## 4 Case Study
### Production Scheduling  
The production scheduling for the next day is established by the Decision-Maker to flatten the energy consumption profile. The combinatorial optimization is used to achieve this objective considering the constraints presented in Section 3.5: the appropriate allocation of the equipment or installations to each activity associated with a technological flow, the working time of each equipment or installation, the phase sequences, and energy consumption balance.  
The input data used in the optimization process are the following: the decision on the operating mode (on or off, quantified by 1 or 0), working time (reduced, normal, or long), see Table 8, working cycles (1, 2, or 3), the dependency between equipment, the start and stop working hours, and technical characteristics, see Table 3, which characterize every equipment and installation associated to each activity.  
Table 11 presents the input data for the equipment and installations from Tuesday (presented in Section 4.4). The activities programmed in this day required mechanical revisions for three cars, dyeing and drying operations for various components of two cars, mechanical repairs for two cars requiring welding and sheet metal straightening operations, and dyeing preparation for three cars. The optimal solution, presented in Table 12 and Figure 16, has been accepted by the Decision-Maker to ensure continuity of the technological flow.  
Table 10 presents the dependence between dyeing operation, drying operation, and the use of the lighting system associated with the dyeing installation. The order inside of the dyeing installation must be the following: use of lighting system to prepare the car inside, dyeing operation, and drying operation. Usually, for the first use of the dyeing installation, preparing the car is done at the end of the working schedule from the previous day. The dependency expresses the association with one equipment or installation identified through the number of the row in the matrix IDM.  
Due to the high energy consumption of dyeing installation and the smaller energy consumptions of other equipment and installations, the flattening of the profile is not very obvious at first glance. However, the analysis of the results presented in Figures 16 and 17 indicates a satisfactory decrease, with 2.95 kWh (16.23%) of the peak value (highlighted with red colour in Figure 17) from 18.17 kWh (recorded at the hour 14) to 15.22 kWh (recorded at the hour 15), after application of the optimization process.  
In addition, the standard deviation of the energy consumption and the load factor can quantify the flattening of the energy consumption profile. Regarding the standard deviation, a decrease was recorded from 7.13 to 5.70 kWh. The load factor expresses a measure of the utilization rate, or efficiency of energy usage, being an indicator that quantifies how steady the energy consumption is over time. It  
Figure 16: The real (blue) and optimal (red) hourly energy consumptions.  
Figure 17: The differences between the real and optimal hourly energy consumptions.  
represents the mean value of the energy consumption divided by the maximum energy consumption in a specified period (in our case, 1 day). The value of this indicator increased with 21.4%, from 0.42 to 0.51.