An extended TOPSIS and entropy measure based on Sugeno integral in Pythagorean fuzzy set setting

1 Introduction

The notion of fuzzy set (FS) was introduced by Zadeh [1] as an extension of a characteristic function of a classical set, and a FS is characterized by a membership function defined on a universal set to the closed interval [0, 1] instead of a characteristic function that only takes values in {0, 1}. In 1986, the notion of FS was expanded to the notion of intuitionistic fuzzy set (IFS) by Atanassov [2] via a membership function with a non-membership function such that the sum of these functions is less than or equal to one. However, in real life applications, some data cannot always be represented by a FS or an IFS. For example, if the membership degree and the membership degree of an element are given with 0.6 and 0.7, respectively, then the sum of these degrees is equal to 1.3 which is larger than 1 and so this case cannot be characterized with a FS or an IFS. With this motivation, Yager [3] presented the notion of Pythagorean fuzzy set (PFS) that is represented by a membership function with a non-membership function as well such that the sum of the squares of these functions is less than or equal to 1. For instance, in the example above, we obtain that 0 . 6² + 0 .7² ≤ 1. Therefore, a PFS is more useful than an IFS as well as a FS. Figure 1 illustrates this fact. Further studies on FS theory and applications may be found in [4–9].

OPEN IN VIEWER

Fig. 1

Comparison of IFSs and PFSs.

The concept of entropy is a crucial measurement method of the information theory and it was proposed by Shannon [10, 11]. This concept is related to the quantity of information and it measures the uncertainty. Zadeh [12] extended (Shannon’s) entropy to the concept of fuzzy entropy. The concept of fuzzy entropy is a fuzzy information measure that measures the vagueness of a fuzzy event or a FS. Since Zadeh introduced the concept of fuzzy entropy, varied kinds of entropy have been introduced in different FS environments. For instance, De Luca and Termini [13] provided a fuzzy entropy that is motivated by Shannon’s entropy. Zhang et al. [14] gave some new entropy measures via distance measures for interval-valued intuitionistic fuzzy sets. Cui and Ye [15] proposed a root entropy for simplified neutrosophic sets. Several entropy measures have been proposed to evaluate the degree of uncertainty of PFSs [16–19]. However, many of these measures are limited in their ability to accurately capture the complex interaction between the different criteria involved in real life applications. In this paper, we propose a new entropy measure for PFSs that utilizes fuzzy measures to model the interaction between criteria. This measure provides a more comprehensive and accurate representation of uncertainty for PFSs, compared to existing measures.

The notion of similarity is one of the most useful notions of classical set theory and statistics. This notion is carried in the fuzzy environment by Wang [20]. A similarity measure provides a formula to determine the similarity among two FSs and is a useful tool that assigns a degree to the similarity of these sets. Several versions of the notion of the similarity measure have been applied to vary fields such as pattern recognition, medical diagnosis, multi-criteria decision making (MCDM) and face recognition systems (see e.g., [21–27]). MCDM helps decision makers to determine an optimal alternative that has the highest grade of satisfaction from a set of possible alternatives. The relationship between PFSs and rough sets has been explored in the context of similarity measures, particularly in the context of multi-granulation probabilistic models. Adjustable multi-granulation Pythagorean fuzzy probabilistic rough sets have been proposed as a tool for handling decision making problems in uncertain environments, allowing for a flexible and efficient analysis of complex decision-making processes [28–30].

Aggregation operators in various fuzzy environments are frequently used to relieve the complicated decision process. In aggregation operators, fuzzy integrals are known as powerful and flexible functions. The most well-known fuzzy integrals are the Choquet integral [31] and the Sugeno [32] integral. Fuzzy integrals evaluate the data supplied by information sources according to a fuzzy measure. Fuzzy measures and fuzzy integrals are studied mostly from a mathematical point of view, particularly in the MCDM field.

