Wireless Network Design: Optimization Models and Solution ...

5 Mathematical Programming Models for Third Generation Wireless Network Design 117
5.4.2 Randomized Greedy Search
Given a set of selected towers ¯L, the algorithms in [7] assign test point m to the
tower ℓ ∈ ¯L with the largest attenuation factor gmℓ. If it turns out that this subscriber
assignment is infeasible, then test points assigned to towers where the SIR is less
than SIRmin are removed (i.e., the corresponding x variables are set to zero) one at a
time, in non-increasing order of dm
g mℓ until the QoS constraints (5.7) are satisfied for
every tower in ¯L. Thus, the subscriber assignment sub-problem can be solved very
quickly for any given tower subset ¯L.
Given ¯L, an iteration of the first procedure proposed in [7], called Add, takes
each unselected tower ℓ ∈ L \ ¯L and evaluates the change in the objective function
(5.13) resulting from adding ℓ to ¯L. The procedure then randomly selects one tower
from the subset of unselected towers whose addition to ¯L leads to the greatest improvement
in (5.13). The procedure starts with no selected towers (i.e., ¯L = /0) and
stops when it reaches a point where the addition of any of the unselected towers to
¯L leads to a decrease in (5.13). That is, it stops when it reaches a local optimum.
The second procedure, called Remove, works in essentially the opposite manner;
it starts with ¯L = L and removes towers in a randomized greedy fashion until no
further improvement in (5.13) is possible.
In [7] the authors compare these procedures against CPLEX—the “gold standard”
for commercial branch-and-bound codes—on a set of 10 small test problems
in which 22 tower locations and 95 test points are randomly placed in a 400 m by
400 m service area. In these problems each test point has a demand of dm = 1 and
the attenuation factors in these problems are such that it is possible to cover all 95
test points. Since the tower locations all have the same usage cost (i.e., aℓ = a for all
ℓ ∈ L), solving these problems amounts to minimizing the number of towers needed
to cover all the test points. Using CPLEX to obtain optimal solutions, it was determined
that in each instance 4 towers were sufficient to cover all test points. Add and
Remove were each run 50 times on each problem instance on a Pentium III/700-
MHz processor. Taking the best solution of the 50 found by each procedure, Add
and Remove found optimal solutions to 5 and 8 out of the 10 instances, respectively.
In the other cases, the best solution found used 5 towers. The average CPU times for
the 50 runs of Add and Remove were 10 and 30 seconds, respectively. By contrast,
CPLEX 7.0 took 5 to 20 minutes to find optimal solutions on a faster machine [7].
At the cost of increased CPU time, the authors were able to improve solution
quality (in terms of the average and standard deviation of the number of towers
used over a set of 50 runs) using the solutions found by Add and Remove as starting
points for a tabu search algorithm. Developed in the seminal papers by Glover
[26, 27], tabu search (TS) is a meta-heuristic for improving the performance of
local search (LS) heuristics. In a LS heuristic one starts with a feasible solution
and considers the “neighborhood” of feasible solutions formed by applying various
problem-specific “moves” to generate new solutions. The LS then moves to the best
solution in the neighborhood of the current solution, or stops if no move produces
an improved solution. LS has been shown to be an effective approach to solving dif-