# Selection of Coal Transportation Company Based on Fuzzy SWARA-COPRAS Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Trapezoidal Fuzzy SWARA-COPRAS Approach

#### 2.1. Fuzzy Set Theory

**Definition**

**1**

**(**[24]

**)**. $X$is a non-empty set. Given a mapping${\mu}_{A}:X\to \left[0,1\right]$,$x\mapsto {\mu}_{A}\left(x\right)$, a fuzzy subset of$X$is determined.${\mu}_{A}$is called the membership function of$A$or the membership degree of$x$to${\mu}_{A}\left(x\right)$.

**Definition**

**2.**

#### 2.2. Trapezoidal Fuzzy SWARA Approach

**Step 1:**Sort the evaluation indicators. According to the corresponding trapezoidal fuzzy number, each decision maker gives the relative importance of each index. Then, the trapezoidal fuzzy number of each index is obtained by using the weight sum Equation (2) of the decision maker, and the defuzzification value of each index is obtained by using Equation (1). According to the defuzzification value of each index, sort from large to small.

**Step 2:**Determine the relative importance-related parameters of two adjacent indicators ${s}_{j}\left(j\ge 2\right)$. From the second index to the last index, the association parameter is determined according to certain rules ${s}_{j}\left(j\ge 2\right)$. In this paper, the difference between the ambiguity resolution values of two adjacent indexes is taken as the correlation parameter.

**Step 3:**According to Equation (7), the comparison coefficient ${k}_{j}$ is calculated.

**Step 4:**According to Equation (8), the relative weight ${q}_{j}$ is calculated.

**Step 5:**According to Equation (9), the final weight ${\lambda}_{j}$ is calculated.

#### 2.3. Trapezoidal Fuzzy COPRAS Approach

**Step 1:**Establish trapezoidal fuzzy decision matrix $R$ according to Equation (10)

**Step 2:**According to Equation (11), obtain the weighted trapezoidal fuzzy decision matrix $Y$:

**Step 3:**Calculate the sum of the benefit index and the cost index according to Equations (12) and (13). The number of indicators is $n$. Let ${{\rm T}}_{1}=\left\{1,2,\cdots ,e\right\}$ represents a collection of benefit indicators, ${{\rm T}}_{2}=\left\{e+1,e+2,\cdots ,n\right\}$ represents a collection of cost indicators; therefore,

**Step 4:**According to Equation (14), calculate the relative importance value ${Q}_{i}\left(i=1,2,\cdots ,m\right)$ of each alternative:

## 3. Method Framework

- (1)
- Implement and understand.
- (2)
- Low transaction costs.
- (3)
- Decision makers have more opportunities to set criteria priorities.

**Stage 1:**Determine decision makers and corresponding weights and select coal transportation companies and evaluation indicators.

**Stage 2:**Use the trapezoidal fuzzy SWARA method to determine the weight of the evaluation index.

**Stage 3:**Using the trapezoidal fuzzy COPRAS method, determine the rank of the alternative coal transportation companies and select the best coal transportation company.

## 4. Case Analysis

_{1}indicates the qualification of the coal transportation company, C

_{2}indicates the quality of the coal transportation company, C

_{3}indicates the service level of the coal transportation company, and C

_{4}indicates the cost of the coal transportation company. Among them, C

_{1}, C

_{2}, and C

_{3}are benefit indicators. The larger the indicator value is, the better; C

_{4}is the cost indicator, and the smaller the indicator value is, the better. A management questionnaire on the types of benefits and costs is established and sent to this independently three experienced decision makers, in order not to be disturbed by other experts when scoring, to ensure the scientific and independent management of the questionnaire. According to their own experience, judgment, and relevant professional knowledge, each decision maker makes opinions on the selection of coal transportation companies. The management questionnaire is designed based on the language variables in Table 1 and Table 2.

**The first stage:**Determine the set of decision makers and the corresponding weights and select the set of coal transportation companies and the set of evaluation indicators.