The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a method that finds an alternative that has the farthest distance from the ideal worst solution and the shortest distance from the ideal best solution [33]. This method has been frequently used to solve MCDM problems in several fuzzy environments. For example, Ashtiani et al. [34] proposed an interval-valued fuzzy TOPSIS, which was presented for solving MCDM problems in which the weights of criteria are unequal. Gündogdu and Kahraman [35] introduced an interval-valued spherical fuzzy TOPSIS to solve a MCDM problem. Akram et al. [36] studied an extension of the TOPSIS to model decision making under complex spherical fuzzy information. Garg et al. [37] proposed a new TOPSIS based on the complex interval-valued q-rung orthopair fuzzy set. Wang et al. [38] have introduced a TOPSIS for interval-valued q-rung dual hesitant fuzzy sets. Huang et al. [39] have presented aggregation operators for spherical fuzzy rough sets, and they have used them to propose a new TOPSIS algorithm in a spherical fuzzy rough environment. Moreover, Zhang and Xu [40] have developed a Pythagorean fuzzy TOPSIS approach to solve the MCDM problems with PFSs, and Yücesan and Gül [41] have used TOPSIS in the Pythagorean fuzzy environment for hospital service quality evaluation. Further applications of TOPSIS for PFSs can be found in [42–44]. In this paper, we propose an extension of the TOPSIS for Pythagorean fuzzy decision-making problems. The TOPSIS method is widely used in decision making problems due to its simplicity, intuitiveness, and ability to provide a ranking of alternatives based on a set of criteria. While there are other methods available for decision-making under uncertainty, TOPSIS is chosen as the base method for our extension due to its widespread use, flexibility, and ability to handle the Pythagorean fuzzy decision making problems.

In this paper, we give a new entropy measure for PFSs via the Sugeno integral. We also prove a theorem to obtain a similarity measure when an entropy measure is provided. Using this new entropy, we define a similarity measure for PFSs and we use this similarity measure to propose an extended TOPSIS in a Pythagorean fuzzy environment. With this respect, the proposed extended TOPSIS is more sensitive than the existing ones since it uses the Sugeno integral, which better considers the interaction among criteria thanks to fuzzy measures. The main contributions of the present study can be listed as follows.

We propose a new entropy measure for PFSs by employing the Sugeno integral. This new entropy measure is more sensitive than the ones in the literature defined via classical aggregations instead of a fuzzy integral thanks to the fuzzy measures.

The rest of this correspondence is organized as follows: In Section 2, we recall some fundamental information. In Section 3, we define a new Pythagorean fuzzy entropy measure. In Section 4, we prove a theorem to obtain a similarity measure from an entropy measure and develop a new similarity measure for PFSs that relying on entropy. In section 5, we provide an extended TOPSIS with the help similarity measures and apply this method to a real life MCDM application from the literature. In Section 6, we conclude the paper.

In this section, we recall some fundamental concepts of FS theory and fuzzy measure theory. Throughout this paper, we assume that X = {x₁, . . . , x _n } is a finite universal set.

Definition 2.1. [1] A FS A in X is given by A = { ( x , μ A ( x ) ) : x ∈ X ) } where μ _A  : X → [0, 1] is a function that is called the membership function of A. For any x ∈ X, μ _A  (x) is called the degree of membership of the element x to FS A.

Definition 2.2. [2] An IFS A in X is given by A = { ( x , μ A ( x ) , ν A ( x ) ) : x ∈ X ) } where μ _A  : X → [0, 1] and ν _A  : X → [0, 1] are functions such that μ _A  (x) + ν _A  (x) ≤1 for any x ∈ X.

Definition 2.3. [3, 45, 46] A PFS A in X is given by A = { ( x , μ A ( x ) , ν A ( x ) ) : x ∈ X ) } where μ _A  : X → [0, 1] and ν _A  : X → [0, 1] are functions such that μ A 2 ( x ) + ν A 2 ( x ) ≤ 1 for any x ∈ X. For a fixed x ∈ X, (x, μ _A  (x) , ν _A  (x)) is called a Pythagorean fuzzy value (PFV). A PFS can be considered as a collection of PFVs, and a PFV is usually represented by a pair (μ, ν) such that μ, ν ∈ [0, 1] and μ² + ν² ≤ 1.

In Definitions 2.2 and 2.3, the functions μ _A and ν _A are called the membership function and non-membership function of A, respectively. For any x ∈ X, μ _A  (x) and ν _A  (x) are called the degree of membership and the degree of non-membership of the element x to IFS/PFS A, respectively. We denote the set of all PFSs of X by PFS (X). Now, we recall some set operations for PFSs.

Definition 2.4. [3, 45, 46] Let A, B ∈ PFS (X). Then

(i) the complement A ^c of A is given by A ^c  = {(x, ν _A  (x)) , μ _A  (x)) |x ∈ X},

(ii) A ⊂ B if μ _A  (x)) ≤ μ _B  (x) and ν _A  (x) ≥ ν _B  (x), for all x ∈ X,

(iii) A∪ B = { (x, max(μ _A  (x) , μ _B  (x)) ,   min(ν _A  (x) ,     ν _B  (x))) : x ∈ X },

(iv) A∩ B = { (x, min(μ _A  (x) , μ _B  (x)) , max(ν _A  (x) ,     ν _B  (x))) : x ∈ X }.

