Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2021.014174

ARTICLE

Multi-Criteria Decision Making Based on Bipolar Picture Fuzzy Operators and New Distance Measures

Muhammad Riaz1, Harish Garg2, Hafiz Muhammad Athar Farid1 and Ronnason Chinram3,4,*

1Department of Mathematics, University of the Punjab, Lahore, 54590, Pakistan
2School of Mathematics, Thapar Institute of Engineering and Technology (Deemed University), Patiala, 147004, India
3Algebra and Applications Research Unit, Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla, 90110, Thailand
4Centre of Excellence in Mathematics, Bangkok, 10400, Thailand
*Corresponding Author: Ronnason Chinram. Email: ronnason.c@psu.ac.th
Received: 06 September 2020; Accepted: 21 December 2020

1  Introduction

In any real-life problem-solving technique, the complexity characterizes the behavior of an object whose components interrelate in multiple ways and follow different logical rules, meaning there is no fixed rule to handle multiple challenges due to various uncertainties in real life circumstances. Many scholars from all over the world have apparently studied MCDM management techniques extensively. This effort resulted in a multitude of innovative solutions to complex real concerns. The frameworks for this objective are largely based on a summary of the issues at hand. To deal with uncertainties the researchers have been proposed various mathematical techniques. Zadeh [1] initiated the idea of fuzzy set (FS) and membership degrees of objects/alternatives. Later, the intuitionistic fuzzy set (IFS) proposed by Atanassov [2] is the direct extension of FS by using membership degrees (MDs) and non-membership degrees (NMDs). Yager et al. [3,4] and Yager et al. [5] introduced Pythagorean fuzzy set and Pythagorean fuzzy membership grades. Zhang et al. [6,7] introduced an independent extension of fuzzy set named as bipolar fuzzy sets (BFSs) and Lee [8] presented some basics operations. A bipolar fuzzy information is used to express a property of an object as well as its counter property.

Alcantud et al. [9] initiated the notion of N-soft set approach to rough sets and introduced the concept of dual extended hesitant fuzzy sets [10]. Akram et al. [11,12] initiated MCDM based on Pythagorean fuzzy TOPSIS method and Pythagorean Dombi fuzzy AOs. Ashraf et al. [13] initiated spherical fuzzy Dombi AOs. Eraslan et al. [14] and Feng et al. [15] proposed new approaches for MCDM. Garg et al. [16–18] introduced some AOs on different sets also their applications to MCDM. Jose et al. [19] proposed AOs for MCDM. Karaaslan [20], Liu et al. [21], Liu et al. [22], Wang et al. [23], Yang et al. [24], Smarandache [25], and Liu et al. [26] initiated many different approaches including AOs on different extension of fuzzy set for MCDM. Naeem et al. [27,28], Peng et al. [29,30], Peng et al. [31] introduced some significant results for Pythagorean fuzzy sets.

Riaz et al. [32], initiated the concept of linear Diophantine fuzzy Set and its applications to MCDM. Riaz et al. [33] introduced some hybrid AOs, Einstein prioritized AOs [34], related to q-ROFSs. Riaz et al. [35] introduced cubic bipolar fuzzy set and related AOs. Cagman et al. [36], and Shabir et al. [37] independently introduced the notion of soft topological spaces.

Cuong [38] presented the idea of a picture fuzzy set (PFS) as a new paradigm distinguished with three functions that assign the positive membership degree (MD), the neutral MD and the negative membership degree (NMD) to each object/alternative. The basic restrictions on these degrees are that they lie in [0, 1] and their sum also lies in [0, 1]. Cuong [39] further introduced the concept of Pythagorean picture fuzzy sets and its basic notions. Garg [40], Jana et al. [41] and Wang et al. [42] proposed some AOs for picture fuzzy information aggregation. Pamucar [43] studied the notion of normalized weighted geometric Dombi Bonferoni mean operator with interval grey numbers: Application in multicriteria decision making. Pamucar et al. [44] proposed an application of the hybrid interval rough weighted Power-Heronian operator in multi-criteria decision making. Ramakrishnan et al. [45] introduced a cloud TOPSIS model for green supplier selection. Riaz et al. also introduced some AOs [46,47] related to green supplier selection. Si et al. [48] and Sinani [49] also presented different AOs in some extension of fuzzy set.

