Three-valued soft set and its multi-criteria group decision making via TOPSIS and ELECTRE

Document Type : Research Article

Authors

1 Department of Accounting and Financial Management, Seydikemer School of Applied Sciences, Mugla Sitki Kocman University, Mugla, Turkey

2 Department of Mathematics, Faculty of Science and Arts, Yozgat Bozok University, Yozgat, Turkey

Abstract

The purpose of this paper is to introduce a generalization of Molodtsov's approach to soft sets obtained by passing from the classical two-valued logic underlying those sets to a three-valued logic, where the third truth value can usually be interpreted as either non-determined or unknown. This extension of soft set approach allows for more intuitive and clearer representation of various decision making problems involving incomplete or uncertain information. In other words, it is a useful way to analyze soft set based multi-criteria group decision making problems under the lack of information resulting from the inability to determine the data.
In this paper, we introduce the concept of three-valued soft set and its some basic operations and products. We propose the formulas to calculate all possible choice values for each object in the (weighted) three-valued soft sets, and thus calculate their respective decision values. By modifying the TOPSIS and ELECTRE methods to deal with multi-criteria group decision problems, three-valued soft set based decision making algorithms are constructed. To demonstrate the practicality of these algorithms, we address the outstanding examples adapted from the decision problems in real-life. Lastly, some aspects of the efficiency of the proposed algorithms are discussed with computational experiments.

