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

Table 5.

Comparison results of RZOA and DDCS 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	DDCS	426	429	2.67	0.00	0.70	2
berlin-52	7542	RZOA	7542	7566	10.70	0.00	0.32	2
berlin-52	7542	DDCS	7542	7542	0.00	0.00	0.00	1
st-70	675	RZOA	675	677	1.43	0.00	0.30	1
st-70	675	DDCS	675	679	3.89	0.00	0.59	2
pr-76	108,159	RZOA	108,159	108,474	85.20	0.00	0.29	1
pr-76	108,159	DDCS	108,159	110,341	1155.50	0.00	2.02	2
eil-76	538	RZOA	538	539	0.70	0.00	0.19	1
eil-76	538	DDCS	538	541	2.86	0.00	0.56	2
rat-99	1211	RZOA	1212	1214	1.33	0.08	0.25	1
rat-99	1211	DDCS	1212	1223	11.07	0.08	0.99	2
kro-A100	21,282	RZOA	21,297	21,393	45.20	0.07	0.52	1
kro-A100	21,282	DDCS	21,282	21,444	156.23	0.00	0.76	2
kro-B100	22,141	RZOA	22,158	22,269	79.30	0.08	0.58	1
kro-B100	22,141	DDCS	22,199	22,357	90.98	0.26	0.98	2
kro-C100	20,749	RZOA	20,770	20,838	42.50	0.10	0.43	1
kro-C100	20,749	DDCS	20,749	21,011	166.59	0.00	1.26	2
kro-D100	21,294	RZOA	21,325	21,405	43.10	0.15	0.52	1
kro-D100	21,294	DDCS	21,374	21,860	316.32	0.38	2.66	2
kro-E100	22,068	RZOA	22,079	22,151	49.10	0.05	0.38	1
kro-E100	22,068	DDCS	22,121	22,227	109.96	0.24	0.72	2
rd-100	7910	RZOA	7921	7948	42.90	0.14	0.48	1
rd-100	7910	DDCS	7916	8046	82.47	0.08	1.72	2
lin-105	14,379	RZOA	14,393	14,453	18.87	0.10	0.51	1
lin-105	14,379	DDCS	14,379	14,543	168.62	0.00	1.14	2
pr-107	44,303	RZOA	44,329	44,452	42.62	0.06	0.34	2
pr-107	44,303	DDCS	44,303	44,437	88.78	0.00	0.30	1
pr-124	59,030	RZOA	59,110	59,298	57.32	0.14	0.45	1
pr-124	59,030	DDCS	59,030	59,300	227.64	0.00	0.46	2
bier-127	118,282	RZOA	118,420	119,893	525.61	0.12	1.36	2
bier-127	118,282	DDCS	118,282	119,885	1067.82	0.00	1.36	1
ch-130	6110	RZOA	6168	6250	51.89	0.95	2.29	1
ch-130	6110	DDCS	6140	6269	64.04	0.49	2.60	2
pr-136	96,772	RZOA	96,890	97,900	459.08	0.12	1.17	1
pr-136	96,772	DDCS	97,144	101,512	1312.63	0.38	4.90	2
pr-144	58,537	RZOA	58,621	59,057	236.48	0.14	0.89	2
pr-144	58,537	DDCS	58,537	58,697	90.85	0.00	0.27	1
ch-150	6528	RZOA	6547	6595	26.31	0.29	1.03	2
ch-150	6528	DDCS	6544	6560	13.93	0.25	0.49	1
d-198	15,780	RZOA	15,820	15,930	58.20	0.25	0.95	1
d-198	15,780	DDCS	15,846	15,933	72.86	0.42	0.97	2
kro-A200	29,368	RZOA	29,520	29,819	138.65	0.52	1.54	1
kro-A200	29,368	DDCS	29,562	29,837	140.30	0.66	1.60	2
kro-B200	29,437	RZOA	29,539	29,920	48.94	0.35	1.64	1
kro-B200	29,437	DDCS	29,927	30,406	277.08	1.66	3.29	2
tsp-225	3916	RZOA	3978	4001	15.60	1.58	2.17	1
tsp-225	3916	DDCS	3957	4016	34.21	1.05	2.55	2
pr-226	80,369	RZOA	80,638	81,540	346.96	0.33	1.46	1
pr-226	80,369	DDCS	80,860	81,549	557.77	0.61	1.47	2