The first objective of this paper is to introduce bipolar picture fuzzy sets (BPFSs) as a new hybrid structure of bipolar fuzzy sets (BFSs) and picture fuzzy sets (PFSs). BPFSs are more efficient for dealing with the real-life situation when modeling needs to address the bipolarity (both positive and negative aspects) to each MD (belonging-ness), neutral membership (not decided), and non-membership degree (refusal). The second objective of BPFSs is propose bipolar picture fuzzy MCDM technique based on bipolar picture fuzzy AOs. The third objective of BPFSs is to define new distance measure and its application towards pattern recognition. Additionally, the proposed methodology can extend to solve various problems of artificial intelligence, computational intelligence and MCDM that involve bipolar picture fuzzy information.

The rest of the paper is as follows. The definitions of IFS, PFS and BFS are discussed in Section 2. Section 3 introduces the definition of BPFS. Section 4 indicates some bipolar picture AOs and new BPFS distance measures. Section 5 shows the generalizability of the suggested paradigm for pattern recognition. Section 6 introduces a new BPF-MCDM approach based on suggested AOs and a numerical example. Finally, Section 7 summarizes the findings of this research study.

2  Preliminaries

In this section, we give some basic definitions to IFSs, BFSs and PFSs.

Definition 2.1 [2] Let Ψ=(δ1,δ2,…δn) be a crisp set, an IFS J in Ψ is defined by

J={〈δ,μJ(δ),νJ(δ)〉:δ∈Ψ}(1)

where 0≤μJ(δ)≤1, 0≤νJ(δ)≤1 and 0≤μJ(δ)+νJ(δ)≤1, ∀δ∈Ψ. πJ(δ)=1-(μJ(δ)+νJ(δ)) is called indeterminacy degree (ID) of J in Ψ. Also 0≤πJ(δ)≤1∀δ∈Ψ.

Definition 2.2 [38] Let W be a crisp set, a PFS A in W is defined as follows:

A={〈δ,μA(δ),λA(δ),νA(δ)〉∣δ∈W}(2)

where, μA(δ)∈[0,1] is called positive MD of δ in A, λA(δ)∈[0,1] is called neutral MD of δ in A, νA(δ)∈[0,1] is called negative MD of δ in A, and μA(δ), λA(δ), νA(δ) satisfy the condition 0≤μA(δ)+λA(δ)+νA(δ)≤1(∀δ∈W) and 1-(μA(δ)+λA(δ)+νA(δ)) is called refusal MD of A.

A basic element 〈δ,μA(δ),λA(δ),νA(δ)〉 in a PFS A is denoted by Ã=〈μA,λA,νA〉, which is called picture fuzzy number (PFN).

Definition 2.3 [38] Some operational laws of picture fuzzy set as follows:

Let P1={〈δ,μP1(δ),λP1(δ),νP1(δ)〉∣δ∈W} and P2={〈δ,μP2(δ),λP2(δ),νP2(δ)〉∣δ∈W} be any two PFS. Then

1. P1⊆P2 iff,

μP1(δ)≤μP2(δ),λP1(δ)≤λP2(δ),νP1(δ)≥νP2(δ).

2. P1 = P2 iff,

μP1(δ)=μP2(δ),λP1(δ)=λP2(δ),νP1(δ)=νP2(δ).

3. The complement of P1 is defined by

P1c={(νP1(δ),λP1(δ),μP1(δ))∣δ∈X}.

4. The union is defined by

P1∪P2={(δ,max(μP1(δ),μP2(δ)),min(λP1(δ),λP2(δ)),min(νP1(δ),νP2(δ))):δ∈W}.

