# Solution of Fully Bipolar Fuzzy Linear Programming Models

Authors: Muhammad Athar Mehmood; Muhammad Akram; Majed G. Alharbi; Shahida Bashir
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The Yin-Yang bipolar fuzzy set is a powerful mathematical tool for depicting fuzziness and vagueness. We first extend the concept of crisp linear programming problem in a bipolar fuzzy environment based on bipolar fuzzy numbers. We first define arithmetic operations of unrestricted bipolar fuzzy numbers and multiplication of an unrestricted trapezoidal bipolar fuzzy number (TrBFN) with non-negative TrBFN. We then propose a method for solving fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints in which the coefficients are unrestricted triangular bipolar fuzzy numbers and decision variables are nonnegative triangular bipolar fuzzy numbers. Furthermore, we present a method for solving FBFLPPs with equality constraints in which the coefficients and decision variables are unrestricted TrBFNs. The FBFLPP is transformed into a crisp linear programming problem, and then, it is solved to achieve the exact bipolar fuzzy optimal solution. We illustrate the proposed methodologies with several numerical examples.


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## Body

## 1. Introduction

Zadeh [1] introduced the concept of fuzzy set (FS) theory which is an extension of classical set theory. FS theory made a swift progress in optimization. Presently, the uncertain problems are dealt in fuzzy mathematics. In a fuzzy set, objects are assigned grades of membership or we can say degree of membership. FS is defined by a membership function which associates to each element a degree of membership varying between 0 and 1. In a few applications, we have to consider truth membership as well as falsity membership supported and opposed by the evident. This gives us the direction towards bipolar fuzzy set theory. In modeling, FS theory is a powerful mathematical tool for analyzing fuzziness and uncertainty. The fundamental arithmetic operations of fuzzy numbers were discussed by Dubois and Prade [2, 3], Mizumoto and Tanaka [4], Mizumoto and Tanaka [5], and Nahmias [6].Linear programming (LP) is showing a significant role in different fields of science and technology. In the last few decades, LP models are being used in the crisp environment as well as in the fuzzy environment because they have vast applications in operation research (OR) methodologies. LP is a powerful mathematical tool for optimal distribution of constrained resources amongst challenging activities. It is a popular method with utilization in different fields such as production, marketing, finance, and advertising [7–10]. In a crisp environment, mathematical models having objective functions with constraints contain no possibility for uncertainty. In real-world problems, LP models contain variables whose values are given by professionals and decision makers, but both professionals and decision makers themselves are unaware of the precise values. In a Fuzzy environment, such types of situations that arise in the real scenarios are solved by fuzzy linear programming (FLP) and such problems under consideration turn into FLP. The classical LP is frequently inadequate in real-world problems, while in recent decades, the FLP is getting popularity in the research community.The FLP problem is first considered by Bellman and Zadeh [11]. Lotfi et al. [12] claimed that no one has solved FFLP problems with equality constraints and suggested the existing method to solve these problems. Tanaka and Asai [13] formulated FLP problems and obtained their solutions. The FLP problems that contain all variables and parameters as fuzzy numbers are said to be FFLP problems. FFLP problems involving inequality constraints were discussed in [14–16]. Kumar et al. [17] pointed out that the solution gained by the existing method does not satisfy the constraints of the FFLP problem, and they presented a method to calculate the solution of FFLP problems which satisfy the constraints of the problem. Allahviranloo et al. [18] obtained the general solution of a fully fuzzy linear system. The main reason to design a new model for solving FBFLPP is that the FFLP problem has not been discussed with bipolar fuzzy numbers yet. The idea of the Yin-Yang bipolar fuzzy set (BFS) was proposed by Zhang [19] which is an extension of FS, and Yin and Yang represent the negative side and positive side of a system, respectively [20]. A large variety of human decision making is based on bipolar judgmental thinking on a positive side and a negative side. Furthermore, in a bipolar fuzzy environment, multicriteria decision making was studied by Alghamdi et al. [21]. Das et al. [22] designed a technique to solve the FFLP problem using trapezoidal fuzzy numbers. Edalatpanah [23] presented a scheme to determine the solution of neutrosophic LPP by using triangular neutrosophic numbers. Najafi et al. [24] obtained a fuzzy optimal solution of the FFLP problem involving variables as unrestricted triangular fuzzy numbers. Two different techniques were used by Behera et al. [25], and they solved the FFLP problems containing nonnegative crisp coefficients and mixed crisp coefficients. In this paper, we propose a method for solving fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints in which the coefficients are unrestricted triangular bipolar fuzzy numbers and decision variables are nonnegative triangular bipolar fuzzy numbers. Furthermore, we present a method for solving FBFLPPs with equality constraints in which the coefficients and decision variables are unrestricted TrBFNs. The FBFLPP is transformed into a crisp linear programming problem, and then, it is solved to achieve the exact bipolar fuzzy optimal solution. We illustrate the proposed methodologies with several numerical examples.We have organized our paper as follows: Section2 presents some necessary preliminaries. Section 3 is devoted to develop new methods for FPFLPP with equality constraints in which the coefficients are unrestricted triangular bipolar fuzzy numbers and decision variables are nonnegative triangular bipolar fuzzy numbers. Section 4 is devoted to develop a method for solving FBFLPPs with equality constraints in which the coefficients and decision variables are unrestricted TrBFNs. Conclusions are given in Section 5.


## 2. Preliminaries

In this section, we review concepts that will be applied in the coming sections.Definition 1 (See [19]).
LetM be a nonempty set. A BFS S˜ in M is an object of the form(1)S˜=ηS˜P,ηS˜N=x,ηS˜Px,ηS˜Nx|x∈M,where ηS˜P:M⟶0,1 and ηS˜N:M⟶−1,0 are mappings. The positive membership degree ηS˜Px denotes the truth or satisfaction degree of an element x to a certain property corresponding to bipolar fuzzy set S˜, and ηS˜Nx represents the satisfaction degree of an element x to some counter property of bipolar fuzzy set S˜. If ηS˜Nx≠0 and ηS˜Px=0, it is the situation that x is not satisfying the property of S˜ but satisfying the counter property to S˜. If ηS˜Px≠0 and ηS˜Nx=0, it is the case when x has only positive satisfaction for S˜. It is possible for x to be such that ηS˜Px≠0 and ηS˜Nx≠0 when x satisfies the property of S˜ as well as its counter property in some part of M.Definition 2 (See [26]).
A TBFN defined on real numbersℝ, denoted by t1,t2,t3,f1,f2,f3, is a BFN, and its satisfaction degree λP and dissatisfaction degree λN are represented by(2)λP=x−t1t2−t1,ifx∈t1,t2,t3−xt3−t2,ifx∈t2,t3,0,otherwise,λN=f1−xf2−f1,ifx∈f1,f2,x−f3f3−f2,ifx∈f3,f2,0,otherwise.Definition 3 (See [26]).
A ranking function is a functionℜ:Fℝ⟶ℝ, where Fℝ is a set of BFNs defined on a set of real numbers, which maps each BFN into real numbers so that they can be ranked. Ranking for TBFN, S˜=t1,t2,t3,f1,f2,f3, is defined as(3)ℜS˜=t1+t2+t33+t3−t12−f1+f2+f33+f3−f12.Definition 4 (See [26]).
A TrBFN defined on real numbersℝ, denoted by(4)S˜=P,N=t1,t2,t3,t4,f1,f2,f3,f4,is a BFN, and its satisfaction degree λP and dissatisfaction degree λN are represented by(5)λP=x−t1t2−t1,ifx∈t1,t2,1,ifx∈t2,t3,t4−xt4−t3,ifx∈t3,t4,0,otherwise,λN=f1−xf2−f1,ifx∈f1,f2,−1,ifx∈f2,f3,x−f4f4−f3,ifx∈f3,f4,0,otherwise.Definition 5 (See [26]).
A ranking function is a functionℜ:Fℝ⟶ℝ, where Fℝ is a set of TrBFNs defined on a set of real numbers, which maps each TrBFN into real numbers so that they can be ranked. Ranking for TrBFN, S˜=t1,t2,t3,t4,f1,f2,f3,f4, is defined as(6)ℜS˜=t1+t2+t3+t44+t4+t3−t2−t12−f1+f2+f3+f44+f4+f3−f2−f12.Definition 6 (See [26]).
LetS˜1 and S˜2 be two BFNs; then,(i)
S˜1≺S˜2 if ℜS˜1<ℜS˜2,(ii)
S˜1≻S˜2 if ℜS˜1>ℜS˜2,(iii)
S˜1≈S˜2 if ℜS˜1=ℜS˜2.
A list of acronyms is given in Table1. For other notations and applications, readers are referred to [27–33].Table 1
List of acronyms.
AcronymsRepresentationBFNBipolar fuzzy numberTBFNTriangular bipolar fuzzy numberTrBFNTrapezoidal bipolar fuzzy numberLPPLinear programming problemFFLPFully fuzzy linear programmingFBFLPPFully bipolar fuzzy linear programming problemBFOSBipolar fuzzy optimal solution


## 3. FBFLPP with Nonnegative Variables

Definition 7.
A TBFNS˜=t1,t2,t3,f1,f2,f3 is called a nonnegative TBFN if and only if t1≥0 and f1≥0. A TBFN S˜=t1,t2,t3,f1,f2,f3 is called an unrestricted TBFN if and only if t1 and f1 are real numbers. Two TBFNs S˜1=α1,β1,γ1,θ1,ϕ1,φ1 and S˜2=α2,β2,γ2,θ2,ϕ2,φ2 are said to be equal if and only if α1=α2, β1=β2, γ1=γ2, θ1=θ2, ϕ1=ϕ2, φ1=φ2.Definition 8.
A TrBFNS˜=t1,t2,t3,t4,f1,f2,f3,f4 is called a nonnegative TrBFN if and only if t1≥0 and f1≥0. A TrBFN S˜=t1,t2,t3,t4,f1,f2,f3,f4 is called an unrestricted TrBFN if and only if t1 and f1 are real numbers. Two TrBFNs A˜1=α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1 and A˜2=α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2 are said to be equal if and only if α1=α2, β1=β2, γ1=γ2, δ1=δ2, θ1=θ2, ϕ1=ϕ2, φ1=φ2, ξ1=ξ2.


### 3.1. Arithmetic Operations for TrBFNs

LetA˜1=α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1 and A˜2=α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2 be two TrBFNs. Then,(1)
A˜1⊕A˜2=α1+α2,β1+β2,γ1+γ2,δ1+δ2,θ1+θ2,ϕ1+ϕ2,φ1+φ2,ξ1+ξ2.(2)
−A˜1=−δ1,−γ1,−β1,−α1,−ξ1,−φ1,−ϕ1,−θ1.(3)
A˜1⊖A˜2=α1−δ2,β1−γ2,γ1−β2,δ1−α2,θ1−ξ2,ϕ1−φ2,φ1−ϕ2,ξ1−θ2.LetA˜=α,β,γ,δ,θ,ϕ,φ,ξ be a TrBFN. Then,(7)ωA˜=ωα,ωβ,ωγ,ωδ,ωθ,ωϕ,ωφ,ωξ,ifω≥0,ωδ,ωγ,ωβ,ωα,ωξ,ωφ,ωϕ,ωθ,ifω<0.We now define multiplication of an unrestricted TrBFN with nonnegative TrBFN.Definition 9.
LetS˜=a,b,c,d,e,f,g,h be an unrestricted TrBFN and U˜=x,y,z,w,t,u,v,r be a nonnegative TrBFN. Then,(8)S˜⊗U˜=ax,by,cz,dw,et,fu,gv,hr,ifa≥0,e≥0,aw,by,cz,dw,er,fu,gv,ht,ifa<0,b≥0,e<0,f≥0,aw,bz,cz,dw,er,fv,gv,hr,ifb<0,c≥0,f<0,g≥0,aw,bz,cy,dw,er,fv,gu,hr,ifc<0,d≥0,g<0,h≥0,aw,bz,cy,dx,er,fv,gu,ht,otherwise.Remark 1.
Ifb=c and f=g in TrBFN S˜=a,b,c,d,e,f,g,h; then, S˜ is called TBFN.
We propose a method for solving FBFLPPs with equality constraints.(9)Maximize/Minimize∑j=1nB˜j⊗Y˜j,subject to(10)∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m,where Y˜j are nonnegative TBFNs and B˜j,F˜ij,E˜i are unrestricted TBFNs.Definition 10.
A bipolar fuzzy optimal solution (BFOS) of FBFLPP (9) will be BFNs Y˜j, if(1)
Y˜j are nonnegative TBFNs(2)
∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m(3)
there exists any nonnegative TBFNY˜j satisfying the constraints, then in case of a maximization problem, ℜ∑j=1nB˜j⊗Y˜j≻ℜ∑j=1nB˜j⊗Y˜j′ and in case of a minimization problem, ℜ∑j=1nB˜j⊗Y˜j≺ℜ∑j=1nB˜j⊗Y˜j′