Keywords

References

References:
1. Lukasiewicz, J. "Philosophische Bemerkungen zu mehrwertigen systemen des Aussagenkalkulls", Algebra C.R. Soc. Sci. Lett. Varsovie, 23, pp. 51-57 (1930).
2. Borowski, L. Ed., Selected Works of J. Lukasiewicz, North Holland Publishing Company, Amsterdam (1970).
3. Hajek, P., Metamathematics of Fuzzy Logic, Kluwer, Dordrecht (1998).
4. Kleene, S.C., Introduction to Metamathematics, North Holland Publishing Company, Amsterdam (1952).
5. Skolem, T.H. "A set theory based on a certain 3- valued logic", Mathematica Scandinavica, 8, pp. 127- 136 (1960).
6. Zadeh, L.A. "Fuzzy sets", Information and Control, 8, pp. 338-353 (1965).
7. Pawlak, Z. "Rough sets", International Journal of Computer and Information Sciences, 11, pp. 341-356 (1982).
8. Molodtsov, D. "Soft set theory-first results", Computers and Mathematics with Applications, 37, pp. 19-31 (1999).
9. Maji, P.K., Biswas, R., and Roy, A.R. "Soft set theory", Computers and Mathematics with Applications, 45, pp. 555-562 (2003).
10. Ali, M.I., Feng, F., Liu, X., et al. "On some new operations in soft set theory", Computers and Mathematics with Applications, 57, pp. 1547-1553 (2009).
11. Aygun, E. and Kamac, H. "Some generalized operations in soft set theory and their role in similarity and decision making", Journal of Intelligent and Fuzzy Systems, 36, pp. 6537-6547 (2019).
12. Sezgin, A. and Atagun, A.O. "On operations of soft sets", Computers and Mathematics with Applications, 61, pp. 1457-1467 (2011).
13. Kamaci, H. "Similarity measure for soft matrices and its applications", Journal of Intelligent and Fuzzy Systems, 36, pp. 3061-3072 (2019).
14. Zhu, P. and Wen, Q. "Operations of soft set revisited", Journal of Applied Mathematics, 2013, 7 pages (2013).
15. Gong, K., Xiao, Z., and Zhang, X. "The bijective soft set with its operations", Computers and Mathematics with Applications, 60, pp. 2270-2258 (2010).
16. Xiao, Z., Gong, K., Xia, S., et al. "Exclusive disjunctive soft sets", Computers and Mathematics with Applications, 59, pp. 2128-2137 (2010).
17. Karaaslan, F. and Karatas, S. "A new approach to bipolar soft sets and its applications", Discrete Mathematics, Algorithms and Applications, 7(4), 1550054 (2015).
18. Cetkin, V., Aygunoglu, A., and Aygun, H. "A new approach in handling soft decision making problems", Journal of Nonlinear Science and Applications, 9, pp. 231-239 (2016).
19. Khalil, A.M. and Hassan, N. "Inverse fuzzy soft set and its application in decision making", International Journal of Information and Decision Sciences, 11(1), pp. 73-92 (2019).
20. Abbasimehr, H. and Tarokh, M.J. "A novel interval type-2 fuzzy AHP-TOPSIS approach for ranking reviewers in online communities", Scientia Iranica, E, 23, pp. 2355-2373 (2016).
21. Adeel, A., Akram, M., and Koam, A.N.A. "Group decision-making based on m-polar hesitant fuzzy linguistic TOPSIS method", Symmetry, 11(6), 735 (2019).
22. Adeel, A., Akram, M., and Koam, A.N.A. "Multi- criteria decision-making under mHF ELECTRE-I and HmF ELECTRE-I", Energies, 12(9), 1661 (2019).
23. Akram, M. and Adeel, A. "TOPSIS approach for MAGDM based on interval-valued hesitant fuzzy N- soft environment", International Journal of Fuzzy Systems, 21(3), pp. 993-1009 (2019).
24. Akram, M., Adeel, A., and Alcantud, J.C.R. "Multi- criteria group decision making using an m-polar hesitant fuzzy TOPSIS approach", Symmetry, 11(6), 795 (2019).
25. Garg, H. and Kaur, G. "Extended TOPSIS method for multi-criteria group decision-making problems under cubic intuitionistic fuzzy environment", Scientia Iranica, E, 25, pp. 1-18 (2018).
26. Jasemi, M. and Ahmadi, E. "A new fuzzy ELECTRE based multiple criteria method for personnel selection", Scientia Iranica, E, 25, pp. 943-953 (2018).
27. Garg, H., Agarwal, N., and Tripathi, A. "Some improved interactive aggregation operators under interval-valued intuitionistic fuzzy environment and its application to decision making process", Scientia Iranica, E, 24, pp. 2581-2604 (2017).
28. Garg, H. and Kumar, K. "A novel correlation coefficient of intuitionistic fuzzy sets based on the connection number of set pair analysis and its application", Scientia Iranica, E, 25, pp. 2373-2388 (2018).
29. Gitinavard, G., Mousavi, S.M., Vahdani, B., et al. "A new distance-based decision model in interval-valued hesitant fuzzy setting for industrial selection problems", Scientia Iranica, E, 23, pp. 1928-1940 (2016).
30. Peng, X. "New operations for interval-valued Pythagorean fuzzy set", Scientia Iranica, E, 26, pp. 1049-1076 (2019).
31. Tang, J. and Meng, F. "An approach to interval-valued intuitionistic fuzzy decision making based on induced generalized symmetrical Choquet Shapley operator", Scientia Iranica, E, 25, pp. 1456-1470 (2018).
32. Maji, P.K., Roy, A.R., and Biswas, R. "An application of soft sets in a decision making problem", Computers and Mathematics with Applications, 44, pp. 1077-1083 (2002).
