Intuitionistic fuzzy interaction Maclaurin symmetric means and their application to multiple-attribute decision-making
Abstract
The Maclaurin symmetric mean (MSM) can capture the interrelationship among the multi-input arguments and it also can generalize most of the existing operators. Now MSM has been extended to intuitionistic fuzzy sets (IFSs) which can easily express the vague information. However, the operational rules of IFSs used in the extended MSM operator didn’t consider the interaction between the membership function and non-membership function, so there are some weaknesses. In this paper, in order to combine the advantages of the MSM and interaction operational rules of IFSs, we propose the intuitionistic fuzzy interaction Maclaurin symmetric mean (IFIMSM) operator, the intuitionistic fuzzy weighted interaction Maclaurin symmetric mean (IFWIMSM) operator, respectively. Furthermore, we research some desirable properties and some special cases of them. Further, we develop a new method to deal with some multi-attribute group decision-making (MAGDM) problems under intuitionistic fuzzy environment based on these operators. Finally, an illustrative example is given to testify the availability of the developed method by comparing with the other existing methods.
Keyword : intuitionistic fuzzy set, Maclaurin symmetric mean operator, multi-attribute group decision-making
How to Cite
Liu, P., & Liu, W. (2018). Intuitionistic fuzzy interaction Maclaurin symmetric means and their application to multiple-attribute decision-making. Technological and Economic Development of Economy, 24(4), 1533-1559. https://doi.org/10.3846/tede.2018.3698
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Atanassov, K. T. (1994). New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets and Systems, 61(2), 137-142. https://doi.org/10.1016/0165-0114(94)90229-1
Celik, E., Gumus, A. T., & Alegoz, M. (2014). A trapezoidal type-2 fuzzy MCDM method to identify and evaluate critical success factors for humanitarian relief logistics management. Journal of Intelligent & Fuzzy Systems, 27(6), 2847-2855.
Chen, S. M., & Tan, J. M. (1994). Handling multi-criteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems, 67(2), 163-172. https://doi.org/10.1016/0165-0114(94)90084-1
Chen, T. Y. (2007). A note on distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Sets and Systems, 158(22), 2523-2525. https://doi.org/10.1016/j.fss.2007.04.024
De Kumar, S., Biswas, R., & Roy, A. R. (2000a). Some operations on intuitionistic fuzzy sets. Fuzzy Sets and Systems, 114(3), 477-484. https://doi.org/10.1016/S0165-0114(98)00191-2
De Kumar, S., Biswas, R., & Roy, A. R. (2000b). Some operations on intuitionistic fussy sets in terms of evidences theory: decision making aspect. Knowledge Based Systems, 23(8), 772-782.
Gürbüz, T., & Albayrak, Y. E. (2014). An engineering approach to human resources performance evaluation: hybrid MCDM application with interactions. Applied Soft Computing, 21, 365-375. https://doi.org/10.1016/j.asoc.2014.03.025
He, Y. D., Chen, H. Y., Zhou, L. G., Han, B., & Zhao, Q. Y. (2014a). Generalized intuitionistic fuzzy geometric interaction operators and their application to decision making. Expert Systems with Applications, 41(5), 2484-2495. https://doi.org/10.1016/j.eswa.2013.09.048
He, Y. D., Chen, H. Y., Zhou, L. G., Liu, J. P., & Tao, Z. F. (2014b). Intuitionistic fuzzy geometric interaction averaging operators and their application to multi-criteria decision making. Information Sciences, 259, 142-159. https://doi.org/10.1016/j.ins.2013.08.018
He, Y. D., He, Z., & Chen, H. (2015). Intuitionistic fuzzy interaction Bonferroni means and its application to multiple attribute decision making. IEEE Transactions on Cybernetics, 45(1), 116-128. https://doi.org/10.1109/TCYB.2014.2320910
Hong, D. H., & Choi, C. H. (2000). Multi-criteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems, 114(1), 103-113. https://doi.org/10.1016/S0165-0114(98)00271-1
Hou, Y. S. (2016). Reflection on the inheritance and development of “Wang PI opera” in Shandong. Dong Yue Tribune, 37(2), 142-146 (in Chinese).
