817-850.  doi: 10.1142/S0219622017500110.  Google Scholar [31] H. G. Peng, J. Q. Wang and P. F. Cheng, A linguistic intuitionistic multi-criteria decision-making method based on the frank heronian mean operator and its application in evaluating coal mine safety, International Journal of Machine Learning and Cybernetics, 9 (2018), 1053-1068.  doi: 10.1007/s13042-016-0630-z.  Google Scholar [32] D. Rani and H. Garg, Distance measures between the complex intuitionistic fuzzy sets and its applications to the decision - making process, International Journal for Uncertainty Quantification, 7 (2017), 423-439.  doi: 10.1615/Int.J.UncertaintyQuantification.2017020356.  Google Scholar [33] D. Rani and H. Garg, Complex intuitionistic fuzzy power aggregation operators and their applications in multi-criteria decision-making, Expert Systems, (2018), e12325, doi: 10.1111/exsy.12325 doi: 10.1111/exsy.12325.  Google Scholar [34] 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.  Google Scholar [35] S. Wan and J. Dong, A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making, Journal of Computer and System Sciences, 80 (2014), 237-256.  doi: 10.1016/j.jcss.2013.07.007.  Google Scholar [36] X. Wang and E. Triantaphyllou, Ranking irregularities when evaluating alternatives by using some electre methods, Omega - International Journal of Management Science, 36 (2008), 45-63.  doi: 10.1016/j.omega.2005.12.003.  Google Scholar [37] S. Xian, N. Jing, W. Xue and J. Chai, A new intuitionistic fuzzy linguistic hybrid aggregation operator and its application for linguistic group decision making, International Journal of Intelligent Systems, 32 (2017), 1332-1352.  doi: 10.1002/int.21902.  Google Scholar [38] Y. J. Xu and H. M. Wang, IFWA and IFWGM methods for MADM under intuitionistic fuzzy environment, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 23 (2015), 263-284.  doi: 10.1142/s0218488515500117.  Google Scholar [39] Z. Xu, Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment, Information Science, 168 (2004), 171-184.  doi: 10.1016/j.ins.2004.02.003.  Google Scholar [40] Z. Xu, An approach based on similarity measure to multiple attribute decision making with trapezoid fuzzy linguistic variables, International Conference on Fuzzy Systems and Knowledge Discovery, 2005,110 - 117. Google Scholar [41] 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.  Google Scholar [42] Z. S. Xu, Algorithm for priority of fuzzy complementary judgment matrix, Journal of Systems Engineering(4), 16 (2001), 311-314.   Google Scholar [43] Z. S. Xu, A method based on linguistic aggregation operators for group decision making under linguistic preference relations, Information Sciences, 166 (2004), 19-30.  doi: 10.1016/j.ins.2003.10.006.  Google Scholar [44] Z. S. Xu, Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making, Control and Decision, 22 (2007), 215-219.   Google Scholar [45] Z. S. Xu, Uncertain Multiple Attribute Decision Making: Methods and Applications, Springer, New York, 2015. doi: 10.1007/978-3-662-45640-8.  Google Scholar [46] Z. S. Xu and Q. L. Da, The uncertain owa operator, International Journal of Intelligent Systems, 17 (2002), 569-575.  doi: 10.1002/int.10038.  