# Fuzzy Demand Vehicle Routing Problem with Soft Time Windows

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## Abstract

**:**

## 1. Introduction

## 2. Problem Description

## 3. Constructing a Fuzzy Chance-Constrained Programming Model

#### Sign Convention

## 4. Designing an Algorithm for Solving the Model

#### 4.1. Random Simulation Operator

#### 4.2. Coding

#### 4.3. Neighborhood Search Algorithm

#### 4.4. Fitness Function

#### 4.5. Selection, Crossover, and Mutation Operators

#### 4.6. Termination Conditions

## 5. Simulation Experiment and Result Analysis

#### 5.1. Description of Instance and Experimental Environment

#### 5.2. Experiment in a Sample Instance

#### 5.3. Comparative Analysis of Algorithms

## 6. Conclusions and Future Works

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Illustration of distribution routes: (

**a**) is a normal distribution route and (

**b**) is a distribution route with replenishment.

Step 1. Randomly generate all customer demand data, which represent the fuzzy demand, as follows: (1) Generate a number γ randomly according to the customer fuzzy demand and calculate its membership degree λ. (2) Randomly generate a number ξ in the range [0, 1]. (3) If λ < ξ, then γ is the customer demand; otherwise, repeat the above steps. (4) Repeat steps 1–3 until all customer demands are generated. |

Step 2. Calculate the additional cost under the condition of customer demand. |

Step 3. Repeat steps 1 and 2 N times. |

Step 4. Take the average value of N simulations as the penalty cost. |

No. | x | y | Demand | No. | x | y | Demand |
---|---|---|---|---|---|---|---|

1 | 38 | 46 | ----- | 16 | 36 | 48 | 5 |

2 | 59 | 46 | 16 | 17 | 45 | 36 | 16 |

3 | 96 | 42 | 18 | 18 | 73 | 57 | 7 |

4 | 47 | 61 | 1 | 19 | 10 | 91 | 4 |

5 | 26 | 15 | 13 | 20 | 98 | 51 | 22 |

6 | 66 | 6 | 8 | 21 | 92 | 62 | 7 |

7 | 96 | 23 | 23 | 22 | 43 | 43 | 23 |

8 | 37 | 25 | 7 | 23 | 53 | 25 | 16 |

9 | 68 | 92 | 27 | 24 | 78 | 65 | 2 |

10 | 78 | 84 | 1 | 25 | 72 | 79 | 2 |

11 | 82 | 28 | 3 | 26 | 37 | 88 | 9 |

12 | 93 | 90 | 6 | 27 | 16 | 73 | 2 |

13 | 74 | 42 | 24 | 28 | 75 | 96 | 12 |

14 | 60 | 20 | 19 | 29 | 11 | 66 | 1 |

15 | 78 | 58 | 2 | 30 | 9 | 49 | 9 |

A | Route Cost | Time Cost | Penalty Cost | Total Cost |
---|---|---|---|---|

0.1 | 3165.33 | 89.75 | 255.45 | 4351.15 |

0.2 | 3246.54 | 90.61 | 257.63 | 4339.37 |

0.3 | 3209.71 | 88.98 | 256.92 | 4277.83 |

0.4 | 3193.26 | 89.25 | 258.77 | 4196.46 |

0.5 | 3006.09 | 88.64 | 260.55 | 4138.41 |

0.6 | 3227.15 | 89.79 | 257.39 | 4375.34 |

0.7 | 3421.48 | 90.53 | 260.93 | 4299.71 |

0.8 | 3568.9 | 91.62 | 265.47 | 4239.53 |

0.9 | 3504.11 | 92.88 | 278.36 | 4447.68 |

1 | 3732.27 | 93.58 | 279.49 | 4585.27 |

e.g., | With Soft Time Windows | Without Soft Time Windows | ||||
---|---|---|---|---|---|---|

k | Time Cost | Total Cost | k | Time Cost | Total Cost | |

C101 | 3 | 90.75 | 4277.01 | 3 | 73.35 | 4369.54 |

C102 | 3 | 88.61 | 4283.68 | 3 | 72.64 | 4354.27 |

C103 | 2 | 90.98 | 4159.93 | 2 | 77.58 | 4342.85 |

C104 | 4 | 87.25 | 4319.72 | 2 | 78.52 | 4521.33 |

C105 | 2 | 91.64 | 4124.55 | 3 | 68.63 | 4238.76 |

C106 | 3 | 88.79 | 4305.72 | 3 | 73.22 | 4395.04 |

C107 | 3 | 88.53 | 4211.4 | 2 | 77.83 | 4321.78 |

C108 | 2 | 90.62 | 4199.23 | 2 | 76.42 | 4253.69 |

C109 | 3 | 89.88 | 4189.87 | 3 | 72.98 | 4283.47 |

C201 | 3 | 90.58 | 4423.27 | 3 | 73.57 | 4498.06 |

e.g., | Optimal | Dual Population Genetic Algorithm | Genetic Simulated Annealing Algorithm | SA-GA | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

k | Total Cost | k | Total Cost | Gap (%) | k | Total Cost | Gap (%) | k | Total Cost | Gap (%) | |

C101 | 3 | 4279.16 | 2 | 4380.68 | 2.41 | 3 | 4332.15 | 1.21 | 3 | 4279.16 | 0 |

C102 | 3 | 4146.58 | 3 | 4200.95 | 1.22 | 2 | 4146.58 | 0 | 3 | 4159.93 | 0.23 |

C103 | 2 | 4397.38 | 3 | 4431.01 | 0.78 | 3 | 4421.06 | 0.55 | 2 | 4397.38 | 0 |

C104 | 3 | 4428.58 | 3 | 4428.58 | 0 | 3 | 4521.13 | 2.09 | 2 | 4435.04 | 0.13 |

C105 | 2 | 4138.29 | 3 | 4188.96 | 1.25 | 2 | 4231.57 | 2.25 | 2 | 4138.29 | 0 |

C106 | 3 | 4438.54 | 3 | 4487.92 | 1.13 | 3 | 4438.54 | 0 | 2 | 4455.34 | 0.36 |

C107 | 2 | 4199.23 | 2 | 4276.13 | 1.84 | 3 | 4243.11 | 1.04 | 2 | 4199.23 | 0 |

C108 | 3 | 4189.86 | 3 | 4198.17 | 0.24 | 3 | 4288.55 | 2.37 | 3 | 4189.86 | 0 |

C109 | 3 | 4423.26 | 3 | 4538.06 | 2.61 | 3 | 4526.23 | 2.3 | 3 | 4423.26 | 0 |

C201 | 3 | 4685.27 | 3 | 4761.73 | 1.65 | 3 | 4695.19 | 0.3 | 3 | 4685.27 | 0 |

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**MDPI and ACS Style**

Yang, T.; Wang, W.; Wu, Q.
Fuzzy Demand Vehicle Routing Problem with Soft Time Windows. *Sustainability* **2022**, *14*, 5658.
https://doi.org/10.3390/su14095658

**AMA Style**

Yang T, Wang W, Wu Q.
Fuzzy Demand Vehicle Routing Problem with Soft Time Windows. *Sustainability*. 2022; 14(9):5658.
https://doi.org/10.3390/su14095658

**Chicago/Turabian Style**

Yang, Tao, Weixin Wang, and Qiqi Wu.
2022. "Fuzzy Demand Vehicle Routing Problem with Soft Time Windows" *Sustainability* 14, no. 9: 5658.
https://doi.org/10.3390/su14095658