Simplified neutrosophic indeterminate decision making method with decision makers’ indeterminate ranges
Abstract
There exists the indeterminate situations of truth, falsity, indeterminacy degrees due to the uncertainty and inconsistency of decision makers’ arguments in a complicated decision making (DM) problem. Then, existing neutrosophic set cannot describe the indeterminate information of truth, falsity, indeterminacy degrees. It is noted that the simplified neutrosophic set (SNS) is depicted by truth, falsity, indeterminacy degrees, while a neutrosophic number (NN) can be flexibly depicted by its determinate part and its indeterminate part. Regarding the indeterminate situations of truth, falsity, indeterminacy degrees in indeterminate DM problems, this study first presents a simplified neutrosophic indeterminate set (SNIS) to express the hybrid information of SNS and NN and defines the score, accuracy, and certainty functions of simplified neutrosophic indeterminate elements (SNIEs) with indeterminate ranges to compare SNIEs. Then, we introduce a SNIE weighted arithmetic averaging (SNIEWAA) operator and a SNIE weighted geometric averaging (SNIEWGA) operator to aggregate simplified neutrosophic indeterminate information. Next, a multi-attribute DM approach with decision makers’ indeterminate ranges is established regarding the SNIEWAA and SNIEWGA operators in SNIS setting. Finally, the proposed DM approach is applied in a DM example on choosing a suitable slope design scheme to indicate the applicability and suitability of the proposed approach.
Keyword : simplified neutrosophic indeterminate set, simplified neutrosophic indeterminate element, simplified neutrosophic indeterminate element weighted arithmetic averaging (SNIEWAA) operator, simplified neutrosophic indeterminate element weighted geometric averaging (SNIEWGA) operator, decision making
How to Cite
Du, S., Ye, J., Yong, R., & Zhang, F. (2020). Simplified neutrosophic indeterminate decision making method with decision makers’ indeterminate ranges. Journal of Civil Engineering and Management, 26(6), 590-598. https://doi.org/10.3846/jcem.2020.12919
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Broumi, S., & Deli, I. (2014). Correlation measure for neutrosophic refined sets and its application in medical diagnosis. Palestine Journal of Mathematics, 3(1), 11–19.
Broumi, S., & Smarandache, F. (2015). Interval neutrosophic rough sets. Neutrosophic Sets and Systems, 7, 23–31. https://doi.org/10.1155/2015/232919
Chen, J. Q., Ye, J., & Du, S. G. (2017). Vector similarity measures between refined simplified neutrosophic sets and their multiple attribute decision making method. Symmetry, 9(8), 153. https://doi.org/10.3390/sym9080153
Can, M. S., & Ozguven, O. F. (2017). PID tuning with neutrosophic similarity measure. International Journal of Fuzzy Systems, 19(2), 489–503. https://doi.org/10.1007/s40815-015-0136-y
Gal, A., Vladareanu, L., Smarandache, F., Yu, H., & Deng, M. (2012). Neutrosophic logic approaches applied to “rabot” real time control. Neutrosophic Theory and Its Applications, 1, 55–60.
Jun, Y. B., Smarandache, F., & Kim, C. S. (2017). Neutrosophic cubic sets. New Mathematics and Natural Computation, 13(1), 41–54. https://doi.org/10.1142/S1793005717500041
Köseoğlu, A., Şahin, R., & Merdan, M. (2019). A simplified neutrosophic multiplicative set‐based TODIM using water‐filling algorithm for the determination of weights. Expert Systems. https://doi.org/10.1111/exsy.12515
Liu, P. D., & Wang, Y. M. (2014). Multiple attribute decision making method based on single-valued neutrosophic normalized weighted Bonferroni mean. Neural Computing and Applications, 25(7–8), 2001–2010. https://doi.org/10.1007/s00521-014-1688-8
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Maji, P. K. (2013). Neutrosophic soft set. Annals of Fuzzy Mathematics and Informatics, 5(1), 157–168.
