key: cord-0073536-52svdgl3
authors: Zararsız, Zarife; Riaz, Muhammad
title: Bipolar fuzzy metric spaces with application
date: 2022-01-13
journal: Comp
DOI: 10.1007/s40314-021-01754-6
sha: d9f1b0281b493de8880ed16e0ed9588366024ef8
doc_id: 73536
cord_uid: 52svdgl3

The concept of a bipolar metric space is presented in this article. In bipolar fuzzy metric space, the notions of continuous t-norms and bipolar continuous symmetry [Formula: see text] -conorms employs important role. Requirements in metric space, triangular norms are used to extrapolate with the probability density function of triangle inequality. Dual processes of triangular norms are known as triangular conorms. Continuous t-norms and continuous symmetry [Formula: see text] -conorms are used to define bipolar metric space in this paper. Besides, a number of topological and functional properties of the bipolar metric space have been studied. Afterwards, Baire Category and Uniform Convergence Theorems for bipolar metric spaces are presented. After that, an application on determining the appropriate type of vaccine in the treatment process of COVID-19 is given using similarity measure between bipolar fuzzy metric spaces. For vaccine selection and supply chain management, an advanced multi-attribute decision-making (MADM) algorithm is being developed. Finally, the validity of the proposed MADM method for selecting the best proper form of vaccine is also established.

Being a member of elements in a set is determined in binary terms according to a bivalent condition in classical set theory, which means that an element either member of the set or does not member of the set. Fuzzy set theory, on the other hand, allows for a gradual assessment Communicated by Eduardo Souza de Cursi. of the membership of elements in a set, which is defined using a membership function with a real unit interval [0, 1]. Since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1, fuzzy sets generalize classical sets. Crisp sets are widely used in fuzzy set theory to refer to classical bivalent sets. Many researchers have been interested in fuzzy sets, which were first advanced by Zadeh (1965) in 1965 to convey indefiniteness in data. A membership function μ : X :→ [0, 1] defines the a fuzzy set description, which assigns a real number in the unit closed interval [0, 1] to each object in the universe. The fuzzy set theory can be applied to a broad variety of domains, including bioinformatics, where knowledge is incomplete or imprecise. Chang (1968) developed the notion of fuzzy topological spaces. Atanassov (1984) , Atanassov and Stoeva (1983) suggested the intuitionistic fuzzy set (IFS) as a direct extension of the fuzzy set, based on the definition of membership grade (MG) and non-membership grade (NMG), with the restriction that the number of MG and NMG not exceed 1. Çoker (1997) introduced the concept of the intuitionistic fuzzy topological space. Smarandache (2005) described the Neutrosophic Set (NS) as a modern version of the classical set definition. Neutrosophy has opened the way for many other modern mathematical ideas that extend both classical and fuzzy equivalents. Balasubramanian and Sundaram (1997) presented the definition of a generalized fuzzy closed set. Pythagorean fuzzy set and Pythagorean fuzzy membership grades were introduced by Yager (2014 Yager ( , 2017 and Yager and Abbasov (2013) , respectively. Yager and Abbasov (2013) , Park (2004) , Yager (2014) , Yager (2017) introduced Pythagorean fuzzy sets (PFSs), also known as intuitionistic fuzzy sets of type-2, as an extension of IFSs and presented Pythagorean membership grades with requirements, by changing the constraint on the parameters. In Gulistana et al. (2021) authors study complex bipolar fuzzy sets. The notion of rough bipolar fuzzy Γ r-hyperideals in Γ r -semihypergroups is introduced by Yaqoob et al. (2018) . Besides, Yaqoob (2014) deal with on bipolar fuzzy sets. The concept of bipolar-valued fuzzification of ordered AG-groupoids is studied in Yousafzai et al. (2012) . The definitions of probabilistic metric space and fuzzy sets are used to construct the fuzzy metric space (FMS) in Kramosil and Michàle (1975) . Afterwards FMS was described by Kaleva and Seikkala (1984) as a non-negative fuzzy number that measures the distance between two points. Following that, FMS has been extended to a variety of fields, fixed point theory, image and signal processing, medical imaging, decision-making, and so on are some of the topics covered. After it was defined, the intuitionistic fuzzy set (IFS) was described and used in all areas where FS theory was explored. In addition these, the concept of intuitionistic fuzzy metric space was invented by Park (2004) . He demonstrated that the topology created by the intuitionistic fuzzy metric (M, N ) coincides with the topology generated by the fuzzy metric M for each intuitionistic fuzzy metric space (X , M, N , * , ). A hesitant fuzzy linear regression model (HFLRM) is investigated in Sultan et al. (2021) to account for multicriteria decision-making (MCDM) problems in a hesitant environment. In Kizielewicz et al. (2021) , a new modified TOPSIS technique for dealing with data uncertainty that is based on TFN similarity metrics is offered.

