On the Flexibility of Constraint Programming Models From Sin(2)

on the cost are not explicitly computed in constraint programming but are rather a consequence of constraint propagation, we cannot easily record the exceeding lower bounds at the leaves of the current search tree. Instead, we base our computation of a new upper bound on the greatest depth achieved in the previous iteration, according to n Bi+1= (1+ i d ) Bii

where B1 is the optimal cost of the tsptw instance obtained by relaxing (6'), di is the maximum depth reached at iteration i and 1= 0:03; 2= 0:06. We allow a maximum of four iterations: the rst one with initial upper bound B1 in case the optimal solution of the relaxed tsptw instance remains feasible for the tspmtw instance; the next two iterations with increasing computed bounds B2 and B3; the nal iteration with no bound at all, to nd a solution if there is one. Of course, as soon as an iteration has successfully terminated with a solution, we do not perform the following ones. The intuition behind the use of ratio n=di 5

1 Centre de recherche sur les transports, Universit'e de Montr'eal,

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1 Centre de recherche sur les transports, Universit'e de Montr'eal,

thus preserving its span.

high density, coarse-grained (type D): randomly pick the number of holes

in the window, between 1 and 4; randomly select their location; randomly determine their size, between 1% and 5% of the original window. medium density, ne-grained (type M): 10 (sub-)windows in all, but 2 must be located respectively at the beginning and the end of the original window; randomly select the location of the 8 remaining (sub-)windows; randomly determine their size, between 3% and 7% of the original window. medium density, coarse-grained (type m): 2 (sub-)windows, leaving one gap; randomly determine their size, between 20% and 30% of the original window. heterogeneous (type H): for each city, choose with equal probability one of the above three types or the original window. The 108 instances thus generated are grouped and identi ed according to the tsptw instance from which they originate and di erentiated by the appropriate su x according to their type. For example, rc205.2-M refers to the type M instance generated from the second route associated with instance rc205 in Solomon's test set.

4 Computational resultsWe report in this section the performance of our algorithm on the problems described in section 3. The tests were carried out on a Sun SS1000 as in 6]. However by the time we ran our experiments, the tsptw algorithm had been re-implemented using ilog Solver 4], a C++ library implementing constraint programming for combinatorial problems. This allows a comparison between the e ciency of the two cp languages used: we observed that the new implementation is roughly 5 times faster. We obtained solutions (or proved infeasibility in 4 cases) for about half of the problems generated (63) within our preset time limit. Most of these (55) were solved to optimality. These statistics are not surprising since about the same proportion of original RC2 tsptw instances had been solved in 6]. The optimal values for the tspmtw instances solved are listed in the appendix. Missing vertical lines in the gures 3 to 6 represent infeasible instances or instances for which feasibility or infeasibility could not be proven within the preset time limit. In gures 3 and 4, the top graphs give the ratios of the solution cost for the tspmtw instance over the solution cost for the corresponding tsptw instance. We can make the following observations: 1. The optimal cost of type D instances is almo

st always the same as that of the original instance. 7

1 Centre de recherche sur les transports, Universit'e de Montr'eal,

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