### 3.2. Methodology

(11)Maximize/Minimize∑j=1nB˜j⊗Y˜j,subject to(12)∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m,where Y˜j are nonnegative TBFNs and B˜j,E˜i,F˜ij are unrestricted TBFNs.Step 1: assumingB˜j=sj,tj,uj,vj,wj,rj, Y˜j=xj,yj,zj,gj,hj,kj, F˜ij=dij,eij,fij,mij,nij,pij, and E˜i=bi,ci,di,ei,fi,ti, the FBFLPP (11) can be transformed as follows:(13)Maximize/Minimize∑j=1nsj,tj,uj,vj,wj,rj⊗xj,yj,zj,gj,hj,kj,subject to(14)∑j=1ndij,eij,fij,mij,nij,pij⊗xj,yj,zj,gj,hj,kj=bi,ci,di,ei,fi,ti,∀i=1,2,3,…,m,wherexj,yj,zj,gj,hj,kj are nonnegative TBFN, ∀j=1,2,3,…,n.Step 2: applying product of TBFNs (3.3) and assuming(15)dij,eij,fij,mij,nij,pij⊗xj,yj,zj,gj,hj,kj=dij′,eij′,fij′,mij′,nij′,pij′,the FBFLPP (3) can be transformed as follows:(16)Maximize/Minimize∑j=1nsj,tj,uj,vj,wj,rj⊗xj,yj,zj,gj,hj,kj,subject to(17)∑j=1ndij′,eij′,fij′,mij′,nij′,pij′=bi,ci,di,ei,fi,ti,∀i=1,2,3,…,m,wherexj,yj,zj,gj,hj,kj are nonnegative TBFN, ∀j=1,2,3,…,n.Step 3: applying arithmetic operations:(18)Maximize/Minimize∑j=1nsj,tj,uj,vj,wj,rj⊗xj,yj,zj,gj,hj,kj,subject to(19)∑j=1ndij′=bi,∑j=1nmij′=ei,∑j=1neij′=ci,∑j=1nnij′=fi,∑j=1nfij′=di,∑j=1npij′=ti,xj≥0,yj−xj≥0,zj−yj≥0,gj≥0,hj−gj≥0,kj−hj≥0,∀j=1,2,3,…,n.Step 4: by applying ranking function, FBFLPP converts as(20)Maximize/Minimizeℜ∑j=1nsj,tj,uj,vj,wj,rj⊗xj,yj,zj,gj,hj,kj,subject to(21)∑j=1ndij′=bi,∑j=1nmij′=ei,∑j=1neij′=ci,∑j=1nnij′=fi,∑j=1nfij′=di,∑j=1npij′=ti,xj≥0,yj−xj≥0,zj−yj≥0,gj≥0,hj−gj≥0,kj−hj≥0,∀j=1,2,3,…,n.Step 5: consideringsj,tj,uj,vj,wj,rj⊗xj,yj,zj,gj,hj,kj=sj′,tj′,uj′,vj′,wj′,rj′, the crisp FBFLPP (6) can be transformed as follows:(22)Maximize/Minimizeℜ∑j=1nsj′,tj′,uj′,vj′,wj′,rj′,subject to(23)∑j=1ndij′=bi,∑j=1nmij′=ei,∑j=1neij′=ci,∑j=1nnij′=fi,∑j=1nfij′=di,∑j=1npij′=ti,xj≥0,yj−xj≥0,zj−yj≥0,gj≥0,hj−gj≥0,kj−hj≥0,∀j=1,2,3,…,n.Step 6: by applying linearity propertyℜ∑j=1nG˜i=∑j=1nℜG˜i, where G˜i is a BFN, the crisp LPP (22) can be transformed as follows:(24)Maximize/Minimize∑j=1nℜsj′,tj′,uj′,vj′,wj′,rj′,subject to(25)∑j=1ndij′=bi,∑j=1nmij′=ei,∑j=1neij′=ci,∑j=1nnij′=fi,∑j=1nfij′=di,∑j=1npij′=ti,xj≥0,yj−xj≥0,zj−yj≥0,gj≥0,hj−gj≥0,kj−hj≥0,∀j=1,2,3,…,n.Step 7: by applying ranking for TBFNs (2.6), the crisp LPP (24) can be transformed as follows:(26)∑j=1nsj′+tj′+uj.′3+uj′−sj.′2−∑j=1nvj′+wj′+rj.′3+rj′−vj.′2,subject to(27)∑j=1ndij′=bi,∑j=1nmij′=ei,∑j=1neij′=ci,∑j=1nnij′=fi,∑j=1nfij′=di,∑j=1npij′=ti,xj≥0,yj−xj≥0,zj−yj≥0,gj≥0,hj−gj≥0,kj−hj≥0,∀j=1,2,3,…,n.Step 8: by solving crisp LPP (26), we find the optimal solution xj∗,yj∗,zj∗,gj∗,hj∗,kj∗,∀j=1,2,3,…,n.Step 9: we find the exact BFOSY˜j.∗ of FBFLPP by putting the values of xj∗,yj∗,zj∗,gj∗,hj∗ and kj∗ in Y˜j∗.=xj∗,yj∗,zj∗,gj∗,hj∗,kj∗.,∀j=1,2,3,…,n.Step 10: we find the bipolar fuzzy optimal value by assigning the values ofY˜j∗ in ∑j=1nB˜j⊗Y˜j.Thus, we state the existence condition for the optimal solution of bipolar fuzzy LPP in the following theorem.Theorem 1.
The solution of FBFLPP:(28)Maximize/MinimizeZ=∑j=1nBj⊗Yjsubject to∑j=1nFij⊗Yj=Ei,∀i=1,…,m.
Yj are nonnegative TBFNs and Fij,Bj,Ei are TBFNs, existing when the solution of the associated crisp LPP,(29)Maximize/MinimizeℜZ=ℜ∑j=1nfj1,fj2,fj3;gj1,gj2,gj3⊗αj,βj,γj;θj,ϕj,φj,subject to(30)∑j=1npij=hi1,∑j=1nqij=hi2,∑j=1nrij=hi3,∑j=1nsij=hi4,∑j=1ntij=hi5,∑j=1nuij=hi6,βj−αj≥0,γj−βj≥0,ϕj−θj≥0,φj−ϕj≥0,αj≥0,θj≥0,∀i=1,…,m exists. Otherwise, there is no guarantee that the BFOS exists.


### 3.3. Numerical Examples

In this section, we illustrate the proposed method with examples.Example 1.
1,5,8,5,10,11⊗Y˜1⊕2,7,9,7,8,12⊗Y˜2 is minimized subject to(31)0,1,3,6,7,9⊗Y˜1⊕1,8,9,2,4,6⊗Y˜2=3,37,93,14,34,63,1,2,3,0,3,4⊗Y˜1⊕4,5,6,5,8,9⊗Y˜2=14,30,69,20,46,66,where Y˜1=χ1,β1,ζ1,α1,φ1,ν1 and Y˜2=χ2,β2,ζ2,α2,φ2,ν2 are nonnegative TBFNs.Step 1:(32)Minimize1,5,8,5,10,11⊗χ1,β1,ζ1,α1,φ1,ν1⊕2,7,9,7,8,12⊗χ2,β2,ζ2,α2,φ2,ν2,subject to(33)0,1,3,6,7,9⊗χ1,β1,ζ1,α1,φ1,ν1⊕1,8,9,2,4,6⊗χ2,β2,ζ2,α2,φ2,ν2,=3,37,93,14,34,631,2,3,0,3,4⊗χ1,β1,ζ1,α1,φ1,ν1⊕4,5,6,5,8,9⊗χ2,β2,ζ2,α2,φ2,ν2,=14,30,69,20,46,66,whereY˜1=χ1,β1,ζ1,α1,φ1,ν1 and Y˜2=χ2,β2,ζ2,α2,φ2,ν2 are nonnegative TBFNs.Step 2:  using the product of TBFNs (3.3), the problem converts as(34)Minimize1χ1,5β1,8ζ1,5α1,10φ1,11ν1⊕2χ2,7β2,9ζ2,7α2,8φ2,12ν2,subject to(35)0χ1,1β1,3ζ1,6α1,7φ1,9ν1⊕1χ2,8β2,9ζ2,2α2,4φ2,6ν2=3,37,93,14,34,63,χ1,2β1,3ζ1,0α1,3φ1,4ν1⊕4χ2,5β2,6ζ2,5α2,8φ2,9ν2=14,30,69,20,46,66χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 3: using arithmetic operations, the problem converts as(36)Minimize1χ1+2χ2,5β1+7β2,8ζ1+9ζ2,5α1+7α2,10φ1+8φ2,11ν1+12ν2,subject to(37)0χ1+1χ2,β1+8β2,3ζ1+9ζ2,6α1+2α2,7φ1+4φ2,9ν1+6ν2=3,37,93,14,34,63,χ1+4χ2,2β1+5β2,3ζ1+6ζ2,0α1+5α2,3φ1+8φ2,4ν1+9ν2=14,30,69,20,46,66χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 4: by applying ranking for TBFNs (2.6), the problem converts as(38)Minimize1χ1+2χ2+5β1+7β2+8ζ1+9ζ23+8ζ1+9ζ2−1χ1−2χ22−5α1+7α2+10φ1+8φ2+11ν1+12ν23+11ν1+12ν2−5α1−7α22,subject to(39)0χ1+1χ2=3,χ1+4χ2=14,β1+8β2=37,2β1+5β2=30,3ζ1+9ζ2=93,3ζ1+6ζ2=69,6α1+2α2=14,0α1+5α2=20,7φ1+4φ2=34,3φ1+8φ2=46,9ν1+6ν2=63,4ν1+9ν2=66,χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0,χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 5: by using software MATLAB, the optimal solution of the crisp LPP isχ1=2,β1=5,ζ1=7,χ2=3,β2=4,ζ2=8,α1=1,φ1=2,ν1=3,α2=4,φ2=5,ν2=6.Step 6: the exact BFOS isY˜1=2,5,7,1,2,3,Y˜2=3,4,8,4,5,6.Step 7: bipolar fuzzy optimal value of FBFLPP is8,53,128,33,60,105.Example 2.
Maximize1,5,9,2,3,6⊗Y˜1⊕3,6,11,4,7,10⊗Y˜2subject to(40)2,4,6,−1,5,8⊗Y˜1⊕1,2,4,4,6,8⊗Y˜2=4,24,52,6,73,120,−1,1,3,2,3,5⊗Y˜1⊕2,4,5,1,5,7⊗Y˜2=0,27,47,7,55,93,where Y˜1=χ1,β1,ζ1,α1,φ1,ν1 and Y˜2=χ2,β2,ζ2,α2,φ2,ν2 are nonnegative TBFNs.Step 1:(41)Maximize1,5,9,2,3,6⊗χ1,β1,ζ1,α1,φ1,ν1⊕3,6,11,4,7,10⊗χ2,β2,ζ2,α2,φ2,ν2,subject to(42)2,4,6,−1,5,8⊗χ1,β1,ζ1,α1,φ1,ν1⊕1,2,4,4,6,8⊗χ2,β2,ζ2,α2,φ2,ν2,=4,24,52,6,73,120−1,1,3,2,3,5⊗χ1,β1,ζ1,α1,φ1,ν1⊕2,4,5,1,5,7⊗χ2,β2,ζ2,α2,φ2,ν2,=0,27,47,7,55,93,whereY˜1=χ1,β1,ζ1,α1,φ1,ν1 and Y˜2=χ2,β2,ζ2,α2,φ2,ν2 are nonnegative TBFNs.Step 2: using the product of TBFNs (3.3), the problem converts as(43)Maximize1χ1,5β1,9ζ1,2α1,3φ1,6ν1⊕3χ2,6β2,11ζ2,4α2,7φ2,10ν2,subject to(44)2χ1,4β1,6ζ1,−1ν1,5φ1,8ν1⊕1χ2,2β2,4ζ2,4α2,6φ2,8ν2=4,24,52,6,73,120,−1ζ1,1β1,3ζ1,2α1,3φ1,5ν1⊕2χ2,4β2,5ζ2,1α2,5φ2,7ν2=0,27,47,7,55,93χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 3: using arithmetic operations, the problem converts as(45)Maximize1χ1+3χ2,5β1+6β2,9ζ1+11ζ2,2α1+4α2,3φ1+7φ2,6ν1+10ν2,subject to(46)2χ1+1χ2,4β1+2β2,6ζ1+4ζ2,−1ν1+4α2,5φ1+6φ2,8ν1+8ν2=4,24,52,6,73,120,−1ζ1+2χ2,1β1+4β2,3ζ1+5ζ2,2α1+1α2,3φ1+5φ2,5ν1+7ν2=0,27,47,7,55,93χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 4: by applying ranking for TBFNs (2.6), the problem converts as(47)Maximize1χ1+3χ2+5β1+6β2+9ζ1+11ζ23+9ζ1+11ζ2−1χ1−3χ22−2α1+4α2+3φ1+7φ2+6ν1+10ν23+6ν1+10ν2−2α1−4α22,subject to(48)2χ1+1χ2=4,−1ζ1+2χ2=0,4β1+2β2=24,1β1+4β2=27,6ζ1+4ζ2=52,3ζ1+5ζ2=47,−1ν1+4α2=6,2α1+1α2=7,5φ1+6φ2=73,3φ1+5φ2=55,8ν1+8ν2=120,5ν1+7ν2=93,χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0,χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 5: by using software MATLAB, the optimal solution of the crisp LPP isχ1=1,β1=3,ζ1=4,χ2=2,β2=6,ζ2=7,α1=2,φ1=5,ν1=6,α2=3,φ2=8,ν2=9.Step 6: the exact BFOS isY˜1=1,3,4,2,5,6,Y˜2=2,6,7,3,8,9.Step 7: bipolar fuzzy optimal value of FBFLPP is7,51,113,16,71,126.Example 3.
A company produces bricks and tiles. It requires two basic raw materialsR1 and R2 to make bricks and tiles. The maximum availability of raw material R1 is 15000,65000,135000,14000,54000,136000 and that of raw material R2 is 17000,89000,185000,32000,94000,254000. The profit per thousand is Rs. 25,31,37,23,30,38 for bricks and Rs.43,49,57,39,52,56 for tiles. The data representing the daily needs of R1 and R2 are shown in Table 2. We find number of bricks and tiles that should be made by the company to get the maximum profit.
LetY˜1 and Y˜2 be units of bricks and tiles, respectively. Then, bipolar fuzzy LPP becomes(49)Maximize25,31,37,23,30,38⊗Y˜1⊕43,49,57,39,52,56⊗Y˜2,subject to(50)3,5,8,1,6,7⊗Y˜1⊕4,7,9,3,4,8⊗Y˜2=15000,65000,135000,14000,54000,136000,2,9,12,4,8,13⊗Y˜1⊕5,7,11,6,9,15⊗Y˜2=17000,89000,185000,32000,94000,254000,where Y˜1=χ1,β1,ζ1,α1,φ1,ν1 and Y˜2=χ2,β2,ζ2,α2,φ2,ν2 are nonnegative TBFNs.Step 1:(51)Maximize25,31,37,23,30,38⊗χ1,β1,ζ1,α1,φ1,ν1⊕43,49,57,39,52,56⊗χ2,β2,ζ2,α2,φ2,ν2,subject to(52)3,5,8,1,6,7⊗χ1,β1,ζ1,α1,φ1,ν1⊕4,7,9,3,4,8⊗χ2,β2,ζ2,α2,φ2,ν2,=15000,65000,135000,14000,54000,1360002,9,12,4,8,13⊗χ1,β1,ζ1,α1,φ1,ν1⊕5,7,11,6,9,15⊗χ2,β2,ζ2,α2,φ2,ν2,=17000,89000,185000,32000,94000,254000,whereY˜1=χ1,β1,ζ1,α1,φ1,ν1 and Y˜2=χ2,β2,ζ2,α2,φ2,ν2 are nonnegative TBFNs.Step 2: using the product of TBFNs (3.3), the problem converts as(53)Maximize25χ1,31β1,37ζ1,23α1,30φ1,38ν1⊕43χ2,49β2,57ζ2,39α2,52φ2,56ν2,subject to(54)3χ1,5β1,8ζ1,α1,6φ1,7ν1⊕4χ2,7β2,9ζ2,3α2,4φ2,8ν2,=15000,65000,135000,14000,54000,1360002χ1,9β1,12ζ1,4α1,8φ1,13ν1⊕5χ2,7β2,11ζ2,6α2,9φ2,15ν2,=17000,89000,185000,32000,94000,254000χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 3: using arithmetic operations, the problem converts as(55)Maximize25χ1+43χ2,31β1+49β2,37ζ1+57ζ2,23α1+39α2,30φ1+52φ2,38ν1+56ν2,subject to(56)3χ1+4χ2,5β1+7β2,8ζ1+9ζ2,α1+3α2,6φ1+4φ2,7ν1+82,=15000,65000,135000,14000,54000,1360002χ1+5χ2,9β1+7β2,12ζ1+11ζ2,4α1+6α2,8φ1+9φ2,13ν1+15ν2,=17000,89000,185000,32000,94000,254000χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 4: by applying ranking for TBFNs (2.6), the problem converts as(57)Maximize25x1+43x2+31y1+49y2+37z1+57z23+37z1+57z2−25x1−43x22−23a1+39a2+30b1+52b2+38c1+56c23+38c1+56c2−23a1−39a22,subject to(58)3χ1+4χ2=15000,2χ1+5χ2=17000,5β1+7β2=65000,9β1+7β2=89000,8ζ1+9ζ2=135000,12ζ1+11ζ2=185000,α1+3α2=14000,4α1+6α2=32000,6φ1+4φ2=54000,8φ1+9φ2=94000,7ν1+8ν2=136000,13ν1+15ν2=254000,χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0,χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 5: by using software MATLAB, the optimal solution of the crisp LPP isχ1=1000,β1=6000,ζ1=9000,χ2=3000,β2=5000,ζ2=7000,α1=2000,φ1=5000,ν1=8000,α2=4000,φ2=6000,ν2=10000.Step 6: the exact BFOS isY˜1=1000,6000,9000,2000,5000,8000,Y˜2=3000,5000,7000,4000,6000,10000.Step 7: the bipolar fuzzy optimal value of FBFLPP is154000,431000,732000,202000,462000,864000.
Hence, the company should produce1000,6000,9000,2000,5000,8000 thousand bricks and 3000,5000,7000,4000,6000,10000 thousand tiles so that the profit could be maximize and the maximum profit will be Rs. 154000,431000,732000,202000,462000,864000.Table 2
Bricks and tiles produced by the manufacturing company.
Raw materialBricksTilesAvailability (in kg)R13,5,8,1,6,74,7,9,3,4,815000,65000,135000,14000,54000,136000R22,9,12,4,8,135,7,11,6,9,1517000,89000,185000,32000,94000,254000


