# Close Interval Approximation of Pentagonal Fuzzy Numbers for Interval Data-Based Transportation Problems

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## Abstract

**:**

## 1. Introduction

- The Interval data-based transportation problem (IDTP) is investigated;
- Two new methods are proposed based on the following two key ideas:
- Proposing a unique pentagonal fuzzification method (PFM) for fuzzifying interval data;
- Proposing two new ranking methods (Method-1 and Method-2) to convert fuzzy numbers of FTPs into crisp values.

## 2. Materials and Methods

#### 2.1. Preliminaries

**Definition**

**1.**

- ${\mathsf{\mu}}_{\tilde{\mathrm{A}}}\left(\mathrm{x}\right):\mathbb{R}\to \left[0,1\right]$, have the following properties:
- ${\mathsf{\mu}}_{\tilde{\mathrm{A}}}\left(\mathrm{x}\right)$ is an upper semi-continuous membership function;
- $\tilde{\mathrm{A}}$ is a convex fuzzy set, i.e., ${\mathsf{\mu}}_{\tilde{\mathrm{A}}}(\mathsf{\delta}\mathrm{is}(1-\mathsf{\delta})\mathrm{y})\ge \mathrm{min}\{{\mathsf{\mu}}_{\tilde{\mathrm{A}}}(\mathrm{x}),{\mathsf{\mu}}_{\tilde{\mathrm{A}}}(\mathrm{y})\}$ for all $\mathrm{x},\mathrm{y}\in \mathbb{R};0\le \mathsf{\delta}\le 1;$
- $\tilde{\mathrm{A}}$ is normal, i.e., $\exists {\mathrm{x}}_{0}\in \mathbb{R}$ for which ${\mathsf{\mu}}_{\tilde{\mathrm{A}}}({\mathrm{x}}_{0})=1$.

**Definition**

**2.**

#### 2.2. Pentagonal Fuzzy Number (PFN)

- µ
_{k*}(X) is a continuous function in the interval [0,1]; - µ
_{k*}(X) is a strictly increasing and continuous function on [p,q] and [q,r]; - µ
_{k*}(X) is a strictly decreasing and continuous function on [r,s] and [s,t].

#### 2.3. Properties of Pentagonal Fuzzy Numbers

_{1}, p

_{2}, p

_{3}, p

_{4}, p

_{5}} and S = {s

_{1}, s

_{2}, s

_{3}, s

_{4}, s

_{5}} be two PFNs. Then, the fuzzy numbers addition, fuzzy numbers subtraction and fuzzy numbers multiplication are given by:

- Addition: P + S = {p
_{1}+ s_{1}, p_{2}+ s_{2}, p_{3}+ s_{3}, p_{4}+ s_{4}, p_{5}+ s_{5}}; - Subtraction: P − S = {p
_{1}− s_{1}, p_{2}− s_{2}, p_{3}− s_{3}, p_{4}− s_{4}, p_{5}− s_{5}}; - Multiplication: P $\times \mathrm{S}=\{\frac{{\mathrm{p}}_{1}}{5}\mathsf{\beta},\frac{{\mathrm{p}}_{2}}{5}\mathsf{\beta},\frac{{\mathrm{p}}_{3}}{5}\mathsf{\beta},\frac{{\mathrm{p}}_{4}}{5}\mathsf{\beta},\frac{{\mathrm{p}}_{5}}{5}\mathsf{\beta}$}, where β = {p
_{1}+ p_{2}+ p_{3}+ p_{4}+ p_{5}}; - Division: P $\xf7$ Q = $\{\frac{5{\mathrm{p}}_{1}}{\mathsf{\beta}},\frac{5{\mathrm{p}}_{2}}{\mathsf{\beta}},\frac{5{\mathrm{p}}_{3}}{\mathsf{\beta}},\frac{5{\mathrm{p}}_{4}}{\mathsf{\beta}},\frac{5{\mathrm{p}}_{5}}{\mathsf{\beta}}\}$, where β = {p
_{1}+ p_{2}+ p_{3}+ p_{4}+ p_{5}} and β $\ne $ 0; - Scalar multiplication: αP = {αp
_{1}+ αp_{2}+ αp + αp_{4}+ αp_{5}} where α $\u03f5$ R.

#### 2.4. The Formulation of the IDTP

#### 2.5. Fuzzy Transportation Problems (FTP)

**Definition 3.**

## 3. The Proposed Pentagonal Fuzzification Methods (PFM) for Solving the IDTP

_{1}, p

_{2}, p

_{3}, p

_{4}, p

_{5}} and S = {s

_{1}, s

_{2}, s

_{3}, s

_{4}, s

_{5}}, then:

- P $\le \mathrm{S}$ ⇔ R(P) $\le $ R(S);
- P $\ge \mathrm{S}$ ⇔ R(P) $\ge $ R(S);
- P $=\mathrm{S}$ ⇔ R(P) $=$ R(S).

