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# A New Kind of Generalized Pythagorean Fuzzy Soft Set and Its Application in Decision-Making

1 College of Big Data, Huanghai University, Qingdao, China

2 Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, 71524, Egypt

* Corresponding Author: Ahmed Mostafa Khalil. Email:

(This article belongs to the Special Issue: Decision making Modeling, Methods and Applications of Advanced Fuzzy Theory in Engineering and Science)

*Computer Modeling in Engineering & Sciences* **2023**, *136*(3), 2861-2871. https://doi.org/10.32604/cmes.2023.026021

**Received** 10 August 2022; **Accepted** 18 November 2022; **Issue published** 09 March 2023

## Abstract

The aim of this paper is to introduce the concept of a generalized Pythagorean fuzzy soft set (GPFSS), which is a combination of the generalized fuzzy soft sets and Pythagorean fuzzy sets. Several of important operations of GPFSS including complement, restricted union, and extended intersection are discussed. The basic properties of GPFSS are presented. Further, an algorithm of GPFSSs is given to solve the fuzzy soft decision-making. Finally, a comparative analysis between the GPFSS approach and some existing approaches is provided to show their reliability over them.## Keywords

In 1965, Zadeh [1] proposed the concept of a fuzzy set (FS) to depict uncertain information in decision-making problems. Atanassov [2] also presented the notion of an intuitionistic fuzzy set (IFS) (i.e., in which the elements of an IFS satisfy the following condition:

In 1999, Molodtsov [6] presented the concept of a soft set (SS) to deal with uncertainties. Many researchers are developing new methods for SS. For example, Maji et al. [7,8] presented several concepts, operations, and examples of SS and gave an application to solve soft decision-making. Maji et al. [9] proposed the notion of the fuzzy soft set, followed by studies on Pythagorean fuzzy soft sets [10], generalized Pythagorean fuzzy soft set [11], the possibility Pythagorean fuzzy soft set [12], and the possibility Pythagorean bipolar fuzzy soft sets [13]. In addition, several expansion models of PFSS are very quickly developed, for example, the decision-making method related to PFSS with infectious diseases application [14], the novel entropy measure of PFSS [15], the parameter-reduction of PFSS and corresponding algorithms [16], the Q-PFS expert set and its application in the multi-criteria decision-making process [17], and the aggregation operators of PFSS with their application for green supplier chain management [18].

There are some shortcomings in the methods used to solve the decision-making problem by using the possibility fuzzy soft set [19] and the PFSS [10]. We will present the concept of generalized Pythagorean fuzzy soft sets (GPFSSs) as a combination of the two above-mentioned models. Furthermore, we study the properties and operations of GPFSSs. We also explore a MADM application under the GPFSS framework. In the end, we provide a comparative analysis between the developed hybrid model and some existing approaches.

This paper is structured as follows: In Section 2, we give several notions of Pythagorean fuzzy sets, soft sets, fuzzy soft sets, and Pythagorean fuzzy soft sets. In Section 3, we present the novel notion of GPFSSs and discuss their properties. In Section 4, we introduce an application of GPFSSs to solve fuzzy soft decision-making. In Section 5, we give a comparison between the proposed approach and some existing approaches. Finally, in Section 6, the conclusion is given.

We will present a short survey of five needed definitions in this paper as indicated below.

2.1 Pythagorean Fuzzy Sets, Soft Sets, and Fuzzy Soft Sets

Definition 2.1. (Cf. [3]). Suppose that

such that

Definition 2.2. (Cf. [3]). Let

and

Then, the subset, equal, union, intersection, and complement, are defined, respectively, as follows:

(1)

(2)

(3)

(4)

(5)

Definition 2.3. (Cf. [6,7,9]). Suppose that

(1)

(2)

2.2 Pythagorean Fuzzy Soft Sets

Definition 2.4. (Cf. [10]). Suppose that

for each

Definition 2.5. (Cf. [10]). Let

(1)

(2)

(3) The intersection of

(4) The union of

for all

(5) The complement of

(6) A PFSS

(7) A PFSS

3 Generalized Pythagorean Fuzzy Soft Sets

In this section, we define the notion of generalized Pythagorean fuzzy soft sets as indicated below:

Definition 3.1. Suppose that

for all

Example 3.2. Let

Definition 3.3. Let

Example 3.4. (Continued from Example 3.2). The GPFSS

Thus,

Definition 3.5. Let

The complement of a GPFSS is elaborated in the Definition

Definition 3.6. Let

where

Example 3.7. (Continued from Example 3.2). The complement

Definition 3.8. (1) A null GPFSS over

(2) An absolute GPFSS over

Example 3.9. (Continued from Example 3.2). The null and absolute of GPFSSs are computed, respectively, as follows:

and

Proposition 3.10. Let

(1)

(2)

(3)

Proof. Follows from Definitions 3.6 and 3.8.

