Multi-Attribute Group Decision-Making Method under Spherical Fuzzy Bipolar Soft Expert Framework with Its Application

1  Introduction

Considering complicated issues like falling economies, fastly spreading epidemics, global carbon footprints, environmental pollution, severe weather conditions due to rapid climate changes, energy crises, and depleting resources, many uncertainties arise in making profitable policy decisions. Concerning these issues, including the decision for an optimum object, ranking the policies concerning their possible outcomes, and predicting based on available data, decision sciences offer tools capable of handling vagueness and uncertainties effectively. Following the very first uncertainty theory of probability that was developed during the 16th century, several research scholars around the globe have made remarkable contributions to the decision-making process by introducing many powerful tools, which are capable of dealing with the uncertainties and complications of vast decision-making situations. Since the group decision-making procedure is getting much more complicated with ever-increasing ambiguities and global issues, continuous development in decision-making tools is necessary analogous to broadening decision requirements.

Concerning the uncertainties and partial information in the problems, classical set theories could not help in decision-making. To tackle these situations, Zadeh [1] revolutionized the decision sciences with his invention of fuzzy sets. These fuzzy sets allowed the solutions to uncertain problems by considering partiality of truth rather than perfect truth and perfect false. In contrast to classical sets with memberships 0 and 1 proclaiming only absolute conditions, Zadeh [1] assumed belongingness degrees for alternatives in the interval [0,1]. Therefore, the fuzzy sets provide information about the partial belongingness of things, contrary to their presence or absence. This revolutionary idea motivated many researchers to develop and base their decision-making models using fuzzy sets [2].

The fuzzy sets discuss only the membership of agreement about an object and therefore fail to solve problems that must consider both the belongingness and non-belongingness degrees. To incorporate disagreement degrees for such issues, Atanassov [3] came up with the intuitionistic fuzzy set (IFS) theory. These IFSs took both belongingness and non-belongingness degrees into consideration, corresponding to the decision-maker’s opinion about agreement and disagreement with a particular perspective, including the restriction that the sum of both these degrees must not exceed unity. Later, this model proved unsuccessful in handling problems with these two membership degrees summing up to values above one. For such problems, Atanassov [4] extended his IFSs to IFSs of the second type (type-2 IFSs). These type-2 IFSs allowed the sums of the belongingness and non-belongingness degrees to exceed unity with the new constraint that the sum of squares of these two degrees must keep from 0 to 1. Identical to these type-2 IFSs, Yager [5] introduced the Pythagorean fuzzy set (PyFS) model as an extension of the IFSs. Later, many decision-makers embraced this generalized form to develop more powerful hybrid decision models. Deveci et al. [6] provided a survey on the applications of PyFSs. Recently, Habib et al. [7] used PyFS-based multi-criteria decision-making (MCDM) methods for diagnosing the risk of eight different types of cancer in children.

Adding to the above scenarios, the situations like elections, surveys, and polls, often ask for a neutral opinion as well. For instance, a survey may ask for positive, negative and neutral opinions about various points. These scenarios require a third degree, i.e., the neutral membership degree in addition to the two prior discussed evaluation degrees. Considering such situations, Coung et al. [8–10] proposed picture fuzzy sets (PFSs) by considering negative, positive, and neutral membership degrees reflecting the disagreement, agreement, and neutrality in the decision-maker’s opinion. Currently, Singh et al. [11] studied applications of picture fuzzy correlation coefficients. A drawback of these PFSs is that they are not capable to tackle problems with negative, positive, and neutral memberships summing above unity. Considering the problems with this sum exceeding unity, Kahraman et al. [12] initiated the notion of spherical fuzzy sets (SFSs) by smoothening the restriction of the picture fuzzy sets with the condition that the sum of squares of positive, neutral, and negative memberships ranges from 0 to 1. Thus, it provided a generalization for the previous models. Later, Gündogdu et al. [13] proposed a TOPSIS (technique of order preference similarity to the ideal solution) approach for dealing with decision-making applications using generalized SFSs. Akram et al. [14] discussed a group decision-making mechanism based on the complex spherical fuzzy VIKOR method. Recently, Donyatalab et al. [15] introduced their new spherical fuzzy similarity and distance measures, and discussed their applicability in the diagnosis of diseases. Wang et al. [16] discussed a detailed site selection application for offshore wind power plants using SFSs under two-stage MCDM. Similarly, in the past few years, SFSs have been employed to tackle many decision-making situations [17–21]. Fig. 1 compares the IFSs, PyFSs, PFSs, and SFSs in a graphical manner.

