With every passing day, the demand for data traffic is increasing, and this urges the research community not only to look for an alternating spectrum for communication but also urges radio frequency planners to use the existing spectrum efficiently. Cell sizes are shrinking with every upcoming communication generation, which makes base station placement planning even more complex and cumbersome. In order to make the next-generation cost-effective, it is important to design a network in such a way that it utilizes the minimum number of base stations while ensuring seamless coverage and quality of service. This paper aims at the development of a new simulation-based optimization approach using a hybrid metaheuristic and metamodel applied in a novel mathematical formulation of the multi-transmitter placement planning (MTPP) problem. We first develop a new mathematical programming model for MTPP that is flexible to design the locations for any number of transmitters. To solve this constrained optimization problem, we propose a hybrid approach using the radial basis function (RBF) metamodel to assist the particle swarm optimizer (PSO) by mitigating the associated computational burden of the optimization procedure. We evaluate the effectiveness and applicability of the proposed algorithm by simulating the MTPP model with two, three, four and five transmitters and estimating the Pareto front for optimal locations of transmitters. The quantitative results show that almost maximum signal coverage can be obtained with four transmitters; thus, it is not a wise idea to use higher number of transmitters in the model. Furthermore, the limitations and future works are discussed.

The demands for data traffic have been increasing at an exponential rate over the last decade, and this trend is expected to continue in the future. Billions of humans and devices require seamless wireless communication in indoor and outdoor environments supporting higher data rates, and this demand has resulted in the evolution of heterogeneous networks (HetNets) with small cell size [

Legacy networks have a huge cell size; however, in next-generation networks, the cell size is dramatically reduced, which creates a major challenge for telecommunication network operators to scrupulously plan their complex networks. Networks of the fifth-generation (5G) and beyond necessitate the massive deployment of BS. If these deployments are unplanned, this can result in a huge cost, higher interference levels, and overall straitened network performance. It is therefore important to perform BS placement planning in such a manner that requires the least number of transmitters to accomplish the desired coverage ratio while maximizing the average received power in conjunction with the quality of service (QoS) [

In various applications of WSNs, the coverage rate is always considered to be an essential indicator because it determines the monitoring capability for the target area [

Metamodeling techniques have been used to avoid intensive computational and numerical simulation models, which might squander time and resources when estimating the model’s parameters. The general overview of a metamodel is illustrated in

In this study, we develop a new mathematical programming model for the MTPP problem. To solve this mathematical optimization model, we propose a new approach combining the PSO metaheuristic and RBF metamodel applied in the practice of simulation–optimization for wireless signal coverage in the MTPP problem. Among metaheuristic techniques, PSO has attracted wide attention in engineering design problems due to its algorithmic simplicity and powerful search performance [

The major contributions of our research can be summarized as follows:

A new mathematical programming model is developed in this paper for coverage optimization in a multi-transmitter placement planning problem. This model has flexibility in the design of optimal placements for any number of transmitters (not limited to an exact number of transmitters in the optimization model).

To solve this new optimization model, this paper also proposes a new hybrid technique using a combination of an RBF metamodel and PSO optimizer. The proposed algorithm can search the whole design space continuously with no need to run a simulation model for all investigated locations (thus incurring a low computational cost).

The rest of this paper is organized as follows. Section 2 provides the preliminaries and follows the algorithmic framework of the proposed algorithm which has been developed for the MTPP problem in Section 3. In Section 4, a numerical case study is presented to show the applicability and effectiveness of the proposed algorithm in the simulation–optimization of the MTPP problem. Finally, this paper is concluded in Section 5.

The RBF is a kind of neural network that employs RBF as a transfer function (see

The simple mathematical forms of RBF can be expressed as

where _{i}_{i}

where _{2} norm and _{ci}_{i}

where _{i}

The canonical PSO algorithm, which simulates the swarm behaviors of social animals, such as bird flocking or fish schooling, was proposed in [

where w is inertia weight factor, _{1} and _{2} represent the speed, regulating the length when flying towards the most optimal particles of the whole swarm, and the most optimal individual particle. The term _{i}_{g}

Here, we explain the algorithmic framework of the proposed simulation-based optimization technique based on a hybrid RBF metamodel and PSO metaheuristic.

