Transportation Problem for Interval-Valued Trapezoidal Intuitionistic Fuzzy Numbers

S. Dhanasekar; J. Jansi Rani; Manivannan Annamalai

doi:10.5391/IJFIS.2022.22.2.155

Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 155-168

Published online June 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.2.155

Transportation Problem for Interval-Valued Trapezoidal Intuitionistic Fuzzy Numbers

S. Dhanasekar , J. Jansi Rani, and Manivannan Annamalai

Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai, India

Correspondence to :
Manivannan Annamalai (manivannan.a@vit.ac.in)

Received: November 30, 2021; Revised: May 2, 2022; Accepted: May 24, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

The aim of the decision-makers in the transportation industry is to maximize profit by minimizing the transportation cost. The transportation structure is the center of economic activity in the business logistics system. However, transportation costs may vary owing to various unpredictable factors. In this study, cost of the transporting unit is considered as an interval-valued trapezoidal intuitionistic fuzzy number to deal with these uncertainties. The transportation problem with interval-valued trapezoidal intuitionistic fuzzy cost is discussed here, and the costs are ordered by score and score expected functions. As a special case, the interval-valued trapezoidal intuitionistic cost is not converted into crisp numbers to solve the transportation problem and derive the initial basic feasible (IBF) solution through interval-valued intuitionistic costs. Furthermore, the optimality of the derived initial basic feasible solution is checked using the modified distribution (MODI) method. The effectiveness and validation of the developed approach were illustrated using numerical examples.

Keywords: Transportation problem, Interval-valued trapezoidal intuitionistic fuzzy number, Arithmetic operations, Interval-valued trapezoidal intuitionistic fuzzy transportation problem

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Transportation problem (TP) was initially developed by Hitchcock [1] in 1941. The optimization requirement was proposed by Koopmans [2] in 1949. The transportation algorithm was introduced by George B. Dantzig in 1963 [3]. Transportation plays a major role in real-world scenarios. The overall economic activity and growth conditions of a country depend on transportation activities. To check the optimality of the initial basic feasible solution, the stepping stone method was invented by Charnes and Cooper [4]. Assigning sources to a destination is the main task in the TP. Decision makers are in a position to maximize the profit of the TP by minimizing transportation costs.

1.1 Fuzzy, Intuitionistic, Interval-Valued Intuitionistic Fuzzy in Optimization Problems

While solving a real-life TP, we need to face many uncertainties owing to different uncontrollable factors. Therefore, the transportation cost may not be always a crisp number each time. Because of various uncertain situations, such as fuel price, traffic jams, and road conditions due to weather, the optimal terms may vary, and the optimality of the TP can be affected. To obtain the optimal solution, we must deal with this uncertainty and vagueness. These uncertainties were modelled by Zadeh [5] in 1965 and named as fuzzy set, which is characterized by membership (belongingness) degree. The introduction of fuzzy set is very useful in solving many real-life decision-making problems and optimization problems with uncertainties. However, non-membership and hesitancy are not considered in the fuzzy set to handle additional uncertainties in real-world problems. Therefore, a fuzzy set with belongingness is insufficient for all types of uncertainties. Atanassov [6] extended fuzzy notions to intuitionistic fuzzy notions, which include non-membership and hesitancy. Moreover, an interval-valued intuitionistic fuzzy set (IVIFS) was developed by Atanassov and [7] in which the degrees of belongingness and non-belongingness are defined as intervals. Because the structure of the IVIFS contains more information, it is applied in various optimization problems for the best decision to be taken by researchers. IVIFS contains more information than the intuitionistic fuzzy set and thus needed in many fields, such as artificial intelligence, data analysis, socio-economic, and decision-making problems where interval analysis is needed.

The intuitionistic fuzzy optimization techniques were invented by Angelov [8]. Fuzzy, intuitionistic, and interval-valued intuitionistic fuzzy optimization techniques have been discussed by researchers [9–11]. Based on the fuzzy environment, many varieties of TPs were solved [12–15]. Kumar [16] solved type-2 and type-4 fuzzy TPs. Recently, Pratihar et al. [17] solved the type-2 fuzzy TP. A fully fuzzy TP involving triangular and trapezoidal fuzzy numbers was presented by [18,19]. In [20,21], the authors solved the TP in a neutrosophic fuzzy environment for Pythagorean fuzzy numbers. Nagoor Gani and Abbas [22] discussed TPs in an intuitionistic fuzzy environment. Kumar and Hussain [23] studied a fully intuitionistic fuzzy TP. The authors [24,25] proposed different types of intuitionistic fuzzy TPs. Malik et al. [26] discussed a fully intuitionistic fuzzy linear programming problem. Recently, Kumar [27] discussed zero-point method to solve intuitionistic fuzzy TP.

The TP with fuzzy and intuitionistic fuzzy parameters has been solved by many researchers using various methods. Although many types of TP have been solved by various researchers based on fuzzy and intuitionistic fuzzy environments to handle different types of uncertainty and vagueness, more realistic approaches are required to handle the uncertainty in real-life TPs. Interval-valued fuzzy numbers play a major role in various domains. Mondal and his colleagues [28–30] discussed the application of differential for interval-valued fuzzy numbers in various topics. Bharati et al. [31–33] developed a TP in an interval-valued intuitionistic fuzzy environment. In addition, Mishra et al. [34] recorded notes on TPs in interval-valued intuitionistic fuzzy environment. Some operational laws have been defined for interval-valued intuitionistic fuzzy environments by [33,35,36]. Algebraic operations of the IVIFS using the extension principle developed by Li [37]. Several ranking methods are available to rank intuitionistic fuzzy numbers. Because each method has some limitations, many researchers still work to produce the best ranking function for intuitionistic fuzzy numbers. Intuitionistic fuzzy ranking was developed in [38]. Bharati [39] defined the ranking methods of intuitionistic fuzzy numbers. Weighted aggregation operators for interval-valued intuitionistic fuzzy numbers (IVIFN) were defined by Xu [40]. Similarity measures for IVIFS were discussed in [41].

1.2 Motivation of the Proposed Study

The main goal of this study is to deal with a TP involving interval-valued trapezoidal intuitionistic fuzzy (IVTrIF) costs. In the above discussions, to the best of our knowledge, there is no TP with cost parameters, such as interval-valued trapezoidal intuitionistic fuzzy numbers (IVTrIFN). There are many decision-making problems, and the ordering functions and optimization problems are defined based on TrIFNs. TrIFNs are particularly popular for characterizing the imprecision and incompleteness of data. Wan and his colleagues [42–44] discussed aggregation operators, such as some power average operators, prioritized aggregation operators using Euclidean, Hamming distances, t-norms, t-conorms, and weighted possibility means for TrIFNs, and applied them to solve various decision-making problems. Therefore, an attempt was made to establish a new strategy for solving the TP with IVTrIF cost parameters. In this study, the TP for IVTrIFN was defined based on the motivation of [31–33]. Moreover, IVTrIFNs are compared using score [40] and the score expected function [45]. An interval-valued trapezoidal intuitionistic fuzzy transportation problem (IVTrIFTP) with IVTrIF costs was developed, in which supply and demand are crisp numbers. In addition, the ordering of the IVTrIFNs was validated through an additional ranking function. Vogel’s approximation method (VAM) [46] is one of the most important methods for determining the initial basic feasible solution for the TP. First, initial basic feasible (IBF) solution was obtained using VAM method. Finally, the optimality of the IVTrIFTP was checked using the modified distribution method for the obtained IBF solution.

1.3 Structure of the Paper

The remainder of this paper is organized as follows. In Section 2, the basic definitions and operational laws of IVIFS and IVTrIFS are provided. In Section 3, the ordering of the IVTrIFNs is discussed. In Section 4, the transportation problem and the algorithm for IVTrIFNs are presented. In Section 5, numerical examples of IVTrIFTP are provided. Finally, Section 6 concludes the paper.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

In this section, basic definitions and operations are provided, which are useful for providing the results.

2.1 Definitions

Definition 2.1 [5]

The number A which is defined in the given interval of real numbers ℝ that has the form A : ℝ → [0, 1] is called a fuzzy number that satisfies the following conditions:

(a) it should be convex
(b) Maximum height is 1, (i.e., normal)
(c) Membership function is piecewise continuous

Definition 2.2 [6]

Let X = {x₁, x₂, x₃, …, x_n} be a fixed set. An IFS A^I in X is defined as A^I = {(x, μ_{A_I} (x), ϑ_{A_I} (x) | x ∈ X)}, where μ_{A_^I} (x) and ϑ_{A_^I} (x) denote the belongingness and non-belongingness degrees of x to A^I that satisfy the condition 0 ≤ μ_{A_^I} (x) + ϑ_{A_^I} (x) ≤ 1, x ∈ X.

Definition 2.3 [7]

For the fixed set X = {x₁, x₂, x₃, …, x_n}, An IVIFS A^IV is defined as A^IV = {(x, μ_{A_^IV} (x), ϑ_{A_^IV} (x) | x ∈ X)} where μ_{A_^IV} (x) : X → ℝ[0, 1], ϑ_{A_^IV} (x) : X → ℝ[0, 1] and μAIV(x)=[μAIVLR(x),μAIVUR(x)],ϑAIV(x)=[ϑAIVLR(x),ϑAIVUR(x)] where μAIVLR(x),μAIVUR(x) and ϑAIVLR(x),ϑAIVUR(x) denote the infimum (inf) and supremum (sup), values of membership and non-membership functions, respectively. The intervals μ_{A_^IV} (x) and ϑ_{A_^IV} (x) denote the degrees of belongingness and non-belongingness of x to A^IV respectively, satisfying the condition μAIVUR(x)+ϑAIVUR(x)≤1,x ∈ X.

