In this section, we first construct a rechargeable cycle model, then transform the formulation into get a near-optimal solution by change-of-variable technology. The entire rechargeable cycle is divided into three parts: the vacation time , the rechargeable time and the traversal time .
4.1. Rechargeable Energy Cycle Construction
We have constructed constraints and lemmas to solve the issue of the dynamic power in the multi-hop WSNs scenario with SIC. In other words, the energy consumption model of each node is complete, and now we start to construct the energy supplement model.
For different rechargeable cycles, node i has the same curve of energy consumption. Each node has to comply with the following energy restrictions: In each cycle of , (i) its remaining battery starts and ends with the same level of energy, and () its remaining battery is always greater than or equal to .
Define a set of node locations as
and define the location of the service station
S as
. Define
as the distance between two nodes. Denote
as the physical path over a trip cycle, which starts from the service station, passes through each node for recharging it, and finally returns to the service station. In each rechargeable cycle, define
as MC arrival time at node
i. Then, we have the constraint as follows:
Define
as the physical path distance of the shortest Hamiltonian cycle, thus denoting
as the traveling time over the shortest Hamiltonian cycle. For one singal rechargeable cycle
, we have the constraint as follows:
During the cycle
, we analyze the power consumption of node
i by utilizing the average power
. The amount of energy consumption of node
i over
is equal to the amount of supplement energy over
. Therefore, we have the energy balance constraint as follows:
During a rechargeable energy cycle, when MC reaches at node
i (i.e.,
), As shown in
Figure 3, there are two rechargeable strategies that can be selected: (i) fully recharge (i.e., MC recharges the battery of node
i to
straightly) and (ii) not fully recharge (i.e., MC leaves before fully recharging). These two choices are not different from the results [
22]. For the sake of convenience, we select the strategy of fully rechargeable case. When the average power
replaces the real-time power
, it is notable that there are only two approximate slopes of the energy consumption curve over one single rechargeable cycle [
]: When MC does not recharge node
i, the energy consumption slope of node
i is
; when MC is recharging node
i, the energy supplement slope of node
i is
, where
u is the power of recharging. The node residual energy cannot exceed the battery capacity
or be less than the minimum energy threshold
, and the energy constraint can be written as follows:
For a rechargeable energy cycle,
, so
. Therefore, the residual energy of the node always needs to be greater than the minimum energy threshold. We have:
The Property 1 in [
22] shows that there always exists at least one "bottleneck" node in the WRSN for an optimal solution, and when MC arrives at these nodes and starts to recharge their batteries, the energy level of these nodes is exactly equal to
(i.e.,
).
Property 1. In an optimal solution, there exists at least one node in the network with its battery energy dropping to when MC arrives at this node and recharges its battery.
4.3. Reformulation
We employ change-of-variable technology to simplify the formulation. For instance, denote , replacing the nonlinear objective .
For constraint (
8), we divide both sides of the formula by
, and this constraint can be rewritten as
. We, respectively, denote
and
to replace the nonlinear terms
and
. Then, constraint (
8) is rewritten as follows:
Through the same method, the energy constraints (
10) and (
11) are reformulated as
By constraints (
12) and (
13), we reformulate (
14) as follows.
By constraint (
13), constraint (
6) is rewritten as
The optimization problem
is reformulated as
, and we show it as follows,
| max | |
| s.t. | Flow balance constraints: (5) |
| | Vaction constraints: (15); |
| | Energy balance constraints: (16); |
| | |
| variables | |
| constants | |
It is not difficult to find that all the variables in are transferred from the variables in .
The constraints (
11) become (
16), which is a linear except, through reformulation (i.e., by change-of-variable technique), but there is still a quadratic term
. In
Section 4.4, we will present how to fit the parabola by a useful technology and show a feasible near-optimal solution to
within a feasible target error.
4.4. A Near-Optimal Solution
In this subsection, we replace the parabola (term
) by piecewise straight lines in the reformulation
. The approximation transforms the corresponding nonlinear constraints into the linear constraints, so that an off-the-shelf solver, like
[
23], etc., can calculate the solution. Then, we use this solution to determine a feasible solution to the initial problem
. Finally, we prove the feasibility of gap between the solution to
and the optimal solution to
.