Google Scholar [47] Z. S. Xu and Q. L. Da, Possibility degree method for ranking interval numbers and its application, Journal of Systems Engineering, 18 (2003), 67-70.   Google Scholar [48] Z. S. Xu and R. R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General Systems, 35 (2006), 417-433.  doi: 10.1080/03081070600574353.  Google Scholar [49] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar [50] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning: Part-1, Information Science, 8 (1975), 199-249.  doi: 10.1016/0020-0255(75)90036-5.  Google Scholar [51] H. Zhang, Linguistic intuitionistic fuzzy sets and application in MAGDM, Journal of Applied Mathematics, 2014 (2014), Article ID 432092, 11 pages. doi: 10.1155/2014/432092.  Google Scholar [52] X. Zhang, G. Yue and Z. Teng, Possibility degree of interval - valued intuitionistic fuzzy numbers and its application, in: Proceedings of the International Symposium on Information Processing (ISIP09), 2009, 33 - 36. Google Scholar

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References:
 [1] R. Arora and H. Garg, Prioritized averaging/geometric aggregation operators under the intuitionistic fuzzy soft set environment, Scientia Iranica, 25 (2018), 466-482.  doi: 10.24200/sci.2017.4410.  Google Scholar [2] R. Arora and H. Garg, A robust correlation coefficient measure of dual hesistant fuzzy soft sets and their application in decision making, Engineering Applications of Artificial Intelligence, 72 (2018), 80-92.  doi: 10.1016/j.engappai.2018.03.019.  Google Scholar [3] 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 [4] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.  doi: 10.1016/S0165-0114(86)80034-3.  Google Scholar [5] Z. Chen, P. Liu and Z. Pei, An approach to multiple attribute group decision making based on linguistic intuitionistic fuzzy numbers, Journal of Computational Intelligence Systems, 8 (2015), 747-760.   Google Scholar [6] F. Dammak, L. Baccour and A. M. Alimi, An exhaustive study of possibility measures of interval-valued intuitionistic fuzzy sets and application to multicriteria decision making, Advances in Fuzzy Systems, 2016 (2016), Article ID 9185706, 10 pages. doi: 10.1155/2016/9185706.  Google Scholar [7] F. Gao, Possibility degree and comprehensive priority of interval numbers, Systems Engineering Theory and Practice, 33 (2013), 2033-2040.   Google Scholar [8] H. Garg, Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision making, Computers and Industrial Engineering, 101 (2016), 53-69.  doi: 10.1016/j.cie.2016.08.017.  Google Scholar [9] 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 [10] 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 [11] H. Garg, Hesitant Pythagorean fuzzy sets and their aggregation operators in multiple attribute decision making, International Journal for Uncertainty Quantification, 8 (2018), 267-289.  doi: 10.1615/Int.J.UncertaintyQuantification.2018020979.  Google Scholar [12] H. Garg, Linguistic Pythagorean fuzzy sets and its applications in multiattribute decision-making process, International Journal of Intelligent Systems, 33 (2018), 1234-1263.  doi: 10.1002/int.21979.  