From this, it is clear that the IVIFS becomes intuitionistic fuzzy set if μAIVLR(x)=μAIVUR(x) and ϑAIVLR(x)=ϑAIVUR(x). Below are some operations of IVIFSs.

Let A^IV and B^IV be two IVIFSs and is defined as AIV={[μAIVLR(x),μAIVUR(x)],[ϑAIVLR(x),ϑAIVUR(x)]} and BIV={[μBIVLR(x),μBIVUR(x)],[ϑBIVLR(x),ϑBIVUR(x)]} Then,

AIV+BIV={[μAIVLR(x)+μBIVLR(x)-μAIVLR(x).μBIVLR(x),μAIVUR(x)+μBIVUR(x)-μAIVR(x).μBIVUR(x)], [ϑAIVLR(x).ϑBIVLR(x);ϑAIVUR(x).ϑBIVUR(x)]}.
AIV.BIV={[μAIVLR(x).μBIVLR(x),μAIVUR(x).μBIVUR(x)],[ϑAIVLR(x)+ϑBIVLR(x)-ϑAIVLR(x).ϑBIVLR(x), ϑAIVUR(x)+ϑBIVUR(x)-ϑAIVUR(x).ϑBIVUR(x)]}.
k(AIV)={[1-(1-μAIVLR(x))k,1-(1-μAIVUR(x))k],[(ϑAIVLR(x))k,(ϑAIVUR(x))k]},k>0.
(AIV)k={[(μAIVLR(x))k,(μAIVUR(x))k],[1-(1-ϑAIVLR(x))k,1-(1-ϑAIVUR(x))k]},k>0.

Definition 2.4 [33]

Membership and non-membership functions with their lower and upper intervals for IVTrIFN AIV={[a1,b1,c1,d1];[μAIVLR(x),μAIVUR(x)];[ϑAIVLR(x),ϑAIVUR(x)]} is defined as follows:

μAIVLR(x)={x-a1b1-a1μAIVLR,a1≤x<b1,μAIVLR,b1≤x≤c1,d1-xd1-c1μAIVLR,c1<x≤d1,0,otherwise,μAIVUR(x)={x-a1b1-a1μAIVUR,a1≤x<b1,μAIVUR,b1≤x≤c1,d1-xd1-c1μAIVUR,c1<x≤d1,0,otherwise,ϑAIVLR(x)={b1-x+ϑAIVLR(x-a1)b1-a1,a1≤x<b1,ϑAIVLR,b1≤x≤c1,x-c1+ϑAIVLR(d1-x)d1-c1,c1<x≤d1,0,otherwise,ϑAIVUR(x)={b1-x+ϑAIVUR(x-a1)b1-a1,a1≤x<b1,ϑAIVUR,b1≤x≤c1,x-c1+ϑAIVUR(d1-x)d1-c1,c1<x≤d1,0,otherwise,

where a₁, b₁, c₁, d₁ are real numbers that satisfy the conditions 0≤μAIVLR≤1, 0≤μAIVUR≤1, 0≤ϑAIVLR≤1,0≤ϑAIVUR≤1 and μAIVUR+ϑAIVUR≤1.

A graphical representation of IVTrIFN AIV={[a1,b1,c1,d1];[μAIVLR(x),μAIVUR(x)];[ϑAIVLR(x),ϑAIVUR(x)]} with its membership and non-membership functions is shown in Figure 1.

2.2 Operations of IVTrIFNs [36]

Let AIV={[a1,b1,c1,d1];[μAIVLR(x),μAIVUR(x)];[ϑAIVLR(x),ϑAIVUR(x)]} and BIV={[a2,b2,c2,d2];[μBIVLR(x),μBIVUR(x)];[ϑBIVLR(x),ϑBIVUR(x)]} are the two IVTrIFNs. Then,

AIV+BIV={([a1+a2,b1+b2,c1+c2,d1+d2];[min(μAIVLR,μBIVLR),min(μAIVUR,μBIVUR)];[max(ϑAIVLR,ϑBIVLR),max(ϑAIVUR,ϑBIVUR)])}.
AIV-BIV={([a1-d2,b1-c2,c1-b2,d1-a2];[min(μAIVLR,μBIVLR),min(μAIVUR,μBIVUR)];[max(ϑAIVLR,ϑBIVLR),max(ϑAIVUR,ϑBIVUR)])}.
AIV.BIV={([a1a2,b1b2,c1c2,d1d2];[μAIVLR.μBIVLR,μAIVUR.μBIVUR];[ϑAIVLR+ϑBIVLR-ϑAIVLR.ϑBIVLR,ϑAIVUR+ϑBIVUR-ϑAIVUR.ϑBIVUR])}.
k(AIV)={([ka1,kb1,kc1,kd1];[1-(1-μAIVLR)k,1-(1-μAIVUR)k];[(ϑAIVLR)k,(ϑAIVUR)k])},k>0.
(AIV)k={([a1k,b1k,c1k,d1k];[(μAIVLR)k,(μAIVUR)k];[1-(1-ϑAIVLR)k,1-(1-ϑAIVUR)k])} k>0.

3. Ordering Methods of IVTrIFNs

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Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Many ranking methods are available for IVIFNs. Because of the limitations of each existing ranking method, there is still no common method for ranking IVIFNs. The most commonly used score function was applied to compare the IVTrIFNs. In addition, the score expected function is applied to strongly validate the comparison of IVTrIF costs in the TP.

Definition 3.1 [40]

Let AIV={([a,b,c,d];[μAIVLR,μAIVUR];[ϑAIVLR,ϑAIVUR])} be an IVTrIFN. Subsequently, the score function is defined as

(1)S(AIV)=μAIVLR(x)+μAIVUR(x)-ϑAIVLR(x)-ϑAIVUR(x)2.

Definition 3.2 [45]

Let (AIV)={([a, b, c, d]; [μAIVLR,μAIVUR];[ϑAIVLR,ϑAIVUR])} be an IVTrIFN. Then, the score expectation function is defined as

(2)I(S(AIV))=S(AIV)2((1-δ)(a+b)+δ(c+d)),

(δ denotes the preference value).

4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem

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Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

The mathematical formulation of IVTrIFTP is in the following form:

minimize ΦIV=∑i=1m∑j=1ncijIV*xij,

subject to

∑i=1mxij≤ai, i=1,2,…,m,∑j=1nxij≥bj, j=1,2,…,n,xij≥0 for all i and j,

where

The IVTrIF cost of sending one unit of the goods from the source (origin) i to destination (end) j is denoted by cijIV.
Here, the number of origins is denoted by m, which is indexed by i.
Here, the number of destinations is denoted by n, which is indexed by j.
x_ij is the quantity of transportation from the ith origin to the jth end.
a_i is the total availability and b_j is the total demand of the goods at the ith source and jth destination, respectively.
∑i=1m∑j=1ncijIV*xij is the total IVTrIF transportation cost.

The IVTrIFTP is balanced if the total availability is equal to the total demand, which can be expressed as ∑i=1mai=∑j=1nbj, where cijIV represents the IVTrIF costs, and a_i and b_j represent the crisp units of the total availability and demand of the goods, respectively.

From Table 1, the number of constraints should be equal to the number of basic variables in a basic solution. In addition, the solution to this problem should have n + m − 1 basic variables that are non-zero, and all the remaining variables will be nonbasic and thus have a value of zero.

Steps to find the IBF solution of IVTrIFTP by IVTrIF VAM and the flow chart of the IVTrIF VAM are given in Figure 2.

Step 1: Establish the IVTrIF transportation table with the costs as IVTrIF numbers.

Step 2: Examine whether the problem is balanced. If ∑i=1mai=∑j=1nbj, then proceed to Step 4. Otherwise go to Step 3.

Step 3: If it is unbalanced, transform the IVTrIFTP into a balanced problem by introducing a dummy source or demand. Then go to Step 4.

Step 4: Identify the smallest and next smallest IVTrIF costs in each row of the transportation table, and find the difference between them for each row. Similarly, we do the same for the columns.

Step 5: Identify the largest penalty among all the rows and columns and use arbitrary tie-breaking choice if tie occurs. Now, we take the maximum penalty corresponding to ith row and let R(cijIV) be the smallest IVTrIF cost in the ith row. Then, allot the maximum feasible amount x_ij in the (i, j)th cell and deletes either the ith row or jth column.

Step 6: Repeat Step 4, to find the row and column penalties for the reduced transportation table. The procedures are repeated until all the demands and supplies are allocated.

Step 7: Finally from the IVTrIF transportation table, the IBF solution is obtained.

The steps to find the optimal solution of IVTrIFTP using the IVTrIF modified distribution (MODI) method and the flowchart of the IVTrIF MODI method are given in Figure 3.

Step 1: After obtaining the IBF solution using the VAM method, compute dual variables uiIV and vjIV using uiIV+vjIV=cijIV.

Step 2: Define the penalty PijIV=uiIV+vjIV-cijIV for all non-basic variables. Two cases arise from the obtained PijIV values.

i) If all PijIV≤0, then the solution obtained from the IBF solution is the optimal solution and the process is stopped.

ii) If PijIV>0 for at least one (i, j), then go to Step 3.

Step 3: The unoccupied cell with the highest positive cost is selected. This is called a new basic cell and draws a closed loop starting from the selected new basic cell and ensures that the path is in a right-angle turn at only occupied cells.