Definition 3.11. Let

(1) The restricted union, denoted by

where

(2) The extended intersection, denoted by

where

Example 3.12. (Continued from Examples 3.2 and 3.4). By Definition 3.11, the restricted union and extended intersection are computed as

and

Proposition 3.13. Let

(1)

(2)

(3)

(4)

Proof. Follows from Definition 3.11.

Proposition 3.14. Let

(1)

(2)

Proof. Follows from Definition 3.11.

4 An Application of GPFSSs to Solve Fuzzy Soft Decision-Making Problems

Based on the notion of GPFSSs and using the comparison tables [20] and the algorithm proposed by Dinda et al. [21], we will give an application of GPFSSs to solve fuzzy soft decision-making problems as indicated below.

Example 4.1. Assume that there are three different universities in universe

Then, we define the following new GPFSSs (i.e., reduced the GPFSSs):

for all

After then, we compute the following in Table 1 (i.e, the reduced membership), Table 4 (i.e., the reduced non-membership), Tables 2 and 5 (i.e., the comparison tables), Tables 3 and 6 (i.e., the comparison scores

Mr. Z will choose the university

In this section, we provide a comparison between (PF)

From Table 8, we can see that the final results between (PF)

We have given the novel model of generalized Pythagorean fuzzy soft sets. We have presented their operations and properties. We have presented an application of GPFSSs in fuzzy soft decision-making. In the future, we will provide a real application with a real data set for lung cancer disease [22] and coronary artery disease [23]. Finally, we will discuss more future studies on the GPFSS information to deal with decision-making problems (for example, [4,5,24,25]).

Acknowledgement: The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

## References

1. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X [Google Scholar] [CrossRef]

2. Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3 [Google Scholar] [CrossRef]

3. Yager, R. R. (2013). Pythagorean fuzzy subsets. IFSA World Congress and NAFIPS Annual Meeting 2013 Joint, pp. 57–61. Edmonton, Canada, IEEE. [Google Scholar]

4. Wan, S. P., Li, S. Q., Dong, J. Y. (2018). A three-phase method for Pythagorean fuzzy multi-attribute group decision making and application to haze management. Computers & Industrial Engineering, 123(1), 348–363. https://doi.org/10.1016/j.cie.2018.07.005 [Google Scholar] [CrossRef]

5. Wan, S. P., Jin, Z., Dong, J. Y. (2020). A new order relation for Pythagorean fuzzy numbers and application to multi-attribute group decision making. Knowledge and Information Systems, 62(2), 751–785. https://doi.org/10.1007/s10115-019-01369-8 [Google Scholar] [CrossRef]

6. Molodtsov, D. (1999). Soft set theory-first results. Computers & Mathematics with Applications, 37(4–5), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5 [Google Scholar] [CrossRef]

7. Maji, P. K., Biswas, R., Roy, A. R. (2003). Soft set theory. Computers and Mathematics with Applications, 45(4–5), 555–562. https://doi.org/10.1016/S0898-1221(03)00016-6 [Google Scholar] [CrossRef]

8. Maji, P. K., Roy, A. R., Biswas, R. (2002). An application of soft sets in a decision making problem. Computers and Mathematics with Applications, 44(8–9), 1077–1083. https://doi.org/10.1016/S0898-1221(02)00216-X [Google Scholar] [CrossRef]

9. Maji, P. K., Biswas, R., Roy, A. R. (2001). Fuzzy soft sets. The Journal of Fuzzy Mathematics, 9(3), 589–602. [Google Scholar]

10. Peng, X., Yang, Y., Song, J. (2015). Pythagorean fuzzy soft set and its application. Computer Engineering, 41(7), 224–229. [Google Scholar]

11. Kirişci, M., Demir, I., Şimşek, N., Topaç, N., Bardak, M. (2022). The novel VIKOR methods for generalized Pythagorean fuzzy soft sets and its application to children of early childhood in COVID-19 quarantine. Neural Computing and Applications, 34(3), 1877–1903. https://doi.org/10.1007/s00521-021-06427-3 [Google Scholar] [PubMed] [CrossRef]

