# Scheduling Algorithms for a Hybrid Flow Shop under Uncertainty
## 4 Computation of Schedules
These considerations are also finally included in Figure 6.

The same procedure of reserving slots for required higher amounts in the schedules can also be applied to other non high-risk job demands, e.g., if products are sensitive about producing less than the required demand. So, for these jobs, this method for overproduction compared to \(demand_{ij}\) can be applied. The demands can be multiplied with another safety factor, namely \(SF1\). This bears the possibility to include the knowledge of experienced production planners, e.g., about sensitive demands. A production planner may decide which value to use for the safety factor. At this point, it is important to note that an earlier completion, which can be the cause of slots being reserved for too long in the schedule, is easy to handle with our algorithms presented in Section 2. Such a schedule causes, at most, only a slightly extended storage cost. In contrast, an unexpected longer slot for jobs would result in drastic delays in the schedule and elaborate rescheduling. With this method, changes in the demand and production quantities can be considered in optimization and simulation models for scheduling. To choose the appropriate scrap rate per job and stage for our application case, different quantile values of the available empirical probability distribution per job and stage are considered. Figure 6 shows a histogram for the scrap rate of an exemplary job. For every job, we use the same quantile value. Afterwards, per job, the scrap rates \(r_{ij}\) are computed as a function of the quantile value and the probability distribution. In practice, it is not realistic that the production volume \(productionvolume_{ij}\) is completely produced without failure. So, for every job, we have to increase the production volume to finally get the demand that is to be produced. All analyzed factors influencing the demand in the production process in our application case result in the following formula to calculate the appropriate production quantity \(productionvolume_{ij}\):

\[productionvolume_{ij}=\frac{demand_{ij}-stock_{ij}}{(1-r_{ij})}\;\;.\;\;SF1_{ij }\;\;.\;\;SF2_{ij} \tag{1}\]

Figure 5: Clusters for identification of high-risk jobs (calculated with the method of [30]). ## 7 Scheduling under Uncertainty

Figure 7 shows the simulation architecture for our application case, which is built in Anylogic 8.5. In contrast to the deterministic optimization model, the discrete event simulation model uses random values, as mentioned in Table 1, for every produced part. Beyond that and the aspects that have already been considered in the optimization model, the following specific characteristics of production are included in the simulation model:

* In the first stage, there are two cranes and one crane needs to be used for every setup process of a machine. The first crane can only reach a defined group of machines in the first stage and the second crane can only reach the other machines. Therefore, the machines in the first stage in the model are divided into two different groups and only two machines can be set up at the same time if they do not belong to the same group, i.e., if they do not need the same crane for the setup process. * Produced products, which are transported by conveyors to a quality check station in the first stage, can cumber each other. This can cause time losses in the production process. * If there are not enough parts available from the first stage for producing the amount of the second stage, the production volume of the second stage has to be reduced. * To transport the products, which were produced at the first stage, to the second stage, they are carried on small load carriers, where a certain number of produced parts fit in one small load carrier and a certain number of small load carriers fits on one trolley, which transports the products to the second stage. Both the packing and the logistics process flow between the two stages are considered. * Detailed modeling of the connection of logistics and production between the first and second stages.