With the explosive advancements in wireless communications and digital electronics, some tiny devices, sensors, became a part of our daily life in numerous fields. Wireless sensor networks (WSNs) is composed of tiny sensor devices. WSNs have emerged as a key technology enabling the realization of the Internet of Things (IoT). In particular, the sensorbased revolution of WSNbased IoT has led to considerable technological growth in nearly all circles of our life such as smart cities, smart homes, smart healthcare, security applications, environmental monitoring, etc. However, the limitations of energy, communication range, and computational resources are bottlenecks to the widespread applications of this technology. In order to tackle these issues, in this paper, we propose an Energyefficient Transmission Range Optimized Model for IoT (ETROMI), which can optimize the transmission range of the sensor nodes to curb the hotspot problem occurring in multihop communication. In particular, we maximize the transmission range by employing linear programming to alleviate the sensor nodes’ energy consumption and considerably enhance the network longevity compared to that achievable using stateoftheart algorithms. Through extensive simulation results, we demonstrate the superiority of the proposed model. ETROMI is expected to be extensively used for various smart city, smart home, and smart healthcare applications in which the transmission range of the sensor nodes is a key concern.
Datadriven wireless sensor networks (WSNs) are widely applied to enhance the Internet of Things (IoT) in terms of the data throughput, energy efficiency, and selfmanagement [
When a WSNbased IoT is deployed over a large application area, the nodes perform multihop communication due to the limited transmission range, and direct data transmission cannot be realized. Furthermore, it has been reported that a larger number of relay nodes on the path of data delivery to the sink corresponds to a higher probability of these nodes closer to the sink suffering from hotspot problem [
Moreover, the battery of the sensor nodes may not be able to be changed or recharged. Therefore, it is necessary to ensure efficient power consumption in a WSNbased IoT [
One solution is to maximize the transmission range between nodes. The key concept of transmission range maximization is that if a sensor initiates a data packet transmission to a sink located 1000 m away, the least number of relay sensors should be selected to forward the packet. The communication range of sensor nodes depends on their transmission power and the volume of the packet to be transmitted. Transmission over long distances requires a higher energy [
Many researchers have attempted to reduce the energy consumption by avoiding the hotspot problem [
Moreover, researchers generally apply the coronabased model to avoid hotspot problems. A survey of the various coronabased approaches has been presented in an existing study [
The review pertaining to the mitigation of hotspot problems indicated that the optimizationbased approach can provide a balanced solution to specific problems. Therefore, in this work, we used linear programming (LP) to compute the maximum data transmission range [
In the context of the aforementioned problems, the key contributions of this work are as follow:
We propose an energyefficient transmission range optimized model for IoT (ETROMI) to optimize the transmission range of the sensor nodes to reduce the hotspot problem in WSNbased IoT.
The mathematical model and formulation using LP is presented.
The simplex method is used to solve the defined problem.
The proposed model’s performance of the proposed model is analyzed in terms of various aspects, and the optimal solution is identified.
The remaining paper is structured as follows. Section 2 presents the background of transmission range adjustment algorithms and describes the existing work pertaining to the hotspot problem in WSNs. Section 3 describes the system model and explains the LP formulation. Section 4 describes the performance evaluation of ETROMI, which is used to compute the maximized distance corresponding to the transmission range of a node. The concluding remarks, along with the limitations and scope for future work, are presented in Section 5.
In this section, we discuss the existing work focused on addressing the hotspot problem through various stateoftheart techniques and on realizing the transmission range adjustment of a sensor node.
In applications involving an extremely large network area, the sensor nodes inevitably perform multihop communication [
Elkamel et al. [
In addition to the static network scenario, certain researchers introduced sink mobility to curb the hotspot problem. Sahoo et al. [
In addition to the network topological changes associated with the introduction of the coronabased model, the characteristics of sensor nodes have been examined. The focus of the present study is to optimize the transmission range. Although certain researchers have attempted to adjust the transmission range to alleviate the hotspot problem, the proposed approaches suffer from the inherent problems, which limit their relevance.
In an existing strategy [
In summary, only a few studies have been focused on addressing the hotspot problem through transmission range adjustment, and this approach exhibits considerable scope for improvement. Furthermore, the use of LP for energyhole mitigation in the transmission range adjustment context is yet to be explored. Therefore, we implement these aspects in our proposed strategy.
In this section, we describe the network assumptions and the system model.
The following network assumptions are considered to implement ETROMI.
The WSN is composed of one sink and several sensor nodes that collect data and transfer them to the sink.
Each sensor has a unique ID.
There is no dispute for medium access, and thus, proportional fair channel access is available to all the sensors.
The minimum cost forwarding approach is employed as the multihoprouting protocol.
The sensor nodes are homogeneous, i.e., all the nodes have the same configuration in terms of energy, computational resources, transmission range, etc.
The entire network is static, including the CHs and the sink.
The entire network has ideal conditions in terms of security, physical medium factors, reflection, refraction, splitting of signals, and presence of other obstacles.
Assume a WSN with
In contrast, the lifetime of the WSNs depends on the remaining energy of the members, i.e., the sensor nodes. Therefore, the number of forwarding nodes must be minimized. In this study, we assume that instead of always selecting the CHs to receive and forward the data packets, the sensors select the farthest sensor node in their transmission range. In other words, the sensor nodes increase their transmission power to transmit a data packet over a longer distance. In this configuration, the number of nodes that are involved in a transmission are minimized, which can considerably improve the energy efficiency. In
In this approach, the nodes send their packets to their corresponding CH, which then forwards the packet to the next CH and so on. Finally, the closest CH delivers the packet to the sink. In contrast, in the approach represented by the green dashed line, the node that initiates the packet sends the packet to the farthest node, and the receiver node follows the same principle and send the packet to the farthest node in its transmission range. Consequently, the number of nodes involved in the transmission procedure is less than that in the clusteringbased method.
Consider a network involving 100 sensor nodes. Sensor 1 initiates a data packet and wants to send it to node 100. As mentioned earlier, in the WSNs, the topology is multihop. In other words, node 1 sends its packet to its neighbor, which receives the packet and forwards it to the neighboring nodes, excluding the node that the packet was received from. This process continues until node 100 receives the packet. The problem then is to determine the number of nodes in the transmission process that receive and forward the packet. This number of the relay nodes should be minimized to abate the energy consumption and, in turn, prolongs the network lifetime.
One solution is to increase the transmission range of the nodes involved in the transmission process from the source to the destination. In this case, a node selects a neighboring node, which is far from it, but in its range, i.e., on the edge of its transmission range, and the number of intermediate nodes is decreased. To this end, we consider the energy consumption accounted to transmission and reception of data packets and also the magnitude of data packets. The total required energy can be expressed as
The list of main symbols used in this paper are listed in
Symbol  Definition 

