August  2019, 2(3): 183-201. doi: 10.3934/mfc.2019013

Triangular picture fuzzy linguistic induced ordered weighted aggregation operators and its application on decision making problems

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, 2320, Pakistan

* Corresponding author: Saleem Abdullah

Received  February 2019 Revised  May 2019 Published  September 2019

The primary goal of this paper is to solve the investment problem based on linguistic picture decision making method under the linguistic triangular picture linguistic fuzzy environment. First to define the triangular picture linguistic fuzzy numbers. Further, we define operations on triangular picture linguistic fuzzy numbers and their aggregation operator namely, triangular picture fuzzy linguistic induce OWA (TPFLIOWA) and triangular picture fuzzy linguistic induce OWG (TPFLIOWG) operators. Multi-criteria group decision making method is developed based on TPFLIOWA and TPFLIOWG operators and solve the uncertainty in the investment problem. We study the applicability of the proposed decision making method under triangular picture linguistic fuzzy environment and construct a descriptive example of investment problem. We conclude from the comparison and sensitive analysis that the proposed decision making method is more effective and reliable than other existing models.

Citation: Muhammad Qiyas, Saleem Abdullah, Shahzaib Ashraf, Saifullah Khan, Aziz Khan. Triangular picture fuzzy linguistic induced ordered weighted aggregation operators and its application on decision making problems. Mathematical Foundations of Computing, 2019, 2 (3) : 183-201. doi: 10.3934/mfc.2019013
References:
[1]

S. AshrafS. Abdullah and A. Qadir, Novel concept of cubic picture fuzzy sets, New Theory, 24 (2018), 69-72.   Google Scholar

[2]

S. AshrafT. MahmoodS. Abdullah and Q. khan, Different approaches to multi-criteria group decision making problems for picture fuzzy environment, Bulletin of the Brazilian Mathematical Society, New Series, 50 (2019), 373-397.  doi: 10.1007/s00574-018-0103-y.  Google Scholar

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

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R. E. Bellman and L. A. Zadeh, Decision-making in a fuzzy environment, Management Science, 17 (1970), B141–B164. doi: 10.1287/mnsc.17.4.B141.  Google Scholar

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C. Bo and X. Zhang, New operations of picture fuzzy relations and fuzzy comprehensive evaluation, Symmetry, 9 (2017), p268. doi: 10.3390/sym9110268.  Google Scholar

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J. J. Buckley, Fuzzy decision making with data: Applications to statistics, Fuzzy Sets and Systems, 16 (1985), 139-147.  doi: 10.1016/S0165-0114(85)80014-2.  Google Scholar

[7]

Y. Chen and B. Li, Dynamic multi-attribute decision making model based on triangular intuitionistic fuzzy numbers, Scientia Iranica, 18 (2011), 268-274.  doi: 10.1016/j.scient.2011.03.022.  Google Scholar

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B. C. Cuong, Picture fuzzy sets-first results. Part 1, Seminar Neuro-Fuzzy Systems with Applications. Preprint 04/2013, Institute of Mathematics, Hanoi, 2013. Google Scholar

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B. C. Cuong, Picture fuzzy sets-first results. Part 2, Seminar Neuro-Fuzzy Systems with Applications. Preprint 04/2013, Institute of Mathematics, Hanoi, 2013. Google Scholar

[10]

B. C. Cuong, Picture fuzzy sets, Journal of Computer Science and Cybernetics, 30 (2014), p409. Google Scholar

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B. C. Cuong and P. V. Hai, Some fuzzy logic operators for picture fuzzy sets, In Knowledge and Systems Engineering (KSE), (2015), 132–137. doi: 10.1109/KSE.2015.20.  Google Scholar

[12]

B. C. Cuong, V. Kreinovitch and R. T. Ngan, A classification of representable t-norm operators for picture fuzzy sets, In Knowledge and Systems Engineering (KSE), (2016), 19–24. doi: 10.1109/KSE.2016.7758023.  Google Scholar

[13]

D. Dubey and A. Mehra, Linear Programming with Triangular Intuitionistic Fuzzy Number, EUSFLAT-LFA, 2011. Google Scholar

[14]

D. J. Dubois, Fuzzy Sets and Systems: Theory and Applications, Academic Press, 1980. Google Scholar

[15]