5. The intersection is defined by

P1∩P2={(δ,min(μP1(δ),μP2(δ)),max(λP1(δ),λP2(δ)),max(νP1(δ),νP2(δ))):δ∈W}.

Definition 2.4 [40] The score function of a PFN δ=〈μA,λA,νA〉 is defined as

ℜ(P)=μA-λA-νA

However, in certain circumstances, the previous score function may not rank any two PFNs. For example, P1 = (0.7, 0.2, 0.1) and P2 = (0.6, 0.1, 0.1) then they have same score function values, i.e., R(P1) = R(P2). For this we use accuracy function given as

ℑ(P)=μA+λA+νA

Let P1=〈μ1,λ1,ν1〉 and P2=〈μ2,λ2,ν2〉 are any two PFNs, and R(P1), R(P2) are the score function of P1 and P2, and ℑ(P1), ℑ(P2) are the accuracy function of P1 and P2, respectively, then

1.    If R(P1) > R(P2), then P1 > P2.

2.    If R(P1) = R(P2), then,

If ℑ(P1)>ℑ(P2) then P1 > P2,

If ℑ(P1)=ℑ(P2), then P1 = P2.

Definition 2.5 [6] Let X be a set, a BFS B inX is defined as follows:

B={〈δ,μB+(δ),μB-(δ)〉∣δ∈X}(3)

where μB+(δ):X→[0,1] and μB-(δ):X→[-1,0]. The positive MD μB+(δ) demonstrates the satisfaction degree of an element δ to the property corresponding to a BFS B, negative MD μB-(δ) denotes the satisfaction degree of an element δ to some implicit counter-property of B.

Definition 2.6 [6,7] Some operational laws of bipolar fuzzy set as follows:

Let B={〈δ,μB+(δ),μB-(δ)〉∣δ∈W}, B1={〈δ,μB1+(δ),μB1-(δ)〉∣δ∈W} and B2={〈δ,μB2+(δ),μB2-(δ)〉∣δ∈W} be three bipolar fuzzy sets. Then

1. B1⊆B2 iff,

μB1+(δ)≤μB2+(δ)andμB1-(δ)≥μB2-(δ)

2. B1 = B2 iff,

μB1+(δ)=μB2+(δ)andμB1-(δ)=μB2-(δ).

3. The complement of B1 is denoted by B1c,

B1c={(1−μB1+(δ),−1−μB1+(δ)∣δ∈W}.

4. The union is defined by

B1∪B2={(δ,max(μB1+(δ),μB2+(δ)),min(μB1+(δ),μB2+(δ)) for all δ∈W}.

5. The intersection is defined by

B1∩B2={(δ,min(μB1+(δ),μB2+(δ)),max(μB1+(δ),μB2+(δ)) for all δ∈W}.

6. α-cut (Bα) of B,

Bα=Bα+∪Bα-Bα+={δ∣μB+(δ)≥α}Bα-={δ∣μB-(δ)≤-α}

Here, Bα+ is called positive α-cut and Bα- is called negative α-cut.

7. Support (shortly,Supp(B)) of B,

Supp(B)=Supp(B)+∪Supp(B)-Supp(B)+={δ∣μB+(δ)>0}Supp(B)-={δ∣μB-(δ)<0}

Here, Supp(B)+ is called positive α-cut and Supp(B)- is called negative α-cut.

3  Bipolar Picture Fuzzy Set

The BFS assign positive and negative grades to the alternatives and PFS is characterized by three functions expressing the MD, the neutral MD and the NMD. Fuzzy set assign a membership grade to each alternatives δ in the unit closed interval [0, 1]. In BFS the positive MD μλ+(δ) represent the satisfaction degree of an alternative δ to the property corresponding to a BFS λ, and negative MD μλ-(δ) represent the satisfaction degree of an element δ to some implicit counter-property of λ. In PFS there are three types of grades, μA(δ)∈[0,1] is called positive MD of δ in A, λA(δ)∈[0,1] is called neutral MD of δ in A, νA(δ)∈[0,1] is called negative MD of δ in A, and where μA, λA, νA satisfy the condition 0≤μA(δ)+λA(δ)+νA(δ)≤1, ∀δ∈X.

