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Solution Algorithms for the Capacitated Single Allocation Hub Location Problem. Ann. Oper. Res. 86, 141–159 (1999) 11. : Telecommunication and Location. In: Facility Location: Applications and Theory, Z. W. ), Springer, 2003, pp. 275–305 12. : The primal-dual method for approximation algorithms and its application to network design problems. S. ), PWS Publishing Company, 1997, pp. 144–191 13. : Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin, 1988 14. : Cover Inequalities for 0-1 Linear Programs: Computation.

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.

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. We also report the number of violated inequalities of each family in Table 10. We observe that the step inequalities are violated when Q is large.

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