# Scheduling Algorithms for a Hybrid Flow Shop under Uncertainty
## 4 Computation of Schedules
Accordingly, for every figure, 280 key indicators are extracted from the simulation per data point. Only for Figure 17 are the SPT results excluded, mainly in order to highlight the possible achievable \(C_{max}\) values. Figure 3:9 Average percentage of unfulfilled orders per quantile over all simulation runs. Figure 5.0 Average makespan of simulation runs over the six metaheuristics per quantile. Figure 16 shows the ratio of orders that produce at least one less than the weekly demand. So, jobs are identified for that the weekly demand did not fulfil. This figure also validates our method to use scrap quantiles to avoid as many unfulfilled orders as possible. This demonstrates that the probability of unfulfilled orders decreases with increasing values for the different quantile values in the expected rates. The effects on the makespan show that the quantile value of 91% is preferable and

Figure 5.1 Average percentage of overproduction in orders producing too many items due to quantile method. Figure 5.2 Average percentage of underproduction in orders producing too few items due to scrap. can already be found in the optimization results of Figures 8-13 and are also validated by simulation with Figure 5.3

Figures 16 and 19, when analyzed together, show that it is not realistic to reduce the probability of unfulfilled orders below 10 % without an extreme overproduction of orders. Furthermore, Figure 19 also confirms what could be expected from Figures 8-13, and Figure 5.4 the makespan increases, starting with the 92% quantile, because of significantly more overproduced parts in orders with overproduction. In contrast, the results in Figure 18 do not indicate a clear correlation. For quantiles 60-91%, the rate of produced jobs decreases by increasing the quantile value, but quantiles 92% and 93% do not validate this assumption. Considering only this figure, quantiles 92% and 93% provide more preferable results. Although Figure 18 indicates that another quantile value should be chosen instead of the 91% quantile, the analyses of Figures 16, 17 and 19 confirm that the 91% quantile should be preferred for production planning in the application case. Because the most important indicators in our application case are the unfulfilled orders, in the following, we use the 91% quantile for our analysis. To summarize the results under almost real conditions and to indicate which method returns the schedules with the best makespan and which schedule is most robust against uncertainties, e.g., in processing times or setup times, Figure 20 shows the makespans for week 12 of different schedules resulting from the seven algorithms analyzed by simulation. The box plots also indicate how the makespan varies in simulation runs of the same schedule. The borders of the boxes describe the first and third quartiles. Upper and lower limits show the minimum and maximum makespan of the simulation runs. Outliers are not included in Figure 5.5 For each schedule, 40 parallel replications are performed, requiring about 9 min. A good schedule should result in a small value for makespan and a small box, indicating robustness. The most robust solution in this week is provided by random descent (swap), but the average makespan is higher than the one provided by Steepest Descent (swap). Since the schedule resulting from Steepest Descent (swap) shows only small differences between the results of the different simulation runs and has a small average makespan for the production program of week 12, it should be selected for this week. Figure 5.6 Comparison of simulation results per optimization method for calendar week 12 with quantile 91%. ## 9 Conclusions

In this paper, we solve a hybrid flow shop scheduling problem under uncertainties derived from a real industrial use case. It is shown that the combination of data analysis to estimate model parameters, heuristic and metaheuristic optimization and detailed stochastic simulation results in robust schedules that can be used in real systems.