封闭与开放场景下两级无人车协同配送路径优化

doi:10.12305/j.issn.1001-506X.2025.12.20

Abstract

Abstract:

To improve the collaborative delivering efficiency of two-echelon unmanned vehicles in closed and open scenes, the delivery network is designed and two-echelon unmanned vehicle routing optimization model is established considering different methods to transport goods between two-echelon unmanned vehicles. Regarding model solving, the improved variable neighborhood search algorithm is designed. In this algorithm, multiple neighborhood search operators are designed, the strategy where the greedy algorithm is used to keep some infeasible solutions into the offspring population and the adaptive strategy are proposed to adjust the process of iterative optimization. Multiple sets of instance experiments show that the proposed algorithm and algorithm strategies can improve the efficiency of problem-solving. This research achievements provide a reference for two-echelon unmanned vehicle delivery routing optimization scheme in closed and open scenes.

Key words: closed and open scenes, unmanned vehicles delivery, two-echelon delivery network, routing optimization model, variable neighborhood search algorithm

CLC Number:

TP 301.6

Xiaolong DIAO, Houming FAN, Yang LI, Zimu BIAN, Shuo WEI, Binghao ZHANG. Two-echelon unmanned vehicles collaborative delivery routing optimization in closed and open scenes[J]. Systems Engineering and Electronics, 2025, 47(12): 3952-3965.

Figures/Tables 16

Table 1

Fig.1

Table 2

Table 3

Table 4

Table 5

Fig.2

Fig.3

Fig.4

Fig.5

Table 6

Solutions result of different algorithms"