Figure 1: Pictorial representation of IFS, PyFS, PFS, and SFS

One common limitation of all the above-discussed models is that they consider opinions respective to only a single parameter. However, generally, the decision-making problems are based on choices keeping multiple parameters under consideration. To integrate multiple parameters in one place, Molodtsov [22] introduced soft sets as the parameterized families of the provided universe of discourse. Maji et al. [23] discussed the decision-making applications of soft sets. Inspired by the parameterizing capability of soft sets, many hybrid structures have been developed in the last two decades by the combination of multiple uncertainty theories with soft sets. For instance, Sun et al. [24] combined linguistic variables with soft set theory to form linguistic valued soft sets with linguistic values. Perveen et al. [25] developed a spherical fuzzy soft set (SFSS) model by combining SFSs with soft sets. Guleria et al. [26] introduced T-spherical fuzzy soft sets (T-SFSSs) as the hybridization of T-SFSs with soft sets. Akram et al. [27] provided TOPSIS approach for dealing with complex spherical fuzzy information. In addition, complex spherical Dombi fuzzy aggregation operators were initiated by Akram et al. [28]. Zahid et al. [29] introduced an ELECTRE-based method using complex spherical fuzzy information for group decision-making. Akram et al. [30] discussed the decision-making applications of complex spherical fuzzy N-soft sets. Recently, Yang et al. [31] introduced T-spherical fuzzy Bonferroni mean operators and discussed their applications in multi-criteria decision-making. Manna et al. [32] presented an MADM method under a complex neutrosophic environment. Adeel et al. [33] provided a decision-making analysis using the ELECTRE-I approach under hesitant fuzzy N-soft sets. Some other recent works include [34–38]. Very recently, Liu et al. [39] presented a fuzzy soft expert prediction system to diagnose the COVID-19 (Coronavirus disease 2019).

The highly demanding problems, particularly those affecting important future considerations, require group decision-making. The previously discussed models are restricted to a single expert’s opinions and are not capable of making group decisions efficiently. To fulfill this requirement, Alkhazaleh et al. [40] developed soft expert sets (SESs), capable of dealing with multiple experts’ opinions in one place. Later, Alkhazaleh et al. [41] combined fuzzy sets with SESs and presented fuzzy soft expert sets (FSESs). Considering the group decision-making capabilities of SESs, lots of recent works focused on the hybridizations and improvements based on these sets. Bashir et al. [42] introduced possibility fuzzy soft expert sets by providing possibility degrees along with the fuzzy memberships. Fuzzy parameterized soft expert sets developed by Bashir et al. [43] allowed preference of parameters by assigning fuzzy values to parameters. Fusing SFSs with SESs, Perveen et al. [44] established spherical fuzzy-SES (SFSES) model and discussed its MAGDM applications. Combining the rating-based decision qualities with SFSs and SESs, Akram et al. [45] presented spherical fuzzy N-SESs and explored their MAGDM problems in survey-based election prediction and product review.

In considering the uncertainties with multiple parameters, many decision-making problems arise with bipolar parameters interrelated symmetrically. For instance, a medical test can be sensitive or specific, a person can be young or old, and a student can be dull or intelligent. Such dual behavior of parameters requires tools eligible to manage and depict this bipolarity. Shabir et al. [46] introduced bipolar soft sets (BSSs) composed of two soft sets; one based on the set of parameters, and the other based upon the not-set of parameters (one set contains the parameters opposite to those in the other set). Ali et al. [47] presented bipolar-SESs (BSESs) as the fusion of BSSs with SESs, and discussed their decision-making applications. Further extending BSESs for uncertain situations, Ali et al. [48] initiated fuzzy BSESs (FBSESs) and discussed their MAGDM applications. Recently, Ali et al. [49] ranked the fish passage designs for a hydropower project based on spherical fuzzy-BSSs (SFBSSs). Many other models based on BSSs came out in the recent years, e.g., q-rung orthopair fuzzy BSSs (qROFBSSs) [50]. Table 1 provides the list of abbreviations used in the paper. Table 2 represents the list of frequently used symbols throughout the work.

The complications arising in the emerging decision problems considering the energy crisis and depleting energy resources ask for more efficient decision-making powered by much stronger and comprehensive tools. For instance, hydropower is a non-depleting carbon-free renewable energy resource bearing maximum potential as compared to all other renewable energy resources. However, all dams and water reservoirs are not dedicated to hydropower generation. Hadjerioua et al. [51] provided a detailed estimation of energy capacity at non-powered dams (NPDs) in the United States and their report signifies the importance of NPDs capable of generating electricity and adding to hydropower generation through proper installations and considerations. The point is that all NPDs cannot generate electricity, and the selection of NPDs capable of hydropower generation requires complicated MAGDM. This demands considering spherical fuzzy opinions of multiple experts (or teams) for tackling the uncertainties governed by the bipolar parameters depicting favoring and symmetrically non-favoring factors. Such a problem is unsolvable by the existing theories, including SFSESs, BSESs, SFBSSs, etc. That is why, this work focuses on the development of a novel hybrid model named spherical fuzzy bipolar soft expert sets or SFBSESs as a combination of SFSs [12] and BSESs [47], and then the ranking of the NPDs’ suitability for hydropower installations under SFBSESs. Table 3 gives an overview of the considerations taken into account by multiple decision-models. It can be clearly observed that the innovative proposed model considers maximum dependencies affecting the decision-making, hence leaving negligible chances of error as compared to all of the pre-existing tools.

1.1 Motivations and Contributions

The motivations of this work are provided as below:

1.   The possible contribution of NPDs in hydropower generation (as discussed in [51]) is very important in minimizing the power crisis. However, a lot of positive and negative factors considering the infrastructure, location, politics, etc., ask for efficient MAGDM decision-making under the expertise of multiple experts.