M. Esmailzadeh and M. Esmailzadeh, New distance between triangular intuitionistic fuzzy numbers, Advances in Computational Mathematics and its Applications, 2 (2013), 310-314.   Google Scholar

[16]

H. Garg, Some picture fuzzy aggregation operators and their applications to multicri-teria decision-making, Arabian Journal for Science and Engineering, 42 (2017), 5275-5290.  doi: 10.1007/s13369-017-2625-9.  Google Scholar

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F. Herrera and E. Herrera-Viedma, Linguistic decision analysis: Steps for solving decision problems under linguistic information, Fuzzy Sets and Systems, 115 (2000), 67-82.  doi: 10.1016/S0165-0114(99)00024-X.  Google Scholar

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D. F. LiJ. K. Nan and M. J. Zhang, A ranking method of triangular intuitionistic fuzzy numbers and application to decision making, International Journal of Computational Intelligence Systems, 3 (2010), 522-530.   Google Scholar

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D. F. Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems, Computers and Mathematics with Applications, 60 (2010), 1557-1570.   Google Scholar

[20]

C. LiangS. Zhao and J. Zhang, Aggregation operators on triangular intuitionistic fuzzy numbers and its application to multi-criteria decision making problems, Foundations of Computing and Decision Sciences, 39 (2014), 189-208.  doi: 10.2478/fcds-2014-0011.  Google Scholar

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L. Marti and F. Herrera, An overview on the 2-tuple linguistic model for computing with words in decision making: Extensions, applications and challenges. Information Sciences, 207 (2012), 1–18. doi: 10.1016/j.ins.2012.04.025.  Google Scholar

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X. Peng and J. Dai, Algorithm for picture fuzzy multiple attribute decision-making based on new distance measure, International Journal for Uncertainty Quantification, 7 (2017), 177-187.  doi: 10.1615/Int.J.UncertaintyQuantification.2017020096.  Google Scholar

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P. H. Phong, D. T. Hieu, R. T. Ngan and P. T. Them, Some compositions of picture fuzzy relations, In Proceedings of the 7th National Conference on Fundamental and Applied Information Technology Research (FAIR?), Thai Nguye, (2014), 19–40. Google Scholar

[24]

P. H. Phong and B. C. Cuong, Multi-criteria group decision making with picture linguistic numbers. VNU Journal of Science, Computer Science and Communication Engineering, 32 (2017). Google Scholar

[25]

P. T. M. Phuong and P. H. Thong, Theoretical analysis of picture fuzzy clustering: Convergence and property, Journal of Computer Science and Cybernetics, 34 (2018), 17-32.  doi: 10.15625/1813-9663/34/1/12725.  Google Scholar

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J. Robinson and H. A. EC, A short primer on the Correlation coefficient of Vague sets, International Journal of Fuzzy System Applications (IJFSA), 1 (2011), 55-69.   Google Scholar

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J. Robinson and H. A. EC, A search for the correlation coefficient of triangular and trapezoidal intuitionistic fuzzy sets for multiple attribute group decision making, In Mathematical Modelling and Scientific Computation, 283 (2012), 333-342.   Google Scholar

[28]

R. RoostaeeM. IzadikhahF. H. Lotfi and M. Rostamy-Malkhalifeh, A multi-criteria intuitionistic fuzzy group decision making method for supplier selection with VIKOR method, International Journal of Fuzzy System Applications (IJFSA), 2 (2012), 1-17.  doi: 10.4018/978-1-4666-2625-6.ch056.  Google Scholar

[29]

M. H. ShuC. H. Cheng and J. R. Chang, Using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly, Microelectronics Reliability, 46 (2006), 2139-2148.  doi: 10.1016/j.microrel.2006.01.007.  Google Scholar

[30]

W. Shuping and D. Jiuying, Multi-attribute decision making based on triangular intuitionistic fuzzy number Choquet integral operator, Chinese Journal of Management Science, 22 (2014), 121-129.   Google Scholar

[31]

P. Singh, Correlation coefficients for picture fuzzy sets, Journal of Intelligent and Fuzzy Systems, 28 (2015), 591-604.   Google Scholar

[32]

L. H. Son, A novel distributed picture fuzzy clustering method on picture fuzzy sets, Expert Syst. Appl, 42 (2015), 51-66.  doi: 10.1016/j.eswa.2014.07.026.  Google Scholar

