# Multiple-Criteria Fuzzy Optimization of the Heat Treatment Processes for Two Steel Rolled Products
## 2 Informal Presentation of the Problem
Note that industrial experiments are usually very expensive, but they are easily paid off after the setting of optimal technological modes. Nevertheless, even the \"precise\" mathematical model obtained after identification, based on a complex system of differential equations in finite differences, is not very convenient for the frequent solution of optimization problems due to the too-high machine-time expenses needed for calculations by the model. To reduce expenses for optimization, the initial \"precise\" model is considered to be a real process, and a certain number of numerical experiments (calculations made for predefined different values of input parameters) are provided according to the defined plan of the numerical experiment. Based on the obtained results and using statistical methods, the simple polynomial regression model representing the dependence of the output parameters on the input ones is finally inferred. This model is simple enough to be used for optimization;
2. Establishment of the membership functions of local criteria based on actual technological instructions, as well as on the experience and intuition of technologists;
3. Calculation of the local criteria's relative importance (weights) based on the linguistic pairwise comparisons matrix using the method developed in [29];
4. Choice of an appropriate method for the local criteria aggregation or a set of relevant methods to build a compromise general criterion;
5. Solution of the optimization task based on the maximization of a general criterion. This methodology can be applied to the optimization of a wide range of technologies, not only to those based on heat transfer processes. The Simulation, Parametric Identification and Fuzzy Multiple-Criteria Optimization of the Heating Steel Ingots for the Rolling Process  
As the first example of the practical implementation of the developed general methodology for the modeling, identification and optimization, we will consider the multiple-criteria fuzzy problem of optimizing the quality of the ingot-heating process before rolling at rolling mill 850 of the Belorussian Mini Metallurgical Plant. Ingots are heated in methodical gas furnaces. The methodical furnace with walking beams of mill 850 is designed for continuously heating cast blooms with a cross-section of 250 \(\times\) 300 and 300 \(\times\) 400 mm with a length of from 2.5 to 5.5 m with a layout step of 150 and 200 mm, respectively. The total length of the furnace is 23 m, according to the location of the burners, and it is divided into seven sections, which can be divided into the methodical, and the so-called welding and homogenizing zones. Due to the widespread use of automated systems for controlling the processes of heating workpieces, it is necessary to obtain a large amount of data on changes in the thermal and thermal deformation states of workpieces in the furnace. A pure experimental approach to developing economical modes of metal-heating in operating furnaces requires a large number of experiments, which are associated with significant labor, time and monetary costs. As a result, the methods for mathematical modeling and the optimization of metal heating processes in pass-through furnaces are of crucial importance. The mathematical model for calculating the quality parameters of heating ingots before rolling includes:  
* Two-dimensional nonlinear equation of the non-stationary heat transfer in partial derivatives with boundary conditions of the third and fourth kind;
* equation for calculating the thickness of the formed scale;
* The system of equations for calculating thermal stresses. The model was implemented numerically using a combination of finite-difference and finite-element methods.