Keywords

1 Introduction

Wireless sensor networks (WSNs) have incited remarkable research interests due to their vast potential in sensors, electronics, and computational fields. They have been exploited for civil as well as defense-related purposes. A WSN typically comprises large numbers of sensor nodes, which are energy constrained with limited computational and communication capabilities [1, 2]. The deployment of WSN nodes is usually based upon its application and could be random or deterministic [3–5]. Random deployment is usually done in hostile scenarios such as battlefield or hazardous environments, whereas amiable scenarios call for the deterministic deployment. In general, WSNs are expected to provide access to information about the physical world, regardless of time and space, this vision poses significant challenges for WSNs. The pervasiveness of WSN’s limits its centralized control and is not practical and calls for capabilities of scalability, self-organization, self-adaptation, and survivability [6].

Energy utilization is a major issue for a WSN as the energy resources are consumed during the operation of nodes. The replacement of batteries or their recharge may sometimes be infeasible. Energy efficiency and utilization of a WSN depends upon the temporal resolution of information being collected, routing strategies, node placements, etc. [7–9]. Another important issues to be taken care of in a WSN are the network lifetime and connectivity. Cluster-based architectures are generally employed, in which the nodes are arranged in their network. These networks communicate with their respective cluster head node. Thus, collected information from the nodes is transmitted to the base station. The network connectivity problems include not only the load handling capability of the sink nodes, but also the ability of the sensor nodes to communicate with the cluster heads. Apart from the above issues, the application-specific design parameters also pose some issues. Several algorithms [3, 10–21] have been reported for the WSN design optimization in terms of scalability, self-organization, self-adaptation, and survivability. However, most of those suggested algorithms do not necessarily address the application-specific issues and make design parameterization and optimization a challenging task.

The design of a WSN system hence calls for simultaneous optimization of multiple nonlinear design parameters. This is a challenging task, as it requires finding pareto-optimal solutions under severe computational limitations. Such problems have been reported to be tackled with the application of computational approaches, such as neural networks, swarm optimization, genetic algorithm (GA), and ant colony optimization [22–28]. Social impact theory (SITO) is a recently introduced approach based on the application of a novel [29]. In this approach, a spatially distributed population of individuals in a two-dimensional lattice networks with each other to generate an optimal solution. In the process, the individuals change their attitude for a particular feature under influence of their neighbors’ number, attitude, strength, and immediacy. The optimizer has been tested on benchmark problems for feature subset selection [29–33]. However, this optimizer has not been attempted for WSN optimization as yet.

In the present work, we have tried to analyze the application of SITO in WSN by integrating the network characteristics according to the application-specific requirements. In general, the algorithm under the constraints of application-specific requirements and energy consumption determines operational modes of the nodes. In particular, the network design has been investigated with respect to the sensor placements, communication range, and clustering. The performance of the proposed approach has been investigated by the study of connectivity and related energy characteristics and application-oriented properties (e.g., uniformity/spatial density of the sensing nodes). The work finally proposes an optimal design in which the mode of operation has been specified for each sensor node.

2 Methodology

2.1 Social Impact Theory-Based Optimization (SITO)

The social impact theory was proposed by Latané [34] wherein the author defined the social impact as any influence on an individual’s feelings, thoughts, or behavior that is exerted by the real, implied, or imagined presence or actions of others. This meta-theory characterized the spatiotemporal variabilities of human opinion formation. This theory was modified by Nowak et al. [35] by taking into consideration the reciprocal influence of the individuals on their environment. Further, Macaš et al. [29] and Bhondekar et al. [30] implemented the above theory for optimal feature extraction and classification. The SITO algorithm is advantageous because of the requirement of few control parameters and capability of analyzing spatially distributed population.

In the SITO algorithm, an individual represents a probable solution for the problem at hand by maintaining a set of spatially distributed population in a two-dimensional lattice. The strength of the individual is estimated by taking in account the fitness value of its opinion. This opinion is subsequently modified at every iteration with respect to number of neighbors, strength, and immediacy. Total societal impact (I) is calculated by difference between the persuasive impact (I p) of individuals holding the opposite opinions and the supportive impact (I s) of individuals with the same opinion. I p and I s are defined as expressed by the following equations.

