# Research on Dynamic Takeout Delivery Vehicle Routing Problem under Time-Varying Subdivision Road Network

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Takeout Delivery VRP Model

#### 2.1. Problem Description

#### 2.2. Solving Method

#### 2.2.1. Dynamic Order Processing Method Based on RHC Strategy

#### 2.2.2. Measurement Methods of Road Traffic Congestion Degree in the Subdivision Road Network

- (1)
- Criteria for the subdivision of road networks

- (2)
- Measurement methods for congestion degree

#### 2.2.3. Measurement Methods of Delivery Penalty Cost

#### 2.3. Problem Description

- (1)
- All delivery vehicles are of identical type and possess equivalent maximum carrying capacity;
- (2)
- Each delivery vehicle is paired with a rider;
- (3)
- The pick-up point and the delivery point in a single order are in a one-to-one correspondence, and the delivery is conducted by the same vehicle;
- (4)
- Upon order placement, the locations of both pick-up and delivery points, the level of demand, as well as the estimated time window for delivery are readily available;
- (5)
- For the same customer position in different orders, the customer is virtualized as multiple customers;
- (6)
- For different orders that have the same merchant location, the merchant is virtualized as multiple merchants.

## 3. Model Solving

#### 3.1. Overall Process

#### 3.2. Algorithm Design

- (1)
- Encoding and decoding of chromosomes

- (2)
- The processes of selection, crossover, and mutation

- (3)
- The enhanced elitist preservation strategy

## 4. Verification and Analysis

#### 4.1. Example Construction

#### 4.2. Test Simulation

#### 4.3. Result Analysis

- (1)
- Comparison and analysis of solution results under different conditions

- (2)
- Sensitivity analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 11.**Optimal driving route of takeout delivery vehicles in different time domains. (

**a**) Takeout delivery route in the first time domain. (

**b**) Takeout delivery route in the second time domain. (

**c**) Takeout delivery route in the third time domain. (

**d**) Takeout delivery route in the fourth time domain. (

**e**) Takeout delivery route in the fifth time domain. (

**f**) Takeout delivery route in the sixth time domain.

**Figure 14.**Total cost under different conditions. (

**a**) Total cost under different numbers of delivery vehicles. (

**b**) Total cost under different numbers of time domains.

Road Grade | Road Description |
---|---|

Express way | A central divider must be installed on urban roads with more than four motorway lanes, all or part of which adopt intersections and control access, for cars to drive at a higher speed. It is also called a car-only road. The design speed of the expressway is 60–100 km/h. |

Main road | The main roads connect the various sections of the city, mainly with traffic functions. The design speed of main roads is 40–60 km/h. |

Secondary road | It serves as traffic agglomeration function between the main roads and each sub-district, and it has service functions. The design speed of secondary roads is 30–50 km/h. |

Branch | The connecting road between the secondary road and the neighborhood road (community road) mainly has service functions. The design speed of branches is 20–40 km/h. |

Speed Limit | Average Driving Speed | |||
---|---|---|---|---|

Smooth | Mild | Moderate | Severe | |

80 | ≥45 | 30 < $v$ ≤ 45 | 20 < $v$ ≤ 30 | 0 < $v$ ≤ 20 |

70 | ≥40 | 30 < $v$ ≤ 40 | 20 < $v$ ≤ 30 | 0 < $v$ ≤ 20 |

60 | ≥35 | 30 < $v$ ≤ 35 | 20 < $v$ ≤ 30 | 0 < $v$ ≤ 20 |

50 | ≥30 | 25 < $v$ ≤ 30 | 15 < $v$ ≤ 25 | 0 < $v$ ≤ 15 |

40 | ≥25 | 20 < $v$ ≤ 25 | 15 < $v$ ≤ 20 | 0 < $v$ ≤ 15 |

<40 | 25 < $v$ ≤ Limited | 20 < $v$ ≤ 25 | 10 < $v$ ≤ 20 | 0 < $v$ ≤ 10 |

Sets | |
---|---|

$O$ | The set of takeout order numbers, $O=\left\{{o}_{1},{o}_{2},{o}_{3},\dots ,{o}_{n}\right\}$, $n$ is the total number of orders. |

${P}^{+}$ | The set of pick-up point numbers, ${P}^{+}=\left\{{p}_{1},{p}_{3},{p}_{5},\dots ,{p}_{2n-1}\right\}$, $n$ is the total number of orders. |

