Wireless Network Design: Optimization Models and Solution ...

5 Mathematical Programming Models for Third Generation Wireless Network Design 109
Observe that if test point m is assigned to tower ℓ, (5.15) ensures that the SIR (5.14)
is at least SIRmin. However, if test point m is not assigned to tower ℓ the constraint
reduces to pm ≥ 0 which is trivially satisfied. The ACMIP with SIR-based power
control is the following mathematical program:
min ∑ aℓyℓ + λ ∑
ℓ∈L m∈M
subject to (5.5), (5.6), (5.15), (5.8), (5.9) and
∑
ℓ∈Lm
dm pmxmℓ, (5.16)
0 ≤ pm ≤ P max ∀m ∈ M. (5.17)
The potential advantage of using SIR-based power control is that mobiles can emit
lower powers and therefore generate less interference with towers they are not assigned
to. Indeed, in their computational study Amaldi et al. found that solutions to
the SIR-based power control model used at least 20% fewer towers than solutions to
their power-based power control model ACMIP [7]. The disadvantage of using this
formulation is that (5.15) is a non-linear constraint which makes the problem extremely
difficult to solve — even more difficult than ACMIP. Consequently Amaldi
et al. did not attempt to find provably optimal solutions to this model. Instead, they
solved it with a heuristic procedure described in Section 5.4.2 to compare it with the
power-based model described in Section 5.2.1.
5.3.2 Profit Maximization with Minimum-Service Restrictions
Building on the work of Amaldi et al. [4, 5, 7], Kalvenes et al. [34] propose a profitmaximization
variant of ACMIP, called KKOIP, in which the objective function is
r ∑
∑
xmℓ − ∑ aℓyℓ
m∈M ℓ∈Lm ℓ∈L
(5.18)
where r is the revenue generated for each unit of demand served in the planning area.
Unlike ACMIP, KKOIP allows for partial demand satisfaction at the test points.
Thus, in this model xmℓ is defined as the maximum number of users at test point
m that can be simultaneously assigned to tower ℓ. Consequently, KKOIP replaces
(5.5), (5.6) and (5.8) with
∑ xmℓ ≤ dm
ℓ∈Lm
∀m ∈ M, (5.19)
xmℓ ≤ dmyℓ ∀m ∈ M,ℓ ∈ Lm, (5.20)
xmℓ ∈ N ∀m ∈ M,ℓ ∈ Lm, (5.21)
and the QoS constraints (5.7) with