TOPSIS Method Based on Entropy Measure for Solving Multiple-Attribute Group Decision-Making Problems with Spherical Fuzzy Soft Information

A spherical fuzzy soft set (SFSS) is a generalized soft set model, which is more sensible, practical, and exact. Being a very natural generalization, introducing uncertainty measures of SFSSs seems to be very important. In this paper, the concept of entropy, similarity, and distance measures are defned for the SFSSs and also, a characterization of spherical fuzzy soft entropy is proposed. Further, the relationship between entropy and similarity measures as well as entropy and distance measures are discussed in detail. As an application, an algorithm is proposed based on the improved technique for order preference by similarity to an ideal solution (TOPSIS) and the proposed entropy measure of SFSSs, to solve the multiple attribute group decision-making problems. Finally, an illustrative example is used to prove the efectiveness of the recommended algorithm.


Introduction
In many real-life situations, individuals have to face diferent kinds of uncertainty problems, and that is unavoidable.Zadeh's fuzzy set theory [1] is a very efective tool to deal with data having uncertainty or vagueness.Since it is very useful and efective, this theory has started to be used in a variety of scientifc felds and also achieved very good acceptance.Many generalizations of fuzzy set theory were developed by researchers such as intuitionistic fuzzy set theory [2], Pythagorean fuzzy set theory [3], picture fuzzy set theory [4], spherical fuzzy set theory [5], and so on.Most of the theories are efectively utilized in a lot of research areas of engineering, business, medicine, natural science, etc.
Te soft set theory developed by Molodtsov [6,7] is considered a general mathematical tool for dealing with uncertainty in a parametric manner.Again, Maji et al. [8] introduced fuzzy soft sets by combining the ideas of fuzzy sets and soft sets.Te fuzzy soft set model is a more generalized concept and is successfully used in a lot of decision-making problems.Many researchers are interested in this new concept, and they developed more generalized versions of the fuzzy soft sets such as generalized fuzzy soft sets [9], group generalized fuzzy soft sets [10], intuitionistic fuzzy soft sets [11], Pythagorean fuzzy soft sets [12], picture fuzzy soft sets [13], interval-valued picture fuzzy soft sets [14], spherical fuzzy soft sets [15], etc.
Te spherical fuzzy soft sets are a novel concept proposed recently by Perveen et al. [15].It is a combined version of spherical fuzzy sets and soft sets.Since it is a generalization of all other existing soft set models, it is more realistic and accurate.Terefore, in certain cases, this new extension can be applied more efectively and precisely in decisionmaking problems, than the other soft set models that are already existing.Tus in the current situation, fnding expressions for entropy measures, similarity measures, and distance measures of SFSSs and studying their properties have great importance.
Te entropy measure helps fnd the measure of fuzziness of objects.An object having less entropy will be more precise.Several researchers have conducted more studies on entropy measures, similarity measures, and distance measures.For example, the concept of entropy between fuzzy sets was initiated by Zadeh [1].Ten, De Luca and Termini [16] introduced entropy by making use of no probabilistic concepts for obtaining the measure of uncertainty.Again, using the distance measure between the membership function of fuzzy sets and its nearest crisp set, Kaufmann [17] defned another entropy measure of fuzzy sets.Higashi and Klir [18] also defned it with the help of the distance between fuzzy sets and their complements.Fan et al. [19] studied the relationship