# Analysis and Visualization of Production Bottlenecks as Part of a Digital Twin in Industrial IoT
## 4 Process Heatmap  
The Active Period Method identifies potential bottlenecks and offers a corresponding visualization for each machine. In this paper, we introduce a process heatmap as additional analysis and visualization technique, displaying the overall production process topology and the stability of the state of each machine, i.e., how likely the machine might turn into a bottleneck when adapting the overall process topology. Thus, the process heatmap will enable one to judge the health of the production system at once. The basis for generating a process heatmap is the Active Period Method, because its metric shows a strong tendency concerning which machine is closest to becoming a bottleneck in the manufacturing line. In addition there are no boundaries in relation to physical layouts, so a lot of machine pools or manufacturing layouts can be regarded in different observation periods. For further investigation, a tangible manufacturing layout given from a planned production line for a certain product is simulated, to demonstrate and validate the process heatmap as intelligent analysis and visualization approach. To simulate the dynamics of the overall production system, a simple model of storage and resistor components is  
Figure 7: An example for the result of the Active Period Method (**top**) visualized with status time series (**bottom**). introduced and all kinds of disturbances are modelled by a Gauss distribution. The cycle times of the machines are adopted from the planned production line. In a first step, the simulation loops over all machines and varies the cycle time from 95 to 120 percent of the calculated mean cycle time for each machine separately as shown in Figure 8. Thus, the cycle times of all other machines are not changed. The density function of the machine cycle times--if a Gauss distribution is used it would be the standard deviation--can be estimated empirically from smart manufacturing data. The cycle time for each machine is varied in 60 steps and for each step the bottleneck scores are calculated with the Active Period Method. The results are then plotted in a diagram as shown in Figure 8. For visualization purposes, a sigmoid function in the form of Equation (1) is plotted additionally, to show the correlation of this function with the calculated data. \[f(x)=\frac{L}{1+e^{-k(x-x_{0})}} \tag{1}\]  
Thereby \(L\) denotes the limiting value, \(k\) is the gradient of the sigmoid and \(x_{0}\) defines the turning point. The variable \(x\) describes the average cycle time of the machine in seconds. For a better visualization of these data, the calculated points of the scatter plot are summarised into a color bar. The color bar groups together percentage ranges of bottleneck likelihoods (Table 2), so that it is clearly visible how far a machine is from being a bottleneck. The color coding in Figure 8 is the same as described in Table 2. In this figure, it can been seen  
Figure 8: Variation of the cycle times from 95% to 120% for each machine. (**a**) Variation of machine M1. (**b**) Variation of machine M2. (**c**) Variation of machine M3. (**d**) Variation of machine M4. (**e**) Variation of machine M5. (**f**) Variation of machine M6. (**g**) Variation of machine M7. (**h**) Variation of machine M8. (**i**) Variation of machine M9. that, if a machine has wide blue and green areas, it means that decreasing the machine's throughput, e.g., by slowing down the machine, will not increase its bottleneck score significantly. Machine M4, for example, has a Bottleneck Score of nearly zero throughout the range of 95% to 120%, whereas machine M1 quickly changes at 101%. This displays that, if a machine has only small blue and green areas, the bottleneck score will increase immediately if the machine's throughput decreases.