Mathematical Problems in Engineering

Research Article

Improved Genetic Algorithm with Gene Recombination for Bus Crew-Scheduling Problem

Table 5

Comparative minimum results and average results of 10 runs for six lines (a) with different for , , (b) with different for , , (c) with different for , , and (d) with different for , , . (The bold number is the minimum shift number for each bus line.)

(a)


						Shifts
	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0	MR

#306	30	26	29	29	29	30	28	31	30	32	33
#695	65	63	65	63	64	64	65	64	64	67	64
#348	22	21	22	22	22	23	24	25	24	27	31
#26	81	80	79	78	81	79	81	79	82	82	80
#43	49	50	50	51	54	53	52	56	52	59	49
#467	36	37	35	36	37	37	38	37	40	42	36
#28	45	45	44	47	46	47	48	49	51	51	52
#34	46	45	45	45	46	47	46	47	49	50	48
#345	128	118	121	124	126	128	132	132	140	139	141
#322	109	109	105	103	104	110	107	110	116	122	139

Avg. RPD (%)	−8.37	−10.78	−10.19	−9.5	−7.84	−6.7	−6.45	−4.47	−2.9	1.69

#306	31.2	30.1	30.7	30.7	30.9	31.4	31.6	32.5	32.2	33.2	33
#695	66.3	66	67	66.4	65.6	66.1	67.1	66.9	68.3	68.3	64
#348	24	23.2	24.1	23.6	23.9	24.8	25	26.1	27.1	29.5	31
#26	81.9	82.5	81.7	81.3	82	82.1	82.7	82.4	83.2	83.4	80
#43	53.6	56.2	55.6	57.3	57	58.1	57.5	59.7	58.8	61.9	49
#467	37.8	38.6	38	38.1	39.1	39.1	40.2	40.3	41.9	44	36
#28	47.9	48.2	48	48.9	50	50.2	50.9	51.8	53.2	55.2	52
#34	47.5	46.9	47.4	47.1	47.8	49	49	49.2	51.5	52.6	48
#345	133.3	127.7	133.1	131.4	133.5	138.1	140.9	141.4	144.6	146.9	141
#322	112	112	110.8	111.9	111.6	113.3	114.4	117.8	120	112	139

Avg. RPD (%)	−4.15	−4.42	−3.82	−3.68	−2.86	−1.37	−0.42	1.11	3.06	5.58

(b)


				Shifts
	0.05	0.1	0.15	0.2	0.25	0.3	MR

#306	26	30	29	26	29	29	33
#695	63	65	63	61	64	64	64
#348	21	22	22	23	24	23	31
#26	80	78	80	80	79	79	80
#43	50	52	53	53	51	53	49
#467	37	37	36	36	37	36	36
#28	45	47	47	47	47	48	52
#34	45	44	45	47	47	46	48
#345	118	122	125	126	127	124	141
#322	109	104	108	107	108	105	139

Avg. RPD (%)	−10.78	−8.68	−8.41	−8.89	−7.3	−7.94

#306	30.1	31	31	30.6	30.7	30.9	33
#695	66	66.7	65.7	65.6	65.8	66.4	64
#348	23.2	23.7	24	25	24.9	24.7	31
#26	82.5	81.2	82.3	81.4	82	81.9	80
#43	56.2	56.1	56.6	56	56	56.8	49
#467	38.6	38.3	37.4	38.4	38	38.3	36
#28	48.2	48.4	49.6	49.1	49.6	49.2	52
#34	46.9	47.3	47.4	48.9	47.9	48.3	48
#345	127.7	131.7	133.6	132.4	132.9	133.2	141
#322	112	110.4	112.4	110.9	110.7	109.5	139

Avg. RPD (%)	−4.42	−3.86	−3.4	−3.14	−3.24	−2.98

(c)


			Shifts
	0.5	0.6	0.7	0.8	0.9	MR

#306	29	29	29	26	26	33
#695	62	62	63	60	63	64
#348	23	21	23	22	21	31
#26	77	79	79	80	80	80
#43	51	51	52	53	50	49
#467	37	36	36	36	37	36
#28	48	45	44	46	45	52
#34	45	47	46	45	45	48
#345	125	126	126	119	118	141
#322	103	108	104	104	109	139

Avg. RPD (%)	−8.91	−9.32	−9	−10.69	−10.78

#306	30.8	31.2	30.3	28.4	30.1	33
#695	67.4	66.5	67	65.7	66	64
#348	24.6	23.8	24.1	23.5	23.2	31
#26	82.1	81.8	81.9	82	82.5	80
#43	56.5	56.5	55.5	55.4	56.2	49
#467	38.5	37.4	37.4	37.8	38.6	36
#28	50	49	47	48.1	48.2	52
#34	47.5	49	47.5	46.9	46.9	48
#345	132	134.3	133	130.5	127.7	141
#322	111.5	113.5	111.5	110	112	139

Avg. RPD (%)	−2.82	−3.01	−4.23	−5.3	−4.42

(d)


			Shifts
	1	10	100	1000	MR

#306	28	29	29	28	33
#695	62	63	65	62	64
#348	21	21	21	22	31
#26	79	80	79	76	80
#43	49	52	49	48	49
#467	36	36	36	35	36
#28	45	46	47	45	52
#34	45	45	46	45	48
#345	121	120	122	122	141
#322	108	105	103	109	139

Avg. RPD (%)	−10.8	−9.7	−9.72	−10.89	−10.8

#306	30.4	30.9	30.6	30	33
#695	66.2	66.3	67	65.3	64
#348	22.8	23.5	24.3	23.8	31
#26	82.2	81.5	81.8	80.3	80
#43	54.1	55.1	56.2	54.6	49
#467	37.5	37.5	37.2	37.2	36
#28	47.8	48.2	49.5	47.8	52
#34	47.4	46.8	47.9	47.8	48
#345	130.1	128.9	128.4	133.1	141
#322	110.4	110.7	109.5	109.6	139

Avg. RPD (%)	−5.12	−4.72	−3.91	−5.04	−5.12