Step 4: Assign plus and minus signs alternately at the corner points of the closed path. From this, we obtain the least negative allocated value. Add this value with the (+) sign and subtract it with the (−) sign.

Step 5: Continuing step 1, until all PijIV≤0. In this manner, an unoccupied cell becomes an occupied cell to obtain the best optimal solution.

5. Numerical Examples

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Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

In this section, two numerical examples are presented to validate the proposed approach.

Example 1: Consider an IVTrIFTP with cost as IVTrIFNs where the supply and demand are crisp numbers.

Steps for determining the IBF solution for Example 1:

Step 1: We express the transportation Table 2 with IVTrIF costs.

Step 2: ∑i=1mai=∑j=1nbj=60. Therefore, the problem was balanced. We then proceed to Step 4.

Step 4: In each row and column, identify the smallest and next smallest IVTrIF costs using Eqs. (1) and (2). Subsequently, the row- and column-wise penalties were obtained.

From Eq. (1), Table 3 was obtained.

From Eq. (2), (preference value δ = 0.5), Table 4 is obtained.

From Tables 3 and 4, the same ordering was obtained in the row and column directions. The row and column wise ordering is shown in Table 5.

Based on the above ordering, the penalties of the row and column are obtained as follows:

Row Penalties:

c12IV-c13IV =([4,5,6,8];[0.3,0.5];[0.2,0.4])-([1,4,5,6];[0.1,0.3];[0.3,0.5]) =([-2,0,2,7];[0.1,0.3];[0.3,0.5]),c21IV-c23IV =([5,6,7,8];[0.3,0.5];[0.2,0.4])-([2,4,6,7];[0.1,0.3];[0.4,0.6]) =([-2,0,3,6];[0.1,0.3];[0.4,0.6]),c32IV-c31IV =([2,4,5,7];[0.5,0.7];[0.2,0.3])-([2,3,4,5];[0.1,0.3];[0.4,0.6]) =([-3,0,2,5];[0.1,0.3];[0.4,0.6]).

Column Penalties:

c21IV-c31IV =([5,6,7,8];[0.3,0.5];[0.2,0.4])-([2,3,4,5];[0.1,0.3];[0.4,0.6]) =([0,2,4,6];[0.1,0.3];[0.4,0.6],c22IV-c12IV =([2,4,6,7];[0.4,0.6];[0.1,0.3])-([4,5,6,8];[0.3,0.5];[0.2,0.4]) =([-6,-2,1,3];[0.3,0.5];[0.2,0.4]),c13IV-c23IV =([1,4,5,6];[0.1,0.3];[0.3,0.5])-([2,4,6,7];[0.1,0.3];[0.4,0.6]) =([-6,-2,1,4];[0.1,0.3];[0.4,0.6]).

Step 5: In Table 6, the allocation of the maximum feasible amount to the smallest IVTrIF cost in the largest penalty column is shown in bold.

Step 6:Table 7 is obtained by repeating the procedure until all demands and supplies are allocated.

Step 7: The solution to this problem is n + m − 1 basic variables. We get the IBF solution as x₁₂ = 19, x₁₃ = 1, x₂₁ = 2, x₂₃ = 13, x₃₁ = 25.

Transportation cost= 19([4,5,6,8];[0.3,0.5];[0.2,0.4]) +1([1,4,5,6];[0.1,0.3];[0.3,0.5]) +2([5,6,7,8];[0.3,0.5];[0.2,0.4]) +13([2,4,6,7];[0.1,0.3];[0.4,0.6]) +25([2,3,4,5];[0.1,0.3];[0.4,0.6])= ([163,238,311,390];[0.1,0.3];[0.3,0.5]).

Steps to determine the optimum solution for the given example:

Before checking the optimality, we need to check whether n + m − 1= number of allocated cells, where m is the number of rows, and n is the number of columns. Here, n + m − 1= number of allocated cells= 5.

Step 1: We start with u1IV=0 to find all dual variables uiIV and vjIV using uiIV+vjIV=cijIV. The values of u1IV,u2IV,u3IV and v1IV,v2IV,v3IV are listed as

u2IV=([-4,-1,2,6];[0.1,0.3];[0.4,0.6]),u3IV=([-10,-5,0,6];[0.1,0.3];[0.4,0.6]),v1IV=([-1,4,8,12];[0.1,0.3];[0.4,0.6]),v2IV=([4,5,6,8];[0.3,0.5];[0.2,0.4]),v3IV=([1,4,5,6];[0.1,0.3];[0.3,0.5]).

Step 2: To find Penalties PijIV using PijIV=uiIV+vjIV-cijIV for all non-basic variables. Optimality is reached if PijIV<0.

P11IV=([-5,1,6,11];[0.1,0.3];[0.4,0.6]),P22IV=([-7,-2,4,12];[0.1,0.3];[0.4,0.6]),P32IV=([-13,-5,2,12];[0.1,0.3];[0.4,0.6]),P33IV=([-17,-7,1,9];[0.1,0.3];[0.4,0.6]).

Here, all PijIV<0. Hence, optimality was reached.

The optimum transportation cost is = {([163, 238, 311, 390]; [0.1, 0.3]; [0.3, 0.5])}.

5.1 Result and Discussion of Example 1

The membership and non-membership functions for the obtained result ([163, 238, 311, 390]; [0.1, 0.3]; [0.3, 0.5]) are

μLR(x)={x-16375(0.1),163≤x<238,0.1,238≤x≤311,390-x79(0.1),311<x≤390,0,otherwise,μUR(x)={x-16375(0.3),163≤x<238,0.3,238≤x≤311,390-x79(0.3),311<x≤390,0,otherwise,ϑLR(x)={238-x+0.3(x-163)75,163≤x<238,0.3,238≤x≤311,x-311+0.3(390-x)79,311<x≤390,0,otherwise,ϑUR(x)={238-x+0.5(x-163)75,163≤x<238,0.5,238≤x≤311,x-311+0.5(390-x)79,311<x≤390,0,otherwise.

According to the optimizer, the minimum transportation cost lies between 163 units and 390 units.
The overall satisfaction level of the optimizer lies between 238 and 311 units within the interval [0.1, 0.3] and the rejection level lies within the interval [0.3, 0.5].

Example 2: Consider another example of IVTrIFTP with cost as IVTrIFNs.

Steps to find the IBF solution, for example, Eq. (2):

Step 1: The transportation table is expressed using the IVTrIF costs in Table 9.

Step 2: ∑i=1mai=∑j=1nbj=60. Therefore, the problem was balanced. We then proceed to Step 4.

Step 4: Based on Eqs. (1) and (2), row- and column-wise ordering are defined as

Row- and column-wise ordering:

c11IV<c13IV<c12IV,c31IV<c11IV<c21IV,c23IV<c22IV<c21IV,c32IV<c22IV<c12IV,c31IV<c33IV<c32IV,c23IV<c33IV<c13IV.

Row Penalties:

c13IV-c11IV =([2,4,5,7];[0.4,0.7];[0.1,0.3])-([3,5,6,8];[0.1,0.3];[0.4,0.6]) =([-6,-2,0,4];[0.1,0.3];[0.4,0.6]),c22IV-c23IV =([1,3,5,8];[0.3,0.5];[0.2,0.4])-([1,2,4,6];[0.1,0.2];[0.3,0.6]) =([-5,-1,3,7];[0.1,0.2];[0.3,0.6]),c33IV-c31IV =([3,4,6,7];[0.3,0.4];[0.4,0.5])-([1,2,3,4];[0.1,0.2];[0.4,0.7]) =([-1,1,4,6];[0.1,0.2];[0.4,0.7]).

Column Penalties:

c11IV-c31IV =([3,5,6,8];[0.1,0.3];[0.4,0.6])-([1,2,3,4];[0.1,0.2];[0.4,0.7]) =([-1,2,4,7];[0.1,0.2];[0.4,0.7]),c22IV-c32IV =([1,3,5,8];[0.3,0.5];[0.2,0.4])-([3,4,5,8];[0.2,0.4];[0.2,0.5]) =([-7,-2,1,5];[0.2,0.4];[0.2,0.5]),c33IV-c23IV =([3,4,6,7];[0.3,0.4];[0.4,0.5])-([1,2,4,6];[0.1,0.2];[0.3,0.6]) =([-3,0,4,6];[0.1,0.2];[0.4,0.6]).

Step 5: The first allocation table is based on the largest penalties of the rows and columns.

Step 6:Table 11 is obtained by repeating the procedure until all demands and supplies are allocated.

Step 7: The solution to this problem is n + m − 1 basic variables. We get the IBF solution as x₁₁ = 20, x₂₁ = 1, x₂₃ = 14, x₃₁ = 6, x₃₂ = 19.

Transportation cost= 20([3,5,6,8];[0.1,0.3];[0.4,0.6]) +1([2,3,4,6];[0.4,0.6];[0.2.0.3]) +14([1,2,4,6];[0.1,0.2];[0.3.0.6]) +6([1,2,3,4];[0.1,0.2];[0.4.0.7]) +19([3,4,5,8];[0.2,0.4];[0.2.0.5])= {([139,219,293,426];[0.4,0.6];[0.2,0.3])}.

To check the optimality of Example 2,

Step 1: All the dual variables uiIV and vjIV using uiIV+vjIV=cijIV are listed as follows:

u2IV=([-6,-3,-1,3];[0.1,0.3];[0.4,0.6]),u3IV=([-7,-4,-2,1];[0.1,0.2];[0.4,0.7]),v1IV=([3,5,6,8];[0.1,0.3];[0.4,0.6]),v2IV=([2,6,9,15];[0.1,0.2];[0.4,0.7]),v3IV=([-2,3,7,12];[0.1,0.2];[0.4,0.6]).