## 3.1. Arithmetic Operations for TrBFNs

LetA˜1=α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1 and A˜2=α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2 be two TrBFNs. Then,(1)
A˜1⊕A˜2=α1+α2,β1+β2,γ1+γ2,δ1+δ2,θ1+θ2,ϕ1+ϕ2,φ1+φ2,ξ1+ξ2.(2)
−A˜1=−δ1,−γ1,−β1,−α1,−ξ1,−φ1,−ϕ1,−θ1.(3)
A˜1⊖A˜2=α1−δ2,β1−γ2,γ1−β2,δ1−α2,θ1−ξ2,ϕ1−φ2,φ1−ϕ2,ξ1−θ2.LetA˜=α,β,γ,δ,θ,ϕ,φ,ξ be a TrBFN. Then,(7)ωA˜=ωα,ωβ,ωγ,ωδ,ωθ,ωϕ,ωφ,ωξ,ifω≥0,ωδ,ωγ,ωβ,ωα,ωξ,ωφ,ωϕ,ωθ,ifω<0.We now define multiplication of an unrestricted TrBFN with nonnegative TrBFN.Definition 9.
LetS˜=a,b,c,d,e,f,g,h be an unrestricted TrBFN and U˜=x,y,z,w,t,u,v,r be a nonnegative TrBFN. Then,(8)S˜⊗U˜=ax,by,cz,dw,et,fu,gv,hr,ifa≥0,e≥0,aw,by,cz,dw,er,fu,gv,ht,ifa<0,b≥0,e<0,f≥0,aw,bz,cz,dw,er,fv,gv,hr,ifb<0,c≥0,f<0,g≥0,aw,bz,cy,dw,er,fv,gu,hr,ifc<0,d≥0,g<0,h≥0,aw,bz,cy,dx,er,fv,gu,ht,otherwise.Remark 1.
Ifb=c and f=g in TrBFN S˜=a,b,c,d,e,f,g,h; then, S˜ is called TBFN.
We propose a method for solving FBFLPPs with equality constraints.(9)Maximize/Minimize∑j=1nB˜j⊗Y˜j,subject to(10)∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m,where Y˜j are nonnegative TBFNs and B˜j,F˜ij,E˜i are unrestricted TBFNs.Definition 10.
A bipolar fuzzy optimal solution (BFOS) of FBFLPP (9) will be BFNs Y˜j, if(1)
Y˜j are nonnegative TBFNs(2)
∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m(3)
there exists any nonnegative TBFNY˜j satisfying the constraints, then in case of a maximization problem, ℜ∑j=1nB˜j⊗Y˜j≻ℜ∑j=1nB˜j⊗Y˜j′ and in case of a minimization problem, ℜ∑j=1nB˜j⊗Y˜j≺ℜ∑j=1nB˜j⊗Y˜j′


## 3.2. Methodology

(11)Maximize/Minimize∑j=1nB˜j⊗Y˜j,subject to(12)∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m,where Y˜j are nonnegative TBFNs and B˜j,E˜i,F˜ij are unrestricted TBFNs.Step 1: assumingB˜j=sj,tj,uj,vj,wj,rj, Y˜j=xj,yj,zj,gj,hj,kj, F˜ij=dij,eij,fij,mij,nij,pij, and E˜i=bi,ci,di,ei,fi,ti, the FBFLPP (11) can be transformed as follows:(13)Maximize/Minimize∑j=1nsj,tj,uj,vj,wj,rj⊗xj,yj,zj,gj,hj,kj,subject to(14)∑j=1ndij,eij,fij,mij,nij,pij⊗xj,yj,zj,gj,hj,kj=bi,ci,di,ei,fi,ti,∀i=1,2,3,…,m,wherexj,yj,zj,gj,hj,kj are nonnegative TBFN, ∀j=1,2,3,…,n.Step 2: applying product of TBFNs (3.3) and assuming(15)dij,eij,fij,mij,nij,pij⊗xj,yj,zj,gj,hj,kj=dij′,eij′,fij′,mij′,nij′,pij′,the FBFLPP (3) can be transformed as follows:(16)Maximize/Minimize∑j=1nsj,tj,uj,vj,wj,rj⊗xj,yj,zj,gj,hj,kj,subject to(17)∑j=1ndij′,eij′,fij′,mij′,nij′,pij′=bi,ci,di,ei,fi,ti,∀i=1,2,3,…,m,wherexj,yj,zj,gj,hj,kj are nonnegative TBFN, ∀j=1,2,3,…,n.Step 3: applying arithmetic operations:(18)Maximize/Minimize∑j=1nsj,tj,uj,vj,wj,rj⊗xj,yj,zj,gj,hj,kj,subject to(19)∑j=1ndij′=bi,∑j=1nmij′=ei,∑j=1neij′=ci,∑j=1nnij′=fi,∑j=1nfij′=di,∑j=1npij′=ti,xj≥0,yj−xj≥0,zj−yj≥0,gj≥0,hj−gj≥0,kj−hj≥0,∀j=1,2,3,…,n.Step 4: by applying ranking function, FBFLPP converts as(20)Maximize/Minimizeℜ∑j=1nsj,tj,uj,vj,wj,rj⊗xj,yj,zj,gj,hj,kj,subject to(21)∑j=1ndij′=bi,∑j=1nmij′=ei,∑j=1neij′=ci,∑j=1nnij′=fi,∑j=1nfij′=di,∑j=1npij′=ti,xj≥0,yj−xj≥0,zj−yj≥0,gj≥0,hj−gj≥0,kj−hj≥0,∀j=1,2,3,…,n.Step 5: consideringsj,tj,uj,vj,wj,rj⊗xj,yj,zj,gj,hj,kj=sj′,tj′,uj′,vj′,wj′,rj′, the crisp FBFLPP (6) can be transformed as follows:(22)Maximize/Minimizeℜ∑j=1nsj′,tj′,uj′,vj′,wj′,rj′,subject to(23)∑j=1ndij′=bi,∑j=1nmij′=ei,∑j=1neij′=ci,∑j=1nnij′=fi,∑j=1nfij′=di,∑j=1npij′=ti,xj≥0,yj−xj≥0,zj−yj≥0,gj≥0,hj−gj≥0,kj−hj≥0,∀j=1,2,3,…,n.Step 6: by applying linearity propertyℜ∑j=1nG˜i=∑j=1nℜG˜i, where G˜i is a BFN, the crisp LPP (22) can be transformed as follows:(24)Maximize/Minimize∑j=1nℜsj′,tj′,uj′,vj′,wj′,rj′,subject to(25)∑j=1ndij′=bi,∑j=1nmij′=ei,∑j=1neij′=ci,∑j=1nnij′=fi,∑j=1nfij′=di,∑j=1npij′=ti,xj≥0,yj−xj≥0,zj−yj≥0,gj≥0,hj−gj≥0,kj−hj≥0,∀j=1,2,3,…,n.Step 7: by applying ranking for TBFNs (2.6), the crisp LPP (24) can be transformed as follows:(26)∑j=1nsj′+tj′+uj.′3+uj′−sj.′2−∑j=1nvj′+wj′+rj.′3+rj′−vj.′2,subject to(27)∑j=1ndij′=bi,∑j=1nmij′=ei,∑j=1neij′=ci,∑j=1nnij′=fi,∑j=1nfij′=di,∑j=1npij′=ti,xj≥0,yj−xj≥0,zj−yj≥0,gj≥0,hj−gj≥0,kj−hj≥0,∀j=1,2,3,…,n.Step 8: by solving crisp LPP (26), we find the optimal solution xj∗,yj∗,zj∗,gj∗,hj∗,kj∗,∀j=1,2,3,…,n.Step 9: we find the exact BFOSY˜j.∗ of FBFLPP by putting the values of xj∗,yj∗,zj∗,gj∗,hj∗ and kj∗ in Y˜j∗.=xj∗,yj∗,zj∗,gj∗,hj∗,kj∗.,∀j=1,2,3,…,n.Step 10: we find the bipolar fuzzy optimal value by assigning the values ofY˜j∗ in ∑j=1nB˜j⊗Y˜j.Thus, we state the existence condition for the optimal solution of bipolar fuzzy LPP in the following theorem.Theorem 1.
The solution of FBFLPP:(28)Maximize/MinimizeZ=∑j=1nBj⊗Yjsubject to∑j=1nFij⊗Yj=Ei,∀i=1,…,m.
Yj are nonnegative TBFNs and Fij,Bj,Ei are TBFNs, existing when the solution of the associated crisp LPP,(29)Maximize/MinimizeℜZ=ℜ∑j=1nfj1,fj2,fj3;gj1,gj2,gj3⊗αj,βj,γj;θj,ϕj,φj,subject to(30)∑j=1npij=hi1,∑j=1nqij=hi2,∑j=1nrij=hi3,∑j=1nsij=hi4,∑j=1ntij=hi5,∑j=1nuij=hi6,βj−αj≥0,γj−βj≥0,ϕj−θj≥0,φj−ϕj≥0,αj≥0,θj≥0,∀i=1,…,m exists. Otherwise, there is no guarantee that the BFOS exists.


## 3.3. Numerical Examples

In this section, we illustrate the proposed method with examples.Example 1.
1,5,8,5,10,11⊗Y˜1⊕2,7,9,7,8,12⊗Y˜2 is minimized subject to(31)0,1,3,6,7,9⊗Y˜1⊕1,8,9,2,4,6⊗Y˜2=3,37,93,14,34,63,1,2,3,0,3,4⊗Y˜1⊕4,5,6,5,8,9⊗Y˜2=14,30,69,20,46,66,where Y˜1=χ1,β1,ζ1,α1,φ1,ν1 and Y˜2=χ2,β2,ζ2,α2,φ2,ν2 are nonnegative TBFNs.Step 1:(32)Minimize1,5,8,5,10,11⊗χ1,β1,ζ1,α1,φ1,ν1⊕2,7,9,7,8,12⊗χ2,β2,ζ2,α2,φ2,ν2,subject to(33)0,1,3,6,7,9⊗χ1,β1,ζ1,α1,φ1,ν1⊕1,8,9,2,4,6⊗χ2,β2,ζ2,α2,φ2,ν2,=3,37,93,14,34,631,2,3,0,3,4⊗χ1,β1,ζ1,α1,φ1,ν1⊕4,5,6,5,8,9⊗χ2,β2,ζ2,α2,φ2,ν2,=14,30,69,20,46,66,whereY˜1=χ1,β1,ζ1,α1,φ1,ν1 and Y˜2=χ2,β2,ζ2,α2,φ2,ν2 are nonnegative TBFNs.Step 2:  using the product of TBFNs (3.3), the problem converts as(34)Minimize1χ1,5β1,8ζ1,5α1,10φ1,11ν1⊕2χ2,7β2,9ζ2,7α2,8φ2,12ν2,subject to(35)0χ1,1β1,3ζ1,6α1,7φ1,9ν1⊕1χ2,8β2,9ζ2,2α2,4φ2,6ν2=3,37,93,14,34,63,χ1,2β1,3ζ1,0α1,3φ1,4ν1⊕4χ2,5β2,6ζ2,5α2,8φ2,9ν2=14,30,69,20,46,66χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 3: using arithmetic operations, the problem converts as(36)Minimize1χ1+2χ2,5β1+7β2,8ζ1+9ζ2,5α1+7α2,10φ1+8φ2,11ν1+12ν2,subject to(37)0χ1+1χ2,β1+8β2,3ζ1+9ζ2,6α1+2α2,7φ1+4φ2,9ν1+6ν2=3,37,93,14,34,63,χ1+4χ2,2β1+5β2,3ζ1+6ζ2,0α1+5α2,3φ1+8φ2,4ν1+9ν2=14,30,69,20,46,66χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 4: by applying ranking for TBFNs (2.6), the problem converts as(38)Minimize1χ1+2χ2+5β1+7β2+8ζ1+9ζ23+8ζ1+9ζ2−1χ1−2χ22−5α1+7α2+10φ1+8φ2+11ν1+12ν23+11ν1+12ν2−5α1−7α22,subject to(39)0χ1+1χ2=3,χ1+4χ2=14,β1+8β2=37,2β1+5β2=30,3ζ1+9ζ2=93,3ζ1+6ζ2=69,6α1+2α2=14,0α1+5α2=20,7φ1+4φ2=34,3φ1+8φ2=46,9ν1+6ν2=63,4ν1+9ν2=66,χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0,χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 5: by using software MATLAB, the optimal solution of the crisp LPP isχ1=2,β1=5,ζ1=7,χ2=3,β2=4,ζ2=8,α1=1,φ1=2,ν1=3,α2=4,φ2=5,ν2=6.Step 6: the exact BFOS isY˜1=2,5,7,1,2,3,Y˜2=3,4,8,4,5,6.Step 7: bipolar fuzzy optimal value of FBFLPP is8,53,128,33,60,105.Example 2.
Maximize1,5,9,2,3,6⊗Y˜1⊕3,6,11,4,7,10⊗Y˜2subject to(40)2,4,6,−1,5,8⊗Y˜1⊕1,2,4,4,6,8⊗Y˜2=4,24,52,6,73,120,−1,1,3,2,3,5⊗Y˜1⊕2,4,5,1,5,7⊗Y˜2=0,27,47,7,55,93,where Y˜1=χ1,β1,ζ1,α1,φ1,ν1 and Y˜2=χ2,β2,ζ2,α2,φ2,ν2 are nonnegative TBFNs.Step 1:(41)Maximize1,5,9,2,3,6⊗χ1,β1,ζ1,α1,φ1,ν1⊕3,6,11,4,7,10⊗χ2,β2,ζ2,α2,φ2,ν2,subject to(42)2,4,6,−1,5,8⊗χ1,β1,ζ1,α1,φ1,ν1⊕1,2,4,4,6,8⊗χ2,β2,ζ2,α2,φ2,ν2,=4,24,52,6,73,120−1,1,3,2,3,5⊗χ1,β1,ζ1,α1,φ1,ν1⊕2,4,5,1,5,7⊗χ2,β2,ζ2,α2,φ2,ν2,=0,27,47,7,55,93,whereY˜1=χ1,β1,ζ1,α1,φ1,ν1 and Y˜2=χ2,β2,ζ2,α2,φ2,ν2 are nonnegative TBFNs.Step 2: using the product of TBFNs (3.3), the problem converts as(43)Maximize1χ1,5β1,9ζ1,2α1,3φ1,6ν1⊕3χ2,6β2,11ζ2,4α2,7φ2,10ν2,subject to(44)2χ1,4β1,6ζ1,−1ν1,5φ1,8ν1⊕1χ2,2β2,4ζ2,4α2,6φ2,8ν2=4,24,52,6,73,120,−1ζ1,1β1,3ζ1,2α1,3φ1,5ν1⊕2χ2,4β2,5ζ2,1α2,5φ2,7ν2=0,27,47,7,55,93χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 3: using arithmetic operations, the problem converts as(45)Maximize1χ1+3χ2,5β1+6β2,9ζ1+11ζ2,2α1+4α2,3φ1+7φ2,6ν1+10ν2,subject to(46)2χ1+1χ2,4β1+2β2,6ζ1+4ζ2,−1ν1+4α2,5φ1+6φ2,8ν1+8ν2=4,24,52,6,73,120,−1ζ1+2χ2,1β1+4β2,3ζ1+5ζ2,2α1+1α2,3φ1+5φ2,5ν1+7ν2=0,27,47,7,55,93χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 4: by applying ranking for TBFNs (2.6), the problem converts as(47)Maximize1χ1+3χ2+5β1+6β2+9ζ1+11ζ23+9ζ1+11ζ2−1χ1−3χ22−2α1+4α2+3φ1+7φ2+6ν1+10ν23+6ν1+10ν2−2α1−4α22,subject to(48)2χ1+1χ2=4,−1ζ1+2χ2=0,4β1+2β2=24,1β1+4β2=27,6ζ1+4ζ2=52,3ζ1+5ζ2=47,−1ν1+4α2=6,2α1+1α2=7,5φ1+6φ2=73,3φ1+5φ2=55,8ν1+8ν2=120,5ν1+7ν2=93,χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0,χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 5: by using software MATLAB, the optimal solution of the crisp LPP isχ1=1,β1=3,ζ1=4,χ2=2,β2=6,ζ2=7,α1=2,φ1=5,ν1=6,α2=3,φ2=8,ν2=9.Step 6: the exact BFOS isY˜1=1,3,4,2,5,6,Y˜2=2,6,7,3,8,9.Step 7: bipolar fuzzy optimal value of FBFLPP is7,51,113,16,71,126.Example 3.
A company produces bricks and tiles. It requires two basic raw materialsR1 and R2 to make bricks and tiles. The maximum availability of raw material R1 is 15000,65000,135000,14000,54000,136000 and that of raw material R2 is 17000,89000,185000,32000,94000,254000. The profit per thousand is Rs. 25,31,37,23,30,38 for bricks and Rs.43,49,57,39,52,56 for tiles. The data representing the daily needs of R1 and R2 are shown in Table 2. We find number of bricks and tiles that should be made by the company to get the maximum profit.
LetY˜1 and Y˜2 be units of bricks and tiles, respectively. Then, bipolar fuzzy LPP becomes(49)Maximize25,31,37,23,30,38⊗Y˜1⊕43,49,57,39,52,56⊗Y˜2,subject to(50)3,5,8,1,6,7⊗Y˜1⊕4,7,9,3,4,8⊗Y˜2=15000,65000,135000,14000,54000,136000,2,9,12,4,8,13⊗Y˜1⊕5,7,11,6,9,15⊗Y˜2=17000,89000,185000,32000,94000,254000,where Y˜1=χ1,β1,ζ1,α1,φ1,ν1 and Y˜2=χ2,β2,ζ2,α2,φ2,ν2 are nonnegative TBFNs.Step 1:(51)Maximize25,31,37,23,30,38⊗χ1,β1,ζ1,α1,φ1,ν1⊕43,49,57,39,52,56⊗χ2,β2,ζ2,α2,φ2,ν2,subject to(52)3,5,8,1,6,7⊗χ1,β1,ζ1,α1,φ1,ν1⊕4,7,9,3,4,8⊗χ2,β2,ζ2,α2,φ2,ν2,=15000,65000,135000,14000,54000,1360002,9,12,4,8,13⊗χ1,β1,ζ1,α1,φ1,ν1⊕5,7,11,6,9,15⊗χ2,β2,ζ2,α2,φ2,ν2,=17000,89000,185000,32000,94000,254000,whereY˜1=χ1,β1,ζ1,α1,φ1,ν1 and Y˜2=χ2,β2,ζ2,α2,φ2,ν2 are nonnegative TBFNs.Step 2: using the product of TBFNs (3.3), the problem converts as(53)Maximize25χ1,31β1,37ζ1,23α1,30φ1,38ν1⊕43χ2,49β2,57ζ2,39α2,52φ2,56ν2,subject to(54)3χ1,5β1,8ζ1,α1,6φ1,7ν1⊕4χ2,7β2,9ζ2,3α2,4φ2,8ν2,=15000,65000,135000,14000,54000,1360002χ1,9β1,12ζ1,4α1,8φ1,13ν1⊕5χ2,7β2,11ζ2,6α2,9φ2,15ν2,=17000,89000,185000,32000,94000,254000χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 3: using arithmetic operations, the problem converts as(55)Maximize25χ1+43χ2,31β1+49β2,37ζ1+57ζ2,23α1+39α2,30φ1+52φ2,38ν1+56ν2,subject to(56)3χ1+4χ2,5β1+7β2,8ζ1+9ζ2,α1+3α2,6φ1+4φ2,7ν1+82,=15000,65000,135000,14000,54000,1360002χ1+5χ2,9β1+7β2,12ζ1+11ζ2,4α1+6α2,8φ1+9φ2,13ν1+15ν2,=17000,89000,185000,32000,94000,254000χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 4: by applying ranking for TBFNs (2.6), the problem converts as(57)Maximize25x1+43x2+31y1+49y2+37z1+57z23+37z1+57z2−25x1−43x22−23a1+39a2+30b1+52b2+38c1+56c23+38c1+56c2−23a1−39a22,subject to(58)3χ1+4χ2=15000,2χ1+5χ2=17000,5β1+7β2=65000,9β1+7β2=89000,8ζ1+9ζ2=135000,12ζ1+11ζ2=185000,α1+3α2=14000,4α1+6α2=32000,6φ1+4φ2=54000,8φ1+9φ2=94000,7ν1+8ν2=136000,13ν1+15ν2=254000,χ1≥0,β1−χ1≥0,ζ1−β1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0,χ2≥0,β2−χ2≥0,ζ2−β2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0.Step 5: by using software MATLAB, the optimal solution of the crisp LPP isχ1=1000,β1=6000,ζ1=9000,χ2=3000,β2=5000,ζ2=7000,α1=2000,φ1=5000,ν1=8000,α2=4000,φ2=6000,ν2=10000.Step 6: the exact BFOS isY˜1=1000,6000,9000,2000,5000,8000,Y˜2=3000,5000,7000,4000,6000,10000.Step 7: the bipolar fuzzy optimal value of FBFLPP is154000,431000,732000,202000,462000,864000.
Hence, the company should produce1000,6000,9000,2000,5000,8000 thousand bricks and 3000,5000,7000,4000,6000,10000 thousand tiles so that the profit could be maximize and the maximum profit will be Rs. 154000,431000,732000,202000,462000,864000.Table 2
Bricks and tiles produced by the manufacturing company.
Raw materialBricksTilesAvailability (in kg)R13,5,8,1,6,74,7,9,3,4,815000,65000,135000,14000,54000,136000R22,9,12,4,8,135,7,11,6,9,1517000,89000,185000,32000,94000,254000