Peng, J. J., Wang, J. Q., Wu, X. H., Wang, J., & Chen, X. H. (2015). Multivalued neutrosophic sets and power aggregation operators with their applications in multi-criteria group decisionmaking problems. International Journal of Computational Intelligence Systems, 8(2), 345–363. https://doi.org/10.1080/18756891.2015.1001957
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Read, J., & Stacey, P. (2009). Guidelines for open pit slope design. CSIRO Publishing. https://doi.org/10.1071/9780643101104
Smarandache, F. (1998). Neutrosophy: neutrosophic probability, set, and logic. American Research Press.
Smarandache, F. (2013a). n-Valued refined neutrosophic logic and its applications in physics. Progress in Physics, 4, 143–146.
Smarandache, F. (2013b). Introduction to neutrosophic measure, neutrosophic integral, and neutrosophic probability. Sitech & Education Publisher.
Smarandache, F. (2014). Introduction to neutrosophic statistics. Sitech & Education Publishing.
Şahin, R. (2018). Normal neutrosophic multiple attribute decision making based on generalized prioritized aggregation operators. Neural Computing & Applications, 30(10), 3095–3115. https://doi.org/10.1007/s00521-017-2896-9
Sahin, R. & Liu, P. D. (2017a). Possibility-induced simplified neutrosophic aggregation operators and their application to multicriteria group decision making. Journal of Experimental & Theoretical Artificial Intelligence, 29(4), 769–785. https://doi.org/10.1080/0952813X.2016.1259266
Sahin, R. & Liu, P. D. (2017b). Some approaches to multi criteria decision making based on exponential operations of simplified neutrosophic numbers. Journal of Intelligent & Fuzzy Systems, 32(3), 2083–2099. https://doi.org/10.3233/JIFS-161695
Thanh, N. D., Ali, M., & Son, L. H. (2017). A novel clustering algorithm on neutrosophic recommender system for medical diagnosis. Cognitive Computation, 9(4), 526–544. https://doi.org/10.1007/s12559-017-9462-8
Thong, N. T., Dat, L. Q., Son, L. H., Hoa, N. D., Ali, M., & Smarandache, F. (2019). Dynamic interval valued neutrosophic set: Modeling decision making in dynamic environments. Computers in Industry, 108, 45–52. https://doi.org/10.1016/j.compind.2019.02.009
Wang, H., Smarandache, F., Zhang, Y. Q., & Sunderraman, R. (2005). Interval neutrosophic sets and logic: Theory and applications in computing. Hexis.
Wang, H., Smarandache, F., Zhang, Y. Q., & Sunderraman, R. (2010). Single valued neutrosophic sets. Multispace and Multistructure, 4, 410–413.
Wu, X. H., Wang, J. Q., Peng, J. J., & Chen, X. H. (2016). Crossentropy and prioritized aggregation operator with simplified neutrosophic sets and their application in multi-criteria decision-making problems. International Journal of Fuzzy Systems, 18(6), 1104–1116. https://doi.org/10.1007/s40815-016-0180-2
Wu, Q., Wu, P., Zhou, L., Chen, H., & Guan, X. (2018). Some new Hamacher aggregation operators under single-valued neutrosophic 2-tuple linguistic environment and their applications to multiattribute group decision making. Computers & Industrial Engineering, 116, 144–162. https://doi.org/10.1016/j.cie.2017.12.024
Wu, Q., Zhou, L., Chen, Y., & Chen, H. (2019a). An integrated approach to green supplier selection based on the interval type-2 fuzzy best-worst and extended VIKOR methods. Information Sciences, 502, 394–417. https://doi.org/10.1016/j.ins.2019.06.049
Wu, Q., Lin, W., Zhou, L., Chen, Y., & Chen, H. (2019b). Enhancing multiple attribute group decision making flexibility based on information fusion technique and hesitant Pythagorean fuzzy sets. Computers & Industrial Engineering, 127, 954–970. https://doi.org/10.1016/j.cie.2018.11.029
Ye, J. (2014a). A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. Journal of Intelligent & Fuzzy Systems, 26, 2459–2466. https://doi.org/10.3233/IFS-130916
Ye, J. (2014b). Clustering methods using distance-based similarity measures of single-valued neutrosophic sets. Journal of Intelligent Systems, 23(4), 379–389. https://doi.org/10.1515/jisys-2013-0091
Ye, J. (2016). Fault diagnoses of steam turbine using the exponential similarity measure of neutrosophic numbers. Journal of Intelligent & Fuzzy Systems, 30, 1927–1934. https://doi.org/10.3233/IFS-151903
Ye, J. (2017). Single valued neutrosophic similarity measures based on cotangent function and their application in the fault diagnosis of steam turbine. Soft Computing, 21(3), 817–825. https://doi.org/10.1007/s00500-015-1818-y
Ye, J. (2018). Neutrosophic number linear programming method and its application under neutrosophic number environments. Soft Computing, 22(14), 4639–4646. https://doi.org/10.1007/s00500-017-2646-z
Ye, S., Fu, J., & Ye, J. (2015). Medical diagnosis using distancebased similarity measures of single valued neutrosophic multisets. Neutrosophic Sets and Systems, 7, 47–52.
