Wireless Network Design: Optimization Models and Solution ...

184 Syam Menon
Variable ρg represents a feasible schedule for a particular cell — for example,
in a three period model, a schedule [0, 1, 1] for cell i indicates that cell i
is connected to the MTSO in the first period and to the hub in the next two periods.
G is the set of all feasible schedules. Parameter aig is 1 if cell i is represented
in variable g ∈ G, and 0 otherwise. Parameter b jg is di j if cell i is connected
to the hub in period j, and 0 otherwise. The coefficient fg in the objective�
function is the cost associated with the schedule; it can be calculated as
T
fg = ∑ hi jxi j +
j=1
T
∑ ri j(1 − xi j) +
j=1
T
∑ si jui j +
j=1
T
�
∑ ti jvi j , where cell i is the cell
j=1
represented by variable ρg. The objective function minimizes total cost while constraints
8.14 ensure a valid schedule for each cell. Constraints 8.15 require the capacity
limitations at the hub to be adhered to in each period.
We assume that all schedules are feasible for every cell. Under this assumption,
there are |N| × 2 |T | feasible schedules; this number grows exponentially with the
number of time periods. Kubat and Smith [8] solve the problem by enumerating all
feasible schedules for small problems (where N = 5 and T = 3), or via heuristic approaches
based on Lagrangian relaxation and cut generation applied to formulation
(M2). While explicit enumeration is reasonable for small problems, this results in
intractable formulations as the size of the problem (in particular, the number of time
periods) increases. In this section, we present a column generation based approach
to solving this problem, which is observed to give gaps of less than 1% even on
problems involving 300 cells and 15 time periods. While we have terminated column
generation when the linear programming optimal solution is identified, we also
outline two valid branching schemes for branch and price.
8.3.2 The Solution Procedure
The problem represented by (M2) is NP-complete as even the single period case
is a binary knapsack problem. Therefore, specialized solution approaches need to
be developed if large instances of the problem are to be solved. Here, we present
a description of the key features of the column generation based algorithm, along
with details on the subproblem formulation used to generate attractive columns.
8.3.2.1 Linear Programming Column Generation
As already discussed, enumerating all feasible schedules is not a reasonable task.
We start with a subset G ′ ⊆ G and solve the linear programming relaxation of the
resulting restricted master problem. The dual prices obtained from solving the restricted
master problem are then used to implicitly price out columns that might be
beneficial to add to the master problem.
A common approach to solving linear programs with a large number of columns
is implicit enumeration (Gilmore and Gomory [5, 6]). We use the standard approach,