4.1 System model
Suppose that sensor nodes are randomly deployed in a rectangular monitoring area to form a multi-hop self-organizing network. The sensor network is fully connected, and all nodes have unique identities and are no longer mobile after deployment. Also, they are powered by limited energy and can not be supplemented. Each node should be configured with the same communication radius r and has the same initial energy E0. Sink node is capable of mobility and has external power supply. To reduce the high transmission overhead caused by long-distance transmission, the sensor node can forward data packets to the destination by multi-hop mode.
The sensor nodes can be classified into two kinds: rendezvous node (RN) and ordinary node, and RNs can communicate directly with sink [
22]. Comparatively, the ordinary node needs to send data to specific RN through a multi-hop mode. When sink node has not moved to the communication range, RN can put the aggregated data into the cache beforehand. Once the sink move along a certain trajectory and stay away with the RN within one hop, it can send to the sink directly. The first-order radio model is applied for energy depletion analysis, and the energy consumption in each round can be expressed as follows:
$$ {E}_{\mathrm{total}}=\sum \limits_{i=1}^n\left({e}_{tr}{k}_t^i+{e}_{\mathrm{rec}}{k}_r^i\right) $$
(1)
In which, etr and erec represents the energy consumption for sending or receiving unit data. \( {k}_t^i \) and \( {k}_r^i \) represent the amount of data being received and sent by node i, respectively.
Assuming that the amount of data generated by each node in each round is
q, there exists a relationship between the data received and the data forwarded by relay node
i, i.e.,
\( {k}_t^i={k}_r^i+q \). Then, it can be deduced that the energy consumption of data transmission in the whole network is related to the number of hops during the data transmission, which can be given as
$$ \sum \limits_{i=1}^n{k}_r^i=\sum \limits_{i=1}^n{h}_iq $$
(2)
where hi represents the minimum hop number that the node i sends data to the fusion node.
Further, the total energy consumption can be expressed as the sum of the minimum hops among all sensor nodes, as shown in formula (
3).
$$ {E}_{\mathrm{total}}=\sum \limits_{i=1}^n\left({e}_{tr}{k}_t^i+{e}_{\mathrm{rec}}{k}_r^i\right)=\sum \limits_{i=1}^n\left[{e}_{tr}\left({k}_r^i+q\right)+{e}_{\mathrm{rec}}{k}_r^i\right]=q\left[{ne}_{tr}+\sum \limits_{i=1}^n\left({e}_{tr}+{e}_{\mathrm{rec}}\right){k}_r^i\right]\kern0.2em $$
(3)
It can be seen that optimization of the total energy consumption is equivalent to minimizing the sum of the hops from the ordinary nodes to fusion nodes. In the process of mobile sink’s data collection, the number of hops depends on to the selection of rendezvous points of mobile sink. Therefore, the choice of sink mobile trajectory directly affects the energy consumption of the whole network [
23].
4.2 Data collection strategy with delay constraint
According to the previous analysis, to achieve the delay requirements and minimize the overall energy consumption of the network, it is necessary to optimize the sink point trajectory in wireless sensor networks [
24]. Furthermore, by considering delay constraints, the solution of optimal sink mobile path reduces the overall energy consumption. On the one hand, it is necessary to solve the optimal set of RNs to satisfy with the minimization the total energy consumption. On the other hand, the optimization of travel path selection by mobile sink needs to be fully considered to accord with the constraints for time delay [
25].
For a given set of sensor nodes S = {s1, s2, ⋯, sn} and the set of RNs V = {v1, v2, ⋯, vK}, an undirected graph G = {Ψ, E} can be constructed with a collection of edges E, and Ψ = S ∪ V. The problem of data collection with mobile sink can be transformed into Traveling Salesman Problem. By constructing the path set PATH(G) for data collection based on the undirected graphs G, it will ensure that the hops from any ordinary node to RN can not exceed the limit value hmax as well as obtaining the minimization of the total energy consumption.
