# A New Optimization Approach Based on Bipolar Type-2 Fuzzy Soft Sets

Authors: Kholood M. Alsager; Sumaiah A. Alfahhad
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The main objective of this research study is to amplify the schematic representation of human reasoning by launching the most generalized fantastic theories of bipolar type-2 fuzzy set (BT2FS) and bipolar type-2 fuzzy soft set (BT2FSf S). These incredible models are exclusively developed for the simultaneous capturing of both polarity and abstruseness inherent in equivocal interpretations. The proposed BT2FS f S theory renders an outstanding parameterized framework that skilfully wipes out the high-order uncertainty of imprecise knowledge-based systems. First, we keenly provide the formal structure of both proposed models along with the deep exploration of their elementary properties. Moreover, we explore rudimentary set-theoretical operations of developed frameworks inclusive of equality, subset, complement, union, and intersection with their noticeable results. We brilliantly formulate a highly proficient algorithm using proposed theory to disentangle the real-world multiattribute decision-making conundrums with two-sided ambiguous information. Finally, we holistically scrutinize an empirical analysis for the selection best way to cultivate the dessert city to demonstrate the remarkable accountability of the proposed methodology.


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## Body

## 1. Introduction

A modern trend in contemporaneous information processing focuses on bipolar information. Bipolarity is important to identify between positive information which acts for what is guaranteed to be possible and negative information which acts for what is impossible or prohibited [1, 2]. This generic domain has recently contributed in several directions, including knowledge representation, preference modeling, multicriteria decision analysis, argumentation, and many others disciplines of science and technology [2–14]. Akram et al. [15] skillfully developed graphical representations for the comprehensive analysis of bipolar fuzzy information and presented novel decision-making techniques for the scrutinization of real-world complexities. Ali et al. [16] introduced competent attributes reduction algorithms for bipolar fuzzy relation decision systems to properly investigate sophisticated decision-making problems. Akram et al. [17] admirably designed multiskilled TOPSIS and ELECTRE-1 methods equipped with bipolar fuzzy information and patently demonstrated its empirical application in medical diagnosis.The theory of bipolar fuzzy sets (BFSs) and its various generalizations were investigated by many researchers and practitioners as they render such brilliant mathematical tools that skillfully address the double-sided or bipolar human vague annotations [14, 18–21]. Akram et al. [22] launched a modified version of PROMETHEE method by employing bipolar fuzzy information for the determination of best green supplier in the supply chain management system. Al-Qudah and Hassan [23] introduced bipolar fuzzy soft expert sets by the amalgamation of BFSs and soft expert sets and defined algebraic operations of union, intersection, and complement for the proposed bipolar fuzzy soft expert sets. Akram et al. [24] presented an advanced trapezoidal bipolar fuzzy TOPSIS method and dexterously illustrated its application for the selection of the best construction project proposal.All the contemporary approaches have a common limitation that arises because of their nonparameterized modeling capabilities. To overcome this problem, Molodtsov [25] developed an advanced novel theory of soft sets (S f Ss) that integrates parametrization for the specification of the expertise of rational alternatives. One of the most important applications that are mentioned in this theory is the evaluation of many practical problems in engineering, economics, medical science, social science, and in various other modern directions. Later, Maji [26] has further studied the theory of S f Ss for the proper examination of some complex realistic problems. Maji [27] explored some fundamental properties of S f Ss and also defined its algebraic operations inclusive of restricted and extended union, intersection, and difference. Chan [28], Zou and Xiao [29], Ali [30], Kong [31], Cagman and Enginoglu [32, 33], and many other pedagogues contributed to the development and improvement of the S f S theory. Moreover, Maji [34] combined the proficient theories of both S f Ss and fuzzy sets and developed an innovative concept of fuzzy soft sets (FS f Ss). Akram et al. [35] admirably proposed a hybrid theory of bipolar fuzzy N-soft sets and established the TOPSIS method under such a versatile environment for the robust inspection of its high potential applications.Zadeh [36] proposed an entirely new theory of fuzzy sets for capturing the imprecision of obscure data with the assistance of a MF (MF) that can grab values from the closed unit interval. The traditional fuzzy sets are also called type-1 fuzzy sets [37]. The most important application in FS f Ss is the investigation of actual decision-making perplexities [38–40]. Later, Zadeh [41–43] introduced the notion of a type-2 fuzzy set (T2FS) in which the membership degree of each element is given by another type-1 fuzzy set defined over the interval [0,1]. It can also be generalized from type-1 FS f Ss to type-2 FS f Ss. Matsumoto and Tanaka [44, 45] discussed some elementary algebraic properties of T2FSs. Dubois and Prade [46] remarkably presented several notions and applications of T2FSs. Later, many researchers conducted a lot of theoretical work on the properties of T2FS and manifested its many pragmatic applications [47–50]. Type-2 fuzzy sets were applied in several domains, especially in decision-making [51], connection admission control [52], mobile robotic control [53], equalization of nonlinear fading channels [54–57], extraction of knowledge from questionnaire surveys [52, 58], forecasting of time-series [57, 59, 60], function approximation [58], learning linguistic membership grades [61], preprocessing radiographic images [62], relational databases [63], solving fuzzy relation equations [64], and transport scheduling [65]. Zhang and Zhang [66] considered type-2 FS f Ss with applications.In several tricky practical applications, we often face an intricate situation during evaluation processes in which stipulated parameters are highly equivocal. It can be represented by fuzzy sets rather than exact numerical values, intervals, and interval-valued intuitionistic fuzzy numbers. Obviously, it is very difficult for the classical Sf S and its contemporary extensions to deal with such arduous problems. Therefore, it is necessary to extend the soft set theory for the accommodation of problematic scenarios in which the specified evaluation parameters are totally ambiguous in nature. The purpose of this paper is to further generalize the concept of type-2 fuzzy sets and type-2 FS f Ss to BT2FSs and bipolar type-2 S f Ss, respectively. Now, we present the highly competent novel theories of BT2FSs and BT2FS f Ss and define some basic operations on them. Moreover, we also examine the applications of BT2FSs and BT2FS f Ss and weighted type-2 FS f Ss in decision-making problems.The rest of this paper is organized as follows. Section2 recollects some relevant studies of the literature. Section 3 explicates the novel theory of BT2FSs, its elementary properties, and some algebraic operations inclusive of union, intersection, and complement. Section 4 presents an advanced concept of BT2FS f Ss and discusses its basic algebraic operations along with some fundamental characteristics. Section 5 illustrates the applications of BT2FSs, BT2FS f Ss, and weighted BT2FS f Ss in decision-making problems. Section 6 explores some concluding remarks and future research directions.


## 2. Preliminaries

In this section, we shall review the concepts of Sf S, fuzzy soft set, BFS and bipolar fuzzy soft set.Definition 1 (see [25]).
LetE be a set of parameters and let H⊆E. Let F be a mapping defined by
(1)F:H⟶Pℚ,where Pℚ is a collocation of all subsets of the initial universal set ℚ. Then, the pair F,H is a S f S.Definition 2 (see [67, 68]).
LetH⊆E be a subset of parameters set. Let PFℚ be a collocation of all fuzzy subset of the initial universal set ℚ and let F be a mapping defined by
(2)F:H⟶PFℚ
Then, we call the pairF,H is a FS f S over ℚ.Definition 3.
A BFSI on U is an object having the form
(3)I=q,μI+q,μI−q;q∈U,where μI+:U⟶0,1,μI−:U⟶−1,0, μI+ represent positive information and μI− represents negative information.Definition 4.
Let pairFb,H be a bipolar FS f S over initial universal set ℚ, where H⊆E is a parameters set and Fb is a mapping such that
(4)Fb:H⟶BFℚ,where BFℚ illustrates all bipolar fuzzy subsets of ℚ.Definition 5.
H is called a T2FS over initial universe ℚ if it is characterized by a secondary MF μHq,k:ℚ×0,1⟶0,1 where k∈Jq is primary membership value of an element q∈ℚ such that Jq:ℚ⟶0,1. Thus we define the set H by
(5)H=q,k,μHq,k;∀q∈ℚ,k∈jq,or by
(6)H=∫q∈ℚ∫k∈Jq⊆0,1μHq,k/k/q.Take into account the symbol∫∫ denotes union over all q and k, and + also denotes union [69].