#### 3.1. Proposed Method-1

- Step 1: Transforming the specified problem in tabular form;
- Step 2: Fuzzifying data using the proposed PFM in Section 3;
- Step 3: Finding the crisp value of the FTP (model 2) using the proposed ranking Method-1;
- Step 4: Formulating a linear programming problem and checking for the balance;
- Step 5: Calculating the initial feasible solution using VAM;
- Step 6: Finally, applying the MODI and obtaining the optimal solution to the problem.

#### 3.2. Proposed Ranking Method-2

_{1}, L

_{2}, and L

_{3}, respectively. Then,

- L
_{1}$=\frac{\mathrm{p}+\mathrm{q}+\mathrm{r}}{3}$, $\frac{\mathrm{w}}{6}$; - L
_{2}$=\frac{\mathrm{r}+\mathrm{s}+\mathrm{t}}{3}$, $\frac{\mathrm{w}}{6}$; - L
_{3}$=\frac{\mathrm{q}+2\mathrm{r}+\mathrm{s}}{4}$, $\frac{\mathrm{w}}{2}$.

_{1}L

_{2}L

_{3}is a triangle, the in-center T of this triangle with vertices L

_{1}, L

_{2}, and L

_{3}can be defined as Centre, T = (x, y), as shown in Figure 7, and formulated in the following.

#### 3.3. Proposed Method-2

- Step 1: Transforming a specified problem into tabular form;
- Step 2: Fuzzifying interval data using the proposed PFM in Section 3;
- Step 3: Finding the crisp value of the FTP (model 2) using the proposed ranking Method-2;
- Step 4: Formulating a linear programming problem and checking for the balance;
- Step 5: Calculating the initial feasible solution using VAM;
- Step 6: Finally, applying the MODI and obtaining the optimal solution to the problem.

## 4. Results and Discussion

#### 4.1. Using the Proposed Method-1

#### 4.2. Using the Proposed Method-2

#### 4.3. Comparative Study

#### 4.4. Statistical Analysis

**H0:**

**H1:**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\tilde{a}}_{i}$ | Fuzzy supply value at i supply node |

${\tilde{b}}_{j}$ | Fuzzy demand per unit time at j demand node |

${\tilde{c}}_{ij}$ | Unit fuzzy transportation cost from i supply node to j demand node |

${x}_{i,j}$ | Number of units transported from i supply node to j demand node |

n | Total number of supply nodes (suppliers) |

m | Total number of demand nodes (buyers) |

$\tilde{\mathrm{A}}$ | convex fuzzy set |

${\mathsf{\mu}}_{\tilde{\mathrm{A}}}\left(\mathrm{x}\right)$ | An upper semi-continuous membership function |

k* | Linear symmetric pentagonal fuzzy number |

µ_{k*}(x) | Pentagonal fuzzy membership function |

$\widehat{S}$ | Interval supply quantity (in units) from _{i} supplier |

$\widehat{d}$ | Interval demand (in units) per unit time to _{j} buyer |

$\underset{\_}{S}$ | Lower bound of supplies from the _{I} supplier |

$\overline{S}$ | Upper bound of supplies from the _{i} supplier |

$\underset{\_}{D}$ | Lower bound of demands of the _{j} buyer |

$\overline{D}$ | Upper bound of demands of the _{j} buyer |

$\widehat{c}$ | Interval unit transportation cost from i supply node to jdemand node |