between fuzzy entropy and Hamming distance.Liu [20] defned the axiomatic defnitions of entropy, similarity measure, and distance measure of fuzzy sets.He also discussed some basic relations between them.Te connection between the similarity measure and entropy of intuitionistic fuzzy sets was investigated by Li et al. [21], and also they were given sufcient conditions to transform the entropy to the similarity measure of intuitionistic fuzzy sets and vice versa.In the case of soft sets and fuzzy soft sets, Majumdar and Samanta [22,23] introduced the uncertainty measures.Tey presented certain similarity measures of soft sets as well as fuzzy soft sets.Based on fuzzy equivalence, a new category of similarity measures and entropies of fuzzy soft sets were proposed by Liu et al. [24].Jiang et al. [25] defned a novel distance measure and entropy measure of intuitionistic fuzzy soft sets and proposed an expression to transform the structure of entropy of intuitionistic fuzzy soft sets to the interval-valued fuzzy soft sets.Later, Athira et al. [26] developed a characterization of the entropy of Pythagorean fuzzy soft sets, and also they proposed the expressions for Hamming distance and Euclidean distance of Pythagorean fuzzy soft sets.Also, a similarity measure of SFSSs is developed by Perveen et al. [27] and applied to a medical diagnosis problem.Recently, masum Raj et al. [28] defned cosine similarity, distance, and entropy measures for fuzzy soft matrices.
In modern society, multiple attribute group decisionmaking (MAGDM) problems are very common, which helps select an optimal solution from a fnite set of available alternatives.Te technique for order preference by similarity to an ideal solution (TOPSIS) was introduced by Hwang and Yoon [29].It is a method of multicriteria decision analysis.To solve the problems involving ranking alternatives, the TOPSIS method is very efective.Chen et al. [30] proposed an extended version of the TOPSIS method to solve a supplier selection problem in a fuzzy environment.Since this method is very fexible, the TOPSIS method is combined with diferent generalizations of fuzzy sets as well as fuzzy soft sets to deal with MAGDM problems, as we can see in [31][32][33].In recent times, a MAGDM method has been used by Qi Han et al. [34] based on Pythagorean fuzzy soft entropy and improved TOPSIS method.
In this paper, we introduce and explore the novel concepts of entropy measure of SFSSs.Tis marks the frst instance in the literature where explicit formulas for these concepts are defned and presented.Here, an axiomatic defnition of the entropy measure of SFSSs is defned, and the relationship between entropy and similarity measure as well as the relationship between entropy and distance measure are discussed.Our contribution enhances the existing body of knowledge in the realm of spherical fuzzy soft environments by laying the groundwork for comprehending the SFSS entropy measure concept through our newly introduced mathematical formulations Tis paper is structured as follows: in Section 2, some fundamental definitions regarding soft sets and fuzzy soft sets are recalled in Section 3, the axiomatic defnition of the entropy measure of SFSSs is defned, and some properties are discussed.In Section 4, the concept of similarity measure of SFSSs is defned and the relationships between entropy and similarity measure of SFSSs are proposed.Section 5 deals with the relationships between entropy and distance measures of SFSSs.In Section 6, an algorithm based on the MAGDM method is presented based on the improved TOPSIS method and the newly proposed entropy measure as an application and illustrated with a numerical example.In Section 7, to prove the accuracy of the suggested algorithm, a comparison with an existing algorithm is made.Finally, conclusions are drawn in Section 8.

