# Multiple-Criteria Fuzzy Optimization of the Heat Treatment Processes for Two Steel Rolled Products
## 2 Informal Presentation of the Problem
The thermal part of the model contains indeterminate parameters: the coefficient of convective heat exchange of the ingot with the heating medium, which, according to the literature data, was assumed to be 30 w/(m\({}^{2}\)\({}^{\circ}\)C), and the reduced radiation coefficient of the ingot surface-\(\alpha\), identified by the production experiment. As a result of the identification, the values of \(\alpha\) were obtained, which vary from zone to zone in the ratesof 1.7-1.9 w/(m\({}^{2}\) \({}^{\circ}\)C\({}^{4}\)) and provide high accuracy in calculating the temperatures in the heated ingot. The equation for calculating the thickness of the formed scale \(\delta\) was taken as  
\[\frac{d\delta}{d\tau}=\frac{\left(exp(-10,125/T(\tau)+7.25)\right)^{2}}{2\delta}, \tag{1}\]  
where \(T(\tau)\) is the surface temperature of the ingot. To assess the risk of cracks in the ingot, along with the stress field, the stress state coefficients \(K_{\sigma}=(\sigma_{i}-\sigma_{T}(T))/\sigma_{T}(T)\) were calculated, where \(\sigma_{i}\) is the stress intensity, and \(\sigma_{T}\) is the destructive stresses. The \(K_{\sigma}\) isolines calculated for the forced mode showed that the tensile stresses that occur in the ingot during heating do not reach the critical values that lead to the appearance of cracks. From the perspective of setting and solving the optimization problem, the obtained result is very important, because it allows us to exclude the stress-state coefficient of the ingot from consideration as a criterion for the quality of the process. As a result, the quality of the process was evaluated according to three main local criteria: minimizing the scale thickness \(\delta\), the accuracy of heating \(T\), and minimizing the maximum temperature difference in the ingot \(\Delta\)\(T\) at the time of unloading. The membership functions of the process local quality criteria are presented in Figure 2.  
Consider the membership function of the heating accuracy criterion (Figure 2b). It can be seen that the optimal temperatures of the wide face of the ingot are in the region of \(1190\)\({}^{\circ}\)C \(\leq T\leq 1200\)\({}^{\circ}\)C. The temperatures \(T\leq 1150\)\({}^{\circ}\)C or \(T\geq 1250\)\({}^{\circ}\)C are unacceptable for technological reasons. Intermediate sections are characterized by a decrease in the membership function when \(T\) approaches the region of invalid values. The linear law of decrease is adopted here only for reasons of simplicity. If there is any additional information about the behavior of the membership function in the intermediate regions, the regularity of decrease can be changed. The membership functions of the criteria for the thickness of scale and the temperature difference over the thickness of the ingot during unloading are constructed on the basis of information obtained in numerical experiments, with variables within the range of acceptable values. The solution to the fuzzy multiple-criteria optimization problem was reduced to finding the optimal thermal operating modes of the furnace, which ensure the maximization of the global quality criterion \(D\) for various furnace capacities \(P\) and the temperatures \(T_{0}\) of the metal loading. The capacity \(P\) was used in the model to calculate the residence time of the ingot in each of the zones. At the same time, in all cases, the furnace temperature was assumed to be \(T=1200\)\({}^{\circ}\)C in the homogenization zone, and at the beginning of the unheated zone \(700\)\({}^{\circ}\)C.  
Figure 2: Membership functions (**a**–**c**) of the local criteria used. As a result, for each fixed pair of \(P\) and \(T_{0}\), the task was reduced to finding the optimal temperatures of the first \(T_{1}\) and second \(T_{2}\) welding zones. The temperature change within each of the zones was assumed to be linear. The direct use of the mathematical model which is the system of second-order partial differential equations to solve the optimization problem requires too much machine time. Therefore, a two-step approach was used.