Step 2: All penalties PijIV using PijIV=uiIV+vjIV-cijIV for all non-basic variables are as follows:

P12IV=([-3,2,6,13];[0.1,0.2];[0.4,0.7]),P13IV=([-9,-2,3,10];[0.1,0.2];[0.4,0.6]),P22IV=([-12,-2,5,17];[0.1,0.2];[0.4,0.7]),P33IV=([-16,-7,1,10];[0.1,0.2];[0.4,0.7]).

All PijIV<0. Optimality is reached.

The optimum transportation cost is = {([139, 219, 293, 426]; [0.4, 0.6]; [0.2, 0.3])}.

5.2 Result and Discussion of Example 2

The membership and non-membership functions for the obtained result ([139, 219, 293, 426]; [0.4, 0.6]; [0.2, 0.3]) are

μLR(x)={x-13980(0.4),139≤x<219,0.4,219≤x≤293,426-x133(0.4),293<x≤426,0,otherwise,μUR(x)={x-13980(0.6),139≤x<219,0.6,219≤x≤293,426-x133(0.6),293<x≤426,0,otherwise,ϑLR(x)={219-x+0.2(x-139)80,139≤x<219,0.2,219≤x≤293,x-293+0.2(426-x)133,293<x≤426,0,otherwise,ϑUR(x)={219-x+0.3(x-139)80,139≤x<219,0.3,219≤x≤293,x-293+0.3(426-x)133,293<x≤426,0,otherwise.

According to the optimizer, the minimum transportation cost lies between 139 units and 426 units.
The overall satisfaction level of the optimizer lies between 219 and 293 units within the interval [0.4, 0.6] and the rejection level lies within the interval [0.2, 0.3].

6. Result and Discussions

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Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

In this study, we focused on TP under IVTrIFNs. Owing to several factors involved in real-life problems, decision makers are in a position to choose the parameters of the TP for better results. For this purpose, the cost parameters are considered as IVTrIFNs. Bharati [33] solved TP in which cost parameters are interval-valued triangular intuitionistic fuzzy numbers; however, there is no method for solving TP under IVTrIFNs. From this perspective, we extend this study. Therefore, it is concluded that our proposed algorithm is an effective and new way to handle uncertainty in real-life scenarios, for example, management and all types of network optimization problems.

6.1 Advantages of the Proposed Method

Because uncertain parameters vary depending upon problem, it is necessary to design such imprecise parameters in decision making problems properly. So, here an attempt is made to solve the TP with IVTrIF cost parameters for the first time. It is more informative than any other parameters in dealing the imprecise parameters in intuitionistic fuzzy environment.

In this paper, as the special case IVTrIFTP is not converted into crisp problem and the IBFS, optimum solution of the given TP are derived as IVTrIFNs.

7. Conclusion

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Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

To deal with uncertain conditions faced by decision makers to predict transportation costs in TPs, the cost of the TP is considered here as IVTrIFNs. In addition, IVTrIFNs were compared with existing ranking methods to obtain an IBF solution using the VAM method. Several researchers, such as Nagoor Gani and Abbas [22] and Kumar and Hussain [23] have discussed TPs in an intuitionistic fuzzy environment. Bharati and his colleague [32,33] discussed transportation problems using interval-valued triangular intuitionistic fuzzy numbers. In this study, a TP is developed under IVTrIFNs and may be implemented for any real-life problem, where the parameters are vague and uncertain in nature. In future, there is a scope to improved ranking methods for a better optimum solution of decision-making problems.

Conflict of Interest

Go to

Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

No potential conflict of interest relevant to this article was reported.

Figures

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Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Fig. 1.

IVTrIFN.

Fig. 2.

Flowchart of IVTrIF VAM.

Fig. 3.

Flowchart of IVTrIF MODI method.

Fig. 4.

Total cost of IVTrIFTP for Example 1.

Fig. 5.

Total cost of IVTrIFTP for Example 2.

Tables

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Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Table. 1.

Table 1. IVTrIF transportation table.

Destinations
Sources	D₁	D₂	⋯	D_n	Supply a_i
S₁	c11IV	c12IV	⋯	c1nIV	a₁
S₂	c21IV	c22IV	⋯	c2nIV	a₂
⋮	⋮	⋮	⋯	⋮	⋮
S_m	cm1IV	cm2IV	⋯	cmnIV	a_m
Demand b_j	b₁	b₂		⋯	b_n

Table. 2.

Table 2. IVTrIF transportation table.

	D₁	D₂	D₃	a_i
S₁	([1, 2, 3, 4]; [0.6, 0.8]; [0.1, 0.2])	([4, 5, 6, 8]; [0.3, 0.5]; [0.2, 0.4])	([1, 4, 5, 6]; [0.1, 0.3]; [0.3, 0.5])	20
S₂	([5, 6, 7, 8]; [0.3, 0.5]; [0.2, 0.4])	([2, 4, 6, 7]; [0.4, 0.6]; [0.1, 0.3])	([2, 4, 6, 7]; [0.1, 0.3]; [0.4, 0.6])	15
S₃	([2, 3, 4, 5]; [0.1, 0.3]; [0.4, 0.6])	([2, 4, 5, 7]; [0.5, 0.7]; [0.2, 0.3])	([3, 4, 6, 8]; [0.4, 0.6]; [0, 0.2])	25
b_j	27	19	14	60

Table. 3.

Table 3. Score matrix.

0.55	0.1	−0.2
0.1	0.3	−0.3
−0.3	0.35	0.4

Table. 4.

Table 4. Score expected matrix.

1.375	0.575	−0.8
0.65	1.425	−1.425
−1.05	1.575	2.1

Table. 5.

Table 5. Ordering table.

Ordering with respect to row and column
c13IV<c12IV<c11IV	c31IV<c21IV<c11IV
c23IV<c21IV<c22IV	c12IV<c22IV<c32IV
c31IV<c32IV<c33IV	c23IV<c13IV<c33IV

Table. 6.

Table 6. Allocated table.

	D₁	D₂	D₃	a_i
S₁	([1, 2, 3, 4]; [0.6, 0.8]; [0.1, 0.2])	([4,5,6,8];[0.3,0.5];[0.2,0.4])(19)	([1, 4, 5, 6]; [0.1, 0.3]; [0.3, 0.5])	20
S₂	([5, 6, 7, 8]; [0.3, 0.5]; [0.2, 0.4])	([2, 4, 6, 7]; [0.4, 0.6]; [0.1, 0.3])	([2, 4, 6, 7]; [0.1, 0.3]; [0.4, 0.6])	15
S₃	([2, 3, 4, 5]; [0.1, 0.3]; [0.4, 0.6])	([2, 4, 5, 7]; [0.5, 0.7]; [0.2, 0.3])	([3, 4, 6, 8]; [0.4, 0.6]; [0, 0.2])	25
b_j	27	19	14

Table. 7.

Table 7. Allocated table.

	D₁	D₂	D₃	a_i
S₁	([1, 2, 3, 4]; [0.6, 0.8]; [0.1, 0.2])	([4,5,6,8];[0.3,0.5];[0.2,0.4])(19)	([1,4,5,6];[0.1,0.3];[0.3,0.5])(1)	20
S₂	([5,6,7,8];[0.3,0.5];[0.2,0.4])(2)	([2, 4, 6, 7]; [0.4, 0.6]; [0.1, 0.3])	([2,4,6,7];[0.1,0.3];[0.4,0.6])(13)	15
S₃	([2,3,4,5];[0.1,0.3];[0.4,0.6])(25)	([2, 4, 5, 7]; [0.5, 0.7]; [0.2, 0.3])	([3, 4, 6, 8]; [0.4, 0.6]; [0, 0.2])	25
b_j	27	19	14

Table. 8.

Table 8. Optimum table for Example 1.

	D₁	D₂	D₃	a_i
S₁	([1, 2, 3, 4]; [0.6, 0.8]; [0.1, 0.2])	([4,5,6,8];[0.3,0.5];[0.2,0.4])(19)	([1,4,5,6];[0.1,0.3];[0.3,0.5])(1)	20
S₂	([5,6,7,8];[0.3,0.5];[0.2,0.4])(2)	([2, 4, 6, 7]; [0.4, 0.6]; [0.1, 0.3])	([2,4,6,7];[0.1,0.3];[0.4,0.6])(13)	15
S₃	([2,3,4,5];[0.1,0.3];[0.4,0.6])(25)	([2, 4, 5, 7]; [0.5, 0.7]; [0.2, 0.3])	([3, 4, 6, 8]; [0.4, 0.6]; [0, 0.2])	25
b_j	27	19	14

Table. 9.

Table 9. IVTrIF transportation table.

	D₁	D₂	D₃	a_i
S₁	([3, 5, 6, 8]; [0.1, 0.3]; [0.4, 0.6])	([2, 3, 4, 5]; [0.6, 0.8]; [0.0, 0.1])	([2, 4, 5, 7]; [0.4, 0.7]; [0.1, 0.3])	20
S₂	([2, 3, 4, 6]; [0.4, 0.6]; [0.2, 0.3])	([1, 3, 5, 8]; [0.3, 0.5]; [0.2, 0.4])	([1, 2, 4, 6]; [0.1, 0.2]; [0.3, 0.6])	15
S₃	([1, 2, 3, 4]; [0.1, 0.2]; [0.4, 0.7])	([3, 4, 5, 8]; [0.2, 0.4]; [0.2, 0.5])	([3, 4, 6, 7]; [0.3, 0.4]; [0.4, 0.5])	25
b_j	27	19	14	60

Table. 10.

