NLP. All decision variables are continuous variables.
There are three sets of those, w
i j
(t) is the amount of
data on the outgoing link (i, j) for time slot t, x
s
(t)
is the amount of data delivered from source f
s
to the
destination d
s
, e
s
(t) represents the amount of energy
expended by a node. The objective function is the
sum of individual node utilities where each of these,
U
s
1
T
∑
T
t=1
x
s
(t)
, is a function of the amount of data
delivered from source node f
s
to destination node d
s
in all T time slots over possibly multiple hops and
multiple paths. Each utility function is assumed to be
a continuous non-linear concave function. There are
two constraint sets, the ﬁrst is conservation of ﬂow
constraint sets, which are linear constraints in w
i j
(t)
and x
s
(t). The second ensures that the sum of ﬂows
emanating from a node i belongs to the set Λ
i
of the
different amounts of data in different time slots un-
der a given replenishment proﬁle vector
−→
r
i
. For any
data vector in Λ
i
, there exists an energy vector e
n
that
achieves that amount of data for a given modulation
and coding scheme. The set Λ
i
was proved to be con-
vex in works earlier to (Chen et al., 2012).
5.2 Solution Method
A heuristic method named DualNet was proposed that
obtains an infeasible upper bound and a feasible lower
bound and iteratively solves the problem until it con-
verges to the optimal solution inﬁnity. First an up-
per bound was obtained on the value of the objective
function at the optimal solution of the problem after
a long period of time (theoretically inﬁnity). The so-
lution that gives the upper bound is obtained by an
infeasible energy allocation i.e. energy allocation that
is higher than the average replenishment rate. The en-
ergy allocation (and hence the routing solution) are
the same over all time slots and is more than the av-
erage replenishment rate, yielding infeasiblity. Using
the energy allocation obtained, a routing sub-problem
that is strictly convex and computationally easier than
the original problem is obtained because of the decou-
pling of the time component. This requires solving
the problem every time slot.
The lower bound solution is obtained by assign-
ing a feasible energy value in each time slot for each
node. The energy assignment for a node is the min-
imum of either the average harvested energy or the
available battery energy (including the instantaneous
replenishment for a given time slot). This assignment
is done by each node on its own and hence is a dis-
tributed energy assignment. Using the energy assign-
ment values the routing subproblem that obtains the
lower bound, is again a similar routing subproblem to
that of the upper bound subproblem.
Dual decomposition was used to solve the pro-
blem which enabled a distributed implementation of
the scheme. Each source node solves two prob-
lems, one to determine the amount of data to inject
in the network at a given time slot t, x
s
(t), the other
subproblem to determine the routes and their ﬂows,
w
i j
(t). All the nodes that are not sources of data, and
only responsible for relaying data over multiple hops,
solve the routing problem only. The dual variables are
computed using the subgradient algorithm.
6 ROUTING WITH DISTANCE
UNCERTAINTIES
Optimization models were considered in (Ye and Or-
donez, 2008) for WSNs subject to distance uncer-
tainty for three different objectives, 1) minimizing the
energy consumed, 2)maximizing the data extracted
and 3) maximizing the network lifetime. Robust opti-
mization was used to take into account the uncertainty
present. In a robust optimization model the uncer-
tainty is represented by considering that the uncertain
parameters belong to a bounded, convex uncertainty
set. A robust solution is the one with best worst case
objective over this set. It was shown in (Ye and Or-
donez, 2008) that solving for the robust solution in
these problems is just as difﬁcult as solving for the
problem without uncertainty. The computational ex-
periments in (Ye and Ordonez, 2008) showed that, as
the uncertainty increases, a robust solution provides a
signiﬁcant improvement in worst case performance at
the expense of a small loss in optimality when com-
pared to the optimal solution of a ﬁxed scenario.
6.1 Problem Statement and Design
Objectives
For the three different types of problems, energy con-
sumption was considered. The transmission and re-
ception energy for each node is accounted for after
normalizing with respect to the radio energy dissi-
pation of the transmitter and receiver circuits. The
expression for the total normalized energy has two
components. One for the normalized received energy
which is equivalent to the number of received bytes,
i.e.
∑
j|( j,i)∈A
f
ji
, and one for the the normalized trans-
mitted energy, which is equivalent to the number of
transmitted bytes times a linear function in the trans-
mission distance, i.e.
∑
j|( j,i)∈A
f
ji
1 + βd
2
i j
, where
• A is the set of nodes in the network,
• f
i j
is the number of transmitted bytes from node j
to node i.
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