算例	INSA					GRASP						本文算法
算例	Avg	Min	Max	Time/s	GAP	Avg	Min	Max	Time/s	GAP	Avg	Min	Max	Time/s	GAP
A1	10523.4	10511	10581	7.3	0.007	10499.7	10487	10563	2.6	0.007	10514.5	10493	10557	5.7	0.006
A2	11935.6	11865	12005	7.3	0.012	11787.2	11759	11823	2.6	0.005	11911.2	11842	11995	5.3	0.013
A3	10502.2	10403	10578	7.9	0.017	10375.8	10344	10469	1.9	0.012	10457.2	10392	10530	5.2	0.013
A4	11311.0	11298	11385	6.2	0.008	11206.3	11198	11262	2.1	0.006	11295.6	11291	11307	4.6	0.001
B1	7623.5	7602	7798	6.7	0.026	7587.6	7549	7603	1.8	0.007	7614.2	7598	7744	4.2	0.019
B2	8147.7	7810	8291	6.2	0.059	8004.2	7989	8044	2.1	0.007	8105.6	7996	8235	4.9	0.029
B3	8802.3	8747	8828	6.4	0.009	8643.3	8604	8695	2.3	0.011	8784.2	8733	8828	4.7	0.011
B4	8884.5	8789	8897	7.1	0.012	8696.7	8621	8758	2.1	0.016	8839	8771	8848	4.7	0.009
C1	6559.2	6463	6681	5.8	0.033	6473.9	6437	6532	1.8	0.015	6534.1	6451	6632	4.0	0.028
C2	6023.1	5982	6107	6.9	0.021	5892.3	5861	5910	1.9	0.008	6009.5	5943	6094	4.2	0.025
C3	7000.5	6991	7205	6.4	0.031	6900.5	6877	6990	1.9	0.016	6989.6	6927	7178	4.4	0.036
C4	6812.8	6776	6904	7.1	0.019	6772.4	6754	6888	2.6	0.020	6785.2	6768	6880	5.4	0.017
A5	11844.5	11673	12245	31.3	0.048	11905.3	11852	12358	16.4	0.043	11788.9	11572	12107	20.3	0.045
A6	12301.9	12246	12672	37.9	0.035	12377.9	12337	12712	17.3	0.030	12239.4	12197	12435	21.7	0.019
A7	12437.8	12306	12731	40.2	0.034	12475.6	12394	12812	19.6	0.034	12367.2	12250	12652	25.0	0.033
A8	12784.5	12677	13031	32.5	0.028	12786.8	12683	13158	16.4	0.037	12704.6	12591	12971	20.0	0.030
B5	9520.2	9412	9945	33.5	0.056	9498.3	9415	9793	12.7	0.040	9496.3	9403	9810	16.0	0.043
B6	10314.7	10213	10712	29.0	0.048	10324.6	10215	10655	15.6	0.043	10271.7	10198	10672	17.6	0.046
B7	9703.2	9572	9997	29.0	0.044	9656.3	9586	9840	16.7	0.026	9654.6	9544	10002	20.7	0.047
B8	9836.0	9677	10211	31.6	0.054	9814.2	9753	10069	15.8	0.032	9787.3	9554	10179	19.9	0.064
C5	8419.4	8299	8732	36.3	0.051	8416.8	8347	8678	15.8	0.039	8320.5	8271	8645	20.6	0.045
C6	9793.2	9702	10313	33.2	0.062	9801.7	9717	10047	15.7	0.034	9776.8	9695	10227	19.4	0.054
C7	10047.2	9964	10341	32.2	0.038	10095.4	9982	10429	15.3	0.044	9945.6	9876	10453	18.5	0.058
C8	10471.6	10254	10819	32.5	0.054	10381.6	10275	10849	15.3	0.055	10347.3	10219	10730	18.7	0.049
A9	15511.8	15465	16122	71.3	0.042	15512.3	15476	16102	45.7	0.040	15446.2	15364	16079	56.4	0.046
A10	15309.5	15132	16109	92.2	0.064	15344.6	15203	16281	47.8	0.070	15221.3	15073	15937	60.8	0.057
A11	15983.8	15799	16555	88.1	0.047	16001.8	15877	16674	47.8	0.050	15890.6	15674	16431	59.3	0.048
A12	13693.5	13527	14460	89.8	0.068	13689.1	13574	14579	45.9	0.073	13654.2	13489	14132	56.8	0.047
B9	15047.1	14839	15785	90.3	0.063	15093.6	14883	16135	43.4	0.083	14988.4	14707	15438	50.7	0.049
B10	14983.4	14305	14908	89.6	0.040	14501.8	14347	15140	45.8	0.055	14425.3	14200	14872	56.0	0.047
B11	13896.0	13704	14556	88.6	0.061	13896.7	13716	14828	45.8	0.080	13800.6	13662	14380	54.0	0.052
B12	14925.9	14753	15591	87.8	0.056	14887.6	14768	15662	47.8	0.060	14845.7	14631	15484	57.7	0.057
C9	14253.2	13916	14904	96.1	0.069	14297.4	14132	14884	47.9	0.053	14186.3	13864	14853	61.0	0.070
C10	13613.2	13492	14232	92.5	0.054	13600.5	13496	14240	47.9	0.055	13590.2	13335	14147	62.8	0.060
C11	13165.4	13005	13826	68.4	0.062	13179.4	13128	13944	42.5	0.062	13083.5	12874	13555	49.4	0.052
C12	12917.6	12621	13607	78.9	0.076	12964.1	12799	13625	45.3	0.064	12746.8	12446	13371	55.8	0.073