Based on the theory of NSs, a new metric space called Neutrosophic metric Spaces (NMS) was defined in Kirişci andŞimşek (2020) . Some properties of NMS are investigated in Kirişci andŞimşek (2020) , including open set, Hausdorff, neutrosophic bounded, compactness and completeness. For NMSs, they also include the Baire Category Theorem and the Uniform Convergence Theorem. As extensions of Partial Algebra, Smarandache (2005) generalized classical Algebraic Structures to NeutroAlgebraic Structures with partially real, partially indeterminate, and partially false operations and axioms. Recently, as a generalization of fuzzy sets, Zhang (1994 Zhang ( , 1998 initiated the concept of bipolar fuzzy sets in 1994. For more information you can see (Abdullah et al. 2014; Hussain et al. 2019; Qurashi and Shabir 2018; Riaz et al. 2020) . In this research, fundamental concepts about the subject was given in the second part. In addition, in this section, the concepts of bipolar continuous symmetry t sconorm and bipolar fuzzy metric space are introduced and relevant examples are discussed. In the third chapter, the topology of bipolar fuzzy sets has been constructed and some theorems and proofs are given. For example, it is showed that BFMS is T 2 -space. Additionally, in this part BF bounded, complete BFMS and convergence of a BFMS concepts are defined. It is demonstrated in Chapter fourth that a BFMS is complete. The Baire Category and Uniform Limit theorems, as well as their proofs, were discovered. In the fifth section, a new similarity measure method for bipolar fuzzy metric spaces is presented. A multiple attribute decision making (MADM) algorithm is also provided, which was designed to determine the best form of vaccine for COVID-19. In the future work, we plan to give definitions of bipolar normed space and statistical convergence on bipolar fuzzy sets. Besides, we will make an application about medicine image processing.

This section contains some definitions for fuzziness, bipolar fuzziness, and bipolar fuzzy metric space.

Definition 1 (Zhang 1994) . BFSs are a type of fuzzy set whose membership value range has been extended from [0, 1] to [−1, 1]. In a bipolar fuzzy set, a membership value of 0 denotes that elements are irrelevant to the corresponding property, a membership value of [−1, 0] denotes that elements satisfy the assumed counter-property, and a membership value of (0, 1) denotes that elements satisfy the property. The accepted representation of a bipolar fuzzy set A on the domain X is as follows:

, μ A (a))) : a ∈ X }. Here, the positive membership value μ r A (a) : X → [0, 1] specifies the value of satisfaction of some element a ∈ X to the property according to the bipolar fuzzy set A, and the negative membership value μ A (a) : X → [−1, 0] describes the value to which a certain element a ∈ X satisfies some implied counter-property corresponding to A. Now let's include some basic definitions defined on bipolar fuzzy sets.