Google Scholar [13] H. Garg, Novel correlation coefficients under the intuitionistic multiplicative environment and their applications to decision - making process, Journal of Industrial and Management Optimization, 14 (2018), 1501-1519.  doi: 10.3934/jimo.2018018.  Google Scholar [14] H. Garg, Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision -making process, Journal of Industrial & Management Optimization, 14 (2018), 283-308.  doi: 10.3934/jimo.2017047.  Google Scholar [15] H. Garg and R. Arora, Dual hesitant fuzzy soft aggregation operators and their application in decision making, Cognitive Computation, (2018), 1-21.  doi: 10.1007/s12559-018-9569-6.  Google Scholar [16] H. Garg and R. Arora, Generalized and group-based generalized intuitionistic fuzzy soft sets with applications in decision-making, Applied Intelligence, 48 (2018), 343-356.  doi: 10.1007/s10489-017-0981-5.  Google Scholar [17] H. Garg and R. Arora, Novel scaled prioritized intuitionistic fuzzy soft interaction averaging aggregation operators and their application to multi criteria decision making, Engineering Applications of Artificial Intelligence, 71 (2018), 100-112.  doi: 10.1016/j.engappai.2018.02.005.  Google Scholar [18] H. Garg and K. Kumar, An advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making, Soft Computing, 22(2018), 4959-4970. doi: 10.1007/s00500-018-3202-1.  Google Scholar [19] H. Garg and K. Kumar, Improved possibility degree method for ranking intuitionistic fuzzy numbers and their application in multiattribute decision-making, Granular Computing, (2018), 1-11.  doi: 10.1007/s41066-018-0092-7.  Google Scholar [20] H. Garg and K. Kumar, Some aggregation operators for linguistic intuitionistic fuzzy set and its application to group decision-making process using the set pair analysis, Arabian Journal for Science and Engineering, 43 (2018), 3213-3227.  doi: 10.1007/s13369-017-2986-0.  Google Scholar [21] Garg and Nancy, Linguistic single-valued neutrosophic prioritized aggregation operators and their applications to multiple-attribute group decision-making, Journal of Ambient Intelligence and Humanized Computing, (2018), 1-23.  doi: 10.1007/s12652-018-0723-5.  Google Scholar [22] H. Garg and D. Rani, Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision-making process, Arabian Journal for Science and Engineering, (2018), 1 - 20, doi: 10.1007/s13369-018-3413-x doi: 10.1007/s13369-018-3413-x.  Google Scholar [23] F. Herrera and L. Martinez, A 2- tuple fuzzy linguistic representation model for computing with words, IEEE Transactions on Fuzzy Systems, 8 (2000), 746-752.   Google Scholar [24] F. Herrera and L. Martínez, A model based on linguistic 2-tuples for dealing with multigranular hierarchical linguistic contexts in multi-expert decision-making, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 31 (2001), 227-234.  doi: 10.1109/3477.915345.  Google Scholar [25] G. Kaur and H. Garg, Cubic intuitionistic fuzzy aggregation operators, International Journal for Uncertainty Quantification, 8 (2018), 405-427.  doi: 10.1615/Int.J.UncertaintyQuantification.2018020471.  Google Scholar [26] G. Kaur and H. Garg, Multi - attribute decision - making based on bonferroni mean operators under cubic intuitionistic fuzzy set environment, Entropy, 20 (2018), Paper No. 65, 26 pp. doi: 10.3390/e20010065.  