Some t-conorm-based distance measures and knowledge measures for Pythagorean fuzzy sets with their application in decision-making

The Pythagorean fuzzy sets are more robust than fuzzy sets and intuitionistic fuzzy sets in dealing with the problems involving uncertainty. To compare two Pythagorean fuzzy sets, distance measures play a crucial role. In this paper, we have proposed some novel distance measures for Pythagorean fuzzy sets using t-conorms. We have also discussed their various desirable properties. With the help of suggested distance measures, we have introduced some new knowledge measures for Pythagorean fuzzy sets. Through numerical comparison and linguistic hedges, we have established the effectiveness of the suggested distance measures and knowledge measures, respectively, over the existing measures in the Pythagorean fuzzy setting. At last, we have demonstrated the application of the suggested measures in pattern analysis and multi-attribute decision-making.


Introduction
Zadeh [1] introduced the concept of fuzzy sets (FSs) for handling deterministic uncertainty. Each element in a fuzzy set (FS) possesses a grade of membership (μ) lying between 0 and 1. However, in an FS, the grade of non-membership (ϑ) is by default taken as 1 − μ. So to assign an independent non-membership grade to an element, a novel extension of FSs known as intuitionistic fuzzy set (IFS) was given by Atanassov [2]. In an IFS, the sum of membership grades is less or equal to one i.e., μ + ϑ ≤ 1. Though there are numerous applications of intuitionistic fuzzy sets (IFSs) but their scope is limited due to the restriction on sum of membership grades. The IFSs fail to handle those uncertain problems where μ + ϑ > 1. So, the concept of Pythagorean fuzzy set (PFS) was proposed by Yager [3] as an extension of intuitionistic fuzzy sets [2] (IFSs) and fuzzy sets [1] (FSs) for solving the problems involving uncertainty more precisely. Each element of a PFS has a membership grade (μ) and a non-membership grade (ϑ) with their square sum at most one μ 2 + ϑ 2 ≤ 1 . The FSs and IFSs form a part of the space B Abdul Haseeb Ganie ahg110605@gmail.com 1 School of Mathematics, Faculty of Sciences, SMVD University, Katra, J & K 182320, India of PFSs and therefore the space of PFSs is wider than the space of FSs and IFSs.So, PFSs are more powerful than FSs and IFSs in handling uncertain problems. The technique for order of preference by similarity to ideal solution (TOPSIS) in the Pythagorean fuzzy (PF) setting and the concept of the Pythagorean fuzzy number were suggested by Zhang and Xu [4]. Various PF aggregation functions with their utility in decision-making were given by Yager [5]. Wei and Lu [6] introduced some power aggregation functions for PFSs. Using Einstein operations, Garg [7] proposed some new aggregation functions in the PF environment. Wei [8] suggested some PF interaction aggregation functions with their utility in multi-attribute decision-making (MADM). Many studies [9][10][11][12] concerning the PF aggregation functions with their various applications are available in the literature. The TODIM (an acronym in Portuguese for Interactive and Multicriteria Decision Making) method for PFSs was introduced by Ren et al. [13]. Peng et al. [14] proposed some information measures for PFSs. A novel PF distance measure was proposed by Peng and Dai [15]. Some PF measures of correlation with their utility were proposed by Singh and Ganie [16]. Various researchers [17][18][19][20][21][22][23][24] have studied PFSs and applied them in distinct uncertain situations. The current study is related to the development of some novel PF distance measures and knowledge measures.
The main contributions of this paper are as:

