A HYBRID HEURISTIC ALGORITHM FOR THE CLUSTERED TRAVELING SALESMAN PROBLEM

Mestria, Mário

doi:10.1590/0101-7438.2016.036.01.0113

Mário Mestria

Instituto Federal do Espírito Santo, Coordenadoria de Engenharia Elétrica, Av. Vitória, 1729, Jucutuquara, 29040-780 Vitória, ES, Brasil. E-mails: mmestria@ifes.edu.br; mmestria@uol.com.br

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Figure 1
A feasible solution for an example of an instance of the CTSP.

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Algorithm 1
Pseudocode of the HHGILS algorithm.

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Algorithm 2
Pseudocode of the nearest insertion heuristic.

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Table 1
Data for an example of the Algorithm 2.

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Table 2
Steps of the Algorithm 2 with data of .

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Table 3
Comparison between HHGILS and CPLEX for the small size instances of Class 1.

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Table 4
Comparison between HHGILS and CPLEX for medium size instances.

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Table 5
The best and average results obtained by HHGILS for large size instances.

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Table 6
Comparison between HHGILS and GPR1R2 Mestria, Ochi & Martins ³³33 MESTRIA M, OCHI LS & MARTINS SL. 2013. Grasp with path relinking for the symmetric euclidean clustered traveling salesman problem. Computers & Operations Research, 40(12): 3218- 3229. for the small instances of Class 1.

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Table 7
Comparison between HHGILS, GPR1R2 Mestria, Ochi & Martins ³³33 MESTRIA M, OCHI LS & MARTINS SL. 2013. Grasp with path relinking for the symmetric euclidean clustered traveling salesman problem. Computers & Operations Research, 40(12): 3218- 3229. and GPR4 Mestria, Ochi & Martins ³³33 MESTRIA M, OCHI LS & MARTINS SL. 2013. Grasp with path relinking for the symmetric euclidean clustered traveling salesman problem. Computers & Operations Research, 40(12): 3218- 3229. for medium size instances.

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Table 8
Comparison between HHGILS, G Mestria, Ochi & Martins ³³33 MESTRIA M, OCHI LS & MARTINS SL. 2013. Grasp with path relinking for the symmetric euclidean clustered traveling salesman problem. Computers & Operations Research, 40(12): 3218- 3229. and GPR1R2 Mestria, Ochi & Martins ³³33 MESTRIA M, OCHI LS & MARTINS SL. 2013. Grasp with path relinking for the symmetric euclidean clustered traveling salesman problem. Computers & Operations Research, 40(12): 3218- 3229. for large size instances.

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Table 9
Comparison between HHGILS and Concorde Applegate, Bixby, Chvátal & Cook ²2 APPLEGATE D, BIXBY R, CHVÁTAL V & COOK W. 2007. Concorde TSP Solver. William Cook, School of Industrial and Systems Engineering, Georgia Tech. http://www.tsp.gatech.edu/concorde/index.html. Accessed 22 December 2007.
http://www.tsp.gatech.edu/concorde/index... for the instances of Class 3.

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(2)

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vertex vk	x-coordinate	y-coordinate	cluster Vi
1	177	311	1
2	924	320	3
3	265	52	1
4	45	500	3
5	233	300	1
6	167	104	3
7	117	275	1
8	190	78	3
9	61	269	2
10	684	539	2
11	380	526	1
12	550	549	2

Steps	vertex vk	cluster Vi	partial solution
1 and 2	1, 5, 7 a	1	{15 7}b
9 and 10	3	1	{1 53 7}
9 and 10	11	1	{1 113 7}
9 and 10	4	3	{154 113 7}
9 and 10	8	3	{1584 113 7}
9 and 10	6	3	{15864 113 7}
9 and 10	2	3	{158642 113 7}
9 and 10	12	2	{158642 12 113 7}
9 and 10	10	2	{158642 10 12 113 7}
13	9	2	{158642 10 12 9 113 7}c

Instances	CPLEX		HHGILS
	Opt	Time (s)	Gap (%)	Time (s)
5-eil51	437	12.31	0.00	3.80
10-eil51	440	74.38	0.00	5.50
15-eil51	437	2.04	0.00	5.00
5-berlin52	7991	201.80	0.11	4.50
10-berlin52	7896	89.17	0.52	5.10
15-berlin52	8049	75.93	0.00	4.00
5-st70	695	13790.11	0.00	6.90
10-st70	691	4581.00	0.43	3.70
15-st70	692	883.50	0.87	6.40
5-eil76	559	83.70	0.00	5.10
10-eil76	561	254.30	1.07	8.60
15-eil76	565	49.66	0.88	6.80
5-pr76	108590	99.29	0.00	6.30
10-pr76	109538	238.13	0.00	8.30
15-pr76	110678	261.94	0.00	9.70
10-rat99	1238	650.67	0.00	12.60
25-rat99	1269	351.15	0.00	21.40
50-rat99	1249	2797.58	0.00	18.60
25-kroA100	21917	3513.57	0.00	21.70
50-kroA100	21453	947.55	0.00	21.70
10-kroB100	22440	4991.44	0.91	12.40
50-kroB100	22355	2579.22	0.00	22.40
25-eil101	663	709.45	1.06	21.10
50-eil101	644	275.33	0.16	20.80
25-lin105	14438	6224.55	1.38	16.70
50-lin105	14379	1577.21	0.00	22.60
75-lin105	14521	15886.77	0.00	23.70
Average values		2266.73	0.27	12.05

