# Energy-Efficient Hybrid Flow-Shop Scheduling under Time-of-Use and Ladder Electricity Tariffs
## 4 Case Study
### Random Instances  
The computational findings for evaluating the modified NSGA-II method for minimizing makespan and total energy consumption in the HFSP issue are described in this part. Comparing their performance with the well-known NSGA-II algorithm indicates where their performance stands. All the algorithms are coded in n Visual Studio2017 on a PC with a 2.8 GHz Intel core i5-7200 processor and 8 GB RAM in Windows 10 system. We need to consider HFSP with various machines and job numbers to evaluate the provided approaches under different problem sizes. Therefore, we generate a set of instances of various sizes that resemble the industrial data. Table 5 identifies and lists the variables that impact the manufacturing system's performance. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Makespan (h)** & **TEC (Yuan)** & **Total Carbon Emission (tCO2)** \\ \hline Efficient & 23.1 & 7498.6 & 4.529 \\ Economical & 24.2 & 5335.9 & 3.222 \\ Trade-off & 23.6 & 6173.6 & 3.729 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of optimal scheduling schemes. This section describes the experiments that have been conducted to compare the results of the improved NSGA-II (I-NSGA-II) and traditional NSGA-II. In this study, experiment case is identified as the form of \"Number of jobs-Number of machines at each stage-Number of stages\". For example, \"20-5-4\" means a case of 20 jobs, 5 stages and 4 parallel machines in each stage. To test the efficiency of the I-NSGA-II algorithm in the environments with different complexity, the test set of instances is classified into the three categories according to the number of jobs (10, 30, and 50). Each category has different stages and parallel machines. Identical factors were generated randomly for each test problem. Genetic factors are important parameters that affect the results. It should be mentioned that algorithm parameters affect the performance of the compared algorithms. Pilot experiments were conducted under a set of potential parameter values to find the best combinations. To establish fair comparisons, as a standard parameter, the maximum population size maxPop, is selected as 200 to represent the standard parameter in all three algorithms. In addition, the performance of NSGA-II is affected by the termination condition, i.e., the number of iterations or the maximum execution time (tmax). These settings take different values in different experiments. In addition, the population size was dynamically determined according to the scale of problem. The mutation probability and the crossover rate were set to be 0.05 and 0.95, respectively. We ran the algorithm ten times, taking into account randomness for each instance, and recorded the minimum (Best) and average values (Avg) of performance metrics in Tables 6 and 7. The overall comparison results of HPSO and NSGA-II on the benchmark instances. The deviation between the I-NSGA-II solutions and the NSGA-II solutions is output, which is defined as follows. \[\text{dev}(\text{TEC})=\frac{\text{TEC}(\text{NSGA}-\text{II})-\text{TEC}( \text{I}-\text{NSGA}-\text{II})}{\text{TEC}(\text{NSGA}-\text{II})} \tag{16}\]  
where TEC(NSGA-II) denotes the TEC objective value of the solutions obtained by algorithm NSGA-II. It can be seen from the Tables 6 and 7, if the processing time is too short without beyond a price interval, there will be no impact on energy consumption under TOU and ladder electricity price. In the case of electricity prices in multiple intervals, the increase of parallel machines reducing the processing time while increasing the energy consumption. However, the existence of parallel machines makes the gap between workpiece processing possible, thereby increasing the space for the right-shift movement. Comparing the best solutions of the two algorithms, it can be found that the two algorithms both have their own advantages.