Li, D. F. (2005). Multi-attribute decision making models and methods using intuitionistic fuzzy sets. Journal of Computer and Systems Sciences, 70(1), 73-85. https://doi.org/10.1016/j.jcss.2004.06.002
Lin, L., Yuan, X. H., & Xia, Z. Q. (2007). Multi-criteria fuzzy decision-making methods based on intuitionistic fuzzy sets. Journal of Computer and Systems Sciences, 73(1), 84-88. https://doi.org/10.1016/j.jcss.2006.03.004
Liu, P. D. (2017). Multiple attribute group decision making method based on interval-valued intuitionistic fuzzy power Heronian aggregation operators. Computers & Industrial Engineering, 108, 199-212. https://doi.org/10.1016/j.cie.2017.04.033
Liu, P. D., & Chen, S. M. (2017). Group decision making based on Heronian aggregation operators of intuitionistic fuzzy numbers. IEEE Transactions on Cybernetics, 47(9), 2514-2530. https://doi.org/10.1109/TCYB.2016.2634599
Liu, P. D., & Chen, S. M. (2018). Multiattribute group decision making based on intuitionistic 2-tuple linguistic information. Information Sciences, 430-431, 599-619. https://doi.org/10.1016/j.ins.2017.11.059
Liu, P. D., Chen, S. M. & Liu, J. L. (2017). Some intuitionistic fuzzy interaction partitioned Bonferroni mean operators and their application to multi-attribute group decision making. Information Sciences, 411, 98-121. https://doi.org/10.1016/j.ins.2017.05.016
Liu, P. D., He, L., & Yu, X. C. (2016). Generalized hybrid aggregation operators based on the 2-dimension uncertain linguistic information for multiple attribute group decision making. Group Decision and Negotiation, 25(1), 103-126. https://doi.org/10.1007/s10726-015-9434-x
Liu, P. D. & Li, H. G. (2017). Interval-valued intuitionistic fuzzy power Bonferroni aggregation operators and their application to group decision making. Cognitive Computation, 9(4), 494-512. https://doi.org/10.1007/s12559-017-9453-9
Liu, P. D., Liu, J. L. & Merigó, J. M. (2018). Partitioned Heronian means based on linguistic intuitionistic fuzzy numbers for dealing with multi-attribute group decision making. Applied Soft Computing, 62, 395-422. https://doi.org/10.1016/j.asoc.2017.10.017
Liu, P. D., Liu, J. L., & Chen, S. M. (2018). Some intuitionistic fuzzy Dombi Bonferroni mean operators and their application to multi-attribute group decision making. Journal of the Operational Research Society, 69(1), 1-24. https://doi.org/10.1057/s41274-017-0190-y
Liu, P. D., & Teng, F. (2016). An extended TODIM Method for Multiple Attribute Group Decision-making Based on 2-dimension uncertain linguistic variable. Complexity, 221(5), 20-30, 8(6), 20-30. https://doi.org/10.1002/cplx.21625
Liu, P. D., Zhang, L. L., Liu, X., & Wang, P. (2016). Multi-valued Neutrosophic number Bonferroni mean operators and their application in multiple attribute group decision making. International Journal of Information Technology & Decision Making, 15(5), 1181-1210. https://doi.org/10.1142/S0219622016500346
Maclaurin, C. (1729). A second letter to Martin Folkes, Esq.; concerning the roots of equations, with demonstration of other rules of algebra. Philosophical Transactions of the Royal Society of London Series A, 36, 59-96. https://doi.org/10.1098/rstl.1729.0011
Meng, F., Zhang, Q., & Zhan, J. (2015). The interval-valued intuitionistic fuzzy geometric choquet aggregation operator based on the generalized banzhaf index and 2-additive measure. Technological and Economic Development of Economy, 21(2), 186-215. https://doi.org/10.3846/20294913.2014.946983
Montajabiha, M. (2016). An extended PROMETHE II multi-criteria group decision making technique based on intuitionistic fuzzy logic for sustainable energy planning. Group Decision and Negotiation, 25(2), 221-2447. https://doi.org/10.1007/s10726-015-9440-z
Mulliner, E., Malys, N., & Maliene, V. (2015). Comparative analysis of MCDM methods for the assessment of sustainable housing affordability. Omega, 59, 146-156. https://doi.org/10.1016/j.omega.2015.05.013
Qin, J. D., & Liu, X. W. (2014). An approach to intuitionistic fuzzy multiple attribute decision making based on Maclaurin symmetric mean operators. Journal of Intelligent & Fuzzy Systems, 27(5), 2177-2190.
