## 3 Cascaded Weighted Sum
### Example of the CWS  
Table 1 gives an example of the usage of weights and thresholds for a problem of scheduling jobs organized in workflows of elementary operations to heterogeneous equipment comparable to the one described in [6]. All operations can be assigned to alternatively usable equipment at different costs and processing times. The task is to produce schedules, where the job processing is as cheap and as fast as possible and each job observes a given budget and a given due date. The rate of utilization of the equipment should be as high and the total makespan of all jobs as low as possible. Additionally, the schedules must be updated frequently, because e.g., new jobs arrive or waiting jobs are cancelled. As described in Section 2.2, all objectives are normalized according to Equation (3). The required limits are obtained as follows: The bounds of job time and costs are calculated by determination of the critical path of the workflow of that job and by the assignment of the fastest/slowest or costliest/cheapest equipment suited for the operations of a job. The user-given due dates and cost limits are checked against these bounds so that the subsequent scheduling is based on goals which are achievable in principle. The lower bound of the makespan is the duration of the longest lasting job using the fastest equipment and the upper bound is the sum of the duration of all jobs using the slowest equipment divided by the smallest number of alternatively usable equipment. As the rate of utilization already yields a value to be maximized between zero and one, there is no need for bounds. Job time and costs are most conflicting, while short processing times support a short makespan and tend to increase the utilization rate. Faster equipment typically is more expensive than slower, and the ratio between costs and duration of the use of equipment will play an important role. Thus, lower costs require the usage of equipment with a lower ratio of costs and duration. This tends to increase the duration and to decrease the workload of less cost-effective equipment. Additionally, shorter job times are also rewarded to some extent by the makespan and possibly by the utilization rate, but costs are not. These considerations are supported by the observation that the processing times are easier to reduce than costs. Thus, job time and costs should compete from the beginning and both should receive a larger portion of the weights. Therefore, they are grouped together at the highest priority so that they always contribute to the weighted sum, as shown in Table 1. As a rule of thumb, further objectives, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Priority** & **Objective** & **Weight [\%]** & **Threshold \(\varepsilon_{i}\)** \\ \hline
1 & job time & 30 & 0.4 \\
1 & job costs & 40 & 0.25 \\
2 & makespan & 20 & - \\
2 & utilization rate & 10 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example of the use of the cascaded weighted sum (CWS) and the effect of objective group weights. The objectives with the highest priority are always active and contribute to the weighted sum. They are marked here by a light green background. which are less conflicting with each other and those of higher priorities, can go into the same group. After having determined the priority and grouping structure based on experience and considerations about the relationships between the objectives, appropriate weights and thresholds must be chosen. Based on these considerations and a representative scheduling task, a schedule can be produced based on Pareto optimality for the identification of the region of interest, from which thresholds and weights can be derived. In the given example, this can be done in a first step for a reduced set of objectives by omitting a less conflicting one, e.g., the utilization rate. The Pareto fronts for good makespans of the two remaining objectives can be plotted.