A dynamic optimal solution approach for solving neutrosophic transportation problem

Abstract

Neutrosophic Set (NS) allows us to handle uncertainty and indeterminacy of the data. Several researchers have investigated the Transportation Problems (TP) with various forms of input data. This paper emphasizes a dynamic optimal solution framework for TPs in a neutrosophic setting. This paper investigates a Neutrosophic Transportation Problem (NTP) in which supply, demand, and transportation cost are considered as Single-Valued Neutrosophic Trapezoidal Numbers (SVNTrNs). The weighted possibilistic mean value of their truth, indeterminacy, and facility membership function are calculated. Then, NTP is modelled as a parametric Linear Programming Problem (LPP) and solved. Further, the drawbacks of the existing approaches and advantages of the developed method are discussed. Finally, the real-time problem and numerical illustrations are presented and compared to existing solutions. This study helps the Decision-Makers (DMs) in budgeting their transportation expenses through strategic distribution.

Keywords

Single valued neutrosophic trapezoidal number transportation problem linear programming problem weighted possibilistic mean

Nomenclature

Decision-Maker

Falsity-Membership

FLP

Fuzzy Linear Programming

FTP

Fuzzy Transportation Problem

IFS

Intuitionistic Fuzzy Set

Indeterminacy-Membership

LPP

Linear Programming Problem

Neutrosophic Set

NTP

Neutrosophic Transportation Problem

Possibilistic Mean

Transportation Problems

Truth-Membership

SVNN

Single-Valued Neutrosophic Number

SVNS

Single-Valued Neutrosophic Set

SVNTrN

Single-Valued Neutrosophic

Trapezoidal Number

1 Introduction

Firms transport their products from the point of production (origins) to the point of consumption (destinations). Each production point has a finite supply and each destination has a certain need that must be satisfied while transporting the goods. In this regard, transportation models are exploited to calculate the minimum shipping cost t-hat will meet the supply and demand. Hitchcock [12] was introduced the concept of a transportation problem in the form of a LPP and solved it by using the Simplex algorithm. It has received a significant amount of interest in the literature. It has been used in a variety of fields, including inventory management, logistics management, supply chain management, etc. In classical TP supply, demand, and costs are all considered as real or crisp numbers.

Recently, there are many situations with many forms of uncertainty that cannot be handled with traditional mathematical theory. Zadeh [35] introduced the fuzzy set theory, which is described by its membership grades, to handle situations involving imprecise or vague information. Many authors have exploited fuzzy numbers (particularly triangular and trapezoidal fuzzy numbers) to represent inexact intervals, unclear boundaries, and imprecise data sets. A Fuzzy Linear Programming (FLP) model was introduced by Zimmermann [36]. It has been used to address a variety of Fuzzy Transportation Problems (FTP) [18 , 24]. For determining the solution of an FTP with fuzzy supply and demand, Chanas et al. [3] proposed an FLP technique, but they considered the cost of the transportation to be real numbers. Dinagar and Palaneivel [3] discussed a solution technique for FTP in which the parameters like demand, supply, and transportation cost are assumed as trapezoidal fuzzy numbers. Kaur and Kumar [14] proposed a new framework for resolving the FTP. The rough set was used by some researchers to deal with the uncertainty of TP. To handle the problem’s ambiguity, Liu [17] proposed the concept of rough variables. One of the best tools for using uncertainty theories in the handling of imperfect information is possibility theory.

However, in many circumstances, the decisions or outputs based on the given data are insufficiently accurate. The notion of an Intuitionistic Fuzzy Set (IFS) was first developed by Atanassov [1], which addresses the problems involving imprecision data because of their membership and non-membership values. As a result, IFS theory has been used to solve a wide range of real-world decision-making issues. To solve TPs, many researchers employed an IFS technique. Other than ambiguity or uncertainty in the current TP model’s restrictions, there is some indeterminacy owing to other causes such as lack of knowledge of the situation, data imperfections, and poor forecasting. The concepts of left and right Possibilistic Mean (PM) values of fuzzy numbers were introduced by Carlsson and Fuller [4]. Fuller and Majlender [10] investigated the footnote of the weighted interval-valued PM value of fuzzy numbers. Wan et al. [32] established the possible mean, variance, and covariance of IFNs. The possibility mean, variance, and covariance of a generalised intuitionistic fuzzy number were proposed by Garg et al. [11]. IFS theory can deal with inadequate data, but not with indeterminate or inconsistent data.

FS and IFS are not applicable for all possible uncertainties in many natural problems. Smarandache [25] proposed Neutrosophic Sets (NSs) as a generalisation of classical sets, FSs, and IFSs to overcome this problem. The Truth-Membership (TM) degree, Indeterminacy-Membership (IM) degree, and Falsity-Membership (FM) degree components of the NS were appropriate to express indeterminacy and inconsistent data. The concept of a Single-Valued Neutrosophic Set (SVNS) has been used to create several real-world decision-making problems. Wang et al. [32] introduced the Single-Valued Neutrosophic Number (SVNN). SVNS is an upgrading generalized version of the FS and IFS. Further, a triplet of membership degrees are used to represents the element into the set. In SVNS, each element is shown by TM, IM, and FM so that their sum lies between 0 and 3. Many researchers gave more attention to SVNNs in the form of triplet structure. Most of the authors are used the ranking method to solve the neutrosophic problems. Few researchers had been worked on various methods such as TPs under neutrosophic environments. For handling the inverse capacitated TPs in a neutrosophic setting. Khalifa et al. [15] proposed the KKM technique. Thamaraiselvi and Santhi [28] developed a new scoring function for solving TP in a neutrosophic environment. Unfortunately, this scoring function does not satisfy the linear property. Singh et al. [26] proposed a modified approach for solving real and NTPs. Dhouib [7] presented a new method based on the Dhouib-TP1 heuristic approach for solving NTP. Singh et al. [27] used a bi-level TP in order to solve the NTP. They modelled a real-time vaccine distribution problem in order to distribute the vaccine from vaccine production companies to the different centres. In this paper, we have proposed a new method for solving NTPs. The proposed method produces dynamic solutions for the DMs interest. Further, we have pointed out the drawbacks of existing methods.

The structuring of this article is abbreviated as follows: Section 2, discusses the basic preliminaries and an overview of SVN numbers, In section 3, we presents the PM and weighted PM of the SVNTrN. The method for solving NTP is explained in section 4. In section 5, discusses the shortcomings of the existing method. The advantages of the proposed method are analysed in the section 6. Numerical examples and comparative studies are given in Section 7. Finally, the conclusion and scope of future work plan affair.

2 Preliminaries

Definition 2.1. [1] Let X be the universe of discourse, then an IFS ${\tilde{A}}^{IFS}$ in X is given by the set of ordered triples ${\tilde{A}}^{IFS} = {< x, μ_{{\tilde{A}}^{IFS}} (x), ϑ_{{\tilde{A}}^{IFS}} (x) >; x ɛ X}$ , where $μ_{{\tilde{A}}^{IFS}}$ (x), $ϑ_{{\tilde{A}}^{IFS}}$ (x): x → [0,1] as functions such that 0 $⩽ μ_{{\tilde{A}}^{IFS}}$ (x) + $ϑ_{{\tilde{A}}^{IFS}}$ (x) ⩽1 ∀ xɛX. For each x the membership $μ_{{\tilde{A}}^{IFS}}$ (x), non-membership $ϑ_{{\tilde{A}}^{IFS}}$ (x), represent the degree of membership and degree of non-membership of the element x ɛ X.

Definition 2.2. [25] Let A be a NS over X is defined by ${\tilde{A}}^{NTN} = {<$ x, ( ${TM}_{{\overset{⌣}{A}}^{NTN}} (x), {IM}_{{\overset{⌣}{A}}^{NTN}} (x), {FM}_{{\overset{⌣}{A}}^{NTN}} (x) >$ ): x ∈ X}, where ${TM}_{{\overset{⌣}{A}}^{NTN}} (x), {IM}_{{\overset{⌣}{A}}^{NTN}} (x), {FM}_{{\overset{⌣}{A}}^{NTN}} (x)$ are called TM function, IM function and FM function, respectively. They are respectively defined by ${TM}_{{\overset{⌣}{A}}^{NTN}}$ : X ⟶]^–0, 1 ⁺ [b, ${IM}_{{\overset{⌣}{A}}^{NTN}}$ : X ⟶]^–0, 1 ⁺ [, ${FM}_{{\overset{⌣}{A}}^{NTN}}$ : X ⟶]^–0, 1 + [such that 0^–≤ ${TM}_{{\overset{⌣}{A}}^{NTN}} (x) + {IM}_{{\overset{⌣}{A}}^{NTN}} (x) + {FM}_{{\overset{⌣}{A}}^{NTN}} (x)$ ≤3⁺

Definition 2.3. [25] Let X be a universe. SVNS over X is a NSover X. The TM function, IM function and FMF are defined by ${TM}_{{\overset{⌣}{A}}^{NTN}}$ : X ⟶ [0, 1], ${IM}_{{\overset{⌣}{A}}^{NTN}}$ : X ⟶ [0, 1], ${FM}_{{\overset{⌣}{A}}^{NTN}}$ : X ⟶ [0, 1]respectively, such that 0≤ ${TM}_{{\overset{⌣}{A}}^{NTN}} (x) + {IM}_{{\overset{⌣}{A}}^{NTN}} (x) + {FM}_{{\overset{⌣}{A}}^{NTN}} (x)$ ≤ 3.

Definition 2.4. [25] A The SVNS ${\tilde{A}}^{NTN}$ = {<x, ( ${TM}_{{\overset{⌣}{A}}^{NTN}} (x), {IM}_{{\overset{⌣}{A}}^{NTN}} (x), {FM}_{{\overset{⌣}{A}}^{NTN}} (x) >$ ): x ∈ X}, is called neutrosophic-normal,if there exist at least three point x₁, x₂, x₃ɛX such that ${TM}_{{\overset{⌣}{A}}^{NTN}} (x_{1}) = 1; {IM}_{{\overset{⌣}{A}}^{NTN}} (x_{2}) = 1, {FM}_{{\overset{⌣}{A}}^{NTN}} (x_{3}) = 1$ .

Definition 2.5. [25] A SVNS ${\tilde{A}}^{NTN}$ = {<x, ( ${TM}_{{\overset{⌣}{A}}^{NTN}} (x), {IM}_{{\overset{⌣}{A}}^{NTN}} (x), {FM}_{{\overset{⌣}{A}}^{NTN}} (x) >$ ): x ∈ X}, is called neutrosophic-convex set on the real line;if the following condition are satisfied ∀x₁, x₂ɛR and λɛ [0, 1].

${TM}_{{\overset{⌣}{A}}^{NTN}} (λ x_{1} + (1 - λ) x_{2}) ⩾ \min ({TM}_{{\overset{⌣}{A}}^{NTN}} (x_{1}), {TM}_{{\overset{⌣}{A}}^{NTN}} (x_{2}))$

${IM}_{{\overset{⌣}{A}}^{NTN}} (λ x_{1} + (1 - λ) x_{2}) ⩽ \max ({IM}_{{\overset{⌣}{A}}^{NTN}} (x_{1}), {IM}_{{\overset{⌣}{A}}^{NTN}} (x_{2}))$

${FM}_{{\overset{⌣}{A}}^{NTN}} (λ x_{1} + (1 - λ) x_{2}) ⩽ \max ({FM}_{{\overset{⌣}{A}}^{NTN}} (x_{1}), {FM}_{{\overset{⌣}{A}}^{NTN}} (x_{2}))$

