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

Table 2

Results for the ILP model running CPLEX with a time limit of 7,200 seconds. the GAP is defined as . Results depicted for the groups of instances 1 and 2. The bolded LB and UB values represent an instance’s optimal solution.

Group 1 instances										Group 2 instances
Instance		Integer model CPLEX				Lagrangean Relaxation				Instance	Integer model CPLEX				Lagrangean Relaxation
		LB	UB	GAP (%)	Time (sec)	LB	UB	GAP (%)	Time (sec)		LB	UB	GAP (%)	Time (sec)	LB	UB	GAP (%)	Time (sec)
5-3-4-1	794		794	0	0	794	799	1	0	5-3-4-2	4,630	4,630	0	1	4,630	4,660	1	0
5-4-4-1	671		671	0	0	671	671	0	0	5-4-4-2	3,222	3,222	0	12	3,222	3,222	0	0
5-5-4-1	842		842	0	0	842	852	1	0	5-5-4-2	3,222	3,222	0	26	3,222	3,305	2	0
5-6-4-1	801		801	0	0	801	812	1	0	5-6-4-2	4,855	4,855	0	26	4,840	4,959	3	0
5-7-4-1	901		901	0	0	901	905	1	0	5-7-4-2	4,954	4,954	0	21	4,954	5,090	3	0
10-6-9-1	939		939	0	1	939	955	2	0	10-6-9-2	5,271	5,272	0.001	7,200	5,221	5,317	2	0
10-8-9-1	983		983	0	5	983	989	1	0	10-8-9-2	4,289	4,289	0	52	4,289	4,289	0	0
10-10-9-1	1,002		1,002	0	10	996	1,024	3	0	10-10-9-2	5,080	5,080	0	3,138	5,080	5,123	1	0
10-12-9-1	1,001		1,001	0	2	1,001	1,013	1	0	10-12-9-2	4,802	4,802	0	70	4,802	4,865	1	0
10-14-9-1	906		906	0	0	906	916	1	0	10-14-9-2	5,151	5,151	0	157	5,151	5,220	2	0
20-12-19-1	926		926	0	28	926	953	3	0	20-12-19-2	5,398	6,124	12	7,200	5,921	6,003	1	1
20-16-19-1	962		962	0	24	954	984	3	0	20-16-19-2	5,248	5,824	10	7,200	5,697	5,963	4	1
20-20-19-1	1,048		1,048	0	43	1,048	1,125	7	0	20-20-19-2	5,394	5,726	6	7,200	5,726	6,124	6	2
20-24-19-1	1,082		1,082	0	51	1,082	1,096	1	0	20-24-19-2	5,313	7,033	24	7,200	5,581	6,201	1	2
20-28-19-1	1,100		1,100	0	189	1,100	1,167	6	0	20-28-19-2	5,647	7,328	23	7,200	6,067	6,617	8	3
40-24-39-1	1,012		1,012	0	2,332	1,006	1,085	7	0	40-24-39-2	5,928	7,941	25	7,200	6,281	7,296	14	6
40-32-39-1	1,096		1,159	5	7,200	1,144	1,266	10	0	40-32-39-2	-	-	-	7,200	6,025	7,132	15	7
40-40-39-1	1,084		1,128	4	7,200	1,092	1,260	13	1	40-40-39-2	-	-	-	7,200	7,296	7,989	9	12
40-48-39-1	1,074		1,146	6	7,200	1,116	1,251	10	1	40-48-39-2	-	-	-	7,200	6,562	8,227	20	15
40-56-39-1	1,051		1,126	7	7,200	1,069	1,155	7	2	40-56-39-2	-	-	-	7,200	7,111	7,976	11	17
60-36-59-1	1,184		1,288	8	7,200	1,233	1,324	7	2	60-36-59-2	-	-	-	7,200	7,349	8,144	10	16
60-48-59-1	1,160		1,505	23	7,200	1,225	1,259	3	3	60-48-59-2	-	-	-	7,200	8,101	9,839	18	34
60-60-59-1	1,194		1,580	24	7,200	1,268	1,381	8	4	60-60-59-2	-	-	-	7,200	7,572	9,609	21	43
60-72-59-1	1,195		1,618	26	7,200	1,289	1,442	10	5	60-72-59-2	-	-	-	7,200	7,150	9,905	28	45
60-84-59-1	1,118		1,100,843	99	7,200	1,283	1,502	14	7	60-84-59-2	-	-	-	7,200	7,388	10,302	28	61