Definition 2 (Zhang 1994 ). Let's admit that A = {a, (μ A (a), μ r A (a)) : a ∈ X } and B = {a, (μ B , μ r B ) : a ∈ X } are two bipolar fuzzy sets defined on X . In this case,

Additionally,

The positive membership degree μ r A (a), in BFSs, denotes the value to which the element a supplies the property of A, while the negative membership value μ A (a) denotes the value to which the element a supplies an implied counter-property of A. However, the membership value μ A (a) denotes the value to which the element a satisfies the property A, while the membership value ν A (a) denotes the value to which a satisfies the not-property of A, in IFSs. BFSs and IFSs are different extensions of FSs since a counter-property is not always equal to not-property. Their differences can be seen in how they view an element a with a membership value of (0, 0). It is considered that the element a does not satisfy both the property A and its implicit counter-property by taking into account the perspective of bipolar fuzzy set A. It implies that it is unaffected by the property and its implicit counterproperty. When this situation is considered in terms of NS A, the variable a is noted as not satisfying the property and its not-property. Menger (1942) established the use of triangular norms (t-norms). Menger discussed using probability distributions instead of numbers to solve the problem of calculating the distance between two elements in space. In metric space conditions, t-norms are used to make generalisations with the probability distribution of triangle inequality. Dual operations of TNs are triangular conorms (t-conorms).

The functions μ r A (u, v, γ ) and μ A (u, v, γ ) represent the value of closeness and the value of distance between u and v with respect to γ , respectively.

In case of μ r A (u) = μ A (u) = 0 for all u ∈ A every fuzzy metric space is bipolar fuzzy metric space.

We can see from the definitions above that if we select δ 1 , δ 2 ∈ (0, 1) taken together δ 1 > δ 2 , there is δ 3 , δ 4 ∈ (0, 1) such that δ 1 Δδ 3 ≥ δ 2 and δ 1 ≥ δ 4 ∇δ 2 . Also, in case of δ 5 ∈ (0, 1) there is δ 6 , δ 7 ∈ (0, 1) such that δ 6 Δδ 6 ≥ δ 5 and δ 7 ∇δ 7 ≤ δ 5 .

Example 1 Let (A, d) be a crisp metric space. Give the functions Δ and ∇ as default uΔv = min{u, v}, u∇v = max{u, v}. μ r

for all u, v ∈ A and γ > 0. Then, (A, B, Δ, ∇) is a bipolar fuzzy metric space.

In this section, the topological structure of bipolar fuzzy sets is examined. In this way, we deal with the structural qualities of BFMS, such as open ball, open set, T 2 space, completeness, and nowhere dense.

is called an open ball because it has a middle point u and a radius δ in relation to γ .

is a open ball together u as the center and δ as the radius. Let us take (u, δ, γ ) . The proof is now complete.

Additionally, if all Cauchy sequence is convergent in respect of τ B , (A, B, Δ, ∇) be a complete bipolar fuzzy metric space.

Remark 4 Any compact set in bipolar fuzzy metric space is closed and bounded.

Proof Assume that X is a compact subset of A, a bipolar fuzzy metric space. Let γ > 0, δ ∈ (0, 1) be the case. Take into account the {φ(u, δ, γ ) : u ∈ X } is a open cover. Because of the fact that X is compact, there exist u 1 , u 2 , ...,

As a result, X is bipolar fuzzy bounded.

Theorem 5 Let (A, B, Δ, ∇) be a BFMS with a convergent subsequence for every Cauchy sequence in A. After that, (A, B, Δ, ∇) is complete.

Proof Let (u k ) be a Cauchy sequence and let (u k n ) be a subsequence of (u k ) that converges to u. We must prove that u k → u. Let γ > 0 and ε ∈ (0, 1). Choose δ ∈ (0, 1) such that

is complete but X is not closed. Take u ∈ X \ X . Then there will be a sequence of points in X called (u k ) that converges to u, and (u k ) is a Cauchy sequence. It means that, for each δ ∈ (0, 1) and

As a result, (u k ) converges to both u and v in (A, B, Δ, ∇). Because of the fact that u / ∈ X and v ∈ X , u = v. This is an inconsistency. (A, B, Δ, ∇) . In the case of γ > 0, δ, ε ∈ (0, 1) and

be an open ball with center u and radius ε. Proof Let ∅ = W ⊂ X be the case. Then there is a open set Z = ∅ in which Z ⊂ W and Z ∩ X = ∅ are valid. Let u ∈ Z . After that, there's δ ∈ (0, 1) and γ > 0 φ(u, δ, γ ) ⊂ Z .

