Wireless Network Design: Optimization Models and Solution ...

106 Eli Olinick
Amaldi et al. [7] propose the following integer programming model, ACMIP, to
minimize the cost of providing service to all test points in the coverage area:
min ∑ aℓyℓ + λ ∑
ℓ∈L m∈M
dm
∑ xmℓ
gmℓ ℓ∈Lm
gmℓ
∑ ∑ dm
m∈M
gm j∈Lm j
(5.4)
∑ xmℓ = 1
ℓ∈Lm
∀m ∈ M, (5.5)
xmℓ ≤ yℓ ∀m ∈ M,ℓ ∈ Lm, (5.6)
xm j ≤ s + (1 − yℓ)βℓ ∀ℓ ∈ L, (5.7)
xmℓ ∈ {0,1} ∀m ∈ M,ℓ ∈ Lm, (5.8)
yℓ ∈ {0,1} ∀ℓ ∈ L. (5.9)
The first term in the objective function (5.4) is the total tower usage cost and
the second term is the total handset transmission power when all connections are
active. The user-specified parameter λ ≥ 0 determines the trade off between infrastructure
cost and power usage (energy). This trade off is discussed in Section 5.3.4.
Constraint set (5.5) forces each test point to be assigned to exactly one tower location,
and constraint set (5.6) links the tower location and subscriber assignment
decision variables so that test points can only be assigned to selected towers. A minimum
acceptable SIR is ensured for all active connections by the quality-of-service
constraints (QoS) (5.7). If tower location ℓ is selected, then yℓ = 1 and (5.7) enforces
the signal-to-interference requirement (5.3). However, if tower location ℓ is
not selected then we do not need to enforce (5.3) at that location. Indeed, doing so
may artificially constrain the solution. Therefore, the constant βℓ is selected to be
large enough so that the constraint is satisfied automatically by any set of binary xmℓ
values when yℓ is zero. A suitable value is
where
�
βℓ = ∑ dm
m∈M
max
j∈Lm\{ℓ}
� gmℓ
gm j
max
j∈Lm\{ℓ}
� gmℓ
gm j
� �
(5.10)
�
= 0 if Lm \ {ℓ} = /0. (5.11)
Constraint sets (5.8) and (5.9) define the domains of the decision variables. Given a
solution to ACMIP, we refer to the set of selected towers as ¯L ⊆ L.