标题：遗传算法优化最大化效应的某些需求点可不配送的vrptw问题
摘要：
在可不配送的车辆路径配送问题（VRPTW）中，我们面临着优化路径规划以最大化效用的挑战。本文提出了一种基于遗传算法的方法，旨在解决具有硬时间窗约束的可不配送VRPTW问题。该方法通过多个遗传算法操作符，如选择、交叉和变异，逐代优化车辆的路径规划，以实现效用的最大化。
引言
可不配送的VRPTW问题是一种经典的组合优化问题，它要求在考虑车辆容量限制和时间窗约束的情况下，将货物从中心仓库配送到多个客户地点。然而，在实际应用中，某些客户可能不愿意接受配送，因此我们需要优化路径规划以最大化效用，同时满足硬时间窗约束。
方法
本文提出的方法基于遗传算法，它是一种启发式算法，通过模拟生物进化过程来求解优化问题。遗传算法操作符包括选择、交叉和变异。
2.1 个体表示
每个个体都表示一个车辆路径规划方案，其中包括从中心仓库出发，经过一系列客户地点，最后返回仓库的路径。
2.2 适应度函数
适应度函数用于评估每个个体的优劣程度。在这里，我们将效用最大化作为目标函数。效用的计算可以根据具体问题进行定义，例如货物价值的总和或客户满意度的加权和。
2.3 选择操作
选择操作用于根据适应度函数的结果选择优秀的个体。在本方法中，我们采用轮盘赌选择策略，根据个体适应度与总适应度的比例进行选择。
2.4 交叉操作
交叉操作用于生成新的个体。在本方法中，我们采用部分映射交叉（PMX）算子，将两个个体的染色体部分交换，以产生具有新路径规划的个体。
2.5 变异操作
变异操作用于引入新的基因组合。在本方法中，我们采用交换变异算子，随机选择两个客户地点，并交换它们在路径中的位置。
实验与结果
我们使用一组模拟数据对提出的方法进行实验。实验结果表明，遗传算法能够有效地优化可不配送的VRPTW问题中的效用最大化。通过多轮迭代，算法逐渐收敛于较优解。
主程序如下:
数据如下:
| 需求地序号 | x坐标(千米) | y坐标(千米) | 需求量 | 时间窗开始 | 时间窗结束 | 利润 | 缺货成本 |
| 0 | 125 | 85 | 0 | 6 | 12 | 0 | 0 |
| 1 | 185 | 35 | 6 | 6 | 12 | 100 | 200 |
| 2 | 153 | 165 | 9 | 6 | 12 | 100 | 200 |
| 3 | 38 | 107 | 10 | 6 | 12 | 100 | 200 |
| 4 | 0 | 0 | 4 | 6 | 12 | 100 | 200 |
| 5 | 8 | 11 | 3 | 6 | 12 | 100 | 200 |
| 6 | 87 | 85 | 2 | 6 | 12 | 100 | 200 |
| 7 | 68 | 160 | 5 | 6 | 12 | 100 | 200 |
| 8 | 118 | 197 | 8 | 6 | 12 | 100 | 200 |
| 9 | 65 | 10 | 2 | 6 | 12 | 100 | 200 |
| 10 | 170 | 120 | 2 | 6 | 12 | 100 | 200 |
| 11 | 190 | 80 | 7 | 6 | 12 | 100 | 200 |
| 12 | 130 | 40 | 9 | 6 | 12 | 100 | 200 |
| 13 | 110 | 121 | 9 | 6 | 12 | 100 | 200 |
| 14 | 188 | 151 | 3 | 6 | 12 | 100 | 200 |
| 15 | 107 | 75 | 1 | 6 | 12 | 100 | 200 |
| 16 | 137 | 51 | 7 | 6 | 12 | 100 | 200 |
| 17 | 149 | 27 | 4 | 6 | 12 | 100 | 200 |
| 18 | 19 | 86 | 7 | 6 | 12 | 100 | 200 |
| 19 | 97 | 149 | 4 | 6 | 12 | 100 | 200 |
| 20 | 131 | 157 | 5 | 6 | 12 | 100 | 200 |
程序结果如下:
程序运行时间(s)
runtime201 =
63.243826
遗传算法优化得到的最优目标函数
ans =
6655.92002859396
遗传算法优化得到的最优染色体
bestChrom =
1 至 28 列
0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 8 20 5 2 14 10 11
29 至 40 列
17 12 15 4 1 9 16 3 18 6 13 19
显示各个路径(遗传算法)
第1辆车的路径
route1 =
0 7 8 20 0
loadline =
0 0 18
18 18 13
13 13 5
5 5 0
0 0 0
运行时间表
outcell01 =
'路径点' '到达时间' '开始服务时间' '结束时间'
[ 0] [ 6] [ 6] [ 6]
[ 7] [7.88403821617291] [7.88403821617291] [7.96737154950624]
[ 8] [9.21139727271296] [9.21139727271296] [9.34473060604629]
