INTEGER FORMULATIONS FOR THE INTEGRATED VEHICLE ROUTING PROBLEM WITH TWO-DIMENSIONAL PACKING CONSTRAINTS

Silva, Lorrany Cristina da; Queiroz, Thiago Alves de; Toledo, Franklina Maria Bragion de

doi:10.1590/0101-7438.2022.042.00248686

Lorrany Cristina da Silva

Institute of Mathematics and Technology, Federal University of Catalão, 75704-020, Catalão-GO, Brazil. - E-mail: lorranyc@usp.br

https://orcid.org/0000-0003-2674-3366

Thiago Alves de Queiroz ^**Corresponding author

Institute of Mathematics and Technology, Federal University of Catalão, 75704-020, Catalão-GO, Brazil. - E-mail: lorranyc@usp.br

Franklina Maria Bragion de Toledo

Institute of Mathematics and Computer Science, University of São Paulo, 13566-590, São Carlos-SP, Brazil. - E-mail: fran@icmc.usp.br

https://orcid.org/0000-0003-2823-7600

*Corresponding author

Group	Authors	Problem	Loading subproblem	Number of routes	Single routes	Cost of edges	Method
Exact	Iori et al. (2007)IORI M, SALAZAR-GONZÁLEZ JJ & VIGO D. 2007. An exact approach for the vehicle routing problem with two-dimensional loading constraints. Transportation Science , 41(2): 253-264.	2L-CVRP	Sequential	= \|K\|	No	Integer	B&C
	Azevedo et al. (2009)AZEVEDO BLP, HOKAMA P, MIYAZAWA FK & XAVIER EC. 2009. A Branch-and-Cut Approach for the Vehicle Routing Problem with Two-dimensional Loading Constraints. In: XLI Simpósio Brasileiro de Pesquisa Operacional. pp. 2491-2502. Brazil.	2L-CVRP	Sequential and Unrestricted	= \|K\|	No	Integer	B&C
	Hokama et al. (2016)HOKAMA P, MIYAZAWA FK & XAVIER EC. 2016. A branch-and-cut approach for the vehicle routing problem with loading constraints. Expert Systems with Applications, 47: 1-13.	2L-CVRP	Sequential	= \|K\|	No	Integer	B&C
	Côté et al. (2017)CÔTÉ JF, GUASTAROBA G & SPERANZA MG. 2017. The value of integrating loading and routing. European Journal of Operational Research, 257(1): 89-105.	2L-CVRP	Sequential	≥ \|K\|	Yes	Integer	B&C
Heuristics	Gendreau et al. (2008)GENDREAU M, IORI M, LAPORTE G & MARTELLO S. 2008. A Tabu search heuristic for the vehicle routing problem with two-dimensional loading constraints. Networks, 51(1): 4-18.	2L-CVRP	Sequential	= \|K\|	No Yes	Integer Real	Tabu Search
		2L-CVRP	Sequential and Unrestricted	≤ \|K\|	No Yes	Integer Real	Tabu Search
	Fuellerer et al. (2009)FUELLERER G, DOERNER KF, HARTL RF & IORI M. 2009. Ant colony optimization for the two-dimensional loading vehicle routing problem. Computers & Operations Research , 36(3): 655-673.	2L-CVRP	Sequential	= \|K\|	No Yes	Integer Real	Ant Colony Algorithm
		2L-CVRP	Sequential and Unrestricted	≤ \|K\|	No Yes	Integer Real	Ant Colony Algorithm
	Zachariadis et al. (2009)ZACHARIADIS EE, TARANTILIS CD & KIRANOUDIS CT. 2009. A Guided Tabu Search for the Vehicle Routing Problem with two-dimensional loading constraints. European Journal of Operational Research , 195(3): 729-743.	2L-CVRP	Sequential and Unrestricted	≤ \|K\|	Yes	Real	Guided Tabu Search
	Strodl et al. (2010)STRODL J, DOERNER KF, TRICOIRE F & HARTL RF. 2010. On Index Structures in Hybrid Metaheuristics for Routing Problems with Hard Feasibility Checks: An Application to the 2- Dimensional Loading Vehicle Routing Problem. In: BLESA MJ, BLUM C, RAIDL G, ROLI A & SAMPELS M (Eds.), Proceedings of the 7th International Workshop on Hybrid Metaheuristics, vol. 6373. pp. 160-173. Springer Berlin Heidelberg.	2L-CVRP	Unrestricted	≤ \|K\|	Yes	Real	Variable Neighborhood Search
	Leung et al. (2010)LEUNG SCH, ZHENG J, ZHANG D & ZHOU X. 2010. Simulated annealing for the vehicle routing problem with two-dimensional loading constraints. Flexible Services and Manufacturing Journal, 22(1-2): 61-82.	2L-CVRP	Sequential and Unrestricted	≤ \|K\|	Yes	Real	Simulated Annealing
	Duhamel et al. (2011)DUHAMEL C, LACOMME P, QUILLIOT A & TOUSSAINT H. 2011. A multi-start evolutionary local search for the two-dimensional loading capacitated vehicle routing problem. Computers & Operations Research , 38(3): 617-640.	2L-CVRP	Unrestricted	≤ \|K\|	Yes	Real	GRASP
	Leung et al. (2011)LEUNG SCH, ZHOU X, ZHANG D & ZHENG J. 2011. Extended guided tabu search and a new packing algorithm for the two-dimensional loading vehicle routing problem. Computers & Operations Research , 38(1): 205-215.	2L-CVRP	Sequential and Unrestricted	≤ \|K\|	Yes	Real	Extended Guided Tabu Search