Consider ε ∈ (0, 1) so that (1 − ε)Δ(1 − ε) ≥ 1 − δ and (ε − 1)∇(ε − 1) ≤ δ − 1. By taking into account Lemma 1 φ (u, ε, γ 2 ) ⊂ φ (u, δ, γ ) . It means that, φ(u, ε, γ 2 ) ⊂ W and φ (u, ε, γ 2 ) ∩ X = ∅. Assume, on the other hand, that X is not dense anywhere. Then, int(X ) = ∅, implying that there is a nonempty open set W such that W ⊂ X . Let φ (u, δ, γ ) be an open ball with the property that φ(u, δ, γ ) is a subset of W . Then φ(u, δ, γ ) ∩ X = ∅ comes into play. This is an inconsistency.

For bipolar fuzzy metric space, we can prove Baire's theorem. 

If we keep going this way, we obtain a sequence (u k ) in A and a sequence (γ k ) such that 0 < γ k < 1 k and φ(u k , δ k , γ k ) ⊂ φ(u k−1 , δ k−1 , γ k−1 ) ∩ W k . Now we will take into account that (u k ) is a Cauchy sequence. For γ > 0 and ε > 0, select k 0 ∈ N such that 1 k 0 < γ and 1 k 0 < ε. Then for k ≥ k 0 and ≥ k,

As a result, the sequence (u k ) is a Cauchy sequence. Because of the fact that A is complete, u ∈ A arises such that u k → u. We offer u ∈ φ(u k , δ k , γ k ) for

Note 1 The first group does not include any complete bipolar fuzzy metric space that cannot be described as the union of a sequence of nowhere dense sets. As a result, any complete bipolar fuzzy metric space belongs to the second group.

Remark 5 Since every metric induces a bipolar fuzzy metric, and bipolar fuzzy metric is a general statement of fuzzy metric, Baire's theorem for complete metric space (Zhang 1998) and Baire's theorem for complete fuzzy metric space (Qurashi and Shabir 2018) are special cases of the above theorem.

The topological structure of bipolar fuzzy sets is investigated in further depth in this section, as well as the Uniform Limit Theorem. (A, B, Δ, ∇) . A set [D k ] k∈N is said to have bipolar fuzzy diameter zero if there exists k 0 ∈ N for each δ ∈ (0, 1) and each γ > 0 so that μ r A (u, v, γ ) > 1 − ε and μ A (u, v, γ ) < ε − 1 for all u, v ∈ D k 0 . Remark 6 If and only if D is a singleton set, a nonempty subset D of a bipolar fuzzy metric space A has bipolar fuzzy diameter zero.

Proof Assume that the specified condition exists first. (A, B, Δ, ∇) , we imply, is complete. In A, consider (u k ) to be a Cauchy sequence. The set N k = {u n : n ≥ k} and D k = N k then [D k ] is said to have a bipolar fuzzy diameter zero. For taken ε ∈ (0, 1) and γ > 0, we select δ ∈ (0, 1) as a result (

has bipolar fuzzy diameter zero and as a result of hypothesis the variable ∩ k∈N D k is nonempty.

