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DOI: 10.32604/cmes.2021.017222


A New Method to Evaluate Linear Programming Problem in Bipolar Single-Valued Neutrosophic Environment

Jamil Ahmed1, Majed G. Alharbi2, Muhammad Akram3,* and Shahida Bashir1

1Department of Mathematics, University of Gujrat, Gujrat, Pakistan
2Department of Mathematics, College of Arts and Sciences, Methnab, Qassim University, Buraydah, Saudi Arabia
3Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan
*Corresponding Author: Muhammad Akram. Email: m.akram@pucit.edu.pk
Received: ; Accepted:

Abstract: A bipolar single-valued neutrosophic set can deal with the hesitation relevant to the information of any decision making problem in real life scenarios, where bipolar fuzzy sets may fail to handle those hesitation problems. In this study, we first develop a new method for solving linear programming problems based on bipolar single-valued neutrosophic sets. Further, we apply the score function to transform bipolar single-valued neutrosophic problems into crisp linear programming problems. Moreover, we apply the proposed technique to solve fully bipolar single-valued neutrosophic linear programming problems with non-negative triangular bipolar single-valued neutrosophic numbers (TBSvNNs) and non-negative trapezoidal bipolar single-valued neutrosophic numbers (TrBSvNNs).

Keywords: Bipolar single-valued neutrosophic numbers; score function; trapezoidal numbers; linear programming

1  Introduction

The origin of linear programming is the 1940s (World War II). Linear programming (LP) has a linear objective function and a group of linear equalities and inequalities. LP was first used in petroleum manufacturing. Well elaborated data with much information is used for LP problems. On the other hand in real life problems the accuracy of data is often deceitful and affects the optimal solution of LP problems. Probability distribution cannot transact with inaccurate and unclear information.

In 1965, Zadeh [1] introduced fuzzy sets to handle imprecise information. Atanassove [2] gave the concept of intuitionistic fuzzy sets. The intuitionistic fuzzy sets consider both truth-membership and falsity-membership. Intuitionistic fuzzy sets can only handle incomplete information and not the indeterminate information and inconsistent information which exist commonly in the belief system. In 1998, Smarandache [3] presented the notion of neutrosophic set theory. Smarandache [3] and Wang et al. [4] defined a single-valued neutrosophic set which takes the value from the subset of [0, 1]. Deli et al. [5] introduced the concept of bipolar single-valued neutrosophic sets as an extension of bipolar fuzzy sets [6].

Bellman et al. [7] first introduced the concept of decision making in a fuzzy environment. Zimmermann [8] proposed the fuzzy programming technique to solve the multiobjective linear programming problem under a fuzzy environment. Tanaka et al. [9] studied fuzzy-mathematical programming. Lotfi et al. [10] discussed full fuzzy linear programming (FFLP) problems in which all parameters and variables are triangular fuzzy numbers. They used the concept of the symmetric triangular fuzzy number and introduced an approach to defuzzify a general fuzzy quantity. Allahviranloo et al. [11] suggested a method to solve FFLP problems by using a ranking function. Veeramani et al. [12] suggested a method to deal with a kind of fuzzy linear programming (FLP) problem involving symmetric trapezoidal fuzzy numbers. Kumar et al. [1315] worked on fuzzy linear programming by using non-negative and unrestricted variables to find optimal solutions of FFLP problems. By using trapezoidal fuzzy numbers Behera et al. [16,17] presented a new method to solve linear programming (LP) problems. Najafi et al. [18,19] proposed an efficient technique for solving FFLP by using unrestricted parameters and variables. Moloudzadeh et al. [20] suggested a simple method to solve an arbitrary fully fuzzy linear system. Akram et al. [21] proposed a method to solve LR-bipolar fuzzy linear systems. Mehmood et al. [22] suggested a method to solve fully bipolar fuzzy linear programming (FBFLP) problems by using non-negative bipolar fuzzy numbers and unrestricted bipolar fuzzy numbers with equality constraints. They transformed FBFLP problem into a crisp linear programming problem and achieved the exact bipolar fuzzy optimal solution.

