Of course, this means that phenomena related to the alternating electromagnetic field, such as conversion forced by Lorentz forces, are not taken into account in the model. However, as shown by simulation results in this paper, the influence of magnetohydrodynamic phenomena can be negligibly small. Another illustrative instance of employing computational numerical modeling to facilitate a comprehensive analysis of metallurgical slag reduction processes within electric furnaces is presented in the study [2]. The paper showcases the outcomes of numerical simulations pertaining to flow and heat transfer within a cylindrical electric furnace dedicated to the processing of metallurgical slag. The results derived from the numerical analyses were subsequently validated by juxtaposing the relationship between slag temperature and electrical potential with empirical data procured from the steelworks. The potential for enhanced slag reduction and copper recovery within crossed electric and magnetic fields is also elucidated in the study [9]. This work encompasses an analysis of the physicochemical phenomena inherent to the copper slag reduction process, leveraging the capabilities of numerical modeling. [PERSON] et al. [10] investigated the influence of electrode immersion depth and furnace operating power on the smelting of platinum group metals (PGMs) from  
Figure 1: Schematic diagram of a resistance furnace with two (**a**) and three (**b**) electrodes. The symbol “x” denotes the Joule heat release center. ores. These analyses indicated significant differences in melt temperature with shallow electrode immersion and a uniform temperature distribution in the case of deep electrode immersion and high power. Analyses that take into account Joule heating are presented in refs. [11; 12; 13], where the influence of electrode shape and immersion depth on the heating of an electric furnace in ferroickel processing is analyzed. They demonstrate the influence of the slag layer height on the amount of Joule heat generated. The literature presents many applications of the finite element method (FEM) in single-criterion optimization. Research [14] including the analysis of the effect of selected process parameters on the temperature field for the Fused Filament Fabrication's optimization was performed. In the work [14], the optimization criterion was determined by a FEM model that calculates the temperature field for three input parameters, such as scanning speed, molding chamber temperature, and nozzle temperature. For that matter, the FEM was used to simulate the temperature field distribution. Another instance is presented in the research [15], where the FEM was used to calculate the stress distribution for various geometries to optimize the topology of aircraft seats. The publication [16] considers using the FEM to optimize magnetic field strength in Electromagnetic Forming (EMF), where the influence of parameters such as current density, gap, and coil size is analyzed. Another study [17] presented the optimization of the geometry of thermoelectric generators, where FEM was used to analyze the temperature distribution. In addition to the presented single-criterion optimization examples, the FEM is also used in multi-criteria optimization. For instance, the publication [18] considered the problem involving a Switched Reluctance Motor (SRM) where both current and torque are optimized by using a Pareto front as well as the FEM method, among others. The literature also provides examples of using the FEM to directly determine criteria for optimization algorithms. Refs. [19; 20] presents an evolutionary algorithm that iteratively uses the FEM to determine the short-circuit impedance and the magnetic field distribution for conductor size optimization. The authors of [21] presented a way to interconnect the independent mesh mode, eXtended Finite Element Method (XFEM), with multi-objective optimization based on the evolution algorithms for electromagnetic problems.