# 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  
The particular MOOP represents a task to design a set of four input parameters: the lot size and the input intervals for arriving supplies P1 and P2 within the specified ranges and production limitations while optimizing the production performance. Optimization involves minimizing the average costs per part unit and the average flow time and maximizing the total number of products and average machine utilization simultaneously. To solve that, we applied two methods belonging to the class of scalarization methods with a priori defined preferences, referring to [PERSON] and [PERSON] (2017). Firstly, we constructed the scalar multi-objective \(U\) function in Equation (3) expressed through the Weighted Sum Method (WSM) as the linear combination of the normalized individual objectives \(F_{i}^{transform}\) using the weight vector \(\mathbf{w}\). All components \(w_{i}\) have the same value 0.25 to not prefer any production goal. \[U=\sum_{i=1}^{k}w_{i}F_{i}^{transform},w_{i}>0\ \forall i,\ \sum_{i=1}^{k}w_{i}=1 \tag{3}\]  
Secondly, we applied the Weighted Product Method (WPM) with the same settings of weights defined by Equation (4). In both cases, minimizing the scalar \(U\) function leads to the single Pareto optimal solution finding. \[U=\prod_{i=1}^{k}\left(F_{i}^{transform}\right)^{w_{i}},w_{i}>0\ \forall i,\ \sum_{i=1}^{k}w_{i}=1 \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Factor** & **Response** \\ \hline A—LotSizeP1 & \(F_{1}\)—Average flow time \\ B—LotSizeP2 & \(F_{2}\)—Average costs per part unit \\ C—IATP1 & \(F_{3}\)—Average machine utilization \\ D—IATP2 & \(F_{4}\)—Total number of products \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Factors and responses for design layouts. There exist many possible transformations described e.g., by [PERSON] and [PERSON] [37] to ensure dimensionless objectives \(F_{i}\) entering the scalar multi-objective function. We employed a robust transformation given by Equation (5). \[F_{i}^{transform}=\frac{F_{i}-F_{i}^{U}}{F_{i}^{N}-F_{i}^{U}} \tag{5}\]  
That relates to a component of a utopia point vector and a nadir point vector, which represent the unreachable solutions for individual production goals as the best one (utopia) and as the worst one (nadir). The utopia point vector \(F^{U}\) can be taken as the optimum of the single-criterion function according to the optimization goal regardless of other objectives. \(F^{N}\) is the vector created by the worst values of the production goals. It can be determined by expert preferences or obtained from single-optimization experiments, too. Table 2 shows the applied values of these reference points. Value \(F_{i}\) in Equation (5) is a component of the production goals (individual objectives) vector that resulted from experiments on a discrete-event driven simulation model. For a numerical computation of the MOOP solution, we use the Evolutionary algorithm in MS Excel module Solver (by Microsoft, Redmond, WA, USA) to perform discrete global optimization. The objective function employing WSM defined by Equation (3) and transformation in Equation (5) with the utopia and nadir points given by Table 2 was expressed via regression models of production goals based on FCD in the following analytical form, and it was minimized:  
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