We present the idea of BPFS as a new hybrid version of BFS and PFS. In this model of BPFS, we assign positive and negative grades for each MD (belonging-ness), neutral membership (not decided), and non-membership degree (refusal). We present specific examples to relate the proposed model with the real life applications. We define some operational laws of BPFS along with its score and accuracy functions.

Definition 3.1 A BPFS Ω on universe W is an abject of the form

Ω={〈δ,μΩ+(δ),λΩ+(δ),νΩ+(δ),μΩ-(δ),λΩ-(δ),νΩ-(δ)〉:δ∈W}(4)

where μ+,λ+,ν+:W→[0,1] and μ-,λ-,ν-:W→[-1,0] with conditions

0≤μΩ++λΩ++νΩ+≤1-1≤μΩ-+λΩ-+νΩ-≤00≤μΩ++λΩ++νΩ+-μΩ--λΩ--νΩ-≤2.

The positive MDs μΩ+(δ), λΩ+(δ), νΩ+(δ) demonstrate the truth MD, indeterminate MD and false MD of an element δ corresponding to a BPFS Ω and the negative MDs μΩ-(δ), λΩ-(δ), νΩ-(δ) demonstrate the truth MD, indeterminate MD and false MD of an element δ to some implicit counter-property corresponding to a BPFS Ω. Absolute BPFS assign (1, 0, 0, −1, 0, 0) to each alternative, denoted by 𝔘 and null BPFS assign (0, 0, 1, 0, 0, −1) to each alternative, denoted by 𝔑. Moreover ρΩ+=1-(μΩ++λΩ++νΩ+) is called the positive degree of refusal membership of δ in Ω and ρΩ-=-1-(μΩ-+λΩ-+νΩ-) is called the negative degree of refusal membership of δ in Ω.

Now we discuss some applications of proposed model to relate it with real life problems.

Business:

In the field of finance and business, we use two terms profit and loss. We can relate the decision-making applications based on business with BPFS. If a person invests some money, then he wants to earn max profit in some interval of time. Bipolar picture fuzzy number (BPFN) can be described as

〈μ+,λ+,ν+,μ-,λ-,ν-〉.

The physical meaning of this structure in business terms is that, what is the satisfactions grade that he earns profit (μ+), what is the dissatisfactions grade that he earns profit (ν+), what is the abstinence grade that he earns profit (μ+), what is the satisfactions grade that he gets loss (μ-), what is the dissatisfactions grade that he gets loss (ν-) and what is the abstinence grade that he gets loss (λ-). We can see how BPFN is useful for the decision-making problems of real life problems. The detail of component of BPFN for finance and business is given in Tab. 1.

Table 1: Tabular representation of BPFN under business related problems

Medication:

In the field of medical, we mostly focus on the effects and side effects of medicines related to every disease. If a patient get some medication according to his type of disease, then we can relate our model to the effects and side effects of that medicine in medical diagnosis, treatment and recovery terms. For the BPFN can be written as

〈μ+,λ+,ν+,μ-,λ-,ν-〉.

μ+ represents the positive effects of recommended medicine to the disease of the patient, ν+ represents the dissatisfaction effects of recommended medicine, λ+ represents the abstinence effects of recommended medicine, μ- represents the negative or bad effects of recommended medicine, ν- represents the dissatisfaction grade of bad effects of recommended medicine and λ- represents the abstinence grades of side effects of recommended medicine. The detail of component of BPFN for medication is given in Tab. 2.

Table 2: Tabular representation of BPFS under medical related problems

The proposed model is superior than these two models, in fact it is hybrid structure of BFS and PFS that assign six grades to the alternative.