Ye, J. & Fu, J. (2016). Multi-period medical diagnosis method using a single valued neutrosophic similarity measure based on tangent function. Computer Methods and Programs in Biomedicine, 123, 142–149. https://doi.org/10.1016/j.cmpb.2015.10.002
Ye, J., Chen, J. Q., Yong, R., & Du, S. G. (2017). Expression and analysis of joint roughness coefficient using neutrosophic number functions. Information, 8(2), 69. https://doi.org/10.3390/info8020069
Yong, R., Ye, J., & Du, S. G. (2019). A Dice similarity measure for TBM penetrability classification in hard rock condition with the intuitionistic fuzzy information of rock mass properties. European Journal of Environmental and Civil Engineering. https://doi.org/10.1080/19648189.2019.1643789
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338– 353. https://doi.org/10.1016/S0019-9958(65)90241-X
Zhang, H. Y., Wang, J. Q., & Chen, X. H. (2014). Interval neutrosophic sets and their application in multicriteria decision making problems. The Science World Journal, Article ID 645953. https://doi.org/10.1155/2014/645953
Zhou, L. P., Dong, J. Y., & Wan, S. P. (2019). Two new approaches for multi-attribute group decision-making with interval-valued neutrosophic Frank aggregation operators and incomplete weights. IEEE Access, 7, 102727–102750. https://doi.org/10.1109/ACCESS.2019.2927133
Atanassov, K., & Gargov, G. (1989). Interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems, 31, 343–349. https://doi.org/10.1016/0165-0114(89)90205-4
Ali, M., & Smarandache, F. (2016). Complex neutrosophic set. Neural Computing and Applications, 28, 1817–1834. https://doi.org/10.1007/s00521-015-2154-y
Ali, M., Deli, I., & Smarandache, F. (2016). The theory of neutrosophic cubic sets and their applications in pattern recognition. Journal of Intelligent & Fuzzy Systems, 30(4), 1957–1963. https://doi.org/10.3233/IFS-151906
Alia, M., Son, L. H., Thanhc, N. D., & Minh, N. V. (2018). A neutrosophic recommender system for medical diagnosis based on algebraic neutrosophic measures. Applied Soft Computing, 71, 1054–1071. https://doi.org/10.1016/j.asoc.2017.10.012
Broumi, S., & Deli, I. (2014). Correlation measure for neutrosophic refined sets and its application in medical diagnosis. Palestine Journal of Mathematics, 3(1), 11–19.
Broumi, S., & Smarandache, F. (2015). Interval neutrosophic rough sets. Neutrosophic Sets and Systems, 7, 23–31. https://doi.org/10.1155/2015/232919
Chen, J. Q., Ye, J., & Du, S. G. (2017). Vector similarity measures between refined simplified neutrosophic sets and their multiple attribute decision making method. Symmetry, 9(8), 153. https://doi.org/10.3390/sym9080153
Can, M. S., & Ozguven, O. F. (2017). PID tuning with neutrosophic similarity measure. International Journal of Fuzzy Systems, 19(2), 489–503. https://doi.org/10.1007/s40815-015-0136-y
Gal, A., Vladareanu, L., Smarandache, F., Yu, H., & Deng, M. (2012). Neutrosophic logic approaches applied to “rabot” real time control. Neutrosophic Theory and Its Applications, 1, 55–60.
