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, 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.

[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.

[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.

[4]

K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. doi: 10.1016/S0165-0114(86)80034-3.

[5]

Z. ChenP. 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.

[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.

[7]

F. Gao, Possibility degree and comprehensive priority of interval numbers, Systems Engineering Theory and Practice, 33 (2013), 2033-2040.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[31]

H. G. PengJ. 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.

[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.

[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.

[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.

[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.

[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.

[37]

S. XianN. JingW. 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.

[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.

[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.

[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.

[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.

[42]

Z. S. Xu, Algorithm for priority of fuzzy complementary judgment matrix, Journal of Systems Engineering(4), 16 (2001), 311-314.

[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.

[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.

[45]

Z. S. Xu, Uncertain Multiple Attribute Decision Making: Methods and Applications, Springer, New York, 2015. doi: 10.1007/978-3-662-45640-8.

[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.

[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.

[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.

[49]

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

[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.

[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.

[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.

show all references

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.

[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.

[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.

[4]

K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. doi: 10.1016/S0165-0114(86)80034-3.

[5]

Z. ChenP. 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.

[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.

[7]

F. Gao, Possibility degree and comprehensive priority of interval numbers, Systems Engineering Theory and Practice, 33 (2013), 2033-2040.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[31]

H. G. PengJ. 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.

[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.

[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.

[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.

[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.

[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.

[37]

S. XianN. JingW. 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.

[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.

[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.

[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.

[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.

[42]

Z. S. Xu, Algorithm for priority of fuzzy complementary judgment matrix, Journal of Systems Engineering(4), 16 (2001), 311-314.

[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.

[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.

[45]

Z. S. Xu, Uncertain Multiple Attribute Decision Making: Methods and Applications, Springer, New York, 2015. doi: 10.1007/978-3-662-45640-8.

[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.

[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.

[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.

[49]

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

[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.

[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.

[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.

Table 1.  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 ]) $$( [s_6 , s_7 ], [ s_1 , s_1])$
Weights0.400.250.200.15
$G_{1}$$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])$
Weights0.400.250.200.15
Table 2.  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 ] ) $$ ( [ s_2 , s_3 ], [ s_1 , s_2])$
Weights0.300.350.250.10
$G_{1}$$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])$
Weights0.300.350.250.10
Table 3.  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}]) $$ ([ s_{5} , s_6 ], [ s_1 , s_2])$
Weights0.350.400.150.10
$G_{1}$$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])$
Weights0.350.400.150.10
Table 4.  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}])$
Table 5.  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}]) $
Table 6.  The characteristic comparisons of different methods
Whether flexibly to express a wider range of informationWhether describe information using linguistic featuresWhether describe information by interval-valued numbersWhether have the characteristic of generalization
Xu and Yager [48]nononono
Xu [44]yesnoyesno
Zhang [51]noyesnoyes
The proposed methodyesyesyesyes
Whether flexibly to express a wider range of informationWhether describe information using linguistic featuresWhether describe information by interval-valued numbersWhether have the characteristic of generalization
Xu and Yager [48]nononono
Xu [44]yesnoyesno
Zhang [51]noyesnoyes
The proposed methodyesyesyesyes
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