## 1 Introduction  
Currently, sensors and sensor networks are widespread in all spheres of human activity. The use of sensors is ensured by their advantages, such as small size, various measured values, ease of software development, low power consumption, mobility and service life. Additionally, many tasks, such as collecting, analyzing, identifying and measuring data, can be solved using sensors [1, 2]. In addition, sensors are an integral part of the Internet of Things and the Internet of Everything [3, 4]. Additionally, sensors can be static and moveable. One of the main problems in the use of sensors and sensor networks is to ensure energy efficiency, as the size of the sensors is reduced, resulting in a decrease in the size of the battery. Most of the tasks of optimizing the energy consumption of sensor networks are formulated in the form of optimization problems. Among the factors affecting the energy consumption of a sensor network can be the energy consumption of each sensor, the number of sensors, the radius of sensor coverage, etc. Additionally, optimization criteria can be the cost of the network, the area of the territory covered by the network and the time of autonomous operation. Some problems are solved using iterative algorithmic procedures. It is also possible to use approximate methods of finding solutions to optimization problems. However, in most cases, the existence of a solution to the problem has not been proven, which calls into question the correctness of such problems. Thus, proving the existence of solutions is an integral part of the process of finding a solution that satisfies the optimality condition. Article [5] introduces a charging planning problem for multiple chargers; namely, the Charging Energy Efficiency Maximization problem for Multi-Chargers in WRSNs is introduced. The problem aims to maximize the charging energy efficiency of the charging process by assigning the charging amount and planning the charging path. To balance the charging consumption among multiple chargers, the authors proposed Ring-Wandering Algorithm and Eight-Wandering Algorithm. In the presented article, NP-hardness of the problem is proved, but the intersection of sensor coverage areas is not taken into account, and the existence of the solution of the problem with the intersection of sensor coverage areas is not proved. Paper [6], approaches to the allocation of resources in the Internet of Things sensor network systems applied to water-quality monitoring for the optimal and more sustainable utilization of resources are presented. An energy efficiency optimization problem is formulated for the successive wireless power sensor network system and solved by exploiting the problem structure and through a meta-heuristic algorithm. However, the existence of the solution of the described optimization problem is not proven. In work [7], a survey on applications of the Markov Decision Process (MDP) for reducing sensor networks' energy consumption is presented by designing the MDP framework which is a powerful tool for decision making and further providing solutions to obtain energy efficient sensor network. Various methods for solutions are discussed, and the methods are compared but do not describe the formulation of the optimization problem and prove the existence of its solution. In article [8], power consumption of the various wireless sensor network components is analyzed. The paper includes a discussion of various factors and the way they impact the level of energy efficiency, but any optimization problem formulation is not presented. In study [9] is a numerical study of the energy consumption and time efficiency of sensor networks with five different structural topologies and four different routing methods, regarding their performances and costs, which might provide some references and guidelines for designing sensor networks under various conditions for possible applications. However, the existence of the solution of the described optimization problem is not proven.