Instances	Id.	#vertices	#V_i	Class		CPLEX		HHGILS gap (%)
					Cost	Lower Bound	Gap (%)	Best	Average
i-50-gil262	J1	262	50	1	135529	135374.68	0.11	0.10	0.15
10-lin318	J2	318	10	1	534640	526412.07	1.54	0.73	0.83
10-pcb442	J3	442	10	1	547152	536478.33	1.95	0.46	0.65
C1k.0	J4	1000	10	2	134025123	131354923.50	1.99	1.45	1.81
C1k.1	J5	1000	10	2	130750874	128540131.50	1.69	0.99	1.28
C1k.2	J6	1000	10	2	144341485	141501445	1.97	1.24	1.41
300-6	J7	300	6	3	8969	8915.18	0.60	0.21	0.31
400-6	J8	400	6	3	9117	9021.51	1.05	0.33	0.46
700-20	J9	700	20	3	41638	41274.00	0.87	0.43	0.51
200-4-h	J10	200	4	4	63429	62244.84	1.87	0.89	1.29
200-4-x1	J11	200	4	4	60797	60242.96	0.91	1.13	1.85
600-8-z	J12	600	8	4	132897	127901.75	3.76	1.17	1.56
600-8-x2	J13	600	8	4	132228	127901.75	3.27	1.04	1.38
300-5-108	J14	300	5	5	68361	67128.93	1.80	1.01	1.18
300-20-111	J15	300	20	5	311286	308595.45	0.86	0.52	0.59
500-15-306	J16	500	15	5	196001	193522.8	1.26	0.86	0.98
500-25-308	J17	500	25	5	367586	364108.13	0.95	0.47	0.54
25-eil101	J18	101	25	6	23671	23668.63	0.01	0.04	0.09
42-a280	J19	280	42	6	130043	129560.53	0.37	0.12	0.21
144-rat783	J20	783	144	6	916174	913715.52	0.27	0.16	0.18
						Average	1.36	0.67	0.86

Instances	Id.	#vertices	#V_i	Class	Best	Average
49-pcb1173	J21	1173	49	6	67043	68260.70
100-pcb1173	J22	1173	100	6	68786	70640.83
144-pcb1173	J23	1173	144	6	66830	69084.25
10-nrw1379	J24	1379	10	6	63620	64643.88
12-nrw1379	J25	1379	12	6	63558	64741.57
1500-10-503	J26	1500	10	5	11986	12109.45
1500-20-504	J27	1500	20	5	17107	17315.72
1500-50-505	J28	1500	50	5	25264	25558.90
1500-100-506	J29	1500	100	5	32260	33760.64
1500-150-507	J30	1500	150	5	37658	38433.09
2000-10-a	J31	2000	10	4	116254	116881.38
2000-10-h	J32	2000	10	4	36447	37305.14
2000-10-z	J33	2000	10	4	37059	37443.69
2000-10-x1	J34	2000	10	4	36752	37704.03
2000-10-x2	J35	2000	10	4	36660	37117.11

Instances	HHG	ILS	GPR1R2
	Gap (%)	Time (s)	Gap (%)	Time (s)
5-eil51	0.00	3.80	0.00	1.00
10-eil51	0.00	5.50	0.00	1.00
15-eil51	0.00	5.00	0.00	1.00
5-berlin52	0.11	4.50	0.00	1.20
10-berlin52	0.52	5.10	0.00	1.10
15-berlin52	0.00	4.00	0.00	1.10
5-st70	0.00	6.90	0.00	2.30
10-st70	0.43	3.70	0.00	2.00
15-st70	0.87	6.40	0.00	2.00
5-eil76	0.00	5.10	0.36	2.70
10-eil76	1.07	8.60	0.53	2.40
15-eil76	0.88	6.80	0.35	2.50
5-pr76	0.00	6.30	0.00	2.70
10-pr76	0.00	8.30	0.00	2.20
15-pr76	0.00	9.70	0.15	2.30
10-rat99	0.00	12.60	0.00	4.90
25-rat99	0.00	21.40	0.63	4.70
50-rat99	0.00	18.60	0.72	4.90
25-kroA100	0.00	21.70	0.00	4.70
50-kroA100	0.00	21.70	0.00	5.20
10-kroB100	0.91	12.40	0.16	4.80
50-kroB100	0.00	22.40	1.33	5.20
25-eil101	1.06	21.10	1.36	4.60
50-eil101	0.16	20.80	1.09	5.40
25-lin105	1.38	16.70	0.00	5.10
50-lin105	0.00	22.60	1.08	5.70
75-lin105	0.00	23.70	0.59	6.40
Average values	0.27	12.05	0.25	3.30