Table 6

Table 7

Table 8

Fig.6

Fig.7

Fig.8

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文献	A	B	C	D	E	F	目标	模型	求解方法
[5]	Y	N	N	N	N	Y	成本最小	混合整数	Gurobi和Spreadsheet求解器
[6]	Y	N	N	N	N	N	成本最小	混合整数	遗传算法和Concorde求解器
[7]	Y	N	N	N	N	N	成本最小与客户满意度最大	混合整数	改进的非支配排序遗传算法-II
[8]	Y	N	N	N	N	N	成本最小	混合整数	分支定价算法
[9]	N	N	Y	N	N	Y	配送时间	混合整数	两阶段启发式算法
[10]	N	Y	Y	N	N	Y	成本最小	混合整数	启发式算法
[11]	N	Y	Y	N	N	Y	成本最小	混合整数	变邻域搜索算法
[13]	N	Y	Y	N	N	Y	成本最小	混合整数	启发式算法
[14]	N	Y	Y	Y	N	Y	成本最小	混合整数	自适应大邻域搜索算法
[21]	N	Y	Y	Y	Y	Y	成本最小	混合整数	改进的变邻域搜索算法
[29]	N	Y	Y	N	N	Y	迟到配送客户数最小	混合整数	局部搜索算法
[30]	N	Y	Y	Y	N	Y	成本最小	混合整数	混合多起点启发式算法
[31]	N	Y	Y	Y	Y	Y	成本最小	混合整数	自适应大邻域搜索算法
[32]	N	Y	Y	Y	Y	Y	成本最小	混合整数	贪婪随机自适应搜索算法
[33]	N	Y	Y	Y	Y	Y	成本最小	混合整数	基于聚类的变邻域搜索算法
本文	Y	Y	Y	Y	Y	Y	成本最小	混合整数	改进的变邻域搜索算法

集合	说明
$N$	封闭小区集合，$ n,{n^ * } $为任一小区，$n,{n^ * } \in N$
${K_1}$	一级无人车集合，${k_1}$为任一一级无人车，${k_1} \in {K_1}$
${K_2}$	二级无人车集合，${k_2}$为任一二级无人车，${k_2} \in {K_2}$，$ \left\|K_{2}\right\|=\|N\| $表示二级无人车数量等于封闭小区数量
$V$	节点集合，$V = \left\{ 0 \right\} \cup {V_1} \cup {V_2} \cup {V_3}$，其中，0为配送中心，${V_1}$为货物存放点集合，${V_2}$为货物转运点集合，${V_3}$为智能柜集合。$V = \left\{ 0 \right\} \cup \sum\limits_{n \in N} {{V_n}} $，${V_n}$为封闭小区$n$的节点集合，$ {V_n} = {V_{n1}} \cup {V_{n2}} \cup {V_{n3}} $，其中，${V_{n1}}$为封闭小区$n$的货物存放点集合，${V_{n2}}$为封闭小区$n$的货物转运点集合，${V_{n3}}$为封闭小区$n$的智能柜集合
$B$	二级无人车配送次序集合，二级无人车从某一货物存放点或货物转运点出发，完成配送任务返回到某一货物存放点或货物转运点视为一次配送。${B_{{k_2}}}$为二级无人车$ {k_2} $配送次序集合，$b$为任一一次配送任务，$b \in {B_{{k_2}}}$

参数	说明
$\left[ {{E_0},{L_0}} \right]$	配送中心的工作时间窗
${d_i}$	智能柜$i$的配货量
${Q_{{k_1}}}$	一级无人车${k_1}$的最大载重
${Q_{{k_2}}}$	二级无人车${k_2}$的最大载重
${t_{{k_1}}}$	一级无人车在货物转运点的最大等待时间
${l_{ij}}$	点$i$到点$j$的曼哈顿距离
${v_1}$	一级无人车的平均速度
${v_2}$	二级无人车的平均速度
$M$	一个比较大的正数
${t_1}$	一级无人车在货物存放点卸下单位货物所需时间
${t_2}$	一级与二级无人车在货物转运点转运单位货物所需时间
${t_3}$	二级无人车在货物存放点装载单位货物所需时间
${t_4}$	二级无人车在智能柜卸下货物所需时间
${c_1}$	单位一级无人车派遣成本
${c_2}$	一级无人车单位距离的运输成本
${c_3}$	货物存放点存放单位货物产生的存储成本
${c_4}$	二级无人车单位距离的运输成本