	Shen & Murata (2012)SHEN Y & MURATA T. 2012. Pick-up Scheduling of Two-dimensional Loading in Vehicle Routing Problem by using GA. In: Proceedings of the 2012 International MultiConference of Engineers and Computer Scientists (IMECS), vol. 2. pp. 1532-1537.	2L-CVRP	Sequential	≤ \|K\|	-	Real	Genetic algorithm
	Bin et al. (2013)BIN W, HONG C & ZHI-YONG C. 2013. Artificial bee colony algorithm for two-dimensional loading capacitated vehicle routing problem. In: 2013 International Conference on Management Science and Engineering (ICMSE). pp. 406-412.	2L-CVRP	Sequential and Unrestricted	≤ \|K\|	Yes	Real	Artificial Bee Colony
	Zachariadis et al. (2013)ZACHARIADIS EE, TARANTILIS CD & KIRANOUDIS CT. 2013. Integrated distribution and loading planning via a compact metaheuristic algorithm. European Journal of Operational Research , 228(1): 56-71.	2L-CVRP	Sequential and Unrestricted	≤ \|K\|	Yes	Real	Local Search
	Domínguez et al. (2014)DOMÍNGUEZ O, JUAN AA & FAULIN J. 2014. A biased-randomized algorithm for the two- dimensional vehicle routing problem with and without item rotations. International Transactions in Operational Research, 21(3): 375-398.	2L-CVRP	Unrestricted	≤ \|K\|	Yes	Real	Multistart Biased-Randomized Algorithm
	Wei et al. (2015)WEI L, ZHANG Z, ZHANG D & LIM A. 2015. A variable neighborhood search for the capacitated vehicle routing problem with two-dimensional loading constraints. European Journal of Operational Research , 243(3): 798-814.	2L-CVRP	Sequential and Unrestricted	≤ \|K\|	Yes	Real	Variable Neighborhood Search
	Wei et al. (2018)WEI L, ZHANG Z, ZHANG D & LIM A. 2018. A simulated annealing algorithm for the capacitated vehicle routing problem with two-dimensional loading constraints. European Journal of Operational Research , 265: 843-859.	2L-CVRP	Sequential and Unrestricted	≤ \|K\|	Yes	Real	Simulated Annealing
Variants	Malapert et al. (2008)MALAPERT A, GUERET C, JUSSIEN N, LANGEVIN A & ROUSSEAU LM. 2008. Two-dimensional pickup and delivery routing problem with loading constraints. In: Proceedings of the First CPAIOR Workshop on Bin Packing and Placement Constraints (BPPC’08). pp. 1-46. 4. Paris, France.	2L-PDP	Sequential and Unrestricted	-	-	-	Constraint Programming
	Munoz (2011)MUNOZ AL. 2011. Solución al problema de ruteo de vehículos con restricciones de capacidad y reordenamiento de carga en los sitios de demanda. Master’s thesis. Escuela de Ingenieria, Pontificia Universidad Catolica de Chile. Santiago, Chile.	2L-CVRP-H	Sequential and Unrestricted	= \|K\|	Yes	Real	Mixed-integer Programming
	Khebbache-Hadji et al. (2013)KHEBBACHE-HADJI S, PRINS C, YALAOUI A & REGHIOUI M. 2013. Heuristics and memetic algorithm for the two-dimensional loading capacitated vehicle routing problem with time windows. Central European Journal of Operations Research , 21(2): 307-336.	2L-CVRPTW	Unrestricted	≤ \|K\|	No	Real	Memetic Algorithm
	Martínez & Amaya (2013)MARTÍNEZ L & AMAYA CA. 2013. A vehicle routing problem with multi-trips and time windows for circular items. Journal of the Operational Research Society, 64(11): 1630-1643.	VRPM-TW-CL	Unrestricted	= \|K\|	No	Real	Tabu Search
	Domínguez et al. (2016)DOMÍNGUEZ O, JUAN AA, BARRIOS B, FAULIN J & AGUSTIN A. 2016. Using biased randomization for solving the two-dimensional loading vehicle routing problem with heterogeneous fleet. Annals of Operations Research , 236(2): 383-404.	2L-HFVRP	Unrestricted	≤ \|K\|	Yes	Real	Multistart Biased-Randomized Algorithm
	Zachariadis et al. (2016)ZACHARIADIS EE, TARANTILIS CD & KIRANOUDIS CT. 2016. The Vehicle Routing Problem with Simultaneous Pick-ups and Deliveries and Two-Dimensional Loading Constraints. European Journal of Operational Research , 251(2): 369-386.	2L-SPD	Sequential and Unrestricted	≤ \|K\|	-	Real	Local Search
	Côté et al. (2020)CÔTÉ JF, GENDREAU M & POTVIN JY. 2020. The Vehicle Routing Problem with Stochastic Two-Dimensional Items. Transportation Science, 54(2): 453-469.	S2L-CVRP	Sequential	= \|K\|	No	Integer	B&C
	Ferreira et al. (2021)FERREIRA KM, QUEIROZ TA & TOLEDO FMB. 2021. An exact approach for the green vehicle routing problem with two-dimensional loading constraints and split delivery. Computers & Operations Research , 136: 105452.	2L-SDVRP	Sequential	≤ \|K\|	Yes	Integer	B&C
		G2L-CVRP
		G2L-SDVRP