Definition 2.6. [25] Let $w_{{\overset{⌣}{A}}^{NTN}}$ , $u_{{\overset{⌣}{A}}^{NTN}}, y_{{\overset{⌣}{A}}^{NTN}} \in [0, 1]$ be any real numbers, a_j∈ R (j = 1, 2, 3, 4) and a₁ ≤ a₂ ≤ a₃ ≤ a₄. Then, a SVNTrN, which is denoted as ${\tilde{A}}^{NTN} = < (a_{1}, a_{2}, a_{3}, a_{4})$ ; $w_{{\tilde{A}}^{NTN}}$ , $u_{{\tilde{A}}^{NTN}}, y_{{\tilde{A}}^{NTN}} >$ is a special neutrosophic set on the set of real numbers R, whose TM function ${TM}_{{\overset{⌣}{A}}^{NTN}}$ : R ⟶ [0, 1], IM function, ${IM}_{{\overset{⌣}{A}}^{NTN}}$ : R ⟶ [0, 1] and FM function ${FM}_{{\overset{⌣}{A}}^{NTN}}$ : R ⟶ [0, 1] respectively, are defined, $\begin{matrix} {TM}_{{\overset{⌣}{A}}^{NTN}} (x) = {\begin{matrix} \frac{(x - a_{1}) w_{{\tilde{A}}^{NTN}}}{a_{2} - a_{1}}, if a_{1} ⩽ x ⩽ a_{2} \\ w_{{\tilde{A}}^{NTN}}, if a_{2} ⩽ x ⩽ a_{3} \\ \frac{(a_{4} - x) w_{{\tilde{A}}^{NTN}}}{a_{4} - a_{3}}, if a_{3} ⩽ x ⩽ a_{4} \\ 0 otherwise \end{matrix} \end{matrix}$ $\begin{matrix} {IM}_{{\overset{⌣}{A}}^{NTN}} (x) = {\begin{matrix} \frac{(a_{2} - x) + (x - a_{1}) u_{{\tilde{A}}^{NTN}}}{a_{2} - a_{1}}, if a_{1} ⩽ x ⩽ a_{2} \\ u_{{\tilde{A}}^{NTN}}, if a_{2} ⩽ x ⩽ a_{3} \\ \frac{x - a_{3} + (a_{4} - x) u_{{\tilde{A}}^{NTN}}}{a_{4} - a_{3}}, if a_{3} ⩽ x ⩽ a_{4} \\ 1 otherwise \end{matrix} \end{matrix}$ $\begin{matrix} {FM}_{{\overset{⌣}{A}}^{NTN}} (x) = {\begin{matrix} \frac{(a_{2} - x) + (x - a_{1}) y_{{\tilde{A}}^{NTN}}}{a_{2} - a_{1}}, {ifa}_{1} ⩽ x ⩽ a_{2} \\ y_{{\tilde{A}}^{NTN}}, {ifa}_{2} ⩽ x ⩽ a_{3} \\ \frac{x - a_{3} + (a_{4} - x) y_{{\tilde{A}}^{NTN}}}{a_{4} - a_{3}}, {ifa}_{3} ⩽ x ⩽ a_{4} \\ 1 otherwise \end{matrix} \end{matrix}$

When a₁ > 0 then ${\tilde{A}}^{NTN}$ =< $(a_{1}, a_{2}, a_{3}, a_{4}); w_{{\tilde{A}}^{NTN}}$ , $u_{{\tilde{A}}^{NTN}}, y_{{\tilde{A}}^{NTN}} >$ is called positive SVNTrN and denoted by ${\tilde{A}}^{NTN}$ >0 and If a₄ ⩽ 0 then ${\tilde{A}}^{NTN}$ = < $(a_{1}, a_{2}, a_{3}, a_{4}); w_{{\tilde{A}}^{NTN}}$ , $u_{{\tilde{A}}^{NTN}}, y_{{\tilde{A}}^{NTN}} >$ is called negative SVNTrN and denoted by ${\tilde{A}}^{NTN}$ <0.

If 0⩽a₁ ≤ a₂ ≤ a₃ ≤ a₄ ⩽ 1, $w_{{\tilde{A}}^{NTN}}$ , $u_{{\tilde{A}}^{NTN}}, y_{{\tilde{A}}^{NTN}} \in$ [0, 1] then ${\tilde{A}}^{NTN}$ is called a normalized SVNTrN.

When ${IM}_{{\overset{⌣}{A}}^{NTN}} = 1 - {TM}_{{\overset{⌣}{A}}^{NTN}} - {FM}_{{\overset{⌣}{A}}^{NTN}}$ then SVNTrN reduce to as intuitionistic fuzzy trapezoidal number. If a₂ = a₃ then SVNTrN reduced to single valued triangular neutrosophic number, which is denoted as a ${\tilde{A}}^{NTN} = < (a_{1}, a_{2}, a_{3}); w_{{\tilde{A}}^{NTN}}$ , $u_{{\tilde{A}}^{NTN}}, y_{{\tilde{A}}^{NTN}} >$

Definition 2.7. [25] Let ${\tilde{A}}^{NTS}$ be a SVNTrN, the (α, β, γ)-cut is defined as ${\tilde{A}}_{(α, β, γ)}^{NTN}$ = ${x : T M_{{\overset{⌣}{A}}^{N T N}} (x) ⩾ α, I M_{{\overset{⌣}{A}}^{N T N}} (x) ⩽ β, F M_{{\overset{⌣}{A}}^{N T N}} (x) ⩽ γ}$ where 0≤ ${TM}_{{\overset{⌣}{A}}^{NTN}} (x) + {IM}_{{\overset{⌣}{A}}^{NTN}} (x) + {FM}_{{\overset{⌣}{A}}^{NTN}} (x)$ ≤3.

(α, β, γ)-cut of SVNTrN ${\tilde{A}}^{NTS}$ , is represented by ${\tilde{A}}_{(α, β, γ)}^{NTN}$ a crisp subset of X, then ${\tilde{A}}_{(α, β, γ)}^{NTN}$ = $A_{(α)}^{\sim NTN}; {\tilde{A}}_{(β)}^{NTN}; {\tilde{A}}_{(γ)}^{NTN}$

α- cut of SVNTrN ${\tilde{A}}^{NTN}$ is a closed interval, which is represented by ${\tilde{A}}_{(α)}^{N T N} = [{\overset{⌣}{A}}^{N T N^{L}} (α), {\overset{⌣}{A}}^{N T N^{R}} (α)] = [I M_{{\overset{⌣}{A}}^{N T N^{L}}} (α), T M_{{\overset{⌣}{A}}^{N T N}} R (α)]$ where $T M_{{\overset{⌣}{A}}^{N T N^{L}}} (α) = a_{1} + \frac{α (a_{2} - a_{1})}{w_{{\tilde{A}}^{N T N}}}$ ; $T M_{{\overset{⌣}{A}}^{N T N^{R}}} (α) = a_{4} - \frac{α (a_{4} - a_{3})}{w_{{\tilde{A}}^{N T N}}}$ .

β- cut of SVNTrN ${\tilde{A}}^{NTN}$ is a closed interval, which is represented by ${\tilde{A}}_{(β)}^{N T N} = [{\overset{⌣}{A}}^{N T N^{L}} (β), {\overset{⌣}{A}}^{N T N^{R}} (β)] = [I M_{{\overset{⌣}{A}}^{N T N}} (β), T M_{{\overset{⌣}{A}}^{N T N^{R}}} (β)]$ , where $I M_{{\overset{⌣}{A}}^{N T N^{L}}} (β) = \frac{a_{1} u_{{\tilde{A}}^{N T N}} - a_{2} + β (a_{2} - a_{1)}}{u_{{\tilde{A}}^{N T N}} - 1}$ ; $I M_{{\overset{⌣}{A}}^{N T N^{R}}} (β) = \frac{a_{3} - a_{4} u_{{\tilde{A}}^{N T N}} + β (a_{4} - a_{3})}{1 - u_{{\tilde{A}}^{N T N}}}$

γ- cut of SVNTrN ${\tilde{A}}^{NTN}$ is a closed interval, which is represented by ${\tilde{A}}_{(γ)}^{N T N} = [{\overset{⌣}{A}}^{N T N^{L}} (γ), {\overset{⌣}{A}}^{N T N^{R}} (γ)] = [F M_{{\overset{⌣}{A}}^{N T N^{L}}} (γ), F M_{{\overset{⌣}{A}}^{N T N^{R}}} (γ)]$ , where $F M_{{\overset{⌣}{A}}^{N T N^{L}}} (γ)$ = $\frac{a_{1} y_{{\tilde{A}}^{NTN}} - a_{2} + γ (a_{2} - a_{1)}}{y_{{\tilde{A}}^{NTN}} - 1}$ ; $F M_{{\overset{⌣}{A}}^{N T N^{R}}} (γ)$ = $= \frac{a_{3} - a_{4} y_{{\tilde{A}}^{NTN}} + γ (a_{4} - a_{3})}{1 - y_{{\tilde{A}}^{NTN}}}$ .

3 PM of SVNNs

This section discusses finding the PM value of the TM function, IM function, and FM function for the SVNN.

3.1 PM of the TM function

Let us, ${\tilde{A}}^{NTN} = < (a_{1}, a_{2}, a_{3}, a_{4})$ ; $w_{{\tilde{A}}^{NTN}}$ , $u_{{\tilde{A}}^{NTN}}, y_{{\tilde{A}}^{NTN}}} >$ be a SVTrNN. The α- cut of ${\tilde{A}}^{NTN}$ is defined as ${\tilde{A}}_{α}^{NTN} = {x | {TM}_{{\overset{⌣}{A}}^{NTN}} (x) ⩾ α : x \in X}$ , $0 ⩽ α ⩽ w_{{\tilde{A}}^{NTN}}$ . The left and right PM of TM function of ${\tilde{A}}^{NTN}$ is denoted $M^{L} [{\tilde{A}}^{NTN} (α)]$ and $M^{R} [{\tilde{A}}^{NTN} (α)]$ respectively, which is defined as follows:

Left PM of TM function of ${\tilde{A}}^{NTN}$ is defined by $\begin{matrix} M^{L} [{\tilde{A}}^{NTN} (α)] \\ = 2 \int_{0}^{w_{{\tilde{A}}^{NTN}}} Poss [{\tilde{A}}^{NTN} ⩽ {\overset{⌣}{A}}^{NTN}^{L} (α)] {\overset{⌣}{A}}^{NTN}^{L} (α) d α \end{matrix}$

Where, poss denotes the possibility, i.e., $\begin{matrix} poss [{\tilde{A}}^{NTN} ⩽ {\overset{⌣}{A}}^{NTN}^{L} (α)] \\ = \prod ((- \infty, {\overset{⌣}{A}}^{NTN}^{L} (α)]) \end{matrix}$

Thus,

$\begin{matrix} M^{L} [{\tilde{A}}^{NTN} (α)] = 2 \int_{0}^{w_{{\tilde{A}}^{NTN}}} α {\overset{⌣}{A}}^{NTN}^{L} (α) d α \\ = (w_{{\tilde{A}}^{NTN}})^{2} [\frac{a_{1} + 2 a_{2}}{3}] \end{matrix}$ (1)

Right PM of the TM function of ${\tilde{A}}^{NTN}$ is defined by $\begin{matrix} M^{R} [{\tilde{A}}^{NTN} (α)] & = 2 \int_{0}^{w_{{\tilde{A}}^{NTN}}} Poss [{\tilde{A}}^{NTN} \\ ⩾ {\overset{⌣}{A}}^{NTN}^{R} (α)] {\overset{⌣}{A}}^{NTN}^{R} (α) d α \end{matrix}$

Where, poss denotes the possibility, i.e., $\begin{matrix} poss [{\tilde{A}}^{NTN} ⩾ {\overset{⌣}{A}}^{NTN}^{L} (α)] = \prod [({\overset{⌣}{A}}^{NTN}^{R} (α), \infty)] \end{matrix}$

Thus,

$\begin{matrix} M^{R} [{\tilde{A}}^{NTN} (α)] = 2 \int_{0}^{w_{{\tilde{A}}^{NTN}}} α {\overset{⌣}{A}}^{NTN}^{R} (α) d α \\ = (w_{{\tilde{A}}^{NTN}})^{2} [\frac{a_{4} + 2 a_{3}}{3}] \end{matrix}$ (2)

PM of the TM for ${\tilde{A}}^{NTN}$ denoted by $M [{\tilde{A}}^{NTN} (α)]$ is defined as follows:

$\begin{matrix} M [{\tilde{A}}^{NTN} (α)] = \frac{M^{L} [{\tilde{A}}^{NTN} (α)] + M^{R} [{\tilde{A}}^{NTN} (α)]}{2} \\ = \frac{(w_{{\tilde{A}}^{NTN}})^{2} (a_{1} + 2 a_{2} + 2 a_{3} + a_{4})}{6} \end{matrix}$ (3)

3.2 PM of the IM function

The β- cut of ${\tilde{A}}^{NTN}$ is defined as ${\tilde{A}}_{β}^{NTN} = {x | {IM}_{{\overset{⌣}{A}}^{NTN}} ⩽ β : x \in X}$ , $u_{{\tilde{A}}^{NTN}} ⩽ β ⩽ 1$ . The left and right PM of IM function of ${\tilde{A}}^{NTN}$ is denoted $M^{L} [{\tilde{A}}^{NTN} (β)]$ and $M^{R} [{\tilde{A}}^{NTN} (β)]$ respectively, which is defined as follows:

