Wireless Network Design: Optimization Models and Solution ...

8 The Design of Partially Survivable Networks 183
The multi-period cellular design problem can be formally described as (M1)
below, after defining parameters N, the set of cells; T , the number of time periods
in the planning horizon; di j, the demand from cell i in period j; ri j, the cost of
connecting cell i to the root (MTSO) in period j; hi j, the cost of connecting cell
i to the hub (backbone) in period j; si j, the cost of setting up a new link between
cell i and the hub in period j; ti j, the cost of terminating an existing link between
cell i and the hub in period j; k j, the capacity available at the hub in period j, and
variables xi j, which is 1 if cell i is assigned to the hub in period j and, 0 otherwise;
ui j which is 1 if a new link is established between cell i and the hub in period j; and
vi j which is 1 if an existing link between cell i and the hub is terminated in period
j. For notational convenience, xi0 is defined to be 0.
min ∑ i∈N
+ ∑ i∈N
T
∑ hi jxi j + ∑
j=1 i∈N
T
∑ si jui j + ∑
j=1 i∈N
T
∑ ri j(1 − xi j)
j=1
T
∑ ti jvi j
j=1
s.t. ∑ di jxi j
i∈N
≤ k j j = 1,...,T (8.9)
(M1) ui j − xi j + xi( j−1) ≥ 0 ∀i ∈ N, j = 1,...,T(8.10)
vi j − x i( j−1) + xi j ≥ 0 ∀i ∈ N, j = 1,...,T(8.11)
xi j
∈ {0, 1} ∀i ∈ N, j = 1,...,T(8.12)
ui j,vi j ≥ 0 ∀i ∈ N, j = 1,...,T(8.13)
The objective is to minimize total cost, comprising the cost of connecting cells to the
hub or the root in each period and the cost of establishing or terminating links from
the cells to the hub in each period. Constraints 8.9 ensure that hub capacity is not
exceeded in any period; constraints 8.10 and 8.11 ensure that the set-up costs and
termination costs, respectively are captured; constraints 8.12 enforce integrality of
the assignment variables. Integrality of the set-up and termination forcing variables
ui j and vi j need not be explicitly enforced — if the xi j are integral, these variables
will be integral as well.
Another formulation for this problem was introduced in Kubat and Smith [8].
This formulation is amenable to column generation, and is presented as (M2) below.
min ∑ fgρg
g∈G
s.t. ∑ aigρg = 1
g∈G
∀i ∈ N (8.14)
(M2) ∑ b jgρg ≤ k j
g∈G
∀ j = 1,...,T (8.15)
ρg ∈ {0, 1} ∀g ∈ G (8.16)