## 4. Unrestricted FBFLPP

In this section, we study FBFLPP with equality constraints in which coefficients and decision variables are unrestricted BFNs. We define multiplication of two unrestricted TrBFNs.Definition 11.
LetA˜=p,q,r,s,t,u,v,w and X˜=a,c,d,f,k,h,m,n be two unrestricted TrBFNs. Then,(59)A˜⊗X˜=minpa,sa,minqc,rc,maxqd,rd,maxpf,sf,mintk,wk,minuh,vh,maxum,vm,maxtn,wn,ifp≥0andt≥0,minpf,sa,minqc,rc,maxqd,rd,maxpa,sf,mintn,wk,minuh,vh,maxum,vm,maxtk,wn,ifp<0,q≥0andt<0,u≥0,minpf,sa,minqd,rc,maxqc,rd,maxpa,sf,mintn,wk,minum,vh,maxuh,vm,maxtk,wn,ifq<0,r≥0andu<0,v≥0,minpf,sa,minqd,qd,maxqc,rc,maxpa,sf,mintn,wk,minum,vm,maxuh,vh,maxtk,wn,ifr<0,s≥0andv<0,w≥0,minpf,sf,minqd,qd,maxqc,rc,maxpa,sa,mintn,wn,minum,vm,maxuh,vh,maxtk,wk,otherwise,where(60)minɛ,κ=ɛ+κ2−ɛ−κ2,maxɛ,κ=ɛ+κ2+ɛ−κ2.
We now present a method for solving FBFLPP:(61)Maximize/Minimize∑j=1nB˜j⊗Y˜j,subject to(62)∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m,where Y˜j, B˜j,E˜i, and F˜ij are unrestricted TrBFNs.Definition 12.
A BFOS of FBFLPP (61) will be TrBFNs Y˜j if(1)
Y˜j are TrBFNs(2)
∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m(3)
there exists any TrBFNY˜j satisfying the constraints, then in case of a maximization problem, ℜ∑j=1nB˜j⊗Y˜j≻ℜ∑j=1nB˜j⊗Y˜j′ and in case of a minimization problem, ℜ∑j=1nB˜j⊗Y˜j≺ℜ∑j=1nB˜j⊗Y˜j′.


### 4.1. Methodology

(63)Maximize/Minimize∑j=1nB˜j⊗Y˜j,subject to(64)∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m,where Y˜j,E˜i,F˜ij, and B˜j are unrestricted TrBFNs.Step 1: we assume(65)F˜ij=eij,fij,gij,hij,aij,bij,cij,dij,E˜i=λi,μi,νi,ϵi,κi,ρi,ωi,τi,B˜j=mj,nj,pj,qj,rj,sj,tj,uj,Y˜j=αj,βj,γj,δj,θj,ϕj,φj,ξj.The FBFLPP (63) can be transformed into problem (66):(66)Maximize/Minimize∑j=1nmj,nj,pj,qj,rj,sj,tj,uj⊗αj,βj,γj,δj,θj,ϕj,φj,ξj,subject to(67)∑j=1neij,fij,gij,hij,aij,bij,cij,dij⊗αj,βj,γj,δj,θj,ϕj,φj,ξj,=λi,μi,νi,ϵi,κi,ρi,ωi,τi,whereαj,βj,γj,δj,θj,ϕj,φj,ξj is an unrestricted TrBFN, ∀j=1,2,3,…,n.Step 2: by applying the product of TrBFNs (4.1), the FBFLPP (66) transforms to problem (68) as(68)Maximize/Minimize∑j=1nmj,nj,pj,qj,rj,sj,tj,uj⊗αj,βj,γj,δj,θj,ϕj,φj,ξj,subject to(69)∑j=1nmineij′,eij″,minfij′,fij″,maxgij′,gij″,maxhij′,hij″,minaij′,aij″,minbij′,bij″,maxcij′,cij″,maxdij′,dij″=λi,μi,νi,ɛi,κi,ρi,ωi,τi,whereαj,βj,γj,δj,θj,ϕj,φj,ξj is an unrestricted TrBFN, ∀j=1,2,3,…,n, and(70)mineij′,eij″,minfij′,fij″,maxgij′,gij″,maxhij′,hij″=mineijαj,hijαj,minfijβj,gijβj,maxfijγj,gijγj,maxeijδj,hijδj,ifeij≥0,mineijδj,hijαj,minfijβj,gijβj,maxfijγj,gijγj,maxeijαj,hijδj,ifeij<0,fij≥0,mineijδj,hijαj,minfijγj,gijβj,maxfijβj,gijγj,maxeijαj,hijδj,iffij<0,gij≥0,mineijδj,hijαj,minfijγj,gijγj,maxfijβj,gijβj,maxeijαj,hijδj,ifgij<0,hij≥0,mineijδj,hijδj,minfijγj,gijγj,maxfijβj,gijβj,maxeijαj,hijαj,otherwise,minaij′,aij″,minbij′,bij″,maxcij′,cij″,maxdij′,dij″=minaijθj,dijθj,minbijϕj,cijϕj,maxbijφj,cijφj,maxaijξj,dijξj,ifaij≥0,minaijξj,dijαj,minbijϕj,cijϕj,maxbijφj,cijφj,maxaijθj,dijξj,ifaij<0,bij≥0,minaijξj,dijαj,minbijφj,cijϕj,maxbijϕj,cijφj,maxaijθj,dijξj,ifbij<0,cij≥0,minaijξj,dijαj,minbijφj,cijφj,maxbijϕj,cijϕj,maxaijθj,dijξj,ifcij<0,dij≥0,minaijξj,dijξj,minbijφj,cijφj,maxbijϕj,cijϕj,maxaijθj,dijθj,otherwise.Step 3: by applying arithmetic operations, the FBFLPP (68) transforms into problem (71) as(71)Maximize/Minimize∑j=1nmj,nj,pj,qj,rj,sj,tj,uj⊗αj,βj,γj,δj,θj,ϕj,φj,ξj,subject to(72)∑j=1nmineij′,eij″=λi,∑j=1nminaij′,aij″=κi,∑j=1nminfij′,fij″=νi,∑j=1nminbij′,bij″=ρi,∑j=1nmaxgij′,gij″=μi,∑j=1nmaxcij′,cij″=ωi,∑j=1nmaxhij′,hij″=ɛi,∑j=1nmaxdij′,hij″=τi,∀i=1,2,3,…,m, where x˜j=αj,βj,γj,δj,θj,ϕj,φj,ξj be unrestricted TrBFNs, ∀j=1,2,3,…,n.Step 4: by applying ranking function, the FBFLPP converts into crisp LPP (73) as(73)Maximize/Minimizeℜ∑j=1nmj,nj,pj,qj,rj,sj,tj,uj⊗αj,βj,γj,δj,θj,ϕj,φj,ξj,subject to(74)∑j=1nmineij′,eij″=λi,∑j=1nminaij′,aij″=κi,∑j=1nminfij′,fij″=νi,∑j=1nminbij′,bij″=ρi,∑j=1nmaxgij′,gij″=μi,∑j=1nmaxcij′,cij″=ωi,∑j=1nmaxhij′,hij″=ɛi,∑j=1nmaxdij′,hij″=τi,∀i=1,2,3,…,m,βj−αj≥0,γj−βj,δj−γj≥0,ϕj−θj,φj−ϕj≥0,ξj−φj≥0,∀j=1,2,3,…,n.Step 5: we consider(75)mj,nj,pj,qj,rj,sj,tj,uj⊗αj,βj,γj,δj,θj,ϕj,φj,ξj=minmij′,mij″,minnij′,nij″,maxpij′,pij″,maxqij′,qij″,minrij′,rij″,minsij′,sij″,maxtij′,tij″,maxuij′,uij″.The crisp LPP (73) can be transformed into the following problem:(76)Maximize/Minimizeℜ∑j=1nminmij′,mij″,minnij′,nij″,maxpij′,pij″,maxqij′,qij″,minrij′,rij″,minsij′,sij″,maxtij′,tij″,maxuij′,uij″,subject to(77)∑j=1nmineij′,eij″=λi,∑j=1nminaij′,aij″=κi,∑j=1nminfij′,fij″=νi,∑j=1nminbij′,bij″=ρi,∑j=1nmaxgij′,gij″=μi,∑j=1nmaxcij′,cij″=ωi,∑j=1nmaxhij′,hij″=ɛi,∑j=1nmaxdij′,hij″=τi,∀i=1,2,3,…,m,βj−αj≥0,γj−βj,δj−γj≥0,ϕj−θj,φj−ϕj≥0,ξj−φj≥0,∀j=1,2,3,…,n.Step 6: by applying the linearity propertyℜ∑j=1nG˜i=∑j=1nℜG˜i, where G˜i be BFN, the crisp LPP (76) can be transformed into the following problem:(78)Maximize/Minimize∑j=1nℜminmij′,mij″,minnij′,nij″,maxpij′,pij″,maxqij′,qij″,minrij′,rij″,minsij′,sij″,maxtij′,tij″,maxuij′,uij″,subject to(79)∑j=1nmineij′,eij″=λi,∑j=1nminaij′,aij″=κi,∑j=1nminfij′,fij″=νi,∑j=1nminbij′,bij″=ρi,∑j=1nmaxgij′,gij″=μi,∑j=1nmaxcij′,cij″=ωi,∑j=1nmaxhij′,hij″=ɛi,∑j=1nmaxdij′,hij″=τi,∀i=1,2,3,…,m,βj−αj≥0,γj−βj,δj−γj≥0,ϕj−θj,φj−ϕj≥0,ξj−φj≥0,∀j=1,2,3,…,n.Step 7: the crisp LLP (78) can be transformed to the following problem:(80)Maximize/Minimize∑j=1nℜmij′+mij″2−mij′−mij″2,nij′+nij″2−nij′−nij″2,pij′+pij″2+pij′−pij″2,qij′+qij″2+qij′−qij″2,rij′+rij″2−rij′−rij″2,sij′+sij″2−sij′−sij″2,tij′+tij″2+tij′−tij″2,uij′+uij″2+uij′−uij″2,subject to(81)∑j=1neij′+eij″2−eij′−eij″2=λi,∑j=1naij′+aij″2−aij′−aij″2=κi,∑j=1nfij′+fij″2−fij′−fij″2=μi,∑j=1nbij′+bij″2−bij′−bij″2=ρi,∑j=1ngij′+gij″2+gij′−gij″2=νi,∑j=1ncij′+cij″2+cij′−cij″2=ωi,∑j=1nhij′+hij″2+hij′−hij″2=ɛi∑j=1ndij′+dij″2+dij′−dij″2=τi,∀i=1,2,3,…,m,βj−αj≥0,γj−βj,δj−γj≥0,ϕj−θj,φj−ϕj≥0,ξj−φj≥0,∀j=1,2,3,…,n.Step 8: by applying ranking for TrBFNs (2.5), the crisp LPP (80) becomes(82)Maximize/Minimize∑j=1n14mij′+mij′′2−mij′−mij′′2+nij′+nij′′2−nij′−nij′′2+pij′+pij′′2+pij′−pij′′2+qij′+qij′′2+qij′−qij′′2+12−mij′+mij′′2+mij′−mij′′2−nij′+nij′′2+nij′−nij′′2+pij′+pij′′2+pij′−pij′′2+qij′+qij′′2+qij′−qij′′2−14rij′+rij′′2−rij′−rij′′2+sij′+sij′′2−sij′−sij′′2+tij′+tij′′2+tij′−tij′′2+uij′+uij′′2+uij′−uij′′2−12−rij′+rij′′2+rij′−rij′′2−sij′+sij′′2+sij′−sij′′2+tij′+tij′′2+tij′−tij′′2+uij′+uij′′2+uij′−uij′′2,subject to(83)∑j=1neij′+eij″2−eij′−eij″2=λi,∑j=1naij′+aij″2−aij′−aij″2=κi,∑j=1nfij′+fij″2−fij′−fij″2=μi,∑j=1nbij′+bij″2−bij′−bij″2=ρi,∑j=1ngij′+gij″2+gij′−gij″2=νi,∑j=1ncij′+cij″2+cij′−cij″2=ωi,∑j=1nhij′+hij″2+hij′−hij″2=ɛi,∑j=1ndij′+dij″2+dij′−dij″2=τi,∀i=1,2,3,…,m,βj−αj≥0,γj−βj,δj−γj≥0,ϕj−θj,φj−ϕj≥0,ξj−φj≥0,∀j=1,2,3,…,n.Step 9: by solving the crisp LPP (82), we get the optimal solution αj∗,βj∗,γj∗,δj∗,θj∗,ϕj∗,φj∗,ξj∗, ∀j=1,2,3,…,n.Step 10: we find the exact BFOSY˜j∗ of FBFLPP by assigning the values of αj∗,βj∗,γj∗,δj∗,θj∗,ϕj∗,φj∗ and ξj∗ in Y˜j∗=αj∗,βj∗,γj∗,δj∗,θj∗,ϕj∗,φj∗,ξj∗, ∀j=1,2,3,…,n.Step 11: we find the bipolar fuzzy optimal value by assigning the values ofY˜j∗ in ∑j=1nB˜j⊗Y˜j, ∀j=1,2,3,…,n.Thus, we state the existence condition for the optimal solution of bipolar fuzzy LPP in the following theorem.Theorem 2.
The solution of FBFLPP:(84)Maximize/MinimizeZ=∑j=1nBj⊗Yjsubject  to∑j=1nFij⊗Yj=Ei,∀i=1,…,m.Bj,Yj,Fij, and Ei are unrestricted TrBFNs and exist when the solution of the associated crisp LPP,(85)Maximize/MinimizeℜZ=ℜ∑j=1nfj1,fj2,fj3,fj4;gj1,gj2,gj3,gj4⊗αj,βj,γj,δj;θj,ϕj,φj,ξj,subject to(86)∑j=1npij=hi1,∑j=1nqij=hi2,∑j=1nrij=hi3,∑j=1nsij=hi4,∑j=1ntij=hi5,∑j=1nuij=hi6,∑j=1nvij=hi7,∑j=1nwij=hi8,βj−αj≥0,γj−βj≥0,δj−γj≥0,ϕj−θj≥0,φj−ϕj≥0,ξj−φj≥0,∀i=1,…,m exists. Otherwise, there is no guarantee that the BFOS exists.