Jun, Y. B., Smarandache, F., & Kim, C. S. (2017). Neutrosophic cubic sets. New Mathematics and Natural Computation, 13(1), 41–54. https://doi.org/10.1142/S1793005717500041
Köseoğlu, A., Şahin, R., & Merdan, M. (2019). A simplified neutrosophic multiplicative set‐based TODIM using water‐filling algorithm for the determination of weights. Expert Systems. https://doi.org/10.1111/exsy.12515
Liu, P. D., & Wang, Y. M. (2014). Multiple attribute decision making method based on single-valued neutrosophic normalized weighted Bonferroni mean. Neural Computing and Applications, 25(7–8), 2001–2010. https://doi.org/10.1007/s00521-014-1688-8
Liu, P. D., & Shi, L. L. (2015). The generalized hybrid weighted average operator based on interval neutrosophic hesitant set and its application to multiple attribute decision making. Neural Computing and Applications, 26(2), 457–471. https://doi.org/10.1007/s00521-014-1736-4
Liu, P. D., & Liu, X. (2018). The neutrosophic number generalized weighted power averaging operator and its application in multiple attribute group decision making. International Journal of Machine Learning and Cybernetics, 9(2), 347–358. https://doi.org/10.1007/s13042-016-0508-0
Liu, P., Khan, Q., & Mahmood, T. (2019). Group decision making based on power Heronian aggregation operators under neutrosophic cubic environment. Soft Computing, 24, 1971– 1997. https://doi.org/10.1007/s00500-019-04025-z
Maji, P. K. (2013). Neutrosophic soft set. Annals of Fuzzy Mathematics and Informatics, 5(1), 157–168.
Peng, J. J., Wang, J. Q., Wu, X. H., Wang, J., & Chen, X. H. (2015). Multivalued neutrosophic sets and power aggregation operators with their applications in multi-criteria group decisionmaking problems. International Journal of Computational Intelligence Systems, 8(2), 345–363. https://doi.org/10.1080/18756891.2015.1001957
Peng, J. J., Wang, J. Q. Wang, J., Zhang, H. Y., & Chen, X. H. (2016). Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems. International Journal of Systems Science, 47(10), 2342–2358. https://doi.org/10.1080/00207721.2014.994050
Read, J., & Stacey, P. (2009). Guidelines for open pit slope design. CSIRO Publishing. https://doi.org/10.1071/9780643101104
Smarandache, F. (1998). Neutrosophy: neutrosophic probability, set, and logic. American Research Press.
Smarandache, F. (2013a). n-Valued refined neutrosophic logic and its applications in physics. Progress in Physics, 4, 143–146.
Smarandache, F. (2013b). Introduction to neutrosophic measure, neutrosophic integral, and neutrosophic probability. Sitech & Education Publisher.
Smarandache, F. (2014). Introduction to neutrosophic statistics. Sitech & Education Publishing.
Şahin, R. (2018). Normal neutrosophic multiple attribute decision making based on generalized prioritized aggregation operators. Neural Computing & Applications, 30(10), 3095–3115. https://doi.org/10.1007/s00521-017-2896-9
Sahin, R. & Liu, P. D. (2017a). Possibility-induced simplified neutrosophic aggregation operators and their application to multicriteria group decision making. Journal of Experimental & Theoretical Artificial Intelligence, 29(4), 769–785. https://doi.org/10.1080/0952813X.2016.1259266
Sahin, R. & Liu, P. D. (2017b). Some approaches to multi criteria decision making based on exponential operations of simplified neutrosophic numbers. Journal of Intelligent & Fuzzy Systems, 32(3), 2083–2099. https://doi.org/10.3233/JIFS-161695
Thanh, N. D., Ali, M., & Son, L. H. (2017). A novel clustering algorithm on neutrosophic recommender system for medical diagnosis. Cognitive Computation, 9(4), 526–544. https://doi.org/10.1007/s12559-017-9462-8
Thong, N. T., Dat, L. Q., Son, L. H., Hoa, N. D., Ali, M., & Smarandache, F. (2019). Dynamic interval valued neutrosophic set: Modeling decision making in dynamic environments. Computers in Industry, 108, 45–52. https://doi.org/10.1016/j.compind.2019.02.009
Wang, H., Smarandache, F., Zhang, Y. Q., & Sunderraman, R. (2005). Interval neutrosophic sets and logic: Theory and applications in computing. Hexis.