## 3. Bipolar Type-2 Fuzzy Set

In this section, we present a new concept of BT2FS, its operations, and some of its basic fundamental properties.Definition 6.
Letℚ be the initial universal set. Then Hb is called a BT2FS if it is characterized by a pair μHb+q,k,μHb−q,k, where μHb+:ℚ×0,1⟶0,1 and μHb−:ℚ×0,1⟶−1,0 represents positive and negative secondary MF of q∈ℚ, respectively, and k∈Jq⊆0,1 denotes the primary membership value of an element q∈ℚ such that Jq:ℚ⟶0,1. Then, the set Hb is defined by
(7)Hb=∫q∈ℚ∫k∈Jq⊆0,1μHb+q,k,μHb−q,kk/q.The set of all BT2FSs inℚ is denoted by BFT2.Remark 7.
The notion∫ denotes union over all admissible q and k, and + also denotes union [70].To clarify the concept of BT2FS, let us consider the following example.Example 1.
Letℚ=q1,q2,q3 be the initial universal set and let Hb be a BT2FS defined as
(8)Hb=0.8,−0.5/0.6+0.9,−0.7/0.3q1+0.7,−0.6/0.3+0.8,−0.3/0.5+0.5,−0.1/0.7q2+0.9,−0.2/0.8q3.Graphical representation is shown in Figure1.Figure 1
Bipolar type-2 fuzzy sets.SupposeHb,Ib∈BFT2 then
(9)Hb=∫q∈ℚ∫k∈Jq⊆0,1μHb+q,k,μHb−q,kk/q,Ib=∫q∈ℚ∫v∈Jq⊆0,1μIb+q,v,μIb−q,vv/q.Definition 8.
For two BT2FSsHbandIb of ℚ, we say that Ib is a bipolar type-2 fuzzy (BT2F) subset of Hb if and only if
(i)
v≤k,(ii)
μIb+q,v≤μHb+q,k for a positive secondary MF(iii)
μIb−q,v≥μHb−q,k for a negative secondary MFwhere k is a primary membership value of Hb and v is a primary membership value of Ib.Example 2.
Consider two sets ofBFT2Hb “used in Example 1” and Ib define
(10)Hb=0.8,−0.4/0.1+0.2,−0.5/0.3q1+0.4,−0.3/0.2+0.6,−0.1/0.3+0.3,−0.1/0.5q2+0.6,−0.2/0.3q3.So,Ib is a BT2F subset of Hb.Remark 9.
Hb and Ib two BFT2 sets of ℚ are said to be equal, if Hb is a BT2F subset of Ib and Ib is a BT2F subset of Hb.


### 3.1. Operations ofBFT2

In this section, we dialogue three essential arithmetic operations inclusive of complement, union, and intersection ofBFT2.(1)
Complement: the complement ofHb a BFT2 is denoted Hbc and defined by
(11)Hbc=∫q∈ℚ∫k∈Jq⊆0,1μHb+q,k,μHb−q,k/1−kq.(2)
Union: letHb,Ib∈BFT2, so we defined the union of Hb and Ib by Jb:
(12)Jb=Hb⊔Ib=∫q∈ℚ∫w∈Jq⊆0,1μJb+q,w,μIb−q,w/wq,where w=k∨v; μJb+q,w=μHb+q,k∧μIb+q,v,μJb−q,w=μHb−q,k∨μIb−q,v,∨ is a maximum and ∧ is a minimum.(3)
Intersection: letHb,Ib∈BFT2, so defined the intersection of Hb and Ib by Jb:
(13)Jb=Hb⊓Ib=∫q∈ℚ∫w∈Jq⊆0,1μJb+q,w,μIb−q,w/wq,where μJb+q,w=μHb+q,k∧μIb+q,v; μJb−q,w=μHb−q,k∨μIb−q,v and where w=k∧v.Example 3.
Let us consider twoBFT2 of the initial universe ℚ as
(14)Hb=0.8,−0.5/0.6+0.9,−0.7/0.3q1+0.7,−0.6/0.3+0.8,−0.3/0.5+0.5,−0.1/0.7q2+0.9,−0.2/0.8q3,Ib=0.9,−0.6/0.8+1.0,−0.8/0.7q1+0.6,−0.2/0.9+0.1,−0.5/0.4q2+0.6,−0.3/0.5+0.2,−0.8/0.4q3,then
(15)Hbc=0.8,−0.5/0.4+0.9,−0.7/0.7q1+0.7,−0.6/0.7+0.8,−0.3/0.5+0.5,−0.1/0.3q2+0.9,−0.2/0.2q3,Hb⊔Ib=0.8,−0.5/0.8+0.8,−0.5/0.7+0.9,−0.6/0.8+0.9,−0.7/0.7q1+0.6,−0.2/0.9+0.1,−0.5/0.4+0.1,−0.3/0.5+0.5,−0.1/0.9+0.1,−0.1/0.7q2+0.6,−0.2/0.8+0.2,−0.2/0.8q3,=0.9,−0.6/0.8+0.9,−0.7/0.7q1+0.6,−0.2/0.9+0.1,−0.5/0.4+0.1,−0.3/0.5+0.1,−0.1/0.7q2+0.6,−0.2/0.8q3,Hb⊓Ib=0.8,−0.5/0.6+0.8,−0.5/0.6+0.9,−0.6/0.3+0.9,−0.7/0.3q1+0.6,−0.2/0.3+0.1,−0.2/0.3+0.6,−0.2/0.5+0.1,−0.3/0.4+0.5,−0.1/0.7+q2+0.6,−0.2/0.5+0.2,−0.2/0.4q3,=0.8,−0.5/0.6+0.9,−0.7/0.3q1+0.6,−0.5/0.3+0.6,−0.2/0.5+0.1,−0.3/0.4+0.5,−0.1/0.7q2+0.6,−0.2/0.5+0.2,−0.2/0.4q3,


### 3.2. Properties

Consider threeBFT2 sets Hb,Ib, and Jb(16)Hb⊔Hb=Hb,Hb⊓Hb=Hb,Hb⊔Ib⊔Jb=Hb⊔Ib⊔Jb,Hb⊓Ib⊓=Hb⊓Ib⊓Jb,Hb⊔Ib⊓Jb=Hb⊔Ib⊓Hb⊔Jb,Hb⊓Ib⊔Jb=Hb⊓Ib⊔Hb⊓Jb,Hb⊔Ibc=Hbc⊓Ibc,Hb⊓Ibc=Hbc⊔Ibc,Hb⊔Ib=Ib⊔Hb,Hb⊓Ib=Ib⊔Hb.Example 4.
Reconsider Example3 and let Jb∈BFT2 defined by
(17)Jb=0.9,−0.3/0.1+0.8,−0.1/0.5q1+0.6,−0.1/0.8q2+0.3,−0.6/1.0+0.4,−0.7/0.2+0.5,−0.5/0.7q3.Thus, we have(18)Hb⊔Ib⊔Jb=0.9,−0.6/0.8+0.9,−0.7/0.7q1+0.6,−0.2/0.9+0.1,−0.5/0.4+0.1,−0.3/0.5+0.1,−0.1/0.7q2+0.6,−0.2/0.8q3⊔0.9,−0.3/0.1+0.8,−0.1/0.5+0.6,−0.1/0.8/q2q1+0.3,−0.6/1.0+0.4,−0.7/0.2+0.5,−0.5/0.7q3,=0.9,−0.3/0.8+0.9,−0.3/0.7+0.6,−0.1/0.9+0.1,−0.1/0.8/q2q1+0.3,−0.2/1.0+0.5,−0.2/0.8q3We also have(19)Hb⊔Ib⊔Jb=0.8,−0.5/0.6+0.9,−0.7/03q1+.7,−0.6/0.3+0.8,−0.3/0.5+0.5,−.1/0.7q2+0.9,−0.2/0.8q3⊔0.9,−0.6/0.8+1.0,−0.8/0.7q1+0.6,−0.2/0.9+0.1,−0.5/0.4q2+0.6,−0.3/0.5+0.2,−0.8/0.4q3⊔0.9,−0.3/0.1+0.8,−0.1/0.5q1+0.6,−0.1/0.8q2+0.3,−0.6/1.0+0.4,−0.7/0.2+0.5,−0.5/0.7q3,=0.9,−0.3/0.8+0.9,−0.3/0.7q1+0.6,−0.1/0.9+0.1,−0.1/0.8q2+0.3,−0.2/1.0+0.5,−0.2/08q3.So, we obtain(20)Hb⊔Ib⊔Jb=Hb⊔Ib⊔JbHb⊔Ibc=0.9,−0.6/0.2+0.9,−0.7/0.3q1+0.6,−0.2/0.1+0.1,−0.5/0.6+0.1,−0.3/0.5+0.1,−0.1/0.3q2+0.6,−0.2/0.2q3,Hbc⊓Ibc=0.9,−0.6/0.2+0.9,−0.7/0.3q1+0.6,−0.2/0.1+0.1,−0.5/0.6+0.1,−0.3/0.5+0.1.−0.1/0.3q2+0.6,−0.2/0.2q3,Then, we deduceHb⊔Ibc=Hbc⊓Ibc.(21)Hb⊓Ibc=Hbc⊔IbcIndeed(22)Hb⊓Ibc=0.8,−0.5/0.4+0.9,−0.7/0.7q1+0.6,−0.5/0.7+0.6,−0.2/0.5+0.1,−0.3/0.6+0.5,−0.1/0.3q2+0.6,−0.2/0.5+0.2,−0.2/0.6q3,Hbc⊔Ibc=0.8,−0.5/0.4+0.9,−0.7/0.7q1+0.7,−0.6/0.7+0.8,−0.3/0.5+0.5,−0.1/0.3q2+0.9,−0.2/0.2q3+0.6,−0.3/0.5+0.2,−0.8/0.6q3=0.8,−0.5/0.4+0.9,−0.7/0.7q1+0.6,−0.5/0.7+0.6,−0.2/0.5+0.1,−0.3/0.6+0.5,−0.1/0.3q2+0.6,−0.2/0.5+0.2,−0.2/0.6q3.Theorem 10.
IfHb BT2F is a subset of Ib and Ib BT2F a subset of Jb, then Hb BT2F is a subset of Jb.Proof.
Suppose thatHb BT2F subset of Ib, thus k≤v for primary membership values and μHb+q,k≤μIb+q,v,μHb−q,k≥μIb−q,v for, respectively, positive and negative secondary membership functions of q∈ℚ, and suppose that Ib BT2F subset of Jb, thus v≤w for primary membership values and μIb+q,v≤μJb+q,w,μIb−q,v≥μJb−q,w for, respectively, positive and negative secondary membership functions of q∈ℚ.Therefore,k≤w,μHb+q,k≤μJb+q,w, and μHb−q,k≥μJb−q,w.Hence,Hb BT2F is a subset of Jb.Theorem 11.
IfHb BT2FS is equal to Ib and Ib BT2FS is equal to Jb, then Hb BT2FS is equal to Jb.Proof.
Same proof as Theorem10.


