# Optimization ofLR-Type Fully Bipolar Fuzzy Linear Programming Problems

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

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## Abstract

In this study, we present a technique to solveLR-type fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints. We define LR-type bipolar fuzzy numbers and their arithmetic operations. We discuss multiplication of LR-type bipolar fuzzy numbers. Furthermore, we develop a method to solve LR-type FBFLPPs with equality constraints involving LR-type bipolar fuzzy numbers as parameters and variables. Moreover, we define ranking for LR-type bipolar fuzzy numbers which transform the LR-type FBFLPP into a crisp linear programming problem. Finally, we consider numerical examples to illustrate the proposed method.


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## Body

## 1. Introduction

Zadeh [1–3]’s fuzzy set (FS) theory has been shown to be a useful tool to describe situations in which the data are imprecise or vague. FSs handle such situations by attributing a degree to which a certain object belongs to a set. In 1994, Zhang [4] initiated the concept of bipolar fuzzy sets (BFSs) as an extension of FSs. The BFS representation is useful when irrelevant elements and contrary elements are needed to be discriminated. Furthermore, Zhang [5] introduced NPN fuzzy sets and NPN qualitative algebra. On the other hand, Akram and Arshad [6] defined bipolar fuzzy numbers and proposed a novel trapezoidal bipolar fuzzy TOPSIS method for group decision making, and Singh [7] discussed two schemes based on the properties of next neighbors and Euclidean distance in a bipolar fuzzy environment. Chakraborty et al. [8] introduced pentagonal neutrosophic numbers and further analyzed their properties. They studied the definitions of score function and accuracy function which transform pentagonal neutrosophic numbers into crisp numbers and also presented transportation models in a neutrosophic environment.In mathematical programming models, the simplest model is a linear programming problem. Linear programming is a mathematical modeling technique in which a linear function is maximized or minimized when subjected to various constraints. Uncertainty is also introduced in the linear programming problem, which is widely studied by many scholars. Bellman and Zadeh [9] studied about objectives and human decision making. Zimmerman [10] presented a scheme to solve the fuzzy linear programming (FLP) problem by using multiobjective function. Tanaka et al. [11] suggested a method and obtained solution of FLP problems. Later, several methods have been developed in [12–17] to solve FLP problems and fuzzy systems of linear equations. Recently, certain methods have been developed in [18–20] to solve bipolar fuzzy linear system (BFLS) of equations.The notion of LR-FN was introduced by Dubois and Prade [21]. Dehghan et al. [22] presented a technique and obtained solution of fully fuzzy linear system D˜X˜=E˜ in which the coefficient matrix D˜ and right hand column vector E˜ contain LR-fuzzy numbers. Kaur and Kumar [15] defined arithmetic operations of LR-FNs and suggested Mehar’s method to solve fully FLP problems by using these numbers as variables and parameters. Buckley [23] gave the idea of fuzzy complex set and fuzzy complex number. Akram et al. [18] proposed a technique to solve LR-BFLS, LR-complex BFLS with real coefficients, and LR-complex BFLS with complex coefficients of equations. Recently, Akram et al. [24, 25] presented methods to solve Pythagorean FLP problems. Mehmood et al. [26] proposed a method for solving fully BFL programming problems. In this research article, we present a technique to solve LR-type fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints. We define LR-type bipolar fuzzy numbers and their arithmetic operations. We discuss multiplication of LR-type bipolar fuzzy numbers. Furthermore, we develop a method to solve LR-type FBFLPPs with equality constraints involving LR-type bipolar fuzzy numbers as parameters and variables. Moreover, we define ranking for LR-type bipolar fuzzy numbers which transform the LR-type FBFLPP into a crisp linear programming problem. Finally, we consider numerical examples to illustrate the proposed method.We have organized the research article as follows: In Section2, some preliminary concepts are presented. In Section 3, arithmetic operations are introduced. Section 4 presents a method to solve LR-type FBFLPPs with equality constraints in which variables and parameters are LR-type BFNs. In Section 5, the numerical example and model are illustrated. Conclusion is given in Section 6. The list of acronyms used in the research article is given in Table 1.Table 1
List of acronyms.
AcronymsRepresentationBFNsBipolar fuzzy numbersLPPsLinear programming problemsFBFLPPsFully bipolar fuzzy linear programming problemsBFOSBipolar fuzzy optimal solution


## 2. Preliminaries

Definition 1.
[4]). Let Z≠ϕ. A BFS K˜ in Z is an object having the form(1)K˜=ηK˜P,ηK˜N=z,ηK˜Pz,ηK˜Nz|z∈Z,which is characterized by the functions ηK˜P:Z⟶0,1 and ηK˜N:Z⟶−1,0, known as the truth membership function and falsity membership function, respectively. Here, truth or positive membership degree ηK˜Pz∈0,1 indicates the satisfaction degree of those elements z∈Z which fulfill a certain property relating to BFS K˜, and negative membership degree ηK˜Nz∈−1,0 indicates the satisfaction degree of those elements z∈Z which fulfill some counter property relating to BFS K˜.Definition 2.
[18]). A BFN, K˜=P,N=p1,p2,p3,p4,n1,n2,n3,n4 is a BFS of the mapping η:ℝ⟶0,1×−1,0, with satisfaction degree ηp and dissatisfaction degree ηn such that(2)ηpx=x−p1p2−p1,ifx∈p1,p2,1,ifx∈p2,p3,p4−xp4−p3,ifx∈p3,p4,0,otherwise,ηnx=n1−x1n2−n1,ifx∈n1,n2,−1,ifx∈n2,n3,x−n4n4−n3,ifx∈n3,n4,0,otherwise.Definition 3.
[6]). Let K˜=p1,p2,p3,p4,n1,n2,n3,n4 be a BFN. Then, its μ,ν−cut is defined as(3)K˜μ,ν=p2−p1μ+p1,−p4−p3μ+p4;−νn2−n1+n1,νn4−n3+n4,where μ∈0,1,ν∈−1,0.Definition 4.
[18]). A BFN is said to be an LR-bipolar fuzzy number of the form K˜=m∗,α∗,β∗LR,m′,α′,β′LR, where m∗,m′ are real numbers and α∗,β∗>0 and α′,β′>0, if its membership function τK˜ and nonmembership function ωK˜, are represented by(4)τK˜x=Lm∗−xα∗,ifx≤m∗,α∗>0,Rx−m∗β∗,ifx≥m∗,β∗>0,ωK˜x=L′m′−xα′,ifx≤m′,α′>0,R′x−m′β′,ifx≥m′,β′>0,where m∗,α∗, and β∗ are called the mean value and left and right spreads of the positive side of K, respectively, while m′,α′, and β′ are called the mean value and left and right spreads of negative side of K, respectively. Also, L and R are continuous and decreasing functions from ℜ+ to 0,1, while L′ and R′ are continuous and increasing functions from ℜ+ to −1,0 such that(1)
L0=R0=1(2)
limx⟶∞Lx=limx⟶∞Rx=0(3)
L′0=R′0=−1(4)
limx⟶∞L′x=limx⟶∞R′x=0Remark 1.
If we put(5)Lx=Rx=1−x,if0≤x≤1,0,otherwise,L′x=R′x=x−1,0≤x≤1,0,otherwisein Definition 4, then K˜=m∗,α∗,β∗LR,m′,α′,β′LR converts to an LR-type bipolar fuzzy number.Definition 5.
AnLR-type BFN K˜=m∗,α∗,β∗LR,m′,α′,β′LR is said to be nonnegative if m∗−α∗≥0 and m′−α′≥0 and is said to be nonpositive if m∗+β∗≤0 and m′+β′≤0.Definition 6.
AnLR-type BFN K˜=m∗,α∗,β∗LR,m′,α′,β′LR is positive if m∗−α∗>0 and m′−α′>0 and is negative if m∗+β∗<0 and m′+β′<0.Definition 7.
AnLR-type BFN K˜=m∗,α∗,β∗LR,m′,α′,β′LR is said to be unrestricted if m∗−α∗ and m′−α′ are real numbers.Definition 8.
AnLR-type BFNs K˜=m∗,α∗,β∗LR,m′,α′,β′LR is said to be zero if and only if m∗=0,α∗=0,β∗=0,m′=0,α′=0, and β′=0.Definition 9.
TwoLR-type BFNs K˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR and K2˜=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR are equal if m1∗=m2∗,α1∗=α2∗,β1∗=β2∗,m1′=m2′,α1′=α2′,β1′=β2′.Definition 10.
LetK˜=m∗,α∗,β∗LR,m′,α′,β′LR be an LR-type BFN; then, their γ−cut and δ−cut are given as follows:(6)K˜γ=m∗−α∗L−1γ,m∗+β∗R−1γ,K˜δ=m′−α′L′−1δ,m′+β′R′−1δ,where γ∈0,1 and δ∈−1,0.Definition 11.
LetK˜=m∗,α∗,β∗LR,m′,α′,β′LR be an LR-type BFN; then, ranking of K˜, represented by ℜK˜, is defined as(7)RankK˜=14∫01m∗−α∗L−1γdγ+∫01m∗+β∗R−1γdγ+14∫0−1m′−α′L′−1δdδ+∫0−1m′+β′R′−1δdδ,where γ∈0,1 and δ∈−1,0.
LetK˜1 and K˜2 be two LR-type BFNs; then,(i)
K˜1<K˜1, if ℜK˜1<ℜK˜2(ii)
K˜1>K˜1, if ℜK˜1>ℜK˜2(iii)
K˜1≈K˜1, if ℜK˜1=ℜK˜2For other concepts and applications, refer to [27–32].