**The second stage**: Determine the weight of the evaluation index.

**The third stage**: Determine the priority of coal transportation companies.

## 5. Sensitivity Analysis

_{1}and C

_{2}be replaced by S

_{1}:C(1,2), and other indicators remain unchanged. There are six scenarios in total, namely S

_{1}: C(1,2), S

_{2}: C(1,3), S

_{3}: C(1,4), S

_{4}: C(2,3), S

_{5}: C(2,4), S

_{6}: C(3,4). In every scenario, the relative importance value ${Q}_{i}$ and utility value ${{\rm N}}_{i}$ of each coal transportation company is calculated. The relative importance values of the six scenarios ${Q}_{i}$ are shown in Table 11 and Figure 2, and the utility degree values ${N}_{i}$ are shown in Table 12 and Figure 3. In the six scenarios, the coal transportation company A1 ranks first in all the scenarios, and the rankings are ${A}_{3}>{A}_{1}>{A}_{4}>{A}_{2}>{A}_{5}$, which are consistent with the results calculated in this paper. As a result, it can be concluded that the trapezoidal fuzzy set is a reliable choice for the selection of coal transportation companies. These tools have good operability and reference value in actual work.

## 6. Discussions and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Linguistic Terms | Trapezoidal Fuzzy Number |
---|---|

Extremely important (EI) | (0.8, 0.9,1.0, 1.0) |

Very important (VI) | (0.7, 0.8, 0.9, 1.0) |

Important (I) | (0.6, 0.7, 0.8, 0.9) |

Middle(M) | (0.4, 0.5, 0.6, 0.7) |

Unimportant (U) | (0.2, 0.3, 0.4, 0.5) |

Very unimportant (VU) | (0.1, 0.2, 0.3, 0.4) |

Extremely unimportant (EU) | (0.0, 0.1, 0.2, 0.3) |

Linguistic Terms | Trapezoidal Fuzzy Number |
---|---|

Extremely Good (EG)/Extremely High (EH) | (0.8, 0.9, 1.0, 1.0) |

Very Very Good (VVG)/Very Very High (VVH) | (0.7, 0.8, 0.9, 1.0) |

Very good (VG)/Very High (VH) | (0.6, 0.7, 0.8, 0.9) |

Good (G)/High (H) | (0.5, 0.6, 0.7, 0.8) |

Medium Good (MG)/Medium-High (MH) | (0.4, 0.5, 0.6, 0.7) |

Fair (F)/Medium (M) | (0.3, 0.4, 0.5, 0.6) |

Medium Bad (MB)/Medium Low (ML) | (0.2, 0.3, 0.4, 0.5) |

Bad (B)/Low (L) | (0.1, 0.2, 0.3, 0.4) |

Very Bad (VB)/Very Low (VL) | (0.0, 0.1, 0.2, 0.3) |

Very Very Bad (VVB)/Very Very Low (VVL) | (0.0, 0.0, 0.1, 0.2) |

No. | Working Years | Education Level | Weights |
---|---|---|---|

Expert 1 | 11 | PhD | 0.3684 |

Expert 2 | 20 | MSc | 0.2961 |

Expert 3 | 25 | MSc | 0.3355 |

Index | E_{1} | E_{2} | E_{3} | Aggregated Fuzzy number | $\mathbf{Crisp}\mathbf{Values}\mathit{P}\left({\mathit{C}}_{\mathit{j}}\right)$ |
---|---|---|---|---|---|

C_{1} | M | S | M | (0.4592, 0.5592, 0.6592, 0.7592) | 0.6092 |

C_{2} | VS | S | VS | (0.6704, 0.7704, 0.8704, 0.9704) | 0.8204 |

C_{3} | S | M | S | (0.5408, 0.6408, 0.7408, 0.8408) | 0.6908 |

C_{4} | ES | VS | VS | (0.7368, 0.8368, 0.9368, 1.0000) | 0.8807 |

Index | ${\mathit{s}}_{\mathit{j}}$ | ${\mathit{k}}_{\mathit{j}}$ | ${\mathit{q}}_{\mathit{j}}$ | ${\mathit{w}}_{\mathit{j}}$ | |
---|---|---|---|---|---|

C_{4} | 0.8807 | - | 1 | 1 | 0.2817 |

C_{2} | 0.8204 | 0.0603 | 1.0603 | 0.9431 | 0.2657 |

C_{3} | 0.6908 | 0.1296 | 1.1296 | 0.8349 | 0.2352 |

C_{1} | 0.6092 | 0.0816 | 1.0816 | 0.7719 | 0.2174 |

Company | A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Experts | E1 | E2 | E3 | E1 | E2 | E3 | E1 | E2 | E3 | E1 | E2 | E3 | E1 | E2 | E3 |