Related work
Distance measures are very powerful in comparing two objects based on their inequality content. For FSs and IFSs there are a lot of studies [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] concerning distance measures along with their various applications. Karmakar et al. [43] suggested a Minkowski distance measure for Type-2 IFSs (T2IFSs) using the Hausdorff metric. Some recent studies related to Type 2 FSs and T2IFSs are given in [44][45][46]. The application of some PF measures of distance and similarity in MADM was shown by Zeng et al. [47]. Hussain and Yang [48] proposed some Hausdorff metric-based PF measures of distance and similarity with their applicability in PF TOPSIS. Some generalized measures of distance and their continuous versions for PFSs were given by Li and Lu [49]. They also proposed set-theoretic-based, matching function-based, and complement-based PF similarity measures. Based on the membership grades, Ejegwa [19] proposed some distance and similarity measures for PFSs. Some cosine functionbased PF similarity measures were suggested by Wei and Wei [50]. Twelve PF measures of distance and similarity with their applicability were given by Peng et al. [14]. For PFSs Zhang [51] introduced a measure of similarity and its utility in decision-making. Some novel measures of similarity and distance for PFSs based on L p norm and level of uncertainty were given by Peng [52]. By combining the Euclidean distance measure and cosine similarity measures, Mohd and Abdullah [53] developed some novel PF similarity measures. Zhang et al. [54] proposed some exponential PF similarity measures and demonstrated their application in MADM, pattern analysis, and medical diagnosis. Some PF Dice similarity measures with application in decision-making were given by Wang et al. [55]. Verma and Merigo [56] developed some trigonometric function-based PF measures of similarity. The application of some multiparametric PF measures of similarity in classification problems was demonstrated by Peng and Garg [57]. Some novel PF similarity measures based on exponential function with their application in classification problems were given by Nguyen et al. [58]. Some recent studies related to PF distance measures along with their applications are given in [59][60][61][62][63][64]. The entropy of an FS is the ambiguous content present in it. Entropy measure is very essential for computing the weight of attributes in a MADM problem involving fuzzy data. The concept of fuzzy entropy was suggested by De Luca and Termini [65]. Some axiomatic requirements for a measure to be a fuzzy entropy measure were given by Ebanks [66]. Some more studies concerning fuzzy entropy measures and their utility are given in [67][68][69][70][71][72]. For IFSs Szmidt and Kacprzyk [73] suggested an entropy measure. Xue et al. [74] introduced the axiomatic definition of PF entropy measure and used the PF entropy measure in decision-making. Some probabilistic and non-probabilistic PF entropy measures were given by Yang and Hussain [75]. With the help of a new PF entropy measure, Thao and Smarandache [76] introduced the COR-PAS MADM method in the PF environment.
Knowledge of an FS is the amount of precision present in it. Knowledge measure (KM) plays a great role in determining the weight of attributes in a MADM problem involving fuzzy data. Singh et al. [77] introduced the axiomatic definition of a fuzzy knowledge measure (FKM) and used it in decision-making. They also proposed a fuzzy accuracy measure and utilized it in image processing. Later on, Singh et al. [78] also introduced a one-parametric generalization of the FKM and discussed its various applications. A twoparametric fuzzy knowledge measure and accuracy measure with their applicability in decision-making and classification problems were given by Singh and Ganie [79]. For IFSs, there are various studies [80][81][82][83][84][85][86] regarding the knowledge measures (KMs) with their practical applications. Lin et al. [87] proposed a knowledge measure (KM) for picture fuzzy sets with its utility in decision-making. Some PF KMs with their various applications were introduced by Singh et al. [88]. For hesitant fuzzy sets, Singh and Ganie [89] introduced a generalized KM.
The following are the primary motivating aspects for this research: • There is no study concerning the generation of the PF distance measures and knowledge measures from t-conorms. • Axiomatic conditions are not met by several existing PF distance metrics. • When computing the distance between distinct PFSs, the majority of the existing PF distance metrics produce unacceptable results. • From the perspective of linguistic hedging, all of the known PF entropy/knowledge measurements are insufficient.
So, keeping the above facts in mind, we suggest some new distance measures and knowledge measures based on PFSs.
The rest of the paper is organized as: Sect. 3 is preliminary. Some novel t-conorm-based PF distance measures along with desirable properties are given in Sect. 4. Section 5 is devoted to the introduction of some distance-based PF knowledge measures. The comparison of the suggested PF distance measures and knowledge measures with the available PF measures of compatibility is shown in Sect. 6. Section 7 demonstrates the applicability of the suggested measures in pattern analysis and MADM. At last, the conclusion and future study are given in Sect. 8.

