A New Approach to Solve Fully Fuzzy Multi-Objective Transportation Problem

Abstract

The transportation problem is the problem of transferring goods from several sources or producers to multiple destinations or consumers in a cost-effective way, which is one of the most important problems in the supply chain management problems. The application of this problem in addition to the distribution of goods in the location and production planning problems is also important. Many real-life transportation problems encounter multiple, conflicting, and incommensurable objective functions. In addition, in real applications, due to lack of information, it is not possible to accurately estimate the parameters of this problem. Therefore, the main goal of this paper is to find the Pareto optimal solutions of fully fuzzy multi-objective transportation problem under the conditions of uncertainty. In accordingly, a new approach based on nearest interval approximation is proposed to solve the problem. Numerical examples are provided to illustrate the proposed approach and results.

KEYWORDS:

• Fully fuzzy transportation problem
• multi-objective problem
• nearest interval approximation
• Pareto optimal solutions

1. Introduction

In today's competitive markets, the pressure on organisations to find better ways to cost production and ship products to customers has increased. How and when to deliver the product to customers is challenging in terms of quantity and in a way that is cost-effective. After the introduction of fuzzy logic in the seventies and its successful applications in the design of control systems, the application of this theory in other fields such as simulation, artificial intelligence, management, operations research, etc. has expanded widely. In many real problems, there may be some uncertainty in some of the model parameters due to insufficient data or lack of information, and this ambiguity may not be probable or the model parameters may be explicitly expressed in fuzzy numbers. It is clear that in this case, the solution of transportation problem must also considered to be fuzzy. In addition, in many real applications of the problem, managerial decision making requirements require that several conflicting objectives be considered. These situations demand fully fuzzy multi-objective transportation (FFMOT) problem.

Transpiration problem presented by Hitchcock in 1941 as a specific case of linear programming problem [1]. Many efficient algorithms were proposed for this problem in the literature ([2,3]).

Furthermore, many researchers considered variants of fuzzy transportation problems. Some of these researches are mentioned below. Fully intuitionistic fuzzy transportation problem was considered in [4], where all variables and parameters are of triangular intuitionistic fuzzy values and a new effective solution method was proposed to solve the problem. Vogel’s Approximation Method was suggested to solve special type of fuzzy optimisation problem in which the values of the demand, supply and cost parameters was taken as pentagonal fuzzy numbers [5]. Also, intuitionistic fuzzy transportation problem of type-2 was studied in [6]. Recent optimisation techniques applied to fuzzy transportation problems was reviewed in [7]. Sahoo considered transportation problem in fuzzy environment, in which supply, demand and transportation costs are Fermatean fuzzy numbers and proposed a new score function to solve the problem [8]. dual-hesitant fuzzy transportation problem including some restrictions was studied in [9].

Despite extensive research on fuzzy multi-objective linear programming problems, researches on fuzzy multi-objective transportation problems are very limited. Evolutionary algorithms to solve fuzzy multi-objective transportation problem in which the value of cost parameters were considered to be fuzzy numbers applied in [10]. This paper investigates fully fuzzy multi-objective transportation (FFMOT) problem. The values of all parameters and variables are considered to be LR fuzzy numbers. In addition, all constraints and objective functions are of fuzzy type. A new approach based on nearest interval approximation is proposed to transform the fuzzy problem. The next section reviews some preliminaries in fuzzy theory, interval approximation and arithmetic operations. The fully fuzzy multi-objective transportation problem is defined and modeled in Section 3. In addition, this section presents the approach of the paper to solve the FFMOT problem. The proposed approach is applied to solve numerical examples in Section 4. Finally, Section 5 ends the paper with a brief conclusions and future directions.

2. Preliminaries

In this section, the basic notions of fuzzy theory, LR fuzzy numbers, closed interval and fuzzy arithmetic operations are reviewed.

Definition 1([11]): Set $A=[a^l,a^u]$ is a closed interval. The center and weight of $A$ is defined as follows: $c(A)=\frac{a^l+a^u}{2},w(A)=\frac{a^u-a^l}{2}.$ In this case the closed interval can be displayed by $A=(c(A),w(A))$.

Definition 2([12]): Let $A=[a^l,a^u]$ and $B=[b^l,b^u]$ be two closed intervals. The following relations define the order relations between $A$ and $B$.

• $A{=}_{mw}B\text{ if and only if c}(A)=c(B)\text{ and }w(A)=w(B),$

• $A{\le }_{mw}B\text{ if and only if c}(A)\le c(B)\text{ and }w(A)\ge w(B),$

• $A{<}_{mw}B\text{ if and only if c}(A)<c(B)\text{ and }w(A)>w(B).$

The set $\tilde{A}=\{(x,{\mu }_{\tilde{A}}(x)):x\in X\}$ in which ${\mu }_{\tilde{A}}:X\to [0,1]$ is the membership function of $\tilde{A}$ is called a fuzzy set. Fuzzy set $\tilde{A}$ is the natural generalisation of the classic set $A\subseteq X$ in which the characteristic function ${\mu }_A$ assigns 0 or 1 to each member of $X$. A fuzzy set $\tilde{A}$ is normal if and only if there exits $x\in X$ such that ${\mu }_{\tilde{A}}(x)=1$ [4].

Definition 3([13]): An LR fuzzy number $\tilde{A}=(m,n,{\delta }_1,{\delta }_2{)}_{LR}$ is defined by the following membership function: ${\mu }_{\tilde{A}}(x)=\{\begin{array}{llll}R\left({\displaystyle \frac{x-n}{{\delta }_2}}\right)& x\ge n,{\delta }_2>01& o.w.L\left({\displaystyle \frac{m-x}{{\delta }_1}}\right)& x\le m,{\delta }_1>0\end{array}$

In which $L,R:[0,\mathrm{\infty })\to [0,1]$ are nonincreasing functions and $R(0)=L(0)=1$.