Subject to:

where _{X}_{Y}

In this section, the applicability of the proposed algorithm in the MTPP problem is studied to obtain the optimal locations of each transmitter to maximize the amount of signal coverage. To continue, first, the application of the proposed algorithm using the hybrid RBF metamodel and PSO metaheuristic integrated by mathematical programming in the current MTPP optimization problem is explained; then, the obtained results are discussed in detail.

In this work, we used a ray-tracing simulator developed in Python. We created a simulation model that simulated the environment of an office room to implement the MTPP problem. The obstacles in the room were modeled according to the layout of the office equipment by creating a polygon of each object and placing it in various positions in the room. The calculations of the coverage area assumed that the signal propagation was omnidirectional, and the transmitter could transmit the signal in two dimensions. To evaluate the coverage rate in a two-dimensional area, we divided the entire monitoring region into

To reduce the number of simulation runs and produced images, we decussated odd locations (odd

This work aimed to develop a new method to investigate the optimal locations of multiple transmitters. However, in the current instance, we applied the proposed algorithm. Note that the proposed algorithm is flexible for any number of transmitters, and we are not limited to the number of transmitters in a model. Here, to show the applicability and effectiveness of the proposed algorithm, the MTPP problem was investigated for two, three, four, and five transmitters, respectively. For this purpose, we first constructed the RBF metamodel over the set of I/O data obtained from the simulation model. We used the “newrbe” Matlab

We constructed the mathematical programming model for the MTPP problem (see

Objective function:

Constraints:

where _{i, j} shows the Euclidean distance between transmitters (e.g.,

In the model with two transmitters (

Transmitter | |||||||||
---|---|---|---|---|---|---|---|---|---|

X | Y | ASC | X | Y | ASC | X | Y | ASC | |

# 1 | 26 | 0 | 0.435 | 0 | 0 | 0.426 | 0 | 0 | 0.426 |

# 2 | 222 | 188 | 0.407 | 222 | 188 | 0.407 | 222 | 188 | 0.407 |

Overall coverage | 0.733 | 0.713 | 0.713 |

Next, we constructed the optimization procedure based on the proposed hybrid RBF and PSO algorithm for a case with three transmitters in the model. The obtained optimization results are shown in

Transmitter | |||||||||
---|---|---|---|---|---|---|---|---|---|

X | Y | ASC | X | Y | ASC | X | Y | ASC | |

# 1 | 26 | 0 | 0.435 | 222 | 0 | 0.384 | 0 | 0 | 0.426 |

# 2 | 222 | 188 | 0.407 | 222 | 188 | 0.407 | 222 | 188 | 0.407 |

# 3 | 222 | 0 | 0.384 | 26 | 0 | 0.435 | 222 | 0 | 0.384 |

Overall coverage | 0.746 | 0.746 | 0.726 |

Transmitter | |||||||||
---|---|---|---|---|---|---|---|---|---|

X | Y | ASC | X | Y | ASC | X | Y | ASC | |

# 1 | 222 | 148 | 0.386 | 222 | 0 | 0.384 | 222 | 188 | 0.407 |

# 2 | 222 | 188 | 0.407 | 74 | 92 | 0.389 | 26 | 0 | 0.435 |

# 3 | 222 | 0 | 0.384 | 0 | 0 | 0.426 | 0 | 222 | 0.277 |

# 4 | 0 | 0 | 0.426 | 222 | 188 | 0.407 | 222 | 0 | 0.384 |

Overall coverage | 0.794 | 0.802 | 0.748 |

Besides, the application of the proposed algorithm was used for the MTPP problem with five transmitters. Optimal locations and relevant obtained results are provided in