## 3. Arithmetic Operations

In this section, we study about arithmetic operations forLR-type BFNs.Theorem 1.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be two LR-type BFNs; then,(8)K˜1⊕K˜2=m1∗+m2∗,α1∗+α2∗,β1∗+β2∗LR,m1′+m2′,α1′+α2′,β1′+β2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be two LR-type BFNs; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(9)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(10)K˜1γ+K˜2γ=m1∗−α1∗L−1γ+m2∗−α2∗L−1γ,m1∗+β1∗R−1γ+m2∗+β2∗R−1γ.
By settingγ=1 in equation (170), we get(11)K˜1+K˜2γ=1=m1∗+m2∗.
By settingγ=0 in equation (170), we get(12)K˜1+K˜2γ=0=m1∗+m2∗−α1∗−α2∗,m1∗+m2∗+β1∗+β2∗.
Now,(13)K˜1δ+K˜2δ=m1′−α1′L′−1δ+m2′−α2′L′−1δ,m1′+β1′R′−1δ+m2′+β2′R′−1δ.
By settingδ=−1 in equation (181), we get(14)K˜1+K˜2δ=−1=m1′+m2′.
By settingδ=0 in equation (181), we get(15)K˜1+K˜2δ=0=m1′+m2′−α1′−α2′,m1′+m2′+β1′+β2′,
On combining the equations (173), (178), (185), and (15), the result follows.Theorem 2.
LetK1˜=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be two LR-type BFNs; then,(16)K˜1⊖K˜2=m1∗−m2∗,α1∗+β2∗,α2∗+β1∗LR,m1′−m2′,α1′+β2′,α2′+β1′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be two LR-type BFNs; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(17)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(18)K˜1γ−K˜2γ=m1∗−α1∗L−1γ−m2∗−β2∗R−1γ,m1∗+β1∗R−1γ−m2∗+α2∗L−1γ.
By settingγ=1 in equation (18), we get(19)K˜1−K˜2γ=1=m1∗−m2∗.
By settingγ=0 in equation (18), we get(20)K˜1−K˜2γ=0=m1∗−m2∗−α1∗−β2∗,m1∗−m2∗+α2∗+β1∗.
Now,(21)K˜1δ−K˜2δ=m1′−α1′L′−1δ−m2′−β2′R′−1δ,m1′+β1′R′−1δ−m2′+α2′L′−1δ.
By settingδ=−1 in equation (21), we get(22)K˜1−K˜2δ=−1=m1′−m2′.
By settingδ=0 in equation (21), we get(23)K˜1−K˜2δ=0=m1′−m2′−α1′−β2′,m1′−m2′+α2′+β1′.
On combining equations (19), (20), (22), and (23), the result follows.Theorem 3.
LetK˜=m∗,α∗,β∗LR,m′,α′,β′LR be an LR-type BFN and ϵ be an arbitrary real number; then,(24)ϵK˜=ϵm∗,ϵα∗,ϵβ∗LR,ϵm′,ϵα′,ϵβ′LR,ifϵ≥0,ϵm∗,−ϵβ∗,−ϵα∗LR,ϵm′,−ϵβ′,−ϵα′LR,ifϵ<0.Proof.
LetK˜=m∗,α∗,β∗LR,m′,α′,β′LR be an LR-type BFN and ϵ be an arbitrary real number; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(25)K˜γ=m∗−α∗L−1γ,m∗+β∗R−1γ,K˜δ=m′−α′L′−1δ,m′+β′R′−1δ.
Now, ifϵ≥0, then(26)ϵK˜γ=ϵm∗−ϵα∗L−1γ,ϵm∗+ϵβ∗R−1γ.
By settingγ=1 in equation (26), we get(27)ϵK˜γ=1=ϵm∗.
By settingγ=0 in equation (26), we get(28)ϵK˜γ=0=ϵm∗−ϵα∗,ϵm∗+ϵβ∗.
Also,(29)ϵK˜δ=ϵm′−ϵα′L′−1δ,ϵm′+ϵβ′R′−1δ.
By settingδ=−1 in equation (29), we get(30)ϵK˜δ=−1=ϵm′.
By settingδ=0 in equation (29), we get(31)ϵK˜δ=0=ϵm′−ϵα′,ϵm′+ϵβ′.
On combining the equations (27), (28), (30), and (31), the case ϵ≥0 follows.
Ifϵ<0, then(32)ϵK˜γ=ϵm∗+ϵβ∗R−1γ,ϵm∗−ϵα∗L−1γ.
By settingγ=1 in equation (32), we get(33)ϵK˜γ=1=ϵm∗.
By settingγ=0 in equation (32), we get(34)ϵK˜γ=0=ϵm∗+ϵβ∗,ϵm∗−ϵα∗.
Also,(35)ϵK˜δ=ϵm′+ϵβ′R′−1δ,ϵm′−ϵα′L′−1δ.
By settingδ=−1 in equation (35), we get(36)ϵK˜δ=−1=ϵm′.
By settingδ=0 in equation (35), we get(37)ϵK˜δ=0=ϵm′+ϵβ′,ϵm′−ϵα′.
On combining equations (33), (34), (36), and (37), the case ϵ<0 follows. Thus result is as required.Theorem 4.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be two nonnegative LR-type BFNs; then,(38)K˜1⊗K˜2=m1∗m2∗,m1∗α2∗+α1∗m2∗−α1∗α2∗,m1∗β2∗+β1∗m2∗+β1∗β2∗LR,m1′m2′,m1′α2′+α1′m2′−α1′α2′,m1′β2′+β1′m2′+β1′β2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be two nonnegative LR-type BFNs; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(39)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(40)K˜1γ×K˜2γ=m1∗−α1∗L−1γm2∗−α2∗L−1γ,m1∗+β1∗R−1γm2∗+β2∗R−1γ.
By settingγ=1 in equation (40), we get(41)K˜1K˜2γ=1=m1∗m2∗.
By settingγ=0 in equation (40), we get(42)K˜1K˜2γ=0=m1∗−α1∗m2∗−α2∗,m1∗+β1∗m2∗+β2∗,(43)K˜1K˜2γ=0=m1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗.
Also,(44)K˜1δ×K˜2δ=m1′−α1′L′−1δm2′−α2′L′−1δ,m1′+β1′R′−1δm2′+β2′R′−1δ.
By settingδ=−1 in equation (44), we get(45)K˜1K˜2δ=−1=m1′m2′.
By settingδ=0 in equation (44), we get(46)K˜1K˜2δ=0=m1′−α1′m2′−α2′,m1′+β1′m2′+β2′,(47)K˜1K˜2δ=0=m1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′.
On combining the equations (41), (43), (45), and (47), the result follows.Theorem 5.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be a nonnegative LR-type BFN andK˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be a non-positive LR-type BFN, then(48)K˜1⊗K˜2=m1∗m2∗,m1∗α2∗−β1∗m2∗+β1∗α2∗,m1∗β2∗−α1∗m2∗−α1∗β2∗LR,m1′m2′,m1′α2′−β1′m2′+β1′α2′,m1′β2′−α1′m2′−α1′β2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be a nonnegative LR-type BFN andK˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be a nonpositive LR-type BFN; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(49)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(50)K˜1γ×K˜2γ=m1∗+β1∗R−1γm2∗−α2∗L−1γ,m1∗−α1∗L−1γm2∗+β2∗R−1γ.
By settingγ=1 in equation (50), we get(51)K˜1K˜2γ=1=m1∗m2∗.
By settingγ=0 in equation (50), we get(52)K˜1K˜2γ=0=m1∗+β1∗m2∗−α2∗,m1∗−α1∗m2∗+β2∗,(53)K˜1K˜2γ=0=m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗,m1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗.
Now,(54)K˜1δ×K˜2δ=m1′+β1′R−1δm2′−α2′L−1δ,m1′−α1′L−1δm2′+β2′R−1δ.
By settingδ=−1 in equation (54), we get(55)K˜1K˜2δ=−1=m1′m2′.
By settingδ=0 in equation (54), we get(56)K˜1K˜2δ=0=m1′+β1′m2′−α2′,m1′−α1′m2′+β2′,(57)K˜1K˜2δ=0=m1′m2′−m1′α2′+β1′m2′−β1′α2′,m1′m2′+m1′β2′−α1′m2′−α1′β2′.
On combining equations (51), (53), (55), and (57), the result follows.Theorem 6.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be a nonpositive LR-type BFN andK˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be a nonnegative LR-type BFN; then,(58)K˜1⊗K˜2=m1∗m2∗,−m1∗β2∗+α1∗m2∗+α1∗β2∗,−m1∗α2∗+β1∗m2∗−β1∗α2∗LR,m1′m2′,−m1′β2′+α1′m2′+α1′β2′,−m1′α2′+β1′m2′−β1′α2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be a nonpositive LR-type BFN andK˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be a nonnegative LR-type BFN; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(59)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(60)K˜1γ×K˜2γ=m1∗−α1∗L−1γm2∗+β2∗R−1γ,m1∗+β1∗R−1γm2∗−α2∗L−1γ.
By settingγ=1 in equation (60), we get(61)K˜1K˜2γ=1=m1∗m2∗.
By settingγ=0 in equation (60), we get(62)K˜1K˜2γ=0=m1∗−α1∗m2∗+β2∗,m1∗+β1∗m2∗−α2∗,(63)K˜1K˜2γ=0=m1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗.
Now,(64)K˜1δ×K˜2δ=m1′−α1′L−1δm2′+β2′R−1δ,m1′+β1′R−1δm2′−α2′L−1δ.
By settingδ=−1 in equation (64), we get(65)K˜1K˜2δ=−1=m1′m2′.
By settingδ=0 in equation (64), we get(66)K˜1K˜2δ=0=m1′−α1′m2′+β2′,m1′+β1′m2′−α2′,(67)K˜1K˜2δ=0=m1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′.
On combining the equations (61), (63), (65), and (67), the result follows.Theorem 7.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be two nonpositive LR-type BFNs; then,(68)K˜1⊗K˜2=m1∗m2∗,−m1∗β2∗−β1∗m2∗−β1∗β2∗,−m1∗α2∗−α1∗m2∗+α1∗α2∗LR,m1′m2′,−m1′β2′−β1′m2′−β1′β2′,−m1′α2′−α1′m2′+α1′α2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be two nonpositive LR-type BFNs; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(69)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(70)K˜1γ×K˜2γ=m1∗+β1∗R−1γm2∗+β2∗R−1γ,m1∗−α1∗L−1γm2∗−α2∗L−1γ.
By settingγ=1 in equation (70), we get(71)K˜1K˜2γ=1=m1∗m2∗.
By settingγ=0 in equation (70), we get(72)K˜1K˜2γ=0=m1∗+β1∗m2∗+β2∗,m1∗−α1∗m2∗−α2∗,(73)K˜1K˜2γ=0=m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗,m1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗.
Now,(74)K˜1γ⊗K˜2δ=m1′+β1′R−1δm2′+β2′R−1δ,m1′−α1′L−1δm2′−α2′L−1δ.