12. Hu, D. J., Zhang, H., Heb, Y. (2019). Possibility Pythagorean fuzzy soft set and its application. Journal of Intelligent & Fuzzy Systems, 36(1), 413–421. https://doi.org/10.3233/JIFS-181649 [Google Scholar] [CrossRef]

13. Palanikumar, M., Arulmozhi, K. (2021). Possibility Pythagorean bipolar fuzzy soft sets and its application. Open Journal of Discrete Applied Mathematics, 4(2), 17–29. https://doi.org/10.30538/psrp-odam2021.0056 [Google Scholar] [CrossRef]

14. Kirişci, M., Şimşek, N. (2022). Decision making method related to Pythagorean fuzzy soft Sets with infectious diseases application. Journal of King Saud University-Computer and Information Sciences, 34(8), 5968–5978. https://doi.org/10.1016/j.jksuci.2021.08.010 [Google Scholar] [CrossRef]

15. Athira, T. M., John, S. J., Garg, H. (2020). A novel entropy measure of Pythagorean fuzzy soft sets. AIMS Mathematics, 5(2), 1050–1061. https://doi.org/10.3934/math.2020073 [Google Scholar] [CrossRef]

16. Athira, T. M., John, S. J. (2021). Parameter-reduction of Pythagorean fuzzy soft sets and corresponding algorithms. AIP Conference Proceedings, 2336(1), 040017. https://doi.org/10.1063/5.0045746 [Google Scholar] [CrossRef]

17. Chinnadurai, V., Arulselvam, A. (2021). Q-Pythagorean fuzzy soft expert set and its application in multi-criteria decision making process. Journal of Physics: Conference Series, 1850(1), 012114. https://doi.org/10.1088/1742-6596/1850/1/012114 [Google Scholar] [CrossRef]

18. Zulqarnain, R. M., Xin, X. L., Garg, H., Khan, W. A. (2021). Aggregation operators of Pythagorean fuzzy soft sets with their application for green supplier chain management. Journal of Intelligent & Fuzzy Systems, 40(3), 5545–5563. https://doi.org/10.3233/JIFS-202781 [Google Scholar] [CrossRef]

19. Alkhazaleh, S., Salleh, A. R., Hassan, N. (2011). Possibility fuzzy soft set. Advances in Decision Sciences, 2011(3), 479756. https://doi.org/10.1155/2011/479756 [Google Scholar] [CrossRef]

20. Maji, P. K., Roy, A. R. (2007). A fuzzy soft set theoretic approach to decision making problems. Journal of Computational and Applied Mathematics, 203(2), 412–418. https://doi.org/10.1016/j.cam.2006.04.008 [Google Scholar] [CrossRef]

21. Dinda, B., Bera, T., Samanta, T. K. Generalised intuitionistic fuzzy soft sets and its application in decision making (in Press). http://arxiv.org/abs/1010.2468 [Google Scholar]

22. Khalil, A. M., Li, S. G., Lin, Y., Li, H. X., Ma, S. (2020). A new expert system in prediction of lung cancer disease based on fuzzy soft sets. Soft Computing, 24(18), 14179–14207. https://doi.org/10.1007/s00500-020-04787-x [Google Scholar] [CrossRef]

23. Hassan, N., Sayed, O. R., Khalil, A. M., Ghany, M. A. (2017). Fuzzy soft expert system in prediction of coronary artery disease. International Journal of Fuzzy Systems, 19(5), 1546–1559. https://doi.org/10.1007/s40815-016-0255-0 [Google Scholar] [CrossRef]

24. Yue, Q. (2022). Bilateral matching decision-making for knowledge innovation management considering matching willingness in an interval intuitionistic fuzzy set environment. Journal of Innovation & Knowledge, 7(3), 100209. https://doi.org/10.1016/j.jik.2022.100209 [Google Scholar] [CrossRef]

25. Zulqarnain, R. M., Siddique, I., Iampan, A., Baleanu, D. (2022). Aggregation operators for interval-valued Pythagorean fuzzy so set with their application to solve multi-attribute group decision making problem. Computer Modeling in Engineering & Sciences, 131(3), 1717–1750. https://doi.org/10.32604/cmes.2022.019408 [Google Scholar] [CrossRef]

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*Computer Modeling in Engineering & Sciences*,

*136*

*(3)*, 2861-2871. https://doi.org/10.32604/cmes.2023.026021

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*Comput. Model. Eng. Sci.*, vol. 136, no. 3, pp. 2861-2871. 2023. https://doi.org/10.32604/cmes.2023.026021

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