Now, we recall the concepts of fuzzy measure and the Sugeno integral. We use these concepts to define an entropy measure for PFSs.

Definition 2.5. [32] A set function σ : P (X) → [0, 1] is said to be a fuzzy measure on X if it satisfies

(i) σ (∅) =0,

(ii) σ (X) =1,

(iii) A ⊂ B implies σ (A) ≤ σ (B) (monotonicity)

where P (X) is the family of all ordinary subsets of X.

Definition 2.6. [32] Let σ be a fuzzy measure over X. Then, the Sugeno integral of a function f : X → [0, 1] is defined by ∫ X fd σ : = ⋁ k = 1 n ( f ( x ( k ) ) ∧ σ ( F ( k ) ) ) where { x ( k ) } k = 0 n is a permutation of X such that 0 = f (x₍₀₎) ≤ f (x₍₁₎) ≤ . . . ≤ f (x_(n)) ≤1 and F _k  : = {x_(k), . . . , x_(n)} .

Note that the Sugeno integral is a generalization of the weighted maximum and weighted minimum which provides a more sensitive analysis in decision making.

3 A Pythagorean fuzzy entropy

Using the (weighted) arithmetic mean Hung and Yang [47], Thao and Smarandache [17] and Thao [48] defined entropy measures for IFSs, PFSs, and picture fuzzy sets, respectively. In this section, instead of the weighted arithmetic mean by using the Sugeno integral, which is a non-linear extension of the (ordered) weighted maximum and (ordered) weighted minimum, we propose a new type of entropy measure for PFSs. In this manner, this new entropy measure takes the preferences of decision makers into account more sensitively thanks to fuzzy measures that consider the interaction among criteria.

We now recall the general definition of the entropy measure for PFSs and the entropy measure given by Thao and Smarandache [17] before introducing the new entropy measure.

Definition 3.1. Let A, B ∈ PFS (X). An entropy measure 𝔼 : PFS ( X ) → [ 0 , 1 ] is a mapping that satisfies

(E1) 𝔼 ( A ) = 0 if A is a crisp set, that is, μ _A  (x) =1 ν _A  (x) =0 or μ _A  (x _i ) =0 ν _A  (x _i ) =1 for any x ∈ X,

(E2) 𝔼 ( A ) = 1 if μ A ( x ) = ν A ( x ) = 1 3 for any x ∈ X,

(E3) 𝔼 ( A ) ≤ 𝔼 ( B ) if A is less fuzzy than B, i.e., if μ A 2 ( x ) ≤ μ B 2 ( x ) ≤ 1 3 , ν A 2 ( x ) ≤ ν B 2 ( x ) ≤ 1 3 or μ A 2 ( x ) ≥ μ B 2 ( x ) ≥ 1 3 , ν A 2 ( x ) ≥ ν B 2 ( x ) ≥ 1 3 for any x ∈ X,

(E4) 𝔼 ( A c ) = 𝔼 ( A ) (see, e.g., [17]).

The function 𝔼 : PFS ( X ) → [ 0 , 1 ] defined by 𝔼 ( A ) = 1 n ∑ i = 1 n E A ( x i ) is an entropy measure for PFSs [17] where E A ( x i ) = 1 - | μ A 2 ( x i ) - 1 3 | - | ν A 2 ( x i ) - 1 3 | for any i = 1, . . . , n .

Using function E _A and the Sugeno integral, we define a new entropy measure for PFSs.

Definition 3.2. Let σ be a fuzzy measure on X. Then we define 𝔼 ( S , σ ) ( A ) : = ∫ X E A d σ = ⋁ i = 1 n ( E A ( x ( i ) ) ∧ σ ( F ( i ) ) ) .(1)

The following theorem shows that the function given in (1) is an entropy measure for PFSs.

Theorem 3.1. Let σ be a fuzzy measure on X. The function 𝔼 ( S , σ ) is an entropy measure on PFS (X).

Proof. (E1) If A is a crisp set, then we get E A ( x i ) = 1 - | μ A 2 ( x i ) - 1 3 | - | ν A 2 ( x i ) - 1 3 | = 0 for any i = 1, . . . , n. Therefore 𝔼 ( S , σ ) ( A ) = 0 .

