A Scheduling Problem to Minimize Total Weighted Tardiness in the Two-Stage Assembly Flowshop

Table 1

The performance summary of the branch-and-bound algorithm.

n	m	B&B with dominance properties				B&B without dominance properties
		Computational time		The number of nodes		Computational time		The number of nodes
		Average	Max	Average	Max	Average	Max	Average	Max
5	2	0.0007	0.004	34.1	109	0.0008	0.004	47.09	176
	4	0.0008	0.005	33.8	96	0.0007	0.001	47.5	106
	6	0.0008	0.003	32.7	105	0.0008	0.003	47.6	121
7	2	0.002	0.007	456.5	2198	0.003	0.008	861.7	3093
	4	0.002	0.01	460.2	3012	0.003	0.01	920.1	3215
	6	0.002	0.01	399.9	1737	0.003	0.01	841.0	2471
9	2	0.05	0.2	11893.4	75986	0.09	0.4	26816.9	162938
	4	0.05	0.3	12159.7	92172	0.1	0.5	27496.8	170525
	6	0.04	0.2	8685.7	39096	0.1	0.3	24986.6	87934
11	2	2.04	33.5	383616.1	8290345	4.9	50.0	1231820.6	15696940
	4	2.3	44.2	401569.1	9476135	5.6	53.2	1281161.9	16173866
	6	2.0	13.7	287125.6	2230298	5.3	27.3	1060709.6	6218744
13	2	170.6	1147.5	26198107.8	257803159	352.1	1180.4	73275993.7	247993027
	4	132.4	1140.9	18033822.7	185741409	376.7	1187.9	69507001.4	249138079
	6	140.0	1120.2	16511705.6	155593682	378.2	1185.2	62695670.7	204011606
15	2	910.2	1194.7	113697715.3	119565585
	4	913.8	1144.9	96860705.2	133536399
	6	907.4	1171.9	86202170.5	87887442

means the problem instances that the B&B algorithm cannot find the optimal solutions.