[33]

L. H. Son, Generalized picture distance measure and applications to picture fuzzy clustering, Applied Soft Computing, 46 (2016), 284-295.  doi: 10.1016/j.asoc.2016.05.009.  Google Scholar

[34]

L. H. Son, Measuring analogousness in picture fuzzy sets: From picture distance mea- sures to picture association measures, Fuzzy Optimization and Decision Making, 16 (2017), 359-378.  doi: 10.1007/s10700-016-9249-5.  Google Scholar

[35]

P. H. Thong, A new approach to multi-variable fuzzy forecasting using picture fuzzy clustering and picture fuzzy rule interpolation method, In Knowledge and Systems Engineering Springer, Cham, 326 (2015), 679-690.  doi: 10.1007/978-3-319-11680-8_54.  Google Scholar

[36]

P. H. Thong, Picture fuzzy clustering for complex data, Engineering Applications of Artificial Intelligence, 56 (2016), 121-130.  doi: 10.1016/j.engappai.2016.08.009.  Google Scholar

[37]

P. H. Thong, A novel automatic picture fuzzy clustering method based on particle swarm optimization and picture composite cardinality, Knowledge-Based Systems, (2016), 86–93. Google Scholar

[38]

P. H. Thong and H. Fujita, Interpolative picture fuzzy rules: A novel forecast method for weather nowcasting, In Fuzzy Systems, 109 (2016), 48-60.   Google Scholar

[39]

P. Van Viet, H. T. M. Chau and P. Van Hai, Some extensions of membership graphs for picture inference systems, In Knowledge and Systems Engineering (KSE), (2015), 192–198. Google Scholar

[40]

P. Van Viet and P. Van Hai, Picture inference system: A new fuzzy inference system on picture fuzzy set, Applied Intelligence, 46 (2017), 652-669.   Google Scholar

[41]

S. P. WanD. F. Li and Z. F. Rui, Possibility mean, variance and covariance of triangular intuitionistic fuzzy numbers, Journal of Intelligent and Fuzzy Systems, 24 (2013), 847-858.   Google Scholar

[42]

S. P. Wan and D. F. Li, Possibility mean and variance based method for multi-attribute decision making with triangular intuitionistic fuzzy numbers, Journal of Intelligent and Fuzzy Systems, 24 (2013), 743-754.   Google Scholar

[43]

S. P. Wan, Multi-attribute decision making method based on possibility variance coefficient of triangular intuitionistic fuzzy numbers, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 21 (2013), 223-243.  doi: 10.1142/S0218488513500128.  Google Scholar

[44]

S. P. Wan and J. Y. Dong, Possibility method for triangular intuitionistic fuzzy multi-attribute group decision making with incomplete weight information, International Journal of Computational Intelligence Systems, 7 (2014), 65-79.   Google Scholar

[45]

J. Q. WangR. NieH. Y. Zhang and X. H. Chen, New operators on triangular intuitionistic fuzzy numbers and their applications in system fault analysis, Information Sciences, 251 (2013), 79-95.  doi: 10.1016/j.ins.2013.06.033.  Google Scholar

[46]

G. Wei, Picture fuzzy cross-entropy for multiple attribute decision making problems, Journal of Business Economics and Management, 17 (2016), 491-502.   Google Scholar

[47]

G. Wei, Picture fuzzy aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems, 33 (2017), 713-724.   Google Scholar

[48]

G. WeiF. E. AlsaadiT. Hayat and A. Alsaedi, Projection models for multiple attribute decision making with picture fuzzy information, International Journal of Machine Learning and Cybernetics, 9 (2018), 713-719.  doi: 10.1007/s13042-016-0604-1.  Google Scholar

[49]

G. Wei and H. Gao, The generalized Dice similarity measures for picture fuzzy sets and their applications, Informatica, 29 (2018), 107-124.  doi: 10.15388/Informatica.2018.160.  Google Scholar

[50]

G. Wei, Some similarity measures for picture fuzzy sets and their applications, Iranian Journal of Fuzzy Systems, 15 (2018), 77-89.   Google Scholar

[51]

S. Xian, Fuzzy linguistic induced ordered weighted averaging operator and its application, Journal of Applied Mathematics, 2012 (2012), Article ID 210392, 10 pages. doi: 10.1155/2012/210392.  Google Scholar