$$ I_{\text{p}} = N_{\text{o}}^{1/2} \left[ {{{\sum {\left( {{{p_{i} } \mathord{\left/ {\vphantom {{p_{i} } {d_{i}^{2} }}} \right. \kern-0pt} {d_{i}^{2} }}} \right)} } \mathord{\left/ {\vphantom {{\sum {\left( {{{p_{i} } \mathord{\left/ {\vphantom {{p_{i} } {d_{i}^{2} }}} \right. \kern-0pt} {d_{i}^{2} }}} \right)} } {N_{\text{o}} }}} \right. \kern-0pt} {N_{\text{o}} }}} \right] $$
(1)
$$ I_{\text{s}} = N_{\text{s}}^{1/2} \left[ {{{\sum {\left( {{{s_{i} } \mathord{\left/ {\vphantom {{s_{i} } {d_{i}^{2} }}} \right. \kern-0pt} {d_{i}^{2} }}} \right)} } \mathord{\left/ {\vphantom {{\sum {\left( {{{s_{i} } \mathord{\left/ {\vphantom {{s_{i} } {d_{i}^{2} }}} \right. \kern-0pt} {d_{i}^{2} }}} \right)} } {N_{\text{s}} }}} \right. \kern-0pt} {N_{\text{s}} }}} \right] $$
(2)

where, p i is the persuasiveness of source i

s i :

denotes the supportiveness of source i

N o :

represents number of sources (individuals with opposing opinion)

N s :

represents the number of individuals with individual opinions and d i refers to the distance between the source i and the recipient

Generally, the individuals’ opinions are modulated by comparing I p and I s, if I p is greater than I s, the of the individual changes with a probability 1 − K. Similarly, the attitude may change with a probability K if I p is lesser than I s. The probability K improves the explorative capability by preventing loss of diversity.

The pseudocode as proposed by Macaš [29] is expressed as under:

2.1.1 Problem Outline

In this work we have assumed a two-dimensional field employing three types of sensors, which monitor parameters related to X, Y, Z. The spatial variability is such that sensor nodes’ density in Z is greater than Y and X and for Y it is greater than X [36].

2.1.2 Network Model

A grid-based Euclidian model has been considered here, wherein the nodes have been placed at the intersections (see Fig. 1).

Fig. 1
figure 1

The layout of a wireless sensor network

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The active sensing nodes considered for this simulation are identical and have the usual features like power control and selection parameter for different sensing modes in X, Y, Z along with power control in transmission. We have assumed a cluster architecture where, the cluster-in-charge are the nodes operating in X-sense, along with Y and Z sensing modes with middle and small transmission ranges, respectively. It should be noted that the nodes present in the X mode can communicate with the base station using a multi-hop protocol and this leads to clustering of nodes in their vicinity [30].

Apart from sensing the X parameter the node in X-sense mode also performs tasks of data collection and its accumulation along with complex computations.

2.1.3 Problem Statement

The design parameters of WSN can be categorized into 3 classes [37]. First category incorporates the parameters of sensor deployment, e.g., uniformity and coverage. The second category deals with the connectivity parameters in a manner that no node remains unconnected. The last category involves the variables or parameters responsible for the survivability of the network, such as operational energy. In the proposed work, we have explored a multi-objective algorithm to optimally select these design parameters by scalarizing them into a single fitness function as given by Eq. 3. The design optimization has been achieved by minimizing constraints such as number of unconnected sensors, operational energy and number of overlapping cluster-in-charge. The parameters namely number of sensors for each cluster-in-charge and field coverage are maximized. The measurement of quality of each probable solution of the optimization problem is given by the objective function in form of a numerical figure [36].

$$ f = { \hbox{min} }\left\{ {\sum\limits_{i = 1}^{5} {k_{i} P_{i} } } \right\} $$
(3)

where, k and P i are the corresponding weight and optimization parameter, respectively [36].

2.2 WSN Representation, Optimization Parameters, and Fitness Function

A square field (L × L length units) has been subdivided into several grids of unit lengths. The nodes are arranged on the grids. A bit-string represents an individual attitude in the society, which is employed for the encoding of the sensor nodes in a row-wise pattern as depicted in Fig. 2. Two bits are needed for the encoding of four states of the sensing nodes, viz. X, Y, Z, and inactive. Thus, the total length of the bit-string is set to be 2 · L2.

