Wireless Network Design: Optimization Models and Solution ...

8 The Design of Partially Survivable Networks 185
and initialize the master problem with every cell being connected to the MTSO. Instead,
one could initialize using a feasible solution obtained via a heuristic procedure.
Solving the master problem provides dual prices which are used to generate
columns by solving a subproblem for each of the cells. We need not go over all the
cells in every iteration — we can switch to the master problem as soon as we have
columns to add to it; the actual number of columns to add at each iteration can be
determined empirically. We iterate between the master and subproblems until no
more columns price out, at which point we have the optimal solution to the linear
programming relaxation of (M2).
8.3.2.2 Representing the Subproblem
One important factor in the success of a column generation based solution procedure
is the ease with which the subproblem can be represented and solved. The
subproblem gets solved many times in the solution process, and it is critical that
the subproblem be easy to solve. A column is attractive to the linear programming
relaxation of the master problem if the associated reduced cost is negative. If we
associate dual variables π∈ R |N| with the constraints 8.14 and λ∈ RT − with the constraints
8.15, a column s could (potentially) improve our objective function value
if and only if
�
fg − πi − T
∑ λ jb jg
j=1
�
< 0. Therefore, the key issue becomes one of
identifying a schedule with this feature if it exists.
At any iteration of the column generation process, we have a subproblem for
each cell i that is best described by Figure 8.2. In this figure, every time period
j is represented by two sets of nodes R j and Hj, corresponding to the cell being
connected to the MTSO and the hub, respectively. In addition, a source node P
and a sink node Q are added, with the source node being connected to the nodes
associated with time period 1 (edges P → R1 and P → H1) and the sink node being
connected to the final period (edges R j → Q and H j → Q). There are edges from
every MTSO node R j, j = 1,..., (T − 1) indicating that the cell is connected to
the root (R j → R ( j+1)) or the hub (R j → H ( j+1)) in the next period. Similarly, there
are edges from every hub node Hj, j = 1,..., (T − 1) indicating that the cell is
connected to the root (Hj → R ( j+1)) or the hub (Hj → H ( j+1)) in the next period.
The P → R1 edge and the R j → R ( j+1) edges have cost ri1 and r i( j+1) associated
with them, respectively; the P → H1 edge and the R j → H ( j+1) edges have cost
(hi1 +si1) and (h i( j+1) +s i( j+1)) associated with them, respectively; the Ht → R ( j+1)
edges have cost (r i( j+1) + t i( j+1)) associated with them, and the cost on the edges
R j → Q and Hj → Q are 0. In addition, there are costs of (−λ j × di j) associated
with each of the Hj nodes as a result of the dual multipliers. Given this description,
the subproblem becomes one of identifying the lowest cost to send one unit of flow
from P to Q.
In the absence of the node weights (−λ j × di j), this problem is equivalent to
one of finding the shortest path from P to Q, and any shortest path algorithm (e.g.,
Dijkstra’s) will provide the solution. In the case of the special network of the sub-