Wireless Network Design: Optimization Models and Solution ...

8 The Design of Partially Survivable Networks 189
existing link between a cell and the hub (ti j) is generated from a uniform distribution
U(100,500). The random number generator used is from Knuth [7]. Ten problem
instances were generated in each of the (8 × 4 × 3) = 96 groups.
Table 8.1 provides the results from applying the solution procedure, for the high,
medium and low capacity levels. |N| is the number of cells and T is the number
of time periods. The columns under each of the capacity categories represent the
average number of variables generated as a fraction of the total (in %) and the average
gaps (in %), where the gaps are obtained as ( UB−LB
LB ) × 100, with UB being the
objective function value corresponding to the best integer feasible solution obtained
after spending a maximum of two minutes on branch and bound, and LB being the
linear programming relaxation value of (M2). The variables and gaps reported are
averages over the 10 data sets of each type. The code was implemented using the
LINDO Callable Library [12] on an Intel Pentium II running at 450 MHz.
IP branch and bound was terminated after a maximum of two minutes of execution.
The gaps between the column generation LP and the integer solution obtained
by switching on the integrality requirements are low in general and decrease with
problem size. The low gaps suggest that the linear programming relaxation of formulation
(M2) is reasonably tight. In most of the realistic cases (|N| > 20), the gaps
are extremely low (well below 1%). This suggests that a branch and price approach
is not necessary to obtain near-optimal solutions. Even in cases where the LP-IP
gaps were high (essentially when |N| = 10), it was noted that this was a result of the
LP bound being relatively weak for the small problems.
In general, as the problem size got larger, the performance of the column generation
approach improved. This could be partly due to the fact that more time is
spent in linear programming column generation, which increases the number of
good quality columns generated. The fraction of variables generated decreased with
problem size, as would be expected. Conditioning on the number of cells, the gaps
increased marginally with the number of time periods.
8.4 The Single Period Problem with Capacity Restrictions
When capacities are introduced at the hubs, the problem becomes more difficult.
Kubat et al. [9] present a problem of designing interconnect networks for a cellular
system that incorporate reliability requirements in the face of capacity constraints,
where the backbone has a tree topology with the MTSO at the root. The links in
the tree have known capacities, and it is possible for two hubs with distinct logical
channels to the MTSO to share the same physical link. In order to provide increased
reliability, it is also desirable to have a cell connect not just to multiple hubs, but to
multiple subtrees that do not have any physical links in common between the hubs
and the MTSO. This is particularly relevant when the backbone topology is a tree, as
the failure of even one high capacity line could affect the entire network. As before,
the number of nodes (hubs + MTSO) on distinct subtrees to which a cell is required
to be connected is its diversity requirement. The objective is to find a least-cost cell-