Lagrangian relaxation algorithm for the truck scheduling problem with products time window constraint in multi-door cross-dock

Binghai Zhou; Yuanrui Lei; Shi Zong

doi:10.3934/jimo.2021151

Article Contents

2022, Volume 18, Issue 6: 4129-4149. Doi: 10.3934/jimo.2021151

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Lagrangian relaxation algorithm for the truck scheduling problem with products time window constraint in multi-door cross-dock

Institute of Industrial Engineering, School of Mechanical Engineering, Tongji University, Shanghai 201804, China

^* Corresponding author: Binghai Zhou
^* Corresponding author: Binghai Zhou

Received: December 2020

Revised: June 2021

Early access: September 2021

Published: September 2022

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Abstract

Cross-docking is a kind of process that products are unloaded in front of the inbound doors, consolidated based on the downstream demand, and then directly transferred to the outbound doors without a long storage process during the transportation. In this paper, a multi-door cross-dock truck scheduling problem is investigated in which the scheduling and sequencing assignment of trucks need to be considered, with the objectives of minimizing the inner transportation cost in the cross-dock and the total truck waiting cost. The major contribution of this paper is that a novel product-related time window constraint and the temporary storage area are firstly introduced to adapt to different physical conditions of goods considering real-world requirements. Then, a Lagrangian relaxation algorithm is proposed which aims to decompose the relaxed problem into several easy-to-be-solved sub-problems. Besides, a subgradient algorithm is used at each iteration to further deal with these sub-problems. Finally, theory analysis and simulation experiments of different problem scales are carried out during the comparison with a Greedy algorithm to evaluate the performance of the proposed algorithm. Results indicate that the Lagrangian relaxation algorithm is able to achieve more satisfactory near-optimal solutions within an acceptable time.

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Mathematics Subject Classification: Primary: 90B35; Secondary: 90C11.

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Figure 1. The scheduling process at cross-dock

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Figure 2. Several values of %gap (group2) based on the different value of slackness

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Figure 3. Values of %gap and %dev in group 2

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Table 1. Computer results for problems in group 1, b = 0.98, 0.95, and 0.90

			p=0.98						p=0.95						p=0.90
				L			G			L			G			L			G
I	#t×#d	s	IP	LR	%gap	CPU	UB	%dev	IP	LR	%gap	CPU	UB	%dev	IP	LR	%gap	CPU	UB	%dev
1		15	1.683	1.683	5.32	0.55	1.815	10.43	1.98	1.98	4.97	0.56	2.187	10.43	1.643	1.643	5.27	0.46	1.99	11.99
2		20	1.701	1.701	4.82	0.5	1.89	8.37	2.1	2.1	4.38	0.33	2.278	8.45	1.785	1.785	5.39	0.42	1.754	8.03
3	4×2	30	1.729	1.729	3.44	0.53	2.004	15.27	2.059	2.059	3.55	0.42	2.358	14.54	1.997	1.997	3.59	0.51	1.769	14.69
4		15	2.56	2.56	9.55	0.66	2.45	14.81	2.56	2.56	9.27	0.68	2.816	14.52	2.765	2.765	9.83	0.84	3.013	14.96
5		20	2.26	2.26	8.15	0.8	3.064	13.02	2.659	2.659	7.03	0.8	3.225	12.76	2.872	2.872	6.89	0.8	2.838	13.4
6	5×3	30	2.498	2.498	10.55	0.65	2.648	12.76	2.602	2.602	9.02	0.8	3.152	12.76	2.862	2.862	11	0.96	3.089	12.89
7		15	2.709	2.709	6.58	1.59	3.954	26.35	3.078	3.078	6.33	1.61	4.163	26.62	2.924	2.924	6.46	1.07	3.83	27.95
8		20	3.108	3.108	4.32	1.64	3.533	25	3.205	3.205	4.32	1.06	4.257	24.27	2.66	2.66	4.62	1.33	4.087	23.06
9	6×3	30	2.495	2.495	7.55	1.6	3.735	26.32	3.08	3.08	6.29	1.02	4.016	26.06	2.957	2.957	6.54	1.27	4.257	27.1
10		15	3.31	3.31	4.76	1.92	4.153	13.37	3.755	3.988	4.21	2.35	4.72	13.37	4.068	4.068	4.04	1.88	4.153	12.84
11		20	3.425	3.624	6.35	1.93	4.337	14.81	3.89	3.897	4.96	1.91	4.518	15.11	4.155	4.17	6.05	2.38	3.433	15.87
12	7×3	30	3.257	3.471	3.69	3.13	3.747	21.32	3.572	3.732	3.55	2.66	4.625	21.98	3.375	3.471	3.41	3.19	4.163	22.2
13		15	3.18	3.381	7.79	5.86	4.848	29.61	3.58	3.636	6.04	7.1	5.051	30.53	3.09	3.09	7.61	7.1	3.788	30.84
14		20	3.155	3.301	6.23	4.89	4.847	29.92	3.42	3.628	6.23	7.41	5.049	30.53	3.171	3.301	6.17	4.94	3.786	29
15	7×4	30	3.178	3.204	6.83	5.38	4.262	32.8	3.595	3.641	5.38	8.41	5.014	31.24	3.285	3.386	6.62	8.41	4.061	32.49
16		15	3.65	3.826	5.76	4.4	5.089	24.08	4.058	4.159	5.7	4.27	5.472	24.08	3.588	3.701	5.81	5.34	5.472	25.28
17		20	3.425	3.425	5.58	5.79	5.329	21.76	4.117	4.228	5.03	4.45	5.438	22.91	4.489	4.608	5.53	5.57	4.894	23.14
18	8×3	30	3.785	3.85	6.41	8.54	4.745	22.79	4.095	4.184	5.57	10.05	5.214	23.49	4.258	4.435	7.19	8.37	4.432	24.19
19		15	3.455	3.485	4.57	8.09	5.375	35.21	3.986	4.1	4.76	7.03	5.973	36.3	3.698	3.772	4.57	7.03	5.973	35.94
20		20	3.576	3.678	5.81	6.36	4.802	34.09	4.168	4.18	4.76	7.64	5.856	33.42	3.259	3.344	5.09	9.17	6.09	33.09
21	8×4	30	3.385	3.428	6.14	9.16	5.446	32.42	4.15	4.18	4.76	11.33	5.856	33.42	3.857	3.929	4.52	11.33	5.739	33.09
Instance is short for $i$, $t$ for trucks, $d$ for doors, $s$ for the slackness of temporary storage capacity (%), $L$ for the Lagrangian relaxation algorithm, $G$ for the Greedy algorithm. The value of $IP, LP$ and $UB$ is in $10^5$.