(E2) If for any x ∈ X, μ A ( x ) = ν A ( x ) = 1 3 , then we obtain

E A ( x i ) = 1 - | μ A 2 ( x i ) - 1 3 | - | ν A 2 ( x i ) - 1 3 | = 1 for any i = 1, . . . , n. So we have 𝔼 ( S , σ ) ( A ) = ⋁ i = 1 n ( 1 ∧ σ ( F ( i ) ) ) = ⋁ i = 1 n σ ( F ( i ) ) = σ ( X ) = 1 . (E3) If μ A 2 ( x ) ≤ μ B 2 ( x ) ≤ 1 3 and ν A 2 ( x ) ≤ ν B 2 ( x ) ≤ 1 3 for any x ∈ X, then we have E A ( x i ) = 1 - | μ A 2 ( x i ) - 1 3 | - | ν A 2 ( x i ) - 1 3 | ≤ 1 - | μ B 2 ( x i ) - 1 3 | - | ν B 2 ( x i ) - 1 3 | = E B ( x i ) . for all i = 1, . . . , n. Since the Sugeno integral is monotone we get 𝔼 ( S , σ ) ( A ) ≤ 𝔼 ( S , σ ) ( B ) . Similarly, if μ A 2 ( x ) ≥ μ B 2 ( x ) ≥ 1 3 and ν A 2 ( x ) ≥ ν B 2 ( x ) ≥ 1 3 for any x ∈ X, then we have

E A ( x i ) = 1 - | μ A 2 ( x i ) - 1 3 | - | ν A 2 ( x i ) - 1 3 | ≤ 1 - | μ B 2 ( x i ) - 1 3 | - | ν B 2 ( x i ) - 1 3 | = E B ( x i ) for all i = 1, . . . , n. Again, using the monotonicity of the Sugeno integral, we conclude that 𝔼 ( S , σ ) ( A ) ≤ 𝔼 ( S , σ ) ( B ) .

(E4) The proof is trivial. □

In this section, we give a theoretical basis to construct a similarity measure with the help of entropy and a particular PFS based on the pairs of PFSs whose similarity is to be measured. A similar idea was used in [27] with the help of G (A, B) ∈ PFS (X) defined by μ G ( A , B ) = ( 1 + min { | μ A 2 ( x ) - μ B 2 ( x ) | , | ν A 2 ( x ) - ν B 2 ( x ) | } 2 ) 1 / 2 and ν G ( A , B ) = ( 1 - max { | μ A 2 ( x ) - μ B 2 ( x ) | , | ν A 2 ( x ) - ν B 2 ( x ) | } 2 ) 1 / 2 where A and B are given PFSs. Peng et al. [27] proved that 𝔼 ( F ( A , B ) ) is a similarity measure for PFSs whenever 𝔼 is an entropy over PFS (X).

Let A, B ∈ PFS (X). Consider the functions μ_N(A,B), μ_N(A,B) : X → [0, 1] defined by μ N ( A , B ) ( x ) = 3 3 ( 1 - | μ A 2 ( x ) - μ B 2 ( x ) | ) ν N ( A , B ) ( x ) = 3 3 ( 1 - | ν A 2 ( x ) - ν B 2 ( x ) | )

that define the PFS N ( A , B ) = { ( x , μ N ( A , B ) ( x ) , ν N ( A , B ) ( x ) ) : x ∈ X } since μ N ( A , B ) 2 ( x i ) + ν N ( A , B ) 2 ( x i ) ≤ 1 .

The following theorem gives a useful idea to construct a similarity measure from an entropy measure for PFSs.

Theorem 4.1. If 𝔼 is an entropy measure on PFS (X), then the function 𝕊 : PFS ( X ) × PFS ( X ) → [ 0 , 1 ] defined by 𝕊 ( A , B ) = 𝔼 ( N ( A , B ) ) satisfies

(S1) 0 ≤ 𝕊 ( A , B ) ≤ 1 ,

(S2) 𝕊 ( A , B ) = 𝕊 ( B , A ) ,

(S3) 𝕊 ( A , B ) = 1 if A = B,

(S4) If A ⊂ B ⊂ C, then 𝕊 ( A , C ) ≤ 𝕊 ( A , B ) and 𝕊 ( A , C ) ≤ 𝕊 ( B , C )

for any A, B, C ∈ PFS (X).

Proof. (S1) The proof is trivial.