Table 10. Allocated table.

	D₁	D₂	D₃	a_i
S₁	([3, 5, 6, 8]; [0.1, 0.3]; [0.4, 0.6])	([2, 3, 4, 5]; [0.6, 0.8]; [0.0, 0.1])	([2, 4, 5, 7]; [0.4, 0.7]; [0.1, 0.3])	20
S₂	([2, 3, 4, 6]; [0.4, 0.6]; [0.2, 0.3])	([1, 3, 5, 8]; [0.3, 0.5]; [0.2, 0.4])	([1, 2, 4, 6]; [0.1, 0.2]; [0.3, 0.6])	15
S₃	([1, 2, 3, 4]; [0.1, 0.2]; [0.4, 0.7])	([3,4,5,8];[0.2,0.4];[0.2,0.5])(19)	([3, 4, 6, 7]; [0.3, 0.4]; [0.4, 0.5])	25
b_j	27	19	14

Table. 11.

Table 11. Allocated table.

	D₁	D₂	D₃	a_i
S₁	([3,5,6,8];[0.1,0.3];[0.4,0.6])(20)	([2, 3, 4, 5]; [0.6, 0.8]; [0.0, 0.1])	([2, 4, 5, 7]; [0.4, 0.7]; [0.1, 0.3])	20
S₂	([2,3,4,6];[0.4,0.6];[0.2,0.3])(1)	([1, 3, 5, 8]; [0.3, 0.5]; [0.2, 0.4])	([1,2,4,6];[0.1,0.2];[0.3,0.6])(14)	15
S₃	([1,2,3,4];[0.1,0.2];[0.4,0.7])(6)	([3,4,5,8];[0.2,0.4];[0.2,0.5])(19)	([3, 4, 6, 7]; [0.3, 0.4]; [0.4, 0.5])	25
b_j	27	19	14

Table. 12.

Table 12. Optimum table for Example 2.

	D₁	D₂	D₃	a_i
S₁	([3,5,6,8];[0.1,0.3];[0.4,0.6])(20)	([2, 3, 4, 5]; [0.6, 0.8]; [0.0, 0.1])	([2, 4, 5, 7]; [0.4, 0.7]; [0.1, 0.3])	20
S₂	([2,3,4,6];[0.4,0.6];[0.2,0.3])(1)	([1, 3, 5, 8]; [0.3, 0.5]; [0.2, 0.4])	([1,2,4,6];[0.1,0.2];[0.3,0.6])(14)	15
S₃	([1,2,3,4];[0.1,0.2];[0.4,0.7])(6)	([3,4,5,8];[0.2,0.4];[0.2,0.5])(19)	([3, 4, 6, 7]; [0.3, 0.4]; [0.4, 0.5])	25
b_j	27	19	14

References

Go to

Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

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Biographies

Go to

Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

S. Dhanasekar received his B.Sc. and M.Sc. degrees in Mathematics from the University of Madras, Chennai, India, in 1996 and 1998, respectively. He obtained his Ph.D. degree in Fuzzy Optimization in the year 2019 from Bharathiar University. He is currently working as an assistant professor in the Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology-Chennai Campus. He published more research articles in the area of fuzzy optimization problems. His research interests include fuzzy logic, fuzzy decision-making problems, and operations research problems in a fuzzy environment.

E-mail: dhanasekar.sundaram@vit.ac.in

J. Jansi Rani received the B.Sc. and M.Sc. degrees in Mathematics from University of Madras, Chennai, India in 2005 and 2007, respectively. She obtained her M.Phil. degree in Mathematics from Sri Venkateswara University, Tirupati, India 2009. Her research interest focuses on the optimization problems in intuitionistic fuzzy and generalized fuzzy environment.

E-mail: jansiranij.2019@vitstudent.ac.in

Manivannan Annamalai received the Ph.D. degree in Mathematics from Madurai Kamaraj University, Madurai, Tamilnadu, India in 2015. He published many research articles in SCI journals. He is currently an assistant professor in the Division of Mathematics at Vellore Institute of Technology - Chennai Campus, India. His research interests include stability theory, neural networks, complex systems, fuzzy functional differential equations, and fractional differential

E-mail: manivannan.a@vit.ac.in

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 155-168

Published online June 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.2.155

Transportation Problem for Interval-Valued Trapezoidal Intuitionistic Fuzzy Numbers

S. Dhanasekar , J. Jansi Rani, and Manivannan Annamalai

Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai, India

Correspondence to:Manivannan Annamalai (manivannan.a@vit.ac.in)

Received: November 30, 2021; Revised: May 2, 2022; Accepted: May 24, 2022

Abstract

Keywords: Transportation problem, Interval-valued trapezoidal intuitionistic fuzzy number, Arithmetic operations, Interval-valued trapezoidal intuitionistic fuzzy transportation problem

1. Introduction

1.1 Fuzzy, Intuitionistic, Interval-Valued Intuitionistic Fuzzy in Optimization Problems

1.2 Motivation of the Proposed Study

1.3 Structure of the Paper

2. Preliminaries

In this section, basic definitions and operations are provided, which are useful for providing the results.

2.1 Definitions

Definition 2.1 [5]

The number A which is defined in the given interval of real numbers ℝ that has the form A : ℝ → [0, 1] is called a fuzzy number that satisfies the following conditions:

(a) it should be convex
(b) Maximum height is 1, (i.e., normal)
(c) Membership function is piecewise continuous

Definition 2.2 [6]

Definition 2.3 [7]

For the fixed set X = {x₁, x₂, x₃, …, x_n}, An IVIFS A^IV is defined as A^IV = {(x, μ_{A_^IV} (x), ϑ_{A_^IV} (x) | x ∈ X)} where μ_{A_^IV} (x) : X → ℝ[0, 1], ϑ_{A_^IV} (x) : X → ℝ[0, 1] and $μ_{A^{I V}} (x) = [μ_{A^{I V}}^{L R} (x), μ_{A^{I V}}^{U R} (x)], ϑ_{A^{I V}} (x) = [ϑ_{A^{I V}}^{L R} (x), ϑ_{A^{I V}}^{U R} (x)]$ where $μ_{A^{I V}}^{L R} (x), μ_{A^{I V}}^{U R} (x)$ and $ϑ_{A^{I V}}^{L R} (x), ϑ_{A^{I V}}^{U R} (x)$ denote the infimum (inf) and supremum (sup), values of membership and non-membership functions, respectively. The intervals μ_{A_^IV} (x) and ϑ_{A_^IV} (x) denote the degrees of belongingness and non-belongingness of x to A^IV respectively, satisfying the condition $μ_{A^{I V}}^{U R} (x) + ϑ_{A^{I V}}^{U R} (x) \leq 1$ ,x ∈ X.

From this, it is clear that the IVIFS becomes intuitionistic fuzzy set if $μ_{A^{I V}}^{L R} (x) = μ_{A^{I V}}^{U R} (x)$ and $ϑ_{A^{I V}}^{L R} (x) = ϑ_{A^{I V}}^{U R} (x)$ . Below are some operations of IVIFSs.

Let A^IV and B^IV be two IVIFSs and is defined as $A^{I V} = {[μ_{A^{I V}}^{L R} (x), μ_{A^{I V}}^{U R} (x)], [ϑ_{A^{I V}}^{L R} (x), ϑ_{A^{I V}}^{U R} (x)]}$ and $B^{I V} = {[μ_{B^{I V}}^{L R} (x), μ_{B^{I V}}^{U R} (x)], [ϑ_{B^{I V}}^{L R} (x), ϑ_{B^{I V}}^{U R} (x)]}$ Then,

$A^{I V} + B^{I V} = {[μ_{A^{I V}}^{L R} (x) + μ_{B^{I V}}^{L R} (x) - μ_{A^{I V}}^{L R} (x) . μ_{B^{I V}}^{L R} (x), μ_{A^{I V}}^{U R} (x) + μ_{B^{I V}}^{U R} (x) - μ_{A^{I V}}^{R} (x) . μ_{B^{I V}}^{U R} (x)], [ϑ_{A^{I V}}^{L R} (x) . ϑ_{B^{I V}}^{L R} (x); ϑ_{A^{I V}}^{U R} (x) . ϑ_{B^{I V}}^{U R} (x)]}$ .
$A^{I V} . B^{I V} = {[μ_{A^{I V}}^{L R} (x) . μ_{B^{I V}}^{L R} (x), μ_{A^{I V}}^{U R} (x) . μ_{B^{I V}}^{U R} (x)], [ϑ_{A^{I V}}^{L R} (x) + ϑ_{B^{I V}}^{L R} (x) - ϑ_{A^{I V}}^{L R} (x) . ϑ_{B^{I V}}^{L R} (x), ϑ_{A^{I V}}^{U R} (x) + ϑ_{B^{I V}}^{U R} (x) - ϑ_{A^{I V}}^{U R} (x) . ϑ_{B^{I V}}^{U R} (x)]}$ .
$k (A^{I V}) = {[1 - {(1 - μ_{A^{I V}}^{L R} (x))}^{k}, 1 - {(1 - μ_{A^{I V}}^{U R} (x))}^{k}], [{(ϑ_{A^{I V}}^{L R} (x))}^{k}, {(ϑ_{A^{I V}}^{U R} (x))}^{k}]}, k > 0$ .
${(A^{I V})}^{k} = {[{(μ_{A^{I V}}^{L R} (x))}^{k}, {(μ_{A^{I V}}^{U R} (x))}^{k}], [1 - {(1 - ϑ_{A^{I V}}^{L R} (x))}^{k}, 1 - {(1 - ϑ_{A^{I V}}^{U R} (x))}^{k}]}, k > 0$ .