Google Scholar [27] K. Kumar and H. Garg, Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making, Applied Intelligence, 48 (2018), 2112-2119.  doi: 10.1007/s10489-017-1067-0.  Google Scholar [28] 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, 37 (2018), 1319-1329.  doi: 10.1007/s40314-016-0402-0.  Google Scholar [29] P. Liu and X. Qin, Power average operators of linguistic intuitionistic fuzzy numbers and their application to multiple - attribute decision making, Journal of Intelligent & Fuzzy Systems, 32 (2017), 1029-1043.   Google Scholar [30] P. Liu and P. Wang, Some improved linguistic intuitionistic fuzzy aggregation operators and their applications to multiple-attribute decision making, International Journal of Information Technology & Decision Making, 16 (2017), 817-850.  doi: 10.1142/S0219622017500110.  Google Scholar [31] H. G. Peng, J. Q. Wang and P. F. Cheng, A linguistic intuitionistic multi-criteria decision-making method based on the frank heronian mean operator and its application in evaluating coal mine safety, International Journal of Machine Learning and Cybernetics, 9 (2018), 1053-1068.  doi: 10.1007/s13042-016-0630-z.  Google Scholar [32] D. Rani and H. Garg, Distance measures between the complex intuitionistic fuzzy sets and its applications to the decision - making process, International Journal for Uncertainty Quantification, 7 (2017), 423-439.  doi: 10.1615/Int.J.UncertaintyQuantification.2017020356.  Google Scholar [33] D. Rani and H. Garg, Complex intuitionistic fuzzy power aggregation operators and their applications in multi-criteria decision-making, Expert Systems, (2018), e12325, doi: 10.1111/exsy.12325 doi: 10.1111/exsy.12325.  Google Scholar [34] 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.  Google Scholar [35] S. Wan and J. Dong, A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making, Journal of Computer and System Sciences, 80 (2014), 237-256.  doi: 10.1016/j.jcss.2013.07.007.  Google Scholar [36] X. Wang and E. Triantaphyllou, Ranking irregularities when evaluating alternatives by using some electre methods, Omega - International Journal of Management Science, 36 (2008), 45-63.  doi: 10.1016/j.omega.2005.12.003.  Google Scholar [37] S. Xian, N. Jing, W. Xue and J. Chai, A new intuitionistic fuzzy linguistic hybrid aggregation operator and its application for linguistic group decision making, International Journal of Intelligent Systems, 32 (2017), 1332-1352.  doi: 10.1002/int.21902.  Google Scholar [38] Y. J. Xu and H. M. Wang, IFWA and IFWGM methods for MADM under intuitionistic fuzzy environment, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 23 (2015), 263-284.  doi: 10.1142/s0218488515500117.  Google Scholar [39] Z. Xu, Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment, Information Science, 168 (2004), 171-184.  doi: 10.1016/j.ins.2004.02.003.  Google Scholar [40] Z. Xu, An approach based on similarity measure to multiple attribute decision making with trapezoid fuzzy linguistic variables, International Conference on Fuzzy Systems and Knowledge Discovery, 2005,110 - 117. Google Scholar [41] 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.  