The Lagrangean Relaxation for the Flow Shop Scheduling Problem with Precedence Constraints, Release Dates and Delivery Times

Table 3

Results obtained with the linear relaxation and the LR on instances of groups 1 and 2. The LB^∗values refer to the lower bounds provided by the Lagrangean Relaxation. Likewise, LB values refer to the lower bounds computed by the linear relaxation.

Group 1 instances						Group 2 instances
Instance	Linear relaxation		Lagrangean Relaxation			Instance	Linear relaxation		Lagrangean Relaxation
	LB	Time (sec)	LB ^∗	(LB^∗/LB)	Time (sec)		LB	Time (sec)	LB ^∗	(LB^∗/LB)	Time (sec)
5-3-4-1	788	0	794	1%	0	5-3-4-2	4,620	0	4,630	0%	0
5-4-4-1	666	0	671	1%	0	5-4-4-2	3,139	0	3,322	6%	0
5-5-4-1	840	0	842	0%	0	5-5-4-2	3,144	0	3,322	6%	0
5-6-4-1	795	0	801	1%	0	5-6-4-2	4,824	0	4,840	0%	0
5-7-4-1	895	0	901	1%	0	5-7-4-2	4,903	4	4,954	1%	0
10-6-9-1	932	0	939	1%	0	10-6-9-2	5,092	9	5,221	3%	0
10-8-9-1	966	0	983	2%	0	10-8-9-2	4,228	7	4,289	1%	0
10-10-9-1	985	0	996	1%	0	10-10-9-2	4,815	25	5,080	6%	0
10-12-9-1	995	0	1,001	1%	0	10-12-9-2	4,777	1	4,802	1%	0
10-14-9-1	905	0	906	0%	0	10-14-9-2	5,028	36	5,151	2%	0
20-12-19-1	909	1	926	2%	0	20-12-19-2	5,394	60	5,921	10%	1
20-16-19-1	941	2	954	1%	0	20-16-19-2	5,247	100	5,697	9%	1
20-20-19-1	1,009	4	1,048	4%	0	20-20-19-2	5,393	97	5,726	6%	2
20-24-19-1	1,044	6	1,082	4%	0	20-24-19-2	5,300	207	5,581	5%	2
20-28-19-1	1,046	5	1,100	5%	0	20-28-19-2	5,646	229	6,067	7%	3
40-24-39-1	978	10	1,006	1%	0	40-24-39-2	5,927	231	6,281	6%	6
40-32-39-1	1,066	10	1,144	7%	0	40-32-39-2	5,940	332	6,025	1%	7
40-40-39-1	1,060	15	1,092	3%	1	40-40-39-2	6,211	596	7,729	24%	12
40-48-39-1	1,064	15	1,116	5%	1	40-48-39-2	6,067	719	6,562	8%	15
40-56-39-1	1,045	26	1,069	2%	2	40-56-39-2	6,181	760	7,111	15%	17
60-36-59-1	1,135	28	1,233	9%	2	60-36-59-2	6,546	724	7,349	12%	16
60-48-59-1	1,118	40	1,225	10%	3	60-48-59-2	-	7,200	8,810	-%	34
60-60-59-1	1,173	75	1,268	8%	4	60-60-59-2	-	7,200	7,757	-%	43
60-72-59-1	1,174	100	1,289	10%	5	60-72-59-2	-	7,200	7,715	-%	45
60-84-59-1	1,162	112	1,283	10%	7	60-84-59-2	-	7,200	7,738	-%	61