## 3.1. Operations ofBFT2

In this section, we dialogue three essential arithmetic operations inclusive of complement, union, and intersection ofBFT2.(1)
Complement: the complement ofHb a BFT2 is denoted Hbc and defined by
(11)Hbc=∫q∈ℚ∫k∈Jq⊆0,1μHb+q,k,μHb−q,k/1−kq.(2)
Union: letHb,Ib∈BFT2, so we defined the union of Hb and Ib by Jb:
(12)Jb=Hb⊔Ib=∫q∈ℚ∫w∈Jq⊆0,1μJb+q,w,μIb−q,w/wq,where w=k∨v; μJb+q,w=μHb+q,k∧μIb+q,v,μJb−q,w=μHb−q,k∨μIb−q,v,∨ is a maximum and ∧ is a minimum.(3)
Intersection: letHb,Ib∈BFT2, so defined the intersection of Hb and Ib by Jb:
(13)Jb=Hb⊓Ib=∫q∈ℚ∫w∈Jq⊆0,1μJb+q,w,μIb−q,w/wq,where μJb+q,w=μHb+q,k∧μIb+q,v; μJb−q,w=μHb−q,k∨μIb−q,v and where w=k∧v.Example 3.
Let us consider twoBFT2 of the initial universe ℚ as
(14)Hb=0.8,−0.5/0.6+0.9,−0.7/0.3q1+0.7,−0.6/0.3+0.8,−0.3/0.5+0.5,−0.1/0.7q2+0.9,−0.2/0.8q3,Ib=0.9,−0.6/0.8+1.0,−0.8/0.7q1+0.6,−0.2/0.9+0.1,−0.5/0.4q2+0.6,−0.3/0.5+0.2,−0.8/0.4q3,then
(15)Hbc=0.8,−0.5/0.4+0.9,−0.7/0.7q1+0.7,−0.6/0.7+0.8,−0.3/0.5+0.5,−0.1/0.3q2+0.9,−0.2/0.2q3,Hb⊔Ib=0.8,−0.5/0.8+0.8,−0.5/0.7+0.9,−0.6/0.8+0.9,−0.7/0.7q1+0.6,−0.2/0.9+0.1,−0.5/0.4+0.1,−0.3/0.5+0.5,−0.1/0.9+0.1,−0.1/0.7q2+0.6,−0.2/0.8+0.2,−0.2/0.8q3,=0.9,−0.6/0.8+0.9,−0.7/0.7q1+0.6,−0.2/0.9+0.1,−0.5/0.4+0.1,−0.3/0.5+0.1,−0.1/0.7q2+0.6,−0.2/0.8q3,Hb⊓Ib=0.8,−0.5/0.6+0.8,−0.5/0.6+0.9,−0.6/0.3+0.9,−0.7/0.3q1+0.6,−0.2/0.3+0.1,−0.2/0.3+0.6,−0.2/0.5+0.1,−0.3/0.4+0.5,−0.1/0.7+q2+0.6,−0.2/0.5+0.2,−0.2/0.4q3,=0.8,−0.5/0.6+0.9,−0.7/0.3q1+0.6,−0.5/0.3+0.6,−0.2/0.5+0.1,−0.3/0.4+0.5,−0.1/0.7q2+0.6,−0.2/0.5+0.2,−0.2/0.4q3,


## 3.2. Properties

Consider threeBFT2 sets Hb,Ib, and Jb(16)Hb⊔Hb=Hb,Hb⊓Hb=Hb,Hb⊔Ib⊔Jb=Hb⊔Ib⊔Jb,Hb⊓Ib⊓=Hb⊓Ib⊓Jb,Hb⊔Ib⊓Jb=Hb⊔Ib⊓Hb⊔Jb,Hb⊓Ib⊔Jb=Hb⊓Ib⊔Hb⊓Jb,Hb⊔Ibc=Hbc⊓Ibc,Hb⊓Ibc=Hbc⊔Ibc,Hb⊔Ib=Ib⊔Hb,Hb⊓Ib=Ib⊔Hb.Example 4.
Reconsider Example3 and let Jb∈BFT2 defined by
(17)Jb=0.9,−0.3/0.1+0.8,−0.1/0.5q1+0.6,−0.1/0.8q2+0.3,−0.6/1.0+0.4,−0.7/0.2+0.5,−0.5/0.7q3.Thus, we have(18)Hb⊔Ib⊔Jb=0.9,−0.6/0.8+0.9,−0.7/0.7q1+0.6,−0.2/0.9+0.1,−0.5/0.4+0.1,−0.3/0.5+0.1,−0.1/0.7q2+0.6,−0.2/0.8q3⊔0.9,−0.3/0.1+0.8,−0.1/0.5+0.6,−0.1/0.8/q2q1+0.3,−0.6/1.0+0.4,−0.7/0.2+0.5,−0.5/0.7q3,=0.9,−0.3/0.8+0.9,−0.3/0.7+0.6,−0.1/0.9+0.1,−0.1/0.8/q2q1+0.3,−0.2/1.0+0.5,−0.2/0.8q3We also have(19)Hb⊔Ib⊔Jb=0.8,−0.5/0.6+0.9,−0.7/03q1+.7,−0.6/0.3+0.8,−0.3/0.5+0.5,−.1/0.7q2+0.9,−0.2/0.8q3⊔0.9,−0.6/0.8+1.0,−0.8/0.7q1+0.6,−0.2/0.9+0.1,−0.5/0.4q2+0.6,−0.3/0.5+0.2,−0.8/0.4q3⊔0.9,−0.3/0.1+0.8,−0.1/0.5q1+0.6,−0.1/0.8q2+0.3,−0.6/1.0+0.4,−0.7/0.2+0.5,−0.5/0.7q3,=0.9,−0.3/0.8+0.9,−0.3/0.7q1+0.6,−0.1/0.9+0.1,−0.1/0.8q2+0.3,−0.2/1.0+0.5,−0.2/08q3.So, we obtain(20)Hb⊔Ib⊔Jb=Hb⊔Ib⊔JbHb⊔Ibc=0.9,−0.6/0.2+0.9,−0.7/0.3q1+0.6,−0.2/0.1+0.1,−0.5/0.6+0.1,−0.3/0.5+0.1,−0.1/0.3q2+0.6,−0.2/0.2q3,Hbc⊓Ibc=0.9,−0.6/0.2+0.9,−0.7/0.3q1+0.6,−0.2/0.1+0.1,−0.5/0.6+0.1,−0.3/0.5+0.1.−0.1/0.3q2+0.6,−0.2/0.2q3,Then, we deduceHb⊔Ibc=Hbc⊓Ibc.(21)Hb⊓Ibc=Hbc⊔IbcIndeed(22)Hb⊓Ibc=0.8,−0.5/0.4+0.9,−0.7/0.7q1+0.6,−0.5/0.7+0.6,−0.2/0.5+0.1,−0.3/0.6+0.5,−0.1/0.3q2+0.6,−0.2/0.5+0.2,−0.2/0.6q3,Hbc⊔Ibc=0.8,−0.5/0.4+0.9,−0.7/0.7q1+0.7,−0.6/0.7+0.8,−0.3/0.5+0.5,−0.1/0.3q2+0.9,−0.2/0.2q3+0.6,−0.3/0.5+0.2,−0.8/0.6q3=0.8,−0.5/0.4+0.9,−0.7/0.7q1+0.6,−0.5/0.7+0.6,−0.2/0.5+0.1,−0.3/0.6+0.5,−0.1/0.3q2+0.6,−0.2/0.5+0.2,−0.2/0.6q3.Theorem 10.
IfHb BT2F is a subset of Ib and Ib BT2F a subset of Jb, then Hb BT2F is a subset of Jb.Proof.
Suppose thatHb BT2F subset of Ib, thus k≤v for primary membership values and μHb+q,k≤μIb+q,v,μHb−q,k≥μIb−q,v for, respectively, positive and negative secondary membership functions of q∈ℚ, and suppose that Ib BT2F subset of Jb, thus v≤w for primary membership values and μIb+q,v≤μJb+q,w,μIb−q,v≥μJb−q,w for, respectively, positive and negative secondary membership functions of q∈ℚ.Therefore,k≤w,μHb+q,k≤μJb+q,w, and μHb−q,k≥μJb−q,w.Hence,Hb BT2F is a subset of Jb.Theorem 11.
IfHb BT2FS is equal to Ib and Ib BT2FS is equal to Jb, then Hb BT2FS is equal to Jb.Proof.
Same proof as Theorem10.