Let
\( {P}_i=\left\{{p}_{i,1},{p}_{i,2},\cdots, {p}_{i,{k}_i}\right\} \) denote the set of all paths from the sensor node
i to its adjacent RN node set, in which
pi, j is the
j-th path from
ki data forwarding paths. Therefore, the location selection of RNs can be formulated as an optimization problem:
$$ {\displaystyle \begin{array}{l}\underset{P\in {P}_s(G)}{\min }{E}_{total}(P)\\ {}s.t.\kern1.7em \forall {P}_i\in PATH,\exists {p}_{i,j}\in PATH, dist\left(i,j\right)+ dist\left(j,v(i)\right)\le {h}_{\mathrm{max}}r\\ {}\underset{i=1,2,\cdots, n}{\max}\left\{h\left(i,v(i)\right)\right\}\le {h}_{\mathrm{max}}\\ {}1\le i\le n,1\le j\le \left|{P}_i\right|\end{array}} $$
(4)
where dist(i, j) is the Euclidean distance between node i and j. v(i) is the next-hop RN being chosen by node i, and the number of the corresponding hops can be defined by h(i, v(i)). Besides, r indicates the communication radius of the sensor node.
For any node j on the path Pi, the sum of the hops to the access path does not exceed the threshold of hmax. Since the Euclidean distance between two adjacent nodes is less than the communication radius r, the sum of the distance from node j to v(i) and the distance between node j and i can be no more than hmaxr.
4.3 Determination of rendezvous points
In order to ensure the overall energy depletion of the network being balanced, the selection of resident points may take into account comprehensively about the factors, including the node’s density, relative residual energy, and uniformity of distribution. As a regional center directly interacts with sink, RN is responsible for data fusion of other nodes within a certain range and has to undertake more energy consumption [
26,
27]. The ratio of residual energy to initial energy of a node is defined as relative energy. In order to achieve better energy consumption balance, the larger the relative energy is, the node can be more competitive. In addition, the node density is defined as the number of nodes in the communication range. The higher the node density, the lower the average energy depletion of nodes will demonstrate during the course of inter-cluster communication. By taking into account of the balanced distribution of RNs in the whole network, it is beneficial to improve the overall transmission efficiency of the network. Therefore, the nodes with larger centroid degree, i.e., the degree of proximity between a node and its neighbors, should be selected as far as possible. Based on the above analysis, the relevant indicators are defined as follows:
$$ \left\{\begin{array}{l}{f}_1=\frac{E_{res}(i)}{E_0}\\ {}{f}_2=\frac{\operatorname{cov}(i)}{\sum \limits_{j\in neighbor(i)}\operatorname{cov}(j)}\\ {}{f}_3=\frac{D(i)}{\max \left\{D(i)\right\}}\\ {}{D}_i=1-\frac{\sqrt{{\left({x}_i-\frac{1}{n}\sum \limits_{j=1}^n{x}_j\right)}^2+{\left({y}_i-\frac{1}{n}\sum \limits_{j=1}^n{y}_j\right)}^2}}{r}\end{array}\right. $$
(5)
where Eres(i) represents the residual energy of node i and E0 is the initial energy. cov(i) denotes the number of nodes within the communication radius of node i, and neighbor(i) represents the neighbor nodes set. In addition, D(i) is defined as the centroid degree of node i.
The weight of each indicator is determined by entropy weight method. First of all, normalize the above indicators as:
$$ {\tilde{f}}_m=\frac{f_m(i)-\min \left\{{f}_m(i)\right\}}{\max \left\{{f}_m(i)\right\}-\min \left\{{f}_m(i)\right\}} $$
(6)
where max{fm(i)} and min{fm(i)} denotes the original minimum and maximum value of the index function, respectively. fm(i) is the m-th index function of node i.
Next, the group vector entropy and weight values can be calculated as:
$$ \left\{\begin{array}{l}{U}_m-\frac{1}{\ln K}\sum {\tilde{f}}_m(i)\ln {\tilde{f}}_m(i)\\ {}{w}_m=\frac{1-{U}_m}{\sum \left(1-{U}_m\right)}\end{array}\right. $$
(7)
According to the index function value and the weight value, the comprehensive evaluation of each node can be determined by formula (
8). The sensor node with higher value will be chosen as the candidate resident point.
$$ Cand\_ RN(i)=\sum {w}_m{\tilde{f}}_m(i) $$
(8)
4.4 Path selection based on ant colony optimization
Ant colony optimization (ACO) is a kind of bionic technique inspired by the behavior of real ants from nature [
28]. By finding optimal paths through graphs, it demonstrates good enough performance for solving combinatorial optimization problems and near-optimal solutions to the travelling salesman problem. Basically, ant colony optimization algorithm is a kind of positive feedback mechanism, which has the characteristics of randomness, adaptability, and distribution. It is suitable for parallel computation and accurate solution [
29]. However, there are two disadvantages in traditional ACO algorithms. One is the operational inefficiency owing to the diversity of ants, the other is that it is easy to fall into local convergence and the obtained path usually can not meet with the optimal solution [
30].