C_{1} | G | VG | G | MG | G | MG | VG | VVG | VG | MG | G | G | F | MG | MG |

C_{2} | MG | G | MG | F | MG | G | VG | G | VG | MG | F | MG | MG | MG | F |

C_{3} | VG | G | VG | G | G | MG | VG | VG | G | VG | G | G | VG | G | MG |

C_{4} | L | ML | M | ML | M | ML | MH | M | M | L | ML | ML | M | MH | MH |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | |
---|---|---|---|---|---|

C_{1} | (0.5296, 0.6296, 0.7296, 0.8296) | (0.4296, 0.5296, 0.6296, 0.7296) | (0.6296, 0.7296, 0.8296, 0.9296) | (0.4632, 0.5632, 0.6632, 0.7632) | (0.3632, 0.4632, 0.5632, 0.6632) |

C_{2} | (0.4296, 0.5296, 0.6296, 0.7296) | (0.3967, 0.4967, 0.5967, 0.6967) | (0.5704, 0.6704, 0.7704, 0.8704) | (0.3704, 0.4704, 0.5704, 0.6704) | (0.3665, 0.4665, 0.5664, 0.6664) |

C_{3} | (0.5704, 0.6704, 0.7704, 0.8704) | (0.4665, 0.5664, 0.6664, 0.7665) | (0.5664, 0.6664, 0.7665, 0.8664) | (0.5704, 0.6704, 0.7704, 0.8704) | (0.5033, 0.6033, 0.7033, 0.8033) |

C_{4} | (0.1967, 0.2967, 0.3967, 0.4967) | (0.2296, 0.3296, 0.4296, 0.5296) | (0.3368,0.4368, 0.5368, 0.6368) | (0.1632, 0.2632, 0.3632, 0.4632) | (0.3632, 0.4632, 0.5632, 0.6632) |

A1 | A2 | A3 | A4 | A5 | |
---|---|---|---|---|---|

C_{1} | (0.1151, 0.1369, 0.1586, 0.1804) | (0.0934, 0.1151, 0.1369, 0.1586) | (0.1369, 0.1586, 0.1804 0.2021) | (0.1007, 0.224, 0.1442, 0.1659) | (0.0790, 0.1007, 0.1224, 0.1442) |

C_{2} | (0.1141, 0.1407, 0.1673, 0.1939) | (0.1054, 0.1320, 0.1585, 0.1851) | (0.1516, 0.1781, 0.2047, 0.2313) | (0.0984, 0.1250, 0.1516, 0.1781) | (0.0974, 0.1239, 0.1505, 0.1771) |

C_{3} | (0.1342, 0.1577, 0.1812, 0.2047) | (0.1097, 0.1332, 0.1567, 0.1803) | (0.1332, 0.1567, 0.1803, 0.2038) | (0.1342, 0.1577, 0.1812, 0.2047) | (0.1184, 0.1419, 0.1654 0.1889) |

C_{4} | (0.0554, 0.0836, 0.1118, 0.1399) | (0.0647, 0.0929, 0.1210, 0.1492) | (0.0949,0.1231, 0.1512, 0.1494) | (0.0460, 0.0741, 0.1023, 0.1305) | (0.1023, 0.1305, 0.1586, 0.1868) |

Company | ${\mathit{\beta}}_{+\mathit{i}}$ | $\mathit{P}\left({\mathit{\beta}}_{+\mathit{i}}\right)$ | ${\mathit{\beta}}_{-\mathit{i}}$ | $\mathit{P}\left({\mathit{\beta}}_{-\mathit{i}}\right)$ | ${\mathit{Q}}_{\mathit{i}}$ | ${\mathit{{\rm N}}}_{\mathit{i}}\left(\mathit{\%}\right)$ | Rank |
---|---|---|---|---|---|---|---|