Preliminaries
In this section, we include some basic defnitions and concepts, which are very helpful for further discussions.All over this paper, denote Z as the universal set and E as the parameter set which is related to the objects in Z and A ⊆ E.
Defnition 1 [6].Let Υ: A ⟶ P(Z) be a mapping from the parameter subset A into P(Z), the power set of Z. Ten, the pair 〈Υ, A〉 is said to be a soft set over Z.
Defnition 2 [8].Let Υ: A ⟶ F(Z) be a mapping, where F(Z) be the set of all fuzzy subsets over Z. Ten, the pair 〈Υ, A〉 is called a fuzzy soft over Z. Tat is, for each a ∈ A, Υ(a) is a fuzzy subset over Z.
Defnition 4 [15].Let SFS(Z) be the set of all spherical fuzzy sets over Z.A pair 〈Υ, A〉 is said to be a spherical fuzzy soft set (SFSS) over Z, where Υ is a mapping given by Υ: A ⟶ SFS(Z).
For each a ∈ A, Υ(a) is a spherical fuzzy set such that 2 Applied Computational Intelligence and Soft Computing where ( Defnition 5 [15].

Entropy Measure of SFSSs
Entropy measures the uncertainty associated with a system.Less entropy implies less uncertainty.Consequently, an element having less uncertainty or vagueness will carry more stable information.SFSSs being one of the most generalized versions of fuzzy soft sets, introducing entropy measures between SFSSs, are highly essential in both theoretical and practical scenarios.In this section, the defnition of the entropy of SFSSs is put forward along with some results and illustrated examples.
Now, Teorem 7 provides an expression to defne different spherical fuzzy soft entropies using the function Ψ that was defned above.

Theorem 7. Let E be a function defned from SFSS(Z) to [0,1] and let
where Ψ is the function that satisfes conditions (1) to ( 4) that were defned above, then E is an entropy measure of SFSS. Proof Terefore, E is an entropy measure of SFSS.

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Example 2. Consider, where ϵ(Υ, e j ) is the spherical fuzzy entropy given by Tat is, To prove E 1 (S), defned above is an entropy measure of SFSS, it is enough to prove that Ten, we get E 1 (S 1 ) = 0.7563 and E 1 (S 2 ) = 0.3441 By examining these two examples, the soundness of our proposed entropy measure for SFSS becomes evident.
Specifcally, when analyzing the SFSSs S 1 and S 2 , it becomes apparent that S 1 exhibits a greater degree of fuzziness compared to S 2 .Tis distinction arises from the fact that the disparity between the positive and negative membership degrees in S 1 is less pronounced than that observed in S 2 .

Entropy and Similarity Measure of SFSSs
A similarity measure is an essential tool for measuring similarities between two or more objects.Objects having high similarity measures are very close to each other.Tus, they have almost similar characteristics.In this section, we defne the similarity measure between SFSSs and propose the relationships between entropy and similarity measures of SFSSs.

Entropy and Distance Measure of SFSSs
We can defne distance measures from the entropy measure, a crucial aspect in the context of spherical fuzzy soft sets (SFSS).
Tis enhancement in our ability to accurately quantify dissimilarity holds particular signifcance in decision-making and pattern recognition.In many cases, similarity measures are inherently connected to distance measures, underscoring the importance of introducing distance measures between SFSSs.In this regard, we not only defne distance measures for SFSSs but also delve into the relationships between entropy measures and distance measures. , then the normalized Hamming distance and the normalized Euclidean distance between SFSSs S 1 and S 2 denoted by l(S 1 , S 2 ) and q(S 1 , S 2 ), respectively, are defned as follows: Example 9. Consider the SFSSs S 1 = 〈Υ, E〉 and S 2 = 〈Ω, E〉 in Example 3. Ten, we get l(S 1 , S 2 ) = 0.3337 and q (S 1 , S 2 ) = 0.3942.

Method to Transform the Entropy Measure into the
Distance Measure of SFSSs.Introducing a method to transform the entropy measure into the distance measure of SFSSs holds signifcant implications for enhancing dissimilarity quantifcation within fuzzy soft sets.Tis approach not only defnes distance measures for SFSSs but also elucidates their relationships with entropy measures, providing valuable insights into the structural nuances of these sets.

Application of the Entropy Measure of Spherical Fuzzy Soft Set Model Based on Multiple-Attribute Group Decision-Making Method
Using the MAGDM method based on the improved TOPSIS method and the novel spherical fuzzy soft entropy, we propose an algorithm for SFSSs.Tis new algorithm helps fnd the optimal object by ranking all objects in our universe.
For that, we frst discuss some fundamental concepts as follows.
be the SFSSs corresponding to each decision-makers D k , k = 1, 2, . .., p. Ten, the overall weighted average spherical fuzzy soft set is given by Entropy is the measure of fuzziness.In the case of parameters, a parameter with smaller entropy decreases uncertainty and evaluation becomes more accurate.Tus, more importance can be given to that parameter.Suppose that, in multiple attribute decision-making problems, decisionmakers give the subjective weight vectors ξ = ξ 1 ,  ξ 2 , . .., ξ n }.Ten, the objective weight and integrated weight of parameters are defned as follows: For each e j ∈ E, j = 1, 2, . .., n, the objective parameter weight ρ j is where, ϵ(Υ, e j ) is the spherical fuzzy entropy given in (5) Te integrated parameter weight vector c j for each e j , j = 1, 2, . .., n is Now, we are proposing a new algorithm for SFSSs based on the multiple-attribute decision-making method as follows.

