# Distribution Path Optimization by an Improved Genetic Algorithm Combined with a Divide-and-Conquer Strategy

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Distribution Path Optimization Model of the MVRP

**Objective function 1.**The total cost is minimized,

**Objective function 2.**The total length of the route is minimized,

**Constraint 1.**The delivered goods shall not exceed the maximum load capacity of a vehicle,

**Constraint 2.**The distribution job of each node is completed by only one vehicle, and all the distribution jobs are completed by ${N}_{\mathrm{v}}$ vehicles,

**Constraint 3.**There is only one vehicle that reaches or leaves a demand node,

**Constraint 4.**There is no duplicate or loop route section on the optimized path,

**Objective function 3.**For any two demand/distribution nodes A and B, assume that there is $n-2$ road intersection nodes between them. A and B are treated as nodes 1 and $n$, respectively, and all the other $n-2$ nodes are permutated and then renamed as 2, 3, …, $n-1$. Then, the path length of any sequence between the two nodes is minimized,

## 3. A Divide-and-Conquer Strategy for the MVRP

## 4. An Improved GA for Distribution Path Optimization of the MVR

- vehicle 1: [0-32-40-22-34-35-6-3-0];
- vehicle 2: [0-16-11-30-33-7-0];
- vehicle 3: [0-38-28-17-14-8-36-29-21-0];
- vehicle 4: [0-25-37-31-27-26-19-15-1-36-23-2-4-0];
- vehicle 5: [0-18-24-39-13-9-20-10-12-0].

Algorithm 1 [Distribution Sequence Search] |

Input: ${N}_{\mathrm{P}}$, ${p}_{\mathrm{c}}$, ${p}_{\mathrm{m}}$, $T$; ${N}_{\mathrm{d}}$, ${N}_{\mathrm{v}}$, $Q$, ${q}_{i}$, node positions, …Output: Distribution sequence of demand nodes.Begin:Load node positions; Code demand nodes into continual integers; Generate initial population; while (TRUE):Generate a population; Preprocess infeasible individuals; Calculate fitness defined by (1) or (3); Perform selection, combination, and mutation; Apply elitist’s strategy; until Termination criterion is met.End |

Algorithm 2 [Route Search Between Two Nodes] |

Input: ${N}_{\mathrm{P}}$, ${p}_{\mathrm{c}}$, ${p}_{\mathrm{m}}$, $T$; node positions, intersection positions, …Output: Optimal route between the two nodes.Begin:while (TRUE):Apply the divide-and-conquer strategy: Find a center between two nodes; Draw a search region using a radius; Identify the road intersections within the range as additional nodes; Generate a population using nodes in the region; Preprocess infeasible individuals; Calculate fitness defined by (12); Perform selection, combination, and mutation; Apply elitist’s strategy; until Termination criterion is met.End |

## 5. Simulation

#### 5.1. Sequence Search Using Algorithm 1

#### 5.2. Route Search Using Algorithm 2

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Determination of the search radius. (

**a**) When node 3 is out of search range, and (

**b**) The new search range.

**Figure 5.**The coordinates of the demand nodes. The $x$- and $y$-coordinates correspond to a position on a map.

**Figure 6.**The distribution sequence, using the improved GA, (

**a**,

**b**) for 5 vehicles, (

**c**,

**d**) for 6 vehicles, and (

**e**,

**f**) for 7 vehicles. For (

**a**,

**c**,

**e**), the $x$- and $y$-coordinates correspond to a position on a map. For (

**b**,

**d**,

**f**), the $x$-coordinate corresponds to the number of generation, and the $y$-coordinate corresponds to the total distance in km.

**Figure 7.**Distribution sequence, using the simple GA, (

**a**,

**b**) for 5 vehicles, (

**c**,

**d**) for 6 vehicles, and (

**e**,

**f**) for 7 vehicles. For (

**a**,

**c**,

**e**), the $x$- and $y$-coordinates correspond to a position on a map. For (

**b**,

**d**,

**f**), the $x$-coordinate corresponds to the number of generation, and the $y$-coordinate corresponds to the total distance in km.

**Figure 8.**Distribution path (

**a**) and the optimization curve (

**b**). For (

**a**), the $x$- and $y$-coordinates correspond to a position on a map. For (

**b**), the $x$-coordinate corresponds to the number of generations, and the $y$-coordinate corresponds to the total route length in km.

No. | $\mathit{x}$ (km) | $\mathit{y}$ (km) | Demand ($\times {10}^{3}$ pcs) | No. | $\mathit{x}$ (km) | $\mathit{y}$ (km) | Demand ($\times {10}^{3}$ pcs) |
---|---|---|---|---|---|---|---|

0 | 50 | 50 | no | 21 | 88 | 51 | 0.3 |

1 | 47 | 42 | 8.1 | 22 | 65 | 29 | 8.4 |

2 | 79 | 26 | 9 | 23 | 44 | 64 | 9.3 |

3 | 99 | 91 | 1.2 | 24 | 91 | 64 | 6.7 |

4 | 30 | 33 | 9.1 | 25 | 74 | 92 | 7.5 |

5 | 65 | 20 | 6.3 | 26 | 1 | 68 | 7.4 |

6 | 53 | 10 | 0.9 | 27 | 53 | 35 | 3.9 |

7 | 40 | 83 | 2.7 | 28 | 44 | 35 | 6.5 |

8 | 83 | 95 | 5.4 | 29 | 58 | 51 | 1.7 |

9 | 53 | 7 | 9.5 | 30 | 88 | 10 | 7.0 |

10 | 35 | 17 | 9.6 | 31 | 49 | 36 | 0.3 |

11 | 72 | 90 | 1.5 | 32 | 71 | 13 | 2.7 |

12 | 35 | 80 | 9.7 | 33 | 39 | 62 | 0.4 |

13 | 57 | 69 | 9.5 | 34 | 84 | 57 | 0.9 |

14 | 43 | 22 | 4.8 | 35 | 30 | 30 | 8.2 |

15 | 44 | 14 | 8.0 | 36 | 17 | 49 | 6.9 |

16 | 46 | 46 | 1.4 | 37 | 10 | 25 | 3.1 |

17 | 94 | 40 | 4.2 | 38 | 18 | 74 | 9.5 |

18 | 42 | 19 | 9.1 | 39 | 18 | 13 | 0.3 |

19 | 66 | 91 | 7.9 | 40 | 8 | 55 | 9.5 |

20 | 70 | 47 | 6.5 | Total | 224.9 |

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**MDPI and ACS Style**

Li, J.; Wang, Y.; Du, K.-L.
Distribution Path Optimization by an Improved Genetic Algorithm Combined with a Divide-and-Conquer Strategy. *Technologies* **2022**, *10*, 81.
https://doi.org/10.3390/technologies10040081

**AMA Style**

Li J, Wang Y, Du K-L.
Distribution Path Optimization by an Improved Genetic Algorithm Combined with a Divide-and-Conquer Strategy. *Technologies*. 2022; 10(4):81.
https://doi.org/10.3390/technologies10040081

**Chicago/Turabian Style**

Li, Jiaqi, Yun Wang, and Ke-Lin Du.
2022. "Distribution Path Optimization by an Improved Genetic Algorithm Combined with a Divide-and-Conquer Strategy" *Technologies* 10, no. 4: 81.
https://doi.org/10.3390/technologies10040081