When sink obtains the set of RNs Ψ = {
v1,
v2, ⋯,
vK}, the planning of the travel path can be approximated to TSP problem, which can be resolved approximately by adopting ant colony algorithm. During the course of the current node ants choosing the next-hop node from the neighboring node set, the probability of selection is crucial. Denote
τij(
t) as the pheromone concentration on the path from node
i to
j. The heuristic function
ηij(
t) can be defined as:
ηij(
t) = 1/|
d(
i,
sink) −
d(
j,
sink)|, which indicates the expected degree of ants from
i to
j.
d(
i,
sink) and
d(
j,
sink) represent the distance from the mobile sink to either the current RN or the next one, respectively.
\( {N}_{k_a} \) denotes the set of RNs to be visited by ant
ka. In addition,
α,
β are parameters to regulate the relative weight of the pheromone trail and heuristic value, which can be satisfied with
α ≥ 0,
β ≥ 0,
α +
β = 1. Hence, the probability of moving from RN
i to
j can be expressed as follows:
$$ {p}_{ij}(t)=\left\{\begin{array}{l}\frac{{\left[{\tau}_{ij}(t)\right]}^{\alpha}\cdotp {\left[{\eta}_{ij}(t)\right]}^{\beta }}{\sum \limits_{u\in {N}_{Ki}}{\left[{\tau}_{iu}(t)\right]}^{\alpha}\cdotp {\left[{\eta}_{iu}(t)\right]}^{\beta }},k\in {N}_i\\ {}0,\kern1.6em other\end{array}\right. $$
(9)
For the distance constraint, the gain of the access points set with minimum energy consumption is obviously optimal. Therefore, it is reasonable to choose the access node based on the energy cost and distance from mobile sink. By defining the effect of RN selection on global energy consumption in utility priority metric path, we introduce the concept of path superiority as a criterion for ant path evaluation after a round. The smaller the superiority is, the better the path quality will be. Then, the path superiority can be defined as:
$$ PR(i)={\lambda}_1\frac{E\left({V}_{k_a}\cup \left\{{v}_i\right\}\right)-E\left({V}_{k_a}\right)}{TSP\left({V}_{k_a}\cup \left\{{v}_i\right\}\right)-L\left({V}_{k_a}\right)}+{\lambda}_2\frac{E\left({V}_{k_a}\right)-E\left({V}_{k_a}-\left\{{v}^{\prime}\right\}\right)}{L\left({V}_{k_a}\right)- TSP\left({V}_{k_a}-\left\{{v}^{\prime}\right\}\right)}, TSP\left({V}_{k_a}\cup \left\{{v}_i\right\}\right)\le {D}_m{V}_m $$
(10)
In which, \( {V}_{k_a} \) represents the RN set being visited, \( L\left({V}_{k_a}\right) \) indicates the corresponding shortest path length, and \( E\left({V}_{k_a}\right) \) represents the total energy consumption for transferring data to RNs.
According to the ratio between energy gain and distance cost, path superiority can evaluate the influence of global optimal solution for candidate node to be selected as next hop. Then, the determination of the relative can impact on the pheromone and the heuristic for the decision of the ant.
Furthermore, after the ants end their tours, the pheromone trail amount on every edge (
i,
j) is updated by
$$ {\tau}_{ij}\left(t+1\right)=\left(1-\rho \right){\tau}_{ij}+\rho \Delta {\tau}_{ij}(t) $$
(11)
where ρ is the local pheromone decay parameter and Δτij(t) is the added pheromone trail amount at the point t.
The forward ants move from access point
i to
j for pheromone updating, and the updating rules are as follows:
$$ \Delta {\tau}_{ij}(t)=\frac{Q_LE(j)\Big)}{E(i)+d\left(i,j\right)} $$
(12)
where QL represent the local pheromone concentration. E(i) and E(j) represent the energy consumption for data being migrated from the member nodes to RN i and j, respectively. d(i, j) is the Euclidian distance.
Subsequently, the backward ants go back to the source along the reverse path and complete global pheromone updating, and the corresponding updating rules can be given as
$$ \Delta {\tau}_{ij}^{k_a}(t)=\frac{Q_G}{PATH\left(G,{k}_a\right)\ast PR(i)} $$
(13)
where QG represents the Global pheromone concentration and PATH(G, ka) is the path length of the ant ka.
Formula (
13) shows that updating pheromones through path superiority ensures the energy requirement of the path and takes into account the energy balance, which is conducive to improving the efficiency of data collection.