Document Type : Article

**Authors**

School of Mathematics, Thapar Institute of Engineering & Technology (Deemed University), Patiala 147004, Punjab, India.

**Abstract**

The objective of this work is to present a novel multi-criteria group decision making (MCGDM) method under cubic intuitionistic fuzzy (CIF) environment by integrating extended TOPSIS method. In the existing studies, the uncertainties which are present in the data are handled either an interval-valued intuitionistic fuzzy sets (IVIFS) or an intuitionistic fuzzy set (IFS) information, which may lose some useful information of alternatives. On the other hand, CIF set (CIFS) handles the uncertainties by considering both the IVIFS and IFS instantaneously. Thus, motivated by this, in the present work, we presented some series of distance measures between the pairs of CIFSs and investigated their various relationship. Further, under this environment, a group decision-making method based on the proposed measure is presented by taking the different priority pairs of the decision makers. A practical example is provided to verify the developed approach and to demonstrate its practicality and feasibility, we compared their results with the several existing approaches results.

**Keywords**

- Cubic intuitionistic fuzzy sets
- IVIFS
- TOPSIS method
- distance measures
- closeness coefficients
- mutlicriteria group decision-making

**Main Subjects**

1. Garg, H. and Arora, R. Dual hesitant fuzzy soft aggregation operators and their application in decision making", Cognitive Computation, 10(5), pp. 769{789 (2018). 2. Garg, H. Some arithmetic operations on the generalized sigmoidal fuzzy numbers and its application", Granular Computing, 3(1), pp. 9{25 (2018). 3. Zadeh, L.A. Fuzzy sets", Information and Control, 8, pp. 338{353 (1965). 4. Atanassov, K.T. Intuitionistic fuzzy sets", Fuzzy Sets and Systems, 20, pp. 87{96 (1986). 5. Atanassov, K. and Gargov, G. Interval-valued intuitionistic fuzzy sets", Fuzzy Sets and Systems, 31, pp. 343{349 (1989). 6.

Arora, R. and Garg, H. Robust aggregation operators for multi-criteria decision making with intuitionistic fuzzy soft set environment", Scientia Iranica, E, 25(2), pp. 931{942 (2018). 7. Bagheri, M., Shojaei, P., and Khorami, M. A comparative survey of the condition of tourism infrastructure in Iranian provinces using VIKOR and TOPSIS", Decision Science Letters, 7(1), pp. 87{102 (2018). 8. Garg, H. and Ansha, Arithmetic operations on generalized parabolic fuzzy numbers and its application", Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 88(1), pp. 15{26 (2018).