## 4. Bipolar Type-2 Fuzzy Soft Set

Throughout this section, we callℚ is an initial universe set, E is a set of all possible parameters under consideration with respect to ℚ, BFT2ℚ is all BT2FSs of ℚ, k,v, and w be primary membership degrees, and μτξ+,μτξ− are, respectively, the positive and negative secondary MF of ℚ.Definition 12.
LetH⊆E be a set of parameters, and τ is a mapping given by
(23)τ:H⟶BFT2ℚ.Then the pairτ,H is called a BT2FS f S over ℚ, and BFST2ℚ refers to all BT2FS f Ss over ℚ.So far a BT2FSf S is a parameterized family of BT2F subset of ℚ.For anyξ∈H,τξ refers to the set of ξ-approximate element of the BT2FS f S τ,H, it is actually a BT2FS on ℚ, and it can be written as
(24)τξ=∫q∈ℚ∫k∈Jq⊆0,1μτξ+q,k,μτξ−q,k/kq,where k is the primary membership degree and μτξ+q,k,μτξ−q,k are, respectively, the positive and negative secondary membership degree that object q holds on parameter ξ.Example 5.
Consider aτ,H∈BFST2ℚ over ℚ, where ℚ is a set of four cars under the consideration of decision-maker to purchase ℚ=q1,q2,q3,q4, and His a parameter set, where H=ξ1,ξ2,ξ3=expensive,beautiful,comfortable. The BT2FSfSτ,H describes the “attractiveness of the cars” to this decision-maker.Assume that(25)τξ1=0.3,−0.5/0.1+1.0,−0.4/0.2+0.7,−0.1/0.3q1+0.4,−0.2/0.5+0.8,−1.0/0.9q2+0.5,−0.2/0.7q3+0.3,−0.6/0.4+0.8,−0.1/0.5+0.2,−0.5/0.6q4,τξ2=0.3,−0.6/0.1+0.6,−0.9/0.4+0.5,−0.8/0.7q1+0.5,−0.1/0.8+0.2,−0.7/0.1+0.6,−0.3/0.2+0.9,−0.6/0.6q2+0.7,−0.2/0.8q3+0.9,−0.6/0.4q4,τξ3=0.2,−0.4/0.6+0.8,−0.3/0.8+0.6,−1.0/0.9q1+0.6,−0.8/0.4+0.8,−0.1/0.7q2+0.8,−0.4/0.9q3+0.7,−0.6/0.3+0.1,−0.4/0.6q4.Definition 13.
LetH,I⊆E. Let τ,H,ψ,I∈BFST2ℚ. If H⊆I and ∀ξ∈H,τξ⊆ψξ then τ,H is called a bipolar type-2 fuzzy soft (BT2FS f) subset of ψ,I.Remark 14.
Forτ,H,ψ,I∈BFST2ℚ are said to be equal, if and only if τ,H is BT2FSf subset of ψ,I and ψ,I is BT2FS f subset of τ,H.