2.   The existing models like PFSs [8–10], SFSSs [25] and SFSESs [44] offer reliable solutions to uncertainties by considering the degrees of positive, negative and neutral opinions. However, all of these are incapable to discuss the bipolarity of decision-parameters, therefore unreliable to solve problems governed by symmetrically opposing factors.

3.   Models based on BSSs [46] like BSESs [47], FBSESs [48] and qROFBSSs [50] efficiently differentiate between the bipolar decision-parameters, however all these models fail to consider the degree of neutrality when solving uncertain problems.

4.   The MAGDM capabilities of SESs [40] for dealing complex situations considering multiple experts offer more reliable and unbiased solutions. Most of the existing works as discussed in Table 3 lack this capability.

The main contributions of this paper are given as below:

1.   Combining the uncertain decision-making properties of SFSs [12] with the group decision-making capabilities of BSESs [47], a hybrid model is presented with higher performance when examining MAGDM problems in spherical fuzzy environment under bipolar parameters. This hybridization allows group decision considerations of 3 dimensional opinions regarding the uncertainties in problems governed by symmetrically opposing parameters.

2.   The algebraic characteristics and operations of the initiated structure are discussed and illustrated through numerical examples. These operations depict inclusion and aggregation techniques in multiple ways for combining and comparing between different sets of SFBSE information.

3.   A powerful MAGDM algorithm under SFBSESs is proposed. This enhanced algorithm combines the previous algorithms (as in [44,48,49]) to tackle their limitations as discussed in the literature. Consequently, many problems unsolvable by the previous works due to the loss of information will now seek their solutions reliably.

4.   Considering the importance of NPDs in improving the power generation and the complicated uncertainties arising in appropriate site selection procedure for hydropower conversion, a model NPDs site selection problem is modeled using SFBSES and solved under the proposed algorithm. This model application gives an insight about how equally complicated actual problems can be solved with the presented method.

1.2 Structure of the Paper

This paper is organized as:

•   Section 2 recalls the preliminary definitions and related notions that will be utilized in the further development of this work.

•   Section 3 presents SFBSESs along with their algebraic properties and fundamental operations, including subset-hood, complement, intersection (restricted and extended), union (restricted and extended), SFBSE-AND operation and SFBSE-OR operation.

•   Section 4 explores an application of the SFBSESs, i.e., ranking of NPDs regarding suitabilities for hydropower generation, and its solution under the novel algorithm proposed for SFBSESs.

•   Section 5 discusses the advantages of proposed MAGDM hybrid model by comparing it with the existing models, and debates the possible limitations.

•   At last, Section 6 gives some conclusive remarks and possible future extensions of the presented method.

2  Preliminaries

This section recalls important notions useful in the development of this paper. Let us start with the definition of spherical fuzzy sets (or SFSs).

Definition 2.1.[12] For 𝒵 being the universe and μ, τ, ν being the fuzzy membership functions representing the positive, negative, and neutral values, respectively. A spherical fuzzy set (SFS) 𝒮 over 𝒵 is represented as:

𝒮={(z,μ𝒮(z),τ𝒮(z),ν𝒮(z))|z∈𝒵}

where

0≤(μ𝒮(z))2+(τ𝒮(z))2+(ν𝒮(z))2≤1,∀z∈𝒵.

The set S(𝒵) represents the family of all SFSs over 𝒵.

Some important properties and operations of the SFSs are listed as follows:

Definition 2.2.[12] Suppose 𝒥 and 𝒦 represent SFSs on the universal set 𝒵. Then

1.   𝒥⊆𝒦 if ∀z∈𝒵, μ𝒥(z)≤μ𝒦(z),τ𝒥(z)≤τ𝒦(z) and ν𝒥(z)≥ν𝒦(z).

2.   𝒥=𝒦⟺ 𝒥⊆𝒦 and 𝒦⊆𝒥.

3.   𝒥∪𝒦={(z,max{μ𝒥(z),μ𝒦(z)},min{τ𝒥(z),τ𝒦(z)},min{ν𝒥(z),ν𝒦(z)})|z∈𝒵}.

4.   𝒥∩𝒦={(z,min{μ𝒥(z),μ𝒦(z)},min{τ𝒥(z),τ𝒦(z)},max{ν𝒥(z),ν𝒦(z)})|z∈𝒵}.

5.   (𝒥)c={(z,ν𝒥(z),τ𝒥(z),μ𝒥(z))|z∈𝒵}.

The next definition calls back the notion of spherical fuzzy soft sets (or SFSSs).

Definition 2.3.[25] Suppose 𝒵 represents the universe of objects, 𝒫 represents the set of parameters, and ℬ⊆𝒫, then the set Θ=(𝒦,ℬ) is called a spherical fuzzy soft set (SFSS) on 𝒵, such that ∀z∈𝒵 and β∈ℬ, the mapping 𝒦: ℬ→S(𝒵) is provided by:

𝒦(β)={(z,μ𝒦(β)(z),τ𝒦(β)(z),ν𝒦(β)(z))}

where μ𝒦(β),τ𝒦(β) and ν𝒦(β) show the degrees of agreement, disagreement, and neutrality of the object z with the condition:

0≤(μ𝒦(β)(z))2+(τ𝒦(β)(z))2+(ν𝒦(β)(z))2≤1,∀z∈𝒵,β∈ℬ.