A trapezoidal fuzzy number denoted by $(a_1,a_2,a_3,a_4)$ where $a_1\le a_2\le a_3\le a_4$ is a special LR number in which $R(x)=L(x)=max(0,1-x)$. Also, the $\alpha$-cut of an LR fuzzy number $\tilde{A}$ is defined by: ${\tilde{A}}_{\alpha }=[{\tilde{A}}_l(\alpha ),{\tilde{A}}_u(\alpha )]=\{x\in X:{\mu }_{\tilde{A}}(x)\ge \alpha \}=\{x\in X|x\in [m-{\delta }_1{L}^{-1}(\alpha ),n+{\delta }_2{R}^{-1}(\alpha )].\}$

Definition 4([14]): If ${\tilde{A}}_1=(m_1,n_1,{\delta }_{11},{\delta }_{12}{)}_{LR}$ and ${\tilde{A}}_2=(m_2,n_2,{\delta }_{21},{\delta }_{22}{)}_{LR}$ are two LR fuzzy numbers. The arithmetic operations on LR fuzzy numbers are defined as follows: ${\tilde{A}}_1\oplus {\tilde{A}}_2=(m_1+m_2,n_1+n_2,{\alpha }_{11}+{\alpha }_{21},{\alpha }_{12}+{\alpha }_{22}{)}_{LR}$ $\lambda {\tilde{A}}_1=\{\begin{array}{ll}{(\lambda m_1,\lambda n_1,\lambda {\alpha }_{11},\lambda {\alpha }_{12})}_{LR}& \lambda \ge 0\\ {(\lambda n_1,\lambda m_1,-\lambda {\alpha }_{12},-\lambda {\alpha }_{11})}_{LR}& \lambda <0\end{array}$

To transform the fully fuzzy multi-objective transportation problem into an interval programming problem, the concept of nearest interval approximation of LR fuzzy numbers was proposed.

Definition 5([15]): The nearest interval approximation of fuzzy number $\tilde{A}$ is defined by: $NIA(\tilde{A})=\left[{\int }_0^1{\tilde{A}}_1(\alpha )d\alpha ,{\int }_0^1{\tilde{A}}_u(\alpha )d\alpha \right]$

Based on the Definition 4 the following order relation can be obtained between LR fuzzy numbers $\tilde{A}$ and $\tilde{B}$:

• $\tilde{A}{\underset{\_}{\prec }}_{mw}\tilde{B}\text{ if and only if c}(NIA(\tilde{A}))\le c(NIA(\tilde{B}))\text{ and }w(NIA(\tilde{A}))\ge w(NIA(\tilde{B})),$

• $\tilde{A}{\prec }_{mw}\tilde{B}\text{ if and only if c}(NIA(\tilde{A}))<c(NIA(\tilde{B}))\text{ and }w(NIA(\tilde{A}))>w(NIA(\tilde{B})),$

• $\tilde{A}{\underset{\_}{\succ }}_{mw}\tilde{B}\text{ if and only if c}(NIA(\tilde{A}))\ge c(NIA(\tilde{B}))\text{ and }w(NIA(\tilde{A}))\le w(NIA(\tilde{B})),$

• $\tilde{A}{\succ }_{mw}\tilde{B}\text{ if and only if c}(NIA(\tilde{A}))>c(NIA(\tilde{B}))\text{ and }w(NIA(\tilde{A}))<w(NIA(\tilde{B})),$

• $\tilde{A}{\approx }_{mw}\tilde{B}\text{ if and only if c}(NIA(\tilde{A}))=c(NIA(\tilde{B}))\text{ and }w(NIA(\tilde{A}))=w(NIA(\tilde{B})).$

3. Fully Fuzzy Multi-Objective Transportation Problem

This section proposes fully fuzzy multi-objective transportation problem in which all parameters, variables, constraints and objective functions of the problem are considred to be fuzzy. It is assumed that the parameters and variables of the problem are LR fuzzy numbers. Accordingly, the mathematical formulation of the FFMOT problem can be staeted as folows: $\begin{array}{l}minz=\left(\sum _{i=1}^m\sum _{j=1}^n{\tilde{c}}_{ij}^1\otimes {\tilde{x}}_{ij}^{},\sum _{i=1}^m\sum _{j=1}^n{\tilde{c}}_{ij}^2\otimes {\tilde{x}}_{ij}^{},\dots ,\sum _{i=1}^m\sum _{j=1}^n{\tilde{c}}_{ij}^h\otimes {\tilde{x}}_{ij}^{}\right)\\ s.t.\\ \begin{array}{ccc}& & \sum _{j=1}^n{\tilde{x}}_{ij}^{}\underset{\_}{\prec }{\tilde{s}}_i\end{array}\begin{array}{ccc}& & i=1,\dots ,m\end{array}\\ \begin{array}{ccc}& & \sum _{i=1}^m{\tilde{x}}_{ij}^{}\underset{\_}{\succ }{\tilde{d}}_j\end{array}\begin{array}{ccc}& & j=1,\dots ,n\end{array}\\ \begin{array}{ccc}& & {\tilde{x}}_{ij}^{}\underset{\_}{\succ }\tilde{0}\end{array}\begin{array}{ccc}& \begin{array}{cc}& \end{array}& i=1,\dots ,m,j=1,\dots ,n\end{array}\end{array}$

In which ${\tilde{s}}_i$ is the amount of inventory at the origin $i(i=1,\dots ,m)$. Also, ${\tilde{d}}_j$ is the amount of demand at the destination $j(j=1,\dots ,n)$. ${\tilde{c}}_{ij}^k(k=1,\dots ,h)$ is the cost of transporting a unit of goods from origin i to destination j related to the k^th objective function. The first constraint ensures that the total amount of goods transferred from the origin i is less than the amount of goods available. Furthermore, the second constraints guaranties that all demands are met. In this problem, all parameters, variables, objective functions and constraints and considered to be fuzzy.