[52]

S. Xian and W. Sun, Fuzzy linguistic induced Euclidean OWA distance operator and its application in group linguistic decision making, International Journal of Intelligent Systems, 29 (2014), 478-491.  doi: 10.1002/int.21648.  Google Scholar

[53]

S. XianW. XueJ. ZhangY. Yin and Q. Xie, Intuitionistic fuzzy linguistic induced ordered weighted averaging operator for group decision making, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 23 (2015), 627-648.  doi: 10.1142/S0218488515500270.  Google Scholar

[54]

Y. YangC. LiangS. Ji and T. Liu, Adjustable soft discernibility matrix based on picture fuzzy soft sets and its applications in decision making, Journal of Intelligent and Fuzzy Systems, 29 (2015), 1711-1722.   Google Scholar

[55]

S. Yu and Z. 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

[56]

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

[57]

X. Zhang and P. Liu, Method for aggregating triangular fuzzy intuitionistic fuzzy information and its application to decision making, Technological and Economic Development of Economy, 16 (2010), 280-290.   Google Scholar

[58]

M. J. ZhangJ. X. NanD. F. Li and Y. X. Li, TOPSIS for MADM with triangular intuitionistic fuzzy numbers, Operations Research and Management Science, 21 (2012), 96-101.   Google Scholar

show all references

References:
[1]

S. AshrafS. Abdullah and A. Qadir, Novel concept of cubic picture fuzzy sets, New Theory, 24 (2018), 69-72.   Google Scholar

[2]

S. AshrafT. MahmoodS. Abdullah and Q. khan, Different approaches to multi-criteria group decision making problems for picture fuzzy environment, Bulletin of the Brazilian Mathematical Society, New Series, 50 (2019), 373-397.  doi: 10.1007/s00574-018-0103-y.  Google Scholar

[3]

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

[4]

R. E. Bellman and L. A. Zadeh, Decision-making in a fuzzy environment, Management Science, 17 (1970), B141–B164. doi: 10.1287/mnsc.17.4.B141.  Google Scholar

[5]

C. Bo and X. Zhang, New operations of picture fuzzy relations and fuzzy comprehensive evaluation, Symmetry, 9 (2017), p268. doi: 10.3390/sym9110268.  Google Scholar

[6]

J. J. Buckley, Fuzzy decision making with data: Applications to statistics, Fuzzy Sets and Systems, 16 (1985), 139-147.  doi: 10.1016/S0165-0114(85)80014-2.  Google Scholar

[7]

Y. Chen and B. Li, Dynamic multi-attribute decision making model based on triangular intuitionistic fuzzy numbers, Scientia Iranica, 18 (2011), 268-274.  doi: 10.1016/j.scient.2011.03.022.  Google Scholar

[8]

B. C. Cuong, Picture fuzzy sets-first results. Part 1, Seminar Neuro-Fuzzy Systems with Applications. Preprint 04/2013, Institute of Mathematics, Hanoi, 2013. Google Scholar

[9]

B. C. Cuong, Picture fuzzy sets-first results. Part 2, Seminar Neuro-Fuzzy Systems with Applications. Preprint 04/2013, Institute of Mathematics, Hanoi, 2013. Google Scholar

[10]

B. C. Cuong, Picture fuzzy sets, Journal of Computer Science and Cybernetics, 30 (2014), p409. Google Scholar

[11]

B. C. Cuong and P. V. Hai, Some fuzzy logic operators for picture fuzzy sets, In Knowledge and Systems Engineering (KSE), (2015), 132–137. doi: 10.1109/KSE.2015.20.  Google Scholar

[12]

B. C. Cuong, V. Kreinovitch and R. T. Ngan, A classification of representable t-norm operators for picture fuzzy sets, In Knowledge and Systems Engineering (KSE), (2016), 19–24. doi: 10.1109/KSE.2016.7758023.  Google Scholar

[13]

D. Dubey and A. Mehra, Linear Programming with Triangular Intuitionistic Fuzzy Number, EUSFLAT-LFA, 2011. Google Scholar

[14]

D. J. Dubois, Fuzzy Sets and Systems: Theory and Applications, Academic Press, 1980. Google Scholar

[15]

