Abstract
Neutrosophic set is a powerful general formal framework which generalizes the concepts of classic set, fuzzy set, interval-valued fuzzy set, intuitionistic fuzzy set, etc. Recent studies have developed systems with complex fuzzy sets, for better designing and modeling real-life applications. The single-valued complex neutrosophic set, which is an extended form of the single-valued complex fuzzy set and of the single-valued complex intuitionistic fuzzy set, presents difficulties to defining a crisp neutrosophic membership degree as in the single-valued neutrosophic set. Therefore, in this paper we propose a new notion, called interval complex neutrosophic set (ICNS), and examine its characteristics. Firstly, we define several set theoretic operations of ICNS, such as union, intersection and complement, and afterward the operational rules. Next, a decision-making procedure in ICNS and its applications to a green supplier selection are investigated. Numerical examples based on real dataset of Thuan Yen JSC, which is a small-size trading service and transportation company, illustrate the efficiency and the applicability of our approach.
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Abbreviations
- NS:
-
Neutrosophic set
- INS:
-
Interval neutrosophic set
- CFS:
-
Complex fuzzy set
- CIFS:
-
Complex intuitionistic fuzzy set
- IVCFS:
-
Interval-valued complex fuzzy set
- CNS:
-
Complex neutrosophic set
- ICNS:
-
Interval-valued complex neutrosophic set, or interval complex neutrosophic set
- SVCNS:
-
Single-valued complex neutrosophic set
- MCDM:
-
Multi-criteria decision-making
- MCGDM:
-
Multi-criteria group decision-making
- \(\vee\) :
-
Maximum operator (t-conorm)
- \(\wedge\) :
-
Minimum operator (t-norm)
References
Ali, M., Smarandache, F.: Complex neutrosophic set. Neural Comput. Appl. 28(7), 1817–1834 (2017)
Biswas, P., Pramanik, S., Giri, B.C.: TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Comput. Appl. 27(3), 727–737 (2016)
Can, M.S., Ozguven, O.F.: PID tuning with neutrosophic similarity measure. Int. J. Fuzzy Syst. 19(2), 489–503 (2017)
Greenfield, S., Chiclana, F., Dick, S.: Interval-valued complex fuzzy logic. In: IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 2014–2019. IEEE (2016)
Li, Y., Liu, P., Chen, Y.: Some single valued neutrosophic number heronian mean operators and their application in multiple attribute group decision-making. Informatica 27(1), 85–110 (2016)
Liu, P.: The aggregation operators based on archimedean t-Conorm and t-Norm for single-valued neutrosophic numbers and their application to decision-making. Int. J. Fuzzy Syst. 18(5), 849–863 (2016)
Liu, P., Tang, G.: Multi-criteria group decision-making based on interval neutrosophic uncertain linguistic variables and Choquet integral. Cognit. Comput. 8(6), 1036–1056 (2016)
Liu, P., Zhang, L., Liu, X., Wang, P.: Multi-valued neutrosophic number bonferronimean operators and their application in multiple attribute group decision-making. Int. J. Inf. Technol. Decis. Mak. 15(5), 1181–1210 (2016)
Ramot, D., Milo, R., Friedman, M., Kandel, A.: Complex fuzzy sets. IEEE Trans. Fuzzy Syst. 10(2), 171–186 (2002)
Ramot, D., Friedman, M., Langholz, G., Kandel, A.: Complex fuzzy logic. IEEE Trans. Fuzzy Syst. 11(4), 450–461 (2003)
Şahin, R., Liu, P.: Maximizing deviation method for neutrosophic multiple attribute decision-making with incomplete weight information. Neural Comput. Appl. 27(7), 2017–2029 (2016)
Smarandache, F.: Neutrosophy. neutrosophic probability, set, and logic, ProQuest information and learning. Ann Arbor, Michigan, USA, 105 p (1998)
Salleh, A.R.: Complex intuitionistic fuzzy sets. In: International Conference on Fundamental and Applied Sciences 2012, vol. 1482(1), pp. 464–470. (2012)
Sahin, R., Yiider, M.: A multi-criteria neutrosophic group decision-making method based TOPSIS for supplier selection. ar Xiv preprint arXiv:1412.5077 (2014)
Son, L.H., Tien, N.D.: Tune up fuzzy C-means for big data: some novel hybrid clustering algorithms based on initial selection and incremental clustering. Int. J. Fuzzy Syst. (2016). doi:10.1007/s40815-016-0260-3
Sahin, R., Yigider, M.: A multi-criteria neutrosophic group decision making metod based TOPSIS for supplier selection. CoRRabs/1412.5077 (2014)
Tian, Z.P., Zhang, H.Y., Wang, J., Wang, J.Q., Chen, X.H.: Multi-criteria decision-making method based on a cross-entropy with interval neutrosophic sets. Int. J. Syst. Sci. 47(15), 3598–3608 (2016)
Wang, H., Smarandache, F., Sunderraman, R., Zhang, Y.Q.: Interval Neutrosophic Sets and Logic: Theory and Applications in Computing, vol. 5. Hexis, Arizona (2005)
Wang, H., Smarandache, F., Zhang, Y., Sunderraman, R.: Single valued neutrosophic sets. Rev. Air Force Acad. 17, 10 (2010)
Ye, J.: Multi-criteria decision-making method using the correlation coefficient under single-valued neutrosophic environment. Int. J. Gen Syst 42(4), 386–394 (2013)
Ye, J.: Similarity measures between interval neutrosophic sets and their applications in multi-criteria decision-making. J. Intell. Fuzzy Syst. 26(1), 165–172 (2014)
Ye, J.: Single valued neutrosophic cross-entropy for multi-criteria decision-making problems. Appl. Math. Model. 38(3), 1170–1175 (2014)
Ye, J.: Vector similarity measures of simplified neutrosophic sets and their application in multi-criteria decision-making. Int. J. Fuzzy Syst. 16(2), 204–215 (2014)
Ye, J.: Multiple attribute group decision-making method with completely unknown weights based on similarity measures under single valued neutrosophic environment. J. Intell. Fuzzy Syst. 27(6), 2927–2935 (2014)
Ye, J., Zhang, Q.S.: Single valued neutrosophic similarity measures for multiple attribute decision-making. Neutrosophic Sets Syst. 2, 48–54 (2014)
Ye, J.: Interval neutrosophic multiple attribute decision-making method with credibility information. Int. J. Fuzzy Syst. 18(5), 914–923 (2016)
Zhang, G., Dillon, T.S., Cai, K.Y., Ma, J., Lu, J.: Operation properties and -equalities of complex fuzzy sets. Int. J. Approx. Reason. 50(8), 1227–1249 (2009)
Zhang, M., Liu, P., Shi, L.: An extended multiple attribute group decision-making TODIM method based on the neutrosophic numbers. J. Intell. Fuzzy Syst. 30(3), 1773–1781 (2016)
Acknowledgement
This research is funded by Graduate University of Science and Technology under grant number GUST.STS.ÐT2017-TT02. The authors are grateful for the support from the Institute of Information Technology, Vietnam Academy of Science and Technology. We received the necessary devices as experiment tools to implement proposed method.
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Ali, M., Dat, L.Q., Son, L.H. et al. Interval Complex Neutrosophic Set: Formulation and Applications in Decision-Making. Int. J. Fuzzy Syst. 20, 986–999 (2018). https://doi.org/10.1007/s40815-017-0380-4
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DOI: https://doi.org/10.1007/s40815-017-0380-4