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. 

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

[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.

[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. 

[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. 

[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.

[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.

[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.

[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.

[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).

[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.

[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. 

[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. 

[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.

[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.

[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. 

[31]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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. 

[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. 

[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. 

[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.

[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. 

[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.

[46]

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

[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. 

[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.

[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.

[50]

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

[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.

[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.

[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.

[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. 

[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.

[56]

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

[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. 

[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. 

show all references

References:
[1]

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

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

[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.

[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. 

[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. 

[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.

[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.

[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.

[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.

[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).

[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.

[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. 

[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. 

[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.

[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.

[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. 

[31]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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. 

[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. 

[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. 

[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.

[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. 

[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.

[46]

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

[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. 

[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.

[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.

[50]

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

[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.

[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.

[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.

[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. 

[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.

[56]

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

[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. 

[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. 

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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