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In Table 12, we report the results for the hub location problems from the OR library. We omit the results for problems with 10 nodes as all of these problems are solved at the root node in less than one second. We also remove problems of type LL, T L with Q = 1/4 and Q = 1/5 since in these problems the capacity constraints are redundant. A branch and cut algorithm for hub location problems with single assignment 403 Table 12. 19 no. of conc 4 4 2 2 2 2 4 4 3 3 2 2 5 5 2 2 2 2 5 5 4 3 4 3 cover 151 47 0 0 0 0 64 39 18 46 0 0 379 1870 6 3 0 0 396 1473 20 15 10 7 spro 755 73 6 1 13 4 165 133 106 62 4 2 829 7757 16 8 11 4 1231 4743 18 17 115 14 step 1 0 0 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 5 32 0 1 0 0 3 3 2 26 0 2 52 15 8 0 0 0 4 4 198 102 358 336 0 0 3 16 32 0 3 2 0 0 10 15 0 0 4 3 102 69 187 99 0 0 The problems of type T T , LT are infeasible for Q = 1 and Q = 1/2 and are not considered.

Results of Tables 8 and 9 allow to conclude that preprocessing is useful when using formulation QHL3 and confirm that formulation QHL3 performs worse than QHL1 even with preprocessing. Similar tests were also performed for formulation QHL2m and yielded to the same conclusion as for QHL3. In the second column of Table 10, we report the number of hubs installed for each problem. When some of the nodes are assigned to themselves and removed from the problem during preprocessing, we add the number of such nodes to the number of hubs installed.

402 M. Labb´e et al. Table 11. 86 no. of conc 5 4 4 3 3 4 4 3 2 2 2 cover 729 4336 783 34 11 3018 457 24 24 4 3 spro 3852 20291 2704 18 11 15577 1380 7 11 3 4 step 0 1 0 0 0 0 0 0 0 0 0 3 3 2 2 2 2 351 61 14 0 0 0 627 37 3 4 3 4 0 0 0 0 0 0 Before moving to other test instances, we test the following strategies for the unsolved problems f t, 17, 15, 1/2 and f t, 17, 20, 2/3: 1. Use Algorithm 3 during the preprocessing and separate the cover inequalities on linear knapsacks (42) to avoid solving quadratic problems.

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