The distance between node 

The initial power  
Total energy consumption of a link  
The receiving power  
Transmission power  
Packet volume  
The distance between node 

The number of nodes 
Consider a WSNbased IoT represented by graph
The objective function is to maximize the transmission distance with respect to the packet volume and the transmission and receiving energies, that is
Constraint
To evaluate the performance of the technique, we consider that a data packet is to be sent from a sensor node to the sink via two intermediate sensor nodes. Therefore, four sensor nodes are involved in the process: one sender, one sink, and two relay nodes. The size of each packet is 50 bits, and the transmission and receiving power are 100 and 70 W, respectively.
The linear problem can be expressed as
By adding slack variables to constraints, the primal problem in standard format is represented as follows:
We use the simplex method to solve the problem. The simplex method is used to solve LP models by using slack variables, tableaus, and pivot variables to determine the optimal solution of an optimization problem [
Obtain the standard form,
Introduce slack variables,
Create the tableau,
Identify the pivot variables,
Create a new tableau,
Check for optimality,
Identify the optimal values.
The procedure starts with an initialization phase, followed by several iterations to determine the optimal solution.
The initialization step for our optimization problem is presented in
Maximize  1  1  1  0  0  0  RHS  








0 
−1  −2  4  1  0  0  50  – 
0 
−2  5  −2  0  1  0  100  – 
0 
4  −2  −1  0  0  1  70  70/4 
1  1  1  0  0  0  0 
Subsequently, we apply the first iteration, as indicated in
0 
0  −5/2  15/4  1  0  1/4  270/4  – 

0 
0  4  −5/2  0  1  1/2  135  135/4 
1 
1  −1/2  −1/4  0  0  1/4  70/4  
0  3/2  5/4  0  0  −1/4  70/4 
We continue by applying the second iteration as indicated in
0 
0  0  35/16  1  5/8  9/16  1215/8  486/7 