Definition 2.4 [33]

Membership and non-membership functions with their lower and upper intervals for IVTrIFN $A^{I V} = {[a_{1}, b_{1}, c_{1}, d_{1}]; [μ_{A^{I V}}^{L R} (x), μ_{A^{I V}}^{U R} (x)]; [ϑ_{A^{I V}}^{L R} (x), ϑ_{A^{I V}}^{U R} (x)]}$ is defined as follows:

\begin{array}{l} μ_{A^{I V}}^{L R} (x) = {\begin{array}{l} \frac{x - a_{1}}{b_{1} - a_{1}} μ_{A^{I V}}^{L R}, & a_{1} \leq x < b_{1}, \\ μ_{A^{I V}}^{L R}, & b_{1} \leq x \leq c_{1}, \\ \frac{d_{1} - x}{d_{1} - c_{1}} μ_{A^{I V}}^{L R}, & c_{1} < x \leq d_{1}, \\ 0, & otherwise, \end{array} \\ μ_{A^{I V}}^{U R} (x) = {\begin{array}{l} \frac{x - a_{1}}{b_{1} - a_{1}} μ_{A^{I V}}^{U R}, & a_{1} \leq x < b_{1}, \\ μ_{A^{I V}}^{U R}, & b_{1} \leq x \leq c_{1}, \\ \frac{d_{1} - x}{d_{1} - c_{1}} μ_{A^{I V}}^{U R}, & c_{1} < x \leq d_{1}, \\ 0, & otherwise, \end{array} \\ ϑ_{A^{I V}}^{L R} (x) = {\begin{array}{l} \frac{b_{1} - x + ϑ_{A^{I V}}^{L R} (x - a_{1})}{b_{1} - a_{1}}, & a_{1} \leq x < b_{1}, \\ ϑ_{A^{I V}}^{L R}, & b_{1} \leq x \leq c_{1}, \\ \frac{x - c_{1} + ϑ_{A^{I V}}^{L R} (d_{1} - x)}{d_{1} - c_{1}}, & c_{1} < x \leq d_{1}, \\ 0, & otherwise, \end{array} \\ ϑ_{A^{I V}}^{U R} (x) = {\begin{array}{l} \frac{b_{1} - x + ϑ_{A^{I V}}^{U R} (x - a_{1})}{b_{1} - a_{1}}, & a_{1} \leq x < b_{1}, \\ ϑ_{A^{I V}}^{U R}, & b_{1} \leq x \leq c_{1}, \\ \frac{x - c_{1} + ϑ_{A^{I V}}^{U R} (d_{1} - x)}{d_{1} - c_{1}}, & c_{1} < x \leq d_{1}, \\ 0, & otherwise, \end{array} \end{array}

where a₁, b₁, c₁, d₁ are real numbers that satisfy the conditions $0 \leq μ_{A^{I V}}^{L R} \leq 1, 0 \leq μ_{A^{I V}}^{U R} \leq 1, 0 \leq ϑ_{A^{I V}}^{L R} \leq 1, 0 \leq ϑ_{A^{I V}}^{U R} \leq 1$ and $μ_{A^{I V}}^{U R} + ϑ_{A^{I V}}^{U R} \leq 1$ .

A graphical representation of IVTrIFN $A^{I V} = {[a_{1}, b_{1}, c_{1}, d_{1}]; [μ_{A^{I V}}^{L R} (x), μ_{A^{I V}}^{U R} (x)]; [ϑ_{A^{I V}}^{L R} (x), ϑ_{A^{I V}}^{U R} (x)]}$ with its membership and non-membership functions is shown in Figure 1.

2.2 Operations of IVTrIFNs [36]

Let $A^{I V} = {[a_{1}, b_{1}, c_{1}, d_{1}]; [μ_{A^{I V}}^{L R} (x), μ_{A^{I V}}^{U R} (x)]; [ϑ_{A^{I V}}^{L R} (x), ϑ_{A^{I V}}^{U R} (x)]}$ and $B^{I V} = {[a_{2}, b_{2}, c_{2}, d_{2}]; [μ_{B^{I V}}^{L R} (x), μ_{B^{I V}}^{U R} (x)]; [ϑ_{B^{I V}}^{L R} (x), ϑ_{B^{I V}}^{U R} (x)]}$ are the two IVTrIFNs. Then,

$A^{I V} + B^{I V} = {([a_{1} + a_{2}, b_{1} + b_{2}, c_{1} + c_{2}, d_{1} + d_{2}]; [m i n (μ_{A^{I V}}^{L R}, μ_{B^{I V}}^{L R}), m i n (μ_{A^{I V}}^{U R}, μ_{B^{I V}}^{U R})]; [m a x (ϑ_{A^{I V}}^{L R}, ϑ_{B^{I V}}^{L R}), m a x (ϑ_{A^{I V}}^{U R}, ϑ_{B^{I V}}^{U R})])}$ .
$A^{I V} - B^{I V} = {([a_{1} - d_{2}, b_{1} - c_{2}, c_{1} - b_{2}, d_{1} - a_{2}]; [m i n (μ_{A^{I V}}^{L R}, μ_{B^{I V}}^{L R}), m i n (μ_{A^{I V}}^{U R}, μ_{B^{I V}}^{U R})]; [m a x (ϑ_{A^{I V}}^{L R}, ϑ_{B^{I V}}^{L R}), m a x (ϑ_{A^{I V}}^{U R}, ϑ_{B^{I V}}^{U R})])}$ .
$A^{I V} . B^{I V} = {([a_{1} a_{2}, b_{1} b_{2}, c_{1} c_{2}, d_{1} d_{2}]; [μ_{A^{I V}}^{L R} . μ_{B^{I V}}^{L R}, μ_{A^{I V}}^{U R} . μ_{B^{I V}}^{U R}]; [ϑ_{A^{I V}}^{L R} + ϑ_{B^{I V}}^{L R} - ϑ_{A^{I V}}^{L R} . ϑ_{B^{I V}}^{L R}, ϑ_{A^{I V}}^{U R} + ϑ_{B^{I V}}^{U R} - ϑ_{A^{I V}}^{U R} . ϑ_{B^{I V}}^{U R}])}$ .
$k (A^{I V}) = {([k a_{1}, k b_{1}, k c_{1}, k d_{1}]; [1 - {(1 - μ_{A^{I V}}^{L R})}^{k}, 1 - {(1 - μ_{A^{I V}}^{U R})}^{k}]; [{(ϑ_{A^{I V}}^{L R})}^{k}, {(ϑ_{A^{I V}}^{U R})}^{k}])}, k > 0$ .
${(A^{I V})}^{k} = {([a_{1}^{k}, b_{1}^{k}, c_{1}^{k}, d_{1}^{k}]; [{(μ_{A^{I V}}^{L R})}^{k}, {(μ_{A^{I V}}^{U R})}^{k}]; [1 - {(1 - ϑ_{A^{I V}}^{L R})}^{k}, 1 - {(1 - ϑ_{A^{I V}}^{U R})}^{k}])} k > 0$ .

3. Ordering Methods of IVTrIFNs

Definition 3.1 [40]

Let $A^{I V} = {([a, b, c, d]; [μ_{A^{I V}}^{L R}, μ_{A^{I V}}^{U R}]; [ϑ_{A^{I V}}^{L R}, ϑ_{A^{I V}}^{U R}])}$ be an IVTrIFN. Subsequently, the score function is defined as

(1)

S (A^{I V}) = \frac{μ_{A^{I V}}^{L R} (x) + μ_{A^{I V}}^{U R} (x) - ϑ_{A^{I V}}^{L R} (x) - ϑ_{A^{I V}}^{U R} (x)}{2} .

Definition 3.2 [45]

Let $(A^{I V}) = {([a, b, c, d]; [μ_{A^{I V}}^{L R}, μ_{A^{I V}}^{U R}]; [ϑ_{A^{I V}}^{L R}, ϑ_{A^{I V}}^{U R}])}$ be an IVTrIFN. Then, the score expectation function is defined as

(2)

I (S (A^{I V})) = \frac{S (A^{I V})}{2} ((1 - δ) (a + b) + δ (c + d)),

(δ denotes the preference value).

4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem

The mathematical formulation of IVTrIFTP is in the following form:

minimize Φ^{I V} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} c_{i j}^{I V} * x_{i j},

subject to

\begin{array}{l} \sum_{i = 1}^{m} x_{i j} \leq a_{i}, i = 1, 2, \dots, m, \\ \sum_{j = 1}^{n} x_{i j} \geq b_{j}, j = 1, 2, \dots, n, \\ x_{i j} \geq 0 for all i and j, \end{array}

where

The IVTrIF cost of sending one unit of the goods from the source (origin) i to destination (end) j is denoted by $c_{i j}^{I V}$ .
Here, the number of origins is denoted by m, which is indexed by i.
Here, the number of destinations is denoted by n, which is indexed by j.
x_ij is the quantity of transportation from the ith origin to the jth end.
a_i is the total availability and b_j is the total demand of the goods at the ith source and jth destination, respectively.
$\sum_{i = 1}^{m} \sum_{j = 1}^{n} c_{i j}^{I V} * x_{i j}$ is the total IVTrIF transportation cost.

The IVTrIFTP is balanced if the total availability is equal to the total demand, which can be expressed as $\sum_{i = 1}^{m} a_{i} = \sum_{j = 1}^{n} b_{j}$ , where $c_{i j}^{I V}$ represents the IVTrIF costs, and a_i and b_j represent the crisp units of the total availability and demand of the goods, respectively.