Preliminaries
Let W {m 1 , m 2 , . . . , m l } be the universe of discourse and P F S(W ) be the set of all PFSs of W .

Definition 1 [3] A PFS M 1 in W is given by
With μ M 1 m j and ϑ M 1 m j representing the membership and non-membership grades of the element m j in Definition 2 [3] For two PFSs M 1 and M 2 in W , some operations are given as: g(x, y) g(y, x); g(x, y) ≤ g(z, t), whenever x ≤ z and y ≤ t; g(x, 1) x; g(x, g(y, z)) g(g(x, y), z). g(x, y) g(y, x); g(x, y) ≤ g(z, t), whenever x ≤ z and y ≤ t; g(x, 0) x; g(x, g(y, z)) g(g(x, y), z).
is called a PF similarity measure if ∀M 1 , M 2 and M 3 ∈ P F S(W ), we have: Definition 8 [14] A function E : P F S(W ) → [0, 1] is called a PF entropy measure if ∀M 1 and M 2 ∈ P F S(W ), we have: In the next section, we introduce some novel t-conormbased distance measures for PFSs along with their properties.

New measures of distance for PFSs
Here, we propose some PF measures of distance based on t-conorms.

Definition 10
Let M 1 , M 2 ∈ P F S(W ), then we define a function.

Example 1
Some examples of PF distance measures are given in Table 1.
In most decision-making problems, the weights w j of the elements m j , j 1, 2, . . . , l are taken into consideration, so we introduce the weighted PF distance measures.
where g is a t-conorm.

Theorem 3
The function D W G given in Eq.
(2) is a valid PF distance measure.
Proof Similar to Theorem 1.

Example 2
Some examples of weighted PF distance measures are given in Table 2.
Next, we propose some novel PF measures of knowledge based on the proposed PF distance measures.

PF distance-based knowledge measures
The entropy measures are used to compute the amount of ambiguity present in a PFS, whereas the knowledge measures acting as the soft duals of entropy measures are used to calculate the amount of precision in a PFS. Here, we introduce a method of constructing PF knowledge measures from the PF distance measures.

Definition 12
Let M 1 ∈ P F S(W ), then we define a function given by where D G is a PF distance measure.
Proof To show that the function K G is a PF measure of knowledge, we show it has the properties of a PF measure of knowledge given in Definition 9.
With the help of Eq. (3) and based on the suggested PF measures of distance, some PF measures of knowledge are given in Table 3 below: Now, we compare the suggested PF measures of distance and knowledge with some available PF measures of information.

Comparative analysis
In this section, we show that our suggested PF measures of distance and knowledge give better results than most of the available PF measures of information.

Comparison of the proposed PF distance measures with various available PF measures of similarity/distance
To contrast the performance of the suggested PF measures of distance, we first list the PF measures of similarity/distance available in the literature as shown in Table 4. Now we compare the suggested measures with the existing ones through some numerical examples related to the computation of the distance/similarity between different PFSs. The computed distance/similarity values for these three cases using the available measures of distance/similarity along with the suggested ones are shown in Table 5.
From Table 5, we have andD PY Y 10 gives the same distance for the two distinct cases (Case II and Case III). 2. The PF distance measure D PY Y 2 gives "0" as the distance between the two different PFSs (Case I) and thus fails to satisfy the axiom (D3) of the PF distance measure given in Definition 7. Distance/similarity measure Expression   Case VI: The computed distance/similarity values for these six cases using the available measures of distance/similarity along with the suggested ones are shown in Table 6.
From Table 6, we have the following: , and D PY Y 10 give the same distance for two distinct cases (Case I and Case II).
2. The PF distance measure D PY Y 2 gives "0" as the distance between the two unequal PFSs (Case III). 7. The PF similarity measure S PY Y 2 gives "1" as the similarity between the two unequal PFSs (Case III). 8. The similarity between the different PFSs comes out to be negative (Case I, Case IV, Case V, and Case V) by the PF similarity measures S PY Y 3 , S PY Y 7 , and S PY Y 8 . So, these similarity measures fail to satisfy the axiom (S1) of Definition 6. 9. The PF similarity measures S PY Y 7 and S PY Y 8 fails to compute the similarity between the PFSs (Case IV). 10. The PF similarity measures S PY Y 7 , S PY Y 8 , S PY Y 9 , and S PY Y 10 gives "0" as the similarity between the PFSs M 1 and M 2 (Case III and Case VI), when neither 11. The suggested PF distance measures D G j , 1 ≤ j ≤ 4 computes the distance of all the PFSs without any counterintuitive results.
Thus, from Examples 3 and 4, we conclude that the suggested distance measures are more robust and effective than most of the available distance/similarity measures in PF theory.
Next, we compare the suggested PF knowledge measures with the available PF measures of entropy/knowledge.