By settingδ=−1 in equation (74), we get(75)K˜1K˜2δ=−1=m1′m2′.
By settingδ=0 in equation (74), we get(76)K˜1K˜2δ=0=m1′+β1′m2′+β2′,m1′−α1′m2′−α2′,(77)K˜1K˜2δ=0=m1′m2′+m1′β2′+β1′m2′+β1′β2′,m1′m2′−m1′α2′−α1′m2′+α1′α2′.
On combining the equations (191), (73), (75), and (77), the result follows.Theorem 8.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗−α1∗≥0, m1′−α1′≥0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then,(78)K˜1⊗K2˜=m1∗m2∗,m1∗m2∗−minm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗maxm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗−m1∗m2∗LRm1′m2′,m1′m2′−minm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′maxm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′−m1′m2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗−α1∗≥0, m1′−α1′≥0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(79)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(80)K˜1γ×K˜2γ=minm1∗−α1∗L−1γm2∗−α2∗L−1γ,m1∗+β1∗R−1γm2∗−α2∗L−1γmaxm1∗−α1∗L−1γm2∗+β2∗R−1γ,m1∗+β1∗R−1γm2∗+β2∗R−1γ.
By settingγ=1 in equation (80), we get(81)K˜1K˜2γ=1=m1∗m2∗.
By settingγ=0 in equation (80), we get(82)K˜1K˜2γ=0=minm1∗−α1∗m2∗−α2∗,m1∗+β1∗m2∗−α2∗maxm1∗−α1∗m2∗+β2∗,m1∗+β1∗m2∗+β2∗.(83)K˜1K˜2γ=0=minm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗,maxm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗.
Now,(84)K˜1δ×K˜2δ=minm1′−α1′L−1δm2′−α2′L−1δ,m1′+β1′R−1δm2′−α2′L−1δ,maxm1′−α1′L−1δm2′+β2′R−1δ,m1′+β1′R−1δm2′+β2′R−1δ.
By settingδ=−1 in equation (84), we get(85)K˜1K˜2δ=−1=m1′m2′.
By settingδ=0 in equation (84), we get(86)K˜1K˜2δ=0=minm1′−α1′m2′−α2′,m1′+β1′m2′−α2′,maxm1′−α1′m2′+β2′,m1′+β1′m2′+β2′.(87)K˜1K˜2δ=0=minm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′maxm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′.
On combining the equations (81), (83), (85), and (87), the result follows.Theorem 9.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗−α1∗≥0, m1′−α1′≤0, m1′+β1′≥0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then,(88)K˜1⊗K˜2=m1∗m2∗,m1∗m2∗−minm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗maxm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗−m1∗m2∗LRm1′m2′,m1′m2′−minm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′maxm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′−m1′m2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗−α1∗≥0, m1′−α1′≤0, m1′+β1′≥0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(89)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(90)K˜1γ×K˜2γ=minm1∗−α1∗L−1γm2∗−α2∗L−1γ,m1∗+β1∗R−1γm2∗−α2∗L−1γmaxm1∗−α1∗L−1γm2∗+β2∗R−1γ,m1∗+β1∗R−1γm2∗+β2∗R−1γ.
By settingγ=1 in equation (90), we get(91)K˜1K˜2γ=1=m1∗m2∗.
By settingγ=0 in equation (90), we get(92)K˜1K˜2γ=0=minm1∗−α1∗m2∗−α2∗,m1∗+β1∗m2∗−α2∗,maxm1∗−α1∗m2∗+β2∗,m1∗+β1∗m2∗+β2∗,(93)K˜1K˜2γ=0=minm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗maxm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗.
Now,(94)K˜1δ×K˜2δ=minm2′+β2′R′−1δ,m1′+β1′R−1δm2′−α2′L−1δmaxm1′−α1′L−1δm2′−α2′L−1δ,m1′+β1′R−1δm2′+β2′R−1δ.
By settingδ=−1 in equation (94), we get(95)K˜1K˜2δ=−1=m1′m2′.
By settingδ=0 in equation (94), we get(96)K˜1K˜2δ=0=minm1′−α1′m2′+β2′,m1′+β1′m2′−α2′maxm1′−α1′m2′−α2′,m1′+β1′m2′+β2′.(97)K˜1K˜2δ=0=minm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′maxm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′.
On combining the equations (91), (93), (95), and (97), the result follows.Theorem 10.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗−α1∗≥0, m1′+β1′≤0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then,(98)K˜1⊗K˜2=m1∗m2∗,m1∗m2∗−minm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗maxm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗−m1∗m2∗LRm1′m2′,m1′m2′−minm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′maxm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′−m1′m2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗−α1∗≥0, m1′−β1′≤0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(99)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(100)K˜1γ×K˜2γ=minm1∗−α1∗L−1γm2∗−α2∗L−1γ,m1∗+β1∗R−1γm2∗−α2∗L−1γ,maxm1∗−α1∗L−1γm2∗+β2∗R−1γ,m1∗+β1∗R−1γm2∗+β2∗R−1γ.
By settingγ=1 in equation (100), we get(101)K˜1K˜2γ=1=m1∗m2∗.
By settingγ=0 in equation (100), we get(102)K˜1K˜2γ=0=minm1∗−α1∗m2∗−α2∗,m1∗+β1∗m2∗−α2∗,maxm1∗−α1∗m2∗+β2∗,m1∗+β1∗m2∗+β2∗,(103)K˜1K˜2γ=0=minm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗maxm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗.
Now,(104)K˜1δ×K˜2δ=minm1′−α1′L′−1δm2′+β2′R′−1δ,m1′+β1′R′−1δm2′+β2′R′−1δ,maxm1′−α1′L′−1δm2′−α2′L′−1δ,m1′+β1′R′−1δm2′−α2′L′−1δ.
By settingδ=−1 in equation (104), we get(105)K˜1K˜2δ=−1=m1′m2′.
By settingδ=0 in equation (104), we get(106)K˜1K˜2δ=0=minm1′−α1′m2′+β2′,m1′+β1′m2′+β2′,maxm1′−α1′m2′−α2′,m1′+β1′m2′−α2′,(107)K˜1K˜2δ=0=minm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′,maxm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′.
On combining equations (101), (103), (105), and (107), the result follows.Theorem 11.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗−α1∗≤0, m1∗+β1∗≥0, m1′−α1′≥0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then,(108)K˜1⊗K˜2=m1∗m2∗,m1∗m2∗−minm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗maxm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗−m1∗m2∗LRm1′m2′,m1′m2′−minm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′maxm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′−m1′m2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗−α1∗≤0, m1∗+β1∗≥0, m1′−α1′≥0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(109)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(110)K˜1γ×K˜2γ=minm1∗−α1∗L−1γm2∗+β2∗R−1γ,m1∗+β1∗R−1γm2∗−α2∗L−1γmaxm1∗−α1∗L−1γm2∗−α2∗L−1γ,m1∗+β1∗R−1γm2∗+β2∗R−1γ.
By settingγ=1 in equation (110), we get(111)K˜1K˜2γ=1=m1∗m2∗.
By settingγ=0 in equation (110), we get(112)K˜1K˜2γ=0=minm1∗−α1∗m2∗+β2∗,m1∗+β1∗m2∗−α2∗,maxm1∗−α1∗m2∗−α2∗,m1∗+β1∗m2∗+β2∗,(113)K˜1K˜2γ=0=minm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗,maxm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗.
Now,(114)K˜1δ×K˜2δ=minm1′−α1′L′−1δm2′−α2′L′−1δ,m1′+β1′R′−1δm2′−α2′L′−1δ,maxm1′−α1′L′−1δm2′+β2′R′−1δ,m1′+β1′R′−1δm2′+β2′R′−1δ.
By settingδ=−1 in equation (114), we get(115)K˜1K˜2δ=−1=m1′m2′.
By settingδ=0 in equation (114), we get(116)K˜1K˜2δ=0=minm1′−α1′m2′−α2′,m1′+β1′m2′−α2′,maxm1′−α1′m2′+β2′,m1′+β1′m2′+β2′,(117)K˜1K˜2δ=0=minm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′maxm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′.
On combining the equations (111), (113), (115), and (117), the result follows.Theorem 12.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗−α1∗≤0, m1∗+β1∗≥0, m1′−α1′≤0, m1′+β1′≥0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then,(118)K˜1⊗K˜2=m1∗m2∗,m1∗m2∗−minm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗maxm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗−m1∗m2∗LRm1′m2′,m1′m2′−minm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′maxm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′−m1′m2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗−α1∗≤0, m1∗+β1∗≥0, m1′−α1′≤0, m1′+β1′≥0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(119)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(120)K˜1γ×K˜2γ=minm1∗−α1∗L−1γm2∗+β2∗R−1γ,m1∗+β1∗R−1γm2∗−α2∗L−1γ,maxm1∗−α1∗L−1γm2∗−α2∗L−1γ,m1∗+β1∗R−1γm2∗+β2∗R−1γ.
By settingγ=1 in equation (120), we get(121)K˜1K˜2γ=1=m1∗m2∗.
