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References 1. : Capacitated Facility Location: Valid Inequalities and Facets. Math. Oper. Res. 20, 562–582 (1995) 2. : Capacitated Facility Location: Separation Algorithms and Computational Experience. Math. Program. 81, 149–175 (1998) 3. : Approximation Algorithms for Access Network Design. Algorithmica 34, 197– 215 (2002) 4. : On the p-Median Polytope. Math. Program. 89, 395–411 (2001) 5. : Facets of the Knapsack Polytope. Math. Program. 8, 146–164 (1975) 6. : OR-Library: distributing test problems by electronic mail.

The number of violated quadratic cover and strengthened projection inequalities also decreases as Q decreases since the capacities become loose. 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.

59% respectively. 56% which are very big. 37% which shows that these were hard problems. The other problems that are not solved to optimality by CPLEX in four hours are solved by the branch and cut algorithm. The longest one f t, 17, 10, 1/3, takes less than one and a half hour. As Q decreases, the duality gap we obtain using the aggregated flow formulation with CPLEX is smaller than the duality gap we obtain using the branch and cut algorithm. For these problems, the two solution methods do not differ much for the CPU time.

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