Vehicle Routing Problem with Transshipment: Mathematical Model and Algorithm

Table 5

Comparison between the best-found solutions from TA-NS and the lower bound (optimal solution of VRP) of set A.

Problem instance	Lower bound (optimal CVRP)	No. of TD	Upper bound CVRPT	Best Sol. CVRPT	%Diff from lower bound (%)	Best sol. CVRPOT	# of TD satisfied	%Diff from lower bound (%)
A n32 k5	784	6	1065	934	19.13	862	3	9.95
A n33 k5	661	6	807	775	17.25	762	5	15.28
A n33 k6	742	6	862	847	14.15	809	5	9.03
A n34 k5	778	7	1015	888	14.14	888	7	14.14
A n36 k5	799	7	1138	1019	27.53	928	6	16.15
A n37 k5	669	7	—	839	25.41	743	4	11.06
A n37 k6	949	7	1395	1141	20.23	989	4	4.21
A n38 k5	730	8	—	948	29.86	847	7	16.03
A n39 k5	822	8	—	935	13.75	909	7	10.58
A n39 k6	831	8	1117	1055	26.96	978	7	17.69
A n44 k6	937	9	—	1175	25.40	1104	7	17.82
A n45 k6	944	9	—	1243	31.67	1073	5	13.67
A n45 k7	1146	9	—	1354	18.15	1331	7	16.14
A n46 k7	914	9	—	1139	24.62	1013	6	10.83
A n48 k7	1073	10	—	1490	38.86	1216	6	13.33
A n53 k7	1010	11	—	1402	38.81	1188	6	17.62
A n54 k7	1167	11	—	1446	23.91	1336	7	14.48
A n55 k9	1073	11	—	1350	25.82	1203	7	12.12
A n60 k9	1354	12	—	1790	32.20	1495	5	10.41
A n61 k9	1035	12	—	1297	25.31	1194	8	15.36
A n62 k8	1288	12	—	1744	35.40	1484	7	15.22
A n63 k9	1634	13	—	1983	21.36	1851	7	13.28
A n63 k10	1315	13	—	1749	33.00	1518	6	15.44
A n64 k9	1402	13	—	1802	28.53	1558	7	11.13
A n65 k9	1177	13	—	1647	39.93	1378	9	17.08
A n69 k9	1159	14	—	1588	37.01	1357	9	17.08
A n80 k10	1764	16	—	2214	25.51	2047	10	16.04