Intuitionistic optimization is an extension of fuzzy optimization. Garg et al. [23] used an Intuitionistic fuzzy optimization method for solving reliability optimization problems in interval environment. Angelov [24] worked in an intuitionistic fuzzy environment. Many researchers solved intuitionistic fuzzy linear programming (IFLP) problems by using triangular intuitionistic fuzzy numbers [2529]. In an interval-valued intuitionistic fuzzy environment, Bharati et al. [30] gave the solution of multiobjective linear programming problems. Parvathi et al. [31] proposed linear regression analysis in an intuitionistic fuzzy environment and also worked on intuitionistic fuzzy linear programming [32].

The first contribution of neutrosophic linear programming theory was studied by Abdel-Basset et al. [33]. They introduced the neutrosophic linear programming (NLP) models in which the parameters are presented with trapezoidal neutrosophic numbers (TrNNs) and presented a technique for solving them. Das et al. [34] worked to solve NLP problems by using mixed constraints. Bera et al. [35] proposed the Big-M simplex technique to solve NLP problems. Edalatpanah [36] proposed a new direct algorithm for solving the LP problems including neutrosophic variables. Hussian et al. [37] proposed LP problems based on neutrosophic environment. Khalifa et al. [38] suggested a method to solve NLP by using single-valued neutrosophic numbers (SvNNs). Recently, Akram et al. [39,40] have presented new methods to solve Pythagorean fuzzy linear programming problems.

In this paper, we are going to extend the NLP problems into bipolar single-valued neutrosophic linear programming (BSvNLP) problems in which all the coefficients, variables and right hand side are represented by bipolar single-valued neutrosophic numbers (BSvNNs). The fully bipolar single-valued neutrosophic linear programming (FBSvNLP) problems are superior to crisp linear programming (CLP) problems and up to our knowledge, there is no work in literature on FBSvNLP. The BSvNLP problems are more appropriate to avert unrealistic modeling. This is unique research that deals with LP problems in a BSvN environment with TBSvNNs and TrBSvNNs. Score function is used to convert BSvNNs to CLP problems.

This paper has been categorized as follows: In Section 2, basic concepts of BSvNs, TBSvNNs, TrBSvNNs and their arithmetic operations are discussed. In Section 3, methodology for solving FBSvNLP problems are explained. In Section 4, some examples and practical models are solved. In Section 5, comparative analysis is depicted and conclusion is given in Section 6. The list of acronyms used in the research paper is given below:

BSvNN Bipolar single-valued neutrosophic number
TBSvNN Triangular Bipolar single-valued neutrosophic number
TrBSvNN Trapezoidal Bipolar single-valued neutrosophic number
LPP Linear programming Problem
FFLP Fully fuzzy linear programming

2  Preliminaries

Definition 2.1. [5] Let X be a non-empty set. A bipolar single-valued Neutrosophic set B̃ in X is an object having the form


where T+(x),I+(x),F+(x):X[0,1] and T-(x),I-(x),F-(x):X[-1,0]. The positive membership degree T+(x), I+(x), F+(x) denotes the truth membership, indeterminate membership and falsity membership of an element xX corresponding to a bipolar neutrosophic set B̃ similarly negative membership degree T(x), I(x), F(x) denotes the truth membership, indeterminate membership and falsity membership of an element xX to some implicit counter-property corresponding to a bipolar neutrosophic set B̃.

Definition 2.2. Based on [15], we define a TBSvNN defined on


is said to be non-negative TBSvNN if and only if ai0 and ei0. where i=1,2,3 such that aibici similarly eifigi also χp,βp,ζp[0,1] and αn,φn,νn[-1,0].

Definition 2.3. Based on [15], we define a TrBSvNN defined on


is said to be non-negative TrBSvNN if and only if ai0 and ei0. where i=1,2,3 and aibicidi similarly eifigihi also χp,βp,ζp[0,1] and αn,φn,νn[-1,0].

