# Multi-Objective Optimization of Production Objectives Based on Surrogate Model
## 2 Background of Multi-Objective Optimization Based on a Surrogate Model  
### Multi-Objective Optimization Problem (MOOP)  
In general, finding the solution of an MOOP can be presented as a procedure of simultaneous optimization of a set of \(k\) individual objective functions under the constraints set. It can be defined in form given in Equation (1) referring to [PERSON] and [PERSON] [7]:  
\[ \begin{array}{ll}\text{Minimizex}\in D&F(\mathbf{x})=[F_{1}(\mathbf{x}),\ F_{2}(\mathbf{x}),\ \dots,\ F_{k}(\mathbf{x})]^{T}\ ;\ k\geq 2\\ \text{subject to}&g_{s}(\mathbf{x})\leq 0,s=1,2,\dots,\ m;\ m\geq 0J_{t}( \mathbf{x})=0,l=1,2,\dots,\ r;\ r\geq 0.\end{array}  \tag{1}\]  
The vector \(\mathbf{x}\in D\), where the set \(D\) is the feasible region in the design space, is constrained by its lower and upper limits \(\mathbf{x}_{L}\leq\mathbf{x}\leq\mathbf{x}_{ll}\). The vector \(\mathbf{F}(\mathbf{x})\in E_{k}\) is the vector of objective functions \(F_{i}(\mathbf{x})\): \(E_{n}\to E_{1}\) in feasible objective space \(Z\), \(Z\subset E_{k}\). Thus, \(Z\) is the forward image of \(D\) under the mapping \(\mathbf{F}\)[8]. The value \(m\) defines the number of inequalities, and \(r\) defines the number of equalities for the constrained problem. If the objectives of \(F_{i}\) are contradictory, there does not exist only one single optimal solution as a MOOP-solving result. This process leads to the set of tradeoffs that are completely equivalent in a mathematical sense. Due to the obvious nonexistence of absolute ordering solutions in the space of objective functions [18], an expert in the role of decision-maker is important as a support for the determination of the most appropriate solution [7]. In general, multi-objective methods can be principally characterized with the respect to the involving decision-maker (DM) preferences, which play a significant role in the selection of an appropriate solution. The a priori defined preferences approach provides one single optimal solution dependent on the parameters specified by the DM. On the contrary, a posteriori defined preferences methods or interactive methods allow generating a set of many optimal solutions that are offered to the DM to select the most suitable one or influence the future direction of the optimization process, respectively [7]. One of the broadly accepted concepts of identifying the optimal solution of an MOOP is an idea of Pareto optimality based on an idea of dominance in the sense of establishing the partial order of solutions. [PERSON] [19] characterizes the Pareto optimum solutions set as a set of such feasible solutions in the objective (criterion) space, where no other feasible solutions for each of them exist that will yield an improvement in one objective without causing worsening in at least one other objective. All solutions in the objective space, which have the property of Pareto optimality mentioned above, are called nondominated solutions, and other solutions in objective space are dominated by them. For precise definitions of properties of dominance and Pareto optimality, we refer to [15]. All Pareto optimal solutions form a so-called Pareto front in an objective space [15; 16]. From the geometric point of view, the Pareto front represents a hypersurface in a \(k\)-dimensional space of objective functions where all nondominated solutions are located. In case of a high-dimensional design space, the hypersurface can have a very complicated structure [8]. Referring to [PERSON] [20], the pre-images of the nondominated outcomes are called efficient solutions located in design space. The concept of Pareto optimality implemented in a method called Pareto ranking is often combined in some modifications with evolutionary algorithms [15] to increase the effectiveness of the computation of the entire set of Pareto optimal solutions. An alternative for defining a partial order in objective space can be found in decomposition methods [8].