# A Study of Community Group Purchasing Vehicle Routing Problems Considering Service Time Windows

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Problem Description

- Each vehicle has only one delivery route to complete the delivery from and back to the distribution centre;
- Only one vehicle per head needs to be satisfied in one distribution mission;
- Vehicles must meet capacity constraints;
- Distribution centre are presumed to have prepared sufficient goods in advance.

## 4. Algorithm Introduction

#### 4.1. Ant Colony Algorithms

#### 4.2. Simulated Annealing Algorithms

#### 4.3. Improved Ant Colony Algorithm

- (1)
- SAA generates initial solution;
- (2)
- The better path leaves the pheromone;
- (3)
- The ACA completes a facilitation to produce a solution;
- (4)
- SAA finds a new solution in the neighbourhood of the solution generated by the ACA;
- (5)
- Determine if it is less than 0. If so, the ant colony updates the pheromone, otherwise return to step 2;
- (6)
- Determine if the ACA has reached the maximum number of iterations, if so, output the optimal solution, otherwise return to step 2 and continue the cycle.

## 5. Case Study

#### 5.1. Parameter Setting

#### 5.2. Algorithm Results

#### 5.3. Comparative Analysis

## 6. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Physical Annealing | Optimization Issues |
---|---|

Physical condition | Solution |

Lowest energy state of matter | Optimal solution |

Annealing process | Solving process |

Temperature | Control parameters |

Energy | Objective function |

Isothermal process | Metropolis Sampling Process |

Parameter | Numerical Value |
---|---|

$M$: Number of ants | 50 |

$\Omega $: number of iterations | 100 |

$\alpha $: initial pheromone factor | 1 |

$\beta $: heuristic factor | 3 |

$\rho $: initial pheromone volatility factor | 0.85 |

$Q$: initial pheromone value | 5 |

${T}_{0}$: initial temperature | 100 |

$\Im $: cooling factor | 0.99 |

${P}_{s}$: probability of choosing a swap structure | 0.2 |

${P}_{r}$: probability of choosing the reversal structure | 0.5 |

${P}_{i}$: probability of choosing the insertion structure | 0.3 |

$\upsilon $: maximum number of iterations (annealing algorithm) | 2000 |

$N$: Number of populations | 100 |

${P}_{c}$: Crossover probability | 0.9 |

${P}_{m}$: Probability of variation | 0.05 |

$G$: Generation gap | 0.9 |

No. | Longitude | Latitude | Coordinates x | Coordinates y | Demand | Time Window | Service Time |
---|---|---|---|---|---|---|---|

