Wireless Network Design: Optimization Models and Solution ...

5 Mathematical Programming Models for Third Generation Wireless Network Design 121
on a series of 300 KKOIP problem instances. The largest problem instances in the
study have 40 candidate tower locations and 250 test points with demands generated
from the discrete uniform distribution on the range [1,32]. Using the default
CPLEX settings and KKOIP without the branching rule or cuts, they solved 10 of
the largest problem instances in an average CPU time of over 7 hours on a Compaq
AlphaServer DS20E with dual EV 6.7(21264A) 667MHz processors and 4,096MB
of RAM. The average and maximum optimality gap of the Phase II solutions were
15.41% and 49.69%, respectively. Using a 1% relative optimality gap as a stopping
criterion for Phase I, the branching rule, and both sets of cuts, they solved a set of
100 of these problems (including the 10 mentioned above) in an average CPU time
of 13 minutes. In these runs the average and maximum relative optimality gap of the
Phase II solutions were reduced to 1.18% and 4.27%, respectively.
Overall, the results in [34] indicate that using the branching rule significantly
reduces CPU time and that both cuts (5.45) and (5.47) improve CPLEX’s performance;
as a general rule, the authors recommend using all three for solving CDMA
network design problems. However, in instances where there are a relatively large
number of tower locations within range of each test point (i.e., |Lm| is close to |L|
for most test points) adding cut set (5.45) to the model can actually increase solution
time rather significantly. Section 5.4.5 describes an additional strategy that has been
shown to be effective in these cases.
The core model presented in this chapter is arguably the most straight-forward, or
“natural”, framework to formulate 3G network design problems, however it is by no
means the only practical approach. Indeed, D’Andreagiovanni [18], D’Andreagiovanni
et al. [19], and Mannino et al. [41] report encouraging computational results with
recently proposed alternative formulations. The details of these formulations are
beyond the scope of this chapter, but in general the idea is to enforce the QoS requirements
without using constraints of the form of (5.7) and thereby avoid the
computational difficulties arising from the use of large βℓ-type constants and from
the potentially wide range of attenuation factors in a given problem instance.
5.4.5 Reducing Problem Size
In their computational study of KKOIP, Kalvenes et al. [34] found that problem
difficulty increases with subscriber density. Subscriber density can be measured in
terms of the number of allowable subscriber-to-tower assignments. In an extreme
case where Lm = M for every test point there are |L||M| subscriber assignment variables
(x’s). This happens, for example, in problem instances generated using the
parameters given in [7]. This leads to integer programs with more decision variables
and denser constraint matrices, both of which contribute to longer solution
times. Thus, a practical strategy for finding high-quality solutions to dense problem
instances is to limit Lm to a relatively small number of the closest tower locations
to test point m. As with using the relative optimality gap as a stopping criterion for
branch-and-bound, restricting the size of Lm also trades solution quality for faster