9. Wang, X. and Triantaphyllou, E. Ranking irregularities when evaluating alternatives by using some ELECTRE methods", Omega - International Journal of Management Science, 36, pp. 45{63 (2008). 10. Peng, X.D. and Garg, H. Algorithms for intervalvalued fuzzy soft sets in emergency decision making based on WDBA and CODAS with new information measure", Computers & Industrial Engineering, 119, pp. 439{452 (2018). 11. Xu, Z.S. and Yager, R.R. Some geometric aggregation operators based on intuitionistic fuzzy sets", International Journal of General Systems, 35, pp. 417{433 (2006). 12. Peng, X.D. and Selvachandran, G. Pythagorean fuzzy set: state of the art and future directions", Arti_cial Intelligence Review, 52, pp. 1873{1927 (2019). 13. Arora, R. and Garg, H. Prioritized averaging/ geometric aggregation operators under the intuitionistic fuzzy soft set environment", Scientia Iranica, E, 25(1), pp. 466{482 (2018). 14. Peng, X.D. New operations for interval-valued Pythagorean fuzzy set", Scientia Iranica, E, 26(2), pp. 1049{1076 (2019). H. Garg and G. Kaur/Scientia Iranica, Transactions E: Industrial Engineering 27 (2020) 396{410 409 15. Garg, H. and Kumar, K. Some aggregation operators for linguistic intuitionistic fuzzy set and its application to group decision-making process using the set pair analysis", Arabian Journal for Science and Engineering, 43(6), pp. 3213{3227 (2018). 16. Garg, H. Linguistic Pythagorean fuzzy sets and its applications in multiattribute decision-making process", International Journal of Intelligent Systems, 33(6), pp. 1234{1263 (2018). 17. Hwang, C.L. and Yoon, K., Multiple Attribute Decision Making Methods and Applications A State-of-the-Art Survey, Springer-Verlag Berlin Heidelberg (1981). 18. Szmidt, E. and Kacprzyk, J. Distances between intuitionistic fuzzy sets", Fuzzy Sets and Systems, 114, pp. 505{518 (2000). 19. Hung, W.L. and Yang, M.S. Similarity measures of intuitionistic fuzzy sets based on hausdor_ distance", Pattern Recognition Letters, 25, pp. 1603{1611 (2004). 20. Boran, F.E., Gen_c S., and Akay, D. Personnel selection based on intuitionistic fuzzy sets", Human Factors and Ergonomics in Manufacturing & Service Industries, 21(5), pp. 493{503 (2011). 21. Dugenci, M. A new distance measure for interval valued intuitionistic fuzzy sets and its application to group decision making problems within complete weights information", Applied Soft Computing, 41, pp. 120{134 (2016). 22. Garg, H. A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems", Applied Soft Computing, 38, pp. 988{999 (2016). 23. Mohammadi, A., Shojaei, P., Kaydan, B., and Akbari, Z. Prioritizing the performance of civil development projects in governmental administration agencies, using gray relational analysis (GRA) and TOPSIS approach", Decision Science Letters, 5(4), pp. 487{498 (2016). 24. Garg, H., Agarwal, N., and Tripathi, A. Generalized intuitionistic fuzzy entropy measure of order _ and degree _ and its applications to multi-criteria decision making problem", International Journal of Fuzzy System Applications, 6(1), pp. 86{107 (2017). 25. Biswas, A. and Kumar, S. An integrated TOPSIS approach to MADM with interval-valued intuitionistic fuzzy settings", in: Advanced Computational and Communication Paradigms, Springer, pp. 533{543 (2018). 26. Vommi, V. TOPSIS with statistical distances: A new approach to MADM", Decision Science Letters, 6(1), pp. 49{66 (2017). 27. Singh, S. and Garg, H. Distance measures between type-2 intuitionistic fuzzy sets and their application to multicriteria decision-making process", Applied Intelligence, 46(4), pp. 788{799 (2017). 28. Li, D.F. TOPSIS- based nonlinear-programming methodology for multiattribute decision making with interval-valued intuitionistic fuzzy sets", IEEE Transactions on Fuzzy Systems, 18, pp. 299{311 (2010). 29. Garg, H. and Arora, R. A nonlinear-programming methodology for multi-attribute decision-making problem with interval-valued intuitionistic fuzzy soft sets information", Applied Intelligence, 48(8), pp. 2031{ 2046 (2018). 30. Lu, Z. and Ye, J. Logarithmic similarity measure between interval-valued fuzzy sets and its fault diagnosis method", Information, 9(2), p. 36 (2018). DOI: 10.3390/info9020036 31. Garg, H. and Kumar, K. An advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making", Soft Computing, 22(15), pp. 4959{ 4970 (2018). 32. Askarifar, K., Mota_ef, Z., and Aazaami, S. An investment development framework in Iran's seashores using TOPSIS and best-worst multi-criteria decision making methods", Decision Science Letters, 7(1), pp. 55{64 (2018). 33. Wang, C.Y. and Chen, S.-M. A new multiple attribute decision making method based on intervalvalued intuitionistic fuzzy sets, linear programming methodology, and the TOPSIS method", in: Advanced Computational Intelligence (ICACI), 2017 Ninth International Conference on, IEEE, pp. 260{263 (2017). 34. Gupta, P., Mehlawat, M.K., Grover, N., and Pedrycz, W. Multi-attribute group decision making based on extended TOPSIS method under interval-valued intuitionistic fuzzy environment", Applied Soft Computing, 69, pp. 554{567 (2018). 35. Kumar, K. and Garg, H. TOPSIS method based on the connection number of set pair analysis under interval-valued intuitionistic fuzzy set environment", Computational and Applied Mathematics, 37(2), pp. 1319{1329 (2018). 36. Kumar, K. and Garg, H. Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making", Applied Intelligence, 48(8), pp. 2112{2119 (2018). 37. Jun, Y.B., Kim, C.S., and Yang, K.O. Cubic sets", Annals of Fuzzy Mathematics and Informatics, 4(1), pp. 83{98 (2012). 38. Khan, M., Abdullah, S., Zeb, A., and Majid, A. Cubic aggregation operators", International Journal of Computer Science and Information Security, 14(8), pp. 670{682 (2016). 39. Mahmood, T., Mehmood, F., and Khan, Q. Some generalized aggregation operators for cubic hesitant fuzzy sets and their applications to multi criteria decision making", Journal of Mathematics, 49(1), pp. 31{49 (2017). 40. Fahmi, A., Abdullah, S., Amin, F., and Ali, A. Precursor selection for sol{gel synthesis of titanium carbide nanopowders by a new cubic fuzzy multiattribute group decision-making model", Journal of Intelligent Systems, 28(5), pp. 699{720 (2019). 410 H. Garg and G. Kaur/Scientia Iranica, Transactions E: Industrial Engineering 27 (2020) 396{410 41. Kaur, G. and Garg, H. Multi-attribute decisionmaking based on bonferroni mean operators under cubic intuitionistic fuzzy set environment", Entropy, 20(1), p. 65 (2018). DOI: 10.3390/e20010065 42. Kaur, G. and Garg, H. Cubic intuitionistic fuzzy aggregation operators", International Journal for Uncertainty Quantification, 8(5), pp. 405{428 (2018). 43. Garg, H. New exponential operational laws and their aggregation operators for interval-valued pythagorean fuzzy multicriteria decision-making", International Journal of Intelligent Systems, 33(3), pp. 653{683 (2018). 44. Garg, H. Hesitant Pythagorean fuzzy sets and their aggregation operators in multiple attribute decision making", International Journal for Uncertainty Quanti _cation, 8(3), pp. 267{289 (2018). 45. Garg, H. and Rani, D. Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision-making process", Arabian Journal for Science and Engineering, 44(3), pp. 2679{2698 (2019).

Transactions on Industrial Engineering (E)

January and February 2020Pages 396-410