Steps to find the IBF solution of IVTrIFTP by IVTrIF VAM and the flow chart of the IVTrIF VAM are given in Figure 2.

Step 1: Establish the IVTrIF transportation table with the costs as IVTrIF numbers.

Step 2: Examine whether the problem is balanced. If $\sum_{i = 1}^{m} a_{i} = \sum_{j = 1}^{n} b_{j}$ , then proceed to Step 4. Otherwise go to Step 3.

Step 3: If it is unbalanced, transform the IVTrIFTP into a balanced problem by introducing a dummy source or demand. Then go to Step 4.

Step 4: Identify the smallest and next smallest IVTrIF costs in each row of the transportation table, and find the difference between them for each row. Similarly, we do the same for the columns.

Step 5: Identify the largest penalty among all the rows and columns and use arbitrary tie-breaking choice if tie occurs. Now, we take the maximum penalty corresponding to ith row and let $R (c_{i j}^{I V})$ be the smallest IVTrIF cost in the ith row. Then, allot the maximum feasible amount x_ij in the (i, j)th cell and deletes either the ith row or jth column.

Step 6: Repeat Step 4, to find the row and column penalties for the reduced transportation table. The procedures are repeated until all the demands and supplies are allocated.

Step 7: Finally from the IVTrIF transportation table, the IBF solution is obtained.

The steps to find the optimal solution of IVTrIFTP using the IVTrIF modified distribution (MODI) method and the flowchart of the IVTrIF MODI method are given in Figure 3.

Step 1: After obtaining the IBF solution using the VAM method, compute dual variables $u_{i}^{I V}$ and $v_{j}^{I V}$ using $u_{i}^{I V} + v_{j}^{I V} = c_{i j}^{I V}$ .

Step 2: Define the penalty $P_{i j}^{I V} = u_{i}^{I V} + v_{j}^{I V} - c_{i j}^{I V}$ for all non-basic variables. Two cases arise from the obtained $P_{i j}^{I V}$ values.

i) If all $P_{i j}^{I V} \leq 0$ , then the solution obtained from the IBF solution is the optimal solution and the process is stopped.

ii) If $P_{i j}^{I V} > 0$ for at least one (i, j), then go to Step 3.

Step 5: Continuing step 1, until all $P_{i j}^{I V} \leq 0$ . In this manner, an unoccupied cell becomes an occupied cell to obtain the best optimal solution.

5. Numerical Examples

In this section, two numerical examples are presented to validate the proposed approach.

Example 1: Consider an IVTrIFTP with cost as IVTrIFNs where the supply and demand are crisp numbers.

Steps for determining the IBF solution for Example 1:

Step 1: We express the transportation Table 2 with IVTrIF costs.

Step 2: $\sum_{i = 1}^{m} a_{i} = \sum_{j = 1}^{n} b_{j} = 60$ . Therefore, the problem was balanced. We then proceed to Step 4.

Step 4: In each row and column, identify the smallest and next smallest IVTrIF costs using Eqs. (1) and (2). Subsequently, the row- and column-wise penalties were obtained.

From Eq. (1), Table 3 was obtained.

From Eq. (2), (preference value δ = 0.5), Table 4 is obtained.

From Tables 3 and 4, the same ordering was obtained in the row and column directions. The row and column wise ordering is shown in Table 5.

Based on the above ordering, the penalties of the row and column are obtained as follows:

Row Penalties:

\begin{array}{l} c_{12}^{I V} - c_{13}^{I V} & = ([4, 5, 6, 8]; [0.3, 0.5]; [0.2, 0.4]) - ([1, 4, 5, 6]; [0.1, 0.3]; [0.3, 0.5]) \\ = ([- 2, 0, 2, 7]; [0.1, 0.3]; [0.3, 0.5]), \\ c_{21}^{I V} - c_{23}^{I V} & = ([5, 6, 7, 8]; [0.3, 0.5]; [0.2, 0.4]) - ([2, 4, 6, 7]; [0.1, 0.3]; [0.4, 0.6]) \\ = ([- 2, 0, 3, 6]; [0.1, 0.3]; [0.4, 0.6]), \\ c_{32}^{I V} - c_{31}^{I V} & = ([2, 4, 5, 7]; [0.5, 0.7]; [0.2, 0.3]) - ([2, 3, 4, 5]; [0.1, 0.3]; [0.4, 0.6]) \\ = ([- 3, 0, 2, 5]; [0.1, 0.3]; [0.4, 0.6]) . \end{array}

Column Penalties:

\begin{array}{l} c_{21}^{I V} - c_{31}^{I V} & = ([5, 6, 7, 8]; [0.3, 0.5]; [0.2, 0.4]) - ([2, 3, 4, 5]; [0.1, 0.3]; [0.4, 0.6]) \\ = ([0, 2, 4, 6]; [0.1, 0.3]; [0.4, 0.6], \\ c_{22}^{I V} - c_{12}^{I V} & = ([2, 4, 6, 7]; [0.4, 0.6]; [0.1, 0.3]) - ([4, 5, 6, 8]; [0.3, 0.5]; [0.2, 0.4]) \\ = ([- 6, - 2, 1, 3]; [0.3, 0.5]; [0.2, 0.4]), \\ c_{13}^{I V} - c_{23}^{I V} & = ([1, 4, 5, 6]; [0.1, 0.3]; [0.3, 0.5]) - ([2, 4, 6, 7]; [0.1, 0.3]; [0.4, 0.6]) \\ = ([- 6, - 2, 1, 4]; [0.1, 0.3]; [0.4, 0.6]) . \end{array}

Step 5: In Table 6, the allocation of the maximum feasible amount to the smallest IVTrIF cost in the largest penalty column is shown in bold.

Step 6:Table 7 is obtained by repeating the procedure until all demands and supplies are allocated.

Step 7: The solution to this problem is n + m − 1 basic variables. We get the IBF solution as x₁₂ = 19, x₁₃ = 1, x₂₁ = 2, x₂₃ = 13, x₃₁ = 25.

\begin{array}{l} Transportation cost = & 19 ([4, 5, 6, 8]; [0.3, 0.5]; [0.2, 0.4]) \\ + 1 ([1, 4, 5, 6]; [0.1, 0.3]; [0.3, 0.5]) \\ + 2 ([5, 6, 7, 8]; [0.3, 0.5]; [0.2, 0.4]) \\ + 13 ([2, 4, 6, 7]; [0.1, 0.3]; [0.4, 0.6]) \\ + 25 ([2, 3, 4, 5]; [0.1, 0.3]; [0.4, 0.6]) \\ = & ([163, 238, 311, 390]; [0.1, 0.3]; [0.3, 0.5]) . \end{array}

Steps to determine the optimum solution for the given example:

Step 1: We start with $u_{1}^{I V} = 0$ to find all dual variables $u_{i}^{I V}$ and $v_{j}^{I V}$ using $u_{i}^{I V} + v_{j}^{I V} = c_{i j}^{I V}$ . The values of $u_{1}^{I V}, u_{2}^{I V}, u_{3}^{I V}$ and $v_{1}^{I V}, v_{2}^{I V}, v_{3}^{I V}$ are listed as

\begin{array}{l} u_{2}^{I V} = ([- 4, - 1, 2, 6]; [0.1, 0.3]; [0.4, 0.6]), \\ u_{3}^{I V} = ([- 10, - 5, 0, 6]; [0.1, 0.3]; [0.4, 0.6]), \\ v_{1}^{I V} = ([- 1, 4, 8, 12]; [0.1, 0.3]; [0.4, 0.6]), \\ v_{2}^{I V} = ([4, 5, 6, 8]; [0.3, 0.5]; [0.2, 0.4]), \\ v_{3}^{I V} = ([1, 4, 5, 6]; [0.1, 0.3]; [0.3, 0.5]) . \end{array}

Step 2: To find Penalties $P_{i j}^{I V}$ using $P_{i j}^{I V} = u_{i}^{I V} + v_{j}^{I V} - c_{i j}^{I V}$ for all non-basic variables. Optimality is reached if $P_{i j}^{I V} < 0$ .

\begin{array}{l} P_{11}^{I V} = ([- 5, 1, 6, 11]; [0.1, 0.3]; [0.4, 0.6]), \\ P_{22}^{I V} = ([- 7, - 2, 4, 12]; [0.1, 0.3]; [0.4, 0.6]), \\ P_{32}^{I V} = ([- 13, - 5, 2, 12]; [0.1, 0.3]; [0.4, 0.6]), \\ P_{33}^{I V} = ([- 17, - 7, 1, 9]; [0.1, 0.3]; [0.4, 0.6]) . \end{array}

Here, all $P_{i j}^{I V} < 0$ . Hence, optimality was reached.

The optimum transportation cost is = {([163, 238, 311, 390]; [0.1, 0.3]; [0.3, 0.5])}.