Rabbani, A., Zamani, M., Yazdani-Chamzini, A., & Zavadskas, E. K. (2014). Proposing a new integrated model based on sustainability balanced scorecard (SBSC) and MCDM approaches by using linguistic variables for the performance evaluation of oil producing companies. Expert Systems with Applications, 41(16), 7316-7327. https://doi.org/10.1016/j.eswa.2014.05.023
Şahin, R., & Liu, P. D. (2017). Possibility-induced simplified neutrosophic aggregation operators and heir application to multi-criteria group decision making. Journal of Experimental & Theoretical Artificial Intelligence, 29(4), 769-785. https://doi.org/10.1080/0952813X.2016.1259266
Straub, J., & Reza, H. (2015). A Blackboard- style decision- making system for multi- tier craft control and its evaluation. Journal of Experimental & Theoretical Artificial Intelligence, 27(6), 763-777. https://doi.org/10.1080/0952813X.2015.1020569
Tian, Z. P., Wang, J., & Wang, J. Q. (2017). Simplified neutrosophic linguistic multi-criteria group decision-making approach to green product development. Group Decision and Negotiation, 26(3), 597-627. https://doi.org/10.1007/s10726-016-9479-5
Tavana, M., Mavi, R. K., Santos-Arteaga, F. J., & Doust, E. R. (2016). An extended VIKOR method using stochastic data and subjective judgments. Computers & Industrial Engineering, 97, 240-247. https://doi.org/10.1016/j.cie.2016.05.013
Uygun, O., & Dede, A. (2016). Performance evaluation of green supply chain management using integrated fuzzy multi-criteria decision making techniques. Computers & Industrial Engineering, 102, 502-511. https://doi.org/10.1016/j.cie.2016.02.020
Wang, F., Zeng, S., & Zhang, C. (2013). A method based on intuitionistic fuzzy dependent aggregation operators for supplier selection. Mathematical Problems in Engineering, 2013, 1-9, Article ID 481202. https://doi.org/10.1155/2013/481202
Wang, J., Wang, J. Q., & Zhang, H. Y. (2016a). A likelihood-based TODIM approach based on multihesitant fuzzy linguistic information for evaluation in logistics outsourcing. Computers & Industrial Engineering, 99, 287-299. https://doi.org/10.1016/j.cie.2016.07.023
Wang, T., Liu, J., Li, J., & Niu C. (2016b). An integrating OWA–TOPSIS framework in intuitionistic fuzzy settings for multiple attribute decision making. Computers & Industrial Engineering, 98, 185-194. https://doi.org/10.1016/j.cie.2016.05.029
Wu, J., Cao, Q. W., & Li, H. (2016). An approach for MADM problems with interval-valued intuitionistic fuzzy sets based on nonlinear functions. Technological and Economic Development of Economy, 22(3), 336-356. https://doi.org/10.3846/20294913.2014.989931
Xu, Y. J., Huang, C., Da, Q. L., & Liu, X. W. (2010). Linear goal programming approach to obtaining the weights of intuitionistic fuzzy ordered weighted averaging operator. Journal of Systems Engineering and Electronics, 21(6), 990-994. https://doi.org/10.3969/j.issn.1004-4132.2010.06.010
Xu, Z. S. (2007). Intuitionistic fuzzy aggregation operators. IEEE Transactions on Fuzzy Systems, 15(6), 1179-1187. https://doi.org/10.1109/TFUZZ.2006.890678
Xu, Z. S. (2011). Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators. Knowledge-Based Systems, 24(6), 749-760. https://doi.org/10.1016/j.knosys.2011.01.011
Xu, Z. S., & Yager, R. R. (2006). Some geometric aggregation operators based on intuitionistic fuzzy sets. International Journal of General Systems, 35(4), 417-433. https://doi.org/10.1080/03081070600574353