${P}^{-}$ | The set of delivery point numbers, ${P}^{-}=\left\{{p}_{2},{p}_{4},{p}_{6},\dots ,{p}_{2n}\right\}$, $n$ is the total number of orders. |

$P$ | The set of all nodes, $P={p}_{0}\cup {P}^{+}\cup {P}^{-}$, ${p}_{0}$ is the distribution center. |

$RS$ | Secondary road $rs\in RS$, $RS$ is the set of all secondary roads. |

$RB$ | Branch $rb\in RB$, $RB$ is the set of all branches. |

$K$ | Delivery vehicle $k\in K$, $K$ is the set of all delivery vehicles. |

Parameters | |

${\theta}_{rs}(t)$ | Traffic congestion degree of secondary road $rs$ at time $t$, $rs\in RS$. |

${\theta}_{rb}(t)$ | Traffic congestion degree of branch $rb$ at time $t$, $rb\in RB$. |

${d}_{ij}$ | The distance between point $i$ and point $j$ (unit: meters), $i,j\in P$. |

${d}_{ij}^{rs}$ | The distance traveled on secondary road $rs$ from point $i$ to point $j$ (unit: meters), $i,j\in P$, $rs\in RS$. |

${d}_{ij}^{rb}$ | The distance traveled on branch $rb$ from point $i$ to point $j$ (unit: meters), $i,j\in P$, $rb\in RB$. |

${v}_{rs}(t)$ | Average driving speed on secondary road $rs$ at time $t$ (unit: meters per second), $rs\in RS$. |

${v}_{rb}(t)$ | Average driving speed on branch $rb$ at time $t$ (unit: meters per second), $rb\in RB$. |

${T}_{ij}^{rs}$ | The driving time on secondary road $rs$ from point $i$ to point $j$ (unit: seconds), $i,j\in P$, $rs\in RS$. |

${T}_{ij}^{rb}$ | The driving time on branch $rb$ from point $i$ to point $j$ (unit: seconds), $i,j\in P$, $rb\in RB$. |

${T}_{ij}$ | Driving time from point $i$ to point $j$ (unit: seconds), $i,j\in P$. |

$A{T}_{i}$ | The arrival time of node $i$ (unit: seconds), $i\in P$. |

$C{T}_{i}$ | The departure time of node $i$ (unit: seconds), $i\in P$. |

$R{T}_{i}$ | For order $i$, the duration of food preparation by the merchant (unit: seconds), $i\in O$. |

$E{T}_{i}$ | For order $i$, the earliest delivery time (unit: seconds), $i\in O$. |

$L{T}_{i}$ | For order $i$, the latest delivery time (unit: seconds), $i\in O$. |

$S{T}_{2i-1}$ | For order $i$, service time at pick-up point ${p}_{2i-1}$ (unit: seconds), $i\in O$. |

$S{T}_{2i}$ | For order $i$, service time at delivery point ${p}_{2i}$ (unit: seconds), $i\in O$. |

${D}_{2i-1}$ | For order $i$, demand for pick-up point ${p}_{2i-1}$ (unit: grams), ${D}_{2i-1}>0$, $i\in O$. |

${D}_{2i}$ | For order $i$, demand for delivery point ${p}_{2i}$ (unit: grams), ${D}_{2i}<0$, $i\in O$. |

${L}_{i}^{k}$ | The load of delivery vehicle $k$ at point $i$ (unit: gram), $i\in P$, $k\in K$. |

${C}_{i}$ | The price of food in takeout order $i$ (Unit: RMB), $i\in O$. |

$Q$ | Load limit for delivery vehicle (unit: grams). |

${C}_{1}$ | The cost of calling delivery vehicle (unit: RMB). |

${C}_{2}$ | Cost of driving distance (unit: RMB per meter). |

Decision Variables | |
---|---|

${x}_{ijk}$ | When ${x}_{ijk}=1$, it means that delivery for points $i$ to $j$ is carried out by the delivery vehicle $k$, and vice versa, $i,j\in P$, $k\in K$. |

Order Number | Order Time | Food Weight (Grams) | Food Price (RMB) | Location of Pick-Up Point | Location of Delivery Point | Distance between Two Points |
---|---|---|---|---|---|---|

1 | 11:10:59 | 841 | 16 | (34.273488, 108.952349) | (34.258614, 108.96526) | 2829 |