Definition 2.4. [30] Let M=(T+(x),I+(x),F+(x),T-(x),I-(x),F-(x)) be a BSvNN then the score function is presented by:


Definition 2.5. Based on [41], we define a BSvNN on is a BSvN set such that:

B̃=P̃,Ñ=([a1,b1,c1,d1];χp),([a2,b2,c2,d2];βp),([a3,b3,c3,d3];ζp),([e1,f1,g1,h1];αn), (5)

([e2,f2,g2,h2];φn),([e3,f3,g3,h3];νn) (6)

where χp,βp,ζp[0,1] and αn,φn,νn[-1,0] whose true membership values are given as:

Tp+(x)={STl(x),a1xb1,χp,b1xc1,STr(x),c1xd1,0, otherwise. Tn-(x)={UTl(x),e1xf1,αn,f1xg1,UTr(x),g1xh1,0, otherwise. 

SlT(x) and UrT(x) are continuous and non-decreasing functions satisfying the following conditions: STl(a1)=0,STl(b1)=χp, and UTr(g1)=αn,UTr(h1)=0, while SrT(x) and UlT(x) are continuous and non-increasing functions and satisfying the following conditions:

STr(c1)=χp,STr(d1)=0,UTl(e1)=0,UTl(f1)=αn, where χp[0,1],αn[-1,0]. The indeterminacy membership functions are given as:

Ip+(x)={SIl(x),a2xb2,βp,b2xc2,SIr(x),c2xd2,1, otherwise. In-(x)={UIl(x),e2xf2,φn,f2xg2,UIr(x),g2xh2,-1, otherwise. 

SrI(x) and UlI(x) are continuous and non-decreasing functions satisfying the following conditions: SIr(c2)=βp,SIr(d2)=1,UIl(e2)=-1,UIl(f2)=φn, while SlI(x) and UrI(x) are continuous and non-increasing functions and are satisfying the following conditions: SIl(a2)=1,SIl(b2)=βp,UIr(g2)=φn,UIr(h2)=-1, where βp[0,1],φn[-1,0]. The falsity membership function are given as:

Fp+(x)={SFl(x),a3xb3,ζp,b3xc3,SFr(x),c3xd3,1, otherwise. Fn-(x)={UFl(x),e3xf3,νn,f3xg3,UFr(x),g3xh3,-1, otherwise. 

SrF(x) and UlF(x) are continuous and non-decreasing functions satisfying the following conditions: SFr(c3)=ζp,SFr(d3)=1,UFl(e3)=-1,UFl(f3)=νn,

SlF(x) and UrF(x) are continuous and non-increasing functions and are satisfying the following conditions: SFl(a3)=1,SFl(b3)=ζp,UFr(g3)=νn,UFr(h3)=-1, where ζp[0,1],νn[-1,0].

Some useful information are given in Eqs. (1)–(6).

Definition 2.6. Based on [41], we define: If [a1, b1, c1, d1] = [a2, b2, c2, d2] = [a3, b3, c3, d3] and [e1, f1, g1, h1] = [e2, f2, g2, h2] = [e3, f3, g3, h3], then the BSvNN is reduced to a TrBSvNN as:

(1) B̃=P̃,Ñ=([a1,b1,c1,d1];χp,βp,ζp),([e1,f1,g1,h1];αn,φn,νn).

(2) B̃=P̃,Ñ=([a1,b1,d1];χp,βp,ζp),([e1,f1,h1];αn,φn,νn) is called a TBSvNN if and only if b1 = c1 and f1 = g1.

Definition 2.7. Based on [42], we define a TBSvNN defined on denoted by:


whose truth, indeterminacy and falsity membership functions are presented by:

Tp+(x)={x-a1b1-a1χp,a1xb1,c1-xc1-b1χp,b1xc1,0, otherwise. Tn-(x)={x-e1f1-e1αn,e1xf1,g1-xg1-f1αn,f1xg1,0, otherwise. 