[ 20] [10.1859202398388] [10.1859202398388] [10.2692535731722]
[ 0] [11.7142449226272] [11.7142449226272] [11.7142449226272]
第2辆车的路径
route1 =
0 5 0
loadline =
0 0 3
3 3 0
0 0 0
运行时间表
outcell01 =
'路径点' '到达时间' '开始服务时间' '结束时间'
[ 0] [ 6] [ 6] [ 6]
[ 5] [8.76875423250241] [8.76875423250241] [8.81875423250241]
[ 0] [11.5875084650048] [11.5875084650048] [11.5875084650048]
第3辆车的路径
route1 =
0 2 14 10 0
loadline =
0 0 14
14 14 5
5 5 2
2 2 0
0 0 0
运行时间表
outcell01 =
'路径点' '到达时间' '开始服务时间' '结束时间'
[ 0] [ 6] [ 6] [ 6]
[ 2] [7.69516960803337] [7.69516960803337] [7.84516960803337]
[ 14] [ 8.5990926810322] [ 8.5990926810322] [ 8.6490926810322]
[ 10] [ 9.3660306141896] [ 9.3660306141896] [9.39936394752293]
[ 0] [10.5395393726221] [10.5395393726221] [10.5395393726221]
第4辆车的路径
route1 =
0 11 17 12 0
loadline =
0 0 20
20 20 13
13 13 9
9 9 0
0 0 0
运行时间表
outcell01 =
'路径点' '到达时间' '开始服务时间' '结束时间'
[ 0] [ 6] [ 6] [ 6]
[ 11] [7.30384048104053] [7.30384048104053] [ 7.4205071477072]
[ 17] [8.76065639312727] [8.76065639312727] [8.82732305979394]
[ 12] [9.28775763712279] [9.28775763712279] [9.43775763712279]
[ 0] [10.3432961509365] [10.3432961509365] [10.3432961509365]
第5辆车的路径
route1 =
0 15 9 16 0
loadline =
0 0 10
10 10 9
9 9 7
7 7 0
0 0 0
运行时间表
outcell01 =
'路径点' '到达时间' '开始服务时间' '结束时间'
[ 0] [ 6] [ 6] [ 6]
[ 15] [6.41182520563948] [6.41182520563948] [6.42849187230615]
[ 9] [7.97626446542181] [7.97626446542181] [8.00959779875515]
[ 16] [9.66670370967334] [9.66670370967334] [ 9.78337037634]
[ 0] [10.5044806314328] [10.5044806314328] [10.5044806314328]
第6辆车的路径
route1 =
0 3 18 6 0
loadline =
0 0 19
19 19 9
9 9 2
2 2 0
0 0 0
运行时间表
outcell01 =
'路径点' '到达时间' '开始服务时间' '结束时间'
[ 0] [ 6] [ 6] [ 6]
[ 3] [7.79477018027379] [7.79477018027379] [7.96143684694045]
[ 18] [8.52782893728071] [8.52782893728071] [8.64449560394737]
[ 6] [10.0046426548209] [10.0046426548209] [10.0379759881543]
[ 0] [10.7979759881543] [10.7979759881543] [10.7979759881543]
第7辆车的路径
route1 =
0 13 19 0
loadline =
0 0 13
13 13 4
4 4 0
0 0 0
运行时间表
outcell01 =
'路径点' '到达时间' '开始服务时间' '结束时间'
[ 0] [ 6] [ 6] [ 6]
[ 13] [ 6.78] [ 6.78] [ 6.93]
[ 19] [7.54741396161733] [7.54741396161733] [7.61408062828399]
[ 0] [9.01122056400983] [9.01122056400983] [9.01122056400983]
punish_early =
0
punish_late =
0
outcell =
'节点编号' '达到时间' '离开时间'
[ 1] [ 0] [ 0]