In multi-objective transportation problem, the decision maker prefers to obtain the optimal value of all objective functions simultaneously, but due to the conflict objective functions, it seems unlikely that all objective functions will reach the desired values simultaneously. Thus, in the literature, the Pareto optimal solutions for multi-objective problems are defined. The set of fuzzy Pareto optimal solutions was defined based on the different fuzzy ranking functions in fuzzy multi-objective programming problems as follows ([11,16–18]):

Definition 6([19]): The vector ${\tilde{X}}^{\ast }=[{\tilde{x}}_{ij}^{\ast }{]}_{m\times n}$ is a Pareto optimal solution of fully fuzzy multi-objective transportation problem if and only if.

1. ${\tilde{X}}^{\ast }$ satisfies the constraints of FFMOT problem.

2. There is not any feasible solution ${\tilde{X}}^{}=[{\tilde{x}}_{ij}^{}{]}_{m\times n}$ such that

   • $\sum _{i=1}^m\sum _{j=1}^n{\tilde{c}}_{ij}^k{\tilde{x}}_{ij}^{}{\underset{\_}{\prec }}_{mw}\sum _{i=1}^m\sum _{j=1}^n{\tilde{c}}_{ij}^k{\tilde{x}}_{ij}^{\ast }$ for all $k=1,2,\dots ,h$

   • $\sum _{i=1}^m\sum _{j=1}^n{\tilde{c}}_{ij}^k{\tilde{x}}_{ij}^{}{\prec }_{mw}\sum _{i=1}^m\sum _{j=1}^n{\tilde{c}}_{ij}^k{\tilde{x}}_{ij}^{\ast }$ for at least one $k=1,2,\dots ,h$

In the following, a new approach to solve fully fuzzy multi-objective transportation problem is proposed. The steps of the proposed algorithm are as follows:

Step 1: Convert fuzzy inequality constraints to fuzzy equality constraints by adding fuzzy surplus and fuzzy slack variables. $\begin{array}{rl}& minz=\left(\sum _{i=1}^m\sum _{j=1}^n{c}_{ij}^{{}^{\sim 1}}\otimes x_{ij}^{{}^{\sim }},\sum _{i=1}^m\sum _{j=1}^n{c}_{ij}^{{}^{\sim 2}}\otimes x_{ij}^{{}^{\sim }},\dots ,\sum _{i=1}^m\sum _{j=1}^n{c}_{ij}^{{}^{\sim h}}\otimes x_{ij}^{{}^{\sim }}\right)\\ & s.t.\\ & \begin{array}{ccc}& & \sum _{j=1}^n{x}_{ij}^{{}^{\sim }}\end{array}\begin{array}{ccc}\oplus l_i^{{}^{\sim }}\cong s_i^{{}^{\sim }}& & i=1,\dots ,m\end{array}\\ & \begin{array}{ccc}& & \sum _{i=1}^m{x}_{ij}^{{}^{\sim }}\end{array}\begin{array}{ccc}\mathrm{\Theta }& e_i^{{}^{\sim }}\cong d_i^{{}^{\sim }}& j=1,\dots ,n\end{array}\\ & \begin{array}{ccc}& & {\tilde{x}}_{ij}^{}\underset{\_}{\succ }\tilde{0}\end{array},l_i\underset{\_}{\succ }\tilde{0},e_j\underset{\_}{\succ }\tilde{0}\begin{array}{ccc}& \begin{array}{cc}& \end{array}& i=1,\dots ,m,j=1,\dots ,n\end{array}\end{array}$ $l_i(i=1,\dots ,m)$ and $e_j(j=1,\dots ,n)$ are fuzzy slack and surplus variables respectively.