It is notable that there is merely one nonlinear term in reformulation that is the quadratic terms , and it lies at the interval ([0, 1]), which is a very small interval. Therefore, we fit the nonlinear term by a piecewise linear approximation.
The essential idea is replacing the parabola by
m piecewise linear segments. For the parabola f(
) =
(
), we connect each point’s (
) to build a piecewise linear approximation. The value of
m can be determined by Lemma 3 in
Section 4.5.
Then, we present a method to mathematically describe (
) (i.e., the piecewise linear segment). For the
th segment, all points on the piecewise linear curve can be described by the following constraints.
in which
and
are two weights, and they satisfy the constraints as follows:
Since the quadratic function
is convex, the piecewise linear curve (
) lies above the convex curve
, and the end of the approximation curve (i.e., the point
and
) falls on the parabola. For the upper boundary of a feasible error
, Lemma 3 can quantify it [
22], if
is divided into
m segment [
22].
Lemma 3. .
Although the above formulas can describe
, the expression of is only suitable for a given linear segment (i.e., known the sequence number
s) and cannot represent a general situation. Now, we present the general mathematical formulas for the entire piecewise linear curve. Denote a binary indicator variable as
(
).
if
; otherwise,
. Because
must fall into only one of the
m segments:
Based on
, we reformulate
for the entire piecewise linear curve. We first present the relationship between
and
(
). After transforming
into
, we have, at most, two positive values (i.e.,
and
), while the others are
. In other words, if
falls only into the first segment at
, we have
; if
falls into the
sth segment
or
, we have
; only if
falls into the last segment
, we have
. These three constraints are formulated as follows:
The constraints (
22)–(
24) indicate that there are at most two positive value
for
. And if there are the two positive
, they must be adjacent. Then, for the piecewise linear fitted curve, we rebuild the mathematical expression as follows,
For constraint (
15), substitute
with
in
; that is,
After reformulating these constraints, we have the following linear relaxed formulation called
as follows:
| max | |
| s.t. | (10), (28), (21)∼(27) |
| | |
| | |
| | |
| | Variables: |
| | Constants: |
The off-the-shelf solver, like Gurobi, is able to calculate the solution to formulation because has been transformed into a linear formulation from a non-linear formulation.
The two solutions to the linear relaxed formulation
and the reformulation
seem to not be same. In fact, we can derive a feasible near-optimal solution to
through the solution to
. Assume the solution to problem
is
. It is notable that (
) satisfies all constraints in
. We attempt to calculate a feasible solution
= (
) to
; we first assume
, and then we rewrite
as follows to meet the constraint (
15) in
.
We can calculate a feasible solution to the original problem after obtaining a solution to .
4.5. Proof of Near-Optimality
We now attempt to quantify the error between the optimal objective to and the relaxed objective to . We hope the gap of the error to be controlled by m (i.e., we can control the gap by tuning the number of the piecewise linear curve). Though Lemma 4, we reverse the value m by a given feasible error ().
Lemma 4. , where is the objective value for the feasible solution Π to the original problem .
Proof. is the result for the solution
to
, and
always is greater than or equal to
(i.e.,
), since the problem
is a relaxation of the convex reformulation problem
. Therefore,
where the first equality proofed in [
22], and the second inequality holds by Lemma 1.
This completes the proof. □
Next, we introduce Theorem 1 and present a method to set an appropriate m for by a given target error .
Theorem 1. For a given target error ), if , then we have .
Proof. Note that the gap is
in Lemma 4. Therefore, set
, and then we have
:
This completes the proof. □
Through Theorem 1, we show the complete procedure on determining the solution to and present its five steps as follows,
Preset a feasible target error .
Set .
Calculate the relaxed linear optimization problem with m linear segment and gain its solution through the solver Gurobi.
Construct a feasible solution for the linear fitted problem by setting and = {1 - - }
Gain a feasible near-optimal solution () to the original optimization problem .