M. Esmailzadeh and M. Esmailzadeh, New distance between triangular intuitionistic fuzzy numbers, Advances in Computational Mathematics and its Applications, 2 (2013), 310-314.   Google Scholar

[16]

H. Garg, Some picture fuzzy aggregation operators and their applications to multicri-teria decision-making, Arabian Journal for Science and Engineering, 42 (2017), 5275-5290.  doi: 10.1007/s13369-017-2625-9.  Google Scholar

[17]

F. Herrera and E. Herrera-Viedma, Linguistic decision analysis: Steps for solving decision problems under linguistic information, Fuzzy Sets and Systems, 115 (2000), 67-82.  doi: 10.1016/S0165-0114(99)00024-X.  Google Scholar

[18]

D. F. LiJ. K. Nan and M. J. Zhang, A ranking method of triangular intuitionistic fuzzy numbers and application to decision making, International Journal of Computational Intelligence Systems, 3 (2010), 522-530.   Google Scholar

[19]

D. F. Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems, Computers and Mathematics with Applications, 60 (2010), 1557-1570.   Google Scholar

[20]

C. LiangS. Zhao and J. Zhang, Aggregation operators on triangular intuitionistic fuzzy numbers and its application to multi-criteria decision making problems, Foundations of Computing and Decision Sciences, 39 (2014), 189-208.  doi: 10.2478/fcds-2014-0011.  Google Scholar

[21]

L. Marti and F. Herrera, An overview on the 2-tuple linguistic model for computing with words in decision making: Extensions, applications and challenges. Information Sciences, 207 (2012), 1–18. doi: 10.1016/j.ins.2012.04.025.  Google Scholar

[22]

X. Peng and J. Dai, Algorithm for picture fuzzy multiple attribute decision-making based on new distance measure, International Journal for Uncertainty Quantification, 7 (2017), 177-187.  doi: 10.1615/Int.J.UncertaintyQuantification.2017020096.  Google Scholar

[23]

P. H. Phong, D. T. Hieu, R. T. Ngan and P. T. Them, Some compositions of picture fuzzy relations, In Proceedings of the 7th National Conference on Fundamental and Applied Information Technology Research (FAIR?), Thai Nguye, (2014), 19–40. Google Scholar

[24]

P. H. Phong and B. C. Cuong, Multi-criteria group decision making with picture linguistic numbers. VNU Journal of Science, Computer Science and Communication Engineering, 32 (2017). Google Scholar

[25]

P. T. M. Phuong and P. H. Thong, Theoretical analysis of picture fuzzy clustering: Convergence and property, Journal of Computer Science and Cybernetics, 34 (2018), 17-32.  doi: 10.15625/1813-9663/34/1/12725.  Google Scholar

[26]

J. Robinson and H. A. EC, A short primer on the Correlation coefficient of Vague sets, International Journal of Fuzzy System Applications (IJFSA), 1 (2011), 55-69.   Google Scholar

[27]

J. Robinson and H. A. EC, A search for the correlation coefficient of triangular and trapezoidal intuitionistic fuzzy sets for multiple attribute group decision making, In Mathematical Modelling and Scientific Computation, 283 (2012), 333-342.   Google Scholar

[28]

R. RoostaeeM. IzadikhahF. H. Lotfi and M. Rostamy-Malkhalifeh, A multi-criteria intuitionistic fuzzy group decision making method for supplier selection with VIKOR method, International Journal of Fuzzy System Applications (IJFSA), 2 (2012), 1-17.  doi: 10.4018/978-1-4666-2625-6.ch056.  Google Scholar

[29]

M. H. ShuC. H. Cheng and J. R. Chang, Using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly, Microelectronics Reliability, 46 (2006), 2139-2148.  doi: 10.1016/j.microrel.2006.01.007.  Google Scholar

[30]

W. Shuping and D. Jiuying, Multi-attribute decision making based on triangular intuitionistic fuzzy number Choquet integral operator, Chinese Journal of Management Science, 22 (2014), 121-129.   Google Scholar

[31]

P. Singh, Correlation coefficients for picture fuzzy sets, Journal of Intelligent and Fuzzy Systems, 28 (2015), 591-604.   Google Scholar

[32]

