Wireless Network Design: Optimization Models and Solution ...

10 Integer Programming Models for Power-optimal Trees in Wireless Networks 237
∑ j:(t, j)∈A xt t j − ∑ j:( j,t)∈A xt jt =
�
∑ j:(t, j)∈A qt t j − qs � + �
t j − ∑ j:( j,t)∈A qt jt − qs � +
jt =
�
−∑j:( j,t)∈A z jt − qs � +
jt = −∑j:(t, j)∈A qs t j = −wt = −1.
The equalities follow from applying the definition of xt , the facts that qt t j = 0 and
qt jt = z jt, (10.20), (10.18), and (10.19), respectively. Thus, xt ∈ F (G,bst). Furthermore,
(10.23) implies (10.8) since xt ≤ qt . Utilizing (10.20) and (10.23), we get
∑ N−1
ℓ=k xt �
πi(ℓ),i = ∑N−1 ℓ=k qt πi(ℓ),i − qs � +
πi(ℓ),i = ∑ N−1
�
ℓ=k −qt (iℓ) + qs � +
(iℓ) ≤ ∑ N−1
ℓ=k qs (iℓ) ≤
∑ N−1
ℓ=k y (iℓ), which implies (10.15). The proof is complete by observing that RAP-F2
and RAP-MT have identical objective functions.
The inequality in Proposition 10.6 may be strict. Some instances where this occurs
are reported in the next section.
10.6 Experimental Results
In this section, we report some experimental results of solving MET and RAP using
the models discussed in the previous sections. Networks of sizes 10, 20, 30, and
40 nodes are used in the experiments. They are obtained by following the instance
generation procedure in [49] with α = 2 in the power formula (Section 10.2.1). For
each network size N, two groups of instances with |D| = N/2 and |D| = N, respectively,
are generated. There are hence eight network groups in total. The number
of instances in each group is 20. For MET, a node in D is randomly chosen as the
source s.
We have applied CPLEX [32] (version 10.1) to all the models. Throughout the
experiments, the CPLEX network optimizer is switched on, and all other solver
parameters follow their default values. All experiments have been conducted on a
server with an Opteron processor at 2.4 GHz and 7 GB RAM. For all models, we
set a time limit of one hour for solving the LP relaxation, and one additional hour
for approaching integer optimum.
For making comparisons among the models, solution time will be used as one
of the performance indicators. However, the aim of the experiments is not to declare
the best model for MET or RAP in terms of solution time. For each of the
models, the time can be indeed improved by, for example, pre-processing, tweaking
the solver parameters, or implementing methods other than using a standard solver.
Instead of focusing on time only, the experiments are intended to also shed light on
other aspects such as LP strength, size of the branch and bound (B&B) tree, and
scalability.