[ 2] [7.69516960803337] [7.84516960803337]
[ 3] [7.79477018027379] [7.96143684694045]
[ 4] [ 0] [ 0]
[ 5] [8.76875423250241] [8.81875423250241]
[ 6] [10.0046426548209] [10.0379759881543]
[ 7] [7.88403821617291] [7.96737154950624]
[ 8] [9.21139727271296] [9.34473060604629]
[ 9] [7.97626446542181] [8.00959779875515]
[ 10] [ 9.3660306141896] [9.39936394752293]
[ 11] [7.30384048104053] [ 7.4205071477072]
[ 12] [9.28775763712279] [9.43775763712279]
[ 13] [ 6.78] [ 6.93]
[ 14] [ 8.5990926810322] [ 8.6490926810322]
[ 15] [6.41182520563948] [6.42849187230615]
[ 16] [9.66670370967334] [ 9.78337037634]
[ 17] [8.76065639312727] [8.82732305979394]
[ 18] [8.52782893728071] [8.64449560394737]
[ 19] [7.54741396161733] [7.61408062828399]
[ 20] [10.1859202398388] [10.2692535731722]
>>
结论
本文提出了一种基于遗传算法的方法，用于解决具有硬时间窗约束的可不配送VRPTW问题。实验结果表明，该方法能够有效地优化路径规划，从而达到效用的最大化。未来的研究可以探索其他启发式算法或改进遗传算法的操作符，以进一步提高问题求解效果。
参考文献：
[1] Goldberg, D. E. (1989). Genetic algorithms in search, optimization, and machine learning. Addison-Wesley.
[2] Braysy, O., & Gendreau, M. (2005). Vehicle routing problem with time windows, part I: Route construction and local search algorithms. Transportation Science, 39(1), 104-118.
程序运行时间(s)
runtime201 =
36.6331173795092
遗传算法优化得到的最优目标函数
ans =
4561.10885596154
遗传算法优化得到的最优染色体
bestChrom =
1 至 25 列
0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 10 14 2 20 3
26 至 40 列
18 6 1 15 4 9 8 12 16 17 11 5 7 19 13
显示各个路径(遗传算法)
第1辆车的路径
route1 =
0 10 14 2 20 0
loadline =
0 0 19
19 19 17
17 17 14
14 14 5
5 5 0
0 0 0
运行时间表
outcell01 =
'路径点' '到达时间' '开始服务时间' '结束时间'
[ 0] [ 6] [ 6] [ 6]
[ 10] [7.14017542509914] [7.14017542509914] [7.17350875843247]
[ 14] [7.89044669158987] [7.89044669158987] [7.94044669158987]
[ 2] [ 8.6943697645887] [ 8.6943697645887] [ 8.8443697645887]
[ 20] [9.31255776101749] [9.31255776101749] [9.39589109435082]
[ 0] [10.8408824438059] [10.8408824438059] [10.8408824438059]
第2辆车的路径
route1 =
0 3 18 6 15 0
loadline =
0 0 20
20 20 10
10 10 3
3 3 1
1 1 0
0 0 0
运行时间表
outcell01 =
'路径点' '到达时间' '开始服务时间' '结束时间'
[ 0] [ 6] [ 6] [ 6]
[ 3] [7.79477018027379] [7.79477018027379] [7.96143684694045]
[ 18] [8.52782893728071] [8.52782893728071] [8.64449560394737]
[ 6] [10.0046426548209] [10.0046426548209] [10.0379759881543]
[ 15] [10.4851895836542] [10.4851895836542] [10.5018562503209]