By settingγ=0 in equation (120), we get(122)K˜1γ×K˜2γ=minm1∗−α1∗m2∗+β2∗,m1∗+β1∗m2∗−α2∗maxm1∗−α1∗m2∗−α2∗,m1∗+β1∗m2∗+β2∗,(123)K˜1K˜2γ=0=minm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗maxm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗.
Now,(124)K˜1δ×K˜2δ=minm1′−α1′L−1γm2′+β2′R−1γ,m1′+β1′R−1γm2′−α2′L−1γmaxm1′−α1′L−1γm2′−α2′L−1γ,m1′+β1′R−1γm2′+β2′R−1γ.
By settingδ=−1 in equation (124), we get(125)K˜1K˜2δ=−1=m1′m2′.
By settingδ=0 in equation (124), we get(126)K˜1K˜2δ=0=minm1′−α1′m2′+β2′,m1′+β1′m2′−α2′maxm1′−α1′m2′−α2′,m1′+β1′m2′+β2′.(127)K˜1K˜2δ=0=minm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′maxm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′.
On combining the equations (121), (123), (125), and (127), the result follows.Theorem 13.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗−α1∗≤0, m1∗+β1∗≥0, m1′+β1′≤0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then,(128)K˜1⊗K˜2=m1∗m2∗,m1∗m2∗−minm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗maxm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗−m1∗m2∗LRm1′m2′,m1′m2′−minm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′maxm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′−m1′m2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗−α1∗≤0, m1∗+β1∗≥0, m1′+β1′≤0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(129)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(130)K˜1γ×K˜2γ=minm1∗−α1∗L−1γm2∗+β2∗R−1γ,m1∗+β1∗R−1γm2∗+β2∗R−1γmaxm1∗−α1∗L−1γm2∗−α2∗L−1γ,m1∗+β1∗R−1γm2∗−α2∗L−1γ.
By settingγ=1 in equation (130), we get(131)K˜1K˜2γ=1=m1∗m2∗.
By settingγ=0 in equation (130), we get(132)K˜1K˜2γ=0=minm1∗−α1∗m2∗+β2∗,m1∗+β1∗m2∗+β2∗maxm1∗−α1∗m2∗−α2∗,m1∗+β1∗m2∗−α2∗,(133)K˜1K˜2γ=0=minm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗maxm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗.
Now,(134)K˜1δ×K˜2δ=minm1′−α1′L′−1δm2′+β2′R′−1δ,m1′+β1′R′−1δm2′+β2′R′−1δmaxm1′−α1′L′−1δm2′−α2′L′−1δ,m1′+β1′R′−1δm2′−α2′L′−1δ.
By settingδ=−1 in equation (134), we get(135)K˜1K˜2δ=−1=m1′m2′.
By settingδ=0 in equation (135), we get(136)K˜1K˜2δ=0=minm1′−α1′m2′+β2′,m1′+β1′m2′+β2′maxm1′−α1′m2′−α2′,m1′+β1′m2′−α2′,(137)K˜1K˜2δ=0=minm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′maxm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′.
On combining the equations (131), (133), (135), and (137), the result follows.Theorem 14.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗+β1∗≤0, m1′−α1′≥0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then,(138)K˜1⊗K˜2=m1∗m2∗,m1∗m2∗−minm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗maxm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗−m1∗m2∗LRm1′m2′,m1′m2′−minm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′maxm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′−m1′m2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗+β1∗≤0, m1′−α1′≥0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(139)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(140)K˜1γ×K˜2γ=minm1∗−α1∗L∗−1γm2∗+β2∗R∗−1γ,m1∗+β1∗R∗−1γm2∗+β2∗R∗−1γmaxm1∗−α1∗L∗−1γm2∗−α2∗L∗−1γ,m1∗+β1∗R∗−1γm2∗−α2∗L∗−1γ.
By settingγ=1 in equation (140), we get(141)K˜1K˜2γ=1=m1∗m2∗.
By settingγ=0 in equation (140), we get(142)K˜1K˜2γ=0=minm1∗−α1∗m2∗+β2∗,m1∗+β1∗m2∗+β2∗maxm1∗−α1∗m2∗−α2∗,m1∗+β1∗m2∗−α2∗,(143)K˜1K˜2γ=0=minm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗maxm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗.
Now,(144)K˜1δ×K˜2δ=minm1′−α1′L−1δm2′−α2′L−1δ,m1′+β1′R−1δm2′−α2′L−1δmaxm1′−α1′L−1δm2′+β2′R−1δ,m1′+β1′R−1δm2′+β2′R−1δ.
By settingδ=−1 in equation (144), we get(145)K˜1K˜2δ=−1=m1′m2′.
By settingδ=0 in equation (144), we get(146)K˜1K˜2δ=0=minm1′−α1′m2′−α2′,m1′+β1′m2′−α2′maxm1′−α1′m2′+β2′,m1′+β1′m2′+β2′,(147)K˜1K˜2δ=0=minm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′maxm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′.
On combining equations (141), (143), (145), and (147), the result follows.Theorem 15.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗+β1∗≤0, m1′−α1′≤0, m1′+β1′≥0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then,(148)K˜1⊗K˜2=m1∗m2∗,m1∗m2∗−minm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗maxm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗−m1∗m2∗LRm1′m2′,m1′m2′−minm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′maxm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′−m1′m2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗+β1∗≤0, m1′−α1′≤0, m1′+β1′≥0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(149)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(150)K˜1γ×K˜2γ=minm1∗−α1∗L−1γm2∗+β2∗R−1γ,m1∗+β1∗R−1γm2∗+β2∗R−1γmaxm1∗−α1∗L−1γm2∗−α2∗L−1γ,m1∗+β1∗R−1γm2∗−α2∗L−1γ.
By settingγ=1 in equation (150), we get(151)K˜1K˜2γ=1=m1∗m2∗.
By settingγ=0 in equation (150), we get(152)K˜1K˜2γ=0=minm1∗−α1∗m2∗+β2∗,m1∗+β1∗m2∗+β2∗maxm1∗−α1∗m2∗−α2∗,m1∗+β1∗m2∗−α2∗,(153)K˜1K˜2γ=0=minm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗maxm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗.
Now,(154)K˜1δ×K˜2δ=minm1′−α1′L−1γm2′+β2′R−1γ,m1′+β1′R−1γm2′−α2′L−1γmaxm1′−α1′L−1γm2′−α2′L−1γ,m1′+β1′R−1γm2′+β2′R−1γ.
By settingδ=−1 in equation (154), we get(155)K˜1K˜2δ=−1=m1′m2′.
By settingδ=0 in equation (154), we get(156)K˜1K˜2δ=0=minm1′−α1′m2′+β2′,m1′+β1′m2′−α2′maxm1′−α1′m2′−α2′,m1′+β1′m2′+β2′,(157)K˜1K˜2δ=0=minm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′maxm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′.
On combining equations (151), (153), (155), and (157), the result follows.Theorem 16.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗+β1∗≤0, m1′+β1′≤0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then,(158)K˜1⊗K˜2=m1∗m2∗,m1∗m2∗−minm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗−m1∗β2∗+β1∗m2∗−β1∗β2∗maxm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗+β1∗α2∗−m1∗m2∗LRm1′m2′,m1′m2′−minm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′−m1′β2′+β1′m2′−β1′β2′maxm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′+m1′α2′+β1′m2′+β1′α2′−m1′m2′LR.Proof.
LetK˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR be an LR-type BFN in which m1∗+β1∗≤0, m1′+β1′≤0 and K˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be an unrestricted LR-type BFN; then, their γ−cut and δ−cut, ∀,γ∈0,1, and δ∈−1,0 are given as follows:(159)K˜1γ=m1∗−α1∗L−1γ,m1∗+β1∗R−1γ,K˜2γ=m2∗−α2∗L−1γ,m2∗+β2∗R−1γ,K˜1δ=m1′−α1′L′−1δ,m1′+β1′R′−1δ,K˜2δ=m2′−α2′L′−1δ,m2′+β2′R′−1δ.
So,(160)K˜1γ×K˜2γ=minm1∗−α1∗L∗−1γm2∗+β2∗R∗−1γ,m1∗+β1∗R∗−1γm2∗+β2∗R∗−1γmaxm1∗−α1∗L∗−1γm2∗−α2∗L∗−1γ,m1∗+β1∗R∗−1γm2∗−α2∗L∗−1γ.
By settingγ=1 in equation (160), we get(161)K˜1K˜2γ=1=m1∗m2∗.
By settingγ=0 in equation (160), we get(162)K˜1K˜2γ=0=minm1∗−α1∗m2∗+β2∗,m1∗+β1∗m2∗+β2∗maxm1∗−α1∗m2∗−α2∗,m1∗+β1∗m2∗−α2∗,(163)K˜1K˜2γ=0=minm1∗m2∗+m1∗β2∗−α1∗m2∗−α1∗β2∗,m1∗m2∗+m1∗β2∗+β1∗m2∗+β1∗β2∗maxm1∗m2∗−m1∗α2∗−α1∗m2∗+α1∗α2∗,m1∗m2∗−m1∗α2∗+β1∗m2∗−β1∗α2∗.
Now,(164)K˜1δ×K˜2δ=minm1′−α1′L′−1δm2′+β2′R′−1δ,m1′+β1′R′−1δm2′+β2′R′−1δmaxm1′−α1′L′−1δm2′−α2′L′−1δ,m1′+β1′R′−1δm2′−α2′L′−1δ.
By settingδ=−1 in equation (164), we get(165)K˜1K˜2δ=−1=m1′m2′.
By settingδ=0 in equation (164), we get(166)K˜1K˜2δ=0=minm1′−α1′m2′+β2′,m1′+β1′m2′+β2′maxm1′−α1′m2′−α2′,m1′+β1′m2′−α2′,(167)K˜1K˜2δ=0=minm1′m2′+m1′β2′−α1′m2′−α1′β2′,m1′m2′+m1′β2′+β1′m2′+β1′β2′,maxm1′m2′−m1′α2′−α1′m2′+α1′α2′,m1′m2′−m1′α2′+β1′m2′−β1′α2′.
On combining equations (161), (163), (165), and (167), the result follows.