### 4.1. Operations onBFST2ℚ

In this section, we debate five operations, namely, complement, union, intersection, AND, and OR of aBFST2ℚ. Let E=ξ1,ξ2,⋯,ξn be a parameter set, and defined ¬E=¬ξ1,¬ξ2,⋯,¬ξn where ¬ξi is not ξi and let H,I⊆E.(1)
ComplementThe complement of BT2FSf S is defined by
(26)τ,Hc=τc,¬H,where τcξ=¬τ¬ξ; for all ξ∈¬H.(2)
UnionLetτ,H,ψ,I∈BFST2ℚ, the union of τ,H and ψ,I is φ,J∈BFST2ℚ, and defined it for all ξ∈J=H∪I(27)φξ=τξifξ∈H−I,ψξifξ∈I−H,τξòψξifξ∈H∩I.(3)
IntersectionLetτ,H,ψ,I∈BFST2ℚ with H∩I≠ϕ, the intersection of τ,H and ψ,I is φ,J∈BFST2ℚ, defined by
(28)φξ=τξ⊓ψξforallξ∈J=H∩I.(4)
ANDτ,H AND ψ,I is defined by τ,H∧ψ,I=φ,H×I, where for all a,b∈H×I,φa,b=τa⊓ψb.(5)
ORτ,H OR ψ,I is defined by τ,H∨ψ,I=φ,H×I, which is φa,b=τa⊔ψb for all a,b∈H×I.Example 6.
LetE=ξ1,ξ2,ξ3 be a parameter set, and let H=ξ1,ξ2 and I=ξ2,ξ3 be subsets of E. Now consider two BT2FS f Ss τ,H and ψ,I over ℚ, where ℚ=q1,q2,q3 and define τ,H by
(29)τξ1=0.3,−0.5/0.1+1.0,−0.4/0.2+0.7,−0.1/0.3q1+0.4,−0.2/0.5+0.8,−0.1/0.9q2+0.5,−0.2/0.7q3,τξ2=0.3,−0.6/0.1+0.6,−0.9/0.4+0.5,−0.8/0.7q1+0.5,−0.1/0.8+0.2,−0.7/0.1+0.6,−0.3/0.2+0.9,−0.6/0.6q2+0.7,−0.2/0.8q3.and ψ,I by
(30)ψξ2=0.2,−0.4/0.6+0.8,−0.3/0.8+0.6,−1.0/0.9q1+0.6,−0.8/0.4+0.8,−0.1/0.7q2+0.8,−0.4/0.9q3,ψξ3=0.3,−0.6/0.4+0.8,−0.1/0.5+0.2,−0.5/0.6q1+0.9,−0.6/0.4q2+0.7,−0.6/0.3+0.1,−0.4/0.6q3.We will compute complement, union, intersection, AND, and OR ofBFST2 sets.
(i)
The complementτ,Hc=τc,¬H of BFST2 set τ,H is given
(31)τ¬ξ1=0.3,−0.5/0.9+1.0,−0.4/0.8+0.7,−0.1/0.7q1+0.4,−0.2/0.5+0.8,−0.1/0.1q2+0.5,−0.2/0.3q3,τ¬ξ2=0.3,−0.6/0.9+0.6,−0.9/0.6+0.5,−0.8/0.3q1+0.5,−0.1/0.2+0.2,−0.7/0.9+0.6,−0.3/0.8+0.9,−0.6/0.4q2+0.7,−0.2/0.2q3.(ii)
The union ofτ,H⊔ψ,I=φ,J, where J=H∪I=ξ1,ξ2,ξ3 is as follows since ξ1∈H−I then
(32)φξ1=τξ1=0.3,−0.5/0.1+1.0,−0.4/0.2+0.7,−0.1/0.3q1+0.4,−0.2/0.5+0.8,−0.1/0.9q2+0.5,−0.2/0.7q3,since ξ2∈H∩I then
(33)φξ2=τξ2⊔ψξ2=0.2,−0.4/0.6+0.6,−0.3/0.8+0.6,−0.9/0.9+0.2,−0.4/0.7q1+0.5,−0.1/0.8+0.6,−0.7/0.4+0.8,−0.1/0.7+0.6,−0.6/0.6q2+0.7,−0.2/0.9q3,since ξ3∈I−H then
(34)φξ3=ψξ3=0.3,−0.6/0.4+0.8,−0.1/0.5+0.2,−0.5/0.6q1+0.9,−0.6/0.4q2+0.7,−0.6/0.3+0.1,−0.4/0.6q3.(iii)
The intersection ofτ,H⊓ψ,I=σ,D, where D=H∩I=ξ2 is given
(35)σξ2=τξ2⊓ψξ2=0.3,−0.6/0.1+0.6,−0.9/0.4+0.2,−0.4/0.6+0.5,−0.8/0.7q1+0.6,−0.6/0.4+0.5,−0.1/0.7+0.2,−0.7/0.1+0.6,−0.3/0.2+0.8,−0.1/0.6q2+0.7,−0.2/0.8q3.(iv)
Aτ,H and ψ,I = τ,H∧ψ,I=φ,H×I are as follows:
(36)φξ1,ξ2=τξ1⊓ψξ2=0.3,−0.5/0.1+0.6,−0.4/0.2+0.7,−0.1/0.3q1+0.6,−0.2/0.4+0.4,−0.1/0.5+0.8,−0.1/0.7q2+0.5,−0.2/0.7q3,φξ1,ξ3=τξ1⊓ψξ3=0.3,−0.5/0.1+0.8,−0.4/0.2+0.7,−0.1/0.3q1+0.8,−0.2/0.4q2+0.5,−0.2/0.3+0.1,−0.2/0.6q3,φξ2,ξ2=τξ2⊓ψξ2=0.3,−0.6/0.1+0.6,−0.9/0.4+0.2,−0.4/0.6+0.5,−0.8/0.7q1+0.6,−0.6/0.4+0.5,−0.1/0.7+0.2,−0.7/0.1+0.6,−0.3/0.2+0.8,−0.1/0.6q2+0.7,−0.2/0.8q3,φξ2,ξ3=τξ2⊓ψξ3=0.3,−0.6/0.1+0.6,−0.6/0.4+0.5,−0.1/0.5+0.2,−0.5/0.6q1+0.9,−0.6/0.4+0.2,−0.6/0.1+0.6,−0.3/0.2q2+0.7,−0.2/0.3+0.1,−0.2/0.6q3.(v)
Aτ,H or ψ,I=τ,H∨ψ,I=σ,H×I is as follows:
(37)σξ1,ξ2=τξ1⊔ψξ2=0.2,−0.4/0.6+0.8,−0.3/0.8+0.6,−0.5/0.9q1+0.4,−0.2/0.5+0.4,−0.1/0.7+0.8,−0.1/0.9q2+0.5,−0.2/0.9q3,σξ1,ξ3=τξ1⊔ψξ3=0.3,−0.5/0.4+0.8,−0.1/0.5+0.2,−0.5/0.6q1+0.4,−0.2/0.5+0.8,−0.1/0.9q2+0.5,−0.2/0.7q3,σξ2,ξ2=τξ2⊔ψξ2=0.2,−0.4/0.6+0.6,−0.3/0.8+0.6,−0.9/0.9+0.2,−0.4/0.7q1+0.5,−0.1/0.8+0.6,−0.7/0.4+0.8,−0.1/0.7+0.6,−0.6/0.6q2+0.7,−0.2/0.9q3,σξ2,ξ3=τξ2⊔ψξ3=0.3,−0.6/0.4+0.6,−0.1/0.5+0.2,−0.5/0.6+0.5,−0.6/0.7q1+0.5,−0.1/0.8+0.6,−0.6/0.4+0.9,−0.6/0.6q2+0.7,−0.2/0.8q3.Proposition 15.
LetH,I,J⊆E and let τ,H,ψ,I,φ,J∈BFST2ℚ. Then, the following results hold:
(1)
τ,H⊔τ,H=τ,H and τ,H⊓τ,H=τ,H,(2)
τ,H⊔ψ,I=ψ,I⊔τ,H and τ,H⊓ψ,I=ψ,I⊓τ,H,(3)
τ,H⊔ψ,I⊓φ,J=τ,H⊔ψ,I⊓φ,J,(4)
τ,H⊓ψ,I⊓φ,J=τ,H⊓ψ,I⊓φ,J,(5)
τ,H⊓ψ,Ic⊆τ,Hc⊔ψ,Ic,(6)
τ,Hc⊓ψ,Ic⊆τ,H⊔ψ,Ic.Proof.
Suppose thatτ,H⊓ψ,I=σ,γ, where σξ=τξ⊓ψξ;γ=H∩I, so we deduce
(38)τ,H⊔ψ,Ic=σ,γc=σc,¬γ,where¬γ=¬H∩¬I.
This imply(39)σcξ=τcξ⊔ψcξ;∀ξ∈¬γ=¬H∩¬I.And if we suppose(40)τ,Hc⊔ψ,Ic=Ω,δ,whereδ=¬H∪¬I,then we obtain
(41)Ωξ=τcξifξ∈¬H−¬I,ψcξifξ∈¬I−¬H,τcξ⊔ψcξifξ∈¬H∩¬I.Henceτ,H⊓ψ,Ic⊆τ,Hc⊔ψ,Ic.Suppose thatτ,Hc⊓ψ,Ic=σ,γ, where σξ=τcξ⊓ψcξandγ=¬H∩¬I.And also suppose(42)τ,H⊔ψ,I=Ω,δwhereδ=H∪I,this imply
(43)Ωξ=τξifξ∈H−I,ψξifξ∈I−H,τξ⊔ψξifξ∈H∩I,so, we deduce
(44)τ,H⊔ψ,Ic=Ω,δc=Ωc,¬δ;¬δ=¬H∪¬I.Therefore(45)Ωcξ=τcξifξ∈¬H−¬I,ψcξifξ∈¬I−¬H,τcξ⊓ψcξifξ∈¬H∩¬I.Hence,τ,Hc⊓ψ,Ic⊆τ,H⊔ψ,Ic.Theorem 16.
LetH,I,J⊆E. For τ,H,ψ,I,φ,J∈BFST2ℚ. If τ,H is BT2FSf subset of ψ,I and ψ,I is BT2FS f subset of φ,J, thus τ,H is BT2FSf subset of φ,J.Proof.
Assume thatτ,H is a BT2FS f subset of ψ,I over ℚ, then we have
(i)
H⊆I(ii)
∀ξ∈H;τξ⊆ψξ
And also assume thatψ,I is a BT2FS f subset of φ,J over ℚ, then we get:
(i)
I⊂J(ii)
∀ξ∈I;ψξ⊆φξ
Hence,τ,H is a BT2FS f subset of φ,J over ℚ.Theorem 17.
LetH,I,J⊆E. For τ,H,ψ,I,φ,J∈BFST2ℚ. If τ,H and ψ,I are equal and at the same time ψ,I and φ,J are also equal, then τ,H and φ,J are equal.Proof.
Same proof as the previous theorem.Definition 18.
A null BT2FSfS denoted ϕT2H is a BT2FS fSτ,H over ℚ which is for all ξ∈H, for all q∈ℚ:
(46)μϕ+q,k=1;k=0,0;k≠0,μϕ−q,k=−1;k=0,0;k≠0.Definition 19.
An absolute BT2FSfS denoted ℚT2H is a BT2FS fSτ,H over ℚ which is for all ξ∈H, and for all q∈ℚ:
(47)μℚ+q,k=1;k=1,0;k≠1,μℚ−q,k=−1;k=1,0;k≠1.Theorem 20.
Letτ,H be a BFST2ℚ. Let ΦT2H and ℚT2H be a null and absolute BT2FS f S over ℚ, then
(i)
ΦT2Hc=ℚT2H and ℚT2Hc=ΦT2H(ii)
τ,H⊓ℚT2H=τ,H,(iii)
τ,H⊔ΦT2H=τ,HButτ,H⊔ℚT2H≠τ,H and τ,H⊓ΦT2H≠τ,H.Proof.
(i)
Suppose thatΦT2H is a null BT2FS f S over ℚ, and define for all ξ∈H(48)μϕ+q,k=1;k=0,0;k≠0,μϕ−q,k=−1;k=0,0;k≠0.
So, we getΦT2Hc defined by
(49)μϕc+q,k=1;k=0,0;k≠0,μϕc−q,k=−1;k=0,0;k≠0.
ThereforeΦT2Hc=ℚT2H.
Same proof forℚT2Hc=ΦT2H.(ii)
Letτ,H be a BFST2ℚ. Let ℚT2H be an absolute BT2FS f S over ℚ. So, we have
(50)τ,H⊓ℚT2H=φ,Hwhereφξ=τξ⊓ℚT2Hξ=τξ.
Thusτ,H⊓ℚT2H=τ,H.(iii)
Letτ,H be a BFST2ℚ. Let ΦT2H be a null BT2FS f S over ℚ. So, we have
(51)τ,H⊔ΦT2H=φ,Hwhereφξ=τξ⊓ℚT2Hξ=τξ.
Thus,τ,H⊓ΦT2H=τ,H.Theorem 21.
Letτ,H and ψ,I be the two BFST2ℚ, where H,I∈E. Then the following equations hold:
(i)
τ,H∧ψ,Ic=τ,Hc∨ψ,Ic(ii)
τ,H∨ψ,Ic=τ,Hc∧ψ,IcProof.
(i)
Suppose thatτ,H and ψ,I are two BFST2ℚ, we have
(52)τ,H∧ψ,I=φ,H×Idefineφbyφa,b=τaóψb∀a,b∈H×I.
Thenτ,H∧ψ,Ic=φ,H×Ic=φc,¬H×¬Iwhichφcα,β=τα⊔ψβc=τcαòψcβ∀α,β∈¬H×¬I.
On the other hand, we have(53)τ,Hc∨ψ,Ic=τc,¬H∨ψc,¬I=Ω,¬H×¬I,whichΩα,β=τcαòψcβ∀α,β∈¬H×¬I.Thereforeτ,H∧ψ,Ic=τ,Hc∨ψ,Ic.(ii)
Suppose thatτ,H and ψ,I are two BFST2ℚ, we have(54)τ,H∨ψ,I=φ,H×Idefineφbyφa,b=τaòψb∀a,b∈H×I.Thenτ,H∨ψ,Ic=φ,H×Ic=φc,¬H×¬Iwhichφcα,β=ταòψβc=τcαóψcβ∀α,β∈¬H×¬I.On the other hand, we obtain(55)τ,Hc∨ψ,Ic=τc,¬H∨ψc,¬I=Ω,¬H×¬I,whichΩα,β=τcαóψcβ∀α,β∈¬H×¬I.Therefore,τ,H∨ψ,Ic=τ,Hc∧ψ,Ic.Proposition 22.
LetH∈E. Let two BFST2ℚ sets τ,H and ψ,H over ℚ. Then
(i)
τ,Hóψ,Hc=τ,Hcòψ,Hc,(ii)
τ,Hòψ,Hc=τ,Hcóψ,HcProof.
(i)
Let twoτ,H,ψ,H∈BFST2ℚ. Then we get
(56)τ,Hóψ,H=φ,H,since H∩H=Hand∀ξ∈Hφξ=τξ⊓ψξ.
Therefore,τ,H⊔ψ,Hc=φ,Hc=φc,¬H, where ∀e∈¬Hφce=τe⊓ψec=τce⊔ψce.
On the other hand, we have(57)τ,Hc⊔ψ,Hc=τc,¬H⊔ψc,¬H=Ω,¬H,where Ωe=τce⊔ψc;∀e∈¬H.(ii)
In the same way.
As we noted earlier, the De Morgan’s laws does not hold. Thus, we can define two conditions under union and intersection operations.Definition 23.
For two BT2FSf Ss τ,H and ψ,I with H∩I≠ϕ over ℚ. Then the restricted union of τ,H and ψ,I is the BT2FS f S φ,J defined by
(58)∀ξ∈J=H∩Iφξ=τξ⊔rψξ.Definition 24.
For two BT2FSf Ss τ,H and ψ,I over ℚ. Then the extended intersection of τ,H and ψ,I is the BT2FS f S φ,J, where ∀ξ∈J=H∪I defined by
(59)φξ=τξ⊔eψξ=τξifξ∈H−I,ψξifξ∈I−H,τξ⊔ψξifξ∈H∩I.Theorem 25.
LetH,I⊂E. Let τ,H,ψ,I∈BFST2ℚ. Then
(i)
τ,H⊔eψ,Ic=τ,Hc⊔ψ,Ic,(ii)
τ,H⊔rψ,Ic=τ,Hc⊔ψ,Ic.Proof.
(i)
Forτ,H,ψ,I∈BFST2ℚ. Then we have(60)τ,H⊔eψ,I=τξifξ∈H−I,ψξifξ∈I−H,τξóψξifξ∈H∩I.
Thus, we obtain(61)τ,Hóeψ,Ic=τc¬ξif¬ξ∈H−I,ψc¬ξif¬ξ∈I−H,τξóψξcifξ∈H∩I.=τc¬ξif¬ξ∈H−I,ψc¬ξif¬ξ∈I−H,τc¬ξòψc¬ξif¬ξ∈H∩I.
On the other hand, we obtain(62)τ,Hc⊔ψ,Ic=τc,¬H⊔ψc,¬I=τc¬ξif¬ξ∈H−I,ψc¬ξif¬ξ∈I−H,τc¬ξòψc¬ξif¬ξ∈H∩I.we conclude that
(63)τ,H⊔eψ,Ic=τ,Hc⊔ψ,Ic.(ii)
Can be done in the same way.