Step 2: Setting $\sum _{i=1}^m\sum _{j=1}^n{\tilde{c}}_{ij}^k\otimes {\tilde{x}}_{ij}^{}=(m_k,n_k,{{\delta }_1}_k,{{\delta }_2}_k{)}_{LR},\sum _{j=1}^n{\tilde{x}}_{ij}^{}\oplus l_i=({m^{\mathrm{\prime }}}_i,{n^{\mathrm{\prime }}}_i,{{\delta }^{\mathrm{\prime }}}_{1i},{{\delta }^{\mathrm{\prime }}}_{2i}{)}_{LR}$ and $\sum _{j=1}^n{\tilde{x}}_{ij}^{}{e}_j=(m_k^{\mathrm{\prime }\mathrm{\prime }},n_k^{\mathrm{\prime }\mathrm{\prime }},{\delta }_{1k}^{\mathrm{\prime }\mathrm{\prime }},{\delta }_{2k}^{\mathrm{\prime }\mathrm{\prime }}{)}_{LR}$ the problem is reformulated as follows: $\begin{array}{l}minz=({(m_1,n_1,{{\delta }_1}_1,{{\delta }_2}_1)}_{LR},{(m_2,n_2,{{\delta }_1}_2,{{\delta }_2}_2)}_{LR},\dots ,{(m_h,n_h,{{\delta }_1}_h,{{\delta }_2}_h)}_{LR})\\ s.t.\\ \begin{array}{ccc}& & {({m^{\mathrm{\prime }}}_i,{n^{\mathrm{\prime }}}_i,{{\delta }^{\mathrm{\prime }}}_{1i},{{\delta }^{\mathrm{\prime }}}_{2i})}_{LR}\cong \text{(}{{\mathrm{a}}^{\mathrm{\prime }}}_i,{b^{\mathrm{\prime }}}_i,{{\beta }^{\mathrm{\prime }}}_{1i},{{\beta }^{\mathrm{\prime }}}_{2i})\end{array}\begin{array}{ccc}& & i=1,\dots ,m\end{array}\\ \begin{array}{ccc}& & {({m^{\mathrm{\prime }\mathrm{\prime }}}_j,{n^{\mathrm{\prime }\mathrm{\prime }}}_j,{{\delta }^{\mathrm{\prime }\mathrm{\prime }}}_{1j},{{\delta }^{\mathrm{\prime }\mathrm{\prime }}}_{2j})}_{LR}\cong \text{(}{{\mathrm{a}}^{\mathrm{\prime }\mathrm{\prime }}}_j,{b^{\mathrm{\prime }\mathrm{\prime }}}_j,{{\beta }^{\mathrm{\prime }\mathrm{\prime }}}_{1j},{{\beta }^{\mathrm{\prime }\mathrm{\prime }}}_{2j})\end{array}\begin{array}{ccc}& & j=1,\dots ,n\end{array}\\ \begin{array}{ccc}& & {\tilde{x}}_{ij}^{}\underset{\_}{\succ }\tilde{0}\end{array},l_i\underset{\_}{\succ }\tilde{0},e_j\underset{\_}{\succ }\tilde{0}\begin{array}{ccc}& \begin{array}{cc}& \end{array}& i=1,\dots ,m,j=1,\dots ,n\end{array}\end{array}$ Step 3: Find the closest interval approximation of the LR fuzzy numbers and reformulate the problem as follows. $\begin{array}{l}minz=([z_1^l,z_1^u],[z_2^l,z_2^u],\dots ,[z_h^l,z_h^u])\\ s.t.\\ \begin{array}{ccc}& & [C_i^l,C_i^u]\text{ = (}[s_i^l,s_i^u]\end{array}\begin{array}{ccc}& & i=1,\dots ,m\end{array}\\ \begin{array}{ccc}& & [D_j^l,D_j^u]\text{ = (}[d_j^l,d_j^u]\end{array}\begin{array}{ccc}& & j=1,\dots ,n\end{array}\\ \begin{array}{ccc}& & {\tilde{x}}_{ij}^{}\underset{\_}{\succ }\tilde{0}\end{array},l_i\underset{\_}{\succ }\tilde{0},e_j\underset{\_}{\succ }\tilde{0}\begin{array}{ccc}& \begin{array}{cc}& \end{array}& i=1,\dots ,m,j=1,\dots ,n\end{array}\end{array}$

Step 4: Finally, by performing computational operations on interval numbers, the fully fuzzy multi-objective transport problem is converted to the following linear programming problem. $\begin{array}{l}minz=\sum _{q=1}^h{\lambda }_{q}m({\tilde{z}}_q)-\sum _{q=1}^h{\mu }_{q}w({\tilde{z}}_q)\\ s.t.\\ \begin{array}{ccc}& & \frac{C_i^l+C_i^u}{2}={\displaystyle \frac{s_i^l+s_i^u}{2}}\end{array}\begin{array}{ccc}& & i=1,\dots ,m\end{array}\begin{array}{ccc}& & {\displaystyle \frac{C_i^u-C_i^l}{2}}={\displaystyle \frac{s_i^u-s_i^l}{2}}\end{array}\begin{array}{ccc}& & i=1,\dots ,m\end{array}\begin{array}{ccc}& & {\displaystyle \frac{D_j^u+D_j^l}{2}}={\displaystyle \frac{d_j^u+d_j^l}{2}}\end{array}\begin{array}{ccc}& & j=1,\dots ,n\end{array}\begin{array}{ccc}& & {\displaystyle \frac{D_j^u-D_j^l}{2}}={\displaystyle \frac{d_j^u-d_j^l}{2}}\end{array}\begin{array}{ccc}& & j=1,\dots ,n\end{array}\begin{array}{ccc}& & 0\le {\lambda }_q,\end{array}{\mu }_q\le 1\begin{array}{ccc}& & q=1,\dots ,h\end{array}\begin{array}{ccc}& & \sum _{q=1}^h{\lambda }_q=\sum _{q=1}^h{\mu }_q=\end{array}1\begin{array}{ccc}& & q=1,\dots ,h\end{array}\begin{array}{ccc}& & {\tilde{x}}_{ij}^{}\underset{\_}{\succ }\tilde{0}\end{array},l_i\underset{\_}{\succ }\tilde{0},e_j\underset{\_}{\succ }\tilde{0}\begin{array}{ccc}& \begin{array}{cc}& \end{array}& i=1,\dots ,m,j=1,\dots ,n\end{array}\end{array}$

Step 5: Solve the final linear programming problem to find the fuzzy pareto-optimal solutions of the fully fuzzy multi-objective transportation problem.

4. Numerical Examples

Consider a fully fuzzy multi-objective transportation problem with two fuzzy objective functions, three supply and four demand locations. The fuzzy parameters of fully fuzzy problem is provided in Table 1.

Table 1. Fuzzy parameters of fully fuzzy multi-objective transportation problem.