Ip+(x)={(b2-x)+βp(x-a2)b2-a2,a2xb2,(x-b2)+βp(c2-x)c2-b2,b2xc2,1, otherwise. In-(x)={(f2-x)+φn(x-e2)f2-e2,e2xf2,(x-f2)+φn(g2-x)g2-f2,f2xg2,-1, otherwise. 

Fp+(x)={(b3-x)+ζp(x-a3)b3-a3,a3xb3,(x-b3)+ζp(c3-x)c3-b3,b3xc3,1, otherwise. Fn-(x)={(f3-x)+νp(x-e3)f3-e3,e3xf3,(x-f3)+νn(g3-x)g3-f3,f3xg3,-1, otherwise. 

where χp,βp,ζp[0,1] and αn,φn,νn[-1,0].

Definition 2.8. Based on [42], we define TrBSvNN defined denoted by:


whose truth, indeterminacy and falsity membership functions are presented by:

Tp+(x)={x-a1b1-a1χp,a1xb1,χp,b1xc1,d1-xd1-c1χp,c1xd1,0, otherwise. Tn-(x)={x-e1f1-e1αn,e1xf1,αn,f1xg1,h1-xh1-g1αn,g1xh1,0, otherwise. 

Ip+(x)={(b2-x)+βp(x-a2)b2-a2,a2xb2,βp,b2xc2,(x-c2)+βp(d2-x)d2-c2,c2xd2,1, otherwise. In-(x)={(f2-x)+φn(x-e2)f2-e2,e2xf2,φn,f2xg2,(x-g2)+φn(h2-x)h2-g2,g2xh2,-1, otherwise. 

Fp+(x)={(b3-x)+ζp(x-a3)b3-a3,a3xb3,ζp,b3xc3,(x-c3)+ζp(d3-x)d3-c3,c3xd3,1, otherwise. Fn-(x)={(f3-x)+νp(x-e3)f3-e3,e3xf3,νn,f3xg3,(x-g3)+νn(h3-x)h3-g3,g3xh3,-1, otherwise. 

where χp,βp,ζp[0,1] and αn,φn,νn[-1,0].

Definition 2.9. Let

A1̃=([a1,b1,c1,d1];χp1),([a2,b2,c2,d2];βp1),([a3,b3,c3,d3];ζp1),([a4,b4,c4,d4];αn1),([a5,b5,c5,d5];φn1),([a6,b6,c6,d6];νn1) and A2̃=([e1,f1,g1,h1];χp2),([e2,f2,g2,h2];βp2),([e3,f3,g3,h3];ζp2),([e4,f4,g4,h4];αn2),([e5,f5,g5,h5];φn2),([e6,f6,g6,h6];νn2).

be two non-negative TrBSvNNs, then

(1) A1̃A2̃=([a1+e1,b1+f1,c1+g1,d1+h1];χp1χp2),([a2+e2,b2+f2,c2+g2,d2+h2];βp1βp2),([a3+e3,b3+f3,c3+g3,d3+h3];ζp1ζp2),([a4+e4,b4+f4,c4+g4,d4+h4];αn1αn2),([a5+e5,b5+f5,c5+g5,d5+h5];φn1φn2),([a6+e6,b6+f6,c6+g6,d6+h6];νn1νn2).

(2) A1̃A2̃=([a1-h1,b1-g1,c1-f1,d1-e1];χp1χp2),([a2-h2,b2-g2,c2-f2,d2-e2];βp1βp2),([a3-h3,b3-g3,c3-f3,d3-e3];ζp1ζp2),([a4-h4,b4-g4,c4-f4,d4-e4];αn1αn2),([a5-h5,b5-g5,c5-f5,d5-e5];φn1φn2),([a6-h6,b6-g6,c6-f6,d6-e6];νn1νn2).