Wang, H., Smarandache, F., Zhang, Y. Q., & Sunderraman, R. (2010). Single valued neutrosophic sets. Multispace and Multistructure, 4, 410–413.
Wu, X. H., Wang, J. Q., Peng, J. J., & Chen, X. H. (2016). Crossentropy and prioritized aggregation operator with simplified neutrosophic sets and their application in multi-criteria decision-making problems. International Journal of Fuzzy Systems, 18(6), 1104–1116. https://doi.org/10.1007/s40815-016-0180-2
Wu, Q., Wu, P., Zhou, L., Chen, H., & Guan, X. (2018). Some new Hamacher aggregation operators under single-valued neutrosophic 2-tuple linguistic environment and their applications to multiattribute group decision making. Computers & Industrial Engineering, 116, 144–162. https://doi.org/10.1016/j.cie.2017.12.024
Wu, Q., Zhou, L., Chen, Y., & Chen, H. (2019a). An integrated approach to green supplier selection based on the interval type-2 fuzzy best-worst and extended VIKOR methods. Information Sciences, 502, 394–417. https://doi.org/10.1016/j.ins.2019.06.049
Wu, Q., Lin, W., Zhou, L., Chen, Y., & Chen, H. (2019b). Enhancing multiple attribute group decision making flexibility based on information fusion technique and hesitant Pythagorean fuzzy sets. Computers & Industrial Engineering, 127, 954–970. https://doi.org/10.1016/j.cie.2018.11.029
Ye, J. (2014a). A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. Journal of Intelligent & Fuzzy Systems, 26, 2459–2466. https://doi.org/10.3233/IFS-130916
Ye, J. (2014b). Clustering methods using distance-based similarity measures of single-valued neutrosophic sets. Journal of Intelligent Systems, 23(4), 379–389. https://doi.org/10.1515/jisys-2013-0091
Ye, J. (2016). Fault diagnoses of steam turbine using the exponential similarity measure of neutrosophic numbers. Journal of Intelligent & Fuzzy Systems, 30, 1927–1934. https://doi.org/10.3233/IFS-151903
Ye, J. (2017). Single valued neutrosophic similarity measures based on cotangent function and their application in the fault diagnosis of steam turbine. Soft Computing, 21(3), 817–825. https://doi.org/10.1007/s00500-015-1818-y
Ye, J. (2018). Neutrosophic number linear programming method and its application under neutrosophic number environments. Soft Computing, 22(14), 4639–4646. https://doi.org/10.1007/s00500-017-2646-z
Ye, S., Fu, J., & Ye, J. (2015). Medical diagnosis using distancebased similarity measures of single valued neutrosophic multisets. Neutrosophic Sets and Systems, 7, 47–52.
Ye, J. & Fu, J. (2016). Multi-period medical diagnosis method using a single valued neutrosophic similarity measure based on tangent function. Computer Methods and Programs in Biomedicine, 123, 142–149. https://doi.org/10.1016/j.cmpb.2015.10.002
Ye, J., Chen, J. Q., Yong, R., & Du, S. G. (2017). Expression and analysis of joint roughness coefficient using neutrosophic number functions. Information, 8(2), 69. https://doi.org/10.3390/info8020069
Yong, R., Ye, J., & Du, S. G. (2019). A Dice similarity measure for TBM penetrability classification in hard rock condition with the intuitionistic fuzzy information of rock mass properties. European Journal of Environmental and Civil Engineering. https://doi.org/10.1080/19648189.2019.1643789
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338– 353. https://doi.org/10.1016/S0019-9958(65)90241-X
Zhang, H. Y., Wang, J. Q., & Chen, X. H. (2014). Interval neutrosophic sets and their application in multicriteria decision making problems. The Science World Journal, Article ID 645953. https://doi.org/10.1155/2014/645953
Zhou, L. P., Dong, J. Y., & Wan, S. P. (2019). Two new approaches for multi-attribute group decision-making with interval-valued neutrosophic Frank aggregation operators and incomplete weights. IEEE Access, 7, 102727–102750. https://doi.org/10.1109/ACCESS.2019.2927133