### 4.2. Numerical Examples

In this section, we illustrate the proposed method with examples.Example 4.
1,5,8,11,3,6,7,12⊗Y˜1⊕2,4,6,9,1,5,6,7⊗Y˜2 is maximized subject to(87)1,4,6,7,2,4,5,8⊗Y˜1⊕0,1,3,6,3,4,7,9⊗Y˜2=2,21,51,109,11,28,72,144,2,3,5,8,1,6,9,10⊗Y˜1⊕1,3,5,6,0,1,2,5⊗Y˜2=5,27,60,116,1,22,66,130,where Y˜1=χ1,β1,ζ1,τ1,α1,φ1,ν1,δ1 and Y˜2=χ2,β2,ζ2,τ2,α2,φ2,ν2,δ2 are nonnegative TrBFNs.Step 1:(88)Maximize1,5,8,11,3,6,7,12⊗χ1,β1,ζ1,τ1,α1,φ1,ν1,δ1⊕2,4,6,9,1,5,6,7⊗χ2,β2,ζ2,τ2,α2,φ2,ν2,δ2,subject to(89)1,4,6,7,2,4,5,8⊗χ1,β1,ζ1,τ1,α1,φ1,ν1,δ1⊕0,1,3,6,3,4,7,9⊗χ2,β2,ζ2,τ2,α2,φ2,ν2,δ2=2,21,51,109,11,28,72,144,2,3,5,8,1,6,9,10⊗χ1,β1,ζ1,τ1,α1,φ1,ν1,δ1⊕1,3,5,6,0,1,2,5⊗χ2,β2,ζ2,τ2,α2,φ2,ν2,δ2=5,27,60,116,1,22,66,130,whereY˜1=χ1,β1,ζ1,τ1,α1,φ1,ν1,δ1 and Y˜2=χ2,β2,ζ2,τ2,α2,φ2,ν2,δ2 are nonnegative TrBFNs.Step 2: using the product of TBFNs (3.3), the problem converts as(90)Maximize1χ1,5β1,8ζ1,11τ1,3α1,6φ1,7ν1,12δ1⊕2χ2,4β2,6ζ2,9τ2,1α2,5φ2,6ν2,7δ2,subject to(91)1χ1,4β1,6ζ1,7τ1,3α1,4φ1,5ν1,8δ1⊕0χ2,1β2,3ζ2,6τ2,3α2,4φ2,7ν2,9δ2,=2,21,51,109,11,28,72,1442χ1,3β1,5ζ1,8τ1,1α1,6φ1,9ν1,10δ1⊕1χ2,3β2,5ζ2,6τ2,0α2,1φ2,2ν2,5δ2,=5,27,60,116,1,22,66,130χ1≥0,β1−χ1≥0,ζ1−β1≥0,τ1−ζ1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0,δ1−ν1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,τ2−ζ2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0,δ2−ν2≥0.Step 3: using arithmetic operations, the problem converts as(92)Maximize1χ1+2χ2,5β1+4β2,8ζ1+6ζ2,11τ1+9τ2,3α1+1α2,6φ1+5φ2,7ν1+6ν2,12δ1+7δ2,subject to(93)1χ1+0χ2,4β1+1β2,6ζ1+3ζ2,7τ1+6τ2,2α1+3α2,4φ1+4φ2,5ν1+7ν2,8δ1+9δ2,=2,21,51,109,11,28,72,1442χ1+1χ2,3β1+3β2,5ζ1+5ζ2,8τ1+6τ2,1α1+0α2,6φ1+1φ2,9ν1+2ν2,10δ1+5δ2,=5,27,60,116,1,22,66,130χ1≥0,β1−χ1≥0,ζ1−β1≥0,τ1−ζ1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0,δ1−ν1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,τ2−ζ2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0,δ2−ν2≥0.Step 4: by applying the definition of ranking function, the problem converts as(94)Maximize1χ1+2χ2+5β1+4β2+8ζ1+6ζ2+11τ1+9τ24+8ζ1+6ζ2+11τ1+9τ2−1χ1−2χ2−5β1−4β22−3α1+1α2+6φ1+5φ2+7ν1+6ν2+12δ1+7δ24+7ν1+6ν2+12δ1+7δ2−3α1−1α2−6φ1−5φ22,subject to(95)1χ1+0χ2=2,2χ1+1χ2=5,4β1+1β2=21,3β1+3β2=27,6ζ1+3ζ2=51,5ζ1+5ζ2=60,7τ1+6τ2=109,8τ1+6τ2=116,2α1+3α2=11,1α1+0α2=1,4φ1+4φ2=28,6φ1+1φ2=22,5ν1+7ν2=72,9ν1+2ν2=66,8δ1+9δ2=144,10δ1+5δ2=130,χ1≥0,β1−χ1≥0,ζ1−β1≥0,τ1−ζ1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0,δ1−ν1≥0,χ2≥0,β2−χ2≥0,ζ2−β2≥0,τ2−ζ2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0,δ2−ν2≥0.Step 5: by using software MATLAB, the optimal solution of the crisp LPP isχ1=2,β1=4,ζ1=5,τ1=7,χ2=1,β2=5,ζ2=7,τ2=10,α1=1,φ1=3,ν1=6,δ1=9α2=3,φ2=4,ν2=6,δ2=8.Step 6: the exact BFOS isY˜1=2,4,5,7,1,3,6,9,Y˜2=1,5,7,10,3,4,6,8.Step 7: bipolar fuzzy optimal value of FBFLPP is4,40,82,167,6,38,78,164.Example 5.
(96)Maximize1,2,3,4,5,6,7,8⊗Y˜1⊕2,4,6,8,9,10,11,12⊗Y˜2,subject to(97)0,1,2,3,4,5,6,7⊗Y˜1⊕1,3,5,7,6,7,8,9⊗Y˜2=−8,2,27,57,−25,7,28,55,2,4,7,9,8,9,10,11⊗Y˜1⊕2,3,5,6,7,9,12,13⊗Y˜2=−25,−8,34,81,−37,9,44,83,where Y˜1=α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1 and Y˜2=α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2 are unrestricted TrBFNs.Step 1:(98)Maximize1,2,3,4,5,6,7,8⊗α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1⊕2,4,6,8,9,10,11,12⊗α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2,subject to(99)0,1,2,3,4,5,6,7⊗α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1⊕1,3,5,7,6,7,8,9⊗α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2=−8,2,27,57,−25,7,28,55,2,4,7,9,8,9,10,11⊗α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1⊕2,3,5,6,7,9,12,13⊗α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2=−25,−8,34,81,−37,9,44,83,whereY˜1=α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1 and Y˜2=α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2are unrestricted TrBFNs.Step 2: using the product of TrBFNs (4.1), the problem converts as(100)Maximizemin1α1,4α1,min2β1,3β1,max2γ1,3γ1,max1δ1,4δ1,min5θ1,8θ1,min6ϕ1,7ϕ1,max6φ1,7φ1,max5ξ1,8ξ1⊕min2α2,8α2,min4β2,6β2,max4γ2,6γ2,max2δ2,8δ2,min9θ2,12θ2,min10ϕ2,11ϕ2,max10φ2,11φ2,max9ξ2,12ξ2,subject to(101)min0α1,3α1,min1β1,2β1,max1γ1,2γ1,max0δ1,3δ1,min4θ1,7θ1,min5ϕ1,6ϕ1,max5φ1,6φ1,max4ξ1,7ξ1⊕min1α2,7α2,min3β2,5β2,max3γ2,5γ2,max1δ2,7δ2,min6θ2,9θ2,min7ϕ2,8ϕ2,max7φ2,8φ2,max6ξ2,9ξ2=−8,2,27,57,−25,7,28,55,min2α1,9α1,min4β1,7β1,max4γ1,7γ1,max2δ1,9δ1,min8θ1,11θ1,min9ϕ1,10ϕ1,max9φ1,10φ1,max8ξ1,11ξ1⊕min2α2,6α2,min3β2,5β2,max3γ2,5γ2,max2δ2,6δ2,min7θ2,13θ2,min9ϕ2,12ϕ2,max9φ2,12φ2,max7ξ2,13ξ2=−25,−8,34,81,−37,9,44,83,where(102)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 3: using arithmetic operations, the problem converts as(103)Maximizemin1α1,4α1⊕min2α2,8α2,min2β1,3β1⊕min4β2,6β2,max2γ1,3γ1⊕max4γ2,6γ2,max1δ1,4δ1⊕max2δ2,8δ2,min5θ1,8θ1⊕min9θ2,12θ2,min6ϕ1,7ϕ1⊕min10ϕ2,11ϕ2,max6φ1,7φ1⊕max10φ2,11φ2,max5ξ1,8ξ1⊕max9ξ2,12ξ2,subject to(104)min0α1,3α1+min1α2,7α2=−8,min2α1,9α1+min2α2,6α2=−25,min1β1,2β1+min3β2,5β2=2,min4β1,7β1+min3β2,5β2=−8,max1γ1,2γ1+max3γ2,5γ2=27,max4γ1,7γ1+max3γ2,5γ2=34,max0δ1,3δ1+max1δ2,7δ2=57,max2δ1,9δ1+max2δ2,6δ2=81,min4θ1,7θ1+min6θ2,9θ2=−25,min8θ1,11θ1+min7θ2,13θ2=−37,min5ϕ1,6ϕ1+min7ϕ2,8ϕ2=7,min9ϕ1,10ϕ1+min9ϕ2,12ϕ2=9,max5φ1,6φ1+max7φ2,8φ2=28,max9φ1,10φ1+max9φ2,12φ2=44,max4ξ1,7ξ1+max6ξ2,9ξ2=55,max8ξ1,11ξ1+max7ξ2,13ξ2=83,where(105)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 4: by applying the ranking function, the problem converts as(106)Maximizeℜmin1α1,4α1⊕min2α2,8α2,min2β1,3β1⊕min4β2,6β2,max2γ1,3γ1⊕max4γ2,6γ2,max1δ1,4δ1⊕max2δ2,8δ2,min5θ1,8θ1⊕min9θ2,12θ2,min6ϕ1,7ϕ1⊕min10ϕ2,11ϕ2,max6φ1,7φ1⊕max10φ2,11φ2,max5ξ1,8ξ1⊕max9ξ2,12ξ2,subject to(107)min0α1,3α1+min1α2,7α2=−8,min2α1,9α1+min2α2,6α2=−25,min1β1,2β1+min3β2,5β2=2,min4β1,7β1+min3β2,5β2=−8,max1γ1,2γ1+max3γ2,5γ2=27,max4γ1,7γ1+max3γ2,5γ2=34,max0δ1,3δ1+max1δ2,7δ2=57,max2δ1,9δ1+max2δ2,6δ2=81,min4θ1,7θ1+min6θ2,9θ2=−25,min8θ1,11θ1+min7θ2,13θ2=−37,min5ϕ1,6ϕ1+min7ϕ2,8ϕ2=7,min9ϕ1,10ϕ1+min9ϕ2,12ϕ2=9,max5φ1,6φ1+max7φ2,8φ2=28,max9φ1,10φ1+max9φ2,12φ2=44,max4ξ1,7ξ1+max6ξ2,9ξ2=55,max8ξ1,11ξ1+max7ξ2,13ξ2=83,where(108)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 5: by applying ranking for TBFNs (2.6), the problem converts as(109)Maximize1452α1−−32α1+5α2−−3α2+52β1−−12β1+5β2−−β2+52γ1+−12γ1+5γ2+−γ2+52δ1+|−32δ1|+5δ2+−3δ2+1252δ1+−32δ1+5δ2+−3δ2+52γ1+−12γ1+5γ2+−γ2−52β1+−12β1−5β2+−β2−52α1+−32α1−5α2+−3α2−14132θ1−−32θ1+212θ2−−32θ2+132ϕ1−−12ϕ1+212ϕ2−−12ϕ2+132φ1+−12φ1+212φ2+−12φ2+132ξ1+−32ξ1+212ξ2+−32ξ2−12132ξ1+−32ξ1+212ξ2+−32ξ2+132φ1+−12φ1+212φ2+−12φ2−132ϕ1+−12ϕ1−212ϕ2+−12ϕ2−132θ1+−32θ1−212θ2+−32θ2,subject to(110)32α1−32α1+4α2−3α2=−8,112α1−72α1+4α2−2α2=−25,32β1−12β1+4β2−β2=2,112β1−32β1+4β2−β2=−8,32γ1+12γ1+4γ2+γ2=27,112γ1+32γ1+4γ2+γ2=34,32δ1+32δ1+4δ2+3δ2=57,112δ1+72δ1+4δ2+2δ2=81,112θ1−32θ1+152θ2−32θ2=−25,192θ1−32θ1+10θ2−3θ2=−37,112ϕ1−12ϕ1+152ϕ2−12ϕ2=7,192ϕ1−12ϕ1+212ϕ2−32ϕ2=9,112φ1+12φ1+152φ2+12φ2=28,192φ1+12φ1+212φ2+32φ2=44,112ξ1+32ξ1+152ξ2+32ξ2=55,192ξ1+32ξ1+10ξ2+3ξ2=83,where(111)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 6: by using software MATLAB, the optimal solution of the crisp LPP isα1=−3,β1=−2,γ1=7/5,δ1=5,θ1=−1,ϕ1=0,φ1=2,ξ1=4,α2=1,β2=2,γ2=121/25,δ2=6,θ2=−2,ϕ2=1,φ2=2,ξ2=3.Step 7: the exact BFOS isY˜1=−3,−2,7/5,5,−1,0,2,4,Y˜2=1,2,121/25,6,−2,1,2,3.Step 8: bipolar fuzzy optimal value of FBFLPP is−10,2,831/25,68,−32,10,36,68.Example 6.
An electrical company produces fans and heaters. Both of them are processed through two different machinesM1 and M2. Working hours for machine M1 per week are −27,17,38,72,−28,17,34,72 and for machine M2 are −25,19,34,67,−25,14,45,83. The profit per hundred for fans and heaters is Rs. 1,3,4,7,2,6,7,9 and Rs. 2,4,5,10,3,6,8,11, respectively. Further details are described in Table 3. How many fans and heaters should be manufactured by the company to get maximum profit is found.
LetY˜1 be hundred number of fans and Y˜2 be hundred number of heaters. Then, FBFLPP becomes(112)Maximize1,3,4,7,2,6,7,9⊗Y˜1⊕2,4,5,10,3,6,8,11⊗Y˜2,subject to(113)1,2,3,9,2,3,4,6⊗Y˜1⊕3,5,8,9,3,5,7,10⊗Y˜2=−27,17,38,72,−28,17,34,72,3,4,5,9,2,3,7,8⊗Y˜1⊕0,5,6,8,1,2,5,9⊗Y˜2=−25,19,34,67,−25,14,45,83,where Y˜1=α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1 and Y˜2=α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2 are unrestricted TrBFNs.Step 1:(114)Maximize1,3,4,7,2,6,7,9⊗α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1⊕2,4,5,10,3,6,8,11⊗α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2,subject to(115)1,2,3,9,2,3,4,6⊗α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1⊕3,5,8,9,3,5,7,10⊗,α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2=−27,17,38,72,−28,17,34,723,4,5,9,2,3,7,8⊗α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1⊕0,5,6,8,1,2,5,9⊗,α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2=−25,19,34,67,−25,14,45,83,whereY˜1=α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1 and Y˜2=α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2are unrestricted TrBFNs.Step 2: using the product of TrBFNs (4.1), the problem converts as(116)Maximizemin1α1,7α1,min3β1,4β1,max3γ1,4γ1,max1δ1,7δ1,min2θ1,9θ1,min6ϕ1,7ϕ1,max6φ1,7φ1,max2ξ1,9ξ1⊕min2α2,10α2,min4β2,5β2,max4γ2,5γ2,max2δ2,10δ2,min3θ2,11θ2,min6ϕ2,8ϕ2,max6φ2,8φ2,max3ξ2,11ξ2,subject to(117)min1α1,9α1,min2β1,3β1,max2γ1,3γ1,max1δ1,9δ1,min2θ1,6θ1,min3ϕ1,4ϕ1,max3φ1,4φ1,max2ξ1,6ξ1⊕min3α2,9α2,min5β2,8β2,max5γ2,8γ2,max3δ2,9δ2,min3θ2,10θ2,min5ϕ2,7ϕ2,max5φ2,7φ2,max3ξ2,10ξ2=−27,17,38,72,−28,17,34,72,min3α1,9α1,min4β1,5β1,max4γ1,5γ1,max3δ1,9δ1,min2θ1,8θ1,min3ϕ1,7ϕ1,max3φ1,7φ1,max2ξ1,8ξ1⊕min0α2,8α2,min5β2,6β2,max5γ2,6γ2,max0δ2,8δ2,min1θ2,9θ2,min2ϕ2,5ϕ2,max2φ2,5φ2,max1ξ2,9ξ2=−25,19,34,67,−25,14,45,83,where(118)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 3: using arithmetic operations, the problem converts as(119)Maximizemin1α1,7α1⊕min2α2,10α2,min3β1,4β1⊕min4β2,5β2,max3γ1,4γ1⊕max4γ2,5γ2,max1δ1,7δ1⊕max2δ2,10δ2,min2θ1,9θ1⊕min3θ2,11θ2,min6ϕ1,7ϕ1⊕min6ϕ2,8ϕ2,max6φ1,7φ1⊕max6φ2,8φ2,max2ξ1,9ξ1⊕max3ξ2,11ξ2,subject to(120)min1α1,9α1+min3α2,9α2=−27,min3α1,9α1+min0α2,8α2=−25,min2β1,3β1+min5β2,8β2=17,min4β1,5β1+min5β2,6β2=19,max2γ1,3γ1+max5γ2,8γ2=38,max4γ1,5γ1+max5γ2,6γ2=34,max1δ1,9δ1+max3δ2,9δ2=72,max3δ1,9δ1+max0δ2,8δ2=67,min2θ1,6θ1+min3θ2,10θ2=−28,min2θ1,8θ1+min1θ2,9θ2=−25,min3ϕ1,4ϕ1+min5ϕ2,7ϕ2=17,min3ϕ1,7ϕ1+min2ϕ2,5ϕ2=14,max3φ1,4φ1+max5φ2,7φ2=34,max3φ1,7φ1+max2φ2,5φ2=45,max2ξ1,6ξ1+max3ξ2,10ξ2=72,max2ξ1,8ξ1+max1ξ2,9ξ2=83,where(121)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 4: by applying the ranking function, the problem converts as(122)Maximizeℜmin1α1,7α1⊕min2α2,10α2,min3β1,4β1⊕min4β2,5β2,max3γ1,4γ1⊕max4γ2,5γ2,max1δ1,7δ1⊕max2δ2,10δ2,min2θ1,9θ1⊕min3θ2,11θ2,min6ϕ1,7ϕ1⊕min6ϕ2,8ϕ2,max6φ1,7φ1⊕max6φ2,8φ2,max2ξ1,9ξ1⊕max3ξ2,11ξ2,subject to(123)min1α1,9α1+min3α2,9α2=−27,min3α1,9α1+min0α2,8α2=−25,min2β1,3β1+min5β2,8β2=17,min4β1,5β1+min5β2,6β2=19,max2γ1,3γ1+max5γ2,8γ2=38,max4γ1,5γ1+max5γ2,6γ2=34,max1δ1,9δ1+max3δ2,9δ2=72,max3δ1,9δ1+max0δ2,8δ2=67,min2θ1,6θ1+min3θ2,10θ2=−28,min2θ1,8θ1+min1θ2,9θ2=−25,min3ϕ1,4ϕ1+min5ϕ2,7ϕ2=17,min3ϕ1,7ϕ1+min2ϕ2,5ϕ2=14,max3φ1,4φ1+max5φ2,7φ2=34,max3φ1,7φ1+max2φ2,5φ2=45,max2ξ1,6ξ1+max3ξ2,10ξ2=72,max2ξ1,8ξ1+max1ξ2,9ξ2=83,where(124)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 5: by applying ranking for TBFNs (2.6), the problem converts as(125)Maximize144α1−−3α1+6α2−−4α2+72β1−−12β1+92β2−−12β2+72γ1+−12γ1+92γ2+−12γ2+4δ1+−3δ1+6δ2+−5δ2+124δ1+−3δ1+6δ2+−5δ2+72γ1+−12γ1+92γ2+−12γ2−72β1+−12β1−92β2+−12β2−4α1+−3α1−6α2+−4α2−14112θ1−−72θ1+7θ2−−4θ2+132ϕ1−−12ϕ1+7ϕ2−−1ϕ2+132φ1+−12φ1+7φ2+−1φ2+112ξ1+−72ξ1+7ξ2+−4ξ2−12112ξ1+−72ξ1+7ξ2+−4ξ2+132φ1+−12φ1+7φ2+−1φ2−132ϕ1+−12ϕ1−7ϕ2+−1ϕ2−112θ1+−72θ1−7θ2+−4θ2,subject to(126)5α1−4α1+6α2−3α2=−27,6α1−3α1+4α2−4α2=−25,52β1−12β1+132β2−32β2=17,92β1−12β1+112β2−12β2=19,52γ1+12γ1+132γ2+32γ2=38,92γ1+12γ1+112γ2+12γ2=34,5δ1+4δ1+6δ2+3δ2=72,6δ1+3δ1+4δ2+4δ2=67,4θ1−2θ1+132θ2−72θ2=−28,5θ1−3θ1+5θ2−4θ2=−25,72ϕ1−12ϕ1+6ϕ2−1ϕ2=17,5ϕ1−2ϕ1+72ϕ2−32ϕ2=14,72φ1+12φ1+6φ2+1φ2=34,5φ1+2φ1+72φ2+32φ2=45,4ξ1+2ξ1+132ξ2+72ξ2=72,5ξ1+3ξ1+5ξ2+4ξ2=83,where(127)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 6: by using software MATLAB, the optimal solution of the crisp LPP isα1=−1,β1=1,γ1=2,δ1=3, θ1=1,ϕ1=4,φ1=5,ξ1=7,α2=−2,β2=3,γ2=4,δ2=5,θ2=−3,ϕ2=1,φ2=2,ξ2=3.Step 7: the exact BFOS isY˜1=−1,1,2,3,1,4,5,7,Y˜2=−2,3,4,5,−3,1,2,3.Step 8: the bipolar fuzzy optimal value of FBFLPP is−27,15,28,71,−31,30,51,96.
Hence, the company should manufacture−1,1,2,3,1,4,5,7 hundred of fans and −2,3,4,5,−3,1,2,3 hundred of heaters to get a maximum profit of Rs. −27,15,28,71,−31,30,51,96.Table 3
Manufacturing problem.
MachinesFansHeatersMaximum hours per weekMachine (M1)1,2,3,9,2,3,4,63,5,8,9,3,5,7,10−27,17,38,72,−28,17,34,72Machine (M2)3,4,5,9,2,3,7,80,5,6,8,1,2,5,9−25,19,34,67,−25,14,45,83