	D1	D2	D3	D4	Supply
S1	(1,1.5,2)	(1,2,3)	(5,7,9)	(4,6,8)	(7,8,9)
	(3,4,5)	(2,4,6)	(2,3,4)	(1,3,5)
S2	(1,1.5,2)	(7,8.5,10)	(2,4,6)	(3,4,5)	(17,19,21)
	(4,5,6)	(7,8,9)	(7,8.5,10)	(9,10,11)
S3	(7,8,9)	(7,9,11)	(3,4,5)	(5,6,7)	(16,17,18)
	(4,6,8)	(1,2,3)	(3,4.5,6)	(1,1.5,2)
Demand	(10,11,12)	(2,3,4)	(13,14,15)	(15,16,17)

The fully fuzzy multi-objective linear programming formulation of the problem is as follows: $\begin{array}{l}min{\tilde{\mathrm{z}}}_1=\left(\begin{array}{l}(1,1.5,2){\tilde{x}}_{11}+(1,2,3){\tilde{x}}_{12}+(5,7,9){\tilde{x}}_{13}+(4,6,8){\tilde{x}}_{14}\\ +(1,1.5,2){\tilde{x}}_{21}+(7,8.5,10){\tilde{x}}_{22}+(2,4,6){\tilde{x}}_{23}+(3,4,5){\tilde{x}}_{24}\\ +(7,8,9){\tilde{x}}_{31}+(7,9,11){\tilde{x}}_{32}+(3,4,5){\tilde{x}}_{33}+(5,6,7){\tilde{x}}_{34}\end{array}\right)\\ min{\tilde{\mathrm{z}}}_2=\left(\begin{array}{l}(3,4,5){\tilde{x}}_{11}+(2,4,6){\tilde{x}}_{12}+(2,3,4){\tilde{x}}_{13}+(1,3,5){\tilde{x}}_{14}\\ +(4,5,6){\tilde{x}}_{21}+(7,8,9){\tilde{x}}_{22}+(7,8.5,10){\tilde{x}}_{23}+(9,10,11){\tilde{x}}_{24}\\ +(4,6,8){\tilde{x}}_{31}+(1,2,3){\tilde{x}}_{32}+(3,4.5,6){\tilde{x}}_{33}+(1,1.5,2){\tilde{x}}_{34}\end{array}\right)\\ s.t.\\ \begin{array}{ccc}& & \end{array}{\tilde{x}}_{11}+{\tilde{x}}_{12}+{\tilde{x}}_{13}+{\tilde{x}}_{14}\underset{\_}{\prec }(7,8,9)\\ \begin{array}{ccc}& & \end{array}{\tilde{x}}_{21}+{\tilde{x}}_{22}+{\tilde{x}}_{23}+{\tilde{x}}_{24}\underset{\_}{\prec }(17,19,21)\\ \begin{array}{ccc}& & \end{array}{\tilde{x}}_{31}+{\tilde{x}}_{32}+{\tilde{x}}_{33}+{\tilde{x}}_{34}\underset{\_}{\prec }(16,17,18)\\ \begin{array}{ccc}& & \end{array}{\tilde{x}}_{11}+{\tilde{x}}_{21}+{\tilde{x}}_{31}\underset{\_}{\succ }(10,11,12)\\ \begin{array}{ccc}& & \end{array}{\tilde{x}}_{12}+{\tilde{x}}_{22}+{\tilde{x}}_{32}\underset{\_}{\succ }(2,3,4)\\ \begin{array}{ccc}& & \end{array}{\tilde{x}}_{13}+{\tilde{x}}_{23}+{\tilde{x}}_{33}\underset{\_}{\succ }(13,14,15)\\ \begin{array}{ccc}& & \end{array}{\tilde{x}}_{14}+{\tilde{x}}_{24}+{\tilde{x}}_{34}\underset{\_}{\succ }(15,16,17)\\ \begin{array}{ccc}& & \end{array}{\tilde{x}}_{ij}\underset{\_}{\succ }\begin{array}{cc}& i=1,2,3,j=1,2,3,4\end{array}\end{array}$

