#### Abstract

Transportation problem (TP) has its uses in real life because it has versatile applications. Real-life problems are often uncertain due to which it is difficult to find the accurate cost. The fuzzy set and intuitionistic fuzzy set are useful for handling the uncertainty, but these also have some limitations. For that reason, in this study, we worked on another set of values called bipolar single-valued neutrosophic set (BSNS) which is the generalization of crisp sets, fuzzy sets, and intuitionistic fuzzy sets to handle the uncertain, unpredictable, and insufficient information in real-life problems. In this study, we develop a new technique for solving transportation problems based on bipolar single-valued neutrosophic sets having nonnegative triangular bipolar single-valued neutrosophic numbers (TBSNNs). A score function is used to transform bipolar single-valued neutrosophic numbers (BSNNs) into crisp numbers. We compare our proposed model with fuzzy transportation and intuitionistic fuzzy transportation models and proved that bipolar single-valued neutrosophic transportation model is more admirable than the existing models. Furthermore, we apply the proposed technique to fully solve the bipolar single-valued neutrosophic transportation (FBSNT) model.

#### 1. Introduction

Nowadays, there is great competition among organizations to find better methods to create and give better services to their customers. They need a cost-effective method to send the products to their customers. This is a more challenging task. The models for transportation provide an effective framework to encounter this challenge. They guarantee the effective movement of raw materials and find products.

##### 1.1. Motivation

In the mathematical programming problem, the linear programming problem is very well-known and it has a wide scope to cover various fields. Among these, transportation problem is the commonly used field. The TP has its own importance in the field of logistics, supply chain management, supplier selection problem, etc. The TPs have great importance in many real-life applications. It works to maintain the supply from source to destination. It is generally considered that transversal costs of supply and demand are expressed in terms of crisp numbers. These values are often not precise. As a result, various researchers have been working on different TPs in fuzzy [1–3] and intuitionistic fuzzy [4, 5] environments, respectively. We proposed a new technique to solve TPs in the bipolar single-valued neutrosophic environment, which is a more generalized form.

##### 1.2. Literature Review

Zadeh [6, 7] proposed the concept of fuzzy sets and fuzzy numbers to reduce uncertainty and incomplete information. Atanassove [8] gave the idea of intuitionistic fuzzy sets (IFSs), which is a generalization of fuzzy sets. In an IFS, we consider the problem in both angles’ positive side and negative side to handle uncertainty. In an IFS, truth membership and falsity membership are independent, while indeterminacy-membership depends on their sum. Smarandache [9] proposed the notion of neutrosophic set theory. Smarandache [9] and Wang et al. [10] defined single-valued neutrosophic sets (SNSs), which are an extension of FSs and IFSs. In an SNS all membership functions truth, falsity, and indeterminacy are independent. Deli et al. [11] suggested the idea of bipolar single-valued neutrosophic set, which is the generalization of single-valued neutrosophic sets [10].

The TPs are related to the transportation of raw material from different sources to different destinations in such a way that the total transportation cost is minimized. Hitchcock [12] was the first to develop a basic transportation problem. The transportation problem can be elaborated as a standard linear programming problem. This can be solved by the simplex method. It was found that the simplex method when applied to the transportation problem becomes more effective when evaluating the simplex method information. Basirzadeh [1] proposed a method to solve fuzzy transportation problems. Ladji et al. [13] proposed a two-step method for solving fuzzy transportation problems where all of the parameters are represented by triangular fuzzy numbers i.e., two interval transportation problems. Pratihar et al. [14] have modified the classical vogel’s approximation method for solving the fuzzy transportation problem. They also worked on the interval type 2 fuzzy set and used it in a fuzzy transportation problem to represent the transportation cost, demand, and supply. Cam et al. [15] worked on the formulation of a linear programming model for the vehicle routing problem to minimize idle time. In fuzzy linear programming problems (FLPPs), many scholars have contributed. By using multiobjective function, Zimmerman [16] gave a technique to solve LP problem. After that, to solve the transportation problems, Zimmerman’s fuzzy linear programming has developed into several fuzzy optimization methods. In a fuzzy environment, Bellman and Zadeh [6] introduced the different concepts of decision making. Lotfi et al. [17] considered FFLP problems in which all parameters and variables are triangular fuzzy numbers. Allahviranloo et al. [18] solved FFLP problem by using a kind of defuzzification approach. Behera et al. [19] suggested two new methods to solve fuzzy linear programming (FLP) problems. They solved two types of problems with two different methods. Kaur and Kumar [20] gave an introduction to FLP problems. Kaur and Kumar [21] suggested an approach to find the exact fuzzy optimal solution to FFLP problems by using unrestricted fuzzy variables. Kaur and Kumar [22] proposed a new method for finding the fuzzy optimal solution to fuzzy transportation problems in which the transportation cost are represented by generalized fuzzy numbers. Najafi and Edalatpanah [23] suggested a better technique to solve the FFLP problem than Kumar et al. [21].

