# 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
As a result of processing data obtained from industrial experiments on one of the steel grades, multiple-linear-regression equations were obtained, which link the input and output variables  
\[ \begin{array}{l}\sigma=1251.74+0.14\cdot T+1336.35\cdot C+195.385\cdot Mn+1.988 \cdot\epsilon-1426.94\cdot\
u-1203.94\cdot\Delta;\\ \delta=0.169-0.00317\cdot T-23.55\cdot C+8.77\cdot Mn+0.0877\cdot\epsilon+29.8 3\cdot\
u+23.97\cdot\Delta;\\ \psi=46.566+3.75\cdot 10^{-3}\cdot T-4.613\cdot C+26.34\cdot Mn+0.0814\cdot\epsilon-16.32 7\cdot\
u-20.51\cdot\Delta;\\ d=0.349-3.749\cdot 10^{-4}\cdot T-0.1156\cdot C+7.429\cdot 10^{-2}\cdot Mn-2.8 \cdot 10^{-4}\cdot\epsilon+0.189\cdot\
u+0.166\cdot\Delta;\\ \theta=1.9-3.383\cdot 10^{-3}\cdot T-1.423\cdot C-0.357\cdot Mn-4.47\cdot 10^{-2} \cdot\epsilon+4.496\cdot\
u+3.487\cdot\Delta.\end{array}  \tag{10}\]  
The above equations are adequate according to the Fisher criterion; the error in the forecast of the main quality indicators \(\sigma\), \(\psi\) and \(\delta\) did not exceed \(10\ldots 15\%\), which indicates that the obtained model is sufficiently accurate. It should be noted that the attempts made to increase the adequacy of the model by constructing regression dependencies of the second- and higher-orders did not lead to a significant reduction in the error. The membership functions were used to formalize local criteria and constraints. In Figure 4, the membership functions of the most important quality indicators are presented. Similarly, the membership functions of the other indicators of the quality \(\mu_{\psi}(\psi)\), \(\mu_{d}(d)\), \(\mu_{\delta}(\delta)\) were built, and the restrictions on the content of \(C\) and \(Mn\): \(\mu_{C}(C)\), \(\mu_{Mn}(Mn)\). The usual constraints of the inequality type were imposed on the variables \(T\), \(\
u\), \(\epsilon\), \(\Delta\). Since the formulated local criteria and constraints are satisfied in practice in various areas of varying process parameters, the optimal solution was found as a compromise between conflicting requirements. To do this, the local criteria and constraints were aggregated into the generalized process-quality criterion, taking the relative importance (weights) of local criteria into account. The generalized process-quality criterion, as seen in the previous subsection, was built using the min operator as follows:  
\[ \begin{array}{l}D(C,Mn,T,\
u,\epsilon,\Delta)=\min(\mu_{\sigma}^{a_{1}}( \sigma_{c}(T,Mn,C,\
u,\epsilon,\Delta)),\mu_{\delta}^{a_{2}}(\delta(T,Mn,C, \
u,\epsilon,\Delta)),\\ \mu_{\psi}^{a_{3}}(\psi(T,Mn,C,\
u,\epsilon,\Delta)),\mu_{\delta}^{a_{4}}( \theta(T,Mn,C,\
u,\epsilon,\Delta)),\mu_{C}^{a_{6}}(d(T,Mn,C,\
u,\epsilon, \Delta)),\mu_{C}^{a_{6}}(C),\mu_{Mn}^{a_{7}}(Mn))\end{array}  \tag{11}\]  
Figure 4: Membership functions of the most important quality indicators. where \(\alpha_{1}\ldots\alpha_{7}\) are the weights of the criteria and constraints. They were determined using the linguistic pairwise comparison matrix, as in the previous subsection. The desired point of the optimum was obtained as follows  
\[(C,Mn,T,\
u,\epsilon,\Delta)_{opt}=\underset{C,Mn,T,\
u,\epsilon,\Delta}{\arg \max}\;(D(C,Mn,T,\
u,\epsilon,\Delta)) \tag{12}\]  
To find the maximum, we used the known direct random-search method. The preliminary results of solving the optimization problem showed the presence of a number of competing local optima, and each of them was achieved at values \(\epsilon\) at the upper limit of the range of acceptable values. This allowed us to reduce the dimension of the factor space by putting in (11) and (12) \(\epsilon=\