中间变量	说明
${q_{{k_1}i}}$	一级无人车${k_1}$在节点$i$的卸货量，$ i \in {V_1} \cup {V_2} $
${q_{{k_2}bi}}$	二级无人车${k_2}$第$b$次配送从点$i$出发时装载的货物量，$ i \in {V_1} \cup {V_2} \cup {V_3} $
$T_{{k_1}i}^a$	一级无人车${k_1}$抵达点$i$的时刻，$ i \in \left\{ 0 \right\} \cup {V_1} \cup {V_2} $
$T_{{k_1}i}^s$	一级无人车${k_1}$在点$i$的开始服务时刻，$ i \in \left\{ 0 \right\} \cup {V_1} \cup {V_2} $
$T_{{k_2}bi}^a$	二级无人车${k_2}$第$b$次配送抵达点$i$的时刻，$ i \in {V_n} $
$T_{{k_2}bi}^s$	二级无人车${k_2}$第$b$次配送在点$i$的开始服务时刻，$ i \in {V_n} $

决策变量	说明
${x_{{k_1}ij}}$	一级无人车${k_1}$是否由点$i$到点$j$，是为1，否为0，$ i,j \in \left\{ 0 \right\} \cup {V_1} \cup {V_2} $
${x_{{k_2}bij}}$	二级无人车${k_2}$第$b$次配送是否由点$i$到点$j$，是为1，否为0，$ i,j \in {V_1} \cup {V_2} \cup {V_3} $
${y_{{k_1}i}}$	一级无人车${k_1}$是否抵达点$i$，是为1，否为0，$ i \in \left\{ 0 \right\} \cup {V_1} \cup {V_2} $
${y_{{k_2}i}}$	二级无人车${k_2}$是否抵达点$i$，是为1，否为0，$ i \in {V_1} \cup {V_2} \cup {V_3} $

Two-echelon unmanned vehicles collaborative delivery routing optimization in closed and open scenes

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算例	方法	不可行解比例
算例	方法	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8
A2	T1	11912.3	11911.2	11912.9	11927.6	11942.7	11943.8	11976	11997.3
A2	T2	11911.6	11911.1	11912	11919.2	11919.4	11929.3	11949.7	11986.5
B1	T1	7631.6	7639.4	7620	7614.2	7639.5	7660.8	7679.1	7684.9
B1	T2	7624.7	7617.6	7615.9	7613	7629.4	7647.3	7662.8	7679
C4	T1	6786.5	6785.2	6782.7	6788.2	6797	6810.4	6854.9	6867.7
C4	T2	6785.7	6784	6781.9	6786.3	6799.5	6790.9	6824	6837.6
A6	T1	12318.5	12312.9	12268.1	12245	12239.4	12251.5	12276.9	12295.4
A6	T2	12246.3	12240.3	12238.7	12237.9	12236.3	12237.4	12254	12278.3
B5	T1	9544.7	9541.2	9518.9	9504.6	8496.3	9537.6	9543.1	9589.2
B5	T2	9542.3	9534	9510.3	9494.7	9494.7	9508.5	9539.8	9561
C8	T1	10348.6	10348.5	10348.5	10348	10347.3	10367.3	10383.5	10407.6
C8	T2	10347	10347	10345.2	10342.5	10343.6	10359.7	10379.4	10393
A10	T1	15344.2	15319.7	15288	15240.4	15221.3	15219.8	15237.2	15297.6
A10	T2	15321.5	15300.8	15244.8	15221	15219.3	15211.4	15224.3	15150.7
B11	T1	13804.1	13803.7	13803.7	13800.8	13797.6	13817.9	13899.7	13912.1
B11	T2	13803.4	13800.7	13799	13798.5	13791.7	13804.5	13819.7	13877
C12	T1	12777.3	12774	12768.1	12750.3	12753.8	12754.2	12779.6	12798.3
C12	T2	12770.4	12770.4	12763.5	12751.6	12746.8	12751.4	12762.8	12781

[1]	FANG Wei, ZHANG Lingzhi. Learning of fuzzy cognitive maps using multi-objective evolutionary algorithm [J]. Systems Engineering and Electronics, 2018, 40(2): 447-455.
[2]	WANG Xing-yuan, ZHANG Peng. Improved particle swarm optimization based on precise bionic metaphor [J]. Journal of Systems Engineering and Electronics, 2012, 34(7): 1484-1492.
[3]	BI Xiao-jun, WANG Yan-jiao. Niche artificial bee colony algorithm for multi-peak function optimization [J]. Journal of Systems Engineering and Electronics, 2011, 33(11): 2564-2568.

