A Single-Machine Two-Agent Scheduling Problem by a Branch-and-Bound and Three Simulated Annealing Algorithms

Table 1

Performance of the branch-and-bound and heuristic algorithms ( = 8~16).

	Pro		Branch-and-bound algorithm					SA1		SA2		SA3		SA4
			Valid sample size	CPU time		Number of nodes		Error percentages
				Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.
8	0.25	0.25	50	0.00	0.01	526	318	0.28	1.46	0.28	1.46	0.28	1.46	0.28	1.46
		0.50	50	0.00	0.01	748	588	1.29	4.12	0.93	3.11	0.86	3.09	0.24	1.69
		0.75	50	0.01	0.01	1383	923	3.42	7.29	2.83	5.73	2.27	4.80	0.62	2.46
	0.50	0.25	50	0.01	0.01	2229	1371	0.01	0.06	0.01	0.06	0.00	0.00	0.00	0.00
		0.50	50	0.01	0.01	2500	1412	0.50	1.78	0.14	0.56	0.20	0.86	0.04	0.20
		0.75	50	0.01	0.01	3183	1139	0.92	2.03	1.27	2.26	1.48	2.76	0.26	1.02
	0.75	0.25	50	0.01	0.01	1091	217	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
		0.50	50	0.01	0.01	1091	217	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
		0.75	50	0.01	0.01	1089	196	0.36	1.18	0.11	0.48	0.49	1.33	0.03	0.24
Average				0.01	0.01	1538	709	0.75	1.99	0.62	1.52	0.62	1.59	0.16	0.79
12	0.25	0.25	50	0.78	0.86	97087	113419	0.29	0.93	0.61	2.30	0.30	1.38	0.05	0.20
		0.50	50	1.36	1.96	175858	261636	1.86	4.41	3.99	7.81	1.45	3.86	0.77	1.95
		0.75	50	3.92	5.24	535992	751273	4.78	7.25	4.40	6.82	5.30	8.43	2.12	4.26
	0.50	0.25	50	19.83	22.75	2278911	2794764	0.02	0.07	0.04	0.20	0.01	0.05	0.00	0.00
		0.50	50	27.91	33.86	3405293	4448908	0.64	1.24	0.53	1.16	0.32	0.73	0.17	0.45
		0.75	50	35.38	21.51	4692766	3096131	2.07	2.16	2.37	2.43	2.37	2.74	0.69	0.92
	0.75	0.25	50	6.30	2.18	794003	304461	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
		0.50	50	6.30	2.18	794439	304555	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
		0.75	50	6.60	1.82	862132	262277	0.44	0.66	0.74	1.22	0.55	1.19	0.15	0.33
Average				12.04	10.26	1515165	1370825	1.12	1.86	1.41	2.44	1.14	2.04	0.44	0.90
16	0.25	0.25	48	846.82	1691.36	62738963	129480137	0.36	0.92	0.39	1.06	0.35	1.16	0.18	0.77
		0.50	46	1432.63	2411.56	105554536	181805362	1.99	3.46	2.49	4.28	1.88	4.38	0.95	2.29
		0.75	33	1446.76	2629.13	106507557	197947458	2.72	3.14	7.79	12.00	4.30	6.64	1.83	2.28
	0.50	0.25	22	8044.31	6058.59	429572906	333869433	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
		0.50	20	7267.99	5697.96	387550386	314387045	0.10	0.29	0.00	0.00	0.11	0.51	0.00	0.00
		0.75	3	7493.01	5776.30	411927486	331822988	0.93	1.07	6.91	6.17	6.65	4.11	0.93	1.07
	0.75	0.25	13	9502.54	3053.15	625205281	214788228	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
		0.50	13	9478.74	3044.52	625205281	214788228	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
		0.75	7	9965.98	3653.10	652692683	258677186	0.41	0.73	0.24	0.63	1.03	1.06	0.00	0.00
Average				6164.31	3779.52	378550564	241951785	0.72	1.07	1.98	2.68	1.59	1.98	0.43	0.71