Xu, Z. S., & Yager, R. R. (2011). Intuitionistic fuzzy Bonferroni means. IEEE Transactions on Systems Man and Cybernetics Part B: Cybernetics, 41(2), 568-578. https://doi.org/10.1109/TSMCB.2010.2072918
Yu, D. J., & Wu, Y. Y. (2012). Interval-valued intuitionistic fuzzy Heronian mean operators and their application in multi-criteria decision making. African Journal of Business Management, 6(11), 4158-4168.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338-356. https://doi.org/10.1016/S0019-9958(65)90241-X
Zha, D., & Kavuri, A. S. (2016). Effects of technical and allocative inefficiencies on energy and nonenergy elasticities: an analysis of energy-intensive industries in China. Chinese Journal of Population Resources and Environment, 14(4), 292-297. https://doi.org/10.1080/10042857.2016.1258805
Zhang, B., He, M., & Pan, H. (2017). A study on the design of a hybrid policy for carbon abatement. Chinese Journal of Population Resources and Environment, 15(1), 50-57. https://doi.org/10.1080/10042857.2016.1258804
Zhang, R., & Shi, G. (2016). Analysis of the relationship between environmental policies and air quality during major social events, Chinese Journal of Population Resources and Environment, 14(3), 167-173. https://doi.org/10.1080/10042857.2016.1177316
Zhang, W. C., Xu, Y. J., & Wang, H. M. (2016). A consensus reaching model for 2-tuple linguistic multiple attribute group decision making with incomplete weight information. International Journal of Systems Science, 47(2), 389-405. https://doi.org/10.1080/00207721.2015.1074761
Zhang, X. L., & Xu, Z. S. (2015). Hesitant fuzzy agglomerative hierarchical clustering algorithms. International journal of systems Science, 46(3), 562-576. https://doi.org/10.1080/00207721.2013.797037
Zhang, Z., & Guo, C. H. (2016). Consistency and consensus models for group decision-making with uncertain 2-tuple linguistic preference relations. International journal of systems Science, 47(11), 2572-2587. https://doi.org/10.1080/00207721.2014.999732
Zhou, H., Wang, J. Q., & Zhang, H. Y. (2016). Linguistic hesitant fuzzy multi-criteria decision-making method based on evidential reasoning. International Journal of Systems Science, 47(2), 314-327. https://doi.org/10.1080/00207721.2015.1042089
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Atanassov, K. T. (1989). More on intuitionistic fuzzy sets. Fuzzy Sets and Systems, 33(1), 37-46. https://doi.org/10.1016/0165-0114(89)90215-7
Atanassov, K. T. (1994). New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets and Systems, 61(2), 137-142. https://doi.org/10.1016/0165-0114(94)90229-1
Celik, E., Gumus, A. T., & Alegoz, M. (2014). A trapezoidal type-2 fuzzy MCDM method to identify and evaluate critical success factors for humanitarian relief logistics management. Journal of Intelligent & Fuzzy Systems, 27(6), 2847-2855.
Chen, S. M., & Tan, J. M. (1994). Handling multi-criteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems, 67(2), 163-172. https://doi.org/10.1016/0165-0114(94)90084-1
Chen, T. Y. (2007). A note on distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Sets and Systems, 158(22), 2523-2525. https://doi.org/10.1016/j.fss.2007.04.024
De Kumar, S., Biswas, R., & Roy, A. R. (2000a). Some operations on intuitionistic fuzzy sets. Fuzzy Sets and Systems, 114(3), 477-484. https://doi.org/10.1016/S0165-0114(98)00191-2
De Kumar, S., Biswas, R., & Roy, A. R. (2000b). Some operations on intuitionistic fussy sets in terms of evidences theory: decision making aspect. Knowledge Based Systems, 23(8), 772-782.
