Wireless Network Design: Optimization Models and Solution ...

112 Eli Olinick
Using bk to denote the cost of using an MTSO at location k, cℓk to denote the cost
of connecting tower location ℓ to MTSO location k, and h jk to represent the cost of
providing a backbone link between locations j and k, CKKOIP is stated as
max r ∑ ∑ xmℓ
m∈M ℓ∈Lm
� �� �
Subscriber
revenue
∑ ∑
− ∑ aℓyℓ
ℓ∈L
� �� �
Tower
cost
cℓksℓk − ∑ h jkw jk
ℓ∈L k∈K
j∈K k∈K0\{ j}
− ∑ bkzk −
k∈K
� �� �
MTSO
cost
� �� � � �� �
Connection Backbone
cost
cost
subject to (5.9), (5.19)-(5.22), (5.27)-(5.34), and the following sets of constraints on
the domains of the decision variables
∑
.
(5.35)
sℓk ∈ {0,1} ∀ℓ ∈ L,k ∈ K, (5.36)
u jk ∈ N ∀ j ∈ K,k ∈ K0 \ { j}, (5.37)
w jk ∈ {0,1} ∀ j ∈ K,k ∈ K0 \ { j}, (5.38)
zk ∈ {0,1} ∀k ∈ K. (5.39)
St-Hilaire et al. [55] propose a generalization of CKKOIP that includes differentiated
service classes. Their model has separate demand values for voice and data
service at each test point. Consequently, the backbone must connect the MTSOs to a
gateway to a public data network for data services in addition to the PSTN gateway.
They use their model to demonstrate the trade-off between solving an integrated
model and using a decomposition approach that first solves the tower location and
subscriber assignment subproblem, then determines which MTSOs to select, and finally
selects the gateway nodes and finds an optimal backbone topology. They point
out that the second approach is faster, but not necessarily optimal, and demonstrate
this with a case study.
Cai et al. [16] extend the CKKOIP model so that it selects at least two PSTN
gateways from a set of candidate locations, and makes the network more reliable
by using a biconnected backbone network for the selected MTSOs and PSTN gateways.
Thus, the backbone in [16] is required to be reliable enough that it will not
be disconnected by any single-node failure. This is the biconnectivity augmentation
problem [25] which is known to belong to the class of NP-hard problems [22] and
has been extensively studied as a fundamental problem in network design. Khuller
[36] presents approximation results for this and a series of related network design
problems. Surveys of the integer programming literature related to the biconnectivity
augmentation problem may be found in Fortz et al. [24] and Grötschel et al.
[28, 29]. In addition to making the backbone biconnected, the model in [16] requires
each tower to be connected to two different MTSOs so that if one MTSO fails the
tower is still connected to the backbone.