Fig. 2
figure 2

Representation of Bit-string network layout [30]

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The optimization parameters listed in Table 1 are derived from following network attributes

n x :

is the X Sensors (cluster-in-charge) in terms of numbers. Similarly,

n y :

Y Sensors

n z :

Z Sensors

n OR :

Out-of-Range Sensors

n inactive :

Inactive Sensors

n total :

total sensing points

n o :

overlaps of cluster heads

Table 1 Correspondences between objectives and optimization parameters [36]
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The parameters to be optimized are derived as follows and are defined in [36]:

$$ {\text{FC}} = \frac{{(n_{x} + n_{y} + n_{z} ) - (n_{\text{OR}} + n_{\text{inactive}} )}}{{n_{\text{total}} }} $$
(4)
$$ {\text{SpCi}} = \frac{{n_{\text{y}} + n_{\text{z}} - n_{\text{OR}} }}{{n_{x} }} $$
(5)
$$ {\text{SORE}} = \frac{{n_{\text{OR}} }}{{n_{\text{total}} - n_{\text{inactive}} }} $$
(6)
$$ {\text{OpCiE}} = \frac{{n_{\text{o}} }}{{n_{x} }} $$
(7)
$$ {\text{NE}}\frac{{4 \cdot n_{x} + 2 \cdot n_{y} + n_{z} }}{{n_{\text{total}} }} $$
(8)

Therefore, there is a unique bit-string sequence for every unique WSN Design whose feature and performance can be estimated using fitness or weighting function. The fitness or weighting function needs to properly signify all the significant design parameters to influence the desired quality/performance of the WSN design. Each of the design parameter is equally important. Therefore for the present problem, the fitness function may be formulated as

$$ f = - \alpha_{1} {\text{FC}} + \alpha_{2} {\text{OpCiE}} + \alpha_{3} {\text{SORE}} - \alpha_{4} {\text{SpCi}} + \alpha_{5} {\text{NE}} $$
(9)

In the above fitness function, the appropriate weighting coefficients α i : i = 1, 2…5 define the significance of each design parameter. Therefore, the SITOs objective is to minimize the value of fitness function, to maximize some parameters their coefficients must be negative. The coefficient values are determined on the basis of design requirements and related experimentation. The desired values of the individual parameter coefficient were manually computed. The well-performing weight are listed in Table 2.

Table 2 Optimized values of the weighing coefficients
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During the analysis (Table 2), the network connectivity variables (weights α 1, α 4) were considered as constraints such that all the sensor nodes are in the limit of a cluster-in-charge, and no cluster-in-charge connects to higher than a predefined number of the sensors nodes.

3 Experimental Results

A network size of 10 × 10 was considered for experimentation. A total of 100 runs were carried out wherein four different machines performed 1000 iterations on four separate segments of 25 samples. Various combinations of society size and neighborhood were experimented and the best results in terms of convergence rate were obtained for society size of 225, and neighborhood size of 2. The convergence results obtained for neighborhood size 2 at different society sizes are shown in Fig. 3. It may be observed that increasing the society size beyond 225 does not improve the performance of the algorithm. Moreover, by further increasing the society size, the convergence time increases exponentially. Similarly, Fig. 4 shows the convergence rate of the fitness value for a society size of 225 with varying neighborhood sizes. It may be observed that the neighborhood size of 2 gives the best convergence rate.

Fig. 3
figure 3

A comparison between the convergence rates obtained at different society sizes with constant neighborhood size of 2

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Fig. 4
figure 4

Comparison of convergence rates obtained at different neighborhood sizes with constant society size of 225

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The optimized network by the algorithm is graphically represented employing a customized MATLAB script. One of the observed designs is shown in Fig. 5 in which the red, blue, and green circle, respectively, denote the X (cluster-in-charge), Y, and Z sensor positions. A cross-mark circle represents the out-of-range sensor node, whereas an inactive sensor node is represented by the empty space.

Fig. 5
figure 5

A graphically represented network as optimized by the algorithm

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Figure 6 shows the progress of average value and minimum value of society fitness of one of the best runs. The three best SITO runs (abbreviated as S1, S2, and S3) that yielded the best results after 3000 iterations were observed and their results are in Table 3.

Fig. 6
figure 6

Evolution progress for the identification of best individual (highest fitness value) and the entire society (average fitness value) using SITO approach

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Table 3 Values of optimized parameters for 3 SITO-generated layouts of the network
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4 Conclusions

A human opinion formation-based strategy (SITO) was used to optimize the nodes deployment of a fixed WSN. A grid-based fixed WSN having nodes of different operating modes was considered. The optimization was based upon various network parameters viz. field coverage, cluster overlapping, out-of-range errors, and network energy. The results showed that human opinion formation-based algorithm such as SITO can be used in WSN applications.