Google Scholar [42] Z. S. Xu, Algorithm for priority of fuzzy complementary judgment matrix, Journal of Systems Engineering(4), 16 (2001), 311-314.   Google Scholar [43] Z. S. Xu, A method based on linguistic aggregation operators for group decision making under linguistic preference relations, Information Sciences, 166 (2004), 19-30.  doi: 10.1016/j.ins.2003.10.006.  Google Scholar [44] Z. S. Xu, Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making, Control and Decision, 22 (2007), 215-219.   Google Scholar [45] Z. S. Xu, Uncertain Multiple Attribute Decision Making: Methods and Applications, Springer, New York, 2015. doi: 10.1007/978-3-662-45640-8.  Google Scholar [46] Z. S. Xu and Q. L. Da, The uncertain owa operator, International Journal of Intelligent Systems, 17 (2002), 569-575.  doi: 10.1002/int.10038.  Google Scholar [47] Z. S. Xu and Q. L. Da, Possibility degree method for ranking interval numbers and its application, Journal of Systems Engineering, 18 (2003), 67-70.   Google Scholar [48] Z. S. Xu and R. R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General Systems, 35 (2006), 417-433.  doi: 10.1080/03081070600574353.  Google Scholar [49] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar [50] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning: Part-1, Information Science, 8 (1975), 199-249.  doi: 10.1016/0020-0255(75)90036-5.  Google Scholar [51] H. Zhang, Linguistic intuitionistic fuzzy sets and application in MAGDM, Journal of Applied Mathematics, 2014 (2014), Article ID 432092, 11 pages. doi: 10.1155/2014/432092.  Google Scholar [52] X. Zhang, G. Yue and Z. Teng, Possibility degree of interval - valued intuitionistic fuzzy numbers and its application, in: Proceedings of the International Symposium on Information Processing (ISIP09), 2009, 33 - 36. Google Scholar
LIVIFNs decision matrix $\widetilde{R}^{(1)}$ of decision maker $D^{(1)}$
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $A_{1}$ $( [s_3 , s_5 ], [ s_2 , s_3 ])$ $( [s_4 , s_5 ], [ s_1 , s_2 ])$ $( [s_4 , s_5 ], [ s_2 , s_3 ])$ $( [s_3 , s_4 ], [ s_1 , s_2])$ $A_{2}$ $( [s_3 , s_5 ], [ s_2 , s_3 ])$ $( [s_2 , s_4 ], [ s_1, s_2 ])$ $( [s_2 , s_4 ], [ s_3 , s_4 ])$ $( [s_1 , s_3 ], [ s_2 , s_3])$ $A_{3}$ $( [s_4 , s_6 ], [ s_1 , s_2 ])$ $( [s_5 , s_6 ], [ s_1 , s_1 ])$ $( [s_3 , s_4 ], [ s_2 , s_3 ])$ $( [s_4 , s_5 ], [ s_1 , s_3])$ $A_{4}$ $( [s_4 , s_5 ], [ s_2 , s_3 ])$ $( [s_1 , s_3 ], [ s_3 , s_4 ])$ $( [s_3 , s_5 ], [ s_1 , s_3 ])$ $( [s_6 , s_7 ], [ s_1 , s_1])$ Weights 0.40 0.25 0.20 0.15
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $A_{1}$ $( [s_3 , s_5 ], [ s_2 , s_3 ])$ $( [s_4 , s_5 ], [ s_1 , s_2 ])$ $( [s_4 , s_5 ], [ s_2 , s_3 ])$ $( [s_3 , s_4 ], [ s_1 , s_2])$ $A_{2}$ $( [s_3 , s_5 ], [ s_2 , s_3 ])$ $( [s_2 , s_4 ], [ s_1, s_2 ])$ $( [s_2 , s_4 ], [ s_3 , s_4 ])$ $( [s_1 , s_3 ], [ s_2 , s_3])$ $A_{3}$ $( [s_4 , s_6 ], [ s_1 , s_2 ])$ $( [s_5 , s_6 ], [ s_1 , s_1 ])$ $( [s_3 , s_4 ], [ s_2 , s_3 ])$ $( [s_4 , s_5 ], [ s_1 , s_3])$ $A_{4}$ $( [s_4 , s_5 ], [ s_2 , s_3 ])$ $( [s_1 , s_3 ], [ s_3 , s_4 ])$ $( [s_3 , s_5 ], [ s_1 , s_3 ])$ $( [s_6 , s_7 ], [ s_1 , s_1])$ Weights 0.40 0.25 0.20 0.15
LIVIFNs decision matrix $\widetilde{R}^{(2)}$ of decision maker $D^{(2)}$
