Wireless Network Design: Optimization Models and Solution ...

10 Integer Programming Models for Power-optimal Trees in Wireless Networks 239
than a few percent in average. One can also observe that the LP gap tends to grow
with instance size for all the three models.
Table 10.2 summarizes the results of approaching integer solutions of MET. For
each network group, the first row shows, within brackets, the numbers of instances
(among a total of 20) solved to integer optimality by the models within the time
limit. Then, the maximum, minimum, and average values of three performance indicators
are displayed. The first indicator is the final gap. If an instance is solved to
optimality, the gap is zero. Otherwise it is the difference between the best known
lower and upper bounds reported by the solver when B&B terminates by time limit.
If all 20 instances are solved to optimality, we use ’–’ to compactly denote zero gap.
The second and third performance indicators are the solution time spent and the size
of the B&B tree, respectively, for reaching the reported gap values.
From Table 10.2, we observe that, for networks of up to 20 nodes, integer optimum
can be computed in a short amount of time by any of the models. The number
of instances solved to optimality drops quickly for MET-Das and MET-F1, when
N ≥ 30. For N = 40, integer optimum becomes out of reach for the two models, and
the remaining gap after running B&B for one hour is large. For MET-F2, integer
optimum is guaranteed for size up to N = 40 and |D| = 20. For the largest network
group with |D| = 40, more than half of the instances are solved to optimality by
MET-F2, and the average remaining gap for the rest of the instances is quite small.
MET-F2 also requires less computing time than the other two models. Thus MET-
F2 scales better than MET-Das and MET-F1 in approaching integer solutions.
What has been observed above is a typical scenario for integer programming
models that differ in size as well as the LP bound. MET-Das and MET-F1 are
much smaller in size than MET-F2, thus solving LP-MET-Das and LP-MET-F1
is very fast. On the other hand, LP-MET-Das and LP-MET-F1 give significantly
weaker bounds than LP-MET-F2. As a result, applying B&B to MET-Das and
MET-F1 requires magnitudes more nodes for reaching optimum in comparison to
MET-F2. For MET, a model with tight LP bound, although requires substantially
more computation per node in the B&B tree, translates into moderate tree size and
better overall scalability.
10.6.2 Results for RAP
In the computational experiments of the RAP models, we have chosen to enhance
the two models RAP-Das and RAP-MG by means of valid inequalities. Without
any valid inequality, the two models behave similarly to MET-Das and MET-F1,
respectively. The valid inequalities originate from the work by Montemanni et al.
[40]. In the reference, three and six sets of valid inequalities are presented for RAP-
Das and RAP-MG, respectively. We refer to [40] for details. All these inequalities
have been included in the experiments. We denote the resulting models by RAP-
Das+ and RAP-MG+. As [40] deals with ARAP (i.e., N = |D|), we have adapted
some of the inequalities in order to generalize them to SRAP.