# Time-Dependent Multi-Depot Heterogeneous Vehicle Routing Problem Considering Temporal–Spatial Distance

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Problem and Mathematical Model

#### 3.1. Problem Description

#### 3.2. Determination of Time-Dependent Function of Vehicle Speed

#### 3.3. Build Mathematical Model

## 4. Solution Methods

#### 4.1. Customer Clustering and Initial Population Generation

#### 4.2. Encoding and Decoding

#### 4.3. Fitness Evaluation

#### 4.4. Selection

#### 4.5. Evolution

Parameters | |

● | $Pop1$, $Pop2$: parental chromosome 1 and 2; |

● | ${i}_{11}$, ${i}_{12}$, ${i}_{21}$, ${i}_{22}$: random nodes; |

● | $newPop1$, $newPop2$: new child 1 and 2; |

1 | begin |

2 | select two parental chromosomes $Pop1$ and $Pop2$; |

3 | randomly generate 2 nodes on $Pop1$, e.g., ${i}_{11}$ and ${i}_{12}$; |

4 | randomly generate 2 nodes on $Pop2$, e.g., ${i}_{21}$ and ${i}_{22}$; |

5 | take the part between ${i}_{11}$ and ${i}_{12}$ of the $Pop1$ as the first part of $newPop1$; |

6 | eliminate points in parent $Pop2$ that existing between point ${i}_{11}$ ${i}_{11}$ and ${i}_{12}$; |

7 | take the eliminated point arrangement as the second part of $newPop1$; |

8 | generate $newPop1$; |

9 | take the part between ${i}_{21}$ and ${i}_{22}$ of the $Pop2$ as the first part of $newPop2$; |

10 | eliminate points in parent $Pop1$ that existing between point ${i}_{21}$ and ${i}_{22}$; |

11 | take the eliminated point arrangement as the second part of $newPop2$; |

12 | generate $newPop2$ |

13 | end |

#### 4.6. Local Search Strategy

- (1)
- Insert: Randomly select two customer points $i$ and $j$, insert $i$ after customer $j$. As shown in Figure 6a, customer 3 and customer 6 are randomly selected and customer 3 is inserted behind customer 6.
- (2)
- Exchange: Randomly select two customer nodes $i$ and $j$ to exchange the positions of the two customer nodes. As shown in Figure 6b, the positions of customer 3 and customer 6 are exchanged.
- (3)
- 2-insert: In the original scheme, two consecutive customer nodes are randomly selected and inserted after the randomly selected customer point $j$. As shown in Figure 6c, customers 3 and 4 are inserted behind customer 6.
- (4)
- 2-opt: Randomly select two customer nodes $i$ and $j$, and exchange the order between customer nodes. As shown in Figure 6d, the position of customer 3 is kept unchanged, and customers 4, 5, 7 and 6 are in reverse order.
- (5)
- or-opt: In the original scheme, two consecutive customer nodes are randomly selected and inserted into the back of randomly selected customer point $j$ in reverse order. As shown in Figure 6e, customers 3 and 4 are inserted behind customer 6 in reverse order.

- (1)
- Setting the initial neighborhood search number ${S}_{n}=1$ and the number of times the optimal solution is continuously unchanged $connum$;
- (2)
- If the optimal solution of the population after this iteration is not improved, let $connum=connum+1$, ${S}_{n}={S}_{n}+1$; If the perturbed solution is improved, let $connum=0$, ${S}_{n}=1$;
- (3)
- When the number of times $connum$ that the optimal solution has not changed continuously reaches the preset value $stopnum$, the algorithm terminates and outputs the optimal solution.

Algorithm (HGAVNS) | |

Parameters | |

● | $popsize$: population size; |

● | $MAXGEN$: maximum number of iterations; |

● | ${N}_{k}=\{{N}_{1},{N}_{2},\dots ,{N}_{l}\}$: neighborhood structure, ${N}_{l}$ is the $l$ neighborhood structure; |

● | ${S}_{n}$: adaptive neighborhood search times; |

● | $max{S}_{n}$: maximum number of neighborhood cycles; |

1. | Initialize $P(t)$; |

2. | $gen=0$; |

3. | while$gen\le MAXGEN$ |

4. | evaluate $P(t)$; |

5. | select $P(t+1)$ from $P(t)$; % elitist preservation+ roulette wheel selection |

6. | evolution in $P(t+1)$; % order crossover |

7. |
for $i=1:popsize$ |

8. |
for $j=1:max{S}_{n}$ |

9. | Individual ${P}_{i}(t+1)$ disturbed from the first neighborhood structure ${N}_{1}$, $iter\leftarrow 1$; |

10. | if $p\ge {p}_{0}$ % ${p}_{0}$ is the acceptance probability of the new solution |

11. | ${P}_{i}(t+1)$←${P}_{i}\prime (t+1)$; |

12. | break |

13. | else $iter\leftarrow iter+1$; |

14. | end |

15. | until ($iter={S}_{n}$); |

16. | Individual ${P}_{i}(t+1)$ continues to be disturbed by the next neighborhood structure ${N}_{l}$, $iter\leftarrow 1$; |

17. | repeat |

18. | until ($iter={S}_{n}$); |

19. | end |

20. | end |

21. | $gen=gen+1$; |

22. | end |

23. | Best solution |

## 5. Numerical Experiments

#### 5.1. Algorithm Test

#### 5.2. The Performance of Proposed Algorithm

#### 5.3. Instance Verification

#### 5.4. Comparison with Different Types of Vehicles

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Instance | n | d | BKS | GRASP/VND | GVNS | CCA | HGAVNS | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