L. H. Son, A novel distributed picture fuzzy clustering method on picture fuzzy sets, Expert Syst. Appl, 42 (2015), 51-66.  doi: 10.1016/j.eswa.2014.07.026.  Google Scholar

[33]

L. H. Son, Generalized picture distance measure and applications to picture fuzzy clustering, Applied Soft Computing, 46 (2016), 284-295.  doi: 10.1016/j.asoc.2016.05.009.  Google Scholar

[34]

L. H. Son, Measuring analogousness in picture fuzzy sets: From picture distance mea- sures to picture association measures, Fuzzy Optimization and Decision Making, 16 (2017), 359-378.  doi: 10.1007/s10700-016-9249-5.  Google Scholar

[35]

P. H. Thong, A new approach to multi-variable fuzzy forecasting using picture fuzzy clustering and picture fuzzy rule interpolation method, In Knowledge and Systems Engineering Springer, Cham, 326 (2015), 679-690.  doi: 10.1007/978-3-319-11680-8_54.  Google Scholar

[36]

P. H. Thong, Picture fuzzy clustering for complex data, Engineering Applications of Artificial Intelligence, 56 (2016), 121-130.  doi: 10.1016/j.engappai.2016.08.009.  Google Scholar

[37]

P. H. Thong, A novel automatic picture fuzzy clustering method based on particle swarm optimization and picture composite cardinality, Knowledge-Based Systems, (2016), 86–93. Google Scholar

[38]

P. H. Thong and H. Fujita, Interpolative picture fuzzy rules: A novel forecast method for weather nowcasting, In Fuzzy Systems, 109 (2016), 48-60.   Google Scholar

[39]

P. Van Viet, H. T. M. Chau and P. Van Hai, Some extensions of membership graphs for picture inference systems, In Knowledge and Systems Engineering (KSE), (2015), 192–198. Google Scholar

[40]

P. Van Viet and P. Van Hai, Picture inference system: A new fuzzy inference system on picture fuzzy set, Applied Intelligence, 46 (2017), 652-669.   Google Scholar

[41]

S. P. WanD. F. Li and Z. F. Rui, Possibility mean, variance and covariance of triangular intuitionistic fuzzy numbers, Journal of Intelligent and Fuzzy Systems, 24 (2013), 847-858.   Google Scholar

[42]

S. P. Wan and D. F. Li, Possibility mean and variance based method for multi-attribute decision making with triangular intuitionistic fuzzy numbers, Journal of Intelligent and Fuzzy Systems, 24 (2013), 743-754.   Google Scholar

[43]

S. P. Wan, Multi-attribute decision making method based on possibility variance coefficient of triangular intuitionistic fuzzy numbers, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 21 (2013), 223-243.  doi: 10.1142/S0218488513500128.  Google Scholar

[44]

S. P. Wan and J. Y. Dong, Possibility method for triangular intuitionistic fuzzy multi-attribute group decision making with incomplete weight information, International Journal of Computational Intelligence Systems, 7 (2014), 65-79.   Google Scholar

[45]

J. Q. WangR. NieH. Y. Zhang and X. H. Chen, New operators on triangular intuitionistic fuzzy numbers and their applications in system fault analysis, Information Sciences, 251 (2013), 79-95.  doi: 10.1016/j.ins.2013.06.033.  Google Scholar

[46]

G. Wei, Picture fuzzy cross-entropy for multiple attribute decision making problems, Journal of Business Economics and Management, 17 (2016), 491-502.   Google Scholar

[47]

G. Wei, Picture fuzzy aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems, 33 (2017), 713-724.   Google Scholar

[48]

G. WeiF. E. AlsaadiT. Hayat and A. Alsaedi, Projection models for multiple attribute decision making with picture fuzzy information, International Journal of Machine Learning and Cybernetics, 9 (2018), 713-719.  doi: 10.1007/s13042-016-0604-1.  Google Scholar

[49]

G. Wei and H. Gao, The generalized Dice similarity measures for picture fuzzy sets and their applications, Informatica, 29 (2018), 107-124.  doi: 10.15388/Informatica.2018.160.  Google Scholar

[50]

G. Wei, Some similarity measures for picture fuzzy sets and their applications, Iranian Journal of Fuzzy Systems, 15 (2018), 77-89.   Google Scholar

[51]