Comparison Analysis:

In this part, we discuss about the terms and characteristics of proposed model and compare it with the existing techniques. There are various objectives to construct this hybrid structure and some of them are listed below:

1.    The first objective to construct this hybrid model is to fill the research gap which exists in previous methodologies. The bipolar fuzzy set and picture fuzzy set can be used together in decision analysis. We can deal with the satisfaction, abstinence and dissatisfaction grades of the alternatives with its counter properties.

2.    The second objective is that we can cover the evaluation space in a different manner. If we compare our model with the existing theories then we find that it is strong, valid and superior to others. The comparison analysis of BPFS with the existing models is given in Tab. 3.

3.    The third objective is to represent the relationship of BPFS to the MCDM problems. We study some real life problems and convert the input data into BPF numeric values and deal it with the proposed aggregation operators. This novel structure is superior, flexible and easy to handle and can deal with the MCDM problems in the field of medical, business, artificial intelligence and engineering etc. The graphical representation of PFS and BPFS is given in the Figs. 1 and 2, respectively.

4.    In bipolar neutrosophic set (see [50]) the conditions are as follows:

0≤T++I++F+≤30≤-T--I--F-≤30≤T++I++F+-T--I--F-≤6

However, in the proposed model BPFS the conditions are as follows:

0≤μΩ++λΩ++νΩ+≤1-1≤μΩ-+λΩ-+νΩ-≤00≤μΩ++λΩ++νΩ+-μΩ--λΩ--νΩ-≤2

Table 3: Comparison of BPFS with the existing set theoretic models

Figure 1: Graphical representation for satisfaction, abstinence and dissatisfaction grades of picture fuzzy set. 0≤x+y+z≤1

Figure 2: Graphical representation for grades of bipolar picture fuzzy set

Definition 3.2 Let Ω1=〈δ,μ1+(δ),λ1+(δ),ν1+(δ),μ1-(δ),λ1-(δ),ν1-(δ)〉 and Ω2=〈δ,μ2+(δ),λ2+(δ),ν2+(δ),μ2-(δ),λ2-(δ),ν2-(δ)〉 be two BPFSs. Then Ω1⊆Ω2 iff

μ1+(δ)≤μ2+(δ),λ1+(δ)≤λ2+(δ),ν1+(δ)≥ν2+(δ) and

μ1-(δ)≥μ2-(δ),λ1-(δ)≥λ2-(δ),ν1-(δ)≤ν2-(δ)

Definition 3.3 Let Ω1=〈δ,μ1+(δ),λ1+(δ),ν1+(δ),μ1-(δ),λ1-(δ),ν1-(δ)〉 and Ω2=〈δ,μ2+(δ),λ2+(δ),ν2+(δ),μ2-(δ),λ2-(δ),ν2-(δ)〉 be two BPFSs. Then Ω1=Ω2 iff

μ1+(δ)=μ2+(δ),λ1+(δ)=λ2+(δ),ν1+(δ)=ν2+(δ)

and

μ1-(δ)=μ2-(δ),λ1-(δ)=λ2-(δ),ν1-(δ)=ν2-(δ).

Definition 3.4 Let Ω1=〈δ,μ1+(δ),λ1+(δ),ν1+(δ),μ1-(δ),λ1-(δ),ν1-(δ)〉 and Ω2=〈δ,μ2+(δ),λ2+(δ),ν2+(δ),μ2-(δ),λ2-(δ),ν2-(δ)〉 be two BPFSs.

The union of these two BPFSs is defined as

(Ω1∪Ω2)(δ)=(max(μ1+(δ),μ2+(δ)),min(λ1+(δ),λ2+(δ)),min(ν1+(δ),ν2+(δ)),min(μ1−(δ),μ2−(δ)),maδ(λ1−(δ),λ2−(δ)),maδ(ν1−(δ),ν2−(δ))).

