Exact Algorithms for Practical Instances of the Railcar Loading Problem at Marine Container Terminals

Table 2

Empirical results.

	CPLEX	Stage 1	Stage 2	Stage 3
Instance 1	#20′ = 30 #40′ = 10 #Car = 20
Objective value	0.600	0.500	0.600	0.600
CPU time (sec)	2562.758	0.039	0.182	0.182
CPU time saved	—	100.00%	99.99%	99.99%
Optimality gap	—	16.67%	0.00%	0.00%
Instance 2	#20′ = 30 #40′ = 10 #Car = 25
Objective value	—	0.400	0.480	0.480
CPU time (sec)	172,800∗	0.086	0.240	0.240
CPU time saved	—	100.00%	100.00%	100.00%
Optimality gap	—	16.67%	0.00%	0.00%
Instance 3	#20′ = 30 #40′ = 10 #Car = 30
Objective value	0.617	0.500	0.600	0.617
CPU time (sec)	111,617.903	0.151	0.774	1.015
CPU time saved	—	100.00%	100.00%	100.00%
Optimality gap	—	18.92%	2.70%	0.00%
Instance 4	#20′ = 60 #40′ = 20 #Car = 30
Objective value	—	0.667	0.800	0.817
CPU time (sec)	172,800∗	0.192	0.564	1.583
CPU time saved	—	100.00%	100.00%	100.00%
Optimality gap	—	18.37%	2.04%	0.00%
Instance 5	#20′ = 60 #40′ = 20 #Car = 35
Objective value	—	0.571	0.700	0.700
CPU time (sec)	172,800∗	0.104	7.518	7.518
CPU time saved	—	100.00%	100.00%	100.00%
Optimality gap	—	18.37%	0.00%	0.00%
Instance 6	#20′ = 10 #40′ = 30 #Car = 15
Objective value	0.900	0.800	0.900	0.900
CPU time (sec)	128,876.324	5.230	5.421	5.421
CPU time saved	—	100.00%	100.00%	100.00%
Optimality gap	—	11.11%	0.00%	0.00%
Instance 7	#20′ = 10 #40′ = 30 #Car = 20
Objective value	—	0.600	0.800	0.800
CPU time (sec)	172,800∗	18.953	19.359	19.359
CPU time saved	—	99.99%	99.99%	99.99%
Optimality gap	—	25.00%	0.00%	0.00%
Instance 8	#20′ = 15 #40′ = 45 #Car = 20
Objective value	—	0.920	0.960	0.960
CPU time (sec)	172,800∗	60.021	1252.628	1252.628
CPU time saved	—	99.97%	99.28%	99.28%
Optimality gap	—	4.17%	0.00%	0.00%

∗The CPU time is limited to about 2 days.