一种无人机配合卡车配送的车辆路径规划模型_无人机-卡车巡检模型求解

TSP 模型

模型参数

决策变量

约束条件

(1) 流入/流出
(1a) 只能使用卡车配送的点
$$\begin{gathered} \sum _{j\in N}{x}_{j,i}=1\forall i\in N\backslash M \\ \sum _{j\in N}{x}_{i,j}=1\forall i\in N\backslash M \end{gathered}$$

(1b) 可以使用无人机配送的点
$$\begin{gathered} \sum _{j\in N}{x}_{j,i}+y_{j,i}=1\forall i\in M \\ \sum _{j\in N}{x}_{i,j}+y_{i,j}=1\forall i\in M \\ \sum _{j\in N}{y}_{j,i}=\sum _{j\in N}{y}_{i,j}\forall i\in M \end{gathered}$$

(2) 去除自循环
$$\begin{gathered} \sum _{i\in N}{x}_{i,i}=1 \\ \sum _{i\in N}{y}_{i,i}=1 \end{gathered}$$

(3) 是否由卡车配送
$$s_i=\sum _{j\in N}{x}_{i,j}\forall i\in N\backslash O$$

(4) 子回路去除
∑ j ∈ N \ { i } a i , j − ∑ j ∈ N \ { O , i } a j , i = s i ∀ i ∈ N \ O a i , j ≤ ∣ N ∣ ⋅ x i , j ∀ i ∈ N \ O ∀ j ∈ N \sum_{j \in N \backslash \{ i \}} a_{i, j} - \sum_{j \in N \backslash \{ O, i \}} a_{j, i} = s_{i} \quad \forall i \in N \backslash O \\ a_{i, j} \leq |N| \cdot x_{i, j} \quad \forall i \in N \backslash O \quad \forall j \in N j∈N\{i}∑​ai,j​−j∈N\{O,i}∑​aj,i​=si​∀i∈N\Oai,j​≤∣N∣⋅xi,j​∀i∈N\O∀j∈N

(5) 无人机配送的流入点和流出点，符合卡车路径的先后关系
$$y_{i,j}+y_{j,k}\le 2\cdot x_{i,k}+1\forall j\in M\forall i\in N\forall k\in N$$

(6) 只有一架无人机
$$\begin{gathered} \sum _{j\in N}{y}_{i,j}\le 1\forall i\in N \\ \sum _{j\in N}{y}_{j,i}\le 1\forall i\in N \end{gathered}$$

目标函数

最小化，加权距离成本
$$min\sum _{i\in N}\sum _{j\in N}{x}_{i,j}\cdot d_{i,j}+y_{i,j}\cdot d_{i,j}\cdot c$$

可视化示例

VRP 模型

模型参数

决策变量

约束条件

(1) 流入/流出
(1a) 只能使用卡车配送的点
∑ j ∈ N x j , i = 1 ∀ i ∈ N \ { O } \ M ∑ j ∈ N x i , j = 1 ∀ i ∈ N \ { O } \ M \sum_{j \in N} x_{j, i} = 1 \quad \forall i \in N \backslash \{O\} \backslash M \\ \sum_{j \in N} x_{i, j} = 1 \quad \forall i \in N \backslash \{O\} \backslash M j∈N∑​xj,i​=1∀i∈N\{O}\Mj∈N∑​xi,j​=1∀i∈N\{O}\M

(1b) 可以使用无人机配送的点
$$\begin{gathered} \sum _{j\in N}{x}_{j,i}+y_{j,i}=1\forall i\in M \\ \sum _{j\in N}{x}_{i,j}+y_{i,j}=1\forall i\in M \\ \sum _{j\in N}{y}_{j,i}=\sum _{j\in N}{y}_{i,j}\forall i\in M \end{gathered}$$

(2) 去除自循环
$$\begin{gathered} \sum _{i\in N}{x}_{i,i}=1 \\ \sum _{i\in N}{y}_{i,i}=1 \end{gathered}$$

(3) 是否由卡车配送
$$s_i=\sum _{j\in N}{x}_{i,j}\forall i\in N\backslash O$$

(4) 载重限制，并消除子回路
z i , j ≥ w j + ∑ k ∈ N \ { j } z j , k + ∑ u ∈ M \ { u } y j , u ⋅ w u + ( x i , j − 1 ) ⋅ B ∀ j ∈ N \ { O } ∀ i ∈ N z i , j ≤ x i , j ⋅ B ∀ j ∈ N \ { O } ∀ i ∈ N x i , j + x j , i ≤ 1 ∀ j ∈ N \ { O } ∀ i ∈ N \ { O } z_{i, j} \geq w_{j} + \sum_{k \in N \backslash \{j\}} z_{j, k} + \sum_{u \in M \backslash \{u\}} y_{j, u} \cdot w_{u} + (x_{i, j} - 1) \cdot B \quad \forall j \in N \backslash \{O\} \quad \forall i \in N \\ z_{i, j} \leq x_{i, j} \cdot B \quad \forall j \in N \backslash \{O\} \quad \forall i \in N \\ x_{i, j} + x_{j, i} \leq 1 \quad \forall j \in N \backslash \{O\} \quad \forall i \in N \backslash \{O\} zi,j​≥wj​+k∈N\{j}∑​zj,k​+u∈M\{u}∑​yj,u​⋅wu​+(xi,j​−1)⋅B∀j∈N\{O}∀i∈Nzi,j​≤xi,j​⋅B∀j∈N\{O}∀i∈Nxi,j​+xj,i​≤1∀j∈N\{O}∀i∈N\{O}

(5) 无人机配送的流入点和流出点，符合卡车路径的先后关系
$$y_{i,j}+y_{j,k}\le 2\cdot x_{i,k}+1\forall j\in M\forall i\in N\forall k\in N$$

(6) 只有一架无人机
$$\begin{gathered} \sum _{j\in N}{y}_{i,j}\le 1\forall i\in N \\ \sum _{j\in N}{y}_{j,i}\le 1\forall i\in N \end{gathered}$$

目标函数

一级目标：最小化，总卡车数；
二级目标：最小化，加权距离成本
m i n ∑ j ∈ N \ { O } x 0 , j ⋅ B + ∑ i ∈ N ∑ j ∈ N ( x i , j ⋅ d i , j + y i , j ⋅ d i , j ⋅ c ) min \quad \sum_{j \in N \backslash \{O\}} x_{0, j} \cdot B + \sum_{i \in N} \sum_{j \in N} (x_{i, j} \cdot d_{i, j} + y_{i, j} \cdot d_{i, j} \cdot c) minj∈N\{O}∑​x0,j​⋅B+i∈N∑​j∈N∑​(xi,j​⋅di,j​+yi,j​⋅di,j​⋅c)

可视化示例

参考文献

2015, The flying sidekick traveling salesman problem Optimization of drone-assisted parcel delivery

代码分享

https://blog.csdn.net/Zhang_0702_China/article/details/119391030