灵活多车场多类型的叫车接送问题的规划模型

陈可嘉; 方云飞; 骆佳艺

doi:10.3963/j.jssn.1674-4861.2021.03.013

灵活多车场多类型的叫车接送问题的规划模型

doi: 10.3963/j.jssn.1674-4861.2021.03.013

福州大学经济与管理学院福州 350116

基金项目:

国家自然科学基金项目 71601050

福建省社会科学规划项目 FJ2020B036

详细信息

通讯作者:
陈可嘉（1978—），博士，教授.研究方向：交通运输系统工程.E-mail：kjchen@fzu.edu.cn

中图分类号: U492.22
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出版历程
- 收稿日期: 2020-07-18

A Programming Model for the Flexible Multi-depot Heterogeneous Dial-a-ride Problems

School of Economics and Management, Fuzhou University, Fuzhou 350116, China

摘要

摘要: 为解决1种新的叫车接送问题（DARP）——灵活多车场多类型的叫车接送问题（MDHDARP-FD），建立以总行驶成本最小为目标的混合整数非线性规划数学模型。由于模型非常复杂，难以直接求解，对模型进行约束线性化、变量聚合和添加有效不等式等处理。在约束线性化过程中，非线性约束被重写为等价的线性约束；在变量聚合过程中，将具有相同性质的决策变量进行聚合，以减少变量数量；还引入了一些有效不等式来强化模型。从而获得1个解空间较小易于求解的新线性规划模型，并结合不同类型的车辆和乘客需求进行叫车接送问题的仿真测试。算例结果表明，相比于传统叫车接送问题，该模型能在满足复杂约束的基础上，得到1组实现乘客接送需求的最优车辆路线，并且能有效降低总行驶成本、车辆行驶时间和求解时间，总行驶成本在不同类型的DARP上可平均降低1.51%~6.69%，证实了叫车接送问题的总行驶成本可以随着灵活车场的引入而减少。
- 交通规划 /
- 叫车接送问题 /
- 灵活车场 /
- 混合整数非线性规划 /
- 模型重构
Abstract: A mathematical model of mixed-integer non-linear programming is firstly constructed to solve a new dial-a-ride problem(DARP), the multi-depot heterogeneous dial-a-ride problem with flexible depots(MDHDARP-FD), as well as minimize the total driving cost.Due to the complexity of the proposed model, some strategies of linearizing constraints, aggregating variables, and strengthening inequalities are performed to reformulate the non-linear model.During linearizing constraints, non-linear constraints are rewritten as equivalent linear constraints.Aggregating variables, decision variables with the same properties, are aggregated to reduce the number of variables, with strengthened inequalities and valid inequalities introduced to strengthen the model.As a result, a new linear programming model with a small solution space is obtained and can be tractable.Then the dial-a-ride problem is simulated by combining different vehicles and passengers'demands.Simulation results show that compared with the traditional dial-a-ride problems, the model can obtain a group of optimal vehicle routes respecting the complex constraints and reduce the total driving cost, vehicle driving time, and solution time.The total driving cost can be reduced by 1.51%to 6.69%for different DARPs on an average, verifying that the total driving cost of dial-a-ride problems can be reduced with the introduced flexible depots.
- transportation planning /
- dial-a-ride problem /
- flexible depots /
- mixed-integer nonlinear programming /
- model reformulation

