Wireless Network Design: Optimization Models and Solution ...

8 The Design of Partially Survivable Networks 193
with the constraints 8.4, λ ∈ R |W|
− with the constraints 8.5, a column p could (potentially)
improve our objective function value if and only if (rp − ∑ πiαip − λw) < 0,
where Tw ∈ W is the subtree involved with this sub-assignment. rp, the cost associated
with column p, is the total connection cost ∑ ci jαip. Any subset of cells form a
feasible column so long as their combined demand does not exceed the capacities of
the hub(s) involved. Therefore, an attractive column exists if and only if the problem
(Sw) has a negative objective function value for some Tw ∈ W.
min ∑ i∈V
(Sw) s.t. ∑ i∈V
∑ (ci j − πi)zi j − λw
j∈w
∑
j∈R
di
si
i∈V
i∈V
a jlzi j ≤ kl ∀l ∈ Tw (8.33)
∑ zi j
j∈w
≤ 1 ∀i ∈ V (8.34)
zi j ∈ {0, 1} ∀i ∈ V, j ∈ w (8.35)
This is a binary multiple knapsack problem, with the objective function minimizing
the reduced cost and constraint 8.33 ensuring that we will remain within capacity
for each hub l contained in Tw. Constraints 8.34 enforces the requirement that a cell
i can home at most once to Tw. Even though it is NP-complete, it can be solved
relatively efficiently using either an integer programming package like LINDO [12]
or a specialized solution approach (Pisinger [16]). When the Tw ∈ W is a single
hub, (Sw) simplifies further, to a binary knapsack problem. For the most part, a
greedy heuristic can be used to solve (Sw), where the knapsacks are packed in greedy
fashion, with an optimal approach being resorted to only if the heuristic fails to
generate a column with negative reduced cost. As in Section 8.3, we have resorted
to the easier approach of switching on the integrality requirements after solving the
LP relaxation.
8.4.1.2 Reducing the Impact of Degeneracy
In the face of degeneracy, basic solutions tend to designate a small subset of the
constraints without slack as “binding”, and assign all the importance to these constraints.
As a result, degenerate basic solutions are often accompanied by extreme
dual prices. This is undesirable when doing column generation, as this could increase
solution times considerably through the generation of a large number of unnecessary
variables. Note that the equality constraints 8.4 can be replaced by inequality
(≥) constraints without loss of generality. If a cell is covered by more than
si columns, a feasible solution to (P2) that is at least as cheap can be obtained by
introducing a new column that is identical to one of the selected columns featuring
cell i, except for the absence of the cell in question. This new column is both feasible
and cheaper (if the connection cost of the discarded cell is positive). Therefore, we