2 | 11:24:13 | 986 | 29 | (34.273384, 108.972752) | (34.25571, 108.966724) | 2431 |

3 | 11:24:40 | 706 | 29 | (34.248861, 108.952353) | (34.271948, 108.952094) | 3698 |

4 | 11:26:26 | 842 | 28 | (34.261158, 108.947717) | (34.279485, 108.945353) | 2536 |

5 | 11:27:59 | 856 | 29 | (34.264237, 108.970589) | (34.264009, 108.945872) | 2450 |

Secondary Roads | 11:00 | 11:15 | 11:20 | 11:25 | 11:30 |
---|---|---|---|---|---|

Gongnong road | 1 | 1 | 1 | 1 | 1 |

Changying west road | 2 | 3 | 3 | 3 | 2 |

North courtyard gate | 1 | 1 | 1 | 2 | 1 |

Xi 1st road | 1 | 1 | 3 | 1 | 1 |

Dongguan street | 1 | 1 | 1 | 2 | 1 |

Cypress grove road | 1 | 2 | 1 | 1 | 1 |

Secondary Roads | 11:00 | 11:15 | 11:20 | 11:25 | 11:30 |
---|---|---|---|---|---|

Lixin street | 1 | 1 | 1 | 2 | 1 |

Tangfang street | 1 | 2 | 2 | 1 | 1 |

East 8th road | 2 | 1 | 1 | 2 | 2 |

Shangde road | 1 | 2 | 2 | 1 | 1 |

Shangqin road | 2 | 1 | 1 | 1 | 1 |

North Guangji street | 3 | 2 | 3 | 2 | 2 |

Time Domain Interval | Orders Generated within the Interval |
---|---|

[0, 1221] | (1, 2, 3, 4, 5, 6, 7, 8, 9) |

[1221, 1542] | (10, 11, 12, 13, 14, 15, 16, 17, 18, 19) |

[1542, 1800] | (20, 21, 22, 23, 24, 25) |

[1800, 2057] | (26, 27, 28, 29, 30, 31) |

[2057, 2378] | (32, 33, 34, 35, 36, 37, 38, 39, 40, 41) |

[2378, 3600] | (42, 43, 44, 45, 46, 47, 48, 49, 50) |

Vehicle Number | Starting Position | Initial Load | Order Taken but Not Delivered | Traveled Route | Planning Route |
---|---|---|---|---|---|

1 | 0 | 0 | — | — | — |

2 | 0 | 0 | — | — | — |

3 | 0 | 0 | — | — | 11-3-17-4-18-9-12-10 |

4 | 0 | 0 | — | — | — |

5 | 0 | 0 | — | — | 1-2-15-13-7-14-16-5-8-6 |

6 | 0 | 0 | — | — | — |

7 | 0 | 0 | — | — | — |

8 | 0 | 0 | — | — | — |

Vehicle Number | Starting Position | Initial Load | Order Taken but Not Delivered | Traveled Route | Planning Route |
---|---|---|---|---|---|

1 | 0 | 0 | — | — | 27-33-3-4-34-15-13-16-28-14 |

2 | 0 | 0 | — | — | — |

3 | 11 | 920 | 12 | 0-11 | 31-9-12-10-32 |

4 | 0 | 0 | — | — | 35-36-19-20 |

5 | 2 | 0 | — | 0-1-2 | 17-21-37-25-26-22-18-38 |

6 | 0 | 0 | — | — | — |

7 | 0 | 0 | — | — | — |

8 | 0 | 0 | — | — | 23-29-7-24-30-5-8-6 |

Vehicle Number | Starting Position | Initial Load | Order Taken but Not Delivered | Traveled Route | Planning Route |
---|---|---|---|---|---|

1 | 27 | 852 | 28 | 0-27 | 3-4-19-43-44-28-20 |

2 | 0 | 0 | — | — | — |

3 | 31 | 1600 | 12, 32 | 0-11-31 | 12-41-9-42-10-32 |

4 | 35 | 541 | 36 | 0-35 | 36-13-49-47-14-50-48 |

5 | 17 | 548 | 18 | 0-1-2-17 | 39-33-15-34-40-18-16 |

6 | 0 | 0 | — | — | — |

7 | 0 | 0 | — | — | 45-21-37-22-46-38 |

8 | 23 | 750 | 24 | 0-23 | 25-29-7-24-30-5-8-26-6 |

Vehicle Number | Starting Position | Initial Load | Order Taken but Not Delivered | Traveled Route | Planning Route |
---|---|---|---|---|---|