Some important definitions for SFSSs are recalled below, which will help in development of the new MAGDM algorithm.

Definition 2.4.[25] Consider a SFSS Θ=(𝒦,ℬ) on the universal set 𝒵. Suppose the mapping λ: ℬ→[0,1]3, where ∀β∈ℬ, λ(β)=(p˚(β),q˚(β),r˚(β)) such that p˚,q˚,r˚: ℬ→[0,1]. Then, a level soft-set L(Θ,λ)=(𝒦λ,ℬ) of Θ represents a classical set, which is given as:

𝒦λ(β)={z∈𝒵|μ𝒦(β)(z)≥p˚(β),τ𝒦(β)(z)≤q˚(β),ν𝒦(β)(z)≤r˚(β)},∀β∈ℬ.

1.   Mid-level Threshold Function (midΘ):
The mapping midΘ: ℬ→[0,1]3 for the SFSS Θ=(𝒦,ℬ) is defined by

midΘ(β)=(p˚midΘ(β),q˚midΘ(β),r˚midΘ(β))∀x∈ℬ,

such that

p˚midΘ(β)=1|𝒵|∑z∈𝒵μ𝒦(β)(z);q˚midΘ(β)=1|𝒵|∑z∈𝒵τ𝒦(β)(z);r˚midΘ(β)=1|𝒵|∑z∈𝒵ν𝒦(β)(z).

The mid-level soft set of Θ with respect to midΘ is represented by L(Θ,midΘ(β)).

2.   Top-bottom-bottom-level Threshold Function (tbbΘ):
The mapping tbbΘ: ℬ→[0,1]3 for the SFSS Θ=(𝒦,ℬ) is provided as:

tbbΘ(β)=(p˚tbbΘ(β),q˚tbbΘ(β),r˚tbbΘ(β))∀β∈ℬ,

such that

p˚tbbΘ(β)=maxz∈𝒵 μ𝒦(β)(z);q˚tbbΘ(β)=minz∈𝒵 τ𝒦(β)(z);r˚tbbΘ(β)=minz∈𝒵 ν𝒦(β)(z).

The top-bottom-bottom-level (tbb-level) soft set of Θ regarding tbbΘ is represented by L(Θ,tbbΘ(β)).

3.   Bottom-bottom-bottom-level Threshold Function (bbbΘ):
The mapping bbbΘ: ℬ→[0,1]3 for the SFSS Θ=(𝒦,ℬ) is defined as:

bbbΘ(β)=(p˚bbbΘ(β),q˚bbbΘ(β),r˚bbbΘ(β))∀β∈ℬ,

such that

p˚bbbΘ(β)=minz∈𝒵 μ𝒦(β)(z);q˚bbbΘ(β)=minz∈𝒵 τ𝒦(β)(z);r˚bbbΘ(β)=minz∈𝒵 ν𝒦(β)(z).

The bottom-bottom-bottom-level (bbb-level) soft set of Θ concerning bbbΘ is represented by L(Θ,bbbΘ(β)).

4.   Med-level Threshold Function (medΘ):
The mapping medΘ: ℬ→[0,1]3 for the SFSS Θ=(𝒦,ℬ) is provided by:

medΘ(β)=(p˚medΘ(β),q˚medΘ(β),r˚medΘ(β))∀β∈ℬ,

such that

p˚medΘ(β)={μ𝒦(β)(z(|𝒵|+12)),if |𝒵| is odd,μ𝒦(β)(z(|𝒵|2))+μ𝒦(β)(z(|𝒵|2+1))2,if |𝒵| is even.q˚medΘ(β)={τ𝒦(β)(z(|𝒵|+12)),if|𝒵| is odd,τ𝒦(β)(z(|𝒵|2))+τ𝒦(β)(z(|𝒵|2+1))2,if |𝒵| is even.r˚medΘ(β)={ν𝒦(β)(z(|𝒵|+12)),if |𝒵| is odd,ν𝒦(β)(z(|𝒵|2))+ν𝒦(β)(z(|𝒵|2+1))2,if |𝒵| is even.

Here p˚medΘ(β),q˚medΘ(β), and r˚medΘ(β) serve as the medians, which are respectively computed by ascending (or descending) arrangement of the positive, negative and neutral values. Note that the med-level soft set of Θ regarding medΘ is represented by L(Θ,medΘ(β)).

To deal with MAGDM problems concerning spherical fuzzy information, the existing notion of spherical fuzzy soft expert set (or SFSES) is defined as below:

Definition 2.6.[44] Suppose 𝒵 is the universe, 𝒫 is the set of parameters, ℰ is the set of experts and 𝒪={1=agree,0=disagree} is the set of their opinions. Consider the set 𝒬=𝒫×ℰ×𝒪 and let ℒ⊆𝒬. Then, the set 𝒮ℰ=(𝒢,ℒ) is called the spherical fuzzy soft expert set (or SFSES) over 𝒵, where 𝒢 is a function from ℒ to S(𝒵).