## 4.1. Operations onBFST2ℚ

In this section, we debate five operations, namely, complement, union, intersection, AND, and OR of aBFST2ℚ. Let E=ξ1,ξ2,⋯,ξn be a parameter set, and defined ¬E=¬ξ1,¬ξ2,⋯,¬ξn where ¬ξi is not ξi and let H,I⊆E.(1)
ComplementThe complement of BT2FSf S is defined by
(26)τ,Hc=τc,¬H,where τcξ=¬τ¬ξ; for all ξ∈¬H.(2)
UnionLetτ,H,ψ,I∈BFST2ℚ, the union of τ,H and ψ,I is φ,J∈BFST2ℚ, and defined it for all ξ∈J=H∪I(27)φξ=τξifξ∈H−I,ψξifξ∈I−H,τξòψξifξ∈H∩I.(3)
IntersectionLetτ,H,ψ,I∈BFST2ℚ with H∩I≠ϕ, the intersection of τ,H and ψ,I is φ,J∈BFST2ℚ, defined by
(28)φξ=τξ⊓ψξforallξ∈J=H∩I.(4)
ANDτ,H AND ψ,I is defined by τ,H∧ψ,I=φ,H×I, where for all a,b∈H×I,φa,b=τa⊓ψb.(5)
ORτ,H OR ψ,I is defined by τ,H∨ψ,I=φ,H×I, which is φa,b=τa⊔ψb for all a,b∈H×I.Example 6.
LetE=ξ1,ξ2,ξ3 be a parameter set, and let H=ξ1,ξ2 and I=ξ2,ξ3 be subsets of E. Now consider two BT2FS f Ss τ,H and ψ,I over ℚ, where ℚ=q1,q2,q3 and define τ,H by
(29)τξ1=0.3,−0.5/0.1+1.0,−0.4/0.2+0.7,−0.1/0.3q1+0.4,−0.2/0.5+0.8,−0.1/0.9q2+0.5,−0.2/0.7q3,τξ2=0.3,−0.6/0.1+0.6,−0.9/0.4+0.5,−0.8/0.7q1+0.5,−0.1/0.8+0.2,−0.7/0.1+0.6,−0.3/0.2+0.9,−0.6/0.6q2+0.7,−0.2/0.8q3.and ψ,I by
(30)ψξ2=0.2,−0.4/0.6+0.8,−0.3/0.8+0.6,−1.0/0.9q1+0.6,−0.8/0.4+0.8,−0.1/0.7q2+0.8,−0.4/0.9q3,ψξ3=0.3,−0.6/0.4+0.8,−0.1/0.5+0.2,−0.5/0.6q1+0.9,−0.6/0.4q2+0.7,−0.6/0.3+0.1,−0.4/0.6q3.We will compute complement, union, intersection, AND, and OR ofBFST2 sets.
(i)
The complementτ,Hc=τc,¬H of BFST2 set τ,H is given
(31)τ¬ξ1=0.3,−0.5/0.9+1.0,−0.4/0.8+0.7,−0.1/0.7q1+0.4,−0.2/0.5+0.8,−0.1/0.1q2+0.5,−0.2/0.3q3,τ¬ξ2=0.3,−0.6/0.9+0.6,−0.9/0.6+0.5,−0.8/0.3q1+0.5,−0.1/0.2+0.2,−0.7/0.9+0.6,−0.3/0.8+0.9,−0.6/0.4q2+0.7,−0.2/0.2q3.(ii)
The union ofτ,H⊔ψ,I=φ,J, where J=H∪I=ξ1,ξ2,ξ3 is as follows since ξ1∈H−I then
(32)φξ1=τξ1=0.3,−0.5/0.1+1.0,−0.4/0.2+0.7,−0.1/0.3q1+0.4,−0.2/0.5+0.8,−0.1/0.9q2+0.5,−0.2/0.7q3,since ξ2∈H∩I then
(33)φξ2=τξ2⊔ψξ2=0.2,−0.4/0.6+0.6,−0.3/0.8+0.6,−0.9/0.9+0.2,−0.4/0.7q1+0.5,−0.1/0.8+0.6,−0.7/0.4+0.8,−0.1/0.7+0.6,−0.6/0.6q2+0.7,−0.2/0.9q3,since ξ3∈I−H then
(34)φξ3=ψξ3=0.3,−0.6/0.4+0.8,−0.1/0.5+0.2,−0.5/0.6q1+0.9,−0.6/0.4q2+0.7,−0.6/0.3+0.1,−0.4/0.6q3.(iii)
The intersection ofτ,H⊓ψ,I=σ,D, where D=H∩I=ξ2 is given
(35)σξ2=τξ2⊓ψξ2=0.3,−0.6/0.1+0.6,−0.9/0.4+0.2,−0.4/0.6+0.5,−0.8/0.7q1+0.6,−0.6/0.4+0.5,−0.1/0.7+0.2,−0.7/0.1+0.6,−0.3/0.2+0.8,−0.1/0.6q2+0.7,−0.2/0.8q3.(iv)
Aτ,H and ψ,I = τ,H∧ψ,I=φ,H×I are as follows:
(36)φξ1,ξ2=τξ1⊓ψξ2=0.3,−0.5/0.1+0.6,−0.4/0.2+0.7,−0.1/0.3q1+0.6,−0.2/0.4+0.4,−0.1/0.5+0.8,−0.1/0.7q2+0.5,−0.2/0.7q3,φξ1,ξ3=τξ1⊓ψξ3=0.3,−0.5/0.1+0.8,−0.4/0.2+0.7,−0.1/0.3q1+0.8,−0.2/0.4q2+0.5,−0.2/0.3+0.1,−0.2/0.6q3,φξ2,ξ2=τξ2⊓ψξ2=0.3,−0.6/0.1+0.6,−0.9/0.4+0.2,−0.4/0.6+0.5,−0.8/0.7q1+0.6,−0.6/0.4+0.5,−0.1/0.7+0.2,−0.7/0.1+0.6,−0.3/0.2+0.8,−0.1/0.6q2+0.7,−0.2/0.8q3,φξ2,ξ3=τξ2⊓ψξ3=0.3,−0.6/0.1+0.6,−0.6/0.4+0.5,−0.1/0.5+0.2,−0.5/0.6q1+0.9,−0.6/0.4+0.2,−0.6/0.1+0.6,−0.3/0.2q2+0.7,−0.2/0.3+0.1,−0.2/0.6q3.(v)
Aτ,H or ψ,I=τ,H∨ψ,I=σ,H×I is as follows:
(37)σξ1,ξ2=τξ1⊔ψξ2=0.2,−0.4/0.6+0.8,−0.3/0.8+0.6,−0.5/0.9q1+0.4,−0.2/0.5+0.4,−0.1/0.7+0.8,−0.1/0.9q2+0.5,−0.2/0.9q3,σξ1,ξ3=τξ1⊔ψξ3=0.3,−0.5/0.4+0.8,−0.1/0.5+0.2,−0.5/0.6q1+0.4,−0.2/0.5+0.8,−0.1/0.9q2+0.5,−0.2/0.7q3,σξ2,ξ2=τξ2⊔ψξ2=0.2,−0.4/0.6+0.6,−0.3/0.8+0.6,−0.9/0.9+0.2,−0.4/0.7q1+0.5,−0.1/0.8+0.6,−0.7/0.4+0.8,−0.1/0.7+0.6,−0.6/0.6q2+0.7,−0.2/0.9q3,σξ2,ξ3=τξ2⊔ψξ3=0.3,−0.6/0.4+0.6,−0.1/0.5+0.2,−0.5/0.6+0.5,−0.6/0.7q1+0.5,−0.1/0.8+0.6,−0.6/0.4+0.9,−0.6/0.6q2+0.7,−0.2/0.8q3.Proposition 15.
LetH,I,J⊆E and let τ,H,ψ,I,φ,J∈BFST2ℚ. Then, the following results hold:
(1)
τ,H⊔τ,H=τ,H and τ,H⊓τ,H=τ,H,(2)
τ,H⊔ψ,I=ψ,I⊔τ,H and τ,H⊓ψ,I=ψ,I⊓τ,H,(3)
τ,H⊔ψ,I⊓φ,J=τ,H⊔ψ,I⊓φ,J,(4)
τ,H⊓ψ,I⊓φ,J=τ,H⊓ψ,I⊓φ,J,(5)
τ,H⊓ψ,Ic⊆τ,Hc⊔ψ,Ic,(6)
τ,Hc⊓ψ,Ic⊆τ,H⊔ψ,Ic.Proof.
Suppose thatτ,H⊓ψ,I=σ,γ, where σξ=τξ⊓ψξ;γ=H∩I, so we deduce
(38)τ,H⊔ψ,Ic=σ,γc=σc,¬γ,where¬γ=¬H∩¬I.
This imply(39)σcξ=τcξ⊔ψcξ;∀ξ∈¬γ=¬H∩¬I.And if we suppose(40)τ,Hc⊔ψ,Ic=Ω,δ,whereδ=¬H∪¬I,then we obtain