Instance					Complete model			Two-index model
Instance					Complete model						Cuts	Time	Cuts	Time	Cuts
Name	Class	n	\|M\|	\|K\|	Time	Sol.	GAP	Time	Sol.	GAP	CVRPSEP	2OPP^W;H	2OPP^W;H	2OPP	2OPP
E016-03m	1	15	15	3	268.44	278.73	0.00	0.07	278.73	0.00	136	0.00	0	0.00	0
	2	15	24	3	3,600.00	285.50	19.09	4.11	278.73	0.00	185	0.07	3	3.75	0
	3	15	31	3	3,600.00	-	-	59.96	284.52	0.00	301	1.28	12	57.82	0
	4	15	37	4	3,600.00	282.95	19.16	34.44	282.95	0.00	328	0.48	16	32.87	0
	5	15	45	4	3,600.00	280.60	21.70	155.30	278.73*	0.00	215	17.19	1	137.36	0
E016-05m	1	15	15	5	547.52	334.96	0.00	0.09	334.96	0.00	203	0.00	0	0.00	0
	2	15	25	5	3,600.00	334.96	14.86	1.04	334.96	0.00	203	0.00	0	0.84	0
	3	15	31	5	3,600.00	352.16	22.43	3.00	352.16	0.00	471	0.04	5	2.20	0
	4	15	40	5	3,600.00	334.96	16.77	3.88	334.96	0.00	203	0.03	0	3.56	0
	5	15	48	5	3,600.00	334.96	14.87	0.70	334.96	0.00	203	0.02	0	0.34	0
E021-04m	1	20	20	4	3,600.00	364.45	20.71	0.55	358.40	0.00	579	0.00	0	0.00	0
	2	20	29	5	3,600.00	387.70	32.49	79.87	387.70	0.00	2,608	0.43	264	18.93	0
	3	20	46	5	3,600.00	-	-	247.18	394.72	0.00	2,410	12.42	276	169.38	0
	4	20	44	5	3,600.00	379.14	30.57	131.20	362.41*	0.00	369	1.77	4	128.55	0
	5	20	49	5	3,600.00	377.23	30.14	7.19	358.40	0.00	340	0.08	1	5.89	0
E021-06m	1	20	20	6	3,600.00	430.88	19.81	1.30	430.88	0.00	938	0.00	0	0.01	0
	2	20	32	6	3,600.00	430.88	29.41	3.39	430.88	0.00	938	0.02	0	1.88	0
	3	20	43	6	3,600.00	445.25	29.56	8.25	430.88	0.00	938	0.06	0	6.52	0
	4	20	50	6	3,600.00	447.37	32.17	8.69	447.37	0.00	1,341	0.07	3	3.71	0
	5	20	62	6	3,600.00	440.94	31.26	4.17	430.88	0.00	938	0.02	0	2.77	0
E022-04g	1	21	21	4	3,600.00	383.52	20.23	0.11	375.28	0.00	114	0.00	0	0.00	0
	2	21	31	4	3,600.00	376.51	27.07	9.38	375.28	0.00	143	0.06	3	8.96	0
	3	21	37	4	3,600.00	402.46	30.18	5.37	381.69	0.00	121	0.14	3	4.77	0
	4	21	41	4	3,600.00	383.87	28.24	2.92	383.87	0.00	174	0.04	2	2.48	0
	5	21	57	5	3,600.00	375.28	25.15	0.70	375.28	0.00	71	0.01	0	0.47	0
E022-06m	1	21	21	6	3,600.00	498.16	29.76	0.17	495.85	0.00	263	0.00	0	0.00	0
	2	21	33	6	3,600.00	499.08	37.60	2.76	495.85	0.00	303	0.02	1	2.36	0
	3	21	40	6	3,600.00	504.68	38.99	114.45	498.16	0.00	610	0.38	5	112.72	1
	4	21	57	6	3,600.00	602.94	50.14	27.04	498.32	0.00	783	0.34	7	24.24	0
	5	21	56	6	3,600.00	495.85	37.07	1.46	495.85	0.00	263	0.01	0	0.70	0
E023-03g	1	22	22	3	1,014.34	568.56	0.00	0.19	568.56	0.00	45	0.00	0	0.01	0
	2	22	32	5	3,600.00	750.64	34.69	3,600.00	750.64*	22.39	15,424	28.22	2,722	168.16	0
	3	22	41	5	3,600.00	729.97	31.76	3,600.00	729.97*	17.31	2,286	113.10	1,284	1,836.35	0
	4	22	51	5	3,600.00	768.78	36.36	3,600.00	727.34*	15.19	1,662	154.76	1,004	3,029.68	1
	5	22	55	6	3,600.00	725.57	32.69	3,600.00	661.41*	9.95	425	1,069.23	191	2,061.02	0
E023-05s	1	22	22	5	1,034.22	568.56	0.00	0.11	568.56	0.00	45	0.00	0	0.01	0
	2	22	29	5	3,600.00	720.77	31.08	456.69	674.55	0.00	3,856	3.08	319	83.41	0
	3	22	42	5	3,600.00	771.89	37.02	3,600.00	756.04*	14.85	7,951	123.32	4,380	392.13	1
	4	22	48	5	3,600.00	730.63	31.35	3,600.00	708.47*	10.25	3,532	195.39	2,089	2,388.38	0
	5	22	52	6	3,600.00	642.22	23.57	3,600.00	612.79*	4.84	158	1,013.08	45	2,006.30	0

