## 2 Short Introduction to Pareto Optimization and Two Aggregation Methods
### Weighted Sum  
One of the probably most often used assessment methods besides Pareto optimality is the weighted sum, which aggregates the objective values to a single quality measure. As the objective functions frequently have different scales, they are usually normalized. This can be done for example by using Equations (3) or (4) when minimizing and maximizing the objectives, respectively:  
\[f_{i}^{nom}=\frac{\max\bigl{(}f_{i}\bigr{)}-f_{i}}{\max\bigl{(}f_{i}\bigr{)}- \min\bigl{(}f_{i}\bigr{)}}\text{ for objectives to be minimized} \tag{3}\]  
\[f_{i}^{nom}=1-\frac{\max\bigl{(}f_{i}\bigr{)}-f_{i}}{\max\bigl{(}f_{i}\bigr{)} -\min\bigl{(}f_{i}\bigr{)}}\text{ for objectives to be maximized} \tag{4}\]  
The bounds of the objective function \(f\!i\) can be estimated or are the result of a maximization of each function individually in case of \(\max(f\!i)\). For the calculation of the weighted sum as shown in Equation (5), a weight \(w_{i}\) has to be chosen for every objective:  
\[\text{maximize}\sum_{i=1}^{k}w_{i}f_{i}^{nom}\bigl{(}x\bigr{)},\ \ x\in S\ \ \text{ where }\ w_{i}>0\ \text{ for all }i=1,\,\ldots,\,k\text{ and }\sum_{i=1}^{k}w_{i}=1 \tag{5}\]  
By varying the weights, any point of a convex Pareto front can be obtained. Figure 2 illustrates this: The straight line corresponding to the chosen weights \(w_{1}\) and \(w_{2}\) is moved towards the border of the feasible objective region during the optimization process and becomes a tangent in point P. The solutions found are Pareto optimal, see [5].  
On the other hand, it is possible that parts of the Pareto front cannot be found in case of a non-convex problem. This is illustrated in Figure 3: the part between points A and B of the Pareto front cannot be obtained for any weights. This is a serious drawback.  
As mentioned above, the weighted sum is often used for practical applications. Reasons are the simplicity of its application and the easy way to integrate restrictions, which are beyond pure limitations of the feasible region. Examples are scheduling tasks, where the jobs to be scheduled have due dates for finalization. Thus, delays can occur and it is not sufficient to tell the search procedure that this constrained violation is an infeasible solution by e.g., rejecting it. Instead, the search must be guided out of the infeasible region by rewarding a reduction of the violation. In the example given, this can be done by counting the number of jobs involved and summing up the amounts of delays, see e.g., [6]. These two key figures can either become new objectives or can be treated as penalty functions. As they do not represent wanted properties and as a low number of objectives is preferable, penalty functions are the method of choice. They can be designed to yield values between zero (maximal violation) and one (no violation). The results of all penalty functions serve as factors by which the weighted sum is multiplied. As a result, the pure weighted sum turns into a raw quality measure, which represents the solution quality of the problem without constraints, while the final  
Figure 3: For non-convex Pareto fronts, it is possible that parts of the front can not be obtained by the weighted sum. The region between points **A** and **B** is an example of this serious draw back of this aggregation method.  
Figure 2: By using appropriate weights, every point of a convex Pareto front can be achieved by the weighted sum. Here, point **P** can be obtained for the weights \(w_{1}\) and \(w_{2}\). The arrows show the movement direction of points where the largest quality gain is obtained.  
product represents the solution for the task with its constraints. Figure 4 shows a typical example of such a penalty function. If the maximum value of constraint violation is hard to predict, as it is often the case, an exponential function can be chosen. A value of delay \(dp\) for poor solutions can usually be estimated roughly and the exponential function is attributed such that it yields a value of \(\
icefrac{{1}}{{\lambda}}\) in this case.