A Memetic Lagrangian Heuristic for the 0-1 Multidimensional Knapsack Problem

Table 2

Results of memetic Lagrangian heuristic on benchmark data.

Instance	Multistart R-CONS-L^1		Memetic Lagrangian heuristic		Number improved/number equalled
	Result	CPU^2	Result	CPU^2
5.100–0.25	2.40	1435	2.32	1466	0/0
5.100–0.50	1.08	1471	1.08	1500	0/0
5.100–0.75	0.72	1394	0.72	1431	0/0
Average (5.100−*)	1.40	1433	1.37	1466	Total 0/0
5.250–0.25	0.92	3513	0.92	3619	0/0
5.250–0.50	0.49	3606	0.49	3748	0/0
5.250–0.75	0.26	3395	0.25	3529	0/0
Average (5.250−*)	0.56	3505	0.56	3632	Total 0/0
5.500–0.25	0.48	7072	0.48	7124	0/0
5.500–0.50	0.20	7318	0.20	7546	0/0
5.500–0.75	0.14	6846	0.14	6985	0/0
Average (5.500−*)	0.27	7079	0.27	7218	Total 0/0
Average (5.*−*)	0.74	4006	0.73	4105	Total 0/0
10.100–0.25	2.45	2679	2.27	2630	0/0
10.100–0.50	1.46	2717	1.14	2650	0/1
10.100–0.75	0.89	2657	0.69	2598	0/0
Average (10.100−*)	1.60	2684	1.37	2626	Total 0/1
10.250–0.25	1.14	6599	0.88	6441	0/0
10.250–0.50	0.57	6718	0.45	6558	0/0
10.250–0.75	0.35	6535	0.24	6408	0-1
Average (10.250−*)	0.69	6617	0.52	6469	Total 0/1
10.500–0.25	0.58	13196	0.50	12811	0/0
10.500–0.50	0.27	13425	0.24	13057	0/0
10.500–0.75	0.21	13006	0.14	12750	0/0
Average (10.500−*)	0.35	13209	0.29	12873	Total 0/0
Average (10.*−*)	0.88	7504	0.73	7323	Total 0/2
30.100–0.25	4.92	6824	3.19	6796	0/3
30.100–0.50	2.67	6870	1.43	7052	0/3
30.100–0.75	1.56	6817	0.89	6780	0/2
Average (30.100−*)	3.05	6837	1.84	6876	Total 0/8
30.250–0.25	4.24	16819	1.28	16799	2/2
30.250–0.50	2.23	16941	0.59	16958	0/0
30.250–0.75	1.40	16742	0.34	16837	0/0
Average (30.250−*)	2.62	16834	0.74	16865	Total 2/2
30.500–0.25	3.81	33630	0.68	33414	2/0
30.500–0.50	2.13	33802	0.29	33737	1/0
30.500–0.75	1.24	33318	0.19	33440	0/0
Average (30.500−*)	2.39	33583	0.39	33530	Total 3/0
Average (30.*−*)	2.69	19085	0.99	19090	Total 5/10
Total average	1.44	10198	0.82	10173	Total 5/12

 ¹Multistart R-CONS-L returns the best result from 5,000 independent runs of R-CONS-L.
²Average CPU seconds on Pentium III 997 MHz.