Solving the Traveling Salesman Problem: A Modified Metaheuristic Algorithm

Table 2

The results of the proposed algorithm for the TSP instances.

	Instance	n	WS (PSO)		AS (PSO)		BS (PSO)		WS (MPSO)		AS (MPSO)		BS (MPSO)		BKS
			Cost	Gap	Cost	Gap	Cost	Gap	Cost	Gap	Cost	Gap	Cost	Gap
1	Burma14	14	3323	0.00	3323	0.00	3323	0.00	3323	0.00	3323	0.00	3323	0.00	3323
2	Gr17	17	2085	0.00	2085	0.00	2085	0.00	2085	0.00	2085	0.00	2085	0.00	2085
3	Gr21	21	2707	0.00	2707	0.00	2707	0.00	2707	0.00	2707	0.00	2707	0.00	2707
4	Gr24	24	1272	0.00	1272	0.00	1272	0.00	1272	0.00	1272	0.00	1272	0.00	1272
5	Fri26	26	937	0.00	937	0.00	937	0.00	937	0.00	937	0.00	937	0.00	937
6	Bayg29	29	1610	0.00	1610	0.00	1610	0.00	1610	0.00	1610	0.00	1610	0.00	1610
7	Bays29	29	2020	0.00	2020	0.00	2020	0.00	2020	0.00	2020	0.00	2020	0.00	2020
8	Arr48	48	10983	3.34	10699	0.67	10628	0.00	10628	0.00	10628	0.00	10628	0.00	10628
9	Eil51	51	454	6.57	429	0.70	426	0.00	426	0.00	426	0.00	426	0.00	426
10	Pr76	76	113545	4.98	115003	6.33	109870	1.58	108159	0.00	108159	0.00	108159	0.00	108159
11	Rat99	99	1397	15.36	1373	13.38	1294	6.85	1211	0.00	1211	0.00	1211	0.00	1211
12	KroA100	100	23342	9.68	21843	2.64	21282	0.00	21352	0.33	21292	0.05	21282	0.00	21282
13	KroB100	100	23424	5.79	22964	3.72	22534	1.77	22141	0.00	22141	0.00	22141	0.00	22141
14	Rd100	100	8601	8.74	8285	4.74	8047	1.73	8135	2.84	8054	1.82	7910	0.00	7910
15	Eil101	101	699	11.13	657	4.45	643	2.23	629	0.00	629	0.00	629	0.00	629
16	Lin105	105	15897	10.56	15142	5.31	14910	3.69	14379	0.00	14379	0.00	14379	0.00	14379
17	Pr107	107	48933	10.45	46761	5.55	45602	2.93	44303	0.00	44303	0.00	44303	0.00	44303
18	Pr124	124	65911	11.66	63660	7.84	61801	4.69	59555	0.89	59113	0.14	59030	0.00	59030
19	Bier127	127	130110	10.00	129379	9.38	127830	8.07	118282	0.00	118282	0.00	118282	0.00	118282
20	Ch130	130	6779	10.95	6454	5.63	6299	3.09	6255	2.37	6176	1.08	6110	0.00	6110
21	Pr136	136	105839	9.37	102183	5.59	99575	2.90	97972	1.24	97543	0.80	96772	0.00	96772
22	Pr144	144	64869	10.82	62129	6.14	60293	3.00	58972	0.74	58612	0.13	58537	0.00	58537
23	Ch150	150	7325	12.21	6974	6.83	6724	3.00	6659	2.01	6579	0.78	6528	0.00	6528
24	KroA150	150	29488	11.17	28328	6.80	27720	4.51	26952	1.61	26711	0.71	26524	0.00	26524
25	Pr152	152	81378	10.44	77350	4.98	75492	2.46	74162	0.65	73915	0.32	73682	0.00	73682
26	Rat195	195	2686	15.63	2498	7.53	2383	2.58	2401	3.36	2351	1.21	2323	0.00	2323
27	D198	198	17828	12.98	16849	6.77	16653	5.53	16222	2.80	16024	1.55	15780	0.00	15780
28	KroA200	200	32799	11.68	31295	6.56	30249	3.00	31052	5.73	29818	1.53	29533	0.56	29368
29	Ts225	225	138929	9.70	133210	5.19	129442	2.21	129157	1.99	128385	1.38	126643	0.00	126643
30	Pr226	226	89207	11.00	85649	6.57	82761	2.98	81279	1.13	80990	0.77	80545	0.22	80369
31	Pr264	264	56140	14.26	53289	8.45	50205	2.18	50945	3.68	50178	2.12	49325	0.39	49135
32	A280	280	2883	11.79	2814	9.11	2658	3.06	2795	8.38	2632	2.06	2598	0.74	2579
33	Pr299	299	53016	10.01	51404	6.67	49582	2.89	49105	1.90	48589	0.83	48332	0.29	48191
34	Lin318	318	50215	19.48	47172	12.24	45311	7.81	45812	9.00	45391	8.00	43710	4.00	42029
35	Linhp318	318	49201	19.00	46356	12.12	44653	8.00	45066	9.00	44653	8.00	41345	0.00	41345
36	Rd400	400	18084	18.34	17215	12.66	16503	8.00	16656	9.00	16503	8.00	15892	4.00	15281
37	Pr439	439	127588	19.00	120083	12.00	115794	8.00	116967	9.09	115994	8.19	111875	4.34	107217
38	Si535	535	57256	18.18	55264	14.06	52326	8.00	52811	9.00	52326	8.00	50388	4.00	48450
39	U574	574	43917	19.00	41134	11.46	39857	8.00	40296	9.19	39757	7.73	36905	0.00	36905
40	Rat575	575	8060	19.00	7586	12.00	7215	6.53	7383	9.01	7355	8.59	7014	3.56	6773