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

Table 6.

Comparison results of RZOA and DKA 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	DKA	427	430	3.14	0.23	0.94	2
berlin-52	7542	RZOA	7542	7566	10.70	0.00	0.32	1
berlin-52	7542	DKA	7542	7646	97.12	0.00	1.38	2
st-70	675	RZOA	675	677	1.43	0.00	0.30	1
st-70	675	DKA	675	684.10	4.81	0.00	1.35	2
pr-76	108,159	RZOA	108,159	108,474	85.20	0.00	0.29	1
pr-76	108,159	DKA	108,304	110,135	1311.37	0.13	1.83	2
eil-76	538	RZOA	538	539	0.70	0.00	0.19	1
eil-76	538	DKA	539	548	5.69	0.19	1.86	2
rat-99	1211	RZOA	1212	1214	1.33	0.08	0.25	1
rat-99	1211	DKA	1211	1225	11.14	0.00	1.16	2
kro-A100	21,282	RZOA	21,297	21,393	45.20	0.07	0.52	1
kro-A100	21,282	DKA	21,282	21,437	143.48	0.00	0.73	2
kro-B100	22,141	RZOA	22,158	22,269	79.30	0.08	0.58	1
kro-B100	22,141	DKA	22,258	22,454	166.77	0.53	1.41	2
kro-C100	20,749	RZOA	20,770	20,838	42.50	0.10	0.43	1
kro-C100	20,749	DKA	20,769	21,081	232.09	0.10	1.60	2
kro-D100	21,294	RZOA	21,325	21,405	43.10	0.15	0.52	1
kro-D100	21,294	DKA	21,398	21,750	223.87	0.49	2.14	2
kro-E100	22,068	RZOA	22,079	22,151	49.10	0.05	0.38	1
kro-E100	22,068	DKA	22,146	22,455	203.42	0.35	1.75	2
eil-101	629	RZOA	630	632.16	1.63	0.16	0.50	1
eil-101	629	DKA	641	653	5.31	1.91	3.82	2
lin-105	14,379	RZOA	14,393	14,453	18.87	0.10	0.51	1
lin-105	14,379	DKA	14,426	14,687	229.62	0.33	2.14	2
pr-107	44,303	RZOA	44,329	44,452	42.62	0.06	0.34	1
pr-107	44,303	DKA	44,303	44,748	219.43	0.00	1.00	2
pr-124	59,030	RZOA	59,110	59,298	57.32	0.14	0.45	1
pr-124	59,030	DKA	59,164	59,985	590.62	0.23	1.62	2
ch-130	6110	RZOA	6168	6250	51.89	0.95	2.29	1
ch-130	6110	DKA	6201	6308	58.11	1.49	3.24	2
pr-136	96,772	RZOA	96,890	97,900	459.08	0.12	1.17	1
pr-136	96,772	DKA	99,692	102,297	1347.85	3.02	5.71	2
pr-144	58,537	RZOA	58,621	59,057	236.48	0.14	0.89	1
pr-144	58,537	DKA	58,603	59,372	732.71	0.11	1.43	2
ch-150	6528	RZOA	6547	6595	26.31	0.29	1.03	1
ch-150	6528	DKA	6553	6654	91.49	0.38	1.93	2
kro-A200	29,368	RZOA	29,520	29,819	138.65	0.52	1.54	1
kro-A200	29,368	DKA	29,688	30,202	326.46	1.09	2.84	2
kro-B200	29,437	RZOA	29,539	29,920	48.94	0.35	1.64	1
kro-B200	29,437	DKA	30,170	30,912	457.45	2.49	5.01	2
tsp-225	3916	RZOA	3978	4001	15.60	1.58	2.17	1
tsp-225	3916	DKA	3989	4096	65.08	1.86	4.60	2
pr-226	80,369	RZOA	80,638	81,540	346.96	0.33	1.46	1
pr-226	80,369	DKA	81,168	83,232	1120.22	0.99	3.56	2
pr-264	49,135	RZOA	49,455	49,855	144.92	0.65	1.47	1
pr-264	49,135	DKA	49,876	52,502	1390.74	1.51	6.85	2
lin-318	42,029	RZOA	42,456	43,661	572.97	1.02	3.88	1
lin-318	42,029	DKA	43,510	44,380	566.16	3.52	5.59	2