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图 1 算例Ea4-16的车辆接送路线图

Figure 1. Route for vehicles of Example Ea4-16

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表 1 算例Ea4-16的乘客需求信息

Table 1. Passengers'demand information of Example Ea4-16

序号	横坐标	纵坐标	服务时间/min	最大乘车时间/ min	位置需求				时间窗下界/min	时间窗上界/min
序号	横坐标	纵坐标	服务时间/min	最大乘车时间/ min	常规座位(员工座位）	常规座位(病人座位）	担架位	轮椅位	时间窗下界/min	时间窗上界/min
1	6.267	0.981	3	30	0	0	0	1	0	1 440
2	—4.718	6.925	3	30	0	1	0	0	0	1 440
3	3.254	7.621	3	30	0	1	0	0	0	1 440
4	9.654	2.799	3	30	0	0	1	0	0	1 440
5	8.575	2.701	3	30	0	1	0	0	0	1 440
6	—9.732	—8.314	3	30	0	1	0	0	0	1 440
7	9.937	4.969	3	30	0	0	0	1	0	1 440
8	—4.604	6.053	3	30	0	0	0	1	0	1 440
9	—5.029	5.070	3	30	0	1	0	0	53	68
10	—9.264	—7.979	3	30	0	1	0	0	105	120
11	8.738	—9.966	3	30	0	1	0	0	74	89
12	—3.420	7.617	3	30	0	0	1	0	3	18
13	—5.865	—4.571	3	30	1	1	0	0	172	187
14	6.300	5.322	3	30	0	0	0	1	61	76
15	—2.219	—7.605	3	30	0	1	0	0	179	194
16	—1.654	—7.838	3	30	0	1	0	0	154	169
17	—1.548	—4.124	3	0	0	0	0	—1	138	153
18	—4.818	6.259	3	0	0	—1	0	0	125	140
19	—7.389	0.376	3	0	0	—1	0	0	95	110
20	6.070	—0.549	3	0	0	0	—1	0	195	210
21	—1.663	6.171	3	0	0	—1	0	0	103	118
22	—0.665	—1.272	3	0	0	—1	0	0	103	118
23	—5.005	0.918	3	0	0	0	0	—1	142	157
24	6.200	—7.679	3	0	0	0	0	—1	157	172
25	5.575	—7.408	3	0	0	—1	0	0	0	1 440
26	—7.754	3.535	3	0	0	—1	0	0	0	1 440
27	2.111	1.439	3	0	0	—1	0	0	0	1 440
28	1.706	—1.733	3	0	0	0	—1	0	0	1 440
29	—1.796	9.104	3	0	—1	—1	0	0	0	1 440
30	—5.500	2.353	3	0	0	0	0	—1	0	1 440
31	7.323	—7.149	3	0	0	—1	0	0	0	1 440
32	—6.413	0.367	3	0	0	—1	0	0	0	1 440

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表 2 算例Ea4-16的模型求解结果

Table 2. Model solution results of Example Ea4-16

约束条件总数	变量总数	整数变量	运行时间/s	总行驶成本
20 385	3 852	3 780	1 661.44	257.00

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表 3 算例Ea4-16的最优接送路线

Table 3. Optimal route of Example Ea4-16

车辆序号	出发车场坐标	最优接送路线	返回车场坐标
1	[-5，-5]	-6-22-16-13-32-29-	[-5，5]
2	[5，5]	-14-3-30-19-10-26-2-18-8-24-4-20-	[5，-5]
3	[-5，5]	-12-28-9-25-11-7-27-23-	[-5，5]
4	[5，-5]	-5-1-21-17-15-31-	[5，-5]

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表 4 系列算例求解结果及比较

Table 4. Comparison results of examples

Dataset	Instance	MDHDARP-FD	CPU/s	MD-H-DARP	G1/%	H-DARP	G2/%	2-index-DARP	G3/%	DARP	G4/%
	Ua2-16	284.18	1.38	284.18	0.00	294.25	3.42	300.17	5.33	294.25	3.42
	Ua2-20	333.57	8.67	343.43	2.87	344.83	3.27	345.74	3.52	344.83	3.27
	Ua2-24	426.75	28.14	427.17	0.10	431.12	1.01	448.07	4.76	431.12	1.01
	Ua3-18	279.05	31.89	289.67	3.67	300.48	7.13	316.78	11.91	300.48	7.13
Dataset U	Ua3-24	344.60	1 062.13	348.30	1.06	344.83	0.07	346.97	0.68	344.83	0.07
	Ua3-30	468.78	45 053	469.16	0.08	494.85	5.27	501.68	6.56	472.17	0.72
	Ua4-16	252.03	3 231.39	262.44	3.97	282.68	10.84	282.68	10.84	282.68	10.84
	Ua4-24	348.26	78 368.40	355.72	2.10	375.02	7.14	386.38	9.87	359.52	3.13
	AVG	342.15	15 973.13	347.51	1.54	358.51	4.56	366.06	6.69	353.734	3.28
	Ea2-16	327.67	2.58	327.67	0.00	331.13	1.04	331.13	1.04	331.13	1.04
	Ea2-20	335.76	8.62	345.59	2.84	347.03	3.25	347.89	3.49	347.03	3.25
	Ea2-24	443.08	56.03	445.88	0.63	450.21	1.58	461.89	4.07	451.27	1.81
	Ea3-18	279.05	41.66	289.67	3.67	300.63	7.18	316.78	11.91	300.63	7.18
Dataset E	Ea3-24	344.67	1 039.07	348.61	1.13	344.91	0.07	347.05	0.69	345.04	0.11
	Ea3-30	471.43	28 200.70	471.43	0.00	500.53	5.81	501.68	6.03	476.28	1.02
	Ea4-16	257.00	1 661.44	262.44	2.07	285.99	10.14	288.7	10.98	286.07	10.16
	Ea4-24	354.97	88 246.60	365.54	2.89	380.48	6.70	386.38	8.13	365.65	2.92
	AVG	351.70	14 907.09	357.10	1.51	367.61	4.33	372.69	5.79	362.88	3.09

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