Instance					Complete model			Two-index model
Instance					Complete model						Cuts	Time	Cuts	Time	Cuts
Name	Class	n	\|M\|	\|K\|	Time	Sol.	GAP	Time	Sol.	GAP	CVRPSEP	2OPP^W;H	2OPP^W;H	2OPP	2OPP
E026-08m	1	25	25	8	3,600.00	620.25	32.10	0.24	607.65	0.00	349	0.00	0	0.00	0
	2	25	40	8	3,600.00	624.94	35.28	2.59	607.65	0.00	349	0.00	0	2.23	0
	3	25	61	8	3,600.00	-	-	2.33	607.65	0.00	349	0.00	0	1.91	0
	4	25	63	8	3,600.00	-	-	16.31	625.10	0.00	2,222	0.07	1	5.04	0
	5	25	91	8	3,600.00	-	-	1.30	607.65	0.00	349	0.00	0	0.74	0
E030-03g	1	29	29	3	3,600.00	535.80	35.98	3.79	535.80	0.00	619	0.00	0	0.01	0
	2	29	43	6	3,600.00	-	-	3,600.00	787.50	28.46	20,917	32.21	3,028	315.91	0
	3	29	49	6	3,600.00	659.18	52.81	3,600.00	647.67*	18.83	8,724	146.34	2,269	1,229.01	1
	4	29	72	7	3,600.00	-	-	3,600.00	-	-	-	-	-	-	-
	5	29	86	7	3,600.00	762.25	60.12	3,600.00	-	-	-	-	-	-	-
E030-04s	1	29	29	4	3,600.00	505.01	37.85	0.14	505.01	0.00	75	0.00	0	0.01	0
	2	29	43	6	3,600.00	880.73	64.98	3,600.00	-	-	-	-	-	-	-
	3	29	62	7	3,600.00	-	-	3,600.00	1,020.76*	47.64	13,987	159.75	3,275	1,846.74	0
	4	29	74	7	3,600.00	-	-	3,600.00	-	-	-	-	-	-	-
	5	29	91	7	3,600.00	-	-	3,600.00	804.92*	36.09	1,255	772.81	378	2,501.07	0
E031-09h	1	30	30	9	3,600.00	632.03	40.70	530.33	610.00	0.00	10,508	0.01	0	0.01	0
	2	30	50	9	3,600.00	-	-	3,600.00	657.57	8.99	28,511	0.08	12	18.67	0
	3	30	56	9	3,600.00	645.02	43.67	801.55	610.00	0.00	12,379	0.02	1	3.98	0
	4	30	82	9	3,600.00	-	-	3,600.00	614.23	0.84	20,449	1.43	20	22.20	0
	5	30	101	9	3,600.00	-	-	567.22	610.00	0.00	10,508	2.43	0	23.54	0
E033-03n	1	32	32	3	3,600.00	2,058.86	16.68	0.77	2,006.34	0.00	252	0.01	0	0.02	0
	2	32	44	7	3,600.00	2,872.95	51.03	3,600.00	-	-	-	-	-	-	-
	3	32	56	7	3,600.00	2,967.01	52.66	3,600.00	-	-	-	-	-	-	-
	4	32	78	7	3,600.00	-	-	3,600.00	-	-	-	-	-	-	-
	5	32	102	8	3,600.00	2,727.78	48.87	3,600.00	2,727.78*	27.18	5,058	796.76	1325	2,509.95	0
E033-04g	1	32	32	4	3,600.00	872.89	56.32	1.73	837.67	0.00	612	0.01	0	0.02	0
	2	32	47	7	3,600.00	1,258.00	70.97	3,600.00	1,258.00*	33.43	22,839	36.41	1,733	639.75	1
	3	32	57	7	3,600.00	1,302.46	71.91	3,600.00	-	-	-	-	-	-	-
	4	32	65	7	3,600.00	1,194.04	69.37	3,600.00	1,194.04*	38.52	2,018	1,334.69	199	2,145.12	0
	5	32	87	8	3,600.00	1,245.73	70.69	3,600.00	1,017.84*	17.70	2,264	847.91	135	2,411.48	0
E033-05s	1	32	32	5	3,600.00	887.15	57.14	0.77	837.67	0.00	454	0.00	0	0.01	0
	2	32	48	6	3,600.00	1,248.55	70.63	3,600.00	1,248.55	33.32	23,278	49.54	1,807	422.96	0
	3	32	59	6	3,600.00	-	-	3,600.00	-	-	-	-	-	-	-
	4	32	84	8	3,600.00	-	-	3,600.00	-	-	-	-	-	-	-
	5	32	114	8	3,600.00	-	-	3,600.00	-	-	-	-	-	-	-
E036-11h	1	35	35	11	3,600.00	704.01	41.50	180.66	698.61	0.00	7,133	0.01	0	0.02	0
	2	35	56	11	3,600.00	-	-	185.81	698.61	0.00	7,133	0.01	0	6.01	0
	3	35	74	11	3,600.00	-	-	239.92	698.61	0.00	8,045	0.04	1	4.30	0
	4	35	93	11	3,600.00	-	-	405.67	703.35	0.00	10,769	0.33	5	13.33	0
	5	35	114	11	3,600.00	-	-	183.22	698.61	0.00	7,133	0.03	0	0.46	0

Instance		Two-index model								Instance		Two-index model
Instance					Cuts	Time	Cuts	Time	Cuts	Instance					Cuts	Time	Cuts	Time	Cuts
Name	Class	Time	Sol.	GAP	CVRPSEP	2OPP^W;H	2OPP^W;H	2OPP	2OPP	Name	Class	Time	Sol.	GAP	CVRPSEP	2OPP^W;H	2OPP^W;H	2OPP	2OPP
E016-03m	1	0.80	278.73	0.00	343	0.00	0	0.02	0	E026-08m	1	7.01	607.65	0.00	2,537	0.00	0	0.01	0
	2	3,600.00	312.69	10.68	6,017	13.29	361	3,475.23	133		2	27.51	611.49	0.00	2,399	0.00	0	20.27	0
	3	3,600.00	285.71*	7.29	1,163	7.44	54	3,244.48	1		3	372.17	613.90*	0.00	2,537	0.01	0	363.01	0
	4	1,536.53	294.25*	0.00	967	3.67	33	1,516.15	0		4	1406.15	625.10*	0.00	11,156	0.18	12	1,084.30	0
	5	1,396.37	278.73*	0.00	599	0.78	4	1,377.52	0		5	195.11	607.65	0.00	2,537	0.01	0	183.12	0
E016-05m	1	0.53	334.96	0.00	379	0.00	0	0.00	0	E030-03g	1	19.63	535.80	0.00	2,165	0.01	0	0.22	0
	2	5.77	347.73	0.00	379	0.00	0	5.06	0		2	3,600.00	-	-	-	-	-	-	-
	3	439.85	353.48	0.00	3,842	0.37	28	369.30	6		3	3,600.00	676.83*	25.28	532	6.04	170	3,517.67	1
	4	122.87	342.00*	0.00	379	0.04	0	120.29	0		4	3,600.00	-	-	-	-	-	-	-
	5	124.54	334.96*	0.00	379	0.05	0	120.42	0		5	3,600.00	-	-	-	-	-	-	-
E021-04m	1	5.73	358.40	0.00	1604	0.00	0	0.03	0	E030-04s	1	0.39	505.01	0.00	105	0.00	0	0.04	0
	2	3,600.00	-	-	-	-	-	-	-		2	3,600.00	-	-	-	-	-	-	-
	3	3,600.00	406.55*	12.23	7,129	1.61	71	1,790.8	0		3	3,,600.00	-	-	-	-	-	-	-
	4	2,260.70	368.56*	0.00	2,440	7.11	22	2,217.75	0		4	3600.00	-	-	-	-	-	-	-
	5	504.27	358.40*	0.00	1,372	0.01	0	493.14	0		5	3,600.00	-	-	-	-	-	-	-
E021-06m	1	5.20	430.88	0.00	2,297	0.00	0	0.01	0	E031-09h	1	3,600.00	648.74	8.84	36,525	0.00	0	0.03	0
	2	220.06	440.94	0.00	2,937	0.06	2	199.02	10		2	3,600.00	-	-	-	-	-	-	-
	3	246.85	440.68*	0.00	2,297	0.04	0	240.11	0		3	3,600.00	637.74	7.00	36,635	1.08	8	86.35	0
	4	146.28	447.37	0.00	3,445	0.15	6	115.97	0		4	3,600.00	675.65*	13.04	2,1675	1.42	30	2,614.28	1
	5	96.49	430.88	0.00	2,297	0.01	0	88.09	0		5	3,600.00	648.74*	8.91	33,029	0.20	0	1,002.26	0
E022-04g	1	0.17	375.28	0.00	135	0.00	0	0.01	0	E033-03n	1	1.20	2,006.34	0.00	203	0.00	0	0.12	0
	2	3,600.00	388.72*	4.22	2,486	3.31	44	3,575.91	69		2	3,600.00	-	-	-	-	-	-	-
	3	1291.53	387.82*	0.00	901	0.71	24	1,276.21	15		3	3,600.00	-	-	-	-	-	-	-
	4	1981.61	383.87*	0.00	1,127	1.93	22	1,963.77	1		4	3,600.00	-	-	-	-	-	-	-
	5	72.32	375.28	0.00	149	0.01	0	69.31	0		5	3,600.00	-	-	-	-	-	-	-
E022-06m	1	1.84	495.85	0.00	914	0.00	0	0.01	0	E033-04g	1	4.70	837.67	0.00	1315	0.00	0	0.11	0
	2	77.45	499.08	0.00	1,700	0.05	3	68.56	2		2	3,600.00	-	-	-	-	-	-	-
	3	964.51	504.68	0.00	6,975	0.82	30	871.19	12		3	3,600.00	1,199.60*	30.26	2,611	11.47	97	1,781.36	1
	4	1,715.21	498.32*	0.00	2,968	0.68	21	1,680.56	0		4	3,600.00	-	-	-	-	-	-	-
	5	111.49	495.85	0.00	914	0.01	0	105.27	0		5	3,600.00	-	-	-	-	-	-	-
E023-03g	1	0.28	568.56	0.00	42	0.00	0	0.04	0	E033-05s	1	2.17	837.67	0.00	650	0.00	0	0.08	0
	2	3,600.00	860.70*	28.04	6,481	14.06	1,682	3,289.44	103		2	3,600.00	-	-	-	-	-	-	-
	3	3,600.00	706.59*	21.61	993	24.53	467	3,452.35	4		3	3,600.00	-	-	-	-	-	-	-
	4	3,600.00	-	-	-	-	-	-	-		4	3,600.00	-	-	-	-	-	-	-
	5	3,600.00	-	-	-	-	-	-	-		5	3,600.00	-	-	-	-	-	-	-
E023-05s	1	0.29	568.56	0.00	42	0.00	0	0.04	0	E036-11h	1	3,600.00	698.61	1.24	35,540	0.01	0	0.05	0
	2	3,600.00	725.91*	30.48	1,663	3.17	204	3,370.95	6		2	3,600.00	740.49	8.31	38,459	0.04	6	98.20	13
	3	3,600.00	818.21*	29.81	454	7.64	152	3,476.83	10		3	3,600.00	698.61	1.32	34702	0.27	1	130.29	0
	4	3,600.00	-	-	-	-	-	-	-		4	3,600.00	779.49*	12.96	25,743	1.42	19	2,165.22	3
	5	3,600.00	657.49*	13.35	213	62.30	50	3,238.68	0		5	3,600.00	698.61	1.25	35363	0.04	0	29.62	0

unrestricted 2L-CVRP										sequential 2L-CVRP
Instance		SA	Two-index		Instance		SA	Two-index		Instance		SA	Two-index		Instance		SA	Two-index
Name	Class	Sol.	Sol.	Dev.	Name	Class	Sol.	Sol.	Dev.	Name	Class	Sol.	Sol.	Dev.	Name	Class	Sol.	Sol.	Dev.
E016-03m	1	278.73	278.73	0.00	E026-08m	1	607.65	607.65	0.00	E016-03m	1	278.73	278.73	0.00	E026-08m	1	607.65	607.65	0.00
	2	278.73	278.73	0.00		2	607.65	607.65	0.00		2	290.84	-	-		2	611.49	611.49	0.00
	3	284.52	284.52	0.00		3	607.65	607.65	0.00		3	284.52	285.71	0.42		3	613.90	613.90	0.00
	4	282.95	282.95	0.00		4	625.10	625.10	0.00		4	294.25	294.25	0.00		4	625.10	625.10	0.00
	5	278.73	278.73	0.00		5	607.65	607.65	0.00		5	278.73	278.73	0.00		5	607.65	607.65	0.00
E016-05m	1	334.96	334.96	0.00	E030-03g	1	535.80	535.80	0.00	E016-05m	1	334.96	334.96	0.00	E030-03g	1	535.80	535.80	0.00
	2	334.96	334.96	0.00		2	689.68	793.41	15.04		2	347.73	347.73	0.00		2	700.20	-	-
	3	352.16	352.16	0.00		3	620.33	754.10	21.56		3	352.16	352.98	0.23		3	639.09	676.83	5.91
	4	334.96	334.96	0.00		4	710.87	-	-		4	342.00	342.00	0.00		4	715.82	-	-
	5	334.96	334.96	0.00		5	678.66	-	-		5	334.96	334.96	0.00		5	690.96	-	-
E021-04m	1	358.40	358.40	0.00	E030-04s	1	505.01	505.01	0.00	E021-04m	1	358.40	358.40	0.00	E030-04s	1	505.01	505.01	0.00
	2	387.70	387.70	0.00		2	684.21	-	-		2	403.93	-	-		2	721.54	-	-
	3	394.72	394.72	0.00		3	706.73	1,020.76	44.43		3	394.72	406.55	3.00		3	719.91	-	-
	4	362.41	362.41	0.00		4	788.54	-	-		4	368.56	368.56	0.00		4	822.61	-	-
	5	358.40	358.40	0.00		5	624.82	804.92	28.82		5	358.40	358.40	0.00		5	624.82	-	-
E021-06m	1	430.89	430.88	0.00	E031-09h	1	610.00	610.00	0.00	E021-06m	1	430.89	430.88	0.00	E031-09h	1	610.00	648.74	6.35
	2	430.89	430.88	0.00		2	610.57	674.96	10.55		2	440.94	440.94	0.00		2	619.63	-	-
	3	430.89	430.88	0.00		3	610.00	610.00	0.00		3	440.68	440.68	0.00		3	610.00	637.74	4.55
	4	447.37	447.37	0.00		4	614.24	623.39	1.49		4	447.37	447.37	0.00		4	618.23	688.82	11.42
	5	430.89	430.88	0.00		5	610.00	610.00	0.00		5	430.89	430.88	0.00		5	610.00	648.74	6.35
E022-04g	1	375.28	375.28	0.00	E033-03n	1	2,006.34	2,006.34	0.00	E022-04g	1	375.28	375.28	0.00	E033-03n	1	2,006.34	2,006.34	0.00
	2	375.28	375.28	0.00		2	2,585.72	-	-		2	388.72	388.72	0.00		2	2,669.39	-	-
	3	381.69	381.69	0.00		3	2,454.37	-	-		3	381.69	387.82	1.61		3	2,486.44	-	-
	4	383.88	383.87	0.00		4	2,548.06	-	-		4	383.88	383.87	0.00		4	2,609.36	-	-
	5	375.28	375.28	0.00		5	2,334.78	2,985.29	27.86		5	375.28	375.28	0.00		5	2,416.04	-	-
E022-06m	1	495.85	495.85	0.00	E033-04g	1	837.67	837.67	0.00	E022-06m	1	495.85	495.85	0.00	E033-04g	1	837.67	837.67	0.00
	2	495.85	495.85	0.00		2	1,038.20	1,275.49	22.86		2	499.08	499.08	0.00		2	1,092.51	-	-
	3	498.16	498.16	0.00		3	1,002.51	-	-		3	504.68	504.68	0.00		3	1,039.06	1,199.60	15.45
	4	498.32	498.32	0.00		4	981.00	1,363.46	38.99		4	498.32	498.32	0.00		4	982.25	-	-
	5	495.75	495.85	0.02		5	871.22	1,017.84	16.83		5	495.85	495.85	0.00		5	922.75	-	-
E023-03g	1	568.56	568.56	0.00	E033-05s	1	837.67	837.67	0.00	E023-03g	1	568.56	568.56	0.00	E033-05s	1	837.67	837.67	0.00
	2	725.46	854.55	17.79		2	1,017.95	1,329.40	30.60		2	734.65	866.98	18.01		2	1,041.75	-	-
	3	678.75	749.88	10.48		3	1,149.99	-	-		3	702.59	706.59	0.57		3	1,181.68	-	-
	4	700.72	716.05	2.19		4	1,184.85	-	-		4	703.49	-	-		4	1,247.39	-	-
	5	657.77	662.04	0.65		5	1,159.94	-	-		5	658.64	-	-		5	1,229.95	-	-
E023-05s	1	568.56	568.56	0.00	E036-11h	1	698.61	698.61	0.00	E023-05s	1	568.56	568.56	0.00	E036-11h	1	698.61	698.61	0.00
	2	674.55	674.55	0.00		2	698.61	698.61	0.00		2	725.91	725.91	0.00		2	698.61	742.83	6.33
	3	740.85	775.36	4.66		3	698.61	698.61	0.00		3	741.12	-	-		3	698.61	698.61	0.00
	4	692.47	735.01	6.14		4	703.35	703.35	0.00		4	697.92	-	-		4	708.20	779.49	10.07
	5	609.90	612.79	0.47		5	698.61	698.61	0.00		5	621.85	657.49	5.73		5	698.61	698.61	0.00

Instances		Hokama et al. (2016)HOKAMA P, MIYAZAWA FK & XAVIER EC. 2016. A branch-and-cut approach for the vehicle routing problem with loading constraints. Expert Systems with Applications, 47: 1-13.	Two-index model			Instances		Hokama et al. (2016)HOKAMA P, MIYAZAWA FK & XAVIER EC. 2016. A branch-and-cut approach for the vehicle routing problem with loading constraints. Expert Systems with Applications, 47: 1-13.	Two-index model
Name	Class	Sol.	Time	Sol.	Dev.	Name	Class	Sol.	Time	Sol.	Dev.
E016-03m	1	273	0.65	273	0.00	E023-03g	1	558	0.44	558	0.00
	2	285	3,600.00	285*	0.00		2	724	3,600.00	724	0.00
	3	280	2658.45	280*	0.00		3	698	3,600.00	698*	0.00
	4	288	57.51	288	0.00		4	714	3,600.00	714*	0.00
	5	279	253.69	279	0.00		5	742	1,533.29	742*	0.00
E016-05m	1	329	1.03	329	0.00	E023-05s	1	657	0.44	657	0.00
	2	342	9.77	342	0.00		2	720	553.63	720*	0.00
	3	347	301.22	347	0.00		3	730	3,600.00	730*	0.00
	4	336	131.32	336*	0.00		4	701	3,600.00	701*	0.00
	5	329	248.69	329*	0.00		5	721	122.67	721*	0.00
E021-04m	1	351	1.58	351	0.00	E026-08m	1	609	0.85	609	0.00
	2	396	988.5	396	0.00		2	612	15.33	612	0.00
	3	387	3,600.00	387*	0.00		3	615	195.93	615*	0.00
	4	374	1030	374*	0.00		4	626	540.25	626*	0.00
	5	369	88.69	369	0.00		5	609	155.38	609	0.00
E021-06m	1	423	0.45	423	0.00	E030-03g	1	524	13.40	524	0.00
	2	434	276.3	434	0.00		2	-	3,600.00	687*	-
	3	432	349.34	432*	0.00		3	637	3,600.00	637*	0.00
	4	438	311.46	438	0.00		4	-	3,600.00	-	-
	5	423	197.84	423*	0.00		5	-	3,600.00	-	-
E022-04g	1	367	0.16	367	0.00	E033-03n	1	1,991	0.79	1,991	0.00
	2	380	1,956.46	380	0.00		2	-	3,600.00	-	-
	3	373	2,351.18	373*	0.00		3	2854	3,600.00	2854*	0.00
	4	377	760.45	377*	0.00		4	-	3,600.00	-	-
	5	389	44.41	389	0.00		5	2,672	3,600.00	2,672*	0.00
E022-06m	1	488	3.39	488	0.00	E036-11h	1	682	772.98	682	0.00
	2	491	64.52	491	0.00		2	682	3,600.00	682	0.00
	3	496	2,099.52	496*	0.00		3	682	937.80	682	0.00
	4	489	392.55	489*	0.00		4	691	3,600.00	691*	0.00
	5	488	62.41	488	0.00		5	-	822.03	682	-

\sum_{j \in S_{R}} \sum_{m \in M_{j}} \sum_{{s \in P_{j m}^{W} : u - w_{j m} + 1 \leq s \leq u}} \sum_{{t \in P_{j m}^{H} : v - h_{j m} + 1 \leq t \leq v}} y_{m s t} \leq 1, \forall u \in P^{W}, \forall v \in P^{H}

\sum_{{i \in S_{R} | o_{i} < o_{j}}} \sum_{l \in M_{i}} \sum_{{u \in P_{i l}^{W} | s - w_{i l} + 1 \leq u \leq s + w_{j m} - 1}} \sum_{{v \in P_{i l}^{H} | v \leq t - h_{i l}}} y_{l u v} \leq (1 - y_{m s t}) \sum_{i \in S_{R}} | M_{i} |, \forall j \in S_{R}, \forall m \in M_{j}, \forall s \in P_{j m}^{W}, \forall t \in P_{j m}^{H}

[1] *Corresponding author

Brasil

Brasil

INTEGER FORMULATIONS FOR THE INTEGRATED VEHICLE ROUTING PROBLEM WITH TWO-DIMENSIONAL PACKING CONSTRAINTS

ABSTRACT