Wireless Network Design: Optimization Models and Solution ...

8 The Design of Partially Survivable Networks 191
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Fig. 8.3 Example Network
home to both of these hubs. The diversity requirements of nodes v, x and w are 1, 2
and 3, respectively. The arrows denote one possible assignment of cells to nodes.
The formulation in Kubat et al. [9] can be expressed as (CP1) below, after defining
the following. H, R, V , ci j, di, and si are as defined in Section 8.2. k j is the
capacity of the path connecting hub j to the MTSO, in number of DS-0 circuits. a jl
is 1 if traffic from hub j flows via hub l, and 0 otherwise. Q is a set of trees such that
Ts ∩ Tt = /0 ∀Ts, Tt ∈ Q. The decision variable xi j is 1 if traffic from cell i is routed
through node j, and 0 otherwise.
min ∑ i∈V
(CP1) ∑ i∈V
∑ ci jxi j
j∈R
s.t. ∑ xi j = si ∀i ∈ V (8.26)
j∈R
∑
j∈R
di
si
a jlxi j ≤ kl ∀l ∈ H (8.27)
∑ xi j
j∈Tq
≤ 1 ∀i ∈ V, ∀Tq ∈ Q (8.28)
xi j ∈ {0, 1} ∀i ∈ V, ∀ j ∈ R (8.29)
The objective function minimizes the total cost of connecting cells to hubs or the
MTSO. Constraints 8.26 ensure that the diversity requirement of each cell is met
by the assignments, while constraints 8.27 ensure that the capacities of the path
connecting the hubs to the MTSO are not exceeded. Constraints 8.28 ensure that
any given cell connects to at most one hub across all the hubs in any given tree.
Kubat et al. [9] solve problem (CP1) via three heuristic techniques. Readers are
referred to Kubat et al. [9] for additional details.
8.4.1 A Partitioning Formulation
The cell assignment problem described in this paper can be viewed as a partitioning
problem where the cells are partitioned into distinct sets and assigned to various
trees (or individual hubs, if there is only one hub in a given tree) in such a way that