Intuitionistic fuzzy linear programming problem is an extension of the fuzzy linear programming problem. Many researchers have worked on different methods to solve LP problems in an intuitionistic fuzzy environment by using intuitionistic fuzzy numbers (IFNs) and LR-type IFNs. Singh and Yadav [4, 5] suggested two different techniques to solve an intuitionistic fuzzy transportation problem (IFTPs) by using triangular intuitionistic fuzzy numbers (TIFNs). Abhishekh and Nishad [24] suggested a novel ranking function for finding an optimal solution to fully LR-intuitionistic fuzzy transportation problem. In an intuitionistic fuzzy environment, Edalatpanah [25] designed a model of data envelopment analysis with triangular intuitionistic fuzzy numbers (TIFNs) and established a strategy to solve it. Kabiraj et al. [26] solved IFLP problems by using a method based on a method suggested by Zimmerman [16]. Malathi and Umadevi [27] worked on IFLP problems in an intuitionistic fuzzy environment. Pythagorean fuzzy linear programming is an extension of intuitionistic fuzzy linear programming. Akram et al. [28, 29] suggested a technique to solve Pythagorean fuzzy linear programming problems by using Pythagorean fuzzy numbers and LR-type Pythagorean fuzzy numbers. Akram et al. [30] used two different techniques to solve Pythagorean fuzzy linear programming problems having mixed constraints.

In daily life routine, we meet a variety of situations depending on multiple factors like uncertainty in judgements. Often it becomes difficult to get relevant data for cost parameters. The data of this type cannot always be represented by random variables obtained from the probability distribution. These data may be represented by bipolar single-valued neutrosophic numbers. So, a bipolar single-valued neutrosophic method to make the decision is needed. Abdel-Basset et al. [31] suggested a technique to solve the fully neutrosophic linear programming (FNLP) problems. Hussian et al. [32] suggested a linear programming model based on neutrosophic environment. Bera and Mahapatra [33] developed the Big-M simplex method to solve the neutrosophic linear programming (NLP) problem. Das and Chakraborty [34] considered a pentagonal NLP problem to solve it. Das and Dash [35] solved NLP problems with mixed constraints. Edalatpanah [36] presented a direct algorithm to solve the linear programming problems. Khalifa et al. [37] solved the NLP problem with single-valued trapezoidal neutrosophic numbers. Ahmed [38] suggested a technique to solve LR-type single-valued neutrosophic linear programming problems by using unrestricted LR-type single-valued neutrosophic numbers. He proposed the ranking function to transform LR-type single-valued neutrosophic problems into crisp problems. Deli et al. [11] gave the idea of bipolar single-valued neutrosophic set. Akram et al. [39] suggested a technique to solve LR-bipolar fuzzy linear system. Mehmood et al. [40, 41] defined LR-type bipolar fuzzy numbers and their arithmetic operations. They also introduced the ranking for LR-type bipolar fuzzy numbers and solved numerical examples. Ahmed et al. [42] suggested a technique to solve bipolar single-valued neutrosophic linear programming problems in which all the coefficients, variables, and right-hand side are presented by bipolar single-valued neutrosophic numbers. Kumar [43] presented the cut of single-valued pentagonal neutrosophic numbers and also introduced the arithmetic operation of single-valued pentagonal neutrosophic numbers. By using two different objective functions, Singh et al. [44] formulated the journey of a vaccine from its manufacture to its delivery using bilevel transportation problems in a neutrosophic environment. Veeranani et al. [45] solved the multiobjective fractional transportation problem by using the neutrosophic goal programming approach [46–48].