Take u ∈ ∩ k∈N D k . We show that u k → u. Then for δ ∈ (0, 1) and γ > 0, there exists

In the opposite case, assume (A, B, Δ, ∇) is complete and [D k ] n∈N is a nested sequence of nonempty closed sets with bipolar fuzzy diameter zero. Get a point u k ∈ D k for each k ∈ N. A Cauchy sequence is defined as (u k ). Because of the fact that [D k ] has bipolar fuzzy diameter zero, for γ > 0 and δ ∈ (0, 1), there is k 0 ∈ N so that μ r A (u, v, γ ) > 1 − δ and μ A (u, v, γ ) < δ −1 for all u, v ∈ D k 0 . Since, [D k ] is nested sequence μ r A (u k , u , γ ) > 1−δ and μ A (u k , u , γ ) < δ −1 for all , k ≥ k 0 . As a consequence, the sequence (u k ) is a Cauchy sequence. Since (A, B, Δ, ∇) is complete, u k → u for some u ∈ A. As a result, for every k, u ∈ D k = D k and u ∈ ∩ k∈N D k . The proof is now finished.

The element u ∈ ∩ k∈N D k is singular. If two elements are present as u, v ∈ ∩ k∈N D k , since [D k ] has bipolar fuzzy diameter zero, for each fixed γ > 0, μ r A (u, v, γ ) > 1 − 1 k and μ A (u, v, γ ) < 1 k − 1 for each k ∈ N. This denotes μ r A (u, v, γ ) = 1 and μ A (u, v, γ ) = −1 and consequently u = −v.

It's worth noting that the regular bipolar fuzzy metric and the corresponding metric produce the same topologies. But here's what we've got:

Theorem 10 Every separable bipolar fuzzy metric space is second countable.

Proof The provided separable bipolar fuzzy metric space is (A, B, Δ, ∇) . A countable dense subset of A will be X = {u k : k ∈ N}. Consider the case of the family K = {φ(u j , 1 n , 1 n ) : j, n ∈ N}. Then K can be counted. We conclude that K is a base for the family of all open sets in A. Take O be any open set in A and u ∈ O. Then there is γ > 0 and δ ∈ (0, 1) such that φ(u, δ, γ ) ⊂ O. Because of the fact that δ ∈ (0, 1), ε ∈ (0, 1) can be selected such that Finally, v ∈ φ(u, δ, γ ) ⊂ O and as a result K is a base.

Since second countability is an inherited property and second countability means separability, we get: A separable bipolar fuzzy metric space has separable subspaces.

Definition 10 Let A be a set that is not empty and (B, B, Δ, ∇) be an bipolar fuzzy metric space. If we take γ > 0 and δ ∈ (0, 1), then a sequence (g k ) of functions from A to B converges uniformly to a function g from A to B and k 0 ∈ N arises so that μ r A (g k (u), g(u), γ ) > 1 − δ and μ A (g k (u), g(u), γ ) > δ − 1 for all k ≥ k 0 and u ∈ A.

Theorem 11 (Uniform limit theorem) Let g k : A → B denote a sequence of continuous functions connecting a topological space A to a bipolar fuzzy metric space(B, B, Δ, ∇). Then, g is continuous, if (g k ) converges uniformly to g : A → B.

Proof Assume Z is an open set of B and allow u 0 ∈ g −1 (Z ). We want to explore a W neighborhood of

Thus g(u) ∈ φ(g(u 0 ), δ, γ ) ⊂ Z for all u ∈ W . From here, g(W ) ⊂ Z and so g is continuous.

The COVID-19 pandemic has emerged as a serious public health emergency and prompted a process to be addressed. As of March 11, 2020, the World Health Organization declared this event as a public health emergency in accordance with the International Health Regulations. It is important to choose the most appropriate vaccine for the groups to be administered COVID-19 vaccine by evaluating the risks of exposure to the disease, the risks of spreading and transmitting the disease, and the negative impact of the disease on the functioning of social life. Until now, serious side effects have not been encountered in both clinical trials conducted for COVID-19 vaccines and current vaccination practices. Post-vaccination side effects are often mild.

These; mild side effects such as fatigue, headache, fever, chills, muscle pain, vomiting, diarrhea, pain, redness, swelling in the area where the vaccine was applied.