Instances	Best gap (%)			Average gap (%)
	HHGILS	GPR1R2	GPR4	HHGILS	GPR1R2	GPR4
10-lin318	0.73	0.76	0.84	0.83	1.18	1.65
10-pcb442	0.46	0.66	0.66	0.65	1.22	1.52
C1k.0	1.45	1.60	1.68	1.81	1.76	1.81
C1k.1	0.99	1.27	1.27	1.28	1.42	1.46
300-6	0.21	0.49	0.50	0.31	0.78	1.04
700-20	0.43	0.64	0.61	0.51	0.72	0.72
200-4-h	0.89	1.19	1.24	1.29	2.30	3.31
600-8-z	1.17	1.96	1.95	1.56	2.54	2.63
300-20-111	0.52	0.43	0.44	0.59	0.63	0.78
500-25-308	0.47	0.58	0.61	0.54	0.73	0.76
25-eil101	0.04	0.03	0.03	0.09	0.18	0.33
144-rat783	0.16	0.20	0.19	0.18	0.24	0.24
Average values	0.67	0.82	0.83	0.80	1.14	1.35

Id.	GPR1R2 Results		Best gap (%)			Average gap (%)
	Best	Average	HHGILS	G	HHGILS	G
J21	70651	73311.92	-5.11	0.40	-6.89	2.94
J22	72512	74871.65	-5.14	0.54	-5.65	2.87
J23	72889	74621.57	-8.31	0.08	-7.42	2.68
J24	66747	68955.78	-4.68	0.66	-6.25	2.33
J25	66444	69141.16	-4.34	0.26	-6.36	2.96
J26	12278	12531.44	-2.38	0.32	-3.37	1.03
J27	17252	17589.12	-0.84	0.06	-1.55	1.36
J28	25124	25761.53	0.56	0.03	-0.79	0.65
J29	33110	33692.73	-2.57	0.52	0.20	1.19
J30	38767	39478.00	-2.86	0.01	-2.65	0.43
J31	116473	118297.46	-0.19	0.86	-1.20	0.53
J32	37529	38861.78	-2.88	0.25	-4.01	1.11
J33	37440	38765.91	-1.02	0.17	-3.41	1.29
J34	37262	39253.08	-1.37	0.91	-3.95	0.49
J35	37704	38699.53	-2.77	0.19	-4.09	1.64
Average values	-	-	-2.93	0.35	-3.83	1.57

Instances	Concorde		HHGILS
	Best	Time (s)	Gap (%)	Time (s)
300-6	774	21.00	1.94	15.00
350-6	894	27.79	5.59	15.00
400-6	885	58.88	4.52	15.00
450-6	945	34.77	6.46	15.00
500-6	1063	41.08	6.49	15.00
550-20	1419	51.55	4.09	45.00
600-20	1475	50.02	6.10	45.00
650-20	1545	71.03	5.57	45.00
700-20	1625	69.47	6.77	45.00
750-25	1694	23.89	7.50	45.00
800-25	1784	30.61	6.39	70.00
850-25	1882	114.19	6.59	70.00
900-25	1949	38.50	9.08	70.00
950-30	2072	121.58	9.89	70.00
1000-30	2144	61.99	9.93	70.00
Average values		54.42	6.46	43.33

[TFN1] a vertices (vl, vl+1, vl+2), b initial solution, c final solution

vertex vk	x-coordinate	y-coordinate	cluster Vi
1	177	311	1
2	924	320	3
3	265	52	1
4	45	500	3
5	233	300	1
6	167	104	3
7	117	275	1
8	190	78	3
9	61	269	2
10	684	539	2
11	380	526	1
12	550	549	2

vertex vk	x-coordinate	y-coordinate	cluster Vi
1	177	311	1
2	924	320	3
3	265	52	1
4	45	500	3
5	233	300	1
6	167	104	3
7	117	275	1
8	190	78	3
9	61	269	2
10	684	539	2
11	380	526	1
12	550	549	2

Brasil

Brasil

A HYBRID HEURISTIC ALGORITHM FOR THE CLUSTERED TRAVELING SALESMAN PROBLEM

ABSTRACT

vertex vk	x-coordinate	y-coordinate	cluster Vi
1	177	311	1
2	924	320	3
3	265	52	1
4	45	500	3
5	233	300	1
6	167	104	3
7	117	275	1
8	190	78	3
9	61	269	2
10	684	539	2
11	380	526	1
12	550	549	2