Comparison of the suggested PF measures of knowledge with the available PF measures of entropy/knowledge
To contrast the performance of the newly introduced PF measures of knowledge, we first list the PF entropy/knowledge measures available in the literature.
Entropy measures due to Peng et al. [14] Entropy measure due to Xue et al. [74] Entropy measure due to Thao and Smarandache [76] Entropy measure due to Yang and Hussain [75] Knowledge measures due to Singh et al. [88] Now, using linguistic hedges, we show the effectiveness of the suggested PF measures of knowledge. Definition 13 [75] For any M 1 ∈ P F S(W ), its modifier (M 1 ) λ , λ > 0 is defined as Then, we have the following PFSs: Since a PF entropy measure,E computes the ambiguous content in a PFS, it has to satisfy the following requirement: Also, a PF knowledge measure K acts as a soft dual of a PF entropy measure and calculates the amount of precision in a PFS, so it has to satisfy the following requirement: We now consider an example related to the ambiguous computation of the above-mentioned PFSs. With the help of Definition 13, we construct the following PFSs: The ambiguous content of these PFSs using the suggested PF knowledge measures and the existing ones is shown in Table 7.
From Table 7, we have the following: Thus, it follows that all the available PF measures of entropy E P L j , 1 ≤ j ≤ 8, E X X Z T , E T S , E Y H , and the PF knowledge measures K SSG j , j 1, 2, 3 do not satisfy the requirements given in Eqs. (4) and (5) respectively. However, all our suggested PF knowledge measures K G j , j 1, 2, 3, 4, follow the desired requirement given in Eq. (5). This shows that from a linguistic hedge perspective, the suggested measures of knowledge are more robust than the available ones. Now, we demonstrate the utility of the proposed PF distance and knowledge measures in pattern identification and decision-making.

Application of the proposed measures
In this section, we show how the suggested metrics can be used in pattern analysis and MCDM.

Pattern analysis
We demonstrate how the suggested PF distance metrics can be employed to solve pattern classification problems. An unfamiliar pattern is classed into one of the known patterns using compatibility measurements such as similarity measures, distance measures, correlation measures, and so on in pattern analysis. We also compare our findings to the existing compatibility measures. Now, we solve some problems related to pattern analysis in the examples given below. We need to find out the nanometer material that N belongs to. The similarity/distance between N and M j , j 1, 2, 3 by various PF similarity/distance measures are shown in Table 8.
From Table 8, we see that the unknown nanometer is assigned to the pattern M 2 as shown by most of the PF distance/similarity measures. We also observe that the PF distance/similarity measures D PY Y 8 , D PY Y 9 , S PY Y 8 , and S PY Y 9 fail to recognize the unknown nanometer N . The following are the typical model data for the shape features of the three gut bacteria: Our aim is to find the bacteria to which N belongs. The similarity/distance between N and M j , j 1, 2, 3 by various PF similarity/distance measures are shown in Table 9.
From Table 9, we observe that the unknown microbe N is assigned to M 1 as shown by most of the PF distance/similarity measures. We also observe that the PF distance/similarity measures D PY Y 2 and S PY Y 2 fail to recognize the unknown microbe N .
Thus from Examples 6 and 7, it is clear that results due to the suggested PF distance measures are consistent with the existing ones and therefore are applicable in classification problems.