No vaccine can be approved or put on the market without conducting phase studies and without transparently sharing the results of these studies with the relevant authorities. The organization conducting vaccine studies must comply with international quality standards during the development and production process. These standards are Good Laboratory Practices (GLP), Good Clinical Practices (GCP) and Good Manufacturing Practices (GMP). In addition, during the phase studies, when a situation that is thought to be related to the vaccine and significantly affects human health is detected, the studies are stopped. Studies are carried out only if it is ensured that the problem is not related to the vaccine, and if vaccination is performed, it is continued from where it left off. However, it is still important to choose the most suitable vaccine for the person.

Several different types of vaccines are being developed for COVID-19. All of these vaccines are designed to teach the body's immune system to safely introduce and destroy the virus that causes COVID-19. We evaluate these vaccines in five categories.

1. Vaccines that do not cause disease but contain inactivated virus (Inactivated vaccines) that produce an immune response. (L 1 ) 2. Vaccines containing attenuated virus that do not cause disease but produce an immune response (Live attenuated vaccines). (L 2 ) 3. Protein-based vaccines that use protein fragments that mimic the structure of the COVID-19 virus to safely induce an immune response.(L 3 ) 4. Viral vector vaccines using non-pathogenic viruses that carry RNA particles of the COVID-19 virus to create a safe immune response. (L 4 ) 5. M-RNA and DNA vaccines, a state-of-the-art approach uses genetically engineered RNA and DNA fragments to produce proteins that induce a safe immune response on its own.

In this section, we have included an application that shows which of these 5 vaccine types is more suitable. Appropriate measurements of similarity were used in this application that allow us to make a comparison between selected bipolar fuzzy sets. Now, let's start by giving a definition of similarity measure on BFS.

Definition 11 (Abdullah et al. 2014) . Let U be the universe of discourse and E be the set of parameters and B ⊆ E. Let ψ : B → B F U be a mapping, then a bipolar fuzzy soft set (BFSS) (ψ, B) or ψ B is defined by 

(μ r 11 , μ 11 ) ψ 2 (μ r 12 , μ 12 ) ψ 2 · · · (μ + 1n , μ 1n ) ψ 2 (μ r 21 , μ 21 ) ψ 2 (μ r 22 , μ 22 ) ψ 2 · · · (μ r 2n , μ 2n ) ψ 2 . . . . . . . . . . . .

are BFS-matrices (ψ 1 , P) and (ψ 2 , P), then similarity measure between (ψ 1 , P) and (ψ 2 , P) is given as below:

Sim(ψ 1 , ψ 2 ) = ψ 1 , ψ 2 ψ 1 ψ 2 .

Here

The similarity measure defined in Definition 12 satisfies the following situations:

1. 0 ≤ Sim(ψ 1 , ψ 2 ) ≤ 1 2. Sim(ψ 1 , ψ 2 ) = 1 ⇔ ψ 1 = ψ 2 3. Sim(ψ 1 , ψ 2 ) = Sim(ψ 2 , ψ 1 ).

Let ψ 1 and ψ 2 be two BFSSs. Suppose that the weights of attributes p j are ϑ j ∈ [0, 1] for j = 1, 2, . . . , n. The weighted similarity measure Sim W between ψ 1 and ψ 2 is defined by

The similarity measure defined in Definition 13 provides the circumstances listed below:

Example 3 Let us take ψ 1 and ψ 2 , BFS-matrices of (ψ 1 , p) and (ψ 2 , p) as given below: Then similarity measure between (ψ 1 , p) and (ψ 2 , p) will be determined as follows:

ψ 1 , ψ 2 = 3.7960

∵ Sim(ψ 1 , ψ 2 ) = 0.8203. Additionally, consider ϑ 1 = 0.12, ϑ 1 = 0.67 and ϑ 1 = 0.54 as weights of p 1 , p 2 and p 3 . After, the subsequent computations can be made:

Step 1: Notice the set of type of vaccines L = {L 1 , L 2 , . . . , L n }.