33. Arora, D. and Garg, H. "Robust aggregation operators for multi-criteria decision-making with intuitionistic fuzzy soft environment", Scientia Iranica, E, 25(2), pp. 931-942 (2018).
34. Arora, R. and Garg, H. "Prioritized averaging/ geometric aggregation operators under the intuitionistic fuzzy soft set environment", Scientia Iranica, E, 25, pp. 466-482 (2018).
35. Feng, F. and Li, Y. "Generalized uni-int decision making schemes based on choice value soft sets", European Journal of Operational Research, 220, pp. 162-170 (2012).
36. Han, B.-H. and Geng, S.-L. "Pruning method for optimal solutions of intm intn decision making", European Journal of Operational Research, 231, pp. 779-783 (2013).
37. Kamaci, H. "Selectivity analysis of parameters in soft set and its effect on decision making", International Journal of Machine Learning and Cybernetics, 11, pp. 313-324 (2020).
38. Eraslan, S. "A decision making method via TOPSIS on soft sets", Journal of New Results in Science, 8, pp. 57-71 (2015).
39. Avron, A. and Konikowska, B. "Rough sets and 3- valued logics", Studia Logica, 90(1), pp. 69-92 (2008).
40. Cagman, N. and Enginoglu, S. "Soft set theory and uni-int decision making", European Journal of Operational Research, 207, pp. 848-855 (2010).
41. D'ottaviano, I.M.L. "The completeness and compactness of a three-valued first-order logic", Revista Colombiana de Matematicas, 19, pp. 77-94 (1985).
42. Jacquette, D. "An internal determinacy metatheorem for Lukasiewicz's Aussagenkalkuls", Bulletin of the Section of Logic, 29(3), pp. 115-124 (2000).
43. Ciucci, D. and Dubois, D. "A map of dependencies among three-valued logics", Information Sciences, 250, 162-177 (2013).
44. Cai, M., Li, Q., and Lang, G. "Shadowed sets of dynamic fuzzy sets", Granular Computing, 2, pp. 85- 94 (2017).
45. Pedrycz, W. "From fuzzy sets to shadowed sets: Interpretation and computing", International Journal of Intelligent Systems, 24(1), pp. 48-61 (2009).
46. Vogel, P., Read, R.W., Hansen, G.M., et al. "Congenital hydrocephalus in genetically engineered mice", Veterinary Pathology, 49(1), pp. 166-181 (2012).
47. Shih, H.-S., Shyur, H.-J., and Lee, E.S. "An extension of TOPSIS for group decision making", Mathematical and Computer Modelling, 45, pp. 801-813 (2007).
48. Hwang, C.L. and Yoon, K., Multiple Attribute Decision Making: Methods and Applications, New York: Springer-Verlag (1981).
49. Milani, A.S., Shanian, A., and Madoliat, R. "The effect of normalization norms in multiple attribute decision making models: A case study in gear material selection", Structural Multidisciplinary Optimization, 29(4), pp. 312-318 (2005).
50. Shih, H.-S., Lin, W.-Y., and Lee, E.S. "Group decision making for TOPSIS", Joint 9th IFSA World Congress and 20th NAFIPS International Conference, IFSA/NAFIPS, Vancouver, Canada, pp. 2712-2717 (2001).
51. Yoon, K.P. and Hwang, C.L., Multiple Attribute Decision Making: An Introduction, Sage Publishing Thousand Oaks, CA (1995).
52. Akram, M., Ilyas, F., and Garg, H. "Multi-criteria group decision making based on ELECTRE I method in Pythagorean fuzzy information", Soft Computing, 24, pp. 3425-3453 (2020). https://doi.org/10.1007/s00500-019-04105-0.
53. Alper, D. and Basdar, C. "A comparison of TOPSIS and ELECTRE methods: An application on the factoring industry", Business and Economics Research Journal, 8(3), pp. 627-646 (2017).
54. Birgun, S. and Cihan, E. "Supplier selection process using ELECTRE method", IEEE 2010 International Conference on Intelligent Systems and Knowledge Engineering, Hangzhou, China, pp. 634-639 (2010). DOI: 10.1109/ISKE.2010.5680767.
55. Chen, T.-Y. "An IVIF-ELECTRE outranking method for multiple criteria decision-making with interval- valued intuitionistic fuzzy sets", Technological and Economic Development of Economy, 22(3), pp. 416- 452 (2016).
56. Devi, K. and Yadav, S.P. "A multicriteria intuitionistic fuzzy group decision making for plant location selection with ELECTRE method", The International Journal of Advanced Manufacturing Technology, 66, pp. 1219-1229 (2013).
57. Xu, J. and Shen, F. "A new outranking choice method for group decision making under Atanassov's intervalvalued intuitionistic fuzzy environment", Knowledge- Based Systems, 70, pp. 177-188 (2014).
58. Zhong, L. and Yao, L. "An ELECTRE I-based multicriteria group decision making method with interval type-2 fuzzy numbers and its application to supplier selection", Applied Soft Computing, 57, pp. 556-576 (2017).
59. Kharal, A. "Soft approximations and uni-int decision making", The Scientific World Journal, 2014, 7 pages (2014).
Scientia Iranica
Volume 28, Issue 6 - Serial Number 6
Transactions on Industrial Engineering (E)
November and December 2021
Pages 3719-3742

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History

  • Receive Date: 24 October 2019
  • Revise Date: 03 January 2020
  • Accept Date: 20 April 2020

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