[ 0] [10.9136814559604] [10.9136814559604] [10.9136814559604]
第3辆车的路径
route1 =
0 9 12 16 0
loadline =
0 0 18
18 18 16
16 16 7
7 7 0
0 0 0
运行时间表
outcell01 =
'路径点' '到达时间' '开始服务时间' '结束时间'
[ 0] [ 6] [ 6] [ 6]
[ 9] [7.92093727122985] [7.92093727122985] [7.95427060456319]
[ 12] [9.38605271089082] [9.38605271089082] [9.53605271089082]
[ 16] [9.79682080709893] [9.79682080709893] [ 9.9134874737656]
[ 0] [10.6345977288584] [10.6345977288584] [10.6345977288584]
第4辆车的路径
route1 =
0 17 11 0
loadline =
0 0 11
11 11 7
7 7 0
0 0 0
运行时间表
outcell01 =
'路径点' '到达时间' '开始服务时间' '结束时间'
[ 0] [ 6] [ 6] [ 6]
[ 17] [7.25538838611802] [7.25538838611802] [7.32205505278468]
[ 11] [8.66220429820476] [8.66220429820476] [8.77887096487143]
[ 0] [ 10.082711445912] [ 10.082711445912] [ 10.082711445912]
第5辆车的路径
route1 =
0 5 0
loadline =
0 0 3
3 3 0
0 0 0
运行时间表
outcell01 =
'路径点' '到达时间' '开始服务时间' '结束时间'
[ 0] [ 6] [ 6] [ 6]
[ 5] [8.76875423250241] [8.76875423250241] [8.81875423250241]
[ 0] [11.5875084650048] [11.5875084650048] [11.5875084650048]
第6辆车的路径
route1 =
0 7 19 13 0
loadline =
0 0 18
18 18 13
13 13 9
9 9 0
0 0 0
运行时间表
outcell01 =
'路径点' '到达时间' '开始服务时间' '结束时间'
[ 0] [ 6] [ 6] [ 6]
[ 7] [7.88403821617291] [7.88403821617291] [7.96737154950624]
[ 19] [8.58769404627708] [8.58769404627708] [8.65436071294374]
[ 13] [9.27177467456107] [9.27177467456107] [9.42177467456107]
[ 0] [10.2017746745611] [10.2017746745611] [10.2017746745611]
punish_early =
0
punish_late =
0
outcell =
'节点编号' '达到时间' '离开时间'
[ 1] [ 0] [ 0]
[ 2] [ 8.6943697645887] [ 8.8443697645887]
[ 3] [7.79477018027379] [7.96143684694045]
[ 4] [ 0] [ 0]
[ 5] [8.76875423250241] [8.81875423250241]
[ 6] [10.0046426548209] [10.0379759881543]
[ 7] [7.88403821617291] [7.96737154950624]
[ 8] [ 0] [ 0]
[ 9] [7.92093727122985] [7.95427060456319]
[ 10] [7.14017542509914] [7.17350875843247]
[ 11] [8.66220429820476] [8.77887096487143]
[ 12] [9.38605271089082] [9.53605271089082]
[ 13] [9.27177467456107] [9.42177467456107]
[ 14] [7.89044669158987] [7.94044669158987]
[ 15] [10.4851895836542] [10.5018562503209]
[ 16] [9.79682080709893] [ 9.9134874737656]
[ 17] [7.25538838611802] [7.32205505278468]
[ 18] [8.52782893728071] [8.64449560394737]
[ 19] [8.58769404627708] [8.65436071294374]
[ 20] [9.31255776101749] [9.39589109435082]
>>
