# Multi-Objective Optimization of Production Objectives Based on Surrogate Model
## 2 Background of Multi-Objective Optimization Based on a Surrogate Model
### The Particular MOOP Definition and Applied Scalar MOO Methods
\begin{array}{l}U=0.25\;\text{*}\;(11.47091+6.53901\;\text{*}\;\text{LotSizeP1 + 7.91294\;\text{*}\;\text{LotSizeP2 - 1.42610\;\text{*}\;\text{IATP1}}}\\ -1.40960\;\text{*}\;\text{IATP2}-0.150095\;\text{*}\;\text{LotSizeP1 *}\;\text{IATP1}-0.176751\;\text{*}\;\text{LotSizeP2 *}\;\text{IATP2 +}\\ 0.030254\;\text{*}\;\text{IATP1}^{2}+0.031313\;\text{*}\;\text{IATP2}^{2}-10.2 09)/(16-10.209)+0.25\;\text{*}\;(91.78175\;\text{+}\\ 0.626283\;\text{*}\;\text{LotSizeP1 + 0.301304\;\text{*}\;\text{LotSizeP2 - 0.923591\;\text{*}\;\text{IATP1}}-0.854483\;\text{*}\;\text{IATP2 +}}\\ 0.048977\;\text{*}\;\text{LotSizeP1 *}\;\text{IATP1 + 0.049608\;\text{*}\;\text{LotSizeP2 *}\;\text{IATP2 - 84.035)}}/{(60-84.035)}\\ 0.25\;\text{*}\;(250.66819+15.89178\;\text{*}\;\text{LotSizeP1 + 14.81886\;\text{*}\;\text{LotSizeP2 - 2.28272\;\text{*}\;\text{IATP1}}-}\\ 2.24722\;\text{*}\;\text{IATP2 + 0.238889\;\text{*}\;\text{LotSizeP1 *}\;\text{IATP1}+0.237500\;\text{*}\;\text{LotSizeP2 *}\;\text{IATP2 -}}\\ 1.47246\;\text{*}\;\text{LotSizeP1}^{2}-1.34746\;\text{*}\;\text{LotSizeP2}^{2}-289 )/(230-289)+0.25\;\text{*}\;(585.01423\;\text{+}\\ 26.38960\;\text{*}\;\text{LotSizeP1 + 24.42893\;\text{*}\;\text{LotSizeP2 - 2.45189\;\text{*}\;\text{IATP1}}-11.04025\;\text{*}\;\text{IATP2 +}}\\ 0.119557\;\text{*}\;\text{IATP1}^{2}+0.097507\;\text{*}\;\text{IATP2}^{2}-296.309 )/(550-296.309).\end{array}  \tag{6}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Production Objective** & **Utopia Point \(F_{i}^{U}\)** & **Nadir Point \(F_{i}^{N}\)** \\ \hline Avg. flow time (min.) & 296.309 & 550 \\ Avg. costs per part unit (€) & 10.209 & 16 \\ Avg. machine utilization (\%) & 84.035 & 60 \\ Total number of products (pcs) & 289 & 230 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Utopia and nadir point values in the transformation of objectives \(F_{i}\). Analogically, the minimized objective function designed on the base WPM with regression models that resulted from the FCD was expressed in the following form:  
\[\begin{array}{l}U=(\text{ABS}(((11.47091+6.53901\ ^{*}\ \text{LotSizeP1}+7.91294\ ^{*}\ \text{ LotSizeP2}-1.42610\ ^{*}\ \text{IATP1}-1.40960\ ^{*}\ \text{IATP2}-0.150095\ ^{*}\ \text{ LotSizeP1}\ ^{*}\ \text{IATP1}-0.176751\ ^{*}\ \text{ LotSizeP2}\ ^{*}\ \text{IATP2}+0.030254\ ^{*}\ \text{IATP1}^{2}+0.031313\ ^{*}\ \text{IATP2}^{2}-10.209)/(16-10.209)\ ^{*}\ ((91.78175+0.626283\)  
\[\ {}^{*}\ \text{LotSizeP1}+0.301304\ ^{*}\ \text{LotSizeP2}-0.923591\ ^{*}\ \text{IATP1}-0.854483\ ^{*}\ \text{IATP2}+0.048977\]  
\[\ {}^{*}\ \text{LotSizeP1}\ ^{*}\ \text{IATP1}+0.049608\ ^{*}\ \text{ LotSizeP2}\ ^{*}\ \text{IATP2}-84.035)/(60-84.035)\ ^{*}\ \text{((250.66819+15.89178\ ^{*}\ \text{ LotSizeP1}+14.81886\ ^{*}\ \text{ LotSizeP2}-2.28272\ ^{*}\ \text{IATP1}-2.24722\)  
\[\ {}^{*}\ \text{LATP2}+0.238889\ ^{*}\ \text{LotSizeP1}\ ^{*}\ \text{IATP1}+0.237500\ ^{*}\ \text{ LotSizeP2}\ ^{*}\ \text{IATP2}-1.47246\]  
\[\ {}^{*}\ \text{LotSizeP1}^{2}-1.34746\ ^{*}\ \text{LotSizeP2}^{2}-289)/(230-289) ^{*}\ ((585.01423+26.38960\)  
\[\ {}^{*}\ \text{LotSizeP1}+24.42893\ ^{*}\ \text{LotSizeP1}-12.45189\ ^{*}\ \text{IATP1}-11.04025\ ^{*}\ \text{IATP2}+0.119557\ ^{*}\]  
Numerical optimization via the \(D\) function demands a range of factors, responses, and optimization goals in the settings. The applied settings of these parameters are listed in Table 4. The limits of functions in the form of ramps for individual responses \(d_{i}\) are depicted in Figure 4. It shows the graphs of \(d_{i}\) functions in Equations (8) and (9) for specified limit values on individual intervals. Two of the notches on each ramp represent the minimum and maximum values of all response values within the experimental space, and two others are lower and upper limits for the given response. They correspond to \(L\) and \(U\) values in Equations (8) and (9), respectively (the lower and upper limit in Table 4). At the same time, they correspond to the utopia and nadir points as reference points in other types of applied scalar multi-objective functions.