Wireless Network Design: Optimization Models and Solution ...

118 Eli Olinick
ficult optimization problems, however it has the potential drawback of converging
to a local optimum. TS is designed to overcome this drawback by allowing nonimproving
moves, making recently used moves “tabu” (i.e., not allowing them) for
a certain number of iterations to avoid cycling back to previously visited neighborhoods,
employing “diversification” strategies to encourage the search to explore new
areas of the feasible region, and using “intensification” strategies to hone in on highquality
solutions. TS has been successfully applied to a wide range of optimization
problems [32]. In this case, the moves in the LS are adding a tower to ¯L, removing
a tower from ¯L, or doing both simultaneously.
The heuristics presented in [7] were also tested on 10 medium (|L| = 120 and
|M| = 400) and 5 large (|L| = 200 and |M| = 750) problem instances. The service
areas for the medium and large instances are 1 km by 1 km, and 1.5 km by 1.5
km, respectively. The demands in the medium size instances are randomly selected
from {1,2,3} and the demands in the larger instances are randomly selected from
{1,2}. Using tabu search, the authors found solutions that satisfy all the demand in
the medium instances using 40 to 48 towers in an average CPU time on the order
of 2 hours and 40 minutes. For the larger instances, the tabu search took on the
order of 12 hours and the best solutions found covered at least 99% of the demand
using 54 to 64 towers. Note that the results described above are for multistart TSs.
That is, the tabu search was run until certain stopping criteria were met and then
restarted with another Add or Remove solution. This process was repeated 50 times
for each problem instance. The authors report that solutions of comparable quality
were found in about half the CPU time by making a single run of Remove followed
by a longer tabu search.
For a given set of selected towers Amaldi et al. [7] determine optimal power
levels in the SIR-based model by assigning each test point to the nearest available
tower and solving the resulting system of linear equations implied by constraint set
(5.15) [7]. This results in considerably longer solution times compared to using the
power-based model, but with a considerable savings in the number of towers needed
to satisfy all the demand. For example, on the medium sized problem instances the
tabu search algorithms took approximately 6 times longer with the SIR-based model
but found solutions that used at at least 20% fewer towers than those found with the
power-based model. Due to the relatively long running time on the small problem
instances, the authors did not attempt to solve the medium or large instances of
ACMIP to provable optimality with CPLEX. In the next section we discuss ways to
improve the running time for CPLEX on power-based models.
5.4.3 Improving Branch-and-Bound Performance
The observation made by Amaldi et al. [7] that optimal subscriber assignments can
be found by only considering assignment of test points to their nearest selected
towers can be exploited in two different ways to improve the performance of the
branch-and-bound process when applied to these types of network design problems.