Wireless Network Design: Optimization Models and Solution ...

5 Mathematical Programming Models for Third Generation Wireless Network Design 111
at location k. The model imposes the following constraint sets to ensure that each
tower is connected to a selected MTSO
∑ sℓk = yℓ
k∈K
∀ℓ ∈ L, (5.27)
sℓk ≤ zk ∀ℓ ∈ L,∀k ∈ K. (5.28)
The model assumes that each MTSO has capacity to connect to at most α towers
and enforces this capacity limit with the constraint set
∑ sℓk ≤ αzk
ℓ∈L
∀k ∈ K. (5.29)
To model the construction of the backbone network, CKKOIP uses a set of network
flow constraints that describe a spanning tree structure that is rooted at the
PSTN gateway and connects all of the selected MTSO locations 1 . The basic idea is
that a unit of flow is generated at MTSO location k if zk = 1, and that unit of flow
either goes directly to the PSTN gateway or to a neighboring MTSO on the backbone.
The model uses the binary variable w jk to indicate whether or not there is a
backbone link between MTSO location j ∈ K and MTSO/PSTN gateway location
k ∈ K0, and integer variable u jk to represent the amount of flow on that link. Note
that u jk does not represent the amount of telecommunications traffic on link ( j,k),
but rather the number of MTSO’s that are connected to the PSTN gateway via a path
using the link between locations j and k. The constraints that construct the backbone
network are
z0 = 1, (5.30)
∑ uk j + uk0 = ∑ u jk + zk ∀k ∈ K, (5.31)
j∈K\{k}
j∈K\{k}
∑ uk0 = ∑ zk, (5.32)
k∈K k∈K
u jk ≤ |K|zk ∀ j ∈ K,k ∈ K0 \ { j}, (5.33)
u jk ≤ |K|w jk ∀ j ∈ K,k ∈ K0 \ { j}. (5.34)
Constraint (5.30) ensures that the backbone network includes the PSTN gateway.
The second set of constraints, (5.31), are for flow conservation. They state, for
MTSO k, that the out-flow is equal to the in-flow plus the generated flow, zk. Constraint
set (5.32) ensures that the total amount of flow reaching the PSTN gateway
is equal to the number of selected MTSOs. Constraint sets (5.33) and (5.34) ensure
that there is flow on the link between MTSO k and MTSO j (or the PSTN gateway)
only if both locations are selected for the design and connected by a direct link in
the backbone network.
1 Editor’s note: Network flow constraints are also used to model spanning tree structures in Chapter
10.