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 this 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
AbstractThe 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.
In 1965, Zadeh  proposed the concept of a fuzzy set (FS) to depict uncertain information in decision-making problems. Atanassov  also presented the notion of an intuitionistic fuzzy set (IFS) (i.e., in which the elements of an IFS satisfy the following condition: where is a membership degree and is a non-membership degree). But there are shortcomings in intuitionistic fuzzy decision-making. For example, if a DM expresses that his/her support for membership of x is 0.9 and the support against membership of x is 0.7, then it can be found that the sum of membership and non-membership is bigger than 1. Hence, the ordered pair (0.9, 0.7) is not allowable for an IFS. To remedy this shortcoming, Yager  proposed the notion of a Pythagorean fuzzy set (PFS) (i.e., in which the elements of a PFS satisfy the following condition: where is a membership degree and is a non-membership degree). Therefore, a PFS has been widely applied to many fields, for example, multi-attribute decision-making  and multi-attribute group decision-making .
In 1999, Molodtsov  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.  proposed the notion of the fuzzy soft set, followed by studies on Pythagorean fuzzy soft sets , generalized Pythagorean fuzzy soft set , the possibility Pythagorean fuzzy soft set , and the possibility Pythagorean bipolar fuzzy soft sets . In addition, several expansion models of PFSS are very quickly developed, for example, the decision-making method related to PFSS with infectious diseases application , the novel entropy measure of PFSS , the parameter-reduction of PFSS and corresponding algorithms , the Q-PFS expert set and its application in the multi-criteria decision-making process , and the aggregation operators of PFSS with their application for green supplier chain management .
There are some shortcomings in the methods used to solve the decision-making problem by using the possibility fuzzy soft set  and the PFSS . 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.
Definition 2.1. (Cf. ). Suppose that be the set. The Pythagorean fuzzy set (PFS) is represented as
such that (i.e., the degree of membership) and (i.e., the degree of non-membership) satisfy the following condition
. The set of all Pythagorean fuzzy sets over is denoted by
Definition 2.2. (Cf. ). Let
Then, the subset, equal, union, intersection, and complement, are defined, respectively, as follows:
(1) if for all and .
(2) if and .
(1) (i.e., is the power set of ) is called soft set (SS) over .
(2) (i.e., is the collection of all fuzzy subsets of ) is called fuzzy soft set (FSS) over .
Definition 2.4. (Cf. ). Suppose that (i.e., be the set) and (i.e., be the set of parameters). For , then is called Pythagorean fuzzy soft set (PFSS or (PF)) over (or ) and we can write
for each (i.e., the degree of membership) and (i.e., the degree of non-membership) satisfies the condition .
Definition 2.5. (Cf. ). Let and Then,
(1) is a soft subset of denoted as if and is a Pythagorean fuzzy subset of .
(2) is a soft equal of denoted as if and .
(3) The intersection of and represented as is a PFSS where and for all
(4) The union of and represented as is a PFSS where
(5) The complement of is denoted by where is a mapping given by for all
(6) A PFSS over is known as a null PFSS represented as if for all where denote the null PFS,
(7) A PFSS over is known as an absolute PFSS represented as if for all where denote the absolute PFS,
In this section, we define the notion of generalized Pythagorean fuzzy soft sets as indicated below:
Definition 3.1. Suppose that (i.e., be the set) and (i.e., be the set of parameters). For an arbitrary , then (i.e., and ) is called generalized Pythagorean fuzzy soft set (for short, GPFSS or (GPF)) over the soft universe and we can write
for all and
Example 3.2. Let be three of elements and be four of parameters. The GPFSS where are defined as
Definition 3.3. Let (GPF) where . Then, is called GPFSS subset of is denote by if and , is fuzzy subset of and is PFS subset of .
Example 3.4. (Continued from Example 3.2). The GPFSS where are defined in the following:
Definition 3.5. Let (GPF) where . Then, is called GPFSS equal of is denote by if and .
The complement of a GPFSS is elaborated in the Definition .
Definition 3.6. Let (GPF) The complement of is defined as
where is defined in Definition 2.5 (4) and
Example 3.7. (Continued from Example 3.2). The complement of is computed as
Definition 3.8. (1) A null GPFSS over , denoted by is a GPFSS, is defined as
(2) An absolute GPFSS over , denoted by is a GPFSS, is defined as
Example 3.9. (Continued from Example 3.2). The null and absolute of GPFSSs are computed, respectively, as follows:
Proposition 3.10. Let and be the null and absolute of GPFSSs over , respectively. Then
Proof. Follows from Definitions 3.6 and 3.8.
Definition 3.11. Let (GPF) over . For , then
(1) The restricted union, denoted by is defined by
(2) The extended intersection, denoted by is defined by
Example 3.12. (Continued from Examples 3.2 and 3.4). By Definition 3.11, the restricted union and extended intersection are computed as
Proposition 3.13. Let (GPF) over . Then the following four properties hold:
Proof. Follows from Definition 3.11.
Proposition 3.14. Let (GPF) over . Then the following two properties hold:
Proof. Follows from Definition 3.11.
Based on the notion of GPFSSs and using the comparison tables  and the algorithm proposed by Dinda et al. , 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 and the parameter set , where stands for “modern”, stands for “international”, stands for “big”, stands for “beautiful”, stands for “full day”, and stands for “high efficiency”. Suppose Mr. Z wants to choose a good university for his daughter on the basis of his wishing parameters between those listed above. Our aim is to find out the most suitable university for his daughter. Consider the GPFSSs defined as follows:
Then, we define the following new GPFSSs (i.e., reduced the GPFSSs):
for all and Therefore, we get the new GPFSSs as
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 and ), and Table 7 (i.e., the final decision scores) as
Mr. Z will choose the university where the score of has a high value from Table 7.
From Table 8, we can see that the final results between (PF) , (PFS) , and our approach (i.e., (GPF)) are different. According to the ranking results of three alternatives in (GPF) is the most accurate and finable. This is due to (PF)  dealing with the (PF)  without fuzzy set. But our method (GPF) depends on the (PF)  with the fuzzy set. (PFS)  is the combination between a fuzzy set and fuzzy set, with fuzzy set and (GPF) is more general than (PFS) , which makes our presented method more reasonable and effective as shown in Fig. 1.
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  and coronary artery disease . 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.
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