## 4. LR-Type FBFLPP

In this section, we study about theLR-type fully bipolar fuzzy linear programming problem with equality constraints in which all the variables are represented by LR-type BFNs. Consider an LR-type FBFLPP with LR-type BFNs(168)Maximize/Minimize∑j=1nB˜j⊗Y˜j,subject to(169)∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m,where Y˜j, B˜j,E˜i, and F˜ij are LR-type BFNs.Definition 12.
A bipolar fuzzy optimal solution ofLR-type FBFLPP (168) will be LR-type BFN Y˜j if(1)
Y˜j are LR-type BFNs(2)
∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m(3)
If there exists anyLR-type BFN Y˜j satisfying the constraints, thenIn case of a maximization problem,ℜ∑j=1nB˜j⊗Y˜j>ℜ∑j=1nB˜j⊗Y˜jIn case of a minimization problem,ℜ∑j=1nB˜j⊗Y˜j<ℜ∑j=1nB˜j⊗Y˜j′


### 4.1. Methodology

(170)Maximize/Minimize∑j=1nB˜j⊗Y˜j,subject to(171)∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m,where Y˜j,E˜i,F˜ij, and B˜jLR-type BFNs.Step 1.
assume that(172)F˜ij=fij∗,ϵij∗,ηij∗LR,fij′,ϵij′,ηij′LR,E˜i=ei∗,ϕi∗,ϑi∗LR,ei′,ϕi′,ϑi′LR,B˜j=bj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR,Y˜j=mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR.
TheLR-type FBFLPP (170) can be transformed into the following problem:(173)Maximize/Minimize∑j=1nbj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR,subject to(174)∑j=1nfij∗,ϵij∗,ηij∗LR,fij′,ϵij′,ηij′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR,=ei∗,ϕi∗,ϑi∗LR,ei′,ϕi′,ϑi′LR,where mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR is an LR-type BFN, ∀j=1,2,3,…,n.Step 2.
By using the product ofLR-type BFNs given in Section 3, we suppose that(175)fij∗,ϵij∗,ηij∗LR,fij′,ϵij′,ηij′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR=tij∗∗,χij∗∗,σij∗∗LR,mij″,χij″,σij″LR.
TheLR-type FBFLPP (173) transforms to a problem as follows:(176)Maximize/Minimize∑j=1nbj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR,subject to(177)tij∗∗,χij∗∗,σij∗∗LR,mij″,χij″,σij″LR=ei∗,ϕi∗,ϑi∗LR,ei′,ϕi′,ϑi′LR,where mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR is an LR-type BFN, ∀j=1,2,3,…,n.Step 3.
By using arithmetic operations, given in Section3, the LR-type FBFLPP (176) transforms into a problem as follows:(178)Maximize/Minimize∑j=1nbj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR,subject to(179)∑j=1ntij∗∗=ei∗,∑j=1ntij″=ei′,∑j=1nχij∗∗=ϕi∗,∑j=1nχij″=ϕi′,∑j=1nσij∗∗=ϑi∗,∑j=1nσij″=ϑi′,∀i=1,2,3,…,m(180)αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,∀j=1,2,3,…,n.Step 4.
By using ranking function, theLR-type FBFLPP converts into a crisp mathematical problem as follows:(181)Maximize/Minimizeℜ∑j=1nbj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR,subject to(182)∑j=1ntij∗∗=ei∗,∑j=1ntij″=ei′,∑j=1nχij∗∗=ϕi∗,∑j=1nχij″=ϕi′,∑j=1nσij∗∗=ϑi∗,∑j=1nσij″=ϑi′,∀i=1,2,3,…,m(183)αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,∀j=1,2,3,…,n.Step 5.
Considering(184)bj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR=uj∗∗,θj∗∗,ςj∗∗LR,uj″,θj″,ςj″LR,the crisp mathematical problem (181) can be transformed into the following problem:(185)Maximize/Minimizeℜ∑j=1nuj∗∗,θj∗∗,ςj∗∗LR,uj″,θj″,ςj″LR,subject to(186)∑j=1ntij∗∗=ei∗,∑j=1ntij″=ei′,∑j=1nχij∗∗=ϕi∗,∑j=1nχij″=ϕi′,∑j=1nσij∗∗=ϑi∗,∑j=1nσij″=ϑi′,∀i=1,2,3,…,m(187)αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,∀j=1,2,3,…,n.Step 6.
By applying the linearity propertyℜ∑j=1nH˜i=∑j=1nℜH˜i, here let H˜i be a BFN, the crisp mathematical problem (185) can be transformed into the following problem:(188)Maximize/Minimize∑j=1nℜuj∗∗,θj∗∗,ςj∗∗LR,uj″,θj″,ςj″LR,subject to(189)∑j=1ntij∗∗=ei∗,∑j=1ntij″=ei′,∑j=1nχij∗∗=ϕi∗,∑j=1nχij″=ϕi′,∑j=1nσij∗∗=ϑi∗,∑j=1nσij″=ϑi′,∀i=1,2,3,…,m(190)αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,∀j=1,2,3,…,n.Step 7.
By applying ranking forLR-type BFNs (2.11), the crisp mathematical problem (188) becomes(191)Maximize/Minimize14∫01uj∗∗−θj∗∗L−1γdγ+∫01uj∗∗+ςj∗∗R−1γdγ+14∫−10uj″−θj″L′−1δdδ+∫−10uj″+ςj″R′−1δdδ,subject to(192)∑j=1ntij∗∗=ei∗,∑j=1ntij″=ei′,∑j=1nχij∗∗=ϕi∗,∑j=1nχij″=ϕi′,∑j=1nσij∗∗=ϑi∗,∑j=1nσij″=ϑi′,∀i=1,2,3,…,m(193)αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,∀j=1,2,3,…,n.Step 8.
By solving the crisp mathematical problem (190), we get the optimal solution mj•,αj•,βj•,mj•′,αj•′,βj•′, ∀j=1,2,3,…,n.Step 9.
find the exactLR-type bipolar fuzzy optimal solution Y˜j• of LR-type FBFLPP by assigning the values of mj•,αj•,βj•,mj•′,αj•′, and βj•′ in Y˜j•=mj•,αj•,βj•LR,mj•′,αj•′,βj•′LR. ∀j=1,2,3,…,n.Step 10.
find theLR-type bipolar fuzzy optimal value by putting the values of Y˜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 17.
The solution ofLR-type FBFLPP.
Maximize/Minimize(194)Z=∑j=1nBj⊗Yj,subject to,∑j=1nFij⊗Yj=Ei,∀i=1,…,m.
Bj,Yj,Fij, and Ei are LR-type BFNs, which exist with the solution of the associated crisp mathematical problem.
Maximize/Minimize(195)ℜZ=ℜ∑j=1nbj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR,subject to(196)∑j=1npij=ei∗,∑j=1nqij=ϕi∗,∑j=1nrij=ϑi∗,∑j=1nsij=ei′,∑j=1ntij=ϕi′,∑j=1nuij=ϑi′αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,∀i=1,…,m exists. Otherwise, there is no guarantee that the LR-type bipolar fuzzy optimal solution exists.Proof.
The proof is straightforward.