5.1 Result and Discussion of Example 1

The membership and non-membership functions for the obtained result ([163, 238, 311, 390]; [0.1, 0.3]; [0.3, 0.5]) are

\begin{array}{l} μ^{L R} (x) = {\begin{array}{l} \frac{x - 163}{75} (0.1), & 163 \leq x < 238, \\ 0.1, & 238 \leq x \leq 311, \\ \frac{390 - x}{79} (0.1), & 311 < x \leq 390, \\ 0, & otherwise, \end{array} \\ μ^{U R} (x) = {\begin{array}{l} \frac{x - 163}{75} (0.3), & 163 \leq x < 238, \\ 0.3, & 238 \leq x \leq 311, \\ \frac{390 - x}{79} (0.3), & 311 < x \leq 390, \\ 0, & otherwise, \end{array} \\ ϑ^{L R} (x) = {\begin{array}{l} \frac{238 - x + 0.3 (x - 163)}{75}, & 163 \leq x < 238, \\ 0.3, & 238 \leq x \leq 311, \\ \frac{x - 311 + 0.3 (390 - x)}{79}, & 311 < x \leq 390, \\ 0, & otherwise, \end{array} \\ ϑ^{U R} (x) = {\begin{array}{l} \frac{238 - x + 0.5 (x - 163)}{75}, & 163 \leq x < 238, \\ 0.5, & 238 \leq x \leq 311, \\ \frac{x - 311 + 0.5 (390 - x)}{79}, & 311 < x \leq 390, \\ 0, & otherwise . \end{array} \end{array}

According to the optimizer, the minimum transportation cost lies between 163 units and 390 units.
The overall satisfaction level of the optimizer lies between 238 and 311 units within the interval [0.1, 0.3] and the rejection level lies within the interval [0.3, 0.5].

Example 2: Consider another example of IVTrIFTP with cost as IVTrIFNs.

Steps to find the IBF solution, for example, Eq. (2):

Step 1: The transportation table is expressed using the IVTrIF costs in Table 9.

Step 2: $\sum_{i = 1}^{m} a_{i} = \sum_{j = 1}^{n} b_{j} = 60$ . Therefore, the problem was balanced. We then proceed to Step 4.

Step 4: Based on Eqs. (1) and (2), row- and column-wise ordering are defined as

Row- and column-wise ordering:

\begin{array}{l} c_{11}^{I V} < c_{13}^{I V} < c_{12}^{I V}, & c_{31}^{I V} < c_{11}^{I V} < c_{21}^{I V}, \\ c_{23}^{I V} < c_{22}^{I V} < c_{21}^{I V}, & c_{32}^{I V} < c_{22}^{I V} < c_{12}^{I V}, \\ c_{31}^{I V} < c_{33}^{I V} < c_{32}^{I V}, & c_{23}^{I V} < c_{33}^{I V} < c_{13}^{I V} . \end{array}

Row Penalties:

\begin{array}{l} c_{13}^{I V} - c_{11}^{I V} & = ([2, 4, 5, 7]; [0.4, 0.7]; [0.1, 0.3]) - ([3, 5, 6, 8]; [0.1, 0.3]; [0.4, 0.6]) \\ = ([- 6, - 2, 0, 4]; [0.1, 0.3]; [0.4, 0.6]), \\ c_{22}^{I V} - c_{23}^{I V} & = ([1, 3, 5, 8]; [0.3, 0.5]; [0.2, 0.4]) - ([1, 2, 4, 6]; [0.1, 0.2]; [0.3, 0.6]) \\ = ([- 5, - 1, 3, 7]; [0.1, 0.2]; [0.3, 0.6]), \\ c_{33}^{I V} - c_{31}^{I V} & = ([3, 4, 6, 7]; [0.3, 0.4]; [0.4, 0.5]) - ([1, 2, 3, 4]; [0.1, 0.2]; [0.4, 0.7]) \\ = ([- 1, 1, 4, 6]; [0.1, 0.2]; [0.4, 0.7]) . \end{array}

Column Penalties:

\begin{array}{l} c_{11}^{I V} - c_{31}^{I V} & = ([3, 5, 6, 8]; [0.1, 0.3]; [0.4, 0.6]) - ([1, 2, 3, 4]; [0.1, 0.2]; [0.4, 0.7]) \\ = ([- 1, 2, 4, 7]; [0.1, 0.2]; [0.4, 0.7]), \\ c_{22}^{I V} - c_{32}^{I V} & = ([1, 3, 5, 8]; [0.3, 0.5]; [0.2, 0.4]) - ([3, 4, 5, 8]; [0.2, 0.4]; [0.2, 0.5]) \\ = ([- 7, - 2, 1, 5]; [0.2, 0.4]; [0.2, 0.5]), \\ c_{33}^{I V} - c_{23}^{I V} & = ([3, 4, 6, 7]; [0.3, 0.4]; [0.4, 0.5]) - ([1, 2, 4, 6]; [0.1, 0.2]; [0.3, 0.6]) \\ = ([- 3, 0, 4, 6]; [0.1, 0.2]; [0.4, 0.6]) . \end{array}

Step 5: The first allocation table is based on the largest penalties of the rows and columns.

Step 6:Table 11 is obtained by repeating the procedure until all demands and supplies are allocated.

Step 7: The solution to this problem is n + m − 1 basic variables. We get the IBF solution as x₁₁ = 20, x₂₁ = 1, x₂₃ = 14, x₃₁ = 6, x₃₂ = 19.

\begin{array}{l} Transportation cost = & 20 ([3, 5, 6, 8]; [0.1, 0.3]; [0.4, 0.6]) \\ + 1 ([2, 3, 4, 6]; [0.4, 0.6]; [0.2.0.3]) \\ + 14 ([1, 2, 4, 6]; [0.1, 0.2]; [0.3.0.6]) \\ + 6 ([1, 2, 3, 4]; [0.1, 0.2]; [0.4.0.7]) \\ + 19 ([3, 4, 5, 8]; [0.2, 0.4]; [0.2.0.5]) \\ = & {([139, 219, 293, 426]; [0.4, 0.6]; [0.2, 0.3])} . \end{array}

To check the optimality of Example 2,

Step 1: All the dual variables $u_{i}^{I V}$ and $v_{j}^{I V}$ using $u_{i}^{I V} + v_{j}^{I V} = c_{i j}^{I V}$ are listed as follows:

\begin{array}{l} u_{2}^{I V} = ([- 6, - 3, - 1, 3]; [0.1, 0.3]; [0.4, 0.6]), \\ u_{3}^{I V} = ([- 7, - 4, - 2, 1]; [0.1, 0.2]; [0.4, 0.7]), \\ v_{1}^{I V} = ([3, 5, 6, 8]; [0.1, 0.3]; [0.4, 0.6]), \\ v_{2}^{I V} = ([2, 6, 9, 15]; [0.1, 0.2]; [0.4, 0.7]), \\ v_{3}^{I V} = ([- 2, 3, 7, 12]; [0.1, 0.2]; [0.4, 0.6]) . \end{array}

Step 2: All penalties $P_{i j}^{I V}$ using $P_{i j}^{I V} = u_{i}^{I V} + v_{j}^{I V} - c_{i j}^{I V}$ for all non-basic variables are as follows:

\begin{array}{l} P_{12}^{I V} = ([- 3, 2, 6, 13]; [0.1, 0.2]; [0.4, 0.7]), \\ P_{13}^{I V} = ([- 9, - 2, 3, 10]; [0.1, 0.2]; [0.4, 0.6]), \\ P_{22}^{I V} = ([- 12, - 2, 5, 17]; [0.1, 0.2]; [0.4, 0.7]), \\ P_{33}^{I V} = ([- 16, - 7, 1, 10]; [0.1, 0.2]; [0.4, 0.7]) . \end{array}

All $P_{i j}^{I V} < 0$ . Optimality is reached.

The optimum transportation cost is = {([139, 219, 293, 426]; [0.4, 0.6]; [0.2, 0.3])}.

5.2 Result and Discussion of Example 2

The membership and non-membership functions for the obtained result ([139, 219, 293, 426]; [0.4, 0.6]; [0.2, 0.3]) are

\begin{array}{l} μ^{L R} (x) = {\begin{array}{l} \frac{x - 139}{80} (0.4), & 139 \leq x < 219, \\ 0.4, & 219 \leq x \leq 293, \\ \frac{426 - x}{133} (0.4), & 293 < x \leq 426, \\ 0, & otherwise, \end{array} \\ μ^{U R} (x) = {\begin{array}{l} \frac{x - 139}{80} (0.6), & 139 \leq x < 219, \\ 0.6, & 219 \leq x \leq 293, \\ \frac{426 - x}{133} (0.6), & 293 < x \leq 426, \\ 0, & otherwise, \end{array} \\ ϑ^{L R} (x) = {\begin{array}{l} \frac{219 - x + 0.2 (x - 139)}{80}, & 139 \leq x < 219, \\ 0.2, & 219 \leq x \leq 293, \\ \frac{x - 293 + 0.2 (426 - x)}{133}, & 293 < x \leq 426, \\ 0, & otherwise, \end{array} \\ ϑ^{U R} (x) = {\begin{array}{l} \frac{219 - x + 0.3 (x - 139)}{80}, & 139 \leq x < 219, \\ 0.3, & 219 \leq x \leq 293, \\ \frac{x - 293 + 0.3 (426 - x)}{133}, & 293 < x \leq 426, \\ 0, & otherwise . \end{array} \end{array}

According to the optimizer, the minimum transportation cost lies between 139 units and 426 units.
The overall satisfaction level of the optimizer lies between 219 and 293 units within the interval [0.4, 0.6] and the rejection level lies within the interval [0.2, 0.3].

6. Result and Discussions

6.1 Advantages of the Proposed Method

In this paper, as the special case IVTrIFTP is not converted into crisp problem and the IBFS, optimum solution of the given TP are derived as IVTrIFNs.

7. Conclusion

Abstract
1. Introduction
2. Preliminaries
3. Ordering Methods of IVTrIFNs
4. Interval-Valued Trapezoidal Intuitionistic Fuzzy Tansportation Problem
5. Numerical Examples
6. Result and Discussions
7. Conclusion

Fig 1.

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