Example 3.5 Let X={δ1,δ2,δ3}. Let us consider are two BPFSs Ω1, Ω2 in X given by

Ω1=〈δ1,0.5,0.2,0.2,-0.1,-0.2,-0.4〉,〈δ2,0.3,0.4,0.3,-0.35,-0.3,-0.3〉,〈δ3,0.3,0.4,0.2,-0.5,-0.1,-0.2〉Ω2=〈δ1,0.4,0.3,0.2,-0.2,-0.4,-0.2〉,〈δ2,0.5,0.1,0.3,-0.3,-0.5,-0.2〉,〈δ3,0.3,0.4,0.1,-0.3,-0.2,-0.4〉

Then their union is

Ω1∪Ω2=〈δ1,0.5,0.2,0.2,-0.2,-0.2,-0.2〉,〈δ2,0.5,0.1,0.3,-0.3,-0.3,-0.2〉,〈δ3,0.3,0.4,0.1,-0.5,-0.1,-0.2〉

Definition 3.6 LetΩ1=〈δ,μ1+(δ),λ1+(δ),ν1+(δ),μ1-(δ),λ1-(δ),ν1-(δ)〉 and Ω2=〈δ,μ2+(δ),λ2+(δ),ν2+(δ),μ2-(δ),λ2-(δ),ν2-(δ)〉 be two BPFSs.

The intersection of these two BPFSs is defined as

(Ω1∩Ω2)(δ)=(min(μ1+(δ),μ2+(δ)),maδ(λ1+(δ),λ2+(δ)),maδ(ν1+(δ),ν2+(δ)),max(μ1−(δ),μ2−(δ)),min(λ1−(δ),λ2−(δ)),min(ν1−(δ),ν2−(δ))).

Example 3.7 Let W={δ1,δ2,δ3}. Let us consider are two bipolar picture fuzzy set Ω1, Ω2 in W given by

Ω1=〈δ1,0.5,0.2,0.2,-0.1,-0.2,-0.4〉,〈δ2,0.3,0.4,0.3,-0.35,-0.3,-0.3〉,〈δ3,0.3,0.4,0.2,-0.5,-0.1,-0.2〉Ω2=〈δ1,0.4,0.3,0.2,-0.2,-0.4,-0.2〉,〈δ2,0.5,0.1,0.3,-0.3,-0.5,-0.2〉,〈δ3,0.3,0.4,0.1,-0.3,-0.2,-0.4〉.

Then their intersection is

Ω1∩Ω2=〈δ1,0.4,0.3,0.2,-0.1,-0.4,-0.4〉,〈δ2,0.3,0.4,0.3,-0.3,-0.5,-0.3〉,〈δ3,0.3,0.4,0.2,-0.3,-0.2,-0.4〉

Definition 3.8 Let Ω=〈δ,μ+(δ),λ+(δ),ν+(δ),μ-(δ),λ-(δ),ν-(δ)〉 be a BPF set in W. Then the compliment of Ω is denoted by Ωc and defined as, for all δ∈W,

Ω=〈δ,ν+(δ),λ+(δ),μ+(δ),ν-(δ),λ-(δ),μ-(δ)〉

Example 3.9 Let W={δ1,δ2,δ3}. Consider a BPFS Ω in W given by

Ω=〈δ1,0.3,0.2,0.4,-0.1,-0.3,-0.4〉,〈δ2,0.3,0.4,0.3,-0.2,-0.2,-0.4〉,〈δ3,0.3,0.4,0.1,-0.2,-0.1,-0.4〉

Then its complement is

Ωc=〈δ1,0.4,0.2,0.3,-0.4,-0.3,-0.1〉,〈δ2,0.3,0.4,0.3,-0.4,-0.2,-0.2〉,〈δ3,0.1,0.4,0.3,-0.4,-0.1,-0.2〉

Now we see that BFS and PFS are special cases of BPFS.

Proposition 3.10 BFS and PFS are special cases of BPFS, i.e., Bipolar fuzzy numbers (BFNs) and picture fuzzy numbers (PFNs) are special cases of the bipolar picture fuzzy numbers (BPFNs).