## 4.1. Methodology

(170)Maximize/Minimize∑j=1nB˜j⊗Y˜j,subject to(171)∑j=1nF˜ij⊗Y˜j=E˜i,∀i=1,2,3,…,m,where Y˜j,E˜i,F˜ij, and B˜jLR-type BFNs.Step 1.
assume that(172)F˜ij=fij∗,ϵij∗,ηij∗LR,fij′,ϵij′,ηij′LR,E˜i=ei∗,ϕi∗,ϑi∗LR,ei′,ϕi′,ϑi′LR,B˜j=bj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR,Y˜j=mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR.
TheLR-type FBFLPP (170) can be transformed into the following problem:(173)Maximize/Minimize∑j=1nbj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR,subject to(174)∑j=1nfij∗,ϵij∗,ηij∗LR,fij′,ϵij′,ηij′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR,=ei∗,ϕi∗,ϑi∗LR,ei′,ϕi′,ϑi′LR,where mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR is an LR-type BFN, ∀j=1,2,3,…,n.Step 2.
By using the product ofLR-type BFNs given in Section 3, we suppose that(175)fij∗,ϵij∗,ηij∗LR,fij′,ϵij′,ηij′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR=tij∗∗,χij∗∗,σij∗∗LR,mij″,χij″,σij″LR.
TheLR-type FBFLPP (173) transforms to a problem as follows:(176)Maximize/Minimize∑j=1nbj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR,subject to(177)tij∗∗,χij∗∗,σij∗∗LR,mij″,χij″,σij″LR=ei∗,ϕi∗,ϑi∗LR,ei′,ϕi′,ϑi′LR,where mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR is an LR-type BFN, ∀j=1,2,3,…,n.Step 3.
By using arithmetic operations, given in Section3, the LR-type FBFLPP (176) transforms into a problem as follows:(178)Maximize/Minimize∑j=1nbj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR,subject to(179)∑j=1ntij∗∗=ei∗,∑j=1ntij″=ei′,∑j=1nχij∗∗=ϕi∗,∑j=1nχij″=ϕi′,∑j=1nσij∗∗=ϑi∗,∑j=1nσij″=ϑi′,∀i=1,2,3,…,m(180)αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,∀j=1,2,3,…,n.Step 4.
By using ranking function, theLR-type FBFLPP converts into a crisp mathematical problem as follows:(181)Maximize/Minimizeℜ∑j=1nbj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR,subject to(182)∑j=1ntij∗∗=ei∗,∑j=1ntij″=ei′,∑j=1nχij∗∗=ϕi∗,∑j=1nχij″=ϕi′,∑j=1nσij∗∗=ϑi∗,∑j=1nσij″=ϑi′,∀i=1,2,3,…,m(183)αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,∀j=1,2,3,…,n.Step 5.
Considering(184)bj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR=uj∗∗,θj∗∗,ςj∗∗LR,uj″,θj″,ςj″LR,the crisp mathematical problem (181) can be transformed into the following problem:(185)Maximize/Minimizeℜ∑j=1nuj∗∗,θj∗∗,ςj∗∗LR,uj″,θj″,ςj″LR,subject to(186)∑j=1ntij∗∗=ei∗,∑j=1ntij″=ei′,∑j=1nχij∗∗=ϕi∗,∑j=1nχij″=ϕi′,∑j=1nσij∗∗=ϑi∗,∑j=1nσij″=ϑi′,∀i=1,2,3,…,m(187)αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,∀j=1,2,3,…,n.Step 6.
By applying the linearity propertyℜ∑j=1nH˜i=∑j=1nℜH˜i, here let H˜i be a BFN, the crisp mathematical problem (185) can be transformed into the following problem:(188)Maximize/Minimize∑j=1nℜuj∗∗,θj∗∗,ςj∗∗LR,uj″,θj″,ςj″LR,subject to(189)∑j=1ntij∗∗=ei∗,∑j=1ntij″=ei′,∑j=1nχij∗∗=ϕi∗,∑j=1nχij″=ϕi′,∑j=1nσij∗∗=ϑi∗,∑j=1nσij″=ϑi′,∀i=1,2,3,…,m(190)αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,∀j=1,2,3,…,n.Step 7.
By applying ranking forLR-type BFNs (2.11), the crisp mathematical problem (188) becomes(191)Maximize/Minimize14∫01uj∗∗−θj∗∗L−1γdγ+∫01uj∗∗+ςj∗∗R−1γdγ+14∫−10uj″−θj″L′−1δdδ+∫−10uj″+ςj″R′−1δdδ,subject to(192)∑j=1ntij∗∗=ei∗,∑j=1ntij″=ei′,∑j=1nχij∗∗=ϕi∗,∑j=1nχij″=ϕi′,∑j=1nσij∗∗=ϑi∗,∑j=1nσij″=ϑi′,∀i=1,2,3,…,m(193)αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,∀j=1,2,3,…,n.Step 8.
By solving the crisp mathematical problem (190), we get the optimal solution mj•,αj•,βj•,mj•′,αj•′,βj•′, ∀j=1,2,3,…,n.Step 9.
find the exactLR-type bipolar fuzzy optimal solution Y˜j• of LR-type FBFLPP by assigning the values of mj•,αj•,βj•,mj•′,αj•′, and βj•′ in Y˜j•=mj•,αj•,βj•LR,mj•′,αj•′,βj•′LR. ∀j=1,2,3,…,n.Step 10.
find theLR-type bipolar fuzzy optimal value by putting the values of Y˜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 17.
The solution ofLR-type FBFLPP.
Maximize/Minimize(194)Z=∑j=1nBj⊗Yj,subject to,∑j=1nFij⊗Yj=Ei,∀i=1,…,m.
Bj,Yj,Fij, and Ei are LR-type BFNs, which exist with the solution of the associated crisp mathematical problem.
Maximize/Minimize(195)ℜZ=ℜ∑j=1nbj∗,ζj∗,ψj∗LR,bj′,ζj′,ψj′LR⊗mj∗,αj∗,βj∗LR,mj′,αj′,βj′LR,subject to(196)∑j=1npij=ei∗,∑j=1nqij=ϕi∗,∑j=1nrij=ϑi∗,∑j=1nsij=ei′,∑j=1ntij=ϕi′,∑j=1nuij=ϑi′αj∗≥0,βj∗≥0,αj′≥0,βj′≥0,∀i=1,…,m exists. Otherwise, there is no guarantee that the LR-type bipolar fuzzy optimal solution exists.Proof.
The proof is straightforward.