Appling slack and surplus fuzzy variables to fuzzy inequality constraints as well as arithmetic operation on fuzzy LR numbers the problem is reformulated as follows: $\begin{array}{rl}& min{\tilde{\mathrm{z}}}_1={\left(\begin{array}{l}{x}_{11}^1+x_{12}^1+5x_{13}^1+4x_{14}^1+x_{21}^1+7x_{22}^1+2x_{23}^1+3x_{24}^1\\ +7x_{31}^1+7x_{32}^1+3x_{33}^1+5x_{34}^1,\\ 1.5x_{11}^2+2x_{12}^2+7x_{13}^2+6x_{14}^2+1.5x_{21}^2\\ +8.5x_{22}^2+4x_{23}^2+4x_{24}^2+8x_{31}^2+9x_{32}^2\\ +4x_{33}^2+6x_{34}^2,\\ 2x_{11}^3+3x_{12}^3+9x_{13}^3+8x_{14}^3+2x_{21}^3+10x_{22}^3+6x_{23}^3\\ +5x_{24}^3+9x_{31}^3+11x_{32}^3+5x_{33}^3+7x_{34}^3\end{array}\right)}_{LR}\\ & \begin{array}{l}min{\tilde{\mathrm{z}}}_2={\left(\begin{array}{l}3x_{11}^1+2x_{12}^1+2x_{13}^1+x_{14}^1+4x_{21}^1+7x_{22}^1+7x_{23}^1\\ +9x_{24}^1+4x_{31}^1+x_{32}^1+3x_{33}^1+x_{34}^1,\\ 4x_{11}^2+4x_{12}^2+3x_{13}^2+3x_{14}^2+5x_{21}^2+8x_{22}^2+8.5x_{23}^2\\ +10x_{24}^2+6x_{31}^2+2x_{32}^2+4.5x_{33}^2+1.5x_{34}^2,\\ 5x_{11}^3+6x_{12}^3+4x_{13}^3+5x_{14}^3+6x_{21}^3+9x_{22}^3+10x_{23}^3\\ +11x_{24}^3+8x_{31}^3+3x_{32}^3+6x_{33}^3+2x_{34}^3\end{array}\right)}_{LR}\\ s.t.\\ \begin{array}{ccc}& & \end{array}(x_{11}^1+x_{12}^1+x_{13}^1+x_{14}^1+l_1^1,x_{11}^2+x_{12}^2+x_{13}^2+x_{14}^2+l_1^2,x_{11}^3+x_{12}^3+x_{13}^3+x_{14}^3+l_1^3)\\ \cong (7,8,9)\\ \begin{array}{ccc}& & \end{array}(x_{21}^1+x_{22}^1+x_{23}^1+x_{24}^1+l_2^1,x_{21}^2+x_{22}^2+x_{23}^2+x_{24}^2+l_2^2,x_{21}^3+x_{22}^3+x_{23}^3+x_{24}^3+l_2^3)\\ \cong (17,19,21)\\ \begin{array}{ccc}& & \end{array}(x_{31}^1+x_{32}^1+x_{33}^1+x_{34}^1+l_3^1,x_{31}^2+x_{32}^2+x_{33}^2+x_{34}^2+l_3^2,x_{31}^3+x_{32}^3+x_{33}^3+x_{34}^3+l_3^3)\\ \cong (16,17,18)\\ \begin{array}{ccc}& & \end{array}(x_{11}^1+x_{21}^1+x_{31}^1-e_1^3,x_{11}^2+x_{21}^2+x_{31}^2-e_1^2,x_{11}^3+x_{21}^3+x_{31}^3-e_1^1)\cong (10,11,12)\\ \begin{array}{ccc}& & \end{array}(x_{12}^1+x_{22}^1+x_{32}^1-e_2^3,x_{12}^2+x_{22}^2+x_{32}^2-e_2^2,x_{12}^3+x_{22}^3+x_{32}^3-e_2^1)\cong (2,3,4)\\ \begin{array}{ccc}& & \end{array}(x_{13}^1+x_{23}^1+x_{33}^1-e_3^3,x_{13}^2+x_{23}^2+x_{33}^2-e_3^2,x_{13}^3+x_{23}^3+x_{33}^3-e_3^1)\cong (13,14,15)\\ \begin{array}{ccc}& & \end{array}(x_{14}^1+x_{24}^1+x_{34}^1-e_4^3,x_{14}^2+x_{24}^2+x_{34}^2-e_4^2,x_{14}^3+x_{24}^3+x_{34}^3-e_4^1)\cong (15,16,17)\\ \begin{array}{ccc}& & \end{array}{\tilde{x}}_{ij}\underset{\_}{\succ }0,{\tilde{l}}_i\underset{\_}{\succ }0,{\tilde{e}}_j\underset{\_}{\succ }0\begin{array}{cc}& i=1,2,3,j=1,2,3,4\end{array}\end{array}\end{array}$

In the next step, the nearest interval approximation of fuzzy LR numbers is calculated and the problems is converted to the following interval multi-objective linear programming problem. $\begin{array}{l}min{\tilde{\mathrm{z}}}_1=[m({\tilde{\mathrm{z}}}_1),w({\tilde{\mathrm{z}}}_1)]\\ min{\tilde{\mathrm{z}}}_2=[m({\tilde{\mathrm{z}}}_2),w({\tilde{\mathrm{z}}}_2)]\\ s.t.\\ \begin{array}{ccc}& & \end{array}[c({\tilde{S}}_1),w({\tilde{S}}_1)]{=}_{mw}[7.5,8.5]\\ \begin{array}{ccc}& & \end{array}[c({\tilde{S}}_2),w({\tilde{S}}_2)]{=}_{mw}[18,20]\\ \begin{array}{ccc}& & \end{array}[c({\tilde{S}}_3),w({\tilde{S}}_3)]{=}_{mw}[16.5,17.5]\\ \begin{array}{ccc}& & \end{array}[c({\tilde{D}}_1),w({\tilde{D}}_1)]{=}_{mw}[10.5,11.5]\\ \begin{array}{ccc}& & \end{array}[c({\tilde{D}}_2),w({\tilde{D}}_2)]{=}_{mw}[2.5,3.5]\\ \begin{array}{ccc}& & \end{array}[c({\tilde{D}}_3),w({\tilde{D}}_3)]{=}_{mw}[13.5,14.5]\\ \begin{array}{ccc}& & \end{array}[c({\tilde{D}}_4),w({\tilde{D}}_4)]{=}_{mw}[15.5,16.5]\\ \begin{array}{ccc}& & \end{array}{x^1}_{ij}\ge 0,{l^1}_i\ge 0,{e^1}_j\ge 0\begin{array}{cc}& i=1,2,3,j=1,2,3,4\end{array}\\ \begin{array}{ccc}& & \end{array}{x^3}_{ij}\ge {x^2}_{ij}\ge {x^3}_{ij},{l^3}_i\ge {l^2}_i\ge {l^1}_i,{e^3}_j\ge {e^2}_j\ge {e^1}_j\begin{array}{cc}& i=1,2,3,j=1,2,3,4\end{array}\end{array}$

In which, calculations of the nearest interval approximation is provided in Table 2 to obtain the values of parameters $c(.)$ and $w(.)$.

Table 2. Calculation of the values of parameters of fully fuzzy multi-objective transportation problem.