(S2) For any A, B ∈ PFS (X) we have 𝕊 ( A , B ) = 𝔼 ( N ( A , B ) ) = 𝔼 ( N ( B , A ) ) = 𝕊 ( B , A ) .

(S3) For any A ∈ PFS (X) we have μ N ( A , A ) ( x i ) = ν N ( A , A ) ( x i ) = 3 3 which yields that 𝕊 ( A , A ) = 𝔼 ( N ( A , A ) ) = 1 .

(S4) If A ⊂ B ⊂ C, then we have μ A 2 ( x i ) ≤ μ B 2 ( x i ) ≤ μ C 2 ( x i ) and ν A 2 ( x i ) ≥ ν B 2 ( x i ) ≥ ν C 2 ( x i ) for all x _i  ∈ X . Thus we get μ N ( A , C ) ( x i ) = 3 3 ( 1 - | μ A 2 ( x i ) - μ C 2 ( x i ) | ) ≤ 3 3 ( 1 - | μ A 2 ( x i ) - μ B 2 ( x i ) | ) = μ N ( A , B ) ( x i ) . Therefore, we get μ N ( A , C ) ( x i ) ≤ μ N ( A , B ) ( x i ) ≤ 1 3 . Similarly, we get ν N ( A , C ) ( x i ) ≤ ν N ( A , B ) ( x i ) ≤ 1 3 . Since 𝔼 is an entropy measure, from (E3), we have 𝕊 ( A , C ) = 𝔼 ( N ( A , C ) ) ≤ 𝔼 ( N ( A , B ) ) = 𝕊 ( A , B ) . □

Remark 4.1. Theorem 4 yields that we can construct a similarity measure once an entropy is provided in the Pythagorean fuzzy environment.

Now, if we consider Theorem 4, we can define the following Sugeno integral and entropy measure ( 𝔼 ( S , σ ) ) based similarity measure as follows:

Definition 4.1. Sugeno similarity measure based on 𝔼 ( S , σ ) is given with 𝕊 ( S , σ ) ( A , B ) : = 𝔼 ( S , σ ) ( N ( A , B ) ) .

Remark 4.2. We have 𝕊 ( S , σ ) ( A , B ) = ∫ X f A , B d σ = ⋁ i = 1 n ( f A , B ( x ( i ) ) ∧ σ ( F ( i ) ) )(2) where f A , B ( x ) = 1 - 1 3 | ( 1 - | μ A 2 ( x ) - μ B 2 ( x ) | ) 2 - 1 | - 1 3 | ( 1 - | ν A 2 ( x ) - ν B 2 ( x ) | ) 2 - 1 | .(3)

Since 𝕊 ( S , σ ) is a similarity measure, the function 𝔻 ( S , σ ) : PFS ( X ) × PFS ( X ) → [ 0 , 1 ] defined by 𝔻 ( S , σ ) = 1 - 𝕊 ( S , σ ) is a distance measure on PFS (X). We use this distance measure to provide the promised extended TOPSIS. The following is a consequence of Theorem 4.

Corollary 4.1. Function 𝔻 ( S , σ ) satisfies

(D1) 0 ≤ 𝔻 ( S , σ ) ( A , B ) ≤ 1 ,

(D2) 𝔻 ( S , σ ) ( A , B ) = 𝔻 ( S , σ ) ( B , A ) ,

(D3) 𝔻 ( S , σ ) ( A , B ) = 0 if A = B,

(D4) If A ⊂ B ⊂ C, then 𝔻 ( S , σ ) ( A , B ) ≤ 𝔻 ( S , σ ) ( A , C ) and 𝔻 ( S , σ ) ( B , C ) ≤ 𝔻 ( S , σ ) ( A , C )

for any A, B, C ∈ PFS (X).

Proof. The proof is trivial. □ We also note that 𝔻 ( S , σ ) ( A , B ) = 1 - ⋁ k = 1 n ( f A , B ( x ( i ) ) ∧ σ ( F ( i ) ) ) .(4)

Example 4.3. Let {x₁, x₂, x₃}. Consider the PFSs on X defined by A = x 1 , 0.4 , 0.7 , x 2 , 0.5 , 0.8 , x 3 , 0.6 , 0.2 and B = x 1 , 0.3 , 0.9 , x 2 , 0.1 , 0.8 , x 3 , 0.5 , 0.4 and the fuzzy measure σ given in Table 1.

