# Multiple-Criteria Fuzzy Optimization of the Heat Treatment Processes for Two Steel Rolled Products
## 2 Informal Presentation of the Problem
### Fuzzy Multiple-Criteria Optimization of the Wire Rod Heat Treatment Process  
Heat treatment is the last operation in the technological chain of wire-rod production at the wire block of the Belorussian Mini Metallurgical plant. The correct choice of heat-treatment mode, taking into account the chemical composition of the metal and its temperature after hot rolling, should provide the required mechanical properties of the wire rod and grain size, with technological restrictions on the values of the operating parameters and the percentage of the main components of the steel composition. The technological scheme of the heat-treatment of rolled products is has two stages. It provides step-by-step water-cooling in four sections, located between the finishing group of stands and the screw-laying machine. To equalize the temperature field along the thickness of the rolled product, gaps (compensation sections) are provided between the sections. The cooling mode is set by turning on the required number of nozzles in the water cooling sections. After cooling from the temperatures of 1010-1070 to 750--900 \({}^{\circ}\)C in the first stage, the rolled product reaches the screw-laying machine, where its temperature \(T\) is measured in a non-contact way, and flat coils are formed. Then, they are less intensive than in the first stage, of air-cooling on a special conveyor. The conveyor has 16 cooling zones; the discharge fans are installed under zones 1-8 and 15, 16. The coils of rolled products are moved by a roller conveyor in a special tunnel equipped with insulating covers, which can be closed and opened to adjust the cooling rate. Thus, the factors that determine the technological mode of the heat-treatment of wire rod are  
Relative water consumption in water-cooling sections \(\gamma_{1}\), \(\gamma_{2}\), \(\gamma_{3}\), \(\gamma_{4}\) measured as a percentage of the maximum flow rate when the injectors are fully engaged;  
Speed of movement of turns on the conveyor \(\
u\);  
The relative amount of air \(\epsilon\) pumped by the fans and measured in fractions of the maximum possible amount;  
The degree of cooling \(\Delta\), equal to the proportion of closed thermal insulation covers. The main indicators of the quality of the process are the following characteristics of the metal of the finished wire rod:  
Tensile strength \(\sigma\);  
The relative narrowing \(\psi\);  
Relative elongation \(\delta\);  
Grain size \(d\);  
The relative thickness of the decarburized layer \(\theta\). In the first stage of solving the optimization problem, a regression mathematical model was constructed, based on the data of the industrial experiment, linking the output quality  
Figure 3: Distribution of the global criterion in the vicinity of the found optimum and the center of the plan. indicators with the factors that determine the technological mode. Since, in the case under consideration, the quality indicators largely depend on the initial chemical composition of the steel, the number of variable factors included the percentage of the main alloying components \(C\) and \(Mn\). The two-stage heat treatment process made it possible to simplify its mathematical description. The mathematical model of the water-cooling section was initially presented in the form \(T=f(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})\), where \(f\) is the regression polynomial. However, during the processing of the experimental data, it was found that the model which is simpler in structure--\(T=950.32-26.79\ \gamma\), where \(\gamma=\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}\)--provides more accurate results than the original model (the maximum error of this simplified model is \(8\%\); the error of the original model is \(30\%\)). The obtained result greatly simplified data-processing, allowing the entire set of factors that determine the water-cooling mode with the single variable \(T\) to be replaced.