1 
0  1  −5/8  0  1/4  1/8  135/4  – 
1 
1  0  −9/16  0  1/8  5/16  275/8  – 
0  0  35/16  0  −3/8  −7/16  545/8 
We proceed to the third iteration, in which all
1 
0  0  1  16/35  2/7  9/35  486/7 

1 
0  1  0  2/7  3/7  2/7  540/7 
1 
1  0  0  9/35  2/7  16/35  514/7 
0  0  0  −1  −1  −1  220 
The duality refers to a specific relationship between an LP problem and another problem, both of which involve the same original data, albeit located differently [
If the primal problem corresponds to “Maximize,” the dual problem corresponds to “Minimize.”
The number of variables in the dual problem is equal to the number of constraints in the primal problem.
The number of constraints in the dual problem, is equal to the number of variables in the primal problem.
The coefficients of the objective function in the dual problem, are equal to the righthand side (RHS) values in the primal problem.
The RHS values in the dual problem are equal to the coefficients of the objective function in the primal problem.
The coefficient variables in the constraints of the dual problem correspond to the transpose matrix of the coefficient variables in the primal problem.
“
The variables in the dual problem are denoted as “
The objective function is denoted as “
Our primal problem is as below:
Coefficient matrix of basic variables in objective function is
By adding surplus variables, the dual problem is as follows:
As indicated in the dual problem, no identity matrix exists for the coefficients of the variables in the constraints; therefore, artificial variables must be introduced. In this case, the dual problem in the standard format is:
By adding artificial variables, an identity matrix can be generated, and the simplex method can be implemented.
As indicated in
50  100  70  0  0  0  − 
− 
− 
RHS  











− 
−1  −2  4  −1  0  0  1  0  0  1  – 
− 
−2  5  −2  0  −1  0  0  1  0  1  – 
− 
4  −2  −1  0  0  −1  0  0  1  1  1/4 
−1  −1  −1  1  1  1  0  0  0  3 
− 
0  −9/4  15/4  1  0  −1/4  1  0  1/4  5/4  – 

− 
0  4  −9/4  0  −1  −1/2  0  1  1/2  3/2  3/8 
50y_{1}  1  −1/2  −1/4  0  0  −1/4  0  0  1/4  1/4  – 
0  −3/2  −5/4  1  1  0  0  1/4  11/4 
− 
0  0  35/16  −1  −5/8  −9/16  1  5/8  9/16  35/16  1 

100 
0  1  −5/8  0  −1/4  −1/8  0  1/8  3/8  –  
50 
1  0  −9/16  0  −1/8  −5/16  0  1/8  5/16  7/16  – 
0  0  −35/16  1  5/8  9/16  0  3/8  7/16  35/16 
70 
0  0  1  −16/35  −2/7  −9/35  16/35  2/7  9/35  1 

100 
0  1  0  −2/7  −3/7  −2/7  2/7  3/7  2/7  1 
50 
1  0  0  −9/35  −2/7  −16/35  9/35  2/7  16/35  1 
0  0  0  0  0  0  1  1  1  0 
70 
0  0  1  −16/35  −2/7  −9/35  1 

100 
0  1  0  −2/7  −3/7  −2/7  1 
1 
1  0  0  −9/35  −2/7  −16/35  1 
0  0  0  514/7  540/7  486/7  220 
Finally, it is observed that the primal solution, presented in
Z  220 

514/7  
540/7  
486/7 

220 

1  
1  
1 
Sensitivity analysis is aimed at examining the influence of changes in the variables, such as the RHS, coefficients of the objective function, and constraints, on the solution. We start with
Maximize  1  1  1  0  0  0  RHS 








1 
0  0  1  16/35  2/7  9/35  486/7 
1 
0  1  0  2/7  3/7  2/7  540/7 
1 
1  0  0  9/35  2/7  16/35  514/7 
0  0  0  −1  −1  −1  220 
In the last iteration, no nonbasic variables of the objective function exist. Therefore, if one of the coefficients is changed, the optimal solution is not influenced.
Suppose the intention is to change the first RHS to
Because,
Now, we continue the tableau with new RHS values;
As indicated in
Maximize  1  1  1  0  0  0  RHS  