## 5. Numerical Examples

In this section, the methodology presented in Section4 is illustrated by solving a numerical example and model.Example 1.
(197)Maximize9,6,3LR,8,5,1LR⊗Y˜1⊕6,3,4LR,5,4,2LR⊗Y˜2,subject to(198)9,5,3LR,4,1,5LR⊗Y˜1⊕8,5,2LR,4,2,3LR⊗Y˜2=94,68,86LR,44,30,100LR,6,2,5LR,6,3,5LR⊗Y˜1⊕8,6,4LR,9,2,1LR⊗Y˜2=76,52,106LR,78,59,111LR,where Y1 and Y2 are LR-type BFNs and Lx=Rx=max0,1−x,L′x=R′x=min0,x−1.Step 1:LetY1˜=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR and Y2˜=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be LR-type BFNs; then,(199)Maximize9,6,3LR,8,5,1LR⊗m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR⊕6,3,4LR,5,4,2LR⊗m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR,subject to(200)9,5,3LR,4,1,5LR⊗m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR⊕8,5,2LR,4,2,3LR⊗m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR=94,68,86LR,44,30,100LR,6,2,5LR,6,3,5LR⊗m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR⊕8,6,4LR,9,2,1LR⊗m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR=76,52,106LR,78,59,111LR,whereY1˜=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR and Y˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR are LR-type BFNsStep 2: by using the product ofLR-type BFNs given in Section 3, the LR-type FBFLPP converts asMaximize(201)9m1∗,9α1∗+6m1∗−6α1∗,9β1∗+3m1∗+3β1∗LR,8m1′,8α1′+5m1′−5α1′,8β1′+1m1′+1β1′LR⊕6m2∗,6α2∗+3m2∗−3α2∗,6β2∗+4m2∗+4β2∗LR,5m2′,5α2′+4m2′−4α2′,5β2′+2m2′+2β2′LR,subject to(202)9m1∗,9α1∗+5m1∗−5α1∗,9β1∗+3m1∗+3β1∗LR,4m1′,4α1′+1m1′−1α1′,4β1′+5m1′+5β1′LR⊕8m2∗,8α2∗+5m2∗−5α2∗,8β2∗+2m2∗+2β2∗LR,4m2′,4α2′+2m2′−2α2′,4β2′+3m2′+3β2′LR=94,68,86LR,44,30,100LR,6m1∗,6α1∗+2m1∗−2α1∗,6β1∗+5m1∗+5β1∗LR,6m1′,6α1′+3m1′−3α1′,6β1′+5m1′+5β1′LR⊕8m2∗,8α2∗+6m2∗−6α2∗,8β2∗+4m2∗+4β2∗LR,9m2′,9α2′+2m2′−2α2′,9β2′+1m2′+1β2′LR=76,52,106LR,78,59,111LR,α1∗≥0,β1∗≥0,α1′≥0,β1′≥0,α2∗≥0,β2∗≥0,α2′≥0,β2′≥0Step 3: using arithmetic operations (12), the LR-type FBFLPP converts asMaximize(203)9m1∗+6m2∗,3α1∗+6m1∗+3α2∗+3m2∗,12β1∗+3m1∗+10β2∗+4m2∗LR8m1′+5m2′,3α1′+5m1′+1α2′+4m2′,9β1′+1m1′+7β2′+2m2′LR,subject to(204)9m1∗+8m2∗,4α1∗+5m1∗+3α2∗+5m2∗,12β1∗+3m1∗+10β2∗+2m2∗LR4m1′+4m2′,3α1′+1m1′+2α2′+2m2′,9β1′+5m1′+7β2′+3m2′LR,=94,68,86LR,44,30,100LR,6m1∗+8m2∗,4α1∗+2m1∗+2α2∗+6m2∗,11β1∗+5m1∗+12β2∗+4m2∗LR6m1′+9m2′,3α1′+3m1′+7α2′+2m2′,11β1′+5m1′+10β2′+1m2′LR=76,52,106LR,78,59,111LR,α1∗≥0,β1∗≥0,α1′≥0,β1′≥0,α2∗≥0,β2∗≥0,α2′≥0,β2′≥0Step 4: by applying ranking function and using (2.8), theLR-type FBFLPP converts asMaximize(205)ℜ9m1∗+6m2∗,3α1∗+6m1∗+3α2∗+3m2∗,12β1∗+3m1∗+10β2∗+4m2∗LR8m1′+5m2′,3α1′+5m1′+1α2′+4m2′,9β1′+1m1′+7β2′+2m2′LR9m1∗+8m2∗=94,4α1∗+5m1∗+3α2∗+5m2∗=68,12β1∗+3m1∗+10β2∗+2m2∗=86,4m1′+4m2′=44,3α1′+1m1′+2α2′+2m2′=30,9β1′+5m1′+7β2′+3m2′=100,6m1∗+8m2∗=76,4α1∗+2m1∗+2α2∗+6m2∗=52,11β1∗+5m1∗+12β2∗+4m2∗=106,6m1′+9m2′=78,3α1′+3m1′+7α2′+2m2′=59,11β1′+5m1′+10β2′+1m2′=111,α1∗≥0,β1∗≥0,α1′≥0,β1′≥0,α2∗≥0,β2∗≥0,α2′≥0,β2′≥0Step 5: using ranking (2.11) forLR-type BFNs, the problem converts to crisp LPP asMaximize(206)338m1∗+258m2∗−38α1∗−38α2∗+64β1∗+54β2∗+144m1′+94m2′−38α1′−18α2′+98β1′+78β2′,subject to(207)9m1∗+8m2∗=94,4α1∗+5m1∗+3α2∗+5m2∗=68,12β1∗+3m1∗+10β2∗+2m2∗=86,4m1′+4m2′=44,3α1′+1m1′+2α2′+2m2′=30,9β1′+5m1′+7β2′+3m2′=100,6m1∗+8m2∗=76,4α1∗+2m1∗+2α2∗+6m2∗=52,11β1∗+5m1∗+12β2∗+4m2∗=106,6m1′+9m2′=78,3α1′+3m1′+7α2′+2m2′=59,11β1′+5m1′+10β2′+1m2′=111,α1∗≥0,β1∗≥0,α1′≥0,β1′≥0,α2∗≥0,β2∗≥0,α2′≥0,β2′≥0Step 6: by using software Maple, an optimal solution of the crisp LPP ism1∗=6,α1∗=1,β1∗=4,m2∗=5,α2∗=3,β2∗=1,m1′=7,α1′=3,β1′=2,m2′=4,α2′=3,β2′=5Step 7: the exactLR-type BFOS is Y˜1=6,1,4,7,3,2,Y˜2=5,3,1,4,3,5Step 8: the bipolar fuzzy optimal value ofLR-type FBFLPP is 84,63,96,76,63,68Example 2.
(Fitness Problem). A chef purchases beef and mutton for the players of a football team. Both types of meat possess plenty of proteins and minerals. The cost of beef and mutton per kilograms is Rs.12,7,5LR,11,9,3LR and Rs. 10,5,2LR,14,8,2LR, respectively. Each player must have to take 101,86,151LR,128,112,55LR grams of proteins and −39,135,141LR,−10,116,74LR grams of minerals daily to maintain the physical fitness standards, and further details are presented in Table 2. How many units of beef and mutton should be used to fulfill the demand of each player at the minimum cost?
We apply the proposed method to solve this problem.
LetY1 and Y2 units be taken of beef and mutton; then, the given problem converts to an LR-type FBFLPP as(208)Minimize12,7,5LR,11,9,3LR⊗Y˜1⊕10,5,2LR,14,8,2LR⊗Y˜2,subject to(209)7,6,5LR,8,4,1LR⊗Y˜1⊕6,2,3LR,8,7,2LR⊗Y˜2=101,86,151LR,128,112,55LR,12,6,2LR,10,8,2LR⊗Y˜1⊕−9,3,1LR,−7,4,2LR⊗Y˜2=−39,135,141LR,−10,116,74LR,where Y1 and Y2 are LR-type BFNs and Lx=Rx=max0,1−x,L′x=R′x=min0,x−1.Step 1: letY˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR and Y˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR be LR-type BFNs; then,(210)Minimize12,7,5LR,11,9,3LR⊗m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR⊕10,5,2LR,14,8,2LR⊗m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR,subject to(211)7,6,5LR,8,4,1LR⊗m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR⊕6,2,3LR,8,7,2LR⊗m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR=101,86,151LR,128,112,55LR,12,6,2LR,10,8,2LR⊗m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR⊕−9,3,1LR,−7,4,2LR⊗m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR=−39,135,141LR,−10,116,74LR,whereY˜1=m1∗,α1∗,β1∗LR,m1′,α1′,β1′LR and Y˜2=m2∗,α2∗,β2∗LR,m2′,α2′,β2′LR are LR-type BFNsStep 2: by using the product ofLR-type BFNs given in Section 3, LR-type FBFLPP converts asMinimize(212)12m1∗,12m1∗−min12m1∗−12α1∗−7m1∗+7α1∗,12m1∗−12α1∗+5m1∗−7α1∗,max12m1∗+12β1∗+5m1∗+5β1∗,12m1∗+12β1∗−7m1∗−7β1∗−12m1∗LR11m1′,11m1′−min11m1′−11α1′−9m1′+9α1′,11m1′−11α1′+3m1′−3α1′max11m1′+11β1′+3m1′+3β1′,11m1′+11β1′−9m1′−9β1′−11m1′LR⊕10m2∗,10m2∗−min10m2∗−10α2∗−5m2∗+5α2∗,10m2∗−10α2∗+2m2∗−2α2∗max10m2∗+10β2∗+2m2∗+2β2∗,10m2∗+10β2∗−5m2∗−5β2∗−10m2∗LR14m2′,14m2′−min14m2′−14α2′−8m2′+8α2′,14m2′−14α2′+2m2′−2α2′max14m2′+14β2′+2m2′+2β2′,14m2′+14β2′−8m2′−8β2′−14m2′LR,subject