Proof. For any δ∈X, consider a BPFN given by, 〈μ+(δ),λ+(δ),ν+(δ),μ-(δ),λ-(δ),ν-(δ)〉. Then by setting the components λ+(δ), ν+(δ), λ-(δ), ν-(δ) equals to zero, we obtain a BFN, 〈μ+(δ),μ-(δ)〉.

Similarly, by setting the components μ-(δ), λ-(δ), ν-(δ) equal to zero, we obtain we obtain a PFN, 〈μ+(δ),λ+(δ),ν+(δ)〉 which can be written as, 〈μ(δ),λ(δ),ν(δ)〉. This complete the proof.

Theorem 3.11 Let Ω1,Ω2 and Ω3 be the BPFSs in a universe X, then we have

1.    Ω1∪Ω1=Ω1 and Ω1∩Ω1=Ω1

2.    Ω1∪Ω2=Ω2∪Ω1 and Ω1∩Ω2=Ω2∩Ω1

3.    (Ω1c)c=Ω1

4.    (Ω1∪Ω2)∪Ω3=Ω1∪(Ω2∪Ω3)

5.    (Ω1∩Ω2)∩Ω3=Ω1∩(Ω2∩Ω3)

6.    Ω1∪(Ω1∩Ω2)=Ω1

7.    Ω1∩(Ω1∪Ω2)=Ω1

8.    Ω1∪(Ω2∩Ω3)=(Ω1∪Ω2)∩(Ω1∪Ω3)

9.    Ω1∩(Ω2∪Ω3)=(Ω1∩Ω2)∪(Ω1∩Ω3)

10.   Ω1∪Ω1c≠𝔘 and Ω1∩Ω1c≠𝔑

Proof. The proof is obvious.

Theorem 3.12 Let O and P be the BPFSs in a universe X, then we have

1.    (O∪P)c≠Oc∩Pc

2.    (O∩P)c≠Oc∪Pc

We will denote the set of all BPFSs in X by 𝔛.

Definition 3.13 Let ϕ1=〈μ1+,λ1+,ν1+,μ1-,λ1-,ν1-〉 and ϕ2=〈μ2+,λ2+,ν2+,μ2-,λ2-,ν2-〉 be two BPFNs then

1.    ϕ1∨ϕ2=(max(μ1+,μ2+),min(λ1+,λ2+),min(ν1+,ν2+),min(μ1-,μ2-),max(λ1-,λ2-),maδ(ν1-,ν2-))

2.    ϕ1∧ϕ2=(min(μ1+,μ2+),max(λ1+,λ2+),max(ν1+,ν2+),max(μ1-,μ2-),min(λ1-,λ2-),min(ν1-,ν2-))

3.    ϕ1c=〈ν1+,λ1+,μ1+,ν1-,λ1-,μ1-〉

4.    ϕ1λ=((μ1++λ1+)λ-(λ1+)λ,(λ1+)λ,1-(1-ν1+)λ,((-μ1-)+(-λ1-))λ-(-λ1-)λ,(-λ1-)λ,1-(1-(-ν1-))λ)

5.    ϕ1⊗ϕ2=((μ1++λ1+)(μ2++λ2+)−λ1+λ2+,λ1+λ2+,1−(1−ν1+)(1−ν2+),((−μ1−)+(−λ1−))((−μ2−)+(−λ2−))−(−λ1−)(−λ2−),(−λ1−)(−λ2−),1−(1−(−ν1−))(1−(−ν2−))

4  Some Bipolar Picture Fuzzy Geometric Operators

In this section, firstly, we introduce score function, accuracy function, and certainty function for BPFNs. Secondly, we introduce BPFWG operator, BPFOWG operator, and BPFHG operator.

Definition 4.1 Let T1=〈μ1+,λ1+,ν1+,μ1-,λ1-,ν1-〉 be (BPFN). Then the score function Φ(T1), accuracy function Υ(T1) and certainty fruition Π(T1) of BPFN are defined as follows: