Presenting a Multi-Start Hybrid Heuristic for Solving the Problem of Two-Echelon Location-Routing Problem with Simultaneous Pickup and Delivery (2E-LRPSPD)

Table 3

Comparison of MHH-PR and other heuristics in Prodhon’s 2E-LRP instances.

Instance	Nguyen et al. [3, 4]		Contardo et al. [45]		Pichka et al. [60]		This paper
		CPU (s)		CPU (s)		CPU (s)		CPU (s)	Gap (%)
20-5-1	89075	2.4	89075	2.8	89075	2.0	89075	4.3	0.00
20-5-1b	61863	2.6	61863	0.2	61838	2.0	61838	4.8	0.00
20-5-2	85290	1.6	84478	2.8	84478	2.0	84478	5.0	0.00
20-5-2b	60838	1.4	60838	0.0	60838	3.0	60838	4.7	0.00
50-5-1	134855	9.2	130843	8.2	135008	47.0	134481	21.7	2.78
50-5-1b	101530	15.8	101530	8.7	101879	32.0	102690	9.4	1.14
50-5-2	132159	12.9	131825	37.2	132131	48.0	133666	32.0	1.40
50-5-2b	110547	18.6	110332	21.1	110395	34.0	111242	12.0	0.82
50-5-2Bis	122654	27.6	122599	43.2	122809	48.0	123286	67.7	0.56
50-5-2bBis	105776	30.0	105696	57.4	105835	34.0	107365	45.2	1.58
50-5-3	128379	16.2	128379	17.8	128379	49.0	129408	29.5	0.80
50-5-3b	104006	17.0	104006	14.2	104006	33.0	104006	22.9	0.00
100-5-1	320130	37.3	319137	445.0	323693	158.0	330437	272.6	3.54
100-5-1b	258205	32.0	257349	607.9	260754	109.0	263363	114.8	2.34
100-5-2	234179	29.6	231305	43.7	235752	157.0	235023	141.0	1.61
100-5-2b	195426	24.9	194729	602.4	195450	107.0	196897	70.3	1.11
100-5-3	245944	68.0	244194	167.9	250992	156.0	249336	111.7	2.11
100-5-3b	195254	54.3	194110	210.9	198343	108.0	199126	105.8	2.58
100-10-1	358939	336.4	358068	65.6	356669	159.0	370088	157.3	3.76
100-10-1b	302584	335.5	297167	118.7	301509	114.0	316786	146.6	6.60
100-10-2	306303	362.0	305402	163.0	308712	157.0	307599	191.9	0.72
100-10-2b	264389	294.1	265138	225.9	266692	116.0	266535	125.4	0.81
100-10-3	313249	370.8	313517	106.0	322096	160.0	321605	287.6	2.67
100-10-3b	266383	129.1	264096	181.6	275951	114.0	272119	190.0	3.04
200-10-1	554598	700.9	552816	648.7	561835	613.0	559353	1878.8	1.18
200-10-1b	452286	723.7	448236	927.6	460476	448.0	458935	906.5	2.39
200-10-2	502173	220.2	498199	339.3	506124	623.0	509329	1766.7	2.23
200-10-2b	425311	267.3	423048	1161.5	429033	424.0	431368	653.4	1.97
200-10-3	533732	632.9	534569	285.2	543312	605.0	545558	1368.0	2.22
200-10-3b	418800	323.6	404284	439.1	409961	413.0	422903	622.4	4.61
Average	246161.9	169.9	244560.9	231.8	248134.2	169.2	249957.8	312.3	1.82

The bold values in the first column is the name of the instance. n denotes number of customers. m denotes number of satellites. Q denotes capacity of secondary vehicle. β denotes number of clusters. The names have the format n-m-β. A suffix “b” signals the instances with . The name of instance can not be set in bold.The bold in other columns are the best objective values, and they need to be saved in bold.