Then we obtain f A , B ( x 1 ) = 0.7883 f A , B ( x 2 ) = 0.8592 f A , B ( x 3 ) = 0.8555 . Thus, using the fuzzy measure σ we have 𝔻 ( A , B ) = 0.2117 .

Remark 4.3. The similarity measure 𝕊 ( S , σ ) and distance measure 𝔻 ( S , σ ) can be calculated via

the Choquet integral instead of the Sugeno integral whenever the fuzzy measure σ is a Boolean fuzzy measure,

In this section, we give an extended (fuzzy) TOPSIS in the Pythagorean fuzzy environment and, we apply this method to a MCDM problem from the literature.

5.2 A medical diagnosis problem

Arora and Naithani [18] proposed novel logarithmic entropy measures under PFSs and, these new entropy measures were use to detect diseases related to post-COVID19 implications through TOPSIS. Now, we apply the extended TOPSIS proposed in this paper to the same MCDM problem. The symptoms (criteria) are cough (C₁), joint aches and muscle pain (C₂), fatigue and dyspnea (C₃), weight loss and poor appetite (C₄), loss of taste and smell (C₅), sleep disorder (C₆) and diseases (alternatives) are cardiac arrest (A₁), diabetic (A₂), lung fibrosis (A₃), pneumonia (A₄), kidney failure (A₅), brain stroke (A₆) that are to be evaluated as PFSs over criteria.

OPEN IN VIEWER

Fig. 2

Flowchart of the extended TOPSIS.

The following steps of the proposed algorithm are implemented as follow.

The ratings related to different alternatives are considered in terms of the decision matrix as mentioned in Table 2 taken from Ref. [18] in terms of PFVs.

Positive and negative ideal solutions are obtained as follows: S + = { ( 1.0 , 0.0 ) , ( 0.7 , 0.1 ) , ( 0.6 , 0.2 ) , ( 0.6 , 0.4 ) , ( 0.8 , 0.1 ) , ( 0.7 , 0.4 ) } S - = { ( 0.4 , 0.8 ) , ( 0.1 , 0.9 ) , ( 0.2 , 0.5 ) , ( 0.2 , 0.8 ) , ( 0.2 , 0.6 ) , ( 0.2 , 0.9 ) } .

To construct a fuzzy measure over the set of criteria, we use the λ-fuzzy measure identification method of [50]. In this method, we use the pairwise comparison matrix based on the expert view given in Table 3, and we obtain the weights of the criteria shown in Table 4.

Now by taking λ = 0.5 we construct the fuzzy measure σ displayed in Table 5.

f_{A i ,S⁺} (x _j ) and f_{A i ,S^-} (x _j ) values are calculated via (3) for i = 1, . . . , 6 and j = 1, . . . , 6. The matrices [f_{A i ,S⁺} (x _j )] _6×6 and [f_{A i ,S^-} (x _j )] _6×6 are given in Tables 6 and 7, respectively.

Using the values of Tables 6 and 7 and the fuzzy measure σ, the distances of the alternatives to the positive ideal solution and negative ideal solution are calculated (see, Table 8).

The closeness coefficients are calculated and given in Table 9.

So, we get A₆ ≻ A₄ ≻ A₂ ≻ A₅ ≻ A₃ ≻ A₁.

OPEN IN VIEWER

Table 2

Decision matrix

	C 1	C 2	C 3	C 4	C 5	C 6
A 1	(0.8, 0.4)	(0.4, 0.9)	(0.2, 0.3)	(0.5, 0.8)	(0.3, 0.6)	(0.4, 0.7)
A 2	(0.8, 0.1)	(0.4, 0.8)	(0.5, 0.4)	(0.6, 0.4)	(0.6, 0.5)	(0.7, 0.5)
A 3	(0.5, 0.8)	(0.4, 0.7)	(0.3, 0.5)	(0.4, 0.7)	(0.8, 0.1)	(0.4, 0.9)
A 4	(0.9, 0.4)	(0.1, 0.8)	(0.4, 0.2)	(0.5, 0.7)	(0.7, 0.2)	(0.7, 0.4)
A 5	(1.0, 0.0)	(0.6, 0.1)	(0.5, 0.4)	(0.4, 0.6)	(0.2, 0.6)	(0.4, 0.5)
A 6	(0.4, 0.5)	(0.7, 0.2)	(0.6, 0.3)	(0.2, 0.5)	(0.8, 0.4)	(0.2, 0.6)