(3) A1̃A2̃={([a1e1,b1f1,c1g1,d1h1];χp1χp2),([a2e2,b2f2,c2g2,d2h2];βp1βp2),([a3e3,b3f3,c3g3,d3h3];ζp1ζp2),([a4e4,b4f4,c4g4,d4h4];αn1αn2),([a5e5,b5f5,c5g5,d5h5];φn1φn2),([a6e6,b6f6,c6g6,d6h6];νn1νn2).

For other notations and applications, readers are referred to [4345].

3  Methodology

In this section, a new method is presented to find the non-negative bipolar single-valued neutrosophic optimal solution of FBSvNLP problems with equality constraints, in which all the parameters are represented by non-negative BSvNNs.

Maximize/Minimize j=1ncj̃Hxj̃H;(7)

subject to


where c̃jH,ãijH,b̃iH and x̃jH are non-negative BSvNNs.

Step 1. Assuming c̃jH=([sj,tj,uj];χj),([sj,tj,uj];βj),([sj,tj,uj];ζj),([vj,wj,rj];αj),([vj,wj,rj];φj),([vj,wj,rj];νj), x̃jH=([xj,yj,zj];ϕj),([xj,yj,zj];γj),([xj,yj,zj];ηj),([gj,hj,kj];θj),([gj,hj,kj];κj),([gj,hj,kj];τj), ãijH=([dij,eij,fij];ξij),([dij,eij,fij];ψij),([dij,eij,fij];Γij),([mij,nij,pij];σij),([mij,nij,pij];ιij),([mij,nij,pij];μij) and b̃iH=([bi,ci,di];ϵj),([bi,ci,di];εj),([bi,ci,di];ϕj),([ei,fi,ti];δj),([ei,fi,ti];ϱj),([ei,fi,ti];ωj), the FBSvNLP problem (7) can be transformed as follows:

 Maximize/Minimize (j=1n([sj,tj,uj];χj),([sj,tj,uj];βj),([sj,tj,uj];ζj),([vj,wj,rj];αj),([vj,wj,rj];φj),([vj,wj,rj];νj)([xj,yj,zj];ϕj),([xj,yj,zj];γj),([xj,yj,zj];ηj),([gj,hj,kj];θj),([gj,hj,kj];κj),([gj,hj,kj];τj));(8)

subject to


where ([xj,yj,zj];ϕj),([xj,yj,zj];γj),([xj,yj,zj];ηj),([gj,hj,kj];θj),

([gj,hj,kj];κj),([gj,hj,kj];τj) are non-negative TBSvFN, j=1,2,3,,n.

Step 2. Using product of non-negative BSvNNs (2.9) and assuming


The FBSvNLP problem (8), can be transformed as follows:

Maximize/Minimize (j=1n(([sj,tj,uj];χj),([sj,tj,uj];βj),([sj,tj,uj];ζj),([vj,wj,rj];αj),([vj,wj,rj];φj),([vj,wj,rj];νj)([xj,yj,zj];ϕj),([xj,yj,zj];γj),([xj,yj,zj];ηj),([gj,hj,kj];θj),([gj,hj,kj];κj),([gj,hj,kj];τj)));(9)

subject to


where ([xj,yj,zj];ϕj),([xj,yj,zj];γj),([xj,yj,zj];ηj),([gj,hj,kj];θj),

([gj,hj,kj];κj),([gj,hj,kj];τj) are non-negative TSvBFN, j=1,2,3,,n.

Step 3. Using arithmetic operations (2.9), above problem becomes:

Maximize/Minimize (j=1n(([sj,tj,uj];χj),([sj,tj,uj];βj),([sj,tj,uj];ζj),([vj,wj,rj];αj),([vj,wj,rj];φj),([vj,wj,rj];νj)([xj,yj,zj];ϕj),([xj,yj,zj];γj),([xj,yj,zj];ηj),([gj,hj,kj];θj),([gj,hj,kj];κj),([gj,hj,kj];τj)));

subject to



and ϕj,γj,ηj[0,1],θj,κj,τj[-1,0].