Parameter	Value
$c({\tilde{\mathrm{z}}}_1)$	$\frac{1}{2}\left(\begin{array}{l}{x}_{11}^1+3x_{11}^2+2x_{11}^3+x_{12}^1+4x_{12}^2+3x_{12}^3+5x_{13}^1+14x_{13}^2+9x_{13}^3+4x_{14}^1+12x_{14}^2+8x_{14}^3\\ +x_{21}^1+3x_{21}^2+2x_{21}^3+7x_{22}^1+17x_{22}^2+10x_{22}^3+2x_{23}^1+8x_{23}^2+6x_{23}^3+3x_{24}^1+8x_{24}^2+5x_{24}^3\\ 7x_{31}^1+16x_{31}^2+9x_{31}^3+7x_{32}^1+18x_{32}^2+11x_{32}^3+3x_{33}^1+8x_{33}^2+5x_{33}^3+5x_{34}^1+12x_{34}^2+7x_{34}^3\end{array}\right)$
$w({\tilde{\mathrm{z}}}_1)$	$\frac{1}{2}\left(\begin{array}{l}2x_{11}^3-x_{11}^1+3x_{12}^3-x_{12}^1+9x_{13}^3-5x_{13}^1+8x_{14}^3-4x_{14}^1+2x_{21}^3-x_{21}^1+10x_{22}^3-7x_{22}^1+6x_{23}^3-2x_{23}^1+5x_{24}^3-3x_{24}^1+9x_{31}^3-7x_{31}^1+11x_{32}^3-7x_{32}^1+5x_{33}^3-3x_{33}^1+7x_{34}^3-5x_{34}^1\end{array}\right)$
$c({\tilde{\mathrm{z}}}_2)$	$\frac{1}{2}\left(\begin{array}{l}3x_{11}^1+8x_{11}^2+5x_{11}^3+2x_{12}^1+8x_{12}^2+6x_{12}^3+2x_{13}^1+6x_{13}^2+4x_{13}^3+1x_{14}^1+6x_{14}^2+5x_{14}^3+4x_{21}^1+10x_{21}^2+6x_{21}^3+7x_{22}^1+16x_{22}^2+9x_{22}^3+7x_{23}^1+17x_{23}^2+10x_{23}^3+9x_{24}^1+20x_{24}^2+11x_{24}^{3}4x_{31}^1+12x_{31}^2+8x_{31}^3+1x_{32}^1+4x_{32}^2+3x_{32}^3+3x_{33}^1+9x_{33}^2+6x_{33}^3+1x_{34}^1+3x_{34}^2+2x_{34}^3\end{array}\right)$
$w({\tilde{\mathrm{z}}}_2)$	$\frac{1}{2}\left(\begin{array}{l}5x_{11}^3-3x_{11}^1+6x_{12}^3-2x_{12}^1+4x_{13}^3-2x_{13}^1+5x_{14}^3-x_{14}^1+6x_{21}^3-4x_{21}^1+9x_{22}^3-7x_{22}^1+10x_{23}^3-7x_{23}^1+11x_{24}^3-9x_{24}^1+8x_{31}^3-4x_{31}^1+3x_{32}^3-x_{32}^1+6x_{33}^3-3x_{33}^1+2x_{34}^3-x_{34}^1\end{array}\right)$
$c({\tilde{S}}_i)$	$\frac{1}{2}(x_{i1}^1+2x_{i1}^2+x_{i1}^3+x_{i2}^1+2x_{i2}^2+x_{i2}^3+x_{i3}^1+2x_{i3}^2+x_{i3}^3+x_{i4}^1+2x_{i4}^2+x_{i4}^3+l_i^1+2l_i^2+l_i^3)$
$w({\tilde{S}}_i)$	$\frac{1}{2}(x_{i1}^3-x_{i1}^1+x_{i2}^3-x_{i2}^1+x_{i3}^3-x_{i3}^1+x_{i4}^3-x_{i4}^1+l_i^3-l_i^1)$
$c({\tilde{D}}_j)$	$\frac{1}{2}(x_{1j}^1+2x_{1j}^2+x_{1j}^3+x_{2j}^1+2x_{2j}^2+x_{2j}^3+x_{3j}^1+2x_{3j}^2+x_{3j}^3-(e_j^1+2e_j^2+e_j^3))$
$w({\tilde{D}}_j)$	$\frac{1}{2}(x_{1j}^3-x_{1j}^1+x_{2j}^3-x_{2j}^1+x_{3j}^3-x_{3j}^1-(e_j^1-e_j^3))$

Finally, setting ${\lambda }_q={\mu }_q=\frac{1}{2}$ the problem is transformed to the following linear programming problem. $\begin{array}{rl}& min{\tilde{\mathrm{z}}}_1=\frac{1}{2}(m({\tilde{\mathrm{z}}}_1)+m({\tilde{\mathrm{z}}}_2)-w({\tilde{\mathrm{z}}}_1)-w({\tilde{\mathrm{z}}}_2))\\ & s.t.\\ & \begin{array}{ccc}& & \end{array}c({\tilde{S}}_1)=8,w({\tilde{S}}_1)=0.5\\ & \begin{array}{ccc}& & \end{array}c({\tilde{S}}_2)=19,w({\tilde{S}}_2)=1\\ & \begin{array}{ccc}& & \end{array}c({\tilde{S}}_3)=17,w({\tilde{S}}_3)=0.5\\ & \begin{array}{ccc}& & \end{array}c({\tilde{D}}_1)=11,w({\tilde{D}}_1)=0.5\\ & \begin{array}{ccc}& & \end{array}c({\tilde{D}}_2)=3,w({\tilde{D}}_2)=0.5\\ & \begin{array}{ccc}& & \end{array}c({\tilde{D}}_3)=14,w({\tilde{D}}_3)=0.5\\ & \begin{array}{ccc}& & \end{array}c({\tilde{D}}_4)=16,w({\tilde{D}}_4)=0.5\\ & \begin{array}{ccc}& & \end{array}{x^1}_{ij}\ge 0,{l^1}_i\ge 0,{e^1}_j\ge 0\begin{array}{cc}& i=1,2,3,j=1,2,3,4\end{array}\\ & \begin{array}{ccc}& & \end{array}{x^3}_{ij}\ge {x^2}_{ij}\ge {x^3}_{ij},{l^3}_i\ge {l^2}_i\ge {l^1}_i,{e^3}_j\ge {e^2}_j\ge {e^1}_j\begin{array}{cc}& i=1,2,3,j=1,2,3,4\end{array}\end{array}$