The following definition reviews the notion of bipolar soft sets:

Definition 2.7.[46] Consider 𝒵 represents the universe of alternatives, 𝒫 represents the universe of parameters, then the set Δ=(ζ,ψ,𝒫) is said to be a bipolar soft set (BSS) on the universe 𝒵, if the functions ζ and ψ are provided as:

ζ: ℬ→P(𝒵),ψ: ¬ℬ→P(𝒵).

The set ¬ℬ serves as the not-set of ℬ with the parameters opposite to the parameters in ℬ, where

ζ(β)∩ψ(¬β)=∅,∀β∈ℬand¬β∈¬ℬ,

whereas P(𝒵) represents the power set of 𝒵.

The following definition recalls the concept of bipolar soft expert sets (or BSESs), which are capable to handle bipolarity in MAGDM situations:

Definition 2.8.[47] Consider 𝒵 represents the universe of alternatives, 𝒫 represents the universe of parameters, ℰ is the set of experts and 𝒪={1=agree,0=disagree} is the set of their estimations. Consider the set 𝒬=𝒫×ℰ×𝒪 and let ℒ⊆𝒬. Then, the triple Γ=(ζ^,ψ^,ℒ) is referred to as a bipolar soft expert set (BSES) over 𝒵, if the functions ζ^ and ψ^ are respectively given by

ζ^: ℒ→P(𝒵),ψ^: ¬ℒ→P(𝒵).

Here, ¬ℒ serves as the not-set of ℒ, which contains the opinions opposite to the opinions in ℒ with the following constraint:

ζ^(ℓ)∩ψ^(¬ℓ)=∅,∀ℓ∈ℒand¬ℓ∈¬ℒ.

3  Spherical Fuzzy Bipolar Soft Expert Sets

This section first introduces the concepts of spherical fuzzy bipolar soft expert sets (or SFBSESs), and then discusses the necessary operations and properties. The following defines the novel SFBSESs.

Definition 3.1.Let 𝒵 be the universe of alternatives, 𝒫 be the set of parameters and let ℰ be the set of experts with the opinions of the form 𝒪={1=agree,0=disagree}. Then for 𝒬=𝒫×ℰ×𝒪, a triple Γ=(ζ^,η^,ℒ) is called a spherical fuzzy bipolar soft expert set (SFBSES), if for ℒ⊆𝒬, the functions ζ^ and η^ are given as:

ζ^: ℒ→SF(𝒵),η^: ¬ℒ→SF(𝒵).

Here, ¬ℒ is the not-set of parameters ℒ with the opinions opposite to the opinions in ℒ, whereas SF(𝒵) is the family of all SFSs on 𝒵.

∀ℓ∈ℒ and ¬ℓ∈¬ℒ,

ζ^(ℓ)={(z,μζ^(ℓ)(z),τζ^(ℓ)(z),νζ^(ℓ)(z))|z∈𝒵},η^(¬ℓ)={(z,μη^(¬ℓ)(z),τη^(¬ℓ)(z),νη^(¬ℓ)(z))|z∈𝒵}.

such that μ, τ, and ν represent the degrees of agreement, disagreement, and neutrality, where ∀z∈𝒵:

0≤(μζ^(ℓ)(z))2+(μη^(¬ℓ)(z))2≤1(1)

0≤(τζ^(ℓ)(z))2+(τη^(¬ℓ)(z))2≤1(2)

0≤(νζ^(ℓ)(z))2+(νη^(¬ℓ)(z))2≤1(3)

The following example illustrates how SFBSESs can be used in modeling and depicting complicated decision-making situations.

Example 3.1.Consider an international bank that is trying to increase its business by effectively implementing the foreign exchange strategies. To ensure it happens, the bank needs a strong analysis as well as a valuable and efficient prediction method for the currency exchanges. However, this is a difficult and complex task. To deal with this, the bank calls its market analysts and currency exchangers to ensure the maximum profit and least risk.

These forex experts propose three strategies already working in the market, comprising the set 𝒵={z1,z2,z3}. To compare these strategies, the experts consider the set ℬ={β1=accurate analysis,β2=strong prediction,β3=least risk} as the collection of favorable qualities (parameters), whereas the not-set ¬ℬ={¬β1=poor analysis,¬β2=weak prediction,¬β3=high risk} as the set of dangerous parameters for the business. Let ℰ={u,v,w} represents the set of experts, and 𝒪={1=agree,0=disagree} the set of opinions. The experts make a SFBSES (ζ^,η^,ℒ) (for ℒ=ℬ×ℰ×𝒪) after a thorough analyzation of the proposed strategies as shown below:

ζ^(β1,u,1)={⟨z1,(0.80,0.05,0.20)⟩,⟨z2,(0.60,0.01,0.30)⟩,⟨z3,(0.70,0.03,0.30)⟩}ζ^(β1,v,1)={⟨z1,(0.76,0.04,0.36)⟩,⟨z2,(0.80,0.03,0.40)⟩,⟨z3,(0.80,0.04,0.25)⟩}ζ^(β1,w,1)={⟨z1,(0.90,0.02,0.15)⟩,⟨z2,(0.70,0.04,0.30)⟩,⟨z3,(0.86,0.04,0.30)⟩}