##### 1.3. Our Contribution

Considering all the available data, there are no methods in literature for TPs under the bipolar single-valued neutrosophic environment. So, there is a need to introduce a technique for BSNTPs. As a result of our facts, there are no optimization models available for TPs under bipolar single-valued neutrosophic environment. This has urged us to develop a new technique to solve TP with the bipolar single-valued neutrosophic environment, which is solved for the first time with the proposed technique. Bipolar single-valued neutrosophic set theory is a well-known technique to deal with uncertainty in the optimization problem. This study has been categorized as follows: In Section 2, basic concepts of BSNS, BSNN, TBSNNs, and their arithmetic operations are discussed. In Section 3, methodology for solving FBSNT problems are explained. In Section 4, a mathematical transportation model is solved. In Section 5, comparative analysis is discussed. In Section 6, the advantages of the proposed method are discussed. In Section 7, the limitations of the proposed method are given, and the conclusion is given in Section 8.

#### 2. Preliminaries

*Definition 1 (see [11]). *Let be a nonempty set. A bipolar single-valued neutrosophic set (BSNS) in is an object having the formwhere and . The positive membership degree denotes the truth membership, indeterminate membership, and falsity membership of an element corresponding to a bipolar neutrosophic set similarly negative membership degree denotes the truth membership, indeterminate membership, and falsity membership of an element to some implicit counter-property corresponding to a bipolar neutrosophic set .

*Definition 2 (see [33]). *A BSNN on is a BSN set such thatwhere and

The truth membership values are given as and are continuous and increasing functions satisfying the following conditions: , and , while and are continuous and decreasing functions and satisfying the following condition:where .

The indeterminacy-membership functions are given as and are continuous and increasing functions satisfying the following conditions: , while and are continuous and decreasing functions and are satisfying the following conditions:where .The falsity membership functions are given as and are continuous and increasing functions satisfying the following conditions: and are continuous and decreasing functions and are satisfying the following conditions:where .

Based on [49], we define some basic definitions.

*Definition 3. *We define a TBSNN defined on is said to be nonnegative TBSNN if and only if where similarly also and .

*Definition 4. *We define a defined on is said to be positive if and only if where . Similarly, ,also and .

*Definition 5. *A TBSNNis said to be zero if , and .

*Definition 6. *Two TBSNNsare said to be equal if , and .

*Definition 7. *Based on [50], we define defined denoted bywhose truth, indeterminacy, and falsity membership functions are presented bywhere and .

*Remark 1. *If we set and in Definition 7, the triangular bipolar single-valued neutrosophic number (TBSNN) is obtained, where .

*Definition 8. *(see [51]). Let be a BSNN then the score function is presented by

*Definition 9. *Based on [52], letbe two nonnegative , then,(1).(2) (3)

#### 3. Methodology to Solve FBSNT Problems

In this section, we present a new method that is based on the formulation of FBSNLP to solve FBSNT problems. We discuss the steps to calculate the bipolar single-valued neutrosophic optimal solution of FBSNT problems with nonnegative TBSNNs:subject towhere , and are all TBSNNs.

*Step 1. *Determine total bipolar single-valued neutrosophic availability and total bipolar single-valued neutrosophic demand.

If a balanced bipolar single-valued neutrosophic transportation problem (BSNTP).

If,An unbalanced BSNTP.

That is,Then one of the following case arise.

*Case 1. *.

*Case 2. *.

*Case 3. *When the above two cases do not hold, then there may exist infinitely many TBSNNs,such thatBut we have to determine such TBSNNswhich satisfy the following conditions.(i) and are nonnegative TBSNNs.(ii)It satisfies(iii)Further, if there exists two nonnegative TBSNNs and such thatthen

*Step 2. *Suppose ,Then FBSNTP (18) can be transformed assubject towhere are nonnegative TBSNNs.

*Step 3. *By applying arithmetic operations as described in Definition 9 and putting , then the FBSNTP (30) can be transformed assubject towhere are nonnegative TBSNNs.

*Step 4. *Now applying the score function, the FBSNTP (32) can be transformed assubject towhere are nonnegative TBSNNs.

*Step 5. *To obtain fuzzy optimal solution, the following crisp linear/nonlinear programming problem is solved:subject to and