Gürbüz, T., & Albayrak, Y. E. (2014). An engineering approach to human resources performance evaluation: hybrid MCDM application with interactions. Applied Soft Computing, 21, 365-375. https://doi.org/10.1016/j.asoc.2014.03.025
He, Y. D., Chen, H. Y., Zhou, L. G., Han, B., & Zhao, Q. Y. (2014a). Generalized intuitionistic fuzzy geometric interaction operators and their application to decision making. Expert Systems with Applications, 41(5), 2484-2495. https://doi.org/10.1016/j.eswa.2013.09.048
He, Y. D., Chen, H. Y., Zhou, L. G., Liu, J. P., & Tao, Z. F. (2014b). Intuitionistic fuzzy geometric interaction averaging operators and their application to multi-criteria decision making. Information Sciences, 259, 142-159. https://doi.org/10.1016/j.ins.2013.08.018
He, Y. D., He, Z., & Chen, H. (2015). Intuitionistic fuzzy interaction Bonferroni means and its application to multiple attribute decision making. IEEE Transactions on Cybernetics, 45(1), 116-128. https://doi.org/10.1109/TCYB.2014.2320910
Hong, D. H., & Choi, C. H. (2000). Multi-criteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems, 114(1), 103-113. https://doi.org/10.1016/S0165-0114(98)00271-1
Hou, Y. S. (2016). Reflection on the inheritance and development of “Wang PI opera” in Shandong. Dong Yue Tribune, 37(2), 142-146 (in Chinese).
Li, D. F. (2005). Multi-attribute decision making models and methods using intuitionistic fuzzy sets. Journal of Computer and Systems Sciences, 70(1), 73-85. https://doi.org/10.1016/j.jcss.2004.06.002
Lin, L., Yuan, X. H., & Xia, Z. Q. (2007). Multi-criteria fuzzy decision-making methods based on intuitionistic fuzzy sets. Journal of Computer and Systems Sciences, 73(1), 84-88. https://doi.org/10.1016/j.jcss.2006.03.004
Liu, P. D. (2017). Multiple attribute group decision making method based on interval-valued intuitionistic fuzzy power Heronian aggregation operators. Computers & Industrial Engineering, 108, 199-212. https://doi.org/10.1016/j.cie.2017.04.033
Liu, P. D., & Chen, S. M. (2017). Group decision making based on Heronian aggregation operators of intuitionistic fuzzy numbers. IEEE Transactions on Cybernetics, 47(9), 2514-2530. https://doi.org/10.1109/TCYB.2016.2634599
Liu, P. D., & Chen, S. M. (2018). Multiattribute group decision making based on intuitionistic 2-tuple linguistic information. Information Sciences, 430-431, 599-619. https://doi.org/10.1016/j.ins.2017.11.059
Liu, P. D., Chen, S. M. & Liu, J. L. (2017). Some intuitionistic fuzzy interaction partitioned Bonferroni mean operators and their application to multi-attribute group decision making. Information Sciences, 411, 98-121. https://doi.org/10.1016/j.ins.2017.05.016
Liu, P. D., He, L., & Yu, X. C. (2016). Generalized hybrid aggregation operators based on the 2-dimension uncertain linguistic information for multiple attribute group decision making. Group Decision and Negotiation, 25(1), 103-126. https://doi.org/10.1007/s10726-015-9434-x
Liu, P. D. & Li, H. G. (2017). Interval-valued intuitionistic fuzzy power Bonferroni aggregation operators and their application to group decision making. Cognitive Computation, 9(4), 494-512. https://doi.org/10.1007/s12559-017-9453-9
Liu, P. D., Liu, J. L. & Merigó, J. M. (2018). Partitioned Heronian means based on linguistic intuitionistic fuzzy numbers for dealing with multi-attribute group decision making. Applied Soft Computing, 62, 395-422. https://doi.org/10.1016/j.asoc.2017.10.017
