. 2026 May 26;16:27069. doi: 10.1038/s41598-026-53013-5

Table 10.

Comparison results of RZOA and DSMO on TSP instances.

Problem	Optimal	Algorithm	Best	Average	SD	PDB (%)	PDA (%)	Rank
eil-51	426	RZOA	426	427	0.83	0.00	0.23	1
eil-51	426	DSMO	429	437	4.73	0.70	2.58	2
berlin-52	7542	RZOA	7542	7566	10.70	0.00	0.32	1
berlin-52	7542	DSMO	7544	7634	85.40	0.03	1.22	2
st-70	675	RZOA	675	677	1.43	0.00	0.30	1
st-70	675	DSMO	677	703	15.04	0.30	4.15	2
pr-76	108,159	RZOA	108,159	108,474	85.20	0.00	0.29	1
pr-76	108,159	DSMO	108,159	109,759	850.64	0.00	1.48	2
eil-76	538	RZOA	538	539	0.70	0.00	0.19	1
eil-76	538	DSMO	559	573	7.56	3.90	6.51	2
rat-99	1211	RZOA	1212	1214	1.33	0.08	0.25	1
rat-99	1211	DSMO	1226	1292	21.07	1.24	6.69	2
kro-A100	21,282	RZOA	21,297	21,393	45.20	0.07	0.52	1
kro-A100	21,282	DSMO	21,298	22,024	508.89	0.08	3.49	2
kro-B100	22,141	RZOA	22,158	22,269	79.30	0.08	0.58	1
kro-B100	22,141	DSMO	22,308	23,022	277.32	0.75	3.98	2
rd-100	7910	RZOA	7921	7948	42.90	0.14	0.48	1
rd-100	7910	DSMO	8041	8378	209.4	1.66	5.92	2
eil-101	629	RZOA	630	632.16	1.63	0.16	0.50	1
eil-101	629	DSMO	649	674	10.97	3.18	7.15	2
lin-105	14,379	RZOA	14,393	14,453	18.87	0.10	0.51	1
lin-105	14,379	DSMO	14,383	15,114	500.76	0.03	5.11	2
pr-107	44,303	RZOA	44,329	44,452	42.62	0.06	0.34	1
pr-107	44,303	DSMO	44,386	45,667	1300.43	0.19	3.08	2
pr-124	59,030	RZOA	59,110	59,298	57.32	0.14	0.45	1
pr-124	59,030	DSMO	60,285	62,443	1644.93	2.13	5.78	2
pr-136	96,772	RZOA	96,890	97,900	459.08	0.12	1.17	1
pr-136	96,772	DSMO	97,539	102,872	2855.28	0.79	6.30	2
d-198	15,780	RZOA	15,820	15,930	58.20	0.25	0.95	1
d-198	15,780	DSMO	15,978	16,270	171.20	1.25	3.11	2
kro-A200	29,368	RZOA	29,520	29,819	138.65	0.52	1.54	1
kro-A200	29,368	DSMO	30,481	31,829	652.32	3.79	8.38	2
kro-B200	29,437	RZOA	29,539	29,920	48.94	0.35	1.64	1
kro-B200	29,437	DSMO	30,716	31,782	487.39	4.34	7.97	2
tsp-225	3916	RZOA	3978	4001	15.60	1.58	2.17	1
tsp-225	3916	DSMO	4014	4163	66.08	2.50	6.31	2
pr-226	80,369	RZOA	80,638	81,540	346.96	0.33	1.46	1
pr-226	80,369	DSMO	83,588	85,936	2105.13	4.01	6.93	2
pr-299	48,191	RZOA	48,358	48,923	195.12	0.35	1.52	1
pr-299	48,191	DSMO	50,580	51,748	863.32	4.96	7.38	2
lin-318	42,029	RZOA	42,456	43,661	572.97	1.02	3.88	1
lin-318	42,029	DSMO	44,119	45,460	660.47	4.97	8.16	2
fl-417	11,861	RZOA	12,020	12,154	103.54	1.34	2.47	1
fl-417	11,861	DSMO	12,219	12,951	360.99	3.02	9.19	2
pr-439	107,217	RZOA	107,985	109,937	1010.35	0.72	2.54	1
pr-439	107,217	DSMO	112,105	116,379	2462.82	4.56	8.55	2