1 | 3 | 1838 | 28, 4 | 0-27-3 | 13-4-9-14-2-10 |

2 | 0 | 0 | — | — | — |

3 | 12 | 680 | 32 | 0-11-31-12 | 5-43-41-44-6-42-32 |

4 | 36 | 0 | — | 0-35-36 | 49-47-61-55-62-50-48-56 |

5 | 39 | 1494 | 18, 40 | 0-1-2-17-39 | 15-19-40-18-16-20 |

6 | 0 | 0 | — | — | 53-21-59-54-60-22 |

7 | 45 | 858 | 46 | 0-45 | 33-34-57-37-38-46-58 |

8 | 25 | 1451 | 24, 26 | 0-23-25 | 29-7-24-30-51-8-26-52 |

Vehicle Number | Starting Position | Initial Load | Order Taken but Not Delivered | Traveled Route | Planning Route |
---|---|---|---|---|---|

1 | 13 | 2694 | 28, 4, 14 | 0-27-3-13 | 4-14-28-75-67-68-57-59-76-60-58 |

2 | 0 | 0 | — | — | 7-55-47-61-8-56-48-62 |

3 | 5 | 1386 | 32, 6 | 0-11-31-12-5 | 79-6-80-69-32-70 |

4 | 49 | 657 | 50 | 0-35-36-49 | 63-50-64-65-77-66-78 |

5 | 15 | 2225 | 18, 40, 16 | 0-1-2-17-39-15 | 19-40-18-16-41-20-42 |

6 | 53 | 594 | 54 | 0-53 | 21-81-54-22-9-10-82 |

7 | 33 | 1589 | 46, 34 | 0-45-33 | 37-34-71-38-72-46 |

8 | 29 | 2217 | 24, 26, 30 | 0-23-25-29 | 24-26-30-73-43-51-74-44-52 |

Vehicle Number | Starting Position | Initial Load | Order Taken but Not Delivered | Traveled Route | Planning Route |
---|---|---|---|---|---|

1 | 14 | 852 | 28 | 0-27-3-13-4-14 | 28-75-65-91-66-76-92-73-74 |

2 | 55 | 1202 | 8, 56 | 0-7-55 | 47-61-62-48-56-51-8-43-44-52 |

3 | 79 | 1901 | 32, 6, 80 | 0-11-37-12-5-79 | 93-6-81-80-94-59-60-82-32 |

4 | 63 | 1646 | 50, 64 | 0-35-36-49-63 | 50-64-77-83-78-84-97-98 |

5 | 19 | 2928 | 18, 40, 16, 20 | 0-1-2-17-39-15-19 | 4-16-18-87-20-95-67-96-68-88-69-70 |

6 | 21 | 1405 | 54, 22 | 0-53-21 | 85-54-22-86 |

7 | 34 | 1798 | 46, 38 | 0-45-33-37-34 | 46-9-38-41-10-42 |

8 | 24 | 1467 | 26, 30 | 0-23-25-29-24 | 26-30-89-99-90-100-57-71-72-58 |

**Table 15.**Total cost and composition of takeout delivery for all orders delivered under different conditions.

Total Cost | Driving Cost | Penalty Cost | |
---|---|---|---|

Under time-varying subdivision network | 411.94 | 241.74 | 30.2 |

Under time-varying overall road network | 428.72 | 238.42 | 50.3 |

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## Share and Cite

**MDPI and ACS Style**

Xie, F.; Chen, Z.; Zhang, Z.
Research on Dynamic Takeout Delivery Vehicle Routing Problem under Time-Varying Subdivision Road Network. *Mathematics* **2024**, *12*, 962.
https://doi.org/10.3390/math12070962

**AMA Style**

Xie F, Chen Z, Zhang Z.
Research on Dynamic Takeout Delivery Vehicle Routing Problem under Time-Varying Subdivision Road Network. *Mathematics*. 2024; 12(7):962.
https://doi.org/10.3390/math12070962

**Chicago/Turabian Style**

Xie, Fengjie, Zhiting Chen, and Zhuan Zhang.
2024. "Research on Dynamic Takeout Delivery Vehicle Routing Problem under Time-Varying Subdivision Road Network" *Mathematics* 12, no. 7: 962.
https://doi.org/10.3390/math12070962