Step 2: Notice the sets of requirement M = {α 1 , α 2 , . . . , α l } and parameters E = {e 1 , e 2 , . . . , e m } and a group of decision makers D = {d 1 , d 2 , . . . , d k }.

Step 3: Create a BFSS (ψ, E) model to compute the similarity measure.

Step 4: Determine the BFS-decision matrix for each vaccine type by each decision maker.

Step 5: Search the unanimous decision matrix and make a table with (ψ, E) and (ψ i , E).

Step 6: Determine the degree of similarity between each BFSS (ψ i , E) and the model BFSS (ψ, E).

Step 7: Determine the threshold measure ξ ∈]0, 1[.

Step 8: The vaccine with the highest similarity measure, ≥ ξ , is the best option.

Step 9: Make a final classification.

The materials required of Algorithm 1 are illustrated in Figs. 1 and 2.

A government intends to purchase vaccines from corporations that have the vaccine due to the COVID-19 outbreak. The government selected two decision makers to determine the best type of vaccines from the five types of vaccines suitable for its conditions along with particularities C = {α 1 , α 2 , α 3 }. Here, which are linguistic identifiers that define the degree of satisfaction with decision-making when evaluating a vaccine. Assume that (ψ, E) is the regular BFSS, which is considered as a model BFSS by the decision maker. A positive grade is given if the effectiveness of the type of vaccine is more appropriate than the country's expectation standard. A negative grade is also granted if the suitability of the type of vaccine is less effective based on certain parameters. The government determines two decision makers from the management committee to assign the best form of vaccine to administer to the population. Conclude that (ψ i , E) is the BFSS model of the vaccine kind, with L i , i = 1, 2, 3, 4, 5 being assigned by each decision manager. Then, by considering the maximum of the positive values and the minimum of the negative values, we arrive at a consensus BFS decision. To determine the SM between these BFSSs, we will compare the vaccines' characteristics with the analysis in Table 3 . BFS decision matrices are shown in Tables 1 and 2. Assume that the threshold value is 0.8. When we class with this value to the determined value of SM in Table 3 , it would seem that the BFSS model (ψ, E) is very similar to (ψ 1 , E). Also, it means that, it is unlike any of the other sets. We acknowledge that the form of vaccine L 1 is the best option for satisfying certain requirements. Since L 1 is the most similar to model, the government should select it as their country's vaccine provider.

In Table 4 , the SMs between BFS-model and vaccine form are listed as below:

To compute the weighted SMs, we use the weights w 1 = 0.119, w 2 = 0.452, and w 3 = 0.817, which correspond to e 1 , e 2 , and e 3 , respectively. By this way we calculate the weighted SMs. Table 5 becomes L 1 L 4 L 5 L 2 L 3 when ranking vaccine types based on weighted similarity measure.

Thus, as shown in the Tables 4, 5, it can be easily said that L 1 is the most suitable and preferable vaccine option. Table 4 becomes L 1 L 5 L 4 L 3 L 1 when ranking vaccine types based on similarity measure. Table 5 shows the weighted SM between the BFSS model and the vaccine type as given in the follows:

The ranking of these two measures for each type of vaccines provider could be seen in Fig. 3 .

The aim of this research is to define bipolar metric spaces and investigate some of their properties. The characteristic features of BFMSs have been developed, including open set, T 2 -space, compactness, completeness and nowhere dense. Following that, Baire Category and Uniform Convergence Theorems for bipolar metric spaces are presented. Afterwards, an application on determining the appropriate type of vaccine in the treatment process of COVID-19 is given using similarity measure between bipolar fuzzy metric spaces. For vaccine selection and supply chain management, an advanced multi-attribute decision-making (MADM) algorithm is being developed using similarity measure between bipolar fuzzy metric spaces. The validity of the proposed MADM method for selecting the best proper form of vaccine is also established.

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