Multi-criteria decision-making
Here, we show that the suggested PF measures of knowledge and distance are useful for solving MCDM problems involving uncertainty and ambiguity. The main hurdle in an MCDM problem is the computation of criteria weights and we use the suggested knowledge measures for this purpose. For determining the best alternative, we take the help of the suggested distance measures. First, we give the algorithm for solving an MCDM problem having n alternatives M j , j 1, 2, . . . , n and k criteria N k , k 1, 2, . . . , m with w k , k 1, 2, . . . , m as criteria weights where 0 ≤ w k ≤ 1 and m k 1 w k 1.

Algorithm
Step 1: Formulate the decision matrix D μ jk , ϑ jk n×m expressing the information of the available alternatives with respect to the criteria.
Step 2: Formulate the normalized decision matrix E μ jk , ϑ jk n×m where, μ jk , ϑ jk μ jk , ϑ jk , i f N k is a bene f it criteria ϑ jk , μ jk , i f N k is a cost criteria .
Here, K is a PF entropy measure.
Step 5: Compute the distance of each alternative M j , j 1, 2, . . . , n from the PF ideal solution M * using the suggested weighted PF distance measures. With the help of Step 3 and using the suggested entropy measure K G1 given in Table 3, we obtain the criteria weights as: The computed values of the distance of each alternative M j , j 1, 2, 3, 4, 5 from the PF ideal solution M * using the suggested weighted distance measures D w G j , j 1, 2, 3, 4 given in Table 2 are shown in Table 10.
The final ranking of alternatives with the help of Step 6 is shown in Table 11.
From Table 11, we conclude that M 2 is the most feasible alternative as all the suggested PF distance measures and the existing PF distance measures D PY Y 1 and D PY Y 4 indicate the same. Further, some existing q-rung orthopair correlation coefficients also indicate the same. This shows that the suggested distance measures are consistent with the existing distance measures.

Conclusion
With the use of t-conorms, this work offered a novel way of building some distance and knowledge metrics for PFSs. First, four new distance measures were presented using t-conorms, and then four new knowledge measures were developed using the proposed distance measures. In terms of the distance/similarity degree between distinct PFSs, the suggested distance measures are more successful than most of the known PF distance/similarity measures. The majority of the existing PF distance/similarity measures produce the same distance/similarity between distinct PFSs, and some of them fail to satisfy all of the axiomatic conditions. The suggested PF distance metrics, on the other hand, are devoid of these flaws. Furthermore, from the linguistic hedging perspective, the suggested measures of knowledge for PFSs are more resilient than the known PF entropy/knowledge measures. The proposed PF distance metrics have shown to be effective in pattern recognition challenges. Finally, in a multi-criteria decision-making situation, the recommended knowledge measures are used to compute the weight of attributes, and the distance measures are utilized to rank the alternatives. The results of the recommended measures are compatible with the available measures in pattern recognition and decision-making situations. The advantages of this study are: 1. The suggested method of constructing the distance measures from t-conorms can be utilized for obtaining the new distance measures for some recent generalizations of fuzzy sets. 2. The distance-based knowledge measures can be used in computing the ambiguity content of Pythagorean fuzzy sets where the existing entropy measures lead to unreasonable results.
3. The suggested distance measures can be applied to bidirectional approximate reasoning.
Our future studies include: • To demonstrate the applicability of the suggested distance measures in clustering and medical diagnosis. • To demonstrate the applicability of the suggested measures in real decision-making problems. • To introduce the t-conorm-based distance measures and knowledge measures for picture fuzzy sets [93], spherical fuzzy sets [94], T-spherical fuzzy sets [94], etc. • To introduce the parametric generalizations of the suggested measures along with their various applications.