Liu, P. D., Liu, J. L., & Chen, S. M. (2018). Some intuitionistic fuzzy Dombi Bonferroni mean operators and their application to multi-attribute group decision making. Journal of the Operational Research Society, 69(1), 1-24. https://doi.org/10.1057/s41274-017-0190-y
Liu, P. D., & Teng, F. (2016). An extended TODIM Method for Multiple Attribute Group Decision-making Based on 2-dimension uncertain linguistic variable. Complexity, 221(5), 20-30, 8(6), 20-30. https://doi.org/10.1002/cplx.21625
Liu, P. D., Zhang, L. L., Liu, X., & Wang, P. (2016). Multi-valued Neutrosophic number Bonferroni mean operators and their application in multiple attribute group decision making. International Journal of Information Technology & Decision Making, 15(5), 1181-1210. https://doi.org/10.1142/S0219622016500346
Maclaurin, C. (1729). A second letter to Martin Folkes, Esq.; concerning the roots of equations, with demonstration of other rules of algebra. Philosophical Transactions of the Royal Society of London Series A, 36, 59-96. https://doi.org/10.1098/rstl.1729.0011
Meng, F., Zhang, Q., & Zhan, J. (2015). The interval-valued intuitionistic fuzzy geometric choquet aggregation operator based on the generalized banzhaf index and 2-additive measure. Technological and Economic Development of Economy, 21(2), 186-215. https://doi.org/10.3846/20294913.2014.946983
Montajabiha, M. (2016). An extended PROMETHE II multi-criteria group decision making technique based on intuitionistic fuzzy logic for sustainable energy planning. Group Decision and Negotiation, 25(2), 221-2447. https://doi.org/10.1007/s10726-015-9440-z
Mulliner, E., Malys, N., & Maliene, V. (2015). Comparative analysis of MCDM methods for the assessment of sustainable housing affordability. Omega, 59, 146-156. https://doi.org/10.1016/j.omega.2015.05.013
Qin, J. D., & Liu, X. W. (2014). An approach to intuitionistic fuzzy multiple attribute decision making based on Maclaurin symmetric mean operators. Journal of Intelligent & Fuzzy Systems, 27(5), 2177-2190.
Rabbani, A., Zamani, M., Yazdani-Chamzini, A., & Zavadskas, E. K. (2014). Proposing a new integrated model based on sustainability balanced scorecard (SBSC) and MCDM approaches by using linguistic variables for the performance evaluation of oil producing companies. Expert Systems with Applications, 41(16), 7316-7327. https://doi.org/10.1016/j.eswa.2014.05.023
Şahin, R., & Liu, P. D. (2017). Possibility-induced simplified neutrosophic aggregation operators and heir application to multi-criteria group decision making. Journal of Experimental & Theoretical Artificial Intelligence, 29(4), 769-785. https://doi.org/10.1080/0952813X.2016.1259266
Straub, J., & Reza, H. (2015). A Blackboard- style decision- making system for multi- tier craft control and its evaluation. Journal of Experimental & Theoretical Artificial Intelligence, 27(6), 763-777. https://doi.org/10.1080/0952813X.2015.1020569
Tian, Z. P., Wang, J., & Wang, J. Q. (2017). Simplified neutrosophic linguistic multi-criteria group decision-making approach to green product development. Group Decision and Negotiation, 26(3), 597-627. https://doi.org/10.1007/s10726-016-9479-5
Tavana, M., Mavi, R. K., Santos-Arteaga, F. J., & Doust, E. R. (2016). An extended VIKOR method using stochastic data and subjective judgments. Computers & Industrial Engineering, 97, 240-247. https://doi.org/10.1016/j.cie.2016.05.013
Uygun, O., & Dede, A. (2016). Performance evaluation of green supply chain management using integrated fuzzy multi-criteria decision making techniques. Computers & Industrial Engineering, 102, 502-511. https://doi.org/10.1016/j.cie.2016.02.020