## 4.1. Methodology

(63)Maximize/Minimize∑j=1nB˜j⊗Y˜j,subject to(64)∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m,where Y˜j,E˜i,F˜ij, and B˜j are unrestricted TrBFNs.Step 1: we assume(65)F˜ij=eij,fij,gij,hij,aij,bij,cij,dij,E˜i=λi,μi,νi,ϵi,κi,ρi,ωi,τi,B˜j=mj,nj,pj,qj,rj,sj,tj,uj,Y˜j=αj,βj,γj,δj,θj,ϕj,φj,ξj.The FBFLPP (63) can be transformed into problem (66):(66)Maximize/Minimize∑j=1nmj,nj,pj,qj,rj,sj,tj,uj⊗αj,βj,γj,δj,θj,ϕj,φj,ξj,subject to(67)∑j=1neij,fij,gij,hij,aij,bij,cij,dij⊗αj,βj,γj,δj,θj,ϕj,φj,ξj,=λi,μi,νi,ϵi,κi,ρi,ωi,τi,whereαj,βj,γj,δj,θj,ϕj,φj,ξj is an unrestricted TrBFN, ∀j=1,2,3,…,n.Step 2: by applying the product of TrBFNs (4.1), the FBFLPP (66) transforms to problem (68) as(68)Maximize/Minimize∑j=1nmj,nj,pj,qj,rj,sj,tj,uj⊗αj,βj,γj,δj,θj,ϕj,φj,ξj,subject to(69)∑j=1nmineij′,eij″,minfij′,fij″,maxgij′,gij″,maxhij′,hij″,minaij′,aij″,minbij′,bij″,maxcij′,cij″,maxdij′,dij″=λi,μi,νi,ɛi,κi,ρi,ωi,τi,whereαj,βj,γj,δj,θj,ϕj,φj,ξj is an unrestricted TrBFN, ∀j=1,2,3,…,n, and(70)mineij′,eij″,minfij′,fij″,maxgij′,gij″,maxhij′,hij″=mineijαj,hijαj,minfijβj,gijβj,maxfijγj,gijγj,maxeijδj,hijδj,ifeij≥0,mineijδj,hijαj,minfijβj,gijβj,maxfijγj,gijγj,maxeijαj,hijδj,ifeij<0,fij≥0,mineijδj,hijαj,minfijγj,gijβj,maxfijβj,gijγj,maxeijαj,hijδj,iffij<0,gij≥0,mineijδj,hijαj,minfijγj,gijγj,maxfijβj,gijβj,maxeijαj,hijδj,ifgij<0,hij≥0,mineijδj,hijδj,minfijγj,gijγj,maxfijβj,gijβj,maxeijαj,hijαj,otherwise,minaij′,aij″,minbij′,bij″,maxcij′,cij″,maxdij′,dij″=minaijθj,dijθj,minbijϕj,cijϕj,maxbijφj,cijφj,maxaijξj,dijξj,ifaij≥0,minaijξj,dijαj,minbijϕj,cijϕj,maxbijφj,cijφj,maxaijθj,dijξj,ifaij<0,bij≥0,minaijξj,dijαj,minbijφj,cijϕj,maxbijϕj,cijφj,maxaijθj,dijξj,ifbij<0,cij≥0,minaijξj,dijαj,minbijφj,cijφj,maxbijϕj,cijϕj,maxaijθj,dijξj,ifcij<0,dij≥0,minaijξj,dijξj,minbijφj,cijφj,maxbijϕj,cijϕj,maxaijθj,dijθj,otherwise.Step 3: by applying arithmetic operations, the FBFLPP (68) transforms into problem (71) as(71)Maximize/Minimize∑j=1nmj,nj,pj,qj,rj,sj,tj,uj⊗αj,βj,γj,δj,θj,ϕj,φj,ξj,subject to(72)∑j=1nmineij′,eij″=λi,∑j=1nminaij′,aij″=κi,∑j=1nminfij′,fij″=νi,∑j=1nminbij′,bij″=ρi,∑j=1nmaxgij′,gij″=μi,∑j=1nmaxcij′,cij″=ωi,∑j=1nmaxhij′,hij″=ɛi,∑j=1nmaxdij′,hij″=τi,∀i=1,2,3,…,m, where x˜j=αj,βj,γj,δj,θj,ϕj,φj,ξj be unrestricted TrBFNs, ∀j=1,2,3,…,n.Step 4: by applying ranking function, the FBFLPP converts into crisp LPP (73) as(73)Maximize/Minimizeℜ∑j=1nmj,nj,pj,qj,rj,sj,tj,uj⊗αj,βj,γj,δj,θj,ϕj,φj,ξj,subject to(74)∑j=1nmineij′,eij″=λi,∑j=1nminaij′,aij″=κi,∑j=1nminfij′,fij″=νi,∑j=1nminbij′,bij″=ρi,∑j=1nmaxgij′,gij″=μi,∑j=1nmaxcij′,cij″=ωi,∑j=1nmaxhij′,hij″=ɛi,∑j=1nmaxdij′,hij″=τi,∀i=1,2,3,…,m,βj−αj≥0,γj−βj,δj−γj≥0,ϕj−θj,φj−ϕj≥0,ξj−φj≥0,∀j=1,2,3,…,n.Step 5: we consider(75)mj,nj,pj,qj,rj,sj,tj,uj⊗αj,βj,γj,δj,θj,ϕj,φj,ξj=minmij′,mij″,minnij′,nij″,maxpij′,pij″,maxqij′,qij″,minrij′,rij″,minsij′,sij″,maxtij′,tij″,maxuij′,uij″.The crisp LPP (73) can be transformed into the following problem:(76)Maximize/Minimizeℜ∑j=1nminmij′,mij″,minnij′,nij″,maxpij′,pij″,maxqij′,qij″,minrij′,rij″,minsij′,sij″,maxtij′,tij″,maxuij′,uij″,subject to(77)∑j=1nmineij′,eij″=λi,∑j=1nminaij′,aij″=κi,∑j=1nminfij′,fij″=νi,∑j=1nminbij′,bij″=ρi,∑j=1nmaxgij′,gij″=μi,∑j=1nmaxcij′,cij″=ωi,∑j=1nmaxhij′,hij″=ɛi,∑j=1nmaxdij′,hij″=τi,∀i=1,2,3,…,m,βj−αj≥0,γj−βj,δj−γj≥0,ϕj−θj,φj−ϕj≥0,ξj−φj≥0,∀j=1,2,3,…,n.Step 6: by applying the linearity propertyℜ∑j=1nG˜i=∑j=1nℜG˜i, where G˜i be BFN, the crisp LPP (76) can be transformed into the following problem:(78)Maximize/Minimize∑j=1nℜminmij′,mij″,minnij′,nij″,maxpij′,pij″,maxqij′,qij″,minrij′,rij″,minsij′,sij″,maxtij′,tij″,maxuij′,uij″,subject to(79)∑j=1nmineij′,eij″=λi,∑j=1nminaij′,aij″=κi,∑j=1nminfij′,fij″=νi,∑j=1nminbij′,bij″=ρi,∑j=1nmaxgij′,gij″=μi,∑j=1nmaxcij′,cij″=ωi,∑j=1nmaxhij′,hij″=ɛi,∑j=1nmaxdij′,hij″=τi,∀i=1,2,3,…,m,βj−αj≥0,γj−βj,δj−γj≥0,ϕj−θj,φj−ϕj≥0,ξj−φj≥0,∀j=1,2,3,…,n.Step 7: the crisp LLP (78) can be transformed to the following problem:(80)Maximize/Minimize∑j=1nℜmij′+mij″2−mij′−mij″2,nij′+nij″2−nij′−nij″2,pij′+pij″2+pij′−pij″2,qij′+qij″2+qij′−qij″2,rij′+rij″2−rij′−rij″2,sij′+sij″2−sij′−sij″2,tij′+tij″2+tij′−tij″2,uij′+uij″2+uij′−uij″2,subject to(81)∑j=1neij′+eij″2−eij′−eij″2=λi,∑j=1naij′+aij″2−aij′−aij″2=κi,∑j=1nfij′+fij″2−fij′−fij″2=μi,∑j=1nbij′+bij″2−bij′−bij″2=ρi,∑j=1ngij′+gij″2+gij′−gij″2=νi,∑j=1ncij′+cij″2+cij′−cij″2=ωi,∑j=1nhij′+hij″2+hij′−hij″2=ɛi∑j=1ndij′+dij″2+dij′−dij″2=τi,∀i=1,2,3,…,m,βj−αj≥0,γj−βj,δj−γj≥0,ϕj−θj,φj−ϕj≥0,ξj−φj≥0,∀j=1,2,3,…,n.Step 8: by applying ranking for TrBFNs (2.5), the crisp LPP (80) becomes(82)Maximize/Minimize∑j=1n14mij′+mij′′2−mij′−mij′′2+nij′+nij′′2−nij′−nij′′2+pij′+pij′′2+pij′−pij′′2+qij′+qij′′2+qij′−qij′′2+12−mij′+mij′′2+mij′−mij′′2−nij′+nij′′2+nij′−nij′′2+pij′+pij′′2+pij′−pij′′2+qij′+qij′′2+qij′−qij′′2−14rij′+rij′′2−rij′−rij′′2+sij′+sij′′2−sij′−sij′′2+tij′+tij′′2+tij′−tij′′2+uij′+uij′′2+uij′−uij′′2−12−rij′+rij′′2+rij′−rij′′2−sij′+sij′′2+sij′−sij′′2+tij′+tij′′2+tij′−tij′′2+uij′+uij′′2+uij′−uij′′2,subject to(83)∑j=1neij′+eij″2−eij′−eij″2=λi,∑j=1naij′+aij″2−aij′−aij″2=κi,∑j=1nfij′+fij″2−fij′−fij″2=μi,∑j=1nbij′+bij″2−bij′−bij″2=ρi,∑j=1ngij′+gij″2+gij′−gij″2=νi,∑j=1ncij′+cij″2+cij′−cij″2=ωi,∑j=1nhij′+hij″2+hij′−hij″2=ɛi,∑j=1ndij′+dij″2+dij′−dij″2=τi,∀i=1,2,3,…,m,βj−αj≥0,γj−βj,δj−γj≥0,ϕj−θj,φj−ϕj≥0,ξj−φj≥0,∀j=1,2,3,…,n.Step 9: by solving the crisp LPP (82), we get the optimal solution αj∗,βj∗,γj∗,δj∗,θj∗,ϕj∗,φj∗,ξj∗, ∀j=1,2,3,…,n.Step 10: we find the exact BFOSY˜j∗ of FBFLPP by assigning the values of αj∗,βj∗,γj∗,δj∗,θj∗,ϕj∗,φj∗ and ξj∗ in Y˜j∗=αj∗,βj∗,γj∗,δj∗,θj∗,ϕj∗,φj∗,ξj∗, ∀j=1,2,3,…,n.Step 11: we find the bipolar fuzzy optimal value by assigning the values ofY˜j∗ in ∑j=1nB˜j⊗Y˜j, ∀j=1,2,3,…,n.Thus, we state the existence condition for the optimal solution of bipolar fuzzy LPP in the following theorem.Theorem 2.
The solution of FBFLPP:(84)Maximize/MinimizeZ=∑j=1nBj⊗Yjsubject  to∑j=1nFij⊗Yj=Ei,∀i=1,…,m.Bj,Yj,Fij, and Ei are unrestricted TrBFNs and exist when the solution of the associated crisp LPP,(85)Maximize/MinimizeℜZ=ℜ∑j=1nfj1,fj2,fj3,fj4;gj1,gj2,gj3,gj4⊗αj,βj,γj,δj;θj,ϕj,φj,ξj,subject to(86)∑j=1npij=hi1,∑j=1nqij=hi2,∑j=1nrij=hi3,∑j=1nsij=hi4,∑j=1ntij=hi5,∑j=1nuij=hi6,∑j=1nvij=hi7,∑j=1nwij=hi8,βj−αj≥0,γj−βj≥0,δj−γj≥0,ϕj−θj≥0,φj−ϕj≥0,ξj−φj≥0,∀i=1,…,m exists. Otherwise, there is no guarantee that the BFOS exists.