OPEN IN VIEWER

Table 3

Pairwise comparison matrix

	C 1	C 2	C 3	C 4	C 5	C 6
C 1	1	1	5	7	4	5
C 2	1	1	4	5	4	5
C 3	0.2	0.25	1	4	0.25	1
C 4	0.142857	0.2	0.25	1	0.2	0.25
C 5	0.25	0.25	4	5	1	3
C 6	0.2	0.2	1	4	0.333333	1

OPEN IN VIEWER

Table 4

Weights of the criteria

W C 1  = 0.345703	W C 4  = 0.0331288
W C 2  = 0.324193	W C 5  = 0.153374
W C 3  = 0.0720693	W C 6  = 0.0715316

OPEN IN VIEWER

Table 5

Fuzzy measure σ

σ (∅) =0	σ ({C1, C3, C5}) =0.521181	σ ({C2, C4, C6}) =0.379838
σ ({C1}) =0.30094	σ ({C2, C3, C5}) =0.499288	σ ({C1, C2, C4, C6}) =0.737932
σ ({C2}) =0.28096	σ ({C1, C2, C3, C5}) =0.875355	σ ({C3, C4, C6}) =0.148575
σ ({C1, C2}) =0.624176	σ ({C4, C5}) =0.157106	σ ({C1, C3, C4, C6}) =0.471872
σ ({C3}) =0.0593055	σ ({C1, C4, C5}) =0.481686	σ ({C2, C3, C4, C6}) =0.450407
σ ({C1, C3}) =0.369169	σ ({C2, C4, C5}) =0.460136	σ ({C1, C2, C3, C4, C6}) =0.819119
σ ({C2, C3}) =0.348596	σ ({C1, C2, C4, C5}) =0.830313	σ ({C5, C6}) =0.190957
σ ({C1, C2, C3}) =0.70199	σ ({C3, C4, C5}) =0.22107	σ ({C1, C5, C6}) =0.520631
σ ({C4}) =0.0270464	σ ({C1, C3, C4, C5}) =0.555275	σ ({C2, C5, C6}) =0.498743
σ ({C1, C4}) =0.332056	σ ({C2, C3, C4, C5}) =0.533086	σ ({C1, C2, C5, C6}) =0.874728
σ ({C2, C4}) =0.311806	σ ({C1, C2, C3, C4, C5}) =0.914239	σ ({C3, C5, C6}) =0.255925
σ ({C1, C2, C4}) =0.659663	σ ({C6}) =0.0588566	σ ({C1, C3, C5, C6}) =0.595374
σ ({C3, C4}) =0.087154	σ ({C1, C6}) =0.368653	σ ({C2, C3, C5, C6}) =0.572837
σ ({C1, C3, C4}) =0.401208	σ ({C2, C6}) =0.348084	σ ({C1, C2, C3, C5, C6}) =0.959972
σ ({C2, C3, C4}) =0.380357	σ ({C1, C2, C6}) =0.701401	σ ({C4, C5, C6}) =0.220586
σ ({C1, C2, C3, C4}) =0.738529	σ ({C3, C6}) =0.119907	σ ({C1, C4, C5, C6}) =0.554718
σ ({C5}) =0.128325	σ ({C1, C3, C6}) =0.43889	σ ({C2, C4, C5, C6}) =0.532534
σ ({C1, C5}) =0.448574	σ ({C2, C3, C6}) =0.417712	σ ({C1, C2, C4, C5, C6}) =0.913604
σ ({C2, C5}) =0.427311	σ ({C1, C2, C3, C6}) =0.781504	σ ({C3, C4, C5, C6}) =0.286433
σ ({C1, C2, C5}) =0.792548	σ ({C4, C6}) =0.086699	σ ({C1, C3, C4, C5, C6}) =0.630472
σ ({C3, C5}) =0.191435	σ ({C1, C4, C6}) =0.400685	σ ({C2, C3, C4, C5, C6}) =0.60763
σ ({C1, C2, C3, C4, C5, C6}) =1

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Table 6

f_{A i ,S⁺} values

	C 1	C 2	C 3	C 4	C 5	C 6
A 1	0.705067	0.4963	0.7883	0.6875	0.541667	0.6326
A 2	0.796567	0.5286	0.8555	1	0.698667	0.9427
A 3	0.397367	0.5731	0.719	0.6963	1	0.5238
A 4	0.787233	0.4691	0.88	0.747	0.8878	1
A 5	1	0.918967	0.8555	0.76	0.5275	0.759
A 6	0.529367	0.9803	0.9675	0.7635	0.9075	0.6475