Left PM of IM function of ${\tilde{A}}^{NTN}$ is defined by $\begin{matrix} M^{L} [{\tilde{A}}^{NTN} (β)] = 2 \int_{u_{{\tilde{A}}^{NTN}}}^{1} Poss [{\tilde{A}}^{NTN} \\ ⩽ {\overset{⌣}{A}}^{NTN}^{L} (β)] {\overset{⌣}{A}}^{NTN}^{L} (β) d β \end{matrix}$ where poss denotes the possibility, i.e.,

Thus $poss [{\tilde{A}}^{NTN} ⩽ {\overset{⌣}{A}}^{NTN}^{L} (β)]$ = $\prod ((- \infty, {\overset{⌣}{A}}^{NTN}^{L} (β)]) = β$

$\begin{matrix} M^{L} [{\tilde{A}}^{NTN} (β)] = 2 \int_{u_{{\tilde{A}}^{NTN}}}^{1} β {\overset{⌣}{A}}^{NTN}^{L} (β) d β \\ = \frac{2 (a_{2} - a_{1}) + (1 + u_{{\tilde{A}}^{NTN}} + {(u_{{\tilde{A}}^{NTN}})}^{2}) + 3 (a_{1} u_{{\tilde{A}}^{NTN}} - a_{2}) (1 + u_{{\tilde{A}}^{NTN}})}{3} \end{matrix}$ (4)

Right PM of the IM function of ${\tilde{A}}^{NTN}$ is defined by $\begin{matrix} M^{R} [{\tilde{A}}^{NTN} (β)] = 2 \int_{u_{{\tilde{A}}^{NTN}}}^{1} Poss [{\tilde{A}}^{NTN} \\ ⩾ {\overset{⌣}{A}}^{NTN}^{R} (β)] {\overset{⌣}{A}}^{NTN}^{R} (β) d β \end{matrix}$ where poss denotes the possibility, i.e., $\begin{matrix} poss [{\tilde{A}}^{NTN} ⩾ {\overset{⌣}{A}}^{NTN}^{R} (β)] \\ = \prod ([{\overset{⌣}{A}}^{NTN}^{R} (β), \infty)) = β \end{matrix}$

Thus

$\begin{matrix} M^{R} [{\tilde{A}}^{NTN} (β)] = 2 \int_{u_{{\tilde{A}}^{NTN}}}^{1} β {\overset{⌣}{A}}^{NTN}^{R} (β) d β \\ = \frac{2 (a_{4} - a_{3}) + (1 + u_{{\tilde{A}}^{NTN}} + {(u_{{\tilde{A}}^{NTN}})}^{2}) - 3 (a_{4} u_{{\tilde{A}}^{NTN}} - a_{3}) (1 + u_{{\tilde{A}}^{NTN}})}{3} \end{matrix}$ (5)

PM of the IM for ${\tilde{A}}^{NTN}$ denoted by $M [{\tilde{A}}^{NTN} (β)]$ is defined as follows:

$\begin{matrix} M [{\tilde{A}}^{NTN} (β)] = \frac{M^{L} [{\tilde{A}}^{NTN} (β)] + M^{R} [{\tilde{A}}^{NTN} (β)]}{2} \\ = \frac{2 (1 + u_{{\tilde{A}}^{NTN}} + {(u_{{\tilde{A}}^{NTN}})}^{2}) (a_{2} - a_{1} + a_{4} - a_{3}) + 3 (1 + u_{{\tilde{A}}^{NTN}}) (a_{1} u_{{\tilde{A}}^{NTN}} - a_{2} + a_{3} - a_{4} u_{{\tilde{A}}^{NTN}})}{6} \\ = \frac{(w_{{\tilde{A}}^{NTN}})^{2} (a_{1} + 2 a_{2} + 2 a_{3} + a_{4})}{6} \end{matrix}$ (6)

3.3 PM of the FM function

The γ- cut of ${\tilde{A}}^{NTN}$ is defined as ${\tilde{A}}_{γ}^{NTN} = {x | {FM}_{{\overset{⌣}{A}}^{NTN}} ⩽ γ : x \in X}$ , $y_{{\tilde{A}}^{NTN}} ⩽ γ ⩽ 1$ . The left and right PM of FM of ${\tilde{A}}^{NTN}$ is denoted $M^{L} [{\tilde{A}}^{NTN} (γ)]$ and $M^{R} [{\tilde{A}}^{NTN} (γ)]$ respectively, which is defined as follows:

Left PM of FM function of ${\tilde{A}}^{NTN}$ is defined by $\begin{matrix} M^{L} [{\tilde{A}}^{NTN} (γ)] \\ = 2 \int_{y_{{\tilde{A}}^{NTN}}}^{1} Poss [{\tilde{A}}^{NTN} ⩽ {\overset{⌣}{A}}^{NTN}^{L} (γ)] {\overset{⌣}{A}}^{NTN}^{L} (γ) d γ \end{matrix}$

Where, poss denotes the possibility, i.e., $\begin{matrix} poss [{\tilde{A}}^{NTN} ⩽ {\overset{⌣}{A}}^{NTN}^{L} (γ)] \\ = \prod ((- \infty, {\overset{⌣}{A}}^{NTN}^{L} (γ)]) = γ \end{matrix}$

Thus

$\begin{matrix} M^{L} [{\tilde{A}}^{NTN} (γ)] = 2 \int_{y_{{\tilde{A}}^{NTN}}}^{1} γ {\overset{⌣}{A}}^{NTN}^{L} (γ) d γ \\ = \frac{3 (a_{2} - a_{1} y_{{\tilde{A}}^{NTN}}) (1 + y_{{\tilde{A}}^{NTN}}) - 2 (a_{2} - a_{1}) (1 + y_{{\tilde{A}}^{NTN}} + {(y_{{\tilde{A}}^{NTN}})}^{2})}{3} \end{matrix}$ (7)

Right PM of the FM function of ${\tilde{A}}^{NTN}$ is defined by $\begin{matrix} M^{R} [{\tilde{A}}^{NTN} (γ)] = 2 \int_{y_{{\tilde{A}}^{NTN}}}^{1} Poss [{\tilde{A}}^{NTN} \\ ⩾ {\overset{⌣}{A}}^{NTN}^{R} (γ)] {\overset{⌣}{A}}^{NTN}^{R} (γ) d γ \end{matrix}$

Where, poss denotes the possibility, i.e., $\begin{matrix} poss [{\tilde{A}}^{NTN} ⩾ {\overset{⌣}{A}}^{NTN}^{R} (γ)] \\ = \prod ([{\overset{⌣}{A}}^{NTN}^{R} (γ), \infty)) = γ \end{matrix}$

Thus

$\begin{matrix} M^{R} [{\tilde{A}}^{NTN} (γ)] = 2 \int_{y_{{\tilde{A}}^{NTN}}}^{1} β {\overset{⌣}{A}}^{NTN}^{R} (γ) d γ \\ = \frac{[2 (a_{4} - a_{3}) (1 + y_{{\tilde{A}}^{NTN}} + {(y_{{\tilde{A}}^{NTN}})}^{2}) - 3 (a_{4} y_{{\tilde{A}}^{NTN}} - a_{3}) (1 + y_{{\tilde{A}}^{NTN}})]}{3} \end{matrix}$ (8)

PM of the FM for ${\tilde{A}}^{NTN}$ denoted by $M [{\tilde{A}}^{NTN} (γ)]$ is defined as follows:

$\begin{matrix} M [{\tilde{A}}^{NTN} (γ)] = \frac{M^{L} [{\tilde{A}}^{NTN} (γ)] + M^{R} [{\tilde{A}}^{NTN} (γ)]}{2} \\ = \frac{3 (1 + y_{{\tilde{A}}^{NTN}}) (a_{2} + a_{3} - a_{1} y_{{\tilde{A}}^{NTN}} - a_{4} y_{{\tilde{A}}^{NTN}}) + 2 (1 + y_{{\tilde{A}}^{NTN}} + {(y_{{\tilde{A}}^{NTN}})}^{2}) (a_{1} - a_{2} + a_{4} - a_{3})}{6} \end{matrix}$ (9)

3.4 Weighted possibilistic means value of SVNTrN

Let ${\tilde{A}}^{NTN} = < (a_{1}, a_{2}, a_{3}, a_{4})$ ; $w_{{\tilde{A}}^{NTN}}$ , $u_{{\tilde{A}}^{NTN}}, y_{{\tilde{A}}^{NTN}}} >$ be a SVNTrN and the PM of TM, IM, and FM denoted by $M [{\tilde{A}}^{NTN} (α)]$ , $M [{\tilde{A}}^{NTN} (β)]$ and $M [{\tilde{A}}^{NTN} (γ)]$ respectively. Generally, the DMs try to maximize TM while minimizing IM and FM. Hence, weighted value of ${\tilde{A}}^{NTN}$ is defined as follows:

$\begin{matrix} V ({\tilde{A}}^{NTN}, λ) = λ M [{\tilde{A}}^{NTN} (α)] \\ + (1 - λ) M [{\tilde{A}}^{NTN} (β)] + (1 - λ) M [{\tilde{A}}^{NTN} (γ)] \end{matrix}$ (10)

Where,

λɛ [0, 0.5] denotes that the DM is a risk seeker who prefers uncertainty over certainty, i.e. the probabilistic mean of indeterminacy and falsity membership.

λ = 0.5.5 indicates that the DM is unbiased (neither risk seeker nor risk-averse investors) about determining the parameters like supply, demand, and transportation cost of NTP

λɛ (0.5, 1] denotes that the DM is a risk-averse investor when it comes to determining the parameters of the NTP problem and prefers certainty.

4 Proposed approach for solving NTP

In this section, we develop a new solution procedure for the NTP. We take into account m number of resource sites from which the commodities will be transferred to n number of demand places. Here, we assume that the transportation cost, supply, and demand are a neutrosophic number. Let ${\tilde{c}}_{ij}^{NTN}$ be the neutrosophic transportation cost form i^th source to j^th destination (i = 1, 2, . . . m, j = 1, 2, . . . , n). Let ${\tilde{S}}_{i}^{NTN}$ be the neutrosophic supply for the i^th company i = 1,2,...m and ${\tilde{D}}_{j}^{NTN}$ is the neutrosophic demand for the j^th destination. The NTP table is given in Table 1.

Table 1
General NTP

1 2 … n Supply

1 ${\tilde{c}}_{11}^{NTN}$ ${\tilde{c}}_{12}^{NTN}$ … ${\tilde{c}}_{1 n}^{NTN}$ ${\tilde{s}}_{1}^{NTN}$

2 ${\tilde{c}}_{21}^{NTN}$ ${\tilde{c}}_{22}^{NTN}$ … ${\tilde{c}}_{2 n}^{NTN}$ ${\tilde{s}}_{2}^{NTN}$

⋮ ⋮ ⋮ ⋮ ⋮ ⋮

m ${\tilde{c}}_{m 1}^{NTN}$ ${\tilde{c}}_{m 2}^{NTN}$ … ${\tilde{c}}_{mn}^{NTN}$ ${\tilde{s}}_{m}^{NTN}$

Demand ${\tilde{D}}_{1}^{NTN}$ ${\tilde{D}}_{2}^{NTN}$ … ${\tilde{D}}_{n}^{NTN}$

	1	2	…	n	Supply
1	${\tilde{c}}_{11}^{NTN}$	${\tilde{c}}_{12}^{NTN}$	…	${\tilde{c}}_{1 n}^{NTN}$	${\tilde{s}}_{1}^{NTN}$
2	${\tilde{c}}_{21}^{NTN}$	${\tilde{c}}_{22}^{NTN}$	…	${\tilde{c}}_{2 n}^{NTN}$	${\tilde{s}}_{2}^{NTN}$
⋮	⋮	⋮	⋮	⋮	⋮
m	${\tilde{c}}_{m 1}^{NTN}$	${\tilde{c}}_{m 2}^{NTN}$	…	${\tilde{c}}_{mn}^{NTN}$	${\tilde{s}}_{m}^{NTN}$
Demand	${\tilde{D}}_{1}^{NTN}$	${\tilde{D}}_{2}^{NTN}$	…	${\tilde{D}}_{n}^{NTN}$

Where, ${\tilde{c}}_{ij}^{NTN} = < (c_{1 ij}, c_{2 ij}, c_{3 ij}, c_{4 ij}); w_{{\tilde{c}}_{ij}^{NTN}}$ , $u_{{\tilde{c}}_{ij}^{NTN}}, y_{{\tilde{c}}_{ij}^{NTN}} >$ , i = 1, 2, . . . m, j = 1, 2, . . . , n

${\tilde{S}}_{i}^{NTN} = < (s_{1 i}, s_{2 i}, s_{3 i}, s_{4 i}); w_{{\tilde{s}}_{i}^{NTN}}$ , $u_{{\tilde{s}}_{i}^{NTN}}, y_{{\tilde{s}}_{i}^{NTN}} >$ , i = 1, 2, . . . m

${\tilde{D}}_{j}^{NTN} = < (d_{1 j}, d_{2 j}, d_{3 j}, d_{4 j}); w_{{\tilde{d}}_{j}^{NTN}}$ , $u_{{\tilde{d}}_{j}^{NTN}}, y_{{\tilde{d}}_{j}^{NTN}} >$ , j = 1, 2, . . . , n are SVNTrN.