to(213)7m1∗,7m1∗−min7m1∗−7α1∗−6m1∗+6α1∗,7m1∗−7α1∗+5m1∗−5α1∗,max7m1∗+7β1∗+5m1∗+5β1∗,7m1∗+7β1∗−6m1∗−6β1∗−7m1∗LR8m1′,8m1′−min8m1′−8α1′−4m1′+4α1′,8m1′−8α1′+1m1′−1α1′,max8m1′+8β1′+1m1′+1β1′,8m1′+8β1′−4m1′−4β1′−8m1′LR⊕6m2∗,6m2∗−min6m2∗−6α2∗−2m2∗+2α2∗,6m2∗−6α2∗+3m2∗−3α2∗,max6m2∗+6β2∗+3m2∗+3β2∗,6m2∗+6β2∗−2m2∗−2β2∗−6m2∗LR,8m2′,8m2′−min8m2′−8α2′−7m2′+7α2′,8m2′−8α2′+2m2′−2α2′,max8m2′+8β2′+2m2′+2β2′,8m2′+8β2′−7m2′−7β2′−8m2′LR12m1∗,12m1∗−min12m1∗−12α1∗−6m1∗+6α1∗,12m1∗−12α1∗+2m1∗−2α1∗max12m1∗+12β1∗+2m1∗+2β1∗,12m1∗+12β1∗−6m1∗−6β1∗−12m1∗LR10m1′,10m1′−min10m1′−10α1′−8m1′+8α1′,10m1′−10α1′+2m1′−2α1′max10m1′+10β1′+2m1′+2β1′,10m1′+10β1′−8m1′−8β1′−10m1′LR⊕−9m2∗,9β2∗+3m2∗+3β2∗,9α2∗+1m2∗−1α2∗LR−7m2′,7β2′+4m2′+4β2′,7α2′+2m2′−2α2′LR=−39,135,141LR,−10,116,74LR,α1∗≥0,β1∗≥0,α1′≥0,β1′≥0,α2∗≥0,β2∗≥0,α2′≥0,β2′≥0Step 3: using arithmetic operations (12), the LR-type FBFLPP converts asMinimize(214)12m1∗+10m2∗,12m1∗−min5m1∗−5α1∗,17m1∗−17α1∗+10m2∗−min5m2∗−5α2∗,12m2∗−12α2∗max17m1∗+17β1∗,5m1∗+5β1∗−12m1∗+max12m2∗+12β2∗,5m2∗+5β2∗−10m2∗LR11m1′+14m2′,11m1′−min2m1′−2α1′,14m1′−14α1′+14m2′−min6m2′−6α2′,16m2′−16α2′max14m1′+14β1′,2m1′+2β1′−11m1′+max16m2′+16β2′,6m2′+6β2′−14m2′LR,subject to(215)7m1∗+6m2∗,7m1∗−min1m1∗−1α1∗,12m1∗−12α1∗+6m2∗−min4m2∗−4α2∗,9m2∗−9α2∗max12m1∗+12β1∗,1m1∗+1β1∗−7m1∗+max9m2∗+9β2∗,4m2∗+4β2∗−6m2∗LR8m1′+8m2′,8m1′−min4m1′−4α1′,9m1′−9α1′+8m2′−min1m2′−1α2′,10m2′−10α2′,max9m1′+9β1′,4m1′+4β1′−8m1′+max10m2′+10β2′,1m2′+1β2′−8m2′LR=101,86,151LR,128,112,55LR,12m1∗−9m2∗,12m1∗−min6m1∗−6α1∗,14m1∗−14α1∗+12β2∗+3m2∗,max14m1∗+14β1∗,6m1∗+6β1∗−12m1∗+8α2∗+1m2∗LR10m1′−7m2′,10m1′−min2m1′−2α1′,12m1′−12α1′+11β2′+4m2′max12m1′+12β1′,2m1′+2β1′−10m1′+5α2′+2m2′LR=−39,135,141LR,−10,116,74LR,α1∗≥0,β1∗≥0,α1′≥0,β1′≥0,α2∗≥0,β2∗≥0,α2′≥0,β2′≥0Step 4: by applying ranking function and usingminp,q=p+q/2−p−q/2 and maxp,q=p+q/2+p−q/2, the LR-type FBFLPP converts asMinimize(216)ℜ12m1∗+10m2∗,12m1∗−22m1∗−22α1∗2+−12m1∗+12α1∗2+10m2∗−17m2∗−17α2∗2+−7m2∗+7α2∗2,22m1∗+22β1∗2+12m1∗+12β1∗2−12m1∗+17m2∗+17β2∗2+7m2∗+7β2∗2−10m2∗LR,11m1′+14m2′,11m1′−16m1′−16α1′2+−12m1′+12α1′2+14m2′−22m2′−22α2′2+−10m2′+10α2′2,16m1′+12β1′2+12m1′+16β1′2−11m1′+22m2′+22β2′2+10m2′+10β2′2−14m2′LR,subject to(217)7m1∗+6m2∗,7m1∗−13m1∗−13α1∗2−−11m1∗+11α1∗2+6m2∗−13m2∗−13α2∗2−−5m2∗+5α2∗2,13m1∗+13β1∗2+11m1∗+11β1∗2−7m1∗+13m2∗+13β2∗2+5m2∗+5β2∗2−6m2∗LR8m1′+8m2′,8m1′−13m1′−13α1′2−−5m1′+5α1′2+8m2′−11m2′−11α2′2−−9m2′+9α2′2,13m1′+13β1′2+5m1′+5β1′2−8m1′+11m2′+11β2′2+9m2′+9β2′2−8m2′LR=101,86,151LR,128,112,55LR,12m1∗−9m2∗,12m1∗−20m1∗−20α1∗2−−8m1∗+8α1∗2+12β2∗+3m2∗20m1∗+20β1∗2+8m1∗+8β1∗2−12m1∗+8α2∗+1m2∗LR10m1′−7m2′,10m1′−14m1′−14α1′2−−10m1′+10α1′2+11β2′+4m2′,14m1′+14β1′2+10m1′+10β1′2−10m1′+5α2′+2m2′LR=−39,135,141LR,−10,116,74LR,α1∗≥0,β1∗≥0,α1′≥0,β1′≥0,α2∗≥0,β2∗≥0,α2′≥0,β2′≥0Step 5: using ranking (2.11) forLR-type BFNs, the LR-type FBFLPP converts to a crisp non-LPP asMinimize(218)234m1∗+378m2∗−58α1∗−58α2∗+178β1∗+32β2∗+194m1′+254m2′−14α1′−34α2′+74β1′+2β2′,subject to(219)7m1∗+6m2∗=101,7m1∗−13m1∗−13α1∗2−−11m1∗+11α1∗2+6m2∗−13m2∗−13α2∗2−−5m2∗+5α2∗2=86,13m1∗+13β1∗2+11m1∗+11β1∗2−7m1∗+13m2∗+13β2∗2+5m2∗+5β2∗2−6m2∗=151,8m1′+8m2′=128,8m1′−13m1′−13α1′2−−5m1′+5α1′2+8m2′−11m2′−11α2′2−−9m2′+9α2′2=112,13m1′+13β1′2+5m1′+5β1′2−8m1′+11m2′+11β2′2+9m2′+9β2′2−8m2′=55,12m1∗−9m2∗=−39,12m1∗−20m1∗−20α1∗2−−8m1∗+8α1∗2+12β2∗+3m2∗=135,20m1∗+20β1∗2+8m1∗+8β1∗2−12m1∗+8α2∗+1m2∗=141,10m1′−7m2′=−10,10m1′−14m1′−14α1′2−−10m1′+10α1′2+11β2′+4m2′=116,14m1′+14β1′2+10m1′+10β1′2−10m1′+5α2′+2m2′=74,α1∗≥0,β1∗≥0,α1′≥0,β1′≥0,α2∗≥0,β2∗≥0,α2′≥0,β2′≥0Step 6: minimize(220)234m1∗+378m2∗−58α1∗−58α2∗+178β1∗+32β2∗+194m1′+254m2′−14α1′−34α2′+74β1′+2β2′,subject to(221)7m1∗+6m2∗=101,12m1∗−12m2∗+132α1∗+132α2∗+−11m1∗+11α1∗2+−5m2∗+5α2∗2=86,−12m1∗+12m2∗+132β1∗+132β2∗+11m1∗+11β1∗2+5m2∗+5β2∗2=151,8m1′+8m2′=128,32m1′+52m2′+132α1′+112α2′+−5m1′+5α1′2+−9m2′+9α2′2=112,−32m1′−52m2′+132β1′+112β2′+5m1′+5β1′2+9m2′+9β2′2=55,12m1∗−9m2∗=−39,2m1∗+3m2∗+10α1∗+12β2∗+−8m1∗+8α1∗2=135,−2m1∗+1m2∗+8α2∗+10β1∗+8m1∗+8β1∗2=141,10m1′−7m2′=−10,3m1′+4m2′+7α1′+11β2′+−10m1′+10α1′2=116,−3m1′+2m2′+5α2′+7β1′+10m1′+10β1′2=74,α1∗≥0,β1∗≥0,α1′≥0,β1′≥0,α2∗≥0,β2∗≥0,α2′≥0,β2′≥0Step 7: by using software Maple, Optimal solution of the crisp non-LPP ism1∗=5,α1∗=2,β1∗=4,m2∗=11,α2∗=8,β2∗=5,m1′=6,α1′=3,β1′=1,m2′=10,α2′=6,β2′=2Step 8: the exactLR-type BFOS is Y˜1=5,2,4,6,3,1,Y˜2=11,8,5,10,6,2Step 9: the bipolar fuzzy optimal value of theLR-type fully bipolar fuzzy linear programming problem is 170,140,175,206,176,84
Thus,5,2,4,6,3,1 units of mutton and 11,8,5,10,6,2 units of beef should be given to each player at a minimum cost of Rs. 170,140,175,206,176,84.Table 2
Fitness problem.
IngredientsMuttonBeefMaximum availability (grams)Protein (grams)7,6,5LR,8,4,1LR6,2,3LR,8,7,2LR101,86,151LR,128,112,55LRMinerals (grams)12,6,2LR,10,8,2LR−9,3,1LR,−7,4,2LR−39,135,141LR,−10,116,74LR


## 6. Conclusions

Linear programming is applied to optimize an objective function subject to constraints. It has vast application in many fields such as science, marketing, industry, business, agriculture, and telecommunication. In this research article, we have definedLR-type BFNs and their arithmetic operations, and particularly by considering different cases, multiplication of LR-type BFNs is discussed. We have introduced the ranking for LR-type BFNs that transform the LR-type FBFLPP into a crisp linear programming problem. The proposed technique is applied to solve LR-type FBFLPPs with equality constraints involving variables and parameters as LR-type BFNs. The given method has been interpreted with a numerical example and a model. The obtained optimal solution satisfies all the constraints of the LR-type FBFLPP and justifies that the proposed scheme is accurately designed. In future, our scheme can be extended to a complex bipolar fuzzy LPP.



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*Source: 1199336-2021-07-02.xml*