Wang, F., Zeng, S., & Zhang, C. (2013). A method based on intuitionistic fuzzy dependent aggregation operators for supplier selection. Mathematical Problems in Engineering, 2013, 1-9, Article ID 481202. https://doi.org/10.1155/2013/481202
Wang, J., Wang, J. Q., & Zhang, H. Y. (2016a). A likelihood-based TODIM approach based on multihesitant fuzzy linguistic information for evaluation in logistics outsourcing. Computers & Industrial Engineering, 99, 287-299. https://doi.org/10.1016/j.cie.2016.07.023
Wang, T., Liu, J., Li, J., & Niu C. (2016b). An integrating OWA–TOPSIS framework in intuitionistic fuzzy settings for multiple attribute decision making. Computers & Industrial Engineering, 98, 185-194. https://doi.org/10.1016/j.cie.2016.05.029
Wu, J., Cao, Q. W., & Li, H. (2016). An approach for MADM problems with interval-valued intuitionistic fuzzy sets based on nonlinear functions. Technological and Economic Development of Economy, 22(3), 336-356. https://doi.org/10.3846/20294913.2014.989931
Xu, Y. J., Huang, C., Da, Q. L., & Liu, X. W. (2010). Linear goal programming approach to obtaining the weights of intuitionistic fuzzy ordered weighted averaging operator. Journal of Systems Engineering and Electronics, 21(6), 990-994. https://doi.org/10.3969/j.issn.1004-4132.2010.06.010
Xu, Z. S. (2007). Intuitionistic fuzzy aggregation operators. IEEE Transactions on Fuzzy Systems, 15(6), 1179-1187. https://doi.org/10.1109/TFUZZ.2006.890678
Xu, Z. S. (2011). Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators. Knowledge-Based Systems, 24(6), 749-760. https://doi.org/10.1016/j.knosys.2011.01.011
Xu, Z. S., & Yager, R. R. (2006). Some geometric aggregation operators based on intuitionistic fuzzy sets. International Journal of General Systems, 35(4), 417-433. https://doi.org/10.1080/03081070600574353
Xu, Z. S., & Yager, R. R. (2011). Intuitionistic fuzzy Bonferroni means. IEEE Transactions on Systems Man and Cybernetics Part B: Cybernetics, 41(2), 568-578. https://doi.org/10.1109/TSMCB.2010.2072918
Yu, D. J., & Wu, Y. Y. (2012). Interval-valued intuitionistic fuzzy Heronian mean operators and their application in multi-criteria decision making. African Journal of Business Management, 6(11), 4158-4168.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338-356. https://doi.org/10.1016/S0019-9958(65)90241-X
Zha, D., & Kavuri, A. S. (2016). Effects of technical and allocative inefficiencies on energy and nonenergy elasticities: an analysis of energy-intensive industries in China. Chinese Journal of Population Resources and Environment, 14(4), 292-297. https://doi.org/10.1080/10042857.2016.1258805
Zhang, B., He, M., & Pan, H. (2017). A study on the design of a hybrid policy for carbon abatement. Chinese Journal of Population Resources and Environment, 15(1), 50-57. https://doi.org/10.1080/10042857.2016.1258804
Zhang, R., & Shi, G. (2016). Analysis of the relationship between environmental policies and air quality during major social events, Chinese Journal of Population Resources and Environment, 14(3), 167-173. https://doi.org/10.1080/10042857.2016.1177316
Zhang, W. C., Xu, Y. J., & Wang, H. M. (2016). A consensus reaching model for 2-tuple linguistic multiple attribute group decision making with incomplete weight information. International Journal of Systems Science, 47(2), 389-405. https://doi.org/10.1080/00207721.2015.1074761
Zhang, X. L., & Xu, Z. S. (2015). Hesitant fuzzy agglomerative hierarchical clustering algorithms. International journal of systems Science, 46(3), 562-576. https://doi.org/10.1080/00207721.2013.797037
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