## Appendix A

R1 | R2 | R3 | Supply | |
---|---|---|---|---|

A | [1,31] | [1,9] | [4,11] | [1,15] |

B | [3,33] | [1,31] | [5,23] | [2,12] |

C | [2,12] | [3,11] | [1,17] | [1,33] |

Demand | [4,22] | [1,9] | [5,23] |

R1 | R2 | R3 | Supply | |
---|---|---|---|---|

A | [4,26] | [4,19] | [1,37] | [5,17] |

B | [3,35] | [2,6] | [1,13] | [4,24] |

C | [2,18] | [4,26] | [2,14] | [2,18] |

Demand | [1,12] | [5,17] | [3,15] |

R1 | R2 | R3 | Supply | |
---|---|---|---|---|

A | [7,22] | [1,33] | [1,31] | [1,9] |

B | [2,34] | [5,23] | [4,19] | [2,29] |

C | [4,11] | [8,23] | [2,34] | [1,18] |

Demand | [1,17] | [5,35] | [1,33] |

R1 | R2 | R3 | Supply | |
---|---|---|---|---|

A | [1,11] | [2,17] | [2,29] | [8,23] |

B | [3,35] | [5,25] | [6,32] | [1,21] |

C | [1,29] | [2,27] | [3,33] | [2,22] |

Demand | [3,31] | [2,29] | [2,24] |

R1 | R2 | R3 | Supply | |
---|---|---|---|---|

A | [1,27] | [7,28] | [4,32] | [1,33] |

B | [3,33] | [1,33] | [1,31] | [2,18] |

C | [7,31] | [1,15] | [1,19] | [2,14] |

Demand | [1,19] | [4,32] | [5,17] |

R1 | R2 | R3 | Supply | |
---|---|---|---|---|

A | [2,8] | [3,15] | [1,13] | [2,13] |

B | [3,11] | [2,10] | [2,13] | [1,19] |

C | [2,13] | [1,9] | [4,8] | [3,10] |

Demand | [1,21] | [2,10] | [3,12] |

R1 | R2 | R3 | Supply | |
---|---|---|---|---|

A | [3,31] | [1,29] | [5,13] | [6,24] |

B | [5,20] | [5,25] | [1,31] | [1,35] |

C | [6,23] | [4,27] | [2,28] | [7,25] |

Demand | [3,28] | [1,33] | [4,32] |

R1 | R2 | R3 | R4 | Supply | |
---|---|---|---|---|---|

A | [4,26] | [1,31] | [5,18] | [4,20] | [1,33] |

B | [2,24] | [5,20] | [2,27] | [1,31] | [4,30] |

C | [7,31] | [4,28] | [2,26] | [4,25] | [4,32] |

Demand | [1,31] | [4,29] | [4,20] | [2,28] |

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R1 | R2 | R3 | Supply | |
---|---|---|---|---|

Step 1: conversion of specified problem in tabular form | ||||

A | [1,19] | [1,9] | [2,18] | [1,9] |

B | [8,26] | [3,12] | [7,28] | [4,10] |

C | [11,27] | [0,15] | [4,11] | [4,11] |

Demand | [3,12] | [4,10] | [2,8] | |

Step 2: Fuzzified interval data | ||||

A | (1,5.5,10,14.5,19) | (1,3,5,7,9) | (2,6,10,14,18) | (1,3,5,7,9) |

B | (8,10,12,14,16) | (3,5.25,7.5,9.75,12) | (7,12.25,17.5,22.75,28) | (4,5.5,7,8.5,10) |

C | (11,15,19,23,27) | (0,3.75,7.5,11.25,15) | 4,5.75,7.5,9.25,11) | (4,5.75,7.5,9.25,11) |

Demand | (3,5.25,7.5,9.75,12) | (4,5.5,7,8.5,10) | (2,3.5,5,6.5,8) | |

Step 3: Defuzzified data | ||||

A | 9.25 | 4.67 | 9.33 | 4.67 |

B | 11.67 | 7.125 | 16.62 | 6.75 |

C | 18.33 | 6.875 | 7.21 | 7.21 |

Demand | 7.125 | 6.92 | 4.75 | |

Step 4: Balanced transportation problem | ||||

A | 9.25 | 4.67 | 9.33 | 4.67 |

B | 11.67 | 7.13 | 16.62 | 6.75 |

C | 18.33 | 6.88 | 7.21 | 7.21 |

D | 0 | 0 | 0 | 0.17 |

Demand | 7.13 | 6.92 | 4.75 | |

Step 5: Initial basic feasible solution for the problem using VAM | ||||

A | 9.25 | 4.67 4.67 | 9.33 | 4.67 |

B | 11.67 6.75 | 7.13 | 16.62 | 6.75 |

C | 18.33 0.21 | 6.88 2.25 | 7.21 4.75 | 7.21 |

D | 0 0.17 | 0 | 0 | 0.17 |

Demand | 7.13 | 6.92 | 4.75 |

H1 | H2 | H3 | H4 | Supply | |
---|---|---|---|---|---|

Step 1: Tabular form—example 2 with interval-based supplies and demands | |||||

L1 | [1,4] | [1,6] | [4,12] | [5,11] | [1,12] |

L2 | [0,4] | [1,4] | [5,8] | [0,3] | [0,3] |

L3 | [3,8] | [5,12] | [12,19] | [7,12] | [5,15.6] |

Demand | [5,10] | [1,10] | [1,6] | [1,4] | |

Step 2: Fuzzified interval data | |||||

L1 | (1,1.75,2.5, 3.25,4) | (1,2.25,3.5, 4.75,6) | (4,6,8, 10,12) | (5,6.5,8, 9.5,11) | (1,3.75,6.5, 9.25,12) |

L2 | (0, 1, 2, 3, 4) | (1,1.75,2.5, 3.25,4) | (5,5.75,6.5, 7.25,8) | (0,0.75,1.5, 2.25,3) | (0,0.75,1.5, 2.25,3) |

L3 | (3,4.25,5.5, 6.75,8) | (5,6.75,8.5, 10.25,12) | (12,13.75,15.5, 17.25,19) | (7,8.25,9.5, 10.75,12) | (5,7.65,10.3, 12.95,15.6) |