0 | 116.347465 | 39.823758 | 24.26 | 2.35 | 0 | [09, 20] | 0 |

1 | 116.361222 | 40.036321 | 26.23 | 36.31 | 40 | [10, 12] | 0.1 |

2 | 116.322584 | 39.986339 | 20.70 | 28.33 | 45 | [11, 13] | 0.2 |

3 | 116.310073 | 39.918547 | 18.91 | 17.49 | 50 | [11, 12] | 0.2 |

4 | 116.342725 | 39.903466 | 23.58 | 15.09 | 40 | [12, 14] | 0.1 |

5 | 116.438126 | 39.872297 | 37.25 | 10.11 | 35 | [11, 15] | 0.1 |

6 | 116.372629 | 39.860298 | 27.87 | 8.19 | 40 | [10, 14] | 0.1 |

7 | 116.424175 | 39.930827 | 35.25 | 19.46 | 45 | [10, 12] | 0.2 |

8 | 116.45413 | 39.97696 | 26.83 | 0.33 | 50 | [11, 14] | 0.2 |

9 | 116.238929 | 39.899818 | 8.72 | 14.50 | 80 | [10, 15] | 0.4 |

10 | 116.471384 | 39.871168 | 42.01 | 9.93 | 45 | [10, 16] | 0.2 |

11 | 116.643538 | 39.887352 | 66.67 | 12.51 | 50 | [15, 17] | 0.2 |

12 | 116.518796 | 39.923086 | 48.80 | 18.22 | 45 | [13, 15] | 0.2 |

13 | 116.324892 | 40.079959 | 21.03 | 43.28 | 75 | [15, 17] | 0.4 |

14 | 116.548831 | 40.112143 | 53.10 | 48.42 | 60 | [16, 17] | 0.2 |

15 | 116.340279 | 39.999152 | 23.23 | 30.37 | 80 | [15, 17] | 0.4 |

16 | 116.265916 | 39.924242 | 12.58 | 18.41 | 35 | [14, 17] | 0.1 |

17 | 116.359216 | 40.086673 | 25.95 | 44.35 | 34 | [16, 17] | 0.1 |

18 | 116.482469 | 40.004037 | 43.60 | 31.15 | 58 | [14, 17] | 0.2 |

19 | 116.46251 | 39.889858 | 40.74 | 12.91 | 89 | [13, 17] | 0.5 |

20 | 116.359177 | 39.964364 | 25.94 | 24.81 | 60 | [15, 17] | 0.2 |

21 | 116.609571 | 39.930805 | 61.80 | 19.45 | 70 | [11, 12] | 0.3 |

22 | 116.178067 | 39.929568 | 0.00 | 19.25 | 35 | [10, 12] | 0.1 |

23 | 116.461371 | 39.914768 | 40.57 | 16.89 | 40 | [11, 13] | 0.1 |

24 | 116.397414 | 39.988556 | 31.41 | 28.68 | 65 | [10, 13] | 0.3 |

25 | 116.528565 | 39.946535 | 50.20 | 21.97 | 50 | [10, 14] | 0.2 |

26 | 116.508315 | 39.811369 | 47.30 | 0.37 | 86 | [10, 12] | 0.5 |

27 | 116.258358 | 39.882154 | 11.50 | 11.68 | 65 | [15, 17] | 0.3 |

28 | 116.250371 | 40.226344 | 10.35 | 66.67 | 75 | [14, 17] | 0.4 |

29 | 116.429564 | 39.809034 | 36.02 | 0.00 | 35 | [10, 14] | 0.1 |

30 | 116.498004 | 39.975869 | 45.82 | 26.65 | 45 | [15, 17] | 0.2 |

$\mathit{T}$ | $\overline{\mathit{D}}$ | ${\mathit{Z}}_{\mathbf{min}}$ | $\overline{\mathit{Z}}$ | $\mathit{l}$ | |
---|---|---|---|---|---|

GA | 2926.25 | 1435 | 720.87 | 732.33 | 9 |

SAA | 63.33 | 1237 | 721.40 | 728.77 | 9 |

ACA | 598.67 | 653 | 808.90 | 818.93 | 9 |

SAAACA | 113.82 | 987 | 700.10 | 712.35 | 9 |

Point of Demand | Position Coordinates x | Position Coordinates y | Demand | Time Window | Service Time |
---|---|---|---|---|---|