## 4.2. Numerical Examples

In this section, we illustrate the proposed method with examples.Example 4.
1,5,8,11,3,6,7,12⊗Y˜1⊕2,4,6,9,1,5,6,7⊗Y˜2 is maximized subject to(87)1,4,6,7,2,4,5,8⊗Y˜1⊕0,1,3,6,3,4,7,9⊗Y˜2=2,21,51,109,11,28,72,144,2,3,5,8,1,6,9,10⊗Y˜1⊕1,3,5,6,0,1,2,5⊗Y˜2=5,27,60,116,1,22,66,130,where Y˜1=χ1,β1,ζ1,τ1,α1,φ1,ν1,δ1 and Y˜2=χ2,β2,ζ2,τ2,α2,φ2,ν2,δ2 are nonnegative TrBFNs.Step 1:(88)Maximize1,5,8,11,3,6,7,12⊗χ1,β1,ζ1,τ1,α1,φ1,ν1,δ1⊕2,4,6,9,1,5,6,7⊗χ2,β2,ζ2,τ2,α2,φ2,ν2,δ2,subject to(89)1,4,6,7,2,4,5,8⊗χ1,β1,ζ1,τ1,α1,φ1,ν1,δ1⊕0,1,3,6,3,4,7,9⊗χ2,β2,ζ2,τ2,α2,φ2,ν2,δ2=2,21,51,109,11,28,72,144,2,3,5,8,1,6,9,10⊗χ1,β1,ζ1,τ1,α1,φ1,ν1,δ1⊕1,3,5,6,0,1,2,5⊗χ2,β2,ζ2,τ2,α2,φ2,ν2,δ2=5,27,60,116,1,22,66,130,whereY˜1=χ1,β1,ζ1,τ1,α1,φ1,ν1,δ1 and Y˜2=χ2,β2,ζ2,τ2,α2,φ2,ν2,δ2 are nonnegative TrBFNs.Step 2: using the product of TBFNs (3.3), the problem converts as(90)Maximize1χ1,5β1,8ζ1,11τ1,3α1,6φ1,7ν1,12δ1⊕2χ2,4β2,6ζ2,9τ2,1α2,5φ2,6ν2,7δ2,subject to(91)1χ1,4β1,6ζ1,7τ1,3α1,4φ1,5ν1,8δ1⊕0χ2,1β2,3ζ2,6τ2,3α2,4φ2,7ν2,9δ2,=2,21,51,109,11,28,72,1442χ1,3β1,5ζ1,8τ1,1α1,6φ1,9ν1,10δ1⊕1χ2,3β2,5ζ2,6τ2,0α2,1φ2,2ν2,5δ2,=5,27,60,116,1,22,66,130χ1≥0,β1−χ1≥0,ζ1−β1≥0,τ1−ζ1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0,δ1−ν1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,τ2−ζ2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0,δ2−ν2≥0.Step 3: using arithmetic operations, the problem converts as(92)Maximize1χ1+2χ2,5β1+4β2,8ζ1+6ζ2,11τ1+9τ2,3α1+1α2,6φ1+5φ2,7ν1+6ν2,12δ1+7δ2,subject to(93)1χ1+0χ2,4β1+1β2,6ζ1+3ζ2,7τ1+6τ2,2α1+3α2,4φ1+4φ2,5ν1+7ν2,8δ1+9δ2,=2,21,51,109,11,28,72,1442χ1+1χ2,3β1+3β2,5ζ1+5ζ2,8τ1+6τ2,1α1+0α2,6φ1+1φ2,9ν1+2ν2,10δ1+5δ2,=5,27,60,116,1,22,66,130χ1≥0,β1−χ1≥0,ζ1−β1≥0,τ1−ζ1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0,δ1−ν1≥0χ2≥0,β2−χ2≥0,ζ2−β2≥0,τ2−ζ2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0,δ2−ν2≥0.Step 4: by applying the definition of ranking function, the problem converts as(94)Maximize1χ1+2χ2+5β1+4β2+8ζ1+6ζ2+11τ1+9τ24+8ζ1+6ζ2+11τ1+9τ2−1χ1−2χ2−5β1−4β22−3α1+1α2+6φ1+5φ2+7ν1+6ν2+12δ1+7δ24+7ν1+6ν2+12δ1+7δ2−3α1−1α2−6φ1−5φ22,subject to(95)1χ1+0χ2=2,2χ1+1χ2=5,4β1+1β2=21,3β1+3β2=27,6ζ1+3ζ2=51,5ζ1+5ζ2=60,7τ1+6τ2=109,8τ1+6τ2=116,2α1+3α2=11,1α1+0α2=1,4φ1+4φ2=28,6φ1+1φ2=22,5ν1+7ν2=72,9ν1+2ν2=66,8δ1+9δ2=144,10δ1+5δ2=130,χ1≥0,β1−χ1≥0,ζ1−β1≥0,τ1−ζ1≥0,α1≥0,φ1−α1≥0,ν1−φ1≥0,δ1−ν1≥0,χ2≥0,β2−χ2≥0,ζ2−β2≥0,τ2−ζ2≥0,α2≥0,φ2−α2≥0,ν2−φ2≥0,δ2−ν2≥0.Step 5: by using software MATLAB, the optimal solution of the crisp LPP isχ1=2,β1=4,ζ1=5,τ1=7,χ2=1,β2=5,ζ2=7,τ2=10,α1=1,φ1=3,ν1=6,δ1=9α2=3,φ2=4,ν2=6,δ2=8.Step 6: the exact BFOS isY˜1=2,4,5,7,1,3,6,9,Y˜2=1,5,7,10,3,4,6,8.Step 7: bipolar fuzzy optimal value of FBFLPP is4,40,82,167,6,38,78,164.Example 5.
(96)Maximize1,2,3,4,5,6,7,8⊗Y˜1⊕2,4,6,8,9,10,11,12⊗Y˜2,subject to(97)0,1,2,3,4,5,6,7⊗Y˜1⊕1,3,5,7,6,7,8,9⊗Y˜2=−8,2,27,57,−25,7,28,55,2,4,7,9,8,9,10,11⊗Y˜1⊕2,3,5,6,7,9,12,13⊗Y˜2=−25,−8,34,81,−37,9,44,83,where Y˜1=α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1 and Y˜2=α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2 are unrestricted TrBFNs.Step 1:(98)Maximize1,2,3,4,5,6,7,8⊗α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1⊕2,4,6,8,9,10,11,12⊗α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2,subject to(99)0,1,2,3,4,5,6,7⊗α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1⊕1,3,5,7,6,7,8,9⊗α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2=−8,2,27,57,−25,7,28,55,2,4,7,9,8,9,10,11⊗α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1⊕2,3,5,6,7,9,12,13⊗α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2=−25,−8,34,81,−37,9,44,83,whereY˜1=α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1 and Y˜2=α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2are unrestricted TrBFNs.Step 2: using the product of TrBFNs (4.1), the problem converts as(100)Maximizemin1α1,4α1,min2β1,3β1,max2γ1,3γ1,max1δ1,4δ1,min5θ1,8θ1,min6ϕ1,7ϕ1,max6φ1,7φ1,max5ξ1,8ξ1⊕min2α2,8α2,min4β2,6β2,max4γ2,6γ2,max2δ2,8δ2,min9θ2,12θ2,min10ϕ2,11ϕ2,max10φ2,11φ2,max9ξ2,12ξ2,subject to(101)min0α1,3α1,min1β1,2β1,max1γ1,2γ1,max0δ1,3δ1,min4θ1,7θ1,min5ϕ1,6ϕ1,max5φ1,6φ1,max4ξ1,7ξ1⊕min1α2,7α2,min3β2,5β2,max3γ2,5γ2,max1δ2,7δ2,min6θ2,9θ2,min7ϕ2,8ϕ2,max7φ2,8φ2,max6ξ2,9ξ2=−8,2,27,57,−25,7,28,55,min2α1,9α1,min4β1,7β1,max4γ1,7γ1,max2δ1,9δ1,min8θ1,11θ1,min9ϕ1,10ϕ1,max9φ1,10φ1,max8ξ1,11ξ1⊕min2α2,6α2,min3β2,5β2,max3γ2,5γ2,max2δ2,6δ2,min7θ2,13θ2,min9ϕ2,12ϕ2,max9φ2,12φ2,max7ξ2,13ξ2=−25,−8,34,81,−37,9,44,83,where(102)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 3: using arithmetic operations, the problem converts as(103)Maximizemin1α1,4α1⊕min2α2,8α2,min2β1,3β1⊕min4β2,6β2,max2γ1,3γ1⊕max4γ2,6γ2,max1δ1,4δ1⊕max2δ2,8δ2,min5θ1,8θ1⊕min9θ2,12θ2,min6ϕ1,7ϕ1⊕min10ϕ2,11ϕ2,max6φ1,7φ1⊕max10φ2,11φ2,max5ξ1,8ξ1⊕max9ξ2,12ξ2,subject to(104)min0α1,3α1+min1α2,7α2=−8,min2α1,9α1+min2α2,6α2=−25,min1β1,2β1+min3β2,5β2=2,min4β1,7β1+min3β2,5β2=−8,max1γ1,2γ1+max3γ2,5γ2=27,max4γ1,7γ1+max3γ2,5γ2=34,max0δ1,3δ1+max1δ2,7δ2=57,max2δ1,9δ1+max2δ2,6δ2=81,min4θ1,7θ1+min6θ2,9θ2=−25,min8θ1,11θ1+min7θ2,13θ2=−37,min5ϕ1,6ϕ1+min7ϕ2,8ϕ2=7,min9ϕ1,10ϕ1+min9ϕ2,12ϕ2=9,max5φ1,6φ1+max7φ2,8φ2=28,max9φ1,10φ1+max9φ2,12φ2=44,max4ξ1,7ξ1+max6ξ2,9ξ2=55,max8ξ1,11ξ1+max7ξ2,13ξ2=83,where(105)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 4: by applying the ranking function, the problem converts as(106)Maximizeℜmin1α1,4α1⊕min2α2,8α2,min2β1,3β1⊕min4β2,6β2,max2γ1,3γ1⊕max4γ2,6γ2,max1δ1,4δ1⊕max2δ2,8δ2,min5θ1,8θ1⊕min9θ2,12θ2,min6ϕ1,7ϕ1⊕min10ϕ2,11ϕ2,max6φ1,7φ1⊕max10φ2,11φ2,max5ξ1,8ξ1⊕max9ξ2,12ξ2,subject to(107)min0α1,3α1+min1α2,7α2=−8,min2α1,9α1+min2α2,6α2=−25,min1β1,2β1+min3β2,5β2=2,min4β1,7β1+min3β2,5β2=−8,max1γ1,2γ1+max3γ2,5γ2=27,max4γ1,7γ1+max3γ2,5γ2=34,max0δ1,3δ1+max1δ2,7δ2=57,max2δ1,9δ1+max2δ2,6δ2=81,min4θ1,7θ1+min6θ2,9θ2=−25,min8θ1,11θ1+min7θ2,13θ2=−37,min5ϕ1,6ϕ1+min7ϕ2,8ϕ2=7,min9ϕ1,10ϕ1+min9ϕ2,12ϕ2=9,max5φ1,6φ1+max7φ2,8φ2=28,max9φ1,10φ1+max9φ2,12φ2=44,max4ξ1,7ξ1+max6ξ2,9ξ2=55,max8ξ1,11ξ1+max7ξ2,13ξ2=83,where(108)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 5: by applying ranking for TBFNs (2.6), the problem converts as(109)Maximize1452α1−−32α1+5α2−−3α2+52β1−−12β1+5β2−−β2+52γ1+−12γ1+5γ2+−γ2+52δ1+|−32δ1|+5δ2+−3δ2+1252δ1+−32δ1+5δ2+−3δ2+52γ1+−12γ1+5γ2+−γ2−52β1+−12β1−5β2+−β2−52α1+−32α1−5α2+−3α2−14132θ1−−32θ1+212θ2−−32θ2+132ϕ1−−12ϕ1+212ϕ2−−12ϕ2+132φ1+−12φ1+212φ2+−12φ2+132ξ1+−32ξ1+212ξ2+−32ξ2−12132ξ1+−32ξ1+212ξ2+−32ξ2+132φ1+−12φ1+212φ2+−12φ2−132ϕ1+−12ϕ1−212ϕ2+−12ϕ2−132θ1+−32θ1−212θ2+−32θ2,subject to(110)32α1−32α1+4α2−3α2=−8,112α1−72α1+4α2−2α2=−25,32β1−12β1+4β2−β2=2,112β1−32β1+4β2−β2=−8,32γ1+12γ1+4γ2+γ2=27,112γ1+32γ1+4γ2+γ2=34,32δ1+32δ1+4δ2+3δ2=57,112δ1+72δ1+4δ2+2δ2=81,112θ1−32θ1+152θ2−32θ2=−25,192θ1−32θ1+10θ2−3θ2=−37,112ϕ1−12ϕ1+152ϕ2−12ϕ2=7,192ϕ1−12ϕ1+212ϕ2−32ϕ2=9,112φ1+12φ1+152φ2+12φ2=28,192φ1+12φ1+212φ2+32φ2=44,112ξ1+32ξ1+152ξ2+32ξ2=55,192ξ1+32ξ1+10ξ2+3ξ2=83,where(111)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 6: by using software MATLAB, the optimal solution of the crisp LPP isα1=−3,β1=−2,γ1=7/5,δ1=5,θ1=−1,ϕ1=0,φ1=2,ξ1=4,α2=1,β2=2,γ2=121/25,δ2=6,θ2=−2,ϕ2=1,φ2=2,ξ2=3.Step 7: the exact BFOS isY˜1=−3,−2,7/5,5,−1,0,2,4,Y˜2=1,2,121/25,6,−2,1,2,3.Step 8: bipolar fuzzy optimal value of FBFLPP is−10,2,831/25,68,−32,10,36,68.Example 6.
