Wireless Network Design: Optimization Models and Solution ...

11 Improving Network Connectivity in MANETs Using PSO and Agents 263
The objective function of Problem ALOC is to maximize the minimum of the
maximum flows between the user node pairs. Given the locations of the user nodes
( ˆxi, ˆyi) ∀i ∈ UN, the decision variables of Problem ALOC include the locations of
agent nodes (i.e, (xi,yi) ∀i ∈ AN). Constraints (11.13)-(11.15) are the node-flow
balance constraints to calculate the maximum flow Ust from the source node s and
to the sink node t for ∀s,t ∈ UN and s < t. Constraints (11.16)-(11.20) are used
to calculate the squared-Euclidian distances sdi j of agent arcs (i, j) for i ∈ N, j ∈
AN,i < j. In constraint (11.20), squared-distances (dx i j )2 and (d y
i j )2 along the x and
y axes, respectively, are approximated by piecewise linear functions while solving
the problem in CPLEX. Constraints (11.21) and (11.22) determine the capacity ui j
of each agent arc (i, j) based on (11.11), which is also approximated by piece-wise
linear functions. Constraint (11.23) ensures that the flow variables do not exceed the
arc capacities.
Problem ALOC:
Mazimize z = U
st.
U ≤ Ust ∀s < t ∈ UN (11.12)
∑ fi jst − ∑ f jist = 0 ∀s < t ∈ UN,∀i ∈ N \ {s,t} (11.13)
{ j:(i, j)∈E} { j:( j,i)∈E}
∑ fs jst − ∑ f jsst = Ust ∀s < t ∈ UN (11.14)
{ j:(s, j)∈E} j:( j,s)∈E}
∑ ft jst − ∑ f jtst = −Ust ∀s < t ∈ UN (11.15)
{ j:(t, j)∈E} { j:( j,t)∈E}
d x i j ≥ xi − x j ∀i ∈ N, j ∈ AN,i < j (11.16)
d x i j ≥ x j − xi ∀i ∈ N, j ∈ AN,i < j (11.17)
d y
i j ≥ yi − y j ∀i ∈ N, j ∈ AN,i < j (11.18)
d y
i j ≥ y j − yi ∀i ∈ N, j ∈ AN,i < j (11.19)
(d x i j) 2 + (d y
i j )2 ≤ sdi j ∀i ∈ N, j ∈ AN,i < j
�
4
ui j − 10 1 + e
(11.20)
10(√sdi j−0.5) �−1 ≤ 0 ∀i ∈ N, j ∈ AN,i < j (11.21)
u ji ≤ ui j ∀i ∈ N, j ∈ AN,i < j (11.22)
fi jst ≤ ui j ∀(i, j) ∈ E (11.23)
xi = ˆxi ∀i ∈ UN (11.24)
yi = ˆyi
ui j,xi,yj, fi jst ≥ 0
∀i ∈ UN (11.25)
Problem ALOC was solved by CPLEX v11 with a limit of eight hours CPU time.
CPLEX was not able to find feasible solutions for problems with more than ten user
nodes and five agents within the allowed CPU time limit. Fig. 11.8 shows the comparisons
on static cases of ten randomly generated 5-user and 10-node problems