算例	S1		S2		S3		S4
算例	Avg	Time/s	Avg	Time/s	Avg	Time/s	Avg	Time/s
A1	10591.2	3.1	10552.1	3.2	10532.9	5.6	10514.5	5.7
A2	11996.2	4.0	11954.1	4.1	11912.3	5.3	11911.2	5.3
A3	10553.0	3.4	10523.8	3.5	10485.6	4.7	10457.2	5.2
A4	11326.2	2.9	11319.2	3.2	11307.2	4.5	11295.6	4.6
B1	7697.9	3.4	7692.2	3.5	7685.3	4.1	7614.2	4.2
B2	8118.1	4.0	8112.6	3.7	8107.3	4.9	8105.6	4.9
B3	8793.4	3.2	8792.4	3.2	8785.3	4.7	8784.2	4.7
B4	8896.2	3.5	8846.0	3.7	8844.6	4.9	8839.0	4.7
C1	6690.4	2.9	6546.3	3.0	6624.3	3.7	6534.1	4.0
C2	6100.1	2.6	6012.9	2.7	6082.0	4.3	6009.5	4.2
C3	7056.4	3.4	7003.5	3.5	6993.7	4.2	6989.6	4.4
C4	6817.5	4.1	6801.3	4.2	6788.1	5.5	6785.2	5.4
A5	12045.1	15.5	11874.3	16.8	11795.7	20.3	11788.9	20.3
A6	12556.9	17.2	12240.5	17.6	12321.7	21.5	12239.4	21.7
A7	12694.2	20.2	12455.9	21.0	12407.5	25.0	12367.2	25.0
A8	13001.4	15.5	12823.7	16.9	12746.1	20.0	12704.6	20.0
B5	9843.1	13.1	9604.5	13.3	9549.1	15.0	9496.3	16.0
B6	10771.3	14.4	10463.7	15.0	10398.1	17.7	10324.6	17.6
B7	9846.3	15.1	9678.9	15.7	9654.3	20.1	9654.6	20.7
B8	9997.3	14.2	9957.6	15.0	9789.6	19.7	9787.3	19.9
C5	8573.2	15.0	8357.2	15.3	8321.6	20.8	8320.5	20.6
C6	10201.0	14.1	9986.7	14.1	9777.9	19.5	9776.8	19.4
C7	10200.0	14.1	10000.7	14.2	9973.2	18.5	9945.6	18.5
C8	10683.5	14.2	10387.9	15.1	10348.6	18.6	10347.3	18.7
A9	15941.3	39.6	15476.2	41.7	15449.3	56.4	15446.2	56.4
A10	15647.2	13.2	15267.3	43.2	15350.6	59.7	15221.3	60.8
A11	15347.8	43.2	15964.3	43.2	15901.7	59.5	15890.6	59.3
A12	14018.6	42.0	13691.5	43.0	13756.1	56.8	13654.2	56.8
B9	15613.0	38.6	15045.9	39.7	14998.6	49.9	14988.4	50.7
B10	14864.2	38.7	14431.0	41.4	14467.2	54.7	14425.3	56.0
B11	14156.7	37.7	13856.0	40.1	13804.6	54.0	13800.6	54.0
B12	15154.3	42.2	14955.3	43.3	14856.1	57.8	14845.7	57.7
C9	14678.5	45.5	14254.6	47.3	14195.2	60.3	14186.3	61.0
C10	13971.2	45.5	13600.2	48.6	13661.8	64.0	13590.2	62.8
C11	13456.3	34.0	13112.6	34.5	13083.6	48.0	13083.5	49.4
C12	13004.1	35.8	12846.3	37.5	12785.2	52.3	12746.8	55.8