(41)Ωξ=τcξifξ∈¬H−¬I,ψcξifξ∈¬I−¬H,τcξ⊔ψcξifξ∈¬H∩¬I.Henceτ,H⊓ψ,Ic⊆τ,Hc⊔ψ,Ic.Suppose thatτ,Hc⊓ψ,Ic=σ,γ, where σξ=τcξ⊓ψcξandγ=¬H∩¬I.And also suppose(42)τ,H⊔ψ,I=Ω,δwhereδ=H∪I,this imply
(43)Ωξ=τξifξ∈H−I,ψξifξ∈I−H,τξ⊔ψξifξ∈H∩I,so, we deduce
(44)τ,H⊔ψ,Ic=Ω,δc=Ωc,¬δ;¬δ=¬H∪¬I.Therefore(45)Ωcξ=τcξifξ∈¬H−¬I,ψcξifξ∈¬I−¬H,τcξ⊓ψcξifξ∈¬H∩¬I.Hence,τ,Hc⊓ψ,Ic⊆τ,H⊔ψ,Ic.Theorem 16.
LetH,I,J⊆E. For τ,H,ψ,I,φ,J∈BFST2ℚ. If τ,H is BT2FSf subset of ψ,I and ψ,I is BT2FS f subset of φ,J, thus τ,H is BT2FSf subset of φ,J.Proof.
Assume thatτ,H is a BT2FS f subset of ψ,I over ℚ, then we have
(i)
H⊆I(ii)
∀ξ∈H;τξ⊆ψξ
And also assume thatψ,I is a BT2FS f subset of φ,J over ℚ, then we get:
(i)
I⊂J(ii)
∀ξ∈I;ψξ⊆φξ
Hence,τ,H is a BT2FS f subset of φ,J over ℚ.Theorem 17.
LetH,I,J⊆E. For τ,H,ψ,I,φ,J∈BFST2ℚ. If τ,H and ψ,I are equal and at the same time ψ,I and φ,J are also equal, then τ,H and φ,J are equal.Proof.
Same proof as the previous theorem.Definition 18.
A null BT2FSfS denoted ϕT2H is a BT2FS fSτ,H over ℚ which is for all ξ∈H, for all q∈ℚ:
(46)μϕ+q,k=1;k=0,0;k≠0,μϕ−q,k=−1;k=0,0;k≠0.Definition 19.
An absolute BT2FSfS denoted ℚT2H is a BT2FS fSτ,H over ℚ which is for all ξ∈H, and for all q∈ℚ:
(47)μℚ+q,k=1;k=1,0;k≠1,μℚ−q,k=−1;k=1,0;k≠1.Theorem 20.
Letτ,H be a BFST2ℚ. Let ΦT2H and ℚT2H be a null and absolute BT2FS f S over ℚ, then
(i)
ΦT2Hc=ℚT2H and ℚT2Hc=ΦT2H(ii)
τ,H⊓ℚT2H=τ,H,(iii)
τ,H⊔ΦT2H=τ,HButτ,H⊔ℚT2H≠τ,H and τ,H⊓ΦT2H≠τ,H.Proof.
(i)
Suppose thatΦT2H is a null BT2FS f S over ℚ, and define for all ξ∈H(48)μϕ+q,k=1;k=0,0;k≠0,μϕ−q,k=−1;k=0,0;k≠0.
So, we getΦT2Hc defined by
(49)μϕc+q,k=1;k=0,0;k≠0,μϕc−q,k=−1;k=0,0;k≠0.
ThereforeΦT2Hc=ℚT2H.
Same proof forℚT2Hc=ΦT2H.(ii)
Letτ,H be a BFST2ℚ. Let ℚT2H be an absolute BT2FS f S over ℚ. So, we have
(50)τ,H⊓ℚT2H=φ,Hwhereφξ=τξ⊓ℚT2Hξ=τξ.
Thusτ,H⊓ℚT2H=τ,H.(iii)
Letτ,H be a BFST2ℚ. Let ΦT2H be a null BT2FS f S over ℚ. So, we have
(51)τ,H⊔ΦT2H=φ,Hwhereφξ=τξ⊓ℚT2Hξ=τξ.
Thus,τ,H⊓ΦT2H=τ,H.Theorem 21.
Letτ,H and ψ,I be the two BFST2ℚ, where H,I∈E. Then the following equations hold:
(i)
τ,H∧ψ,Ic=τ,Hc∨ψ,Ic(ii)
τ,H∨ψ,Ic=τ,Hc∧ψ,IcProof.
(i)
Suppose thatτ,H and ψ,I are two BFST2ℚ, we have
(52)τ,H∧ψ,I=φ,H×Idefineφbyφa,b=τaóψb∀a,b∈H×I.
Thenτ,H∧ψ,Ic=φ,H×Ic=φc,¬H×¬Iwhichφcα,β=τα⊔ψβc=τcαòψcβ∀α,β∈¬H×¬I.
On the other hand, we have(53)τ,Hc∨ψ,Ic=τc,¬H∨ψc,¬I=Ω,¬H×¬I,whichΩα,β=τcαòψcβ∀α,β∈¬H×¬I.Thereforeτ,H∧ψ,Ic=τ,Hc∨ψ,Ic.(ii)
Suppose thatτ,H and ψ,I are two BFST2ℚ, we have(54)τ,H∨ψ,I=φ,H×Idefineφbyφa,b=τaòψb∀a,b∈H×I.Thenτ,H∨ψ,Ic=φ,H×Ic=φc,¬H×¬Iwhichφcα,β=ταòψβc=τcαóψcβ∀α,β∈¬H×¬I.On the other hand, we obtain(55)τ,Hc∨ψ,Ic=τc,¬H∨ψc,¬I=Ω,¬H×¬I,whichΩα,β=τcαóψcβ∀α,β∈¬H×¬I.Therefore,τ,H∨ψ,Ic=τ,Hc∧ψ,Ic.Proposition 22.
LetH∈E. Let two BFST2ℚ sets τ,H and ψ,H over ℚ. Then
(i)
τ,Hóψ,Hc=τ,Hcòψ,Hc,(ii)
τ,Hòψ,Hc=τ,Hcóψ,HcProof.
(i)
Let twoτ,H,ψ,H∈BFST2ℚ. Then we get
(56)τ,Hóψ,H=φ,H,since H∩H=Hand∀ξ∈Hφξ=τξ⊓ψξ.
Therefore,τ,H⊔ψ,Hc=φ,Hc=φc,¬H, where ∀e∈¬Hφce=τe⊓ψec=τce⊔ψce.
On the other hand, we have(57)τ,Hc⊔ψ,Hc=τc,¬H⊔ψc,¬H=Ω,¬H,where Ωe=τce⊔ψc;∀e∈¬H.(ii)
In the same way.
As we noted earlier, the De Morgan’s laws does not hold. Thus, we can define two conditions under union and intersection operations.Definition 23.
For two BT2FSf Ss τ,H and ψ,I with H∩I≠ϕ over ℚ. Then the restricted union of τ,H and ψ,I is the BT2FS f S φ,J defined by
(58)∀ξ∈J=H∩Iφξ=τξ⊔rψξ.Definition 24.
For two BT2FSf Ss τ,H and ψ,I over ℚ. Then the extended intersection of τ,H and ψ,I is the BT2FS f S φ,J, where ∀ξ∈J=H∪I defined by
(59)φξ=τξ⊔eψξ=τξifξ∈H−I,ψξifξ∈I−H,τξ⊔ψξifξ∈H∩I.Theorem 25.
LetH,I⊂E. Let τ,H,ψ,I∈BFST2ℚ. Then
(i)
τ,H⊔eψ,Ic=τ,Hc⊔ψ,Ic,(ii)
τ,H⊔rψ,Ic=τ,Hc⊔ψ,Ic.Proof.
(i)
Forτ,H,ψ,I∈BFST2ℚ. Then we have(60)τ,H⊔eψ,I=τξifξ∈H−I,ψξifξ∈I−H,τξóψξifξ∈H∩I.
Thus, we obtain(61)τ,Hóeψ,Ic=τc¬ξif¬ξ∈H−I,ψc¬ξif¬ξ∈I−H,τξóψξcifξ∈H∩I.=τc¬ξif¬ξ∈H−I,ψc¬ξif¬ξ∈I−H,τc¬ξòψc¬ξif¬ξ∈H∩I.
On the other hand, we obtain(62)τ,Hc⊔ψ,Ic=τc,¬H⊔ψc,¬I=τc¬ξif¬ξ∈H−I,ψc¬ξif¬ξ∈I−H,τc¬ξòψc¬ξif¬ξ∈H∩I.we conclude that
(63)τ,H⊔eψ,Ic=τ,Hc⊔ψ,Ic.(ii)
Can be done in the same way.