ζ^(β2,u,1)={⟨z1,(0.30,0.10,0.79)⟩,⟨z2,(0.50,0.09,0.60)⟩,⟨z3,(0.34,0.05,0.50)⟩}ζ^(β2,v,1)={⟨z1,(0.50,0.08,0.70)⟩,⟨z2,(0.60,0.03,0.40)⟩,⟨z3,(0.70,0.09,0.39)⟩}ζ^(β2,w,1)={⟨z1,(0.35,0.02,0.75)⟩,⟨z2,(0.40,0.04,0.70)⟩,⟨z3,(0.62,0.08,0.40)⟩}

ζ^(β3,u,1)={⟨z1,(0.70,0.06,0.40)⟩,⟨z2,(0.86,0.06,0.30)⟩,⟨z3,(0.90,0.03,0.15)⟩}ζ^(β3,v,1)={⟨z1,(0.67,0.03,0.25)⟩,⟨z2,(0.73,0.07,0.10)⟩,⟨z3,(0.75,0.04,0.35)⟩}ζ^(β3,w,1)={⟨z1,(0.80,0.10,0.30)⟩,⟨z2,(0.90,0.01,0.10)⟩,⟨z3,(0.70,0.10,0.40)⟩}

ζ^(β1,u,0)={⟨z1,(0.15,0.04,0.66)⟩,⟨z2,(0.45,0.02,0.50)⟩,⟨z3,(0.30,0.10,0.70)⟩}ζ^(β1,v,0)={⟨z1,(0.30,0.03,0.46)⟩,⟨z2,(0.23,0.03,0.70)⟩,⟨z3,(0.20,0.07,0.73)⟩}ζ^(β1,w,0)={⟨z1,(0.10,0.05,0.80)⟩,⟨z2,(0.30,0.04,0.60)⟩,⟨z3,(0.15,0.09,0.60)⟩}

ζ^(β2,u,0)={⟨z1,(0.65,0.02,0.40)⟩,⟨z2,(0.60,0.07,0.50)⟩,⟨z3,(0.65,0.02,0.20)⟩}ζ^(β2,v,0)={⟨z1,(0.40,0.01,0.60)⟩,⟨z2,(0.55,0.01,0.66)⟩,⟨z3,(0.30,0.03,0.45)⟩}ζ^(β2,w,0)={⟨z1,(0.70,0.06,0.50)⟩,⟨z2,(0.38,0.10,0.40)⟩,⟨z3,(0.40,0.05,0.36)⟩}

ζ^(β3,u,0)={⟨z1,(0.30,0.10,0.60)⟩,⟨z2,(0.23,0.06,0.60)⟩,⟨z3,(0.22,0.10,0.80)⟩}ζ^(β3,v,0)={⟨z1,(0.25,0.03,0.80)⟩,⟨z2,(0.10,0.02,0.91)⟩,⟨z3,(0.33,0.06,0.77)⟩}ζ^(β3,w,0)={⟨z1,(0.20,0.07,0.76)⟩,⟨z2,(0.24,0.01,0.40)⟩,⟨z3,(0.40,0.07,0.35)⟩}

η^(¬β1,u,1)={⟨z1,(0.20,0.02,0.70)⟩,⟨z2,(0.40,0.03,0.60)⟩,⟨z3,(0.32,0.01,0.60)⟩}η^(¬β1,v,1)={⟨z1,(0.30,0.03,0.60)⟩,⟨z2,(0.25,0.04,0.20)⟩,⟨z3,(0.30,0.02,0.70)⟩}η^(¬β1,w,1)={⟨z1,(0.15,0.04,0.70)⟩,⟨z2,(0.13,0.03,0.73)⟩,⟨z3,(0.20,0.05,0.65)⟩}

η^(¬β2,u,1)={⟨z1,(0.60,0.03,0.36)⟩,⟨z2,(0.42,0.04,0.39)⟩,⟨z3,(0.62,0.06,0.42)⟩}η^(¬β2,v,1)={⟨z1,(0.40,0.07,0.54)⟩,⟨z2,(0.70,0.01,0.42)⟩,⟨z3,(0.30,0.09,0.36)⟩}η^(¬β2,w,1)={⟨z1,(0.30,0.08,0.65)⟩,⟨z2,(0.60,0.10,0.40)⟩,⟨z3,(0.30,0.02,0.70)⟩}

η^(¬β3,u,1)={⟨z1,(0.50,0.06,0.70)⟩,⟨z2,(0.40,0.03,0.70)⟩,⟨z3,(0.12,0.06,0.68)⟩}η^(¬β3,v,1)={⟨z1,(0.32,0.01,0.72)⟩,⟨z2,(0.40,0.05,0.64)⟩,⟨z3,(0.22,0.07,0.73)⟩}η^(¬β3,w,1)={⟨z1,(0.30,0.02,0.70)⟩,⟨z2,(0.15,0.16,0.89)⟩,⟨z3,(0.25,0.02,0.80)⟩}

η^(¬β1,u,0)={⟨z1,(0.80,0.03,0.15)⟩,⟨z2,(0.65,0.06,0.46)⟩,⟨z3,(0.70,0.01,0.30)⟩}η^(¬β1,v,0)={⟨z1,(0.65,0.04,0.30)⟩,⟨z2,(0.70,0.07,0.24)⟩,⟨z3,(0.80,0.01,0.15)⟩}η^(¬β1,w,0)={⟨z1,(0.78,0.06,0.20)⟩,⟨z2,(0.70,0.03,0.30)⟩,⟨z3,(0.60,0.05,0.28)⟩}