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $A_{1}$ $( [s_2 , s_4 ], [ s_1 , s_3 ] )$ $( [s_4 , s_5 ], [ s_1 , s_2 ] )$ $([ s_ 4 , s_5 ], [ s_1 , s_3 ] )$ $( [s_3 , s_6 ], [ s_1 , s_2])$ $A_{2}$ $( [s_3 , s_5 ], [ s_1 , s_3] )$ $( [s_1 , s_2 ], [ s_1 , s_4] )$ $( [s_2 , s_3 ], [ s_3 , s_4 ] )$ $( [ s_3 , s_5 ], [ s_1 , s_3])$ $A_{3}$ $( [s_3 , s_4 ], [ s_2 , s_3 ] )$ $( [s_5 , s_6 ], [ s_1 , s_2 ] )$ $([ s_3 , s_5 ], [ s_2 , s_3 ] )$ $( [s_3 , s_4 ], [ s_2 , s_3])$ $A_{4}$ $( [s_4 , s_5 ], [ s_1 , s_2 ] )$ $( [s_1 , s_2 ], [ s_3 , s_5 ] )$ $( [s_3 , s_3 ], [ s_2 , s_3 ] )$ $( [ s_2 , s_3 ], [ s_1 , s_2])$ Weights 0.30 0.35 0.25 0.10
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $A_{1}$ $( [s_2 , s_4 ], [ s_1 , s_3 ] )$ $( [s_4 , s_5 ], [ s_1 , s_2 ] )$ $([ s_ 4 , s_5 ], [ s_1 , s_3 ] )$ $( [s_3 , s_6 ], [ s_1 , s_2])$ $A_{2}$ $( [s_3 , s_5 ], [ s_1 , s_3] )$ $( [s_1 , s_2 ], [ s_1 , s_4] )$ $( [s_2 , s_3 ], [ s_3 , s_4 ] )$ $( [ s_3 , s_5 ], [ s_1 , s_3])$ $A_{3}$ $( [s_3 , s_4 ], [ s_2 , s_3 ] )$ $( [s_5 , s_6 ], [ s_1 , s_2 ] )$ $([ s_3 , s_5 ], [ s_2 , s_3 ] )$ $( [s_3 , s_4 ], [ s_2 , s_3])$ $A_{4}$ $( [s_4 , s_5 ], [ s_1 , s_2 ] )$ $( [s_1 , s_2 ], [ s_3 , s_5 ] )$ $( [s_3 , s_3 ], [ s_2 , s_3 ] )$ $( [ s_2 , s_3 ], [ s_1 , s_2])$ Weights 0.30 0.35 0.25 0.10
LIVIFNs decision matrix $\widetilde{R}^{(3)}$ of decision maker $D^{(3)}$
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $A_{1}$ $( [s_2 , s_4 ], [ s_1 , s_2 ] )$ $( [s_2 , s_3 ], [ s_2 , s_4 ] )$ $([ s_ 3 , s_5 ], [ s_2 , s_3 ] )$ $( [s_5 , s_6 ], [ s_1 , s_2])$ $A_{2}$ $( [s_1 , s_4 ], [ s_2 , s_3] )$ $( [s_4 , s_5 ], [ s_1 , s_2] )$ $( [s_2 , s_4 ], [ s_1 , s_3 ] )$ $( [ s_3 , s_4 ], [ s_2 , s_4])$ $A_{3}$ $( [s_2 , s_3 ], [ s_1 , s_3 ] )$ $( [s_3 , s_5 ], [ s_2 , s_3 ] )$ $([ s_3 , s_5 ], [ s_1 , s_3 ] )$ $( [s_3 , s_5 ], [ s_2 , s_3])$ $A_{4}$ $( [s_3 , s_4 ], [ s_2 , s_3 ] )$ $( [s_1 , s_2 ], [ s_3 , s_4 ] )$ $( [s_3 , s_5 ], [ s_1 , s_{2}])$ $([ s_{5} , s_6 ], [ s_1 , s_2])$ Weights 0.35 0.40 0.15 0.10
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $A_{1}$ $( [s_2 , s_4 ], [ s_1 , s_2 ] )$ $( [s_2 , s_3 ], [ s_2 , s_4 ] )$ $([ s_ 3 , s_5 ], [ s_2 , s_3 ] )$ $( [s_5 , s_6 ], [ s_1 , s_2])$ $A_{2}$ $( [s_1 , s_4 ], [ s_2 , s_3] )$ $( [s_4 , s_5 ], [ s_1 , s_2] )$ $( [s_2 , s_4 ], [ s_1 , s_3 ] )$ $( [ s_3 , s_4 ], [ s_2 , s_4])$ $A_{3}$ $( [s_2 , s_3 ], [ s_1 , s_3 ] )$ $( [s_3 , s_5 ], [ s_2 , s_3 ] )$ $([ s_3 , s_5 ], [ s_1 , s_3 ] )$ $( [s_3 , s_5 ], [ s_2 , s_3])$ $A_{4}$ $( [s_3 , s_4 ], [ s_2 , s_3 ] )$ $( [s_1 , s_2 ], [ s_3 , s_4 ] )$ $( [s_3 , s_5 ], [ s_1 , s_{2}])$ $([ s_{5} , s_6 ], [ s_1 , s_2])$ Weights 0.35 0.40 0.15 0.10
Collective individual performance of each decision maker
 $D^{(1)}$ $D^{(2)}$ $D^{(3)}$ $A_{1}$ $( [ s_{3.4146 }, s_{ 4.8354 }], [s_{ 1.6184 }, s_{ 2.6217 }])$ $([ s_{ 3.1569 }, s_{ 4.7623 }], [s_{ 1.0000 }, s_{ 2.5725 }])$ $([ s_{2.3294 }, s_{ 3.8391 }], [s_{ 1.5690 }, s_{ 3.0359}])$ $A_{2}$ $( [s_{ 2.1199 }, s_{ 4.1887 }], [s_{ 1.9875 }, s_{ 2.9952 }])$ $([ s_{ 1.8455 }, s_{ 3.1932 }], [s_{ 1.5647 }, s_{ 3.6266 }])$ $([ s_{ 2.1562 }, s_{ 4.3734 }], [s_{ 1.4691 }, s_{ 2.7404}])$ $A_{3}$ $( [s_{ 3.9930 }, s_{ 5.3834 }], [s_{ 1.2125 }, s_{ 2.1497 }])$ $([ s_{ 3.5873 }, s_{ 4.8744 }], [s_{ 1.6674 }, s_{ 2.6705 }])$ $([ s_{ 2.6031 }, s_{ 4.1814 }], [s_{ 1.5193 }, s_{ 3.0000}])$ $A_{4}$ $( [s_{ 2.8377 }, s_{ 4.6284 }], [s_{ 1.9496 }, s_{ 3.0265 }])$ $([ s_{ 2.1380 }, s_{ 3.0342 }], [s_{ 2.0129 }, s_{ 3.5022 }])$ $([ s_{ 2.0345 }, s_{ 3.2643 }], [s_{ 2.2028 }, s_{ 3.2137}])$