An electrical company produces fans and heaters. Both of them are processed through two different machinesM1 and M2. Working hours for machine M1 per week are −27,17,38,72,−28,17,34,72 and for machine M2 are −25,19,34,67,−25,14,45,83. The profit per hundred for fans and heaters is Rs. 1,3,4,7,2,6,7,9 and Rs. 2,4,5,10,3,6,8,11, respectively. Further details are described in Table 3. How many fans and heaters should be manufactured by the company to get maximum profit is found.
LetY˜1 be hundred number of fans and Y˜2 be hundred number of heaters. Then, FBFLPP becomes(112)Maximize1,3,4,7,2,6,7,9⊗Y˜1⊕2,4,5,10,3,6,8,11⊗Y˜2,subject to(113)1,2,3,9,2,3,4,6⊗Y˜1⊕3,5,8,9,3,5,7,10⊗Y˜2=−27,17,38,72,−28,17,34,72,3,4,5,9,2,3,7,8⊗Y˜1⊕0,5,6,8,1,2,5,9⊗Y˜2=−25,19,34,67,−25,14,45,83,where Y˜1=α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1 and Y˜2=α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2 are unrestricted TrBFNs.Step 1:(114)Maximize1,3,4,7,2,6,7,9⊗α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1⊕2,4,5,10,3,6,8,11⊗α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2,subject to(115)1,2,3,9,2,3,4,6⊗α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1⊕3,5,8,9,3,5,7,10⊗,α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2=−27,17,38,72,−28,17,34,723,4,5,9,2,3,7,8⊗α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1⊕0,5,6,8,1,2,5,9⊗,α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2=−25,19,34,67,−25,14,45,83,whereY˜1=α1,β1,γ1,δ1,θ1,ϕ1,φ1,ξ1 and Y˜2=α2,β2,γ2,δ2,θ2,ϕ2,φ2,ξ2are unrestricted TrBFNs.Step 2: using the product of TrBFNs (4.1), the problem converts as(116)Maximizemin1α1,7α1,min3β1,4β1,max3γ1,4γ1,max1δ1,7δ1,min2θ1,9θ1,min6ϕ1,7ϕ1,max6φ1,7φ1,max2ξ1,9ξ1⊕min2α2,10α2,min4β2,5β2,max4γ2,5γ2,max2δ2,10δ2,min3θ2,11θ2,min6ϕ2,8ϕ2,max6φ2,8φ2,max3ξ2,11ξ2,subject to(117)min1α1,9α1,min2β1,3β1,max2γ1,3γ1,max1δ1,9δ1,min2θ1,6θ1,min3ϕ1,4ϕ1,max3φ1,4φ1,max2ξ1,6ξ1⊕min3α2,9α2,min5β2,8β2,max5γ2,8γ2,max3δ2,9δ2,min3θ2,10θ2,min5ϕ2,7ϕ2,max5φ2,7φ2,max3ξ2,10ξ2=−27,17,38,72,−28,17,34,72,min3α1,9α1,min4β1,5β1,max4γ1,5γ1,max3δ1,9δ1,min2θ1,8θ1,min3ϕ1,7ϕ1,max3φ1,7φ1,max2ξ1,8ξ1⊕min0α2,8α2,min5β2,6β2,max5γ2,6γ2,max0δ2,8δ2,min1θ2,9θ2,min2ϕ2,5ϕ2,max2φ2,5φ2,max1ξ2,9ξ2=−25,19,34,67,−25,14,45,83,where(118)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 3: using arithmetic operations, the problem converts as(119)Maximizemin1α1,7α1⊕min2α2,10α2,min3β1,4β1⊕min4β2,5β2,max3γ1,4γ1⊕max4γ2,5γ2,max1δ1,7δ1⊕max2δ2,10δ2,min2θ1,9θ1⊕min3θ2,11θ2,min6ϕ1,7ϕ1⊕min6ϕ2,8ϕ2,max6φ1,7φ1⊕max6φ2,8φ2,max2ξ1,9ξ1⊕max3ξ2,11ξ2,subject to(120)min1α1,9α1+min3α2,9α2=−27,min3α1,9α1+min0α2,8α2=−25,min2β1,3β1+min5β2,8β2=17,min4β1,5β1+min5β2,6β2=19,max2γ1,3γ1+max5γ2,8γ2=38,max4γ1,5γ1+max5γ2,6γ2=34,max1δ1,9δ1+max3δ2,9δ2=72,max3δ1,9δ1+max0δ2,8δ2=67,min2θ1,6θ1+min3θ2,10θ2=−28,min2θ1,8θ1+min1θ2,9θ2=−25,min3ϕ1,4ϕ1+min5ϕ2,7ϕ2=17,min3ϕ1,7ϕ1+min2ϕ2,5ϕ2=14,max3φ1,4φ1+max5φ2,7φ2=34,max3φ1,7φ1+max2φ2,5φ2=45,max2ξ1,6ξ1+max3ξ2,10ξ2=72,max2ξ1,8ξ1+max1ξ2,9ξ2=83,where(121)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 4: by applying the ranking function, the problem converts as(122)Maximizeℜmin1α1,7α1⊕min2α2,10α2,min3β1,4β1⊕min4β2,5β2,max3γ1,4γ1⊕max4γ2,5γ2,max1δ1,7δ1⊕max2δ2,10δ2,min2θ1,9θ1⊕min3θ2,11θ2,min6ϕ1,7ϕ1⊕min6ϕ2,8ϕ2,max6φ1,7φ1⊕max6φ2,8φ2,max2ξ1,9ξ1⊕max3ξ2,11ξ2,subject to(123)min1α1,9α1+min3α2,9α2=−27,min3α1,9α1+min0α2,8α2=−25,min2β1,3β1+min5β2,8β2=17,min4β1,5β1+min5β2,6β2=19,max2γ1,3γ1+max5γ2,8γ2=38,max4γ1,5γ1+max5γ2,6γ2=34,max1δ1,9δ1+max3δ2,9δ2=72,max3δ1,9δ1+max0δ2,8δ2=67,min2θ1,6θ1+min3θ2,10θ2=−28,min2θ1,8θ1+min1θ2,9θ2=−25,min3ϕ1,4ϕ1+min5ϕ2,7ϕ2=17,min3ϕ1,7ϕ1+min2ϕ2,5ϕ2=14,max3φ1,4φ1+max5φ2,7φ2=34,max3φ1,7φ1+max2φ2,5φ2=45,max2ξ1,6ξ1+max3ξ2,10ξ2=72,max2ξ1,8ξ1+max1ξ2,9ξ2=83,where(124)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 5: by applying ranking for TBFNs (2.6), the problem converts as(125)Maximize144α1−−3α1+6α2−−4α2+72β1−−12β1+92β2−−12β2+72γ1+−12γ1+92γ2+−12γ2+4δ1+−3δ1+6δ2+−5δ2+124δ1+−3δ1+6δ2+−5δ2+72γ1+−12γ1+92γ2+−12γ2−72β1+−12β1−92β2+−12β2−4α1+−3α1−6α2+−4α2−14112θ1−−72θ1+7θ2−−4θ2+132ϕ1−−12ϕ1+7ϕ2−−1ϕ2+132φ1+−12φ1+7φ2+−1φ2+112ξ1+−72ξ1+7ξ2+−4ξ2−12112ξ1+−72ξ1+7ξ2+−4ξ2+132φ1+−12φ1+7φ2+−1φ2−132ϕ1+−12ϕ1−7ϕ2+−1ϕ2−112θ1+−72θ1−7θ2+−4θ2,subject to(126)5α1−4α1+6α2−3α2=−27,6α1−3α1+4α2−4α2=−25,52β1−12β1+132β2−32β2=17,92β1−12β1+112β2−12β2=19,52γ1+12γ1+132γ2+32γ2=38,92γ1+12γ1+112γ2+12γ2=34,5δ1+4δ1+6δ2+3δ2=72,6δ1+3δ1+4δ2+4δ2=67,4θ1−2θ1+132θ2−72θ2=−28,5θ1−3θ1+5θ2−4θ2=−25,72ϕ1−12ϕ1+6ϕ2−1ϕ2=17,5ϕ1−2ϕ1+72ϕ2−32ϕ2=14,72φ1+12φ1+6φ2+1φ2=34,5φ1+2φ1+72φ2+32φ2=45,4ξ1+2ξ1+132ξ2+72ξ2=72,5ξ1+3ξ1+5ξ2+4ξ2=83,where(127)β1−α1≥0,γ1−β1,δ1−γ1≥0,ϕ1−θ1,φ1−ϕ1≥0,ξ1−φ1≥0.Step 6: by using software MATLAB, the optimal solution of the crisp LPP isα1=−1,β1=1,γ1=2,δ1=3, θ1=1,ϕ1=4,φ1=5,ξ1=7,α2=−2,β2=3,γ2=4,δ2=5,θ2=−3,ϕ2=1,φ2=2,ξ2=3.Step 7: the exact BFOS isY˜1=−1,1,2,3,1,4,5,7,Y˜2=−2,3,4,5,−3,1,2,3.Step 8: the bipolar fuzzy optimal value of FBFLPP is−27,15,28,71,−31,30,51,96.
Hence, the company should manufacture−1,1,2,3,1,4,5,7 hundred of fans and −2,3,4,5,−3,1,2,3 hundred of heaters to get a maximum profit of Rs. −27,15,28,71,−31,30,51,96.Table 3
Manufacturing problem.
MachinesFansHeatersMaximum hours per weekMachine (M1)1,2,3,9,2,3,4,63,5,8,9,3,5,7,10−27,17,38,72,−28,17,34,72Machine (M2)3,4,5,9,2,3,7,80,5,6,8,1,2,5,9−25,19,34,67,−25,14,45,83


## 5. Conclusions

The LP problem is mostly used in different fields of science and engineering for modeling real-world problems. By using fuzzy set theory, the vagueness of data is modeled in a mathematical form. The LP parameters are represented by fuzzy numbers rather than crisp numbers in many applications. A bipolar fuzzy model is an extension of the fuzzy model. We have extended the concept of crisp linear programming problem in a bipolar fuzzy environment based on bipolar fuzzy numbers. We have defined the multiplication of two TBFNs as well as TrBFNs. We have solved FBFLPPs based the TrBFNs and TBFNs and have obtained bipolar fuzzy optimal solutions. The obtained bipolar fuzzy optimal solutions satisfy the given constraints of the FBFLPPs, showing that the proposed scheme is correct and reliable. In future, this work can be extended to (1) bipolar fuzzy integer LPP, (2) bipolar fuzzy transportation problems, and (3) spherical fuzzy LPP.



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