Algorithm Based on Multiple-Attribute Group Decision
be the universal set and E = e 1 , e 2 , . .., e n   be the parameter set.Let D k , k = 1, 2, . .., p be the "p" decision-makers.Decisionmakers can choose parameters that are very much familiar to them.It is not necessary to give evaluation values to all the parameters because all the parameters are independent.

Comparison Analysis
Tis section seeks to compare the proposed algorithm with the already existing algorithm for SFSSs to prove the validity, reliability, and dependability of the novel algorithm.Here, we are considering the algorithm for SFSSs based on the group decision-making method and the extension of the TOPSIS (Technique of Order Preference by Similarity to an Ideal Solution) approach developed by Garg et al. [35].While examining Example 12 using this approach, we get  C 1 = 0.5041,  C 2 = 0.6434,  C 3 = 0.4059, and  C 4 = 0.4372.Tat is, the order relation between the alternatives is given by It can be seen that the order relation and the optimal solution for the previously known method are the same as the proposed algorithm based on the multiple-attribute group decision-making method.
Te following is a succinct summary of the benefts of the work illustrated in earlier sections: (i) Te entropy, distance, and similarity measures of SFSSs have a lot of applications in real-life situations including pattern recognition, group decisionmaking, image processing, medical diagnosis, etc. (ii) SFSS is one of the perfect generalized forms of fuzzy soft sets and it is certainly the more realistic, useful, and accurate.(iii) Introducing the entropy, distance, and similarity measures to SFSS appears to be crucial in both theoretical and practical contexts.(iv) Te proposed approach is more consistent and reliable for dealing with SFSS multi-attribute group decision-making problems.

Conclusions
In this paper, we introduced an axiomatic defnition of the entropy measure of SFSSs and proposed a characterization of spherical fuzzy soft entropy.Also, an expression for calculating the entropy measure of SFSSs has been put forward.Again, the notions of similarity measure and distance measure of SFSSs are defned, and the relationships between them with the entropy measure of SFSSs are studied in detail, which includes the transformations between entropy and similarity measure of SFSSs as well as the transformations between entropy and distance measure of SFSSs.For the application, we generalized the TOPSIS method to cope with MAGDM problems for SFSSs.Te newly proposed spherical fuzzy soft entropy measure is used to obtain the weights of the parameters.Finally, an algorithm based on the MAGDM method is presented and applied in a numerical example to show the usefulness of the proposed algorithm.More applications can be found out to solve the decision-making problems of SFSSs, and also, the algebraic and topological structures can be introduced for SFSSs as future work.
Example 1.Let Z = ζ 1 , ζ 2 , ζ 3  be the set of three motor bikes under consideration and let A = a 1 , a 2 , a 3 , a 4   be the set of parameters, a 1 � Good looking, a 2 � Cheap, a 3 � Quality of parts, and a 4 � Ride quality.Ten, the SFSS 〈Υ, A〉 describes the "user-friendly motor bikes" as given as follows: 〈Υ, A〉 and 〈Ω, B〉 be two SFSSs over Z, where A, B ⊆ E. 〈Υ, A〉 is called the spherical fuzzy soft subset of 〈Ω, B〉, denoted by 〈Υ, A〉 ⊆ 〈Ω, B〉, if

Table 3 :
Te weights of parameters.