Best | %Dev | Best | %Dev | Best | %Dev | Best | %Dev | ||||

p01 | 50 | 4 | 576.87 | 592.21 | 2.66 | 582.34 | 0.95 | 576.87 | 0 | 576.87 | 0.00 |

p02 | 50 | 4 | 473.53 | 529.64 | 11.85 | 473.87 | 0.07 | 473.87 | 0.07 | 473.53 | 0.00 |

p03 | 75 | 5 | 641.19 | 648.68 | 1.17 | 641.19 | 0 | 641.19 | 0 | 646.33 | 0.80 |

p04 | 100 | 2 | 1001.04 | 1055.26 | 5.42 | 1008.66 | 0.76 | 1007.40 | 0.64 | 1001.54 | 0.05 |

p05 | 100 | 2 | 750.03 | 769.37 | 2.58 | 752.97 | 0.39 | 750.11 | 0.01 | 751.26 | 0.16 |

p06 | 100 | 3 | 876.50 | 924.68 | 5.50 | 878.02 | 0.17 | 876.50 | 0 | 876.70 | 0.02 |

p07 | 100 | 4 | 881.97 | 925.80 | 4.97 | 890.46 | 0.96 | 888.41 | 0.73 | 884.43 | 0.28 |

p12 | 80 | 2 | 1318.95 | 1326.85 | 0.60 | 1318.95 | 0 | 1318.95 | 0 | 1318.95 | 0 |

p15 | 160 | 4 | 2505.42 | 2553.80 | 1.93 | 2525.85 | 0.82 | 2526.06 | 0.82 | 2505.42 | 0 |

p18 | 240 | 6 | 3702.85 | 4209.56 | 13.68 | 3796.04 | 2.52 | 3771.35 | 1.85 | 3780.42 | 2.09 |

Ave | - | - | - | - | 5.04 | - | 0.66 | - | 0.41 | - | 0.34 |

Vehicle | Capacity | Fixed Cost | Transportation Cost |
---|---|---|---|

1 | 60 | 200 | 1.2 |

2 | 100 | 300 | 1.9 |

3 | 120 | 400 | 2.25 |

Vehicle | Capacity | Vehicle Route | Total Cost |
---|---|---|---|

1 | 100 | 53-16-29-20-35-3-28-22-52 | 1599.97 |

2 | 100 | 53-38-2-50-34-9-49-5-53 | |

3 | 60 | 51-27-46-11-32-1-8-52 | |

4 | 100 | 51-6-18-41-4-47-12-53 | |

5 | 120 | 52-26-13-7-25-43-44-24-10-36-48-40-17-33-39-45-23-14-31-42-19-30-21-15-37-53 |

Vehicle | Capacity | Vehicle Route | Total Cost |
---|---|---|---|

1 | 100 | 51-47-17-44-49-30-34-21-20-52 | 1828.71 |

2 | 100 | 51-25-18-41-42-5-12-53 | |

3 | 60 | 52-3-35-22-28-8-52 | |

4 | 100 | 51-27-23-1-32-46-16-50-11-2-52 | |

5 | 120 | 51-6-9-29-14-45-37-48-38-10-40-4-36-13-7-26-39-31-33-19-43-24-15-53 |

Q | Vehicle Route | n | Total Cost |
---|---|---|---|

60 | 53-5-10-33-37-12-53 53-9-16-32-1-22-11-53 51-13-41-19-42-44-4-46-51 53-50-38-47-18-5153-49-34-2-52 52-3-36-28-31-8-7-27-51 51-6-48-35-20-52 53-21-29-30-17-14-45-26-23-15-24-25-39-43-40-53 | 8 | 1950.23 |

100 | 53-9-49-37-44-33-39-30-34-2-52 51-27-48-8-31-3-35-36-20-52 51-4-18-19-41-17-12-53 52-28-42-45-1-6-22-10-40-21-29-26-23-15-14-24-7-25-13-43-46-51 | 4 | 1905.98 |

120 | 53-38-16-35-29-21-34-30-9-50-2-52 52-8-27-32-1-11-49-5-12-53 51-47-42-41-18-46-3-28-20-52 52-36-48-44-45-15-19-7-14-6-17-31-37-33-25-43-13-10-24-26-23-39-4-40-22-52 | 4 | 2035.52 |

Heterogeneous vehicles | 53-16-29-20-35-3-28-22-52 53-38-2-50-34-9-49-5-53 51-27-46-11-32-1-8-52 51-6-18-41-4-47-12-53 51-6-9-29-14-45-37-48-38-10-40-4-36-13-7-26-39-31-33-19-43-24-15-53 | 5 | 1599.97 |

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**MDPI and ACS Style**

Hou, D.; Fan, H.; Ren, X.; Tian, P.; Lv, Y.
Time-Dependent Multi-Depot Heterogeneous Vehicle Routing Problem Considering Temporal–Spatial Distance. *Sustainability* **2021**, *13*, 4674.
https://doi.org/10.3390/su13094674

**AMA Style**

Hou D, Fan H, Ren X, Tian P, Lv Y.
Time-Dependent Multi-Depot Heterogeneous Vehicle Routing Problem Considering Temporal–Spatial Distance. *Sustainability*. 2021; 13(9):4674.
https://doi.org/10.3390/su13094674

**Chicago/Turabian Style**

Hou, Dengkai, Houming Fan, Xiaoxue Ren, Panjun Tian, and Yingchun Lv.
2021. "Time-Dependent Multi-Depot Heterogeneous Vehicle Routing Problem Considering Temporal–Spatial Distance" *Sustainability* 13, no. 9: 4674.
https://doi.org/10.3390/su13094674