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Table 7

f_{A i ,S^-} values

	C 1	C 2	C 3	C 4	C 5	C 6
A 1	0.5136	0.9075	0.901867	0.8747	0.9675	0.7456
A 2	0.4691	0.8038	0.8174	0.5776	0.7515	0.4987
A 3	0.9427	0.7283	0.9675	0.8323	0.5275	0.9248
A 4	0.4643	0.8963	0.7995	0.7822	0.5883	0.475
A 5	0.385067	0.4875	0.8174	0.764267	1	0.656
A 6	0.7907	0.4411	0.722667	0.7907	0.6	0.7675

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Table 8

Similarities and distances of the alternatives to the positive and negative ideal solutions

	𝕊 ( S , σ ) ( A i , S + )	𝕊 ( S , σ ) ( A i , S - )	𝔻 ( S , σ ) ( A i , S + )	𝔻 ( S , σ ) ( A i , S - )
A 1	0.541667	0.60763	0.458333	0.39237
A 2	0.630472	0.533086	0.369528	0.466914
A 3	0.533086	0.7283	0.466914	0.2717
A 4	0.630472	0.533086	0.369528	0.466914
A 5	0.759	0.4875	0.241	0.5125
A 6	0.60763	0.6	0.39237	0.4

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Table 9

Closeness coefficients and rankings

	Closeness coefficients (s i )	Ranking	Ranking of [18]
A 1	0.46123	4	6
A 2	0.558214	2	3
A 3	0.367851	5	5
A 4	0.558214	2	4
A 5	0.680159	1	2
A 6	0.504815	3	1

5.3 Sensitivity analysis

In this sub-section, we analyse the impact of the selection of the parameter λ in the λ-fuzzy measure and compare the results with each other and with those in the literature. To conduct the sensitivity analysis, we vary the parameter λ between 0.5 and 4 by 0.5 change. We show the results in Table 10.

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Table 10

Rankings for different λ values

λ	Ranking
0.5	A5 ≻ A4 = A2 ≻ A6 ≻ A1 ≻ A3
1	A5 ≻ A4 ≻ A2 ≻ A6 ≻ A1 ≻ A3
1.5	A5 ≻ A4 ≻ A2 ≻ A1 ≻ A6 ≻ A3
2	A5 ≻ A4 ≻ A2 ≻ A1 ≻ A6 ≻ A3
2.5	A5 ≻ A4 ≻ A2 ≻ A1 ≻ A6 ≻ A3
3	A5 ≻ A4 ≻ A2 ≻ A6 ≻ A1 ≻ A3
3.5	A5 ≻ A6 ≻ A2 ≻ A1 = A4 ≻ A3
4	A5 ≻ A2 ≻ A6 ≻ A4 ≻ A1 ≻ A3

For each λ the best alternative is A₅ and the worst alternative is A₃. Actually, the second best alternative of [18] is the best alternative of the present study. The reason for this difference is that the method used in the present study uses the Sugeno integral, which makes a more sensitive evaluation. Figure 3 also illustrates the comparison of the ranking of [18] with the rankings of the present study.

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Fig. 3

Sensitivity analysis for different values of λ.

In this paper, we propose a new entropy measure for PFSs using the Sugeno integral and fuzzy measure theory. The proposed entropy measure is more sensitive in modeling the interaction between criteria compared to traditional measures. We also present a theorem to obtain a similarity measure from a given entropy, which is then used to define a new distance measure for PFSs. The extended TOPSIS is also introduced for PFSs using the proposed distance measure. The technique is applied to a medical diagnosis problem and shows more sensitive results compared to previous literature. Moreover, there are several promising areas of research that could further advance the field of PFSs and uncertainty modeling, including: exploring the use of deep learning techniques to improve the accuracy and efficiency of PFS models, investigating the integration of PFSs with other types of FSs and uncertainty models, and developing new decision-making frameworks that incorporate PFSs into multi-criteria decision analysis.

In the future, the aspiration of the authors is to extend the present methodology for different extensions of the fuzzy as well as in terms of utilizing fuzzy integrals to define entropy measures for PFSs. The applicability of the proposed measures can also be extended to diverse application of threshold-based value-driven methods to support consensus reaching in multi-criteria group sorting problems, specifically from a minimum adjustment perspective [51–53] etc. Also, we shall extend our work to define some more generalized information measures to rank the different alternative. Finally, we shall extend our approach to analyze the different application related to Emergency supply, and different tools of artificial intelligence such as optimization or neural network.