The NTP is solved by using the weighted possibility mean value approach. The detailed procedure is explained as follows:

Step: 1 Using equation (1), calculate the left PM of TM function of ${\tilde{c}}_{ij}^{NTN}$ , ${\tilde{S}}_{i}^{NTN}$ , and ${\tilde{D}}_{j}^{NTN}$ which is denoted $M^{L} [{\tilde{c}}_{ij}^{NTN} (α)]$ , $M^{L} [{\tilde{S}}_{i}^{NTN} (α)]$ , and $M^{L} [{\tilde{D}}_{j}^{NTN} (α)]$ , respectively as follows: $\begin{matrix} M^{L} [{\tilde{c}}_{ij}^{NTN} (α)] = {(w_{{\tilde{c}}_{ij}^{NTN}})}^{2} [\frac{c_{1 ij} + 2 c_{2 ij}}{3}], \end{matrix}$ $\begin{matrix} M^{L} [{\tilde{S}}_{i}^{NTN} (α)] = {(w_{{\tilde{s}}_{i}^{NTN}})}^{2} [\frac{s_{1 i} + 2 s_{2 i}}{3}], \end{matrix}$ $\begin{matrix} M^{L} [{\tilde{D}}_{j}^{NTN} (α)] = {(w_{{\tilde{d}}_{j}^{NTN}})}^{2} [\frac{d_{1 j} + 2 d_{2 j}}{3}], \end{matrix}$ where i = 1,2,...m, j = 1,2,...,n.

Step: 2 Using equation (2), calculate the right PM of TM function of ${\tilde{c}}_{ij}^{NTN}$ , ${\tilde{S}}_{i}^{NTN}$ , and ${\tilde{D}}_{j}^{NTN}$ which is denoted $M^{R} [{\tilde{c}}_{ij}^{NTN} (α)]$ , $M^{R} [{\tilde{S}}_{i}^{NTN} (α)]$ , and $M^{R} [{\tilde{D}}_{j}^{NTN} (α)]$ , respectively as follows: $\begin{matrix} M^{R} [{\tilde{c}}_{ij}^{NTN} (α)] = (w_{{\tilde{c}}_{ij}^{NTN}})^{2} [\frac{c_{4 ij} + 2 c_{3 ij}}{3}] \end{matrix}$ $\begin{matrix} M^{R} [{\tilde{S}}_{i}^{NTN} (α)] = (w_{{\tilde{s}}_{i}^{NTN}})^{2} [\frac{s_{4 i} + 2 s_{3 i}}{3}] \end{matrix}$ $\begin{matrix} M^{R} [{\tilde{D}}_{j}^{NTN} (α)] = {(w_{{\tilde{d}}_{j}^{NTN}})}^{2} [\frac{d_{4 j} + 2 d_{3 j}}{3}], \end{matrix}$ where i = 1,2,...m, j = 1,2,...,n.

Step: 3 Compute the left PM of IM function of ${\tilde{c}}_{ij}^{NTN}$ , ${\tilde{S}}_{i}^{NTN}$ , and ${\tilde{D}}_{j}^{NTN}$ which is denoted $M^{L} [{\tilde{c}}_{ij}^{NTN} (β)]$ , $M^{L} [{\tilde{S}}_{i}^{NTN} (β)]$ , and $M^{L} [{\tilde{D}}_{j}^{NTN} (β)]$ respectively. By using equation (4) as follows: $\begin{matrix} M^{L} [{\tilde{c}}_{ij}^{NTN} (β)] = \frac{2 (c_{2 ij} - c_{1 ij}) + (1 + u_{{\tilde{c}}_{ij}^{NTN}} + {(u_{{\tilde{c}}_{ij}^{NTN}})}^{2}) + 3 (c_{1 ij} u_{{\tilde{c}}_{ij}^{NTN}} - c_{2 ij}) (1 + u_{{\tilde{c}}_{ij}^{NTN}})}{3} \end{matrix}$ $\begin{matrix} M^{L} [{\tilde{s}}_{i}^{NTN} (β)] = \frac{2 (s_{2 i} - s_{1 i}) + (1 + u_{{\tilde{s}}_{i}^{NTN}} + {(u_{{\tilde{s}}_{i}^{NTN}})}^{2}) + 3 (s_{1 i} u_{{\tilde{s}}_{i}^{NTN}} - s_{2 i}) (1 + u_{{\tilde{s}}_{i}^{NTN}})}{3} \end{matrix}$ $\begin{matrix} M^{L} [{\tilde{d}}_{j}^{NTN} (β)] = \frac{2 (d_{2 j} - d_{1 j}) + (1 + u_{{\tilde{d}}_{j}^{NTN}} + {(u_{{\tilde{d}}_{j}^{NTN}})}^{2}) + 3 (d_{1 j} u_{{\tilde{d}}_{j}^{NTN}} - d_{2 j}) (1 + u_{{\tilde{d}}_{j}^{NTN}})}{3}, \end{matrix}$

Where, i = 1, 2, . . . m, j = 1, 2, . . . , n.

Step: 4 Compute the PM of IM function of ${\tilde{c}}_{ij}^{NTN}$ , ${\tilde{S}}_{i}^{NTN}$ , and ${\tilde{D}}_{j}^{NTN}$ which is denoted $M^{R} [{\tilde{c}}_{ij}^{NTN} (β)]$ , $M^{R} [{\tilde{S}}_{i}^{NTN} (β)]$ , and $M^{R} [{\tilde{D}}_{j}^{NTN} (β)]$ , respectively.By using equation (5) as follows: $\begin{matrix} M^{R} [{\tilde{c}}_{ij}^{NTN} (β)] = \frac{2 (c_{4 ij} - c_{3 ij}) + (1 + u_{{\tilde{c}}_{ij}^{NTN}} + {(u_{{\tilde{c}}_{ij}^{NTN}})}^{2}) - 3 (c_{4 ij} u_{{\tilde{c}}_{ij}^{NTN}} - c_{3 ij}) (1 + u_{{\tilde{c}}_{ij}^{NTN}})}{3} \end{matrix}$ $\begin{matrix} M^{R} [{\tilde{s}}_{i}^{NTN} (β)] = \frac{2 (s_{4 i} - s_{3 i}) + (1 + u_{{\tilde{s}}_{i}^{NTN}} + {(u_{{\tilde{s}}_{i}^{NTN}})}^{2}) - 3 (s_{4 i} u_{{\tilde{s}}_{i}^{NTN}} - s_{3 i}) (1 + u_{{\tilde{s}}_{i}^{NTN}})}{3} \end{matrix}$ $\begin{matrix} M^{R} [{\tilde{d}}_{j}^{NTN} (β)] = \frac{2 (d_{4 j} - d_{3 j}) + (1 + u_{{\tilde{d}}_{j}^{NTN}} + {(u_{{\tilde{d}}_{j}^{NTN}})}^{2}) - 3 (d_{4 j} u_{{\tilde{d}}_{j}^{NTN}} - d_{3 j}) (1 + u_{{\tilde{d}}_{j}^{NTN}})}{3}, \end{matrix}$

Where, i = 1, 2, . . . m, j = 1, 2, . . . , n.

Step: 5 Compute the left PM of FM function of ${\tilde{c}}_{ij}^{NTN}$ , ${\tilde{S}}_{i}^{NTN}$ , and ${\tilde{D}}_{j}^{NTN}$ which is denoted $M^{L} [{\tilde{c}}_{ij}^{NTN} (γ)]$ , $M^{L} [{\tilde{S}}_{i}^{NTN} (γ)]$ , and $M^{L} [{\tilde{D}}_{j}^{NTN} (γ)]$ , respectively by using equation (7) as follows: $\begin{matrix} M^{L} [{\tilde{c}}_{ij}^{NTN} (γ)] = \frac{3 (c_{2 ij} - c_{1 ij} y_{{\tilde{c}}_{ij}^{NTN}}) (1 + y_{{\tilde{c}}_{ij}^{NTN}}) - 2 (c_{2 ij} - c_{1 ij}) (1 + y_{{\tilde{c}}_{ij}^{NTN}} + {(y_{{\tilde{c}}_{ij}^{NTN}})}^{2})}{3} \end{matrix}$ $\begin{matrix} M^{L} [{\tilde{s}}_{i}^{NTN} (γ)] = \frac{3 (s_{2 i} - s_{1 i} y_{{\tilde{s}}_{i}^{NTN}}) (1 + y_{{\tilde{s}}_{i}^{NTN}}) - 2 (s_{2 i} - s_{1 i}) (1 + y_{{\tilde{s}}_{i}^{NTN}} + {(y_{{\tilde{s}}_{i}^{NTN}})}^{2})}{3} \end{matrix}$ $\begin{matrix} M^{L} [{\tilde{d}}_{j}^{NTN} (γ)] = \frac{3 (d_{2 j} - d_{1 j} y_{{\tilde{d}}_{j}^{NTN}}) (1 + y_{{\tilde{d}}_{j}^{NTN}}) - 2 (d_{2 j} - d_{1 j}) (1 + y_{{\tilde{d}}_{j}^{NTN}} + {(y_{{\tilde{d}}_{j}^{NTN}})}^{2})}{3}, \end{matrix}$

where i = 1,2,...m, j = 1,2,...,n.

Step: 6 Compute the right PM of FM function of ${\tilde{c}}_{ij}^{NTN}$ , ${\tilde{S}}_{i}^{NTN}$ , and ${\tilde{D}}_{j}^{NTN}$ which is denoted $M^{L} [{\tilde{c}}_{ij}^{NTN} (γ)]$ , $M^{L} [{\tilde{S}}_{i}^{NTN} (γ)]$ , and $M^{L} [{\tilde{D}}_{j}^{NTN} (γ)]$ , respectively by using equation (8) as follows: $\begin{matrix} M^{R} [{\tilde{c}}_{ij}^{NTN} (γ)] = \frac{[2 (c_{4 ij} - c_{3 ij}) (1 + y_{{\tilde{c}}_{ij}^{NTN}} + {(y_{{\tilde{c}}_{ij}^{NTN}})}^{2}) - 3 (c_{4 ij} y_{{\tilde{c}}_{ij}^{NTN}} - c_{3 ij}) (1 + y_{{\tilde{c}}_{ij}^{NTN}})]}{3} \end{matrix}$ $\begin{matrix} M^{R} [{\tilde{s}}_{i}^{NTN} (γ)] = \frac{[2 (s_{4 i} - s_{3 i}) (1 + y_{{\tilde{s}}_{i}^{NTN}} + {(y_{{\tilde{s}}_{i}^{NTN}})}^{2}) - 3 (s_{4 i} y_{{\tilde{s}}_{i}^{NTN}} - s_{3 i}) (1 + y_{{\tilde{s}}_{i}^{NTN}})]}{3} \end{matrix}$ $\begin{matrix} M^{R} [{\tilde{d}}_{j}^{NTN} (γ)] = \frac{[2 (d_{4 j} - d_{3 j}) (1 + y_{{\tilde{d}}_{j}^{NTN}} + {(y_{{\tilde{d}}_{j}^{NTN}})}^{2}) - 3 (d_{4 j} y_{{\tilde{d}}_{j}^{NTN}} - d_{3 j}) (1 + y_{{\tilde{d}}_{j}^{NTN}})]}{3}, \end{matrix}$

Where, i = 1,2,...m, j = 1,2,...,n.