S. Xian, Fuzzy linguistic induced ordered weighted averaging operator and its application, Journal of Applied Mathematics, 2012 (2012), Article ID 210392, 10 pages. doi: 10.1155/2012/210392.  Google Scholar

[52]

S. Xian and W. Sun, Fuzzy linguistic induced Euclidean OWA distance operator and its application in group linguistic decision making, International Journal of Intelligent Systems, 29 (2014), 478-491.  doi: 10.1002/int.21648.  Google Scholar

[53]

S. XianW. XueJ. ZhangY. Yin and Q. Xie, Intuitionistic fuzzy linguistic induced ordered weighted averaging operator for group decision making, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 23 (2015), 627-648.  doi: 10.1142/S0218488515500270.  Google Scholar

[54]

Y. YangC. LiangS. Ji and T. Liu, Adjustable soft discernibility matrix based on picture fuzzy soft sets and its applications in decision making, Journal of Intelligent and Fuzzy Systems, 29 (2015), 1711-1722.   Google Scholar

[55]

S. Yu and Z. 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

[56]

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

[57]

X. Zhang and P. Liu, Method for aggregating triangular fuzzy intuitionistic fuzzy information and its application to decision making, Technological and Economic Development of Economy, 16 (2010), 280-290.   Google Scholar

[58]

M. J. ZhangJ. X. NanD. F. Li and Y. X. Li, TOPSIS for MADM with triangular intuitionistic fuzzy numbers, Operations Research and Management Science, 21 (2012), 96-101.   Google Scholar

Table 1.   
Table 2.   
Table 3.   
Table 4.   
Method Ranking
TPFLIOWA$\overset{\sim }{e}_{1}\succ _{PF}\overset{\sim }{e}_{2}\succ _{PF}\overset{\sim }{e}_{4}\succ _{PF}\overset{\sim }{e}_{5}\succ _{PF} \overset{\sim }{e}_{3}$
TPFLIOWG$\overset{\sim }{e}_{1}\succ _{PF}\overset{\sim }{e}_{2}\succ _{PF}\overset{\sim }{e}_{5}\succ _{PF}\overset{\sim }{e}_{4}\succ _{PF} \overset{\sim }{e}_{3}$
IFLIOWA [53]ş$_{\overset{\sim }{e}_{1}}\succ _{IF}$ş$_{ \overset{\sim }{e}_{4}}\succ _{IF}$ş$_{\overset{\sim }{e}_{2}}\succ _{IF} $ş$_{\overset{\sim }{e}_{5}}\succ _{IF}$ş$_{\overset{\sim }{e} _{3}}$
IFLIOWG [53]ş$_{\overset{\sim }{e}_{1}}\succ _{IF}$ş$_{ \overset{\sim }{e}_{5}}\succ _{IF}$ş$_{\overset{\sim }{e}_{4}}\succ _{IF} $ş$_{\overset{\sim }{e}_{2}}\succ _{IF}$ş$_{\overset{\sim }{e} _{3}}$
Method Ranking
TPFLIOWA$\overset{\sim }{e}_{1}\succ _{PF}\overset{\sim }{e}_{2}\succ _{PF}\overset{\sim }{e}_{4}\succ _{PF}\overset{\sim }{e}_{5}\succ _{PF} \overset{\sim }{e}_{3}$
TPFLIOWG$\overset{\sim }{e}_{1}\succ _{PF}\overset{\sim }{e}_{2}\succ _{PF}\overset{\sim }{e}_{5}\succ _{PF}\overset{\sim }{e}_{4}\succ _{PF} \overset{\sim }{e}_{3}$
IFLIOWA [53]ş$_{\overset{\sim }{e}_{1}}\succ _{IF}$ş$_{ \overset{\sim }{e}_{4}}\succ _{IF}$ş$_{\overset{\sim }{e}_{2}}\succ _{IF} $ş$_{\overset{\sim }{e}_{5}}\succ _{IF}$ş$_{\overset{\sim }{e} _{3}}$
IFLIOWG [53]ş$_{\overset{\sim }{e}_{1}}\succ _{IF}$ş$_{ \overset{\sim }{e}_{5}}\succ _{IF}$ş$_{\overset{\sim }{e}_{4}}\succ _{IF} $ş$_{\overset{\sim }{e}_{2}}\succ _{IF}$ş$_{\overset{\sim }{e} _{3}}$
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