Transmitter | |||||||||
---|---|---|---|---|---|---|---|---|---|

X | Y | ASC | X | Y | ASC | X | Y | ASC | |

# 1 | 0 | 0 | 0.426 | 222 | 188 | 0.407 | 222 | 0 | 0.384 |

# 2 | 222 | 188 | 0.407 | 0 | 0 | 0.426 | 222 | 188 | 0.407 |

# 3 | 222 | 0 | 0.384 | 74 | 92 | 0.389 | 90 | 110 | 0.389 |

# 4 | 26 | 32 | 0.427 | 222 | 0 | 0.384 | 0 | 0 | 0.426 |

# 5 | 32 | 0 | 0.425 | 62 | 0 | 0.381 | 2 | 222 | 0.276 |

Overall coverage | 0.759 | 0.802 | 0.799 |

Finally, we compared all obtained results based on

Here, we developed a new mathematical programming model for the MTPP problem. This model has flexibility in terms of the design of optimal placements for any number of transmitters (not limited to an exact number of transmitters in the optimization model). To solve this new optimization model, we also propose a new hybrid technique using a combination of the RBF metamodel and PSO optimizer. In comparison with common model-based optimization techniques, two main advantages of the proposed hybrid algorithm are as follows:

The proposed algorithm can search in the whole design space continuously. In other words, in the current instance in this paper, we are not restricted to investigating optimal locations only by discrete areas when searching among the 12,544 locations (as aforementioned, the signal coverage for 12,544 locations in the two-dimensional design space is modeled and computed through the simulation model); thus, any area within these 12,544 locations can be investigated as well because we APPLY the interpolation method (the well-known RBF metamodel) that can be trained on the whole design space. In contrast, the model-based optimization techniques are limited to searching only within a limited number of locations.

In the current paper, for the MTPP problem, one way to tackle the previous shortcomings in the case of using model-based techniques for coverage optimization is that the number of locations modeled in the simulation can be increased. However, the simulation model needs to run with a larger number of locations (more than 12,544 locations in the current study) and compute the signal coverage for locations with smaller distances. However, this procedure can increase the computational cost of optimization significantly.

There are two main limitations in the current study, as stated below:

The proposed method employs the RBF metamodel to train the black-box simulation model. Therefore, the approximate errors cannot be ignored when solving inequality-constrained simulation-based optimization problems. A challenge for optimization under restricted budgets (limited number of sample points) will be to find the right degree of approximation (smoothing factor) from only a relatively few samples [

In this study, we use the common death penalty method to deal with inequality constraints in the optimization model. However, other constraint handling methods [

Simulation-based optimization is a viable solution for solving complex real-world problems, such as the base station placement problem in wireless sensor networks. The multi-transmitter placement planning problem of guaranteeing coverage while meeting some application requirements quite often gives rise to NP-hard optimization problems. In this paper, we developed a new mathematical programming model to handle the design of optimal locations for any number of transmitters in a two-dimensional design space. To solve the mentioned MTPP model, we proposed a new hybrid method combining the PSO metaheuristic and RBF metamodel. The applicability of the proposed method was shown using a simulation-based MTPP optimization problem with two, three, four, and five transmitters. The results showed the effectiveness of the proposed algorithm for solving a mathematical model of the MTPP optimization problem with a computationally less expensive method. The results also showed that, in the current case study, cost-effective signal coverage concerning considered obstacles could be obtained with four transmitters (with an average signal coverage equal to 82% of the area in the model), and more transmitters did not provide a significant change in the amount of average signal coverage. This approach can continuously search the whole of the design space to obtain optimal locations for any number of transmitters in the model without requiring the simulation of signal coverage for all locations via a simulation model. The RBF metamodel can train the model using a black-box simulation model and a limited amount of input-output data to approximate all locations in the design space. Finally, the limitations of the current study were discussed. Regarding the mentioned limitations of the current study, future research can be addressed.