Step: 7 Evaluate the PM of the TM for ${\tilde{c}}_{ij}^{NTN}$ , ${\tilde{S}}_{i}^{NTN}$ , and ${\tilde{D}}_{j}^{NTN}$ which is denoted $M [{\tilde{c}}_{ij}^{NTN} (α)]$ , $M [{\tilde{s}}_{i}^{NTN} (α)]$ , and $M [{\tilde{d}}_{j}^{NTN} (α)]$ , respectively by using equation (3) as follows: $\begin{matrix} M [{\tilde{c}}_{ij}^{NTN} (α)] \\ = \frac{1}{6} [(w_{{\tilde{c}}_{ij}^{NTN}})^{2} (c_{1 ij} + 2 c_{2 ij} + 2 c_{3 ij} + c_{4 ij})], \end{matrix}$ $\begin{matrix} M [{\tilde{s}}_{i}^{NTN} (α)] \\ = \frac{1}{6} [(w_{{\tilde{s}}_{i}^{NTN}})^{2} (s_{1 i} + 2 s_{2 i} + 2 s_{3 i} + s_{4 i})], \end{matrix}$ $\begin{matrix} M [{\tilde{d}}_{j}^{NTN} (α)] \\ = \frac{1}{6} [(w_{{\tilde{d}}_{j}^{NTN}})^{2} (d_{1 j} + 2 d_{2 j} + 2 d_{3 j} + d_{4 j})], \end{matrix}$

Where, i = 1,2,...m, j = 1,2,...,n.

Step: 8 Evaluate the PM of the IM for ${\tilde{c}}_{ij}^{NTN}$ , ${\tilde{S}}_{i}^{NTN}$ , and ${\tilde{D}}_{j}^{NTN}$ which is denoted $M [{\tilde{c}}_{ij}^{NTN} (β)]$ , $M [{\tilde{s}}_{i}^{NTN} (β)]$ , and $M [{\tilde{d}}_{j}^{NTN} (β)]$ , respectively, by using equation (6) as follows: $\begin{matrix} M [{\tilde{c}}_{ij}^{NTN} (β)] = \frac{1}{6} [(2 c_{1 ij} + c_{2 ij} + c_{3 ij} + 2 c_{4 ij}) \\ - (c_{1 ij} - c_{2 ij} - c_{3 ij} + c_{4 ij}) u_{{\tilde{c}}_{ij}^{NTN}} \\ - (c_{1 ij} + 2 c_{2 ij} + 2 c_{3 ij} + c_{4 ij}) {(u_{{\tilde{c}}_{ij}^{NTN}})}^{2}] \end{matrix}$

$\begin{matrix} M [{\tilde{s}}_{i}^{NTN} (β)] = \frac{1}{6} [(2 s_{1 i} + s_{2 i} + s_{3 i} + 2 s_{4 i}) \\ - (s_{1 i} - s_{2 i} - s_{3 i} + s_{4 i}) u_{{\tilde{s}}_{i}^{NTN}} \\ - (s_{1 i} + 2 s_{2 i} + 2 s_{3 j} + s_{4 j}) (u_{{\tilde{s}}_{i}^{NTN}})^{2}] \end{matrix}$ $\begin{matrix} M [{\tilde{d}}_{j}^{NTN} (β)] = \frac{1}{6} [(2 d_{1 j} + d_{2 j} + d_{3 j} + d_{4 j}) \\ - (d_{1 j} - d_{2 j} - d_{3 j} + d_{4 j}) u_{{\tilde{d}}_{j}^{NTN}} \\ - (d_{1 j} + 2 d_{2 j} + 2 d_{3 j} + d_{4 j}) (u_{{\tilde{d}}_{j}^{NTN}})^{2}] \end{matrix}$

Where, i = 1,2,...m, j = 1,2,...,n.

Step: 9 Evaluate the PM of FM for ${\tilde{c}}_{ij}^{NTN}$ , ${\tilde{S}}_{i}^{NTN}$ , and ${\tilde{D}}_{j}^{NTN}$ which is denoted $M [{\tilde{c}}_{ij}^{NTN} (γ)]$ , $M [{\tilde{s}}_{i}^{NTN} (γ)]$ , and $M [{\tilde{d}}_{j}^{NTN} (γ)]$ respectively, by using equation (9) as follows: $\begin{matrix} M [{\tilde{c}}_{ij}^{NTN} (γ)] = \frac{1}{6} [(2 c_{1 ij} + c_{2 ij} + c_{3 ij} + 2 c_{4 ij}) \\ - (c_{1 ij} - c_{2 ij} - c_{3 ij} + c_{4 ij}) \\ y_{{\tilde{c}}_{ij}^{NTN}} - (c_{1 ij} + 2 c_{2 ij} + 2 c_{3 ij} + c_{4 ij}) {(y_{{\tilde{c}}_{ij}^{NTN}})}^{2}] \end{matrix}$ $\begin{matrix} M [{\tilde{s}}_{i}^{NTN} (γ)] = \frac{1}{6} [(2 s_{1 i} + s_{2 i} + s_{3 i} + 2 s_{4 i}) \\ - (s_{1 i} - s_{2 i} - s_{3 i} + s_{4 i}) y_{{\tilde{s}}_{i}^{NTN}} \\ - (s_{1 i} + 2 s_{2 i} + 2 s_{3 j} + s_{4 j}) (y_{{\tilde{s}}_{i}^{NTN}})^{2}] \end{matrix}$

$\begin{matrix} M [{\tilde{d}}_{j}^{NTN} (γ)] = \frac{1}{6} [(2 d_{1 j} + d_{2 j} + d_{3 j} + d_{4 j}) \\ - (d_{1 j} - d_{2 j} - d_{3 j} + d_{4 j}) y_{{\tilde{d}}_{j}^{NTN}} \\ - (d_{1 j} + 2 d_{2 j} + 2 d_{3 j} + d_{4 j}) (u_{y})^{2}] \end{matrix}$ where i = 1,2,...m, j = 1,2,...,n,

Step: 10 Calculate the weighted PM value of ${\tilde{c}}_{ij}^{NTN}$ , ${\tilde{S}}_{i}^{NTN}$ , and ${\tilde{D}}_{j}^{NTN}$ which is denoted $V ({\tilde{c}}_{ij}^{NTN}, λ)$ , $V ({\tilde{s}}_{i}^{NTN}, λ)$ , and $V ({\tilde{d}}_{j}^{NTN}, λ)$ , respectively, by using Equation (10) as follows:

$\begin{matrix} \begin{matrix} V ({\tilde{c}}_{ij}^{NTN}, λ) \\ = λ (\frac{1}{6} [(w_{{\tilde{c}}_{ij}^{NTN}})^{2} (c_{1 ij} + 2 c_{2 ij} + 2 c_{3 ij} + c_{4 ij})]) \\ + (1 - λ) (\frac{1}{6} [(2 c_{1 ij} + c_{2 ij} + c_{3 ij} + 2 c_{4 ij}) \\ - (c_{1 ij} - c_{2 ij} - c_{3 ij} + c_{4 ij}) u_{{\tilde{c}}_{ij}^{NTN}} \\ - (c_{1 ij} + 2 c_{2 ij} + 2 c_{3 ij} + c_{4 ij}) {(u_{{\tilde{c}}_{ij}^{NTN}})}^{2}]) \\ + (1 - λ) (\frac{1}{6} [(2 c_{1 ij} + c_{2 ij} + c_{3 ij} + 2 c_{4 ij}) \\ - (c_{1 ij} - c_{2 ij} - c_{3 ij} + c_{4 ij}) y_{{\tilde{c}}_{ij}^{NTN}} \\ - (c_{1 ij} + 2 c_{2 ij} + 2 c_{3 ij} + c_{4 ij}) {(y_{{\tilde{c}}_{ij}^{NTN}})}^{2}]) \end{matrix} \end{matrix}$

$\begin{matrix} \begin{matrix} V ({\tilde{s}}_{i}^{NTN}, λ) \\ = λ (\frac{1}{6} [(w_{{\tilde{s}}_{i}^{NTN}})^{2} (s_{1 i} + 2 s_{2 i} + 2 s_{3 i} + s_{4 i})]) \\ + (1 - λ) (\frac{1}{6} [(2 s_{1 i} + s_{2 i} + s_{3 i} + 2 s_{4 i}) \\ - (s_{1 i} - s_{2 i} - s_{3 i} + s_{4 i}) u_{{\tilde{s}}_{i}^{NTN}} \\ - (s_{1 i} + 2 s_{2 i} + 2 s_{3 j} + s_{4 j}) (u_{{\tilde{s}}_{i}^{NTN}})^{2}]) \\ + (1 - λ) (\frac{1}{6} [(2 s_{1 i} + s_{2 i} + s_{3 i} + 2 s_{4 i}) \\ - (s_{1 i} - s_{2 i} - s_{3 i} + s_{4 i}) y_{{\tilde{s}}_{i}^{NTN}} \\ - (s_{1 i} + 2 s_{2 i} + 2 s_{3 j} + s_{4 j}) (y_{{\tilde{s}}_{i}^{NTN}})^{2}) \end{matrix} \end{matrix}$

$\begin{matrix} \begin{matrix} V ({\tilde{d}}_{j}^{NTN}, λ) \\ = λ (\frac{1}{6} [(w_{{\tilde{d}}_{j}^{NTN}})^{2} (d_{1 j} + 2 d_{2 j} + 2 d_{3 j} + d_{4 j})]) \\ + (1 - λ) (\frac{1}{6} [(2 d_{1 j} + d_{2 j} + d_{3 j} + d_{4 j}) \\ - (d_{1 j} - d_{2 j} - d_{3 j} + d_{4 j}) u_{{\tilde{d}}_{j}^{NTN}} \\ - (d_{1 j} + 2 d_{2 j} + 2 d_{3 j} + d_{4 j}) (u_{{\tilde{d}}_{j}^{NTN}})^{2}]) \\ + (1 - λ) (\frac{1}{6} [(2 d_{1 j} + d_{2 j} + d_{3 j} + d_{4 j}) \\ - (d_{1 j} - d_{2 j} - d_{3 j} + d_{4 j}) y_{{\tilde{d}}_{j}^{NTN}} \\ - (d_{1 j} + 2 d_{2 j} + 2 d_{3 j} + d_{4 j}) (u_{y})^{2}]) \end{matrix} \end{matrix}$

Where, i = 1,2,...m, j = 1,2,...,n.

Step: 11 The NTP Table 1 can be converted in to the weighted PM transportation table, which is shown in Table 2.

Table 2

General weighted possibilistic mean transportation

	1	2	…	n	Supply
1	$V ({\tilde{c}}_{11}^{NTN}, λ)$	$V ({\tilde{c}}_{12}^{NTN}, λ)$	…	$V ({\tilde{c}}_{1 n}^{NTN}, λ)$	$V ({\tilde{s}}_{1}^{NTN}, λ)$
2	$V ({\tilde{c}}_{21}^{NTN}, λ)$	$V ({\tilde{c}}_{22}^{NTN}, λ)$	…	$V ({\tilde{c}}_{2 n}^{NTN}, λ)$	$V ({\tilde{s}}_{2}^{NTN}, λ)$
⋮	⋮	⋮	⋮	⋮	⋮
M	$V ({\tilde{c}}_{m 1}^{NTN}, λ)$	$V ({\tilde{c}}_{m 2}^{NTN}, λ)$	…	$V ({\tilde{c}}_{mn}^{NTN}, λ)$	$V ({\tilde{s}}_{m}^{NTN}, λ)$
Demand	$V ({\tilde{D}}_{1}^{NTN}, λ)$	$V ({\tilde{D}}_{2}^{NTN}, λ)$	…	$V ({\tilde{D}}_{n}^{NTN}, λ)$

Step: 12 Convert NTP in to the following LPP $\begin{matrix} Min Z = \sum_{i = 1}^{m} \sum_{j = 1}^{n} V ({\tilde{c}}_{ij}^{NTN}, λ) x_{ij} \end{matrix}$

Subject to $\begin{matrix} \sum_{j = 1}^{n} x_{ij} ⩽ V ({\tilde{s}}_{i}^{NTN}, λ), i = 1, 2, \dots, m, \end{matrix}$ $\begin{matrix} \sum_{i = 1}^{m} x_{ij} ⩽ V ({\tilde{D}}_{j}^{NTN}, λ), j = 1, 2, \dots, n, \end{matrix}$ $\begin{matrix} x_{ij} ⩾ 0, i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}$

Step: 13 According to step 10, transform the problem obtained by step 12 into the following LPP.

