Wireless Network Design: Optimization Models and Solution ...

8 The Design of Partially Survivable Networks 181
lem (e.g., Balas and Zemel [1], Pisinger [15]). Most of these approaches are based
on dynamic programming and are pseudo-polynomial in the capacity of the knapsack.
As we shall see, the capacity of the binary knapsack problem to be solved is
very small, and this makes its solution through one of these specialized approaches
almost instantaneous. The solution technique presented here relies on the following
result.
Lemma 8.1. Every cell i ∈ V will be connected to the cheapest (Si − 1) hubs in the
optimal solution.
Proof: At most one connection can be to the MTSO. Therefore, at least (Si − 1)
connections have to be to the hubs. There are no capacity restrictions specific to the
hubs. Therefore, these (Si − 1) connections to the ring can be to any of the hubs.
The cheapest way to achieve this is to connect to the (Si − 1) cheapest hubs. ⊳
For each cell i, let c i j represent the costs ci j in ascending order (i.e., c i 1 ≤ ci 2 ≤
··· ≤ c i |V | ). Based on Lemma 8.1, all the variables associated with ci 1 , ci 2 ,··· , ci (Si−1)
will be set to 1. Similarly, all the variables associated with c i (Si+1) , ci (Si+2) ,··· , ci |V |
will be set to 0.
So, the only decision to be made concerns whether to connect each cell i to the
Si-th cheapest hub, or to the MTSO directly. Let ji be the index of the Si-th cheapest
hub for cell i. Incorporating this into (P1), we get
(P2) ∑
i∈V
Si
The second term ( ∑
i∈V
min ∑(ci0xi0 + ci ji
i∈V
xi ji ) + ∑ ∑ c
i∈V j< ji
i j
s.t. xi0 + xi ji = 1 ∀i ∈ V (8.4)
Di
xi ji ≤ 2K − ∑(Si − 1) ×
i∈V
Di
Si
(8.5)
xi0,xi ji ∈ {0, 1} ∀i ∈ V (8.6)
∑ c
j< ji
i j ) in the objective function represents the cost of assigning
each cell i to its cheapest (Si − 1) hubs. The right hand side of constraint 8.5 is the
remaining capacity of the ring after these assignments have been made. This value
has to be non-negative in order for a feasible solution to exist — i.e., the problem is
feasible only if ∑ (Si − 1) ×
i∈V
Di ≤ 2K. Si
Constraints 8.4 imply that xi ji = (1 − xi0). Incorporating this into the objective
function of (P2) and re-arranging the terms, we get the equivalent formulation (P3)
below.
∑ ci0 + ∑ ∑ c
i∈V i∈V j< ji
i j − max ∑(ci0 − ci ji
i∈V
)xi ji
Di
(P3) s.t. ∑
i∈V
xi ji
Si
≤ 2K − ∑ Di + ∑
i∈V i∈V
Si
(8.7)
xi ji ∈ {0, 1} ∀i ∈ V (8.8)
Di