Wireless Network Design: Optimization Models and Solution ...

180 Syam Menon
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Fig. 8.1 Example Network
min ∑ i∈V
(P1) ∑ i∈V
∑ ci jxi j
j∈R
st ∑ xi j = Si ∀i ∈ V (8.1)
j∈R
∑
j∈H
Di
Si
xi j ≤ 2K (8.2)
xi j ∈ {0, 1} ∀i ∈ V, ∀ j ∈ R (8.3)
The objective function minimizes the total cost of connecting cells to nodes on the
ring. Constraint set 8.1 ensures that the diversity requirement of every cell is met,
while constraint 8.2 ensures that the ring capacity is not exceeded. All cell demands
will ultimately terminate at the MTSO, and therefore these flows must enter the
MTSO in either the clockwise or the counter-clockwise direction. Consequently, the
total traffic in the ring cannot exceed 2K when the ring capacity is K. Note that if
traffic is allowed to flow just in one direction, the only alteration to the formulation
would be to limit the total traffic on the ring to K. As already mentioned, Dutta
and Kubat [4] solve problem (P1) via a heuristic technique based on Lagrangian
relaxation.
8.2.2 Reduction to a Binary Knapsack Problem
Formulation (P1) involves |V | × |R| variables and |V | + 1 constraints. In this section,
we show how (P1) can be reduced to a significantly smaller binary knapsack
problem (with at most |V | variables). This is convenient, as the size of the binary
knapsack problem to be solved is independent of the values of |H| and Si. A variety
of specialized solution approaches are available to solve the binary knapsack prob-