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Table 2. Computer results for problems in group 2 (slackness = 10)

Instances	#trucks×#doors	Lagrangian relaxation algorithm			Greedy algorithm
		LR	%gap	CPU time	UB	%dev
1	9×3	4.759	5.21	18	5.965	18.8
2	9×4	4.856	5.28	19.76	6.382	24.5
3	9×5	4.661	4.38	30.13	6.8	39.5
4	10×3	5.527	5.25	42.36	6.876	14.43
5	10×4	5.595	4.76	15.57	7.014	21.56
6	10×5	5.055	5.75	44.57	7.609	43.35
7	11×5	6.017	7.86	77.65	8.618	31.41
8	11×6	5.997	9.09	80.49	9.149	38.69
9	15×6	8.223	9.09	177.84	12.43	37.39
10	15×7	8.21	7.42	124.02	13.09	47.64
11	20×10	10.89	9.91	265.06	20.73	71.41

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Table 3. Computer results for problems in group 2 (slackness = 15)

Instances	#trucks×#doors	Lagrangian relaxation algorithm			Greedy algorithm
		LR	%gap	CPU time	UB	%dev
1	9×3	4.742	5.17	24.5	5.892	19.3
2	9×4	4.813	4.97	22.69	6.382	24.5
3	9×5	4.559	5.65	36.25	6.66	40.72
4	10×3	5.456	5.09	40.71	6.674	16.1
5	10×4	5.195	4.76	31.97	7.142	30.92
6	10×5	5.355	5.3	42.97	7.609	35.32
7	11×5	6.017	6.54	55.32	8.618	33.86
8	11×6	5.6	7.45	91.46	9.149	41.4
9	15×6	8.403	8.26	125.77	12.43	35.68
10	15×7	8.237	6.65	81.64	13.18	49.35
11	20×10	10.64	8.26	164.22	20.69	68.47

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Table 4. Computer results for problems in group 2 (slackness = 20)

Instances	#trucks×#doors	Lagrangian relaxation algorithm			Greedy algorithm
		LR	%gap	CPU time	UB	%dev
1	9×3	5.035	4.53	18.16	6.003	13.12
2	9×4	4.764	5.32	27.23	6.173	29.83
3	9×5	4.758	6.02	26.69	6.899	34.3
4	10×3	5.466	4.92	29.03	6.674	16.1
5	10×4	5.087	5.56	27.07	7.142	32.45
6	10×5	5.075	4.9	40.41	7.609	42.79
7	11×5	6.037	7.41	62.67	8.618	32.18
8	11×6	5.777	7.41	62.09	9.139	46.64
9	15×6	8.012	7.41	133.29	12.4	36.94
10	15×7	8.045	5.82	159.65	13.22	45.71
11	20×10	10.85	7.41	226.27	20.73	66.9

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Table 5. Computer results for problems in group 2 (slackness = 30)

Instances	#trucks×#doors	Lagrangian relaxation algorithm			Greedy algorithm
		LR	%gap	CPU time	UB	%dev
1	9×3	4.76	4.8	17.01	5.965	19.3
2	9×4	4.647	5.08	22.76	6.108	30.38
3	9×5	4.684	3.9	22.59	6.8	39.5
4	10×3	5.943	4.9	35.42	6.344	6.74
5	10×4	5.375	4.76	42.1	7.082	26.54
6	10×5	5.315	4.76	47.71	7.609	36.34
7	11×5	6.057	4.76	50.29	8.618	35.51
8	11×6	5.817	6.54	77.44	9.149	46.99
9	15×6	8.221	6.54	114.18	12.37	40.67
10	15×7	8.527	4.29	148.43	13.3	47.33
11	20×10	10.79	5.66	302.54	23.68	70.76

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