η^(¬β2,u,0)={⟨z1,(0.40,0.05,0.60)⟩,⟨z2,(0.60,0.04,0.32)⟩,⟨z3,(0.40,0.03,0.66)⟩}η^(¬β2,v,0)={⟨z1,(0.50,0.01,0.65)⟩,⟨z2,(0.36,0.05,0.55)⟩,⟨z3,(0.70,0.04,0.32)⟩}η^(¬β2,w,0)={⟨z1,(0.70,0.01,0.32)⟩,⟨z2,(0.30,0.01,0.72)⟩,⟨z3,(0.65,0.09,0.40)⟩}

η^(¬β3,u,0)={⟨z1,(0.50,0.02,0.60)⟩,⟨z2,(0.29,0.02,0.70)⟩,⟨z3,(0.80,0.06,0.21)⟩}η^(¬β3,v,0)={⟨z1,(0.67,0.10,0.30)⟩,⟨z2,(0.60,0.10,0.30)⟩,⟨z3,(0.63,0.08,0.35)⟩}η^(¬β3,w,0)={⟨z1,(0.59,0.03,0.41)⟩,⟨z2,(0.80,0.06,0.10)⟩,⟨z3,(0.70,0.05,0.28)⟩}

Tables 4 and 5 represent the formulated SFBSES in terms of ζ^ and η^, where (ζ^,ℒ) and (η^,¬ℒ) are SFSESs. For ℓi∈ℒ, ¬ℓi∈¬ℒ, and zj∈𝒵, Table 6 represents the SFBSES with entries.

kij=(ζ^(ℓi)(zj)η^(¬ℓi)(zj)),

whereas Table 7 represents the general form of the single tabular format for SFBSESs. It can be observed from Table 6, how the opinions of the experts differ from each other according to their expertise. This flexible opinion set provides better insight and analysis for the complex scenario and hence offers unbiased and more reliable handling of the decision problem under spherical fuzzy information, while considering the bipolarity of the affecting parameters.

Remark 3.1.The above example demonstrates a problem that could not be fully represented by any of the previous tools. SFSESs fail to consider the bipolarity of parameters, BSESs fail to tackle the complicated spherical fuzzy uncertainties in the problem and SFBSSs cannot manage opinions of multiple experts in a single place. This highlights the need for the proposed work which unifies all these models into a single model providing superior uncertainty handling under multiple experts while considering the bipolar behavior of governing decision-parameters.

The coming definition defines the SFBSE subset.

Definition 3.2.A SFBSES Γ=(ζ^,η^,ℒ) is called the SFBSE subset of a SFBSES Λ=(π^,ψ^,ℳ), if and only if:

1.   ℒ⊆ℳ.

2.   (ζ^,ℒ) is a spherical fuzzy soft expert subset of (π^,ℳ).

3.   (ψ^,¬ℳ) is a spherical fuzzy soft expert subset of (η^,¬ℒ).

Γ⊆^Λ represents the above subset relation. In the same way, Λ is called the superset of Γ and the relation is represented by Λ⊇^Γ.

Remark 3.2. Analytically, a SFBSE-subset shows lesser suitability in a decision scenario as compared to that shown by the respective super-set. This lesser suitability or deprived favorability is expressed in the form of missing considerations (relatively fewer decision-parameters) accompanied by lower supporting membership degrees. For instance, two SFBSESs are reported for a firm’s performance at different times with the second set being the subset of the first set. The subset relation then indicates a decline in the firm’s performance with increasing time.

Example 3.2.Recall Example 3.1. Suppose that two SFBSESs Γ=(ζ^,η^,ℳ) and Λ=(π^,ψ^,𝒩) are respectively presented by Tables 8 and 9 where ℳ={(β1,u,1),(β3,v,1),(β2,w,0)} and 𝒩={(β1,u,1),(β3,v,1),(β2,u,0),(β2,w,0)}.

Clearly, Γ⊆^Λ.

Definition 3.3.Any two SFBSESs Γ and Λ are equal, if and only if they are both SFBSE subsets of each other. That is,

Γ=Λ ⟺ Γ⊆^ΛandΛ⊆^Γ.

Based on the concept of complement of SFSs, the complement of a SFBSES is defined as:

ζ^c(ℓ)={(z,νζ^(ℓ)(z),τζ^(ℓ)(z),μζ^(ℓ)(z))|z∈𝒵},η^c(¬ℓ)={(z,νη^(¬ℓ)(z),τη^(¬ℓ)(z),μη^(¬ℓ)(z))|z∈𝒵}.

Example 3.3.Consider the SFBSES Γ=(ζ^,η^,ℳ) as in Example 3.2. By Definition 3.4, its complement Γc=(ζ^c,η^c,ℳ) is given in Table 10.