Demand | (5,6.25,7.5, 8.75,10) | (1,3.25,5.5, 7.75,10) | (1,2.25,3.5, 4.75,6) | (1,1.75,2.5, 3.25,4) | |

Step 3: Defuzzified data | |||||

A | 2.38 | 3.29 | 7.67 | 7.75 | 6.04 |

B | 1.83 | 2.38 | 6.38 | 1.38 | 1.38 |

C | 5.29 | 8.21 | 15.21 | 9.29 | 9.86 |

Demand | 7.29 | 5.13 | 3.29 | 2.38 | |

Step 4: Balanced transportation problem | |||||

A | 2.38 | 3.29 | 7.67 | 7.75 | 6.04 |

B | 1.83 | 2.38 | 6.38 | 1.38 | 1.38 |

C | 5.29 | 8.21 | 15.21 | 9.29 | 9.86 |

D | 0 | 0 | 0 | 0 | 0.81 |

Demand | 7.29 | 5.13 | 3.29 | 2.38 | |

Step 5: Initial basic feasible solution for the problem using VAM | |||||

A | 2.38 | 3.29 3.56 | 7.67 2.48 | 7.75 | 6.04 |

B | 1.83 | 2.38 | 6.38 | 1.38 1.38 | 1.38 |

C | 5.29 7.29 | 8.21 1.57 | 15.21 | 9.29 1 | 9.86 |

D | 0 | 0 | 0 0.81 | 0 | 0.81 |

Demand | 7.29 | 5.13 | 3.29 | 2.38 |

R1 | R2 | R3 | Supply | |
---|---|---|---|---|

Step 1: conversion of specified problem in tabular form | ||||

A | [1,19] | [1,9] | [2,18] | [1,9] |

B | [8,26] | [3,12] | [7,28] | [4,10] |

C | [11,27] | [0,15] | [4,11] | [4,11] |

Demand | [3,12] | [4,10] | [2,8] | |

Step 5: Initial basic feasible solution for the problem using VAM | ||||

A | 20 | 5 5 | 10 | 5 |

B | 17 7 | 7.5 | 17.5 | 7 |

C | 19 0.5 | 7.5 2 | 7.5 5 | 7.5 |

Demand | 7.5 | 7 | 5 |

Example | Bisht and Srivastava [13] | Proposed Method-2 | Proposed Method-1 |
---|---|---|---|

1 | 154.93 | 206 | 152.25 |

2 | 93.39 | 103.9 | 93.38 |

3 | 272.5 | 272.5 | 237.89 |

4 | 190.5 | 190.5 | 169.61 |

5 | 337 | 337 | 294.71 |

6 | 453.75 | 453.75 | 397.61 |

7 | 483 | 483 | 422.2 |

8 | 140.75 | 140.75 | 125.84 |

9 | 612.75 | 612.75 | 545.51 |

10 | 676.25 | 676.25 | 589.55 |

**Table 5.**Statistical significance of the average performance between the proposed methods and the existing methods.

Comparison | Mean Difference | Standard Error | 95% Confidence Interval for the Mean Difference | p-Value |
---|---|---|---|---|

Bisht and Srivastava [13] Proposed Method-1 | 38.63 | 9.20 | (17.81, 59.44) | 0.002 |

Proposed Method-2 Proposed Method-1 | 44.79 | 7.80 | (27.14, 62.43) | <0.001 |

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## Share and Cite

**MDPI and ACS Style**

Juman, Z.A.M.S.; Mostafa, S.A.; Batuwita, A.P.; AlArjani, A.; Sharif Uddin, M.; Jaber, M.M.; Alam, T.; Attia, E.-A.
Close Interval Approximation of Pentagonal Fuzzy Numbers for Interval Data-Based Transportation Problems. *Sustainability* **2022**, *14*, 7423.
https://doi.org/10.3390/su14127423

**AMA Style**

Juman ZAMS, Mostafa SA, Batuwita AP, AlArjani A, Sharif Uddin M, Jaber MM, Alam T, Attia E-A.
Close Interval Approximation of Pentagonal Fuzzy Numbers for Interval Data-Based Transportation Problems. *Sustainability*. 2022; 14(12):7423.
https://doi.org/10.3390/su14127423

**Chicago/Turabian Style**

Juman, Z. A. M. S., Salama A. Mostafa, A. P. Batuwita, Ali AlArjani, Md Sharif Uddin, Mustafa Musa Jaber, Teg Alam, and El-Awady Attia.
2022. "Close Interval Approximation of Pentagonal Fuzzy Numbers for Interval Data-Based Transportation Problems" *Sustainability* 14, no. 12: 7423.
https://doi.org/10.3390/su14127423