$\begin{matrix} \begin{matrix} MinZ \\ = \sum_{i = 1}^{m} \sum_{j = 1}^{n} (λ (\frac{1}{6} [(w_{{\tilde{c}}_{ij}^{NTN}})^{2} (c_{1 ij} + 2 c_{2 ij} + 2 c_{3 ij} + c_{4 ij})]) + \\ (1 - λ) (\frac{1}{6} [(2 c_{1 ij} + c_{2 ij} + c_{3 ij} + 2 c_{4 ij}) \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} - (c_{1 ij} - c_{2 ij} - c_{3 ij} + c_{4 ij}) u_{{\tilde{c}}_{ij}^{NTN}} \\ - (c_{1 ij} + 2 c_{2 ij} + 2 c_{3 ij} + c_{4 ij}) {(u_{{\tilde{c}}_{ij}^{NTN}})}^{2}]) \\ + (1 - λ) (\frac{1}{6} [(2 c_{1 ij} + c_{2 ij} + c_{3 ij} + 2 c_{4 ij}) \\ - (c_{1 ij} - c_{2 ij} - c_{3 ij} + c_{4 ij}) y_{{\tilde{c}}_{ij}^{NTN}} \\ - (c_{1 ij} + 2 c_{2 ij} + 2 c_{3 ij} + c_{4 ij}) {(y_{{\tilde{c}}_{ij}^{NTN}})}^{2}])) x_{ij} \end{matrix} \end{matrix}$

Subject to

$\begin{matrix} \begin{matrix} \sum_{j = 1}^{n} x_{ij} ⩽ λ (\frac{1}{6} [(w_{{\tilde{s}}_{i}^{NTN}})^{2} (s_{1 i} + 2 s_{2 i} + 2 s_{3 i} + s_{4 i})]) \\ + (1 - λ) (\frac{1}{6} [(2 s_{1 i} + s_{2 i} + s_{3 i} + 2 s_{4 i}) \\ - (s_{1 i} - s_{2 i} - s_{3 i} + s_{4 i}) u_{{\tilde{s}}_{i}^{NTN}} \\ - (s_{1 i} + 2 s_{2 i} + 2 s_{3 j} + s_{4 j}) (u_{{\tilde{s}}_{i}^{NTN}})^{2}]) \\ + (1 - λ) (\frac{1}{6}), \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} \sum_{i = 1}^{m} x_{ij} ⩽ λ (\frac{1}{6} [(w_{{\tilde{d}}_{j}^{NTN}})^{2} (d_{1 j} + 2 d_{2 j} + 2 d_{3 j} + d_{4 j})]) \\ + (1 - λ) (\frac{1}{6} [(2 d_{1 j} + d_{2 j} + d_{3 j} + d_{4 j}) \\ - (d_{1 j} - d_{2 j} - d_{3 j} + d_{4 j}) u_{{\tilde{d}}_{j}^{NTN}} \\ - (d_{1 j} + 2 d_{2 j} + 2 d_{3 j} + d_{4 j}) (u_{{\tilde{d}}_{j}^{NTN}})^{2}]) \\ + (1 - λ) (\frac{1}{6} [(2 d_{1 j} + d_{2 j} + d_{3 j} + d_{4 j}) \\ - (d_{1 j} - d_{2 j} - d_{3 j} + d_{4 j}) y_{{\tilde{d}}_{j}^{NTN}} \\ - (d_{1 j} + 2 d_{2 j} + 2 d_{3 j} + d_{4 j}) (u_{y})^{2}]), \end{matrix} \end{matrix}$ $\begin{matrix} x_{ij} ⩾ 0, i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}$

Step: 14 Solving the LPP obtained in step 13 for different values of λ (0≤λ≤1), we get the dynamic optimal transportation cost. The framework of the proposed method is explained in the following diagram, which is shown in Fig. 1.

Fig. 1

Framework of the proposed method.

5 Incorrect assumptions of the existing methods

This section discusses the incorrect assumptions of the existing methods.

Tamaraiselvi and Santhi [28] have used the following ranking function to convert SVNTrN into crisp value. Let ${\tilde{A}}^{NTN} = < (a_{1}, a_{2}, a_{3}, a_{4})$ ; $w_{{\tilde{A}}^{NTN}}$ , $u_{{\tilde{A}}^{NTN}}, y_{{\tilde{A}}^{NTN}}} >$ be a SVNTrN. The scoring function $S ({\tilde{A}}^{NTN}) = (\frac{1}{16}) (a_{1} + a_{2} + a_{3} + a_{4}) \times (w_{{\tilde{A}}^{NTN}} + (1 - u_{{\tilde{A}}^{NTN}}) + (1 - y_{{\tilde{A}}^{NTN}})$ . They assume that scoring function satisfied the linear property. That is, If ${\tilde{A}}^{NTN}$ and ${\tilde{B}}^{NTN}$ are two SVTrNNs, then $S ({\tilde{A}}^{NTN} + {\tilde{B}}^{NTN}) = S ({\tilde{A}}^{NTN}) + S ({\tilde{B}}^{NTN})$ . But, Singh et al. [26] showed that this assumption is incorrect. They proved $S ({\tilde{A}}^{NTN} + {\tilde{B}}^{NTN}) \neq S ({\tilde{A}}^{NTN}) + S ({\tilde{B}}^{NTN})$ .

Singh et al. [28] used a modified scoring function, in order to adding two SVNTrNs. That is, If ${\tilde{A}}^{NTN} = < (a_{1}, a_{2}, a_{3}, a_{4})$ ; $w_{{\tilde{A}}^{NTN}}$ , $u_{{\tilde{A}}^{NTN}}, y_{{\tilde{A}}^{NTN}}} >$ and ${\tilde{B}}^{NTN} = < (b_{1}, b_{2}, b_{3}, b_{4})$ ; $w_{{\tilde{B}}^{NTN}}$ , $u_{{\tilde{B}}^{NTN}}, y_{{\tilde{B}}^{NTN}}} >$ are two SVNTrNs, then the addition of two SVTrNNs based on the modified scoring function, which is defined by

\begin{matrix} \begin{matrix} M ({\tilde{A}}^{NTN} + {\tilde{B}}^{NTN}) = \\ (min (w_{{\tilde{A}}^{NTN}}, w_{{\tilde{B}}^{NTN}}) + \max ((1 - u_{{\tilde{A}}^{NTN}}), (1 - u_{{\tilde{B}}^{NTN}})) + \max ((1 - y_{{\tilde{A}}^{NTN}}), (1 - y_{{\tilde{B}}^{NTN}}))) \\ (\frac{S ({\tilde{A}}^{NTN})}{(w_{{\tilde{A}}^{NTN}} + (1 - u_{{\tilde{A}}^{NTN}}) + (1 - y_{{\tilde{A}}^{NTN}})} + \frac{S ({\tilde{B}}^{NTN})}{(w_{{\tilde{B}}^{NTN}} + (1 - u_{{\tilde{B}}^{NTN}}) + (1 - y_{{\tilde{B}}^{NTN}})}) \end{matrix} \end{matrix}

Where, $S ({\tilde{A}}^{NTN}) = (\frac{1}{16}) (a_{1} + a_{2} + a_{3} + a_{4}) \times (w_{{\tilde{A}}^{NTN}} + (1 - u_{{\tilde{A}}^{NTN}}) + (1 - y_{{\tilde{A}}^{NTN}})$ and $S ({\tilde{B}}^{NTN}) = (\frac{1}{16}) (b_{1} + b_{2} + b_{3} + b_{4}) \times (w_{{\tilde{B}}^{NTN}} + (1 - u_{{\tilde{B}}^{NTN}}) + (1 - y_{{\tilde{B}}^{NTN}})$ .

Which is also not satisfied the assumption $M ({\tilde{A}}^{NTN} + {\tilde{B}}^{NTN}) = M ({\tilde{A}}^{NTN}) + M ({\tilde{B}}^{NTN})$ .

To overcome the above drawbacks, the proposed method NTP is converted as a set of LPPs. After that, the problem can be solved using any one of the linear programming solvers.

6 Advantages of the proposed method

This section demonstrates the major advantages of the suggested method over existing NTP-solving methods.

Since DMs have no specific knowledge regarding the data connected to its various transportation factors, the parameters (supply capacities, unit transportation costs, and so on) of the TP are uncertain values. The DMs perceive more flexibility in their decision-making process when they employ the NS approximation to this feasible zone. As a result, we provide these components in NS form, with the source, demand, and conveyances all being NSs.

Neutrosophic numbers are a generalization of fuzzy numbers and intuitionistic fuzzy numbers. Hence, the recommended approach can be exploited for solving FTPs, IFTPs, and NTPs.

To overcome the drawbacks of the existing methods, the proposed dynamic approach has given a superior solution in a neutrosophic environment. As a result, the developed method can be used to solve a variety of neutrosophic optimization problems.

Since the TP is solved using conventional linear programming algorithms, the suggested method is easier to employ in real-world applications than existing methods for obtaining the neutrosophic optimal solution of NTP.

In previous methods, the solution of the NTP is a single value, but the proposed method gives the range of value based on the DMs interest. Hence the proposed method is more flexible to handle uncertain situations.

7 Numerical illustration

In this section, we solve two real-time transportation problems in a neutrosophic environment using the proposed framework. Further, the obtained dynamic optimum transportation cost compare with existing methods.

7.1 Numerical Example 1

Consider the NTP has three sources and four destinations, which is taken form [28]. The neutrosophic supply, demand and transportation cost are given in Table 3.

Table 3
Transportation table for Example 1

D ₁ D ₂ D ₃ D ₄ Supply

O ₁ < (3,5,6,8)0.6,0.5,0.4> < (5,8,10,14)0.3,0.6,0.6> < (12,15,19,22)0.6,0.4,0.5> < (14,17,21,28)0.8,0.2,0.6> < (22,26,28,32)0.7,0.3,0.4>

O ₂ < (0.1.3,6)0.7,0.5,0.3> < (5,7,9,11)0.9,0.7,0.5) < (15,17,19,21)0.4,0.8,0.4> <9,11,1416)0.5,0.40.7) < (17,22,27,31)0.6,0.4,0.5>

O ₃ < (4,8,11,15)0.6,0.3,0.2> < (1,3,4,6)0.6,0.3,0.5> < (5,7,8,10)0.5,0.4,0.7> < (5,9,14,19)0.3,0.7,0.6> < (21,28,32,37),0.8,0.2,0.4>

Demand <13,16,18,21)0.5,0.5,0.6> < (17,21,24,28)0.8,0.2,0.4> < (24,29,32,34)0.9,0.5,0.3> < (6,10,13,15)0.7,0.3,0.4>

	D ₁	D ₂	D ₃	D ₄	Supply
O ₁	< (3,5,6,8)0.6,0.5,0.4>	< (5,8,10,14)0.3,0.6,0.6>	< (12,15,19,22)0.6,0.4,0.5>	< (14,17,21,28)0.8,0.2,0.6>	< (22,26,28,32)0.7,0.3,0.4>
O ₂	< (0.1.3,6)0.7,0.5,0.3>	< (5,7,9,11)0.9,0.7,0.5)	< (15,17,19,21)0.4,0.8,0.4>	<9,11,1416)0.5,0.40.7)	< (17,22,27,31)0.6,0.4,0.5>
O ₃	< (4,8,11,15)0.6,0.3,0.2>	< (1,3,4,6)0.6,0.3,0.5>	< (5,7,8,10)0.5,0.4,0.7>	< (5,9,14,19)0.3,0.7,0.6>	< (21,28,32,37),0.8,0.2,0.4>
Demand	<13,16,18,21)0.5,0.5,0.6>	< (17,21,24,28)0.8,0.2,0.4>	< (24,29,32,34)0.9,0.5,0.3>	< (6,10,13,15)0.7,0.3,0.4>

Weighted PM value of the neutrosophic transportation cost, supply and demand are calculated by using step 1 to step 9, which is given in the following Table 4.

Table 4

Weighted possibilistic mean transportation table

	D ₁	D ₂	D ₃	D ₄	Supply
O ₁	1.98λ + (1 - λ)4.125+(1-λ)4.62	0.825λ + (1 - λ)5.98+(1-λ)5.93	6.125λ + (1 - λ)14.28+(1-λ)12.75	12.59λ + (1 - λ)19.01+(1-λ)12.85	13.23λ + (1 - λ)24.57+(1-λ)22.68
O ₂	1.14λ + (1 - λ)1.85+(1-λ)2.36	6.48λ + (1 - λ)4.08+(1-λ)6	2.91λ + (1 - λ)6.64+(1-λ)15.64	3.13λ + (1 - λ)10.5+(1-λ)6.38	8.76λ + (1 - λ)20.37+(1-λ)18.17
O ₃	3.42λ + (1 - λ)8.65+(1-λ)9.12	1.26λ + (1 - λ)3.19+(1-λ)2.63	1.88λ + (1 - λ)6.3+(1-λ)3.83	1.05λ + (1 - λ)6.07+(1-λ)7.53	18.99λ + (1 - λ)28.41+(1-λ)24.72
Demand	4.25λ + (1 - λ)12.75+(1-λ)10.88	14.44λ + (1 - λ)21.6+(1-λ)18.9	24.44λ + (1 - λ)22.59+(1-λ)127.22	5.47λ + (1 - λ)10.06+(1-λ)9.18

For different values of λ, (0 ⩽ λ ⩽ 1) the weighted PM values of the transportation cost, supply and demand are evaluated by using step 10, which is given in Table 5.