Proposition 3.1.Let Γ=(ζ^,η^,ℒ) be a SFBSES on 𝒵, then

1.   (Γc)c=Γ

1.   By Definition 3.4, Γc=(ζ^,η^,ℒ)c=(ζ^c,η^c,ℒ) such that

ζ^c(ℓ)={(z,νζ^(ℓ)(z),τζ^(ℓ)(z),μζ^(ℓ)(z))|z∈𝒵},η^c(¬ℓ)={(z,νη^(¬ℓ)(z),τη^(¬ℓ)(z),μη^(¬ℓ)(z))|z∈𝒵},

for all ℓ∈ℒ and ¬ℓ∈¬ℒ. This infers that

(ζ^c(ℓ))c={(z,μζ^(ℓ)(z),τζ^(ℓ)(z),νζ^(ℓ)(z))|z∈𝒵}=ζ^,(η^c(¬ℓ))c={(z,μη^(¬ℓ)(z),τη^(¬ℓ)(z),νη^(¬ℓ)(z))|z∈𝒵}=η^.

Thus, ((ζ^,η^,ℒ)c)c=(ζ^,η^,ℒ), or (Γc)c=Γ.

The relative null and relative absolute SFBSESs over 𝒵 are provided in the following Definition 3.5.

Definition 3.5.A SFBSES Φ^=(ϕ^,U^,ℒ) is called a relative null SFBSES on the universe 𝒵, if ∀ℓ∈ℒ,

ϕ^(ℓ)={(z,0,0,1)|z∈𝒵},

and ∀¬ℓ∈¬ℒ,

U^(¬ℓ)={(z,1,0,0)|z∈𝒵}.

Similarly, a SFBSES U^=(U^,ϕ^,ℒ) is called a relative absolute SFBSES on universe 𝒵, if ∀ℓ∈ℒ,

U^(ℓ)={(z,1,0,0)|z∈𝒵},

and ∀¬ℓ∈¬ℒ,

ϕ^(¬ℓ)={(z,0,0,1)|z∈𝒵}.

Proposition 3.2.Consider the relative null and relative absolute SFBSESs, i.e., Φ^ and U^, respectively, then

1.   Φ^c=U^

2.   U^c=Φ^

Proof. The proofs are obvious from the Definitions 3.4 and 3.5.

The following Definition 3.6 explains the concepts of agree and disagree SFBSESs:

Definition 3.6.The agree SFBSES (ζ^,η^,ℒ)1 over the universe 𝒵 is the SFBSE subset of (ζ^,η^,ℒ) defined as:

(ζ^,η^,ℒ)1=ζ^(ℓ)∪η^(¬ℓ),

for all ℓ∈ℬ×ℰ×{1} and ¬ℓ∈¬ℬ×ℰ×{1}. Similarly, the disagree SFBSES (ζ^,η^,ℒ)0 over the universe 𝒵 is the SFBSE subset of (ζ^,η^,ℒ) defined as:

(ζ^,η^,ℒ)0=ζ^(ℓ)∪η^(¬ℓ),

for all ℓ∈ℬ×ℰ×{0} and ¬ℓ∈¬ℬ×ℰ×{0}.

Example 3.4.Consider the SFBSES (ζ^,η^,ℒ) in Example 3.1. The corresponding agree and disagree SFBSESs are respectively shown by Tables 11 and 12.

The following definition provide the idea of AND operation between SFBSESs:

Definition 3.7.For two SFBSESs Γ=(ζ^,η^,ℒ) and Λ=(π^,ψ^,ℳ) on the universal set 𝒵, the AND operation denoted as a SFBSES Γ∧^Λ, is defined by:

Γ∧^Λ=(ρ^,θ^,ℒ×ℳ),

such that ∀(ℓ,m)∈ℒ×ℳ and (¬ℓ,¬m)∈¬ℒ׬ℳ, ρ^: ℒ×ℳ→SF(𝒵) and θ^: ¬ℒ׬ℳ→SF(𝒵) are functions defined as:

ρ^(ℓ,m)=ζ^(ℓ)∩π^(m),θ^(¬ℓ,¬m)=η^(¬ℓ)∪ψ^(¬m).

The next definition gives the OR operation between SFBSESs.

Definition 3.8.For two SFBSESs Γ=(ζ^,η^,ℒ) and Λ=(π^,ψ^,ℳ) on the universal set 𝒵, the OR operation represented as a SFBSES Γ∨^Λ is defined by:

Γ∨^Λ=(γ^,λ^,ℒ×ℳ),

such that ∀(ℓ,m)∈ℒ×ℳ and (¬ℓ,¬m)∈¬ℒ׬ℳ, γ^: ℒ×ℳ→SF(𝒵) and λ^: ¬ℒ׬ℳ→SF(𝒵) are functions defined as:

γ^(ℓ,m)=ζ^(ℓ)∪π^(m),λ^(¬ℓ,¬m)=η^(¬ℓ)∩ψ^(¬m).

The following example provides an illustration of the AND and OR operations between two SFBSESs:

Example 3.5.Reconsider Example 3.1. For ℳ={(β1,u,1),(β2,v,1),(β3,u,0),(β3,v,0)}, and 𝒩={(β1,u,1),(β3,v,1)}, the corresponding SFBSESs are shown in Tables 13 and 14, respectively. Using Definition 3.7, the AND operation Γ∧^Λ between these two SFBSESs is displayed in Table 15. Similarly, using Definition 3.8, the OR operation Γ∨^Λ between the given SFBSESs is calculated as shown in Table 16.