AMPL (A Mathematical Programming Language) is applied to solve the transformed problem. The optimal values of fuzzy variables is reported in Table 3. Also, the optimal values of the fuzzy objective function is provided in Table 4.

Table 3. The optimal values of fuzzy variables.

Variable	Value
${\tilde{x}}_{11}$	$(0,0,1)$
${\tilde{x}}_{12}$	$(1.25,1.25,1.25)$
${\tilde{x}}_{13}$	$(0,0,0)$
${\tilde{x}}_{14}$	$(2.5,2.5,2.5)$
${\tilde{x}}_{21}$	$(5.25,5.25,5.25)$
${\tilde{x}}_{22}$	$(0,0,1)$
${\tilde{x}}_{23}$	$(3.75,3.75,4.75)$
${\tilde{x}}_{24}$	$(0,0,0)$
${\tilde{x}}_{31}$	$(0,0,0)$
${\tilde{x}}_{32}$	$(0,0,0)$
${\tilde{x}}_{33}$	$(3,3,3)$
${\tilde{x}}_{34}$	$(5.25,5.25,6.25)$

Table 4. The optimal values of fuzzy objective functions.

Objective function	Value
${\tilde{\mathrm{z}}}_1$	$(59.25,83.875,133.5)$
${\tilde{\mathrm{z}}}_2$	$(66.5,92,143.5)$

In addition, the set of Pareto optimal solutions of the problem is depicted in Figure 1.

Figure 1. The set of Pareto optimal solutions.

To compare the results of the proposed method with the available approaches, the example provided in [20] is considered. Consider an online dairy store to buy fresh dairy products in three cities. They must deliver their products in eight cities within 12 hours of online purchase. The main objective is to maximise profit while minimising delivery time and loss during transportation through a given route. Delivery time, transportation losses, total transportation profit per unit, supply and demand were given as triangular fuzzy numbers [20]. Solving the problem with the proposed approach, the fuzzy optimal solution is shown in Table 5. In addition, the optimal value of fuzzy objective functions is proposed in Table 6.

Table 5. The optimal values of fuzzy variables of the fuzzy transportation problem instance in [20].

	1	2	3	4	5	6	7	8
1	$(0,0,0)$	$(0,0,0)$	$(34,36,36)$	$(37,43,43)$	$(12.1,17.1,17.1)$	$(0,0,0)$	$(27,29.5,29.5)$	$(0,0,0)$
2	$(27.13,30.63,30.63)$	$(19,21,21)$	$(0,0,0)$	$(0,0,0)$	$(39.1,39.1,39.1)$	$(0,3.5,3.5)$	$(0,0,0)$	$(0,0,0)$
3	$(0,0,0)$	$(2.9,7.4,7.4)$	$(0,0,0)$	$(0,0,0)$	$(0,0,0)$	$(22.63,22.63,22.63)$	$(0,3.5,3.5)$	$(37.5,41.5,41.5)$

Table 6. The optimal values of fuzzy objective functions of the fuzzy transportation problem instance in [20].

Objective function	Value
${\tilde{\mathrm{z}}}_1$	$(1877.32,2377.9,2655.28)$
${\tilde{\mathrm{z}}}_2$	$(543.16,798.1,987.07)$
${\tilde{\mathrm{z}}}_3$	$(16402.4,21449.12,23998.56)$

The both optimal solutions obtained using the approach proposed in this paper and the approach proposed in [20] are Pareto optimal solutions of the problem. The efficiency score of the transportation plan of the two approaches are $(109.3,162.9,270.14)$ and $(105.5,166.9,269.54)$, respectively. From the point of efficiency score presented in [20], the proposed solution in this paper obtains the better optimal solution.

5. Conclusions and Future Directions

In this paper, fully fuzzy multi-objective transportation problem is considered. The concept of nearest interval approximation is applied to transform the fully fuzzy problem to an equivalent interval programming problem. Finally, using arithmetic operations on closed intervals and scalarization technique, the interval problem is transformed into a linear programming problem. Numerical examples is proposed to obtain the Pareto optimal solutions of fully fuzzy multi-objective transportation problem. In addition, the results shows that, the efficiency score of the Pareto optimal solution using the proposed approach is improved. Proposing appropriate algorithms to solve large-scale fully fuzzy multi-objective transportation problems will be followed in future researches.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Malihe Niksirat

Malihe Niksirat received her PhD degree in 2016 on Applied Mathematics and Computer Sciences from Amirkabir University of Technology, and since 2018 she has been a faculty member at the Department of Computer Sciences in Birjand University of Technology, Birjand, Iran. She was a member of the Scientific Committee of the 9th International Conference on Fuzzy Information and Engineering in 2018. Now, she is a member of Iranian Operations Research Society. Her research interests are in the areas of Fuzzy Mathematical Models and Methods, Fuzzy Arithmetic, Fuzzy Optimization and Decision Making, Operations Research, Transportation problems, Meta-heuristic optimization, Gray Systems, Neural Networks, Logistic and Uncertainty Analysis.