## 5. Application ofBFST2 Set in Decision Making

Decision making is one of the most popular applications of Sf S theory which has a lot of applications in many practical problems. Maji et al. introduced the idea of S f Ss for the evaluation of parameterized decision-making problems. Moreover, Roy and Maji presented an algorithm to solve the recognition problem by employing FS f Ss. Subsequently, Kong et al. gave a counterexample to illustrate a fact that the optimal choice could not be obtained in general by using Roy and Maji’s algorithm and conferred a modified version. Feng et al. gave deeper insights into decision-making methods based on FSf Ss. They conferred an adjustable approach for fuzzy-soft-set-based decision making by employing level Sf Ss. By generalizing the adjustable approach for fuzzy-soft-set-based decision making, Jiang et al. presented an adjustable approach for intuitionistic fuzzy-soft-set-based decision making and explore some illustrative examples.In this section, we shall present an adjustable approach for BT2FSf S-based decision-making problems. This technique is based on the concept of level Sf Ss.Definition 26.
LetH⊂E.Letτ,H be a BT2FSfS over the initial universe ℚ. For γ∈0,1 and β∈0,1, the γ,β-level (the threshold) SfS of τ,His crisp soft set Tτ,H,γ,β=τγ,β,H for all ξ∈H, defined by
(64)τγ,βξ=τξβγ=q∈ℚ∣kq≥β;∀kq∈Jqγ,where Jqγ=k∣μτξ+q,k−μτξ−q,k≥γ;k∈Jq.Definition 27.
LetH⊂E.Letτ,Hbe aBFST2set overℚ. Letγ:H⟶0,1andβ:H⟶0,1be two fuzzy sets inHwhich are two threshold fuzzy sets. Then, the level SfS ofτ,Hwith respect toγandβis crisp soft setTτ,H,γ,β=τγ,β,Hfor allξ∈H,defined by(65)τγ,βξ=τξβγ=q∈ℚkq≥βξ;∀kq∈Jqγ,where Jqγ=k∣μτξ+q,k−μτξ−q,k≥γξ;k∈Jq.Definition 28.
LetH⊂Ebe a parameter set. Letτ,Hbe aBFST2set overℚ, and letΛ:H⟶0,1be the weight function specifying the weightΛi=Λξifor allξi∈H.Then, the tripleτ,H,Λis a weighted BT2FSfS.


### 5.1. Algorithm

Consider the following.Step 1: Input a weighted BT2FSf S Γ=τ,H,Λ.Step 2: Input two threshold fuzzy setsγ:H⟶0,1 and β:H⟶0,1 for decision making.Step 3: Compute the level Sf S Tτ,H,γ,β of Γ w.r.t the two threshold fuzzy sets γ and β.Step 4: Present the level Sf S Tτ,H,γ,β in tabular form. For all qi∈ℚ, compute the weighted choice value c of qi, where c=∑ξ∈Hτξqi.Γξ.Step 5: The optimal decision is to selectqi if ck=maxqi∈ℚci.Step 6: If k has more than one value then any one ofhk may be chosen.Example 7.
Suppose that a company is planning to increase its green area in the center of a desert city, so by raising a large number of trees and plants, it will improve generally. There are two ways for the cultivation of area, one is the traditional way and the other one is by groasis technologyℚ=q1,q2.Let there exist are four parameters E=e1,e2,e3,e4=Temperature,Soilfertility,Humidity,Sunlight. Assume that weight for the parameters in E:Λ1=Λξ1=0.2,Λ2=Λξ2=0.9,Λ3=Λξ3=0.7andΛ4=Λξ4=0.4; and two threshold fuzzy sets γ:E⟶0,1 and β:E⟶,1 defined by
(66)γ=0.90ξ1+0.80ξ2+0.85ξ3+0.60ξ4;β=0.75ξ1+0.60ξ2+0.55ξ3+0.40ξ4Consider bipolar type-2 FSf S as follows:
(67)τ,E=e1,0.30,−0.65/0.30+0.50,−0.75/0.60q1+0.45,−0.30/0.60+0.70,−0.25/0.85q2,e2,0.50,−0.40/0.25q1+0.55,−0.20/0.45+0.80,−0.35/0.75q2,e3,0.15,−0.25/0.15+0.40,−0.60/0.75q1+0.70,−0.30/0.50q2,e4,0.75,−0.60/0.85q1+0.80,−0.30/0.80q2.The values are shown in Tables1, 2Table 1
Tabular representation ofΓ=τ,H,Λ..
ℚξ1,Λ1=0.2ξ2,Λ2=0.9ξ3,Λ3=0.7ξ4,Λ4=0.4q10.30,−0.650.30+0.50,−0.750.600.50,−0.400.250.15,−0.250.15+0.40,−0.600.750.75,−0.600.85q20.45,−0.300.60+0.70,−0.250.850.55,−0.200.45+0.80,−0.350.750.70,−0.300.500.80,−0.300.80Table 2
Tabular representation ofΓ=τ,H,Λ with weighted choice value.
ℚξ1,Λ1=0.2ξ2,Λ2=0.9ξ3,Λ3=0.7ξ4,Λ4=0.4Weighted choice valueciq10011c1=1.1q21101c2=1.40We deduced that weighted optimal choice valuemax1≤i≤2ci=c2. Hence, q2 be the required optimal choice.


## 5.1. Algorithm

Consider the following.Step 1: Input a weighted BT2FSf S Γ=τ,H,Λ.Step 2: Input two threshold fuzzy setsγ:H⟶0,1 and β:H⟶0,1 for decision making.Step 3: Compute the level Sf S Tτ,H,γ,β of Γ w.r.t the two threshold fuzzy sets γ and β.Step 4: Present the level Sf S Tτ,H,γ,β in tabular form. For all qi∈ℚ, compute the weighted choice value c of qi, where c=∑ξ∈Hτξqi.Γξ.Step 5: The optimal decision is to selectqi if ck=maxqi∈ℚci.Step 6: If k has more than one value then any one ofhk may be chosen.Example 7.
Suppose that a company is planning to increase its green area in the center of a desert city, so by raising a large number of trees and plants, it will improve generally. There are two ways for the cultivation of area, one is the traditional way and the other one is by groasis technologyℚ=q1,q2.Let there exist are four parameters E=e1,e2,e3,e4=Temperature,Soilfertility,Humidity,Sunlight. Assume that weight for the parameters in E:Λ1=Λξ1=0.2,Λ2=Λξ2=0.9,Λ3=Λξ3=0.7andΛ4=Λξ4=0.4; and two threshold fuzzy sets γ:E⟶0,1 and β:E⟶,1 defined by
(66)γ=0.90ξ1+0.80ξ2+0.85ξ3+0.60ξ4;β=0.75ξ1+0.60ξ2+0.55ξ3+0.40ξ4Consider bipolar type-2 FSf S as follows:
(67)τ,E=e1,0.30,−0.65/0.30+0.50,−0.75/0.60q1+0.45,−0.30/0.60+0.70,−0.25/0.85q2,e2,0.50,−0.40/0.25q1+0.55,−0.20/0.45+0.80,−0.35/0.75q2,e3,0.15,−0.25/0.15+0.40,−0.60/0.75q1+0.70,−0.30/0.50q2,e4,0.75,−0.60/0.85q1+0.80,−0.30/0.80q2.The values are shown in Tables1, 2Table 1
Tabular representation ofΓ=τ,H,Λ..
ℚξ1,Λ1=0.2ξ2,Λ2=0.9ξ3,Λ3=0.7ξ4,Λ4=0.4q10.30,−0.650.30+0.50,−0.750.600.50,−0.400.250.15,−0.250.15+0.40,−0.600.750.75,−0.600.85q20.45,−0.300.60+0.70,−0.250.850.55,−0.200.45+0.80,−0.350.750.70,−0.300.500.80,−0.300.80Table 2
Tabular representation ofΓ=τ,H,Λ with weighted choice value.
ℚξ1,Λ1=0.2ξ2,Λ2=0.9ξ3,Λ3=0.7ξ4,Λ4=0.4Weighted choice valueciq10011c1=1.1q21101c2=1.40We deduced that weighted optimal choice valuemax1≤i≤2ci=c2. Hence, q2 be the required optimal choice.


## 6. Conclusion

In this research article, we have rapturously consummated our goal by establishing the ground-breaking incredible theories of BT2FS and BT2FSf S. These flexible models are highly dexterous for addressing a comprehensive range of ambiguity embedded in bipolar human interpretations. The newly developed BT2FS f S provides an outstanding mathematical framework that admirably integrates the specialities of novel BT2FS with the parametric nature of S f S. The proposed theories have extrapolated the contemporary unipolar fuzzy structures because of their fantastic representational and reasoning potentialities. Firstly, we have explicated the systematic definitions of presented notions and discussed some of their fundamental properties. We have thrown light on some elementary set-theoretical operations for these emerging theories inclusive of equality, subset, complement, union and intersection. Another noteworthy contribution of this study is the development of a highly proficient MCDM algorithm for the specification of the best pragmatic alternative on the grounds of multiple rational parameters. Finally, we have skilfully implemented the proposed methodology on real-life heuristic application to demonstrate the versatility of our developed strategy. The presented frameworks have an edge over the already existing notions, as they remarkably decipher all the faults and shortcomings of recent approaches by capturing both polarity and abstruseness of paradoxical data set. Admittedly, our proposed models have some difficulties because of their limited one-dimensional modeling capabilities. Thus in the future, we are envisioning to amplify our research study for the two dimensional vague information by presenting the excellent frameworks of complex BT2FS and complex BT2FS f S in order to design more advanced decision-making strategies. We intend to deliberate the scope of these decision-making approaches in the disciplines of artificial intelligence, medical sciences, logistics and aeronautical industries. Another goal is to unfold significant fusions of ELECTRE methods with the proposed theories to develop more practical decision-making approaches.



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*Source: 8472982-2022-04-30.xml*