A_{1} | (0.3634, 0.4353, 0.5071, 0.5789) | 0.4712 | (0.0554, 0.0836, 0.1118, 0.1399) | 0.0977 | 0.6015 | 96.67 | 2 |

A_{2} | (0.3085, 0.3803, 0.4522, 0.5240) | 0.4163 | (0.0647, 0.0929, 0.1210, 0.1492) | 0.1069 | 0.5353 | 86.03 | 4 |

A_{3} | (0.4217, 0.4935, 0.5653, 0.6371) | 0.5294 | (0.0949,0.1231, 0.1512, 0.1494) | 0.1371 | 0.6222 | 100.00 | 1 |

A_{4} | (0.3333, 0.4051, 0.4769, 0.5487) | 0.4410 | (0.0460, 0.0741, 0.1023, 0.1305) | 0.0882 | 0.5853 | 94.07 | 3 |

A_{5} | (0.3095, 0.3814, 0.4532, 0.5250) | 0.4173 | (0.1023, 0.1305, 0.1586, 0.1868) | 0.1446 | 0.5053 | 81.22 | 5 |

Proposed Method | TOPSIS | CoCoSo | MOORA | MABAC | |
---|---|---|---|---|---|

A_{1} | 2 | 2 | 2 | 3 | 2 |

A_{2} | 4 | 4 | 4 | 4 | 4 |

A_{3} | 1 | 1 | 1 | 1 | 1 |

A_{4} | 3 | 3 | 3 | 2 | 3 |

A_{5} | 5 | 5 | 5 | 5 | 5 |

Scenarios | A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | Sort | |
---|---|---|---|---|---|---|---|

S_{1} | C(1,2) | 0.6063 | 0.5369 | 0.6251 | 0.5898 | 0.5052 | A3 > A1 > A4 > A2 > A5 |

S_{2} | C(1,3) | 0.6008 | 0.5346 | 0.6233 | 0.5834 | 0.5017 | A3 > A1 > A4 > A2 > A5 |

S_{3} | C(1,4) | 0.6155 | 0.5454 | 0.6512 | 0.5918 | 0.5182 | A3 > A1 > A4 > A2 > A5 |

S_{4} | C(2,3) | 0.6058 | 0.5374 | 0.6221 | 0.5914 | 0.5114 | A3 > A1 > A4 > A2 > A5 |

S_{5} | C(2,4) | 0.6034 | 0.5373 | 0.6285 | 0.5854 | 0.5086 | A3 > A1 > A4 > A2 > A5 |

S_{6} | C(3,4) | 0.6135 | 0.5443 | 0.6402 | 0.5950 | 0.5241 | A3 > A1 > A4 > A2 > A5 |

Scenarios | A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | Sort | |
---|---|---|---|---|---|---|---|

S_{1} | C(1,2) | 97.00 | 85.89 | 100 | 94.35 | 80.82 | A3 > A1 > A4 > A2 > A5 |

S_{2} | C(1,3) | 96.38 | 85.77 | 100 | 93.59 | 80.49 | A3 > A1 > A4 > A2 > A5 |

S_{3} | C(1,4) | 94.52 | 83.76 | 100 | 90.88 | 79.59 | A3 > A1 > A4 > A2 > A5 |

S_{4} | C(2,3) | 97.38 | 86.39 | 100 | 95.06 | 82.21 | A3 > A1 > A4 > A2 > A5 |

S_{5} | C(2,4) | 96.01 | 85.49 | 100 | 93.15 | 80.92 | A3 > A1 > A4 > A2 > A5 |

S_{6} | C(3,4) | 95.83 | 85.02 | 100 | 92.93 | 81.87 | A3 > A1 > A4 > A2 > A5 |

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**MDPI and ACS Style**

Xiang, Z.; Naseem, M.H.; Yang, J.
Selection of Coal Transportation Company Based on Fuzzy SWARA-COPRAS Approach. *Logistics* **2022**, *6*, 7.
https://doi.org/10.3390/logistics6010007

**AMA Style**

Xiang Z, Naseem MH, Yang J.
Selection of Coal Transportation Company Based on Fuzzy SWARA-COPRAS Approach. *Logistics*. 2022; 6(1):7.
https://doi.org/10.3390/logistics6010007

**Chicago/Turabian Style**

Xiang, Ziquan, Muhammad Hamza Naseem, and Jiaqi Yang.
2022. "Selection of Coal Transportation Company Based on Fuzzy SWARA-COPRAS Approach" *Logistics* 6, no. 1: 7.
https://doi.org/10.3390/logistics6010007