Table 5

Weighted PM values of transportation cost, supply and demand different values of λ

	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
${\tilde{C_{11}}}^{NTL}$	9	8	7	7	6	5	5	4	3	3	2
${\tilde{C_{12}}}^{NTL}$	12	11	10	9	7	6	5	4	3	2	1
${\tilde{C_{13}}}^{NTL}$	27	25	23	21	19	17	14	12	10	8	6
${\tilde{C_{14}}}^{NTL}$	32	30	28	26	24	22	20	18	16	15	13
${\tilde{C_{21}}}^{NTL}$	4	4	4	3	3	3	2	2	2	1	1
${\tilde{C_{22}}}^{NTL}$	10	10	9	9	9	8	8	8	7	7	7
${\tilde{C_{23}}}^{NTL}$	22	20	18	16	14	12	11	9	7	5	3
${\tilde{C_{24}}}^{NTL}$	17	16	14	13	11	10	9	7	6	5	3
${\tilde{C_{31}}}^{NTL}$	18	16	15	13	12	11	9	8	6	5	3
${\tilde{C_{32}}}^{NTL}$	6	5	5	4	4	4	3	3	2	2	1
${\tilde{C_{33}}}^{NTL}$	10	9	8	8	7	6	5	4	4	3	2
${\tilde{C_{34}}}^{NTL}$	14	12	11	10	9	7	6	5	4	2	1
${\tilde{S_{1}}}^{NTL}$	47	44	40	37	34	30	27	23	20	17	13
${\tilde{S_{2}}}^{NTL}$	39	36	33	30	27	24	21	18	15	12	9
${\tilde{S_{3}}}^{NTL}$	53	50	46	43	39	36	33	29	26	22	19
${\tilde{D_{1}}}^{NTL}$	24	22	20	18	16	14	12	10	8	6	4
${\tilde{D_{2}}}^{NTL}$	41	38	35	33	30	27	25	22	20	17	14
${\tilde{D_{3}}}^{NTL}$	50	47	45	42	40	37	35	32	30	27	24
${\tilde{D_{4}}}^{NTL}$	19	18	16	15	14	12	11	10	8	7	5

Further, the NTP is converted as a crisp LPP for different values of λ and solved by using LIGO software. The dynamic transportation allotment for different values of λ are given in the Table 6.

Table 6

Dynamic solution of the NTP for Example 1

λ	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
x ₁₁	1	4	2	0	3	3	2	5	5	6	0
x ₁₂	41	38	35	25	30	27	25	22	20	17	12
x ₁₃	0	0	0	0	0	0	2	0	0	0	1
x ₁₄	0	0	0	0	0	0	0	0	0	0	0
x ₂₁	23	18	18	8	13	11	10	5	3	0	0
x ₂₂	0	0	0	7	0	0	0	0	0	0	0
x ₂₃	0	0	0	0	0	1	0	3	4	12	9
x ₂₄	16	17	15	15	14	12	11	10	8	0	0
x ₃₁	0	0	0	0	0	0	0	0	0	0	0
x ₃₂	0	0	0	1	0	0	0	0	0	0	0
x ₃₃	50	49	45	42	40	36	33	29	26	15	14
x ₃₄	3	1	0	0	0	0	0	0	0	7	5
Z	1407	1247	1006	847	701	558	447	331	261	171	78

From the table, we observe that the optimal transportation cost is 1407 for λ=0, and 78 for λ=1. That is, the optimum cost decreases whenλ is increased.λ=0 indicates that the data have more indeterminacy and falsity degrees and it skips the truth membership degree, which is indicating that the expert likes uncertainty. When the uncertainties are high then the transportation cost is high, which is proven in the table. When, the value ofλ increases, the DMs have decreased the indeterminacy and falsity degrees of the data, and increase the truth membership degree knowledge of the parameters. Hence the transportation cost reduces, which is shown in the Table. Whenλ=1 means the DM has confidence in data, which indicates that the information is truth. So, indeterminacy and falsity degrees of the transportation parameters cost, supply, and demand are ignored. Hence, the transportation cost is minimum, which is shown in the table.

7.2 Numerical Example-2

Canned peas are one of the primary products of Sailco Corporation. Three processing plants (in Bellingham, Washington; Eugene, Oregon; and Albert Lea, Minnesota) prepare the peas, which are then trucked to three distributing warehouses in the western United States (Sacramento, California; Salt Lake City, Utah; and Albuquerque, New Mexico). Since transportation expenses are a significant burden, the administration has interested to minimize the transportation cost. Each cannery’s output has been estimated for the succeeding season, and each warehouse has been assigned a specific quantity from the total supply of peas. This data (in truckloads) is provided in Table 7, as well as the shipping cost per truckload for each cannery-warehouse combination given in Table 7. Determine which plan for assigning these shipments to the various cannery-warehouse combinations would minimize the total shipping cost. By using step 1 to step 9, we calculate the weighted PM, which given in the Table 8.

Table 7
Neutrosophic transportation table for Example 2

Shipping cost($) per truckload

Warehouse

D ₁ D ₂ D ₃ Output

O ₁ < (14, 17, 21, 28); 0.8, 0.2, 0.6> 13, 18, 20, 24); 0.6, 0.4, 0.5 14, 16, 21, 23); 0.7, 0.5, 0.3 15, 18, 23, 30); 0.9, 0.2, 0.3

O ₂ 15, 18, 23, 30); 0.9, 0.2, 0.3 6, 10, 13, 15);0.7, 0.3, 0.4 15, 18, 23, 30); 0.9, 0.2, 0.3 20, 25, 30, 35); 0.8, 0.4, 0.2

O ₃ 13, 18, 20, 24); 0.6, 0.4, 0.5 14, 17, 21, 28); 0.8, 0.2, 0.6 14, 16, 21, 23); 0.7, 0.5, 0.3 28, 32, 35, 40); 0.9, 0.3, 0.2

Allocation 20, 25, 30, 35); 0.8, 0.4, 0.2 28, 32, 35, 40); 0.9, 0.3, 0.2 15, 18, 23, 30); 0.9, 0.2, 0.3

Shipping cost($) per truckload
O ₁	< (14, 17, 21, 28); 0.8, 0.2, 0.6>	13, 18, 20, 24); 0.6, 0.4, 0.5	14, 16, 21, 23); 0.7, 0.5, 0.3	15, 18, 23, 30); 0.9, 0.2, 0.3
O ₂	15, 18, 23, 30); 0.9, 0.2, 0.3	6, 10, 13, 15);0.7, 0.3, 0.4	15, 18, 23, 30); 0.9, 0.2, 0.3	20, 25, 30, 35); 0.8, 0.4, 0.2
O ₃	13, 18, 20, 24); 0.6, 0.4, 0.5	14, 17, 21, 28); 0.8, 0.2, 0.6	14, 16, 21, 23); 0.7, 0.5, 0.3	28, 32, 35, 40); 0.9, 0.3, 0.2
Allocation	20, 25, 30, 35); 0.8, 0.4, 0.2	28, 32, 35, 40); 0.9, 0.3, 0.2	15, 18, 23, 30); 0.9, 0.2, 0.3

Table 8

Weighted possibility mean transportation table

	D ₁	D ₂	D ₃	SUPPLY
O ₁	12.58λ + (1 - λ)19.01+(1-λ)12.85	6.78λ + (1 - λ)15.72+(1-λ)14.04	9.1λ + (1 - λ)13.9+(1-λ)16.8	17.2λ + (1 - λ)20.4+(1-λ)19.7
O ₂	17.2λ + (1 - λ)20.4+(1-λ)19.7	5.5λ + (1 - λ)10.1+(1-λ)9.2	17.2λ + (1 - λ)20.4+(1-λ)19.7	17.5λ + (1 - λ)23.1+(1-λ)26.4
O ₃	6.8λ + (1 - λ)15.8+(1-λ)14.1	12.6λ + (1 - λ)19+(1-λ)28.9	9.1λ + (1 - λ)13.9+(1-λ)16.8	27.3λ + (1 - λ)30.7+(1-λ)32.45
DEMAND	17.6λ + (1 - λ)23.1+(1-λ)26.4	27.3λ + (1 - λ)30.7+(1-λ)32.5	17.2λ + (1 - λ)20.4+(1-λ)19.7

Further, for different values of λ (0 ⩽ λ ⩽ 1), we calculate the weighted possibilistic mean of supply, demand and cost, which is given in the Table 9.

Table 9

Weighted PM values of cost, supply and demand different values of λ

	V(0,1)	V(0.1, N)	V(0.2,N)	V(0.3, N)	V(0.4,N)	V(0.5, N)	V(0.6,N)	V(0.7, N)	V(0.8,)	V(0.9,N)	V(1,N)
${\tilde{C_{11}}}^{NTN}$	32	30	28	26	24	22	20	18	16	15	13
${\tilde{C_{12}}}^{NTN}$	30	28	25	23	21	19	16	14	11	9	7
${\tilde{C_{13}}}^{NTN}$	31	29	27	24	22	20	18	16	13	11	9
${\tilde{C_{21}}}^{NTN}$	40	38	36	33	31	29	26	24	22	19	17
${\tilde{C_{22}}}^{NTN}$	19	18	17	15	14	12	11	10	8	7	6
${\tilde{C_{23}}}^{NTN}$	40	38	36	33	31	29	26	24	22	19	17
${\tilde{C_{31}}}^{NTN}$	30	28	25	23	21	18	16	14	11	9	7
${\tilde{C_{32}}}^{NTN}$	32	30	28	26	24	22	20	18	16	15	13
${\tilde{C_{33}}}^{NTN}$	31	29	26	24	22	20	18	16	13	11	9
${\tilde{S_{1}}}^{NTN}$	40	38	36	33	31	29	26	24	22	19	17
${\tilde{S_{2}}}^{NTN}$	50	46	43	40	37	34	30	27	24	21	18
${\tilde{S_{3}}}^{NTN}$	63	60	56	52	49	45	42	38	34	31	27
${\tilde{D_{1}}}^{NTN}$	50	46	43	40	37	34	30	27	24	21	18
${\tilde{D_{2}}}^{NTN}$	63	60	56	52	49	45	42	38	34	31	27
${\tilde{D_{3}}}^{NTN}$	40	38	36	33	31	29	26	24	22	19	21

By using the weighted possibility mean values of cost, supply and demand in Table 9, we formulate the linear programming problems for different values of λ. The linear programming problems are solved by using LINGO software and the solution are depicted in Table 10.

Table 10

Dynamic solution of NTP 1

λ	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
Min z	4630	4078	3032	2611	2299	1832	1470	1186	852	635	450
x ₁₁	0	0	0	0	0	0	0	0	0	0	0
x ₁₂	13	14	13	13	12	11	12	11	10	10	9
x ₁₃	27	24	20	20	19	18	14	13	12	9	8
x ₂₁	0	0	0	0	0	0	0	0	0	0	0
x ₂₂	50	46	43	40	37	34	30	27	24	21	18
x ₂₃	0	0	0	0	0	0	0	0	0	0	0
x ₃₁	50	38	40	40	37	34	30	27	24	1	18
x ₃₂	0	0	0	0	0	0	0	0	0	0	0
x ₃₃	13	22	16	13	12	11	12	11	10	10	9

7.3 Comparative study

In this section, we compare the proposed method within Thamaraiselvi et al. [28], Singh et al. [27], and Dhouib [6] approach. Suppose the numerical example 1 is solved by deterministic supply and demand as given in Table 11. For different values of λ, the weighted values of the transportation cost are evaluated by using step 10. The transportation cost is the same value, which is mentioned in the Table. Further, the NTP is modelled as a crisp LPP for different values of λ with deterministic supply and demand values and solved by using LIGO software. The solutions are given in Table 12.

