#### 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.

#### 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: Section 2 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]). *Let be a nonempty set. A BFS in *M* is an object of the formwhere and are mappings. The positive membership degree denotes the truth or satisfaction degree of an element to a certain property corresponding to bipolar fuzzy set , and represents the satisfaction degree of an element to some counter property of bipolar fuzzy set . If and , it is the situation that is not satisfying the property of but satisfying the counter property to . If and , it is the case when has only positive satisfaction for . It is possible for to be such that and when satisfies the property of as well as its counter property in some part of .

*Definition 2 (See [26]). *A TBFN defined on real numbers , denoted by , is a BFN, and its satisfaction degree and dissatisfaction degree are represented by

*Definition 3 (See [26]). *A ranking function is a function , where 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, , is defined as

*Definition 4 (See [26]). *A TrBFN defined on real numbers , denoted byis a BFN, and its satisfaction degree and dissatisfaction degree are represented by

*Definition 5 (See [26]). *A ranking function is a function , where 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, , is defined as

*Definition 6 (See [26]). *Let and be two BFNs; then,(i) if ,(ii) if ,(iii) if .A list of acronyms is given in Table 1. For other notations and applications, readers are referred to [27–33].

#### 3. FBFLPP with Nonnegative Variables

*Definition 7. *A TBFN is called a nonnegative TBFN if and only if and . A TBFN is called an unrestricted TBFN if and only if and are real numbers. Two TBFNs and are said to be equal if and only if , , , , , .

*Definition 8. *A TrBFN is called a nonnegative TrBFN if and only if and . A TrBFN is called an unrestricted TrBFN if and only if and are real numbers. Two TrBFNs and are said to be equal if and only if , , , , , , , .

##### 3.1. Arithmetic Operations for TrBFNs

Let and be two TrBFNs. Then,(1).(2).(3).

Let be a TrBFN. Then,

We now define multiplication of an unrestricted TrBFN with nonnegative TrBFN.

*Definition 9. *Let be an unrestricted TrBFN and be a nonnegative TrBFN. Then,

*Remark 1. *If and in TrBFN ; then, is called TBFN.

We propose a method for solving FBFLPPs with equality constraints.subject towhere are nonnegative TBFNs and are unrestricted TBFNs.

*Definition 10. *A bipolar fuzzy optimal solution (BFOS) of FBFLPP (9) will be BFNs , if(1) are nonnegative TBFNs(2)(3)there exists any nonnegative TBFN satisfying the constraints, then in case of a maximization problem, and in case of a minimization problem,

##### 3.2. Methodology

subject towhere are nonnegative TBFNs and are unrestricted TBFNs. Step 1: assuming , , , and , the FBFLPP (11) can be transformed as follows: subject to where are nonnegative TBFN, . Step 2: applying product of TBFNs (3.3) and assuming the FBFLPP (3) can be transformed as follows: subject to where are nonnegative TBFN, . Step 3: applying arithmetic operations: subject to Step 4: by applying ranking function, FBFLPP converts as subject to Step 5: considering , the crisp FBFLPP (6) can be transformed as follows: subject to Step 6: by applying linearity property , where is a BFN, the crisp LPP (22) can be transformed as follows: subject to Step 7: by applying ranking for TBFNs (2.6), the crisp LPP (24) can be transformed as follows: subject to Step 8: by solving crisp LPP (26), we find the optimal solution . Step 9: we find the exact BFOS of FBFLPP by putting the values of and in . Step 10: we find the bipolar fuzzy optimal value by assigning the values of in .

Thus, we state the existence condition for the optimal solution of bipolar fuzzy LPP in the following theorem.

Theorem 1. *The solution of FBFLPP:** are nonnegative TBFNs and are TBFNs, existing when the solution of the associated crisp LPP,subject to 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. * is minimized subject towhere and are nonnegative TBFNs. Step 1: subject to where and are nonnegative TBFNs. Step 2: using the product of TBFNs (3.3), the problem converts as subject to Step 3: using arithmetic operations, the problem converts as subject to Step 4: by applying ranking for TBFNs (2.6), the problem converts as subject to Step 5: by using software MATLAB, the optimal solution of the crisp LPP is . Step 6: the exact BFOS is . Step 7: bipolar fuzzy optimal value of FBFLPP is .

*Example 2. *Maximize subject towhere and are nonnegative TBFNs. Step 1: subject to where and are nonnegative TBFNs. Step 2: using the product of TBFNs (3.3), the problem converts as subject to Step 3: using arithmetic operations, the problem converts as subject to Step 4: by applying ranking for TBFNs (2.6), the problem converts as subject to Step 5: by using software MATLAB, the optimal solution of the crisp LPP is . Step 6: the exact BFOS is . Step 7: bipolar fuzzy optimal value of FBFLPP is .

*Example 3. *A company produces bricks and tiles. It requires two basic raw materials and to make bricks and tiles. The maximum availability of raw material