1 
0  0  1  16/35  2/7  9/35  278/7  
1 
0  1  0  2/7  3/7  2/7  220/7  
1 
1  0  0  9/35  2/7  16/35  226/7  
0  0  0  −1  −1  −1  220 
The initialization step, as the first iteration, is presented in
1 
0  0  1  16/35  2/7  9/35  278/7 

1 
0  1  0  2/7  3/7  2/7  220/7 
1 
1  0  0  9/35  2/7  16/35  226/7 
0  0  0  −1  −1  −1  220  
–  0  –  −7/2  −7/3  −7/2 
Subsequently, we implement the second iteration, as indicated in
1 
0  −2/3  1  28/105  0  7/105  394/21  – 

0 
0  7/3  0  2/3  1  2/3  220/3  220/7 
1 
1  −2/3  0  7/105  0  28/105  34/3  – 
0  7  0  −1/3  0  −1/3  30.1 
We then implement the third iteration as indicated in
1 
0  0  1  48/105  6/21  37/105  278/7 

1 
0  1  0  2/7  3/7  2/7  220/7 
1 
1  0  0  27/105  6/21  48/105  226/7 
0  0  0  −1  −1  −115/105  724/7 
We consider the case in which the coefficient of
Any change in the coefficient of the basic variables of the objective function affects the value of
If
In this case, the range of
Maximize  −4  1  1  0  0  0  RHS  








1 
0  0  1  16/35  2/7  9/35  486/7  270 
1 
0  1  0  2/7  3/7  2/7  540/7  270 
−4 
1  0  0  9/35  2/7  16/35  514/7  1285/8 
0  0  0  2/7  3/7  9/7  −1030/7 
Upon completing the first iteration, the leaving variable is
The second iteration is presented in
1 
0  0  35/9  16/9  10/9  1  270 

1 
0  1  −10/9  −2/9  1/9  0  0 
−4 
1  0  −16/9  −7/9  −2/9  0  −50 
0  0  −80/9  −42/9  −19/9  −1  470 
In the last iteration, no nonbasic variable of the objective function exists. Therefore, if one of the coefficients is changed, the optimal solution is not influenced.
Consider a new variable
In this case, we perform three iterations as indicated in
Maximize  1  1  1  0  0  0  RHS  








1 
0  0  1  16/35  2/7  9/35  486/7  
1 
0  1  0  2/7  3/7  2/7  540/7  
1 
1  0  0  9/35  2/7  16/35  514/7  
0  0  0  −1  −1  −1  220 
Maximize  1  1  1  0  0  0  12  RHS  









1 
0  0  1  16/35  2/7  9/35  212/35  486/7  104/9 
1 
0  1  0  2/7  3/7  2/7  12/7  540/7  45 
1 
1  0  0  9/35  2/7  16/35  61/35  514/7  2570/61 
0  0  0  −1  −1  −1  137/105 
Maximize  1  1  1  0  0  0  12  RHS  









12 
0  0  3/18  8/9  15/9  27/18  1  104/18  
1 
1  0  37/127  68/319  65/319  67/319  0  36/18  
1 
0  1  −18/63  −78/63  −153/63  −144/63  0  235/18  
0  0  −1  −87/9  −160/9  −143/9  0  115 
Upon completing the first iteration, the leaving variable is
The third iteration is presented in
To examine the influence of the addition of a new constraint to the problem, we consider
As indicated in
Maximize  1  1  1  0  0  0  0  RHS 









1 
0  0  1  16/35  2/7  9/35  0  486/7 
1 
0  1  0  2/7  3/7  2/7  0  540/7 
1 
1  0  0  9/35  2/7  16/35  0  514/7 
0 
0  0  1  0  0  0  1  40 
0  0  0  −1  −1  −1  0  220 
The transmission range of a sensor node defines whether the communication mode is singlehop or multihop. In this paper, we proposed the use of ETROMI, which can determine the maximum distance to which a sensor node can transmit data with the least possible number of relay nodes. We presented an LPbased analytical model to determine the transmission range of the sensor node. Moreover, we explained the mathematical model associated with the ETROMI to reduce the energy consumption of WSNbased IoT. A key concern about the ETROMI is that it considers the ideal conditions involving no obstacles between the sensor nodes and the sink. Therefore, the model performance is specific to the circumstances. Furthermore, the network is assumed to be homogeneous, whereas homogeneity does not exist in an actual network due to the different factors associated with network deployment. In future work, we aim to extend our work to address the aforementioned scenarios.