 $D^{(1)}$ $D^{(2)}$ $D^{(3)}$ $A_{1}$ $( [ s_{3.4146 }, s_{ 4.8354 }], [s_{ 1.6184 }, s_{ 2.6217 }])$ $([ s_{ 3.1569 }, s_{ 4.7623 }], [s_{ 1.0000 }, s_{ 2.5725 }])$ $([ s_{2.3294 }, s_{ 3.8391 }], [s_{ 1.5690 }, s_{ 3.0359}])$ $A_{2}$ $( [s_{ 2.1199 }, s_{ 4.1887 }], [s_{ 1.9875 }, s_{ 2.9952 }])$ $([ s_{ 1.8455 }, s_{ 3.1932 }], [s_{ 1.5647 }, s_{ 3.6266 }])$ $([ s_{ 2.1562 }, s_{ 4.3734 }], [s_{ 1.4691 }, s_{ 2.7404}])$ $A_{3}$ $( [s_{ 3.9930 }, s_{ 5.3834 }], [s_{ 1.2125 }, s_{ 2.1497 }])$ $([ s_{ 3.5873 }, s_{ 4.8744 }], [s_{ 1.6674 }, s_{ 2.6705 }])$ $([ s_{ 2.6031 }, s_{ 4.1814 }], [s_{ 1.5193 }, s_{ 3.0000}])$ $A_{4}$ $( [s_{ 2.8377 }, s_{ 4.6284 }], [s_{ 1.9496 }, s_{ 3.0265 }])$ $([ s_{ 2.1380 }, s_{ 3.0342 }], [s_{ 2.0129 }, s_{ 3.5022 }])$ $([ s_{ 2.0345 }, s_{ 3.2643 }], [s_{ 2.2028 }, s_{ 3.2137}])$
Worse alternative $A_{4}^{\prime}$ for each decision maker
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $D^{(1)}$ $( [s_{2}, s_{ 3 }], [s_{ 3 }, s_{ 4}])$ $( [s_{ 0 }, s_{ 3 }], [s_{ 4 }, s_{ 5 }])$ $( [s_{ 2 }, s_{ 4 }], [s_{ 2 }, s_{ 4 }])$ $( [s_{ 3 }, s_{ 4 }], [s_{ 2 }, s_{3}])$ $D^{(2)}$ $( [s_{ 2 }, s_{ 3 }], [s_{ 3 }, s_{ 4 }])$ $( [s_{ 0 }, s_{ 1 }], [s_{ 4 }, s_{ 6 }])$ $( [s_{ 1 }, s_{ 2 }], [s_{ 3 }, s_{ 4 }])$ $( [s_{ 1 }, s_{ 2 }], [s_{ 3 }, s_{ 4}])$ $D^{(3)}$ $( [s_{ 1 }, s_{ 2 }], [s_{ 3 }, s_{ 5 }])$ $( [s_{ 1 }, s_{ 1 }], [s_{ 4 }, s_{ 5 }])$ $( [s_{ 2 }, s_{ 4 }], [s_{ 3 }, s_{ 4 }])$ $( [s_{ 2 }, s_{ 3 }], [s_{ 2 }, s_{3}])$
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $D^{(1)}$ $( [s_{2}, s_{ 3 }], [s_{ 3 }, s_{ 4}])$ $( [s_{ 0 }, s_{ 3 }], [s_{ 4 }, s_{ 5 }])$ $( [s_{ 2 }, s_{ 4 }], [s_{ 2 }, s_{ 4 }])$ $( [s_{ 3 }, s_{ 4 }], [s_{ 2 }, s_{3}])$ $D^{(2)}$ $( [s_{ 2 }, s_{ 3 }], [s_{ 3 }, s_{ 4 }])$ $( [s_{ 0 }, s_{ 1 }], [s_{ 4 }, s_{ 6 }])$ $( [s_{ 1 }, s_{ 2 }], [s_{ 3 }, s_{ 4 }])$ $( [s_{ 1 }, s_{ 2 }], [s_{ 3 }, s_{ 4}])$ $D^{(3)}$ $( [s_{ 1 }, s_{ 2 }], [s_{ 3 }, s_{ 5 }])$ $( [s_{ 1 }, s_{ 1 }], [s_{ 4 }, s_{ 5 }])$ $( [s_{ 2 }, s_{ 4 }], [s_{ 3 }, s_{ 4 }])$ $( [s_{ 2 }, s_{ 3 }], [s_{ 2 }, s_{3}])$
The characteristic comparisons of different methods
 Whether flexibly to express a wider range of information Whether describe information using linguistic features Whether describe information by interval-valued numbers Whether have the characteristic of generalization Xu and Yager [48] no no no no Xu [44] yes no yes no Zhang [51] no yes no yes The proposed method yes yes yes yes
 Whether flexibly to express a wider range of information Whether describe information using linguistic features Whether describe information by interval-valued numbers Whether have the characteristic of generalization Xu and Yager [48] no no no no Xu [44] yes no yes no Zhang [51] no yes no yes The proposed method yes yes yes yes
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