0 | 40 | 50 | 0 | [09, 20] | 0 |

1 | 62 | 80 | 54 | [12, 16] | 0.2 |

2 | 84 | 25 | 41 | [12, 15] | 0.2 |

3 | 55 | 15 | 27 | [13, 17] | 0.1 |

4 | 17 | 32 | 60 | [13, 15] | 0.2 |

5 | 69 | 91 | 49 | [10, 13] | 0.2 |

6 | 57 | 10 | 57 | [10, 14] | 0.2 |

7 | 49 | 45 | 60 | [11, 15] | 0.2 |

8 | 13 | 15 | 34 | [13, 15] | 0.1 |

9 | 72 | 75 | 37 | [12, 16] | 0.1 |

10 | 58 | 34 | 25 | [11, 14] | 0.1 |

11 | 44 | 85 | 36 | [12, 16] | 0.1 |

12 | 29 | 86 | 25 | [10, 14] | 0.1 |

13 | 74 | 92 | 52 | [14, 17] | 0.2 |

14 | 19 | 49 | 34 | [14, 17] | 0.1 |

15 | 95 | 66 | 49 | [10, 12] | 0.2 |

16 | 69 | 96 | 53 | [10, 12] | 0.2 |

17 | 60 | 95 | 44 | [10, 12] | 0.2 |

18 | 33 | 64 | 58 | [12, 16] | 0.2 |

19 | 19 | 53 | 29 | [12, 15] | 0.1 |

20 | 30 | 70 | 35 | [11, 14] | 0.1 |

21 | 67 | 39 | 29 | [10, 13] | 0.1 |

22 | 69 | 10 | 56 | [10, 12] | 0.2 |

23 | 84 | 68 | 58 | [10, 12] | 0.2 |

24 | 48 | 91 | 38 | [11, 13] | 0.1 |

25 | 91 | 25 | 35 | [10, 14] | 0.1 |

26 | 44 | 70 | 34 | [11, 13] | 0.1 |

27 | 79 | 18 | 33 | [10, 14] | 0.1 |

28 | 84 | 59 | 44 | [12, 14] | 0.2 |

29 | 52 | 60 | 58 | [12, 14] | 0.2 |

30 | 73 | 15 | 60 | [10, 13] | 0.2 |

31 | 13 | 93 | 31 | [09, 12] | 0.1 |

32 | 45 | 46 | 60 | [10, 13] | 0.2 |

33 | 23 | 43 | 36 | [13, 16] | 0.1 |

34 | 45 | 13 | 57 | [11, 15] | 0.2 |

35 | 98 | 69 | 55 | [10, 12] | 0.2 |

36 | 64 | 20 | 40 | [11, 14] | 0.1 |

37 | 93 | 23 | 34 | [10, 14] | 0.1 |

38 | 15 | 27 | 44 | [12, 16] | 0.2 |

39 | 29 | 51 | 49 | [11, 13] | 0.2 |

40 | 31 | 63 | 44 | [12, 14] | 0.2 |

41 | 96 | 87 | 57 | [11, 13] | 0.2 |

42 | 16 | 79 | 40 | [12, 17] | 0.1 |

43 | 30 | 93 | 31 | [14, 17] | 0.1 |

44 | 99 | 13 | 27 | [15, 17] | 0.1 |

45 | 18 | 82 | 56 | [14, 17] | 0.2 |

46 | 21 | 29 | 38 | [13, 17] | 0.1 |

47 | 36 | 24 | 35 | [14, 17] | 0.1 |

48 | 83 | 83 | 44 | [10, 13] | 0.2 |

49 | 36 | 73 | 47 | [10, 14] | 0.2 |

50 | 60 | 83 | 32 | [12, 16] | 0.1 |

$\mathit{T}$ | $\overline{\mathit{D}}$ | ${\mathit{Z}}_{\mathbf{min}}$ | $\overline{\mathit{Z}}$ | $\mathit{l}$ | |
---|---|---|---|---|---|

GA | 5724.87 | 1337 | 1309.98 | 1349.33 | 12 |

SAA | 80.52 | 1234 | 1308.03 | 1388.65 | 12 |

ACA | 1244.76 | 1036 | 1659.80 | 1733.25 | 12 |

SAAACA | 157.37 | 538 | 1272.21 | 1312.22 | 12 |

$\mathit{T}$ | $\overline{\mathit{D}}$ | ${\mathit{Z}}_{\mathbf{min}}$ | $\overline{\mathit{Z}}$ | $\mathit{l}$ | |
---|---|---|---|---|---|

GA | 550.44 | 70 | 828.94 | 852.32 | 10 |

SAA | 87.57 | 1200 | 1004.48 | 1223.35 | 12 |

ACA | 4285.57 | 1571 | 1255.81 | 1523.92 | 12 |

SAAACA | 323.22 | 538 | 828.94 | 878.17 | 10 |

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## Share and Cite

**MDPI and ACS Style**

Song, W.; Yuan, S.; Yang, Y.; He, C.
A Study of Community Group Purchasing Vehicle Routing Problems Considering Service Time Windows. *Sustainability* **2022**, *14*, 6968.
https://doi.org/10.3390/su14126968

**AMA Style**

Song W, Yuan S, Yang Y, He C.
A Study of Community Group Purchasing Vehicle Routing Problems Considering Service Time Windows. *Sustainability*. 2022; 14(12):6968.
https://doi.org/10.3390/su14126968

**Chicago/Turabian Style**

Song, Wei, Shuailei Yuan, Yun Yang, and Chufeng He.
2022. "A Study of Community Group Purchasing Vehicle Routing Problems Considering Service Time Windows" *Sustainability* 14, no. 12: 6968.
https://doi.org/10.3390/su14126968