Table 11
NTP with crisp supply and demand

D ₁ D ₂ D ₃ D ₄ SUPPLY

O ₁ < (3,5,6,8)0.6,0.5,0.4> < (5,8,10,14)0.3,0.6,0.6> < (12,15,19,22)0.6,0.4,0.5> < (14,17,21,28)0.8,0.2,0.6> 26

O ₂ < (0.1.3,6)0.7,0.5,0.3> < (5,7,9,11)0.9,0.7,0.5) < (15,17,19,21)0.4,0.8,0.4> <9,11,1416)0.5,0.40.7) 24

O ₃ < (4,8,11,15)0.6,0.3,0.2> < (1,3,4,6)0.6,0.3,0.5> < (5,7,8,10)0.5,0.4,0.7> < (5,9,14,19)0.3,0.7,0.6> 30

DEMAND 17 23 28 12

	D ₁	D ₂	D ₃	D ₄	SUPPLY
O ₁	< (3,5,6,8)0.6,0.5,0.4>	< (5,8,10,14)0.3,0.6,0.6>	< (12,15,19,22)0.6,0.4,0.5>	< (14,17,21,28)0.8,0.2,0.6>	26
O ₂	< (0.1.3,6)0.7,0.5,0.3>	< (5,7,9,11)0.9,0.7,0.5)	< (15,17,19,21)0.4,0.8,0.4>	<9,11,1416)0.5,0.40.7)	24
O ₃	< (4,8,11,15)0.6,0.3,0.2>	< (1,3,4,6)0.6,0.3,0.5>	< (5,7,8,10)0.5,0.4,0.7>	< (5,9,14,19)0.3,0.7,0.6>	30
DEMAND	17	23	28	12

Table 12

Solution of the comparative example

λ	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
x ₁₁	3	3	3	3	3	3	3	3	3	3	3
x ₁₂	23	23	23	23	23	23	23	23	23	23	23
x ₁₃	0	0	0	0	0	0	0	0	0	0	0
x ₁₄	0	0	0	0	0	0	0	0	0	0	0
x ₂₁	14	14	14	14	14	14	14	14	14	14	14
x ₂₂	0	0	0	0	0	0	0	0	0	0	0
x ₂₃	0	0	0	0	0	0	0	0	0	0	0
x ₂₄	10	10	10	10	10	10	10	10	10	10	10
x ₃₁	0	0	0	0	0	0	0	0	0	0	0
x ₃₂	0	0	0	0	0	0	0	0	0	0	0
x ₃₃	28	28	28	28	28	28	28	28	28	28	18
x ₃₄	2	2	2	2	2	2	2	2	2	2	12
Z	837	769	693	644	545	477	400	324	286	197	124

This problem is solved by Thamaraiselvi et al. [28] approach the transportation cost is 140.118, Singh et al. [27] approach the optimum cost is 275.25, and Dhouib [6] approach optimum transportation cost is 272, whereas the proposed approach, the optimal transportation cost is 837 for λ=0, and 124 for λ=1. From the Table 13, we see that the optimum cost decreases when λ is increased. λ=0 indicates that the data have more indeterminacy and falsity degrees and it skips the truth membership degree, which is indicating that the expert likes uncertainty. When the uncertainties are high then the transportation cost is high, which is proven in the Table 13. When, the value of λ increases, the DMs have decreased the indeterminacy and falsity degrees of the data, and increase the truth membership degree knowledge of the parameters. Hence the transportation cost reduces, which is shown in the Table. When λ=1 means the DM has confidence in data, which indicates that the information is truth only. So, indeterminacy and falsity degrees of the transportation parameters cost, supply, and demand are ignored. Hence, the transportation cost is minimum, which is shown in the table. These dynamic solutions cannot be obtained in the existing approach.

Table 13

Comparison of proposed method vs existing method

S.No.	Thamaraiselvi et al. Approach [28]	Singh et al. Approach [27]	Dhouib Approach [6]
1	140.118	127.25	272

8 Conclusion

Most of the decision-making problems involve not only instability but also ambiguity, uncertainty, inaccuracies, error, contradiction, undefined, unknown, incompleteness, redundancy data. Since the NS can handle uncertain data, many characteristic decision-making problems can be easily solved. In this paper, we have presented a solution framework for solving NTPs. As a result, for the first time in the field of NTP, the idea of the PM has been introduced. The developed method also analyses the DM’s risk attitude/preference parameter and evaluates how it influences the optimal transportation cost. Finally, a numerical example and real-life example are shown to demonstrate the method’s application and performance. It is observed that applying probabilistic means and experts’ risk attitudes, optimum solutions to the neutrosophic optimization issue can meet the objective function to a greater extent. The study also shows that experts will have adequate knowledge about the various possible solutions ranging from the best alternative solution to the worst alternative solution to make proper decisions based on the condition when evaluating the varying degrees of the feasibility of k in the scenario of vagueness, insufficient, and imprecise parameters. Future developments will cover multi-objective NTP and neutrosophic fractional TP, as well as the implications of duality theory on them.

References

Atanassov

, K, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986), 87–96.

Chanas

, Kołodziejczyk

and Machaj

, A fuzzy approach to the transportation problem, Fuzzy Sets System 13(3) (1984), 211–221.

Dinagar

D.S.

and Palanivel

, The transportation problem in fuzzy environment, International Journal of Algorithms Computing and Mathematics 2(3) (2009), 65–71.

Carlsson

and Fullér

, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems 122(2) (2001), 315–326.

Christer

and Robert

, On Possibilistic Mean Value and Variance of Fuzzy Numbers, Science Direct Working Paper No S1574-0358(04)(2001), 70515–8.

Dhouib

, Solving the Single-Valued Trapezoidal Neutrosophic Transportation Problems through the Novel Dhouib-Matrix-TP1 Heuristic, Mathematical Problems in Engineering 2021 (2021), 1–11 Article ID 3945808.

Dhouib

and Dhouib

, Optimizing the trapezoidal fuzzy travelling salesman problem through dhouib-matrix-TSP1 method based on magnitude technique, International Journal of Scientific Research in Mathematical and Statistical Sciences 8(2) (2021), 1–4.

Dubois

and Prade

, Possibility theory: an approach to computerized processing of uncertainty, Plenum, NewYork, 1988.

Edalatpanah

S.A.

, A nonlinear approach for neutrosophic linear programming, Journal of Applied Research on Industrial Engineering 6(4) (2019), 367–373.

10.

Fuller

and Majlender

, On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets System 136 (2003), 363–374.

11.

Garg

and Arora

, Bonferroni mean aggregation operators under intuitionistic fuzzy soft set environment and their applications to decision-making, Journal of Operations Research Society 69 (2018), 1711–1724.

12.

Hitchcock

F.L.

, The distribution of a product from several sources to numerous localities, Journal of Mathematics and Physics 20 (1941), 224–230.

13.

Hussain

R.J.

and Senthil Kumar

, Algorithmic approach for solving intuitionistic fuzzy transportation problem, Applied Mathematical Sciences, 6(80) (2012), 3981–3989.

14.

Kaur

and Kumar

, A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers, Applied Soft Computing 12(3) (2012), 1201–1213.

15.

Khalifa

K.A.E.W.

, Kumar

and Mirjalili

, A KKM approach for inverse capacitated transportation problem in neutrosophic environment, Sādhanā 46 (2021), 166.

16.

Kumar

, Edalatpanah

S.A.

, Jha

and Singh

, A Pythagorean fuzzy approach to the transportation problem, Complex & Intelligent Systems 5(2) (2019), 255–263.

17.

Liu

, Uncertainty theory: an introduction to its axiomatic foundations, Springer, Berlin, 2004.

18.

Maity

, Mardanya

, Roy

S.K.

and Weber

G.W.

, A new approach for solving dual-hesitant fuzzy transportation problem with restrictions, Sādhanā 44(4) (2019), 75.

19.

Majumder

, Kundu

, Kar

and Pal

, Uncertain multiobjective multi-item fxed charge solid transportation problem with budget constraint, Soft Computing 23(10) (2019), 3279–3301.

20.

Miledi

, Dhouib

and Loukil

, Dhouib-Matrix-TSP1 method to optimize octagonal fuzzy travelling salesman problem using α-cut technique, International Journal of Computer and Information Technology 10(3) (2021), 130–133.

21.

NagoorGani

and Abbas

, Solving intuitionistic fuzzy transportation problem using zero sufx algorithm, International, Journal of Mathematical Sciences & Engineering Applications 6 (2012), 73–82.

22.

Pratihar

, Kumar

, Edalatpanah

S.A.

and Dey

, Modified Vogel’s approximation method for transportation problem under uncertain environment, Complex & Intelligent Systems 7(1) (2021), 29–40.

23.

Paul

, Sarma

and Bera

U.K.

, A Neutrosophic Solid Transportation Model with Insufficient Supply, 2019 IEEE Region 10 Symposium (TENSYMP), (2019), pp. 379–384.

24.

Roy

S.K.

and Midya

, Multi-objective fixed-charge solid transportation problem with product blending under intuitionistic fuzzy environment, Applied Intelligence 49(10) (2019), 3524–3538.

25.

Smarandache

, A Unifying Field in Logics, Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press, Rehoboth, NM, USA, 1998.

26.

Singh

, Kumar

and Appadoo

S.S.

, Modified Approach for Optimization of Real Life Transportation Problem in Neutrosophic Environment, Mathematical Problems in Engineering 2017 (2017), 1–9, Article ID 2139791.

27.

Singh

, Arora

and Arora

, Bilevel transportation problem in neutrosophic environment, Computational Applied Mathematics 41 (2022), 44.

28.

Tamaraiselvi

and Santhi

, A new approach for optimization of real life transportation problem in neutrosophic environment,Mathematical Problems in Engineering 2016, Article ID 5950747, 2016.

29.

Veeramani

, Robinson

M.J.

and Vasanthi

, Value- and ambiguity-based approach for solving intuitionistic fuzzy transportation problem with total quantity discounts and incremental quantity discounts, Mathematical Problems in Engineering 2020 (2020), 21 pages, Article ID 8891713.

30.

Veeramani

, Edalatpanah

S.A.

and Sharanya

, Solving the Multi-objective Fractional Transportation Problem through the Neutrosophic Goal Programming Approach, Discrete Dynamics in Nature and Society 2021 (2021), 17 pages, Article ID 7308042.

31.

Wan

P.S.

, Li

F.D.

and Rui

F.Z.

, Possibility mean, variance and covariance of triangular intuitionistic fuzzy numbers, Journal of Intelligent Fuzzy System 24 (2013), 847–858.

32.

Wang

, Smarandache

, Zhang

Y.Q.

and Sunderraman

, Single valued neutrosophic sets, Multispace and Multi-Structure, 4 (2010), 410–413.

33.

Yager

R.R.

, On the specificity of a possibility distribution, Fuzzy Sets System 50 (1992), 279–292.

34.

Zadeh

L.A.

, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets System 1 (1978), 3–28.

35.

Zadeh

L.A.

, Fuzzy sets, Information Sciences 8 (1965), 338–356.

36.

Zimmermann

H.J.

, Fuzzy programming and linear programming with several objective function, Fuzzy Sets and Systems 1(1) (1978), 45–55.

Shipping cost($) per truckload
Warehouse
	D ₁	D ₂	D ₃	Output
O ₁	< (14, 17, 21, 28); 0.8, 0.2, 0.6>	13, 18, 20, 24); 0.6, 0.4, 0.5	14, 16, 21, 23); 0.7, 0.5, 0.3	15, 18, 23, 30); 0.9, 0.2, 0.3
O ₂	15, 18, 23, 30); 0.9, 0.2, 0.3	6, 10, 13, 15);0.7, 0.3, 0.4	15, 18, 23, 30); 0.9, 0.2, 0.3	20, 25, 30, 35); 0.8, 0.4, 0.2
O ₃	13, 18, 20, 24); 0.6, 0.4, 0.5	14, 17, 21, 28); 0.8, 0.2, 0.6	14, 16, 21, 23); 0.7, 0.5, 0.3	28, 32, 35, 40); 0.9, 0.3, 0.2
Allocation	20, 25, 30, 35); 0.8, 0.4, 0.2	28, 32, 35, 40); 0.9, 0.3, 0.2	15, 18, 23, 30); 0.9, 0.2, 0.3

A dynamic optimal solution approach for solving neutrosophic transportation problem

Abstract

Keywords

Nomenclature

1 Introduction

2 Preliminaries

3 PM of SVNNs

3.1 PM of the TM function

Table 1 General NTP 1 2 … n Supply 1 c ˜ 11 NTN c ˜ 12 NTN … c ˜ 1 n NTN s ˜ 1 NTN 2 c ˜ 21 NTN c ˜ 22 NTN … c ˜ 2 n NTN s ˜ 2 NTN ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ m c ˜ m 1 NTN c ˜ m 2 NTN … c ˜ mn NTN s ˜ m NTN Demand D ˜ 1 NTN D ˜ 2 NTN … D ˜ n NTN

6 Advantages of the proposed method

7 Numerical illustration

7.1 Numerical Example 1

References