## 3 Results and Discussion
### Results of the Two-Level System of Multi-Objective GAs
#### 3.2.2 Results of Level 2: Multi-Objective GA for Measuring the Forecasting Accuracy  
The outcome from the first level, specifically for each generation for each population, indicates the forecasting accuracy for each cost object. For each generation, the multi-objective GA based on multiple fitness functions was used to find the best fitness value through five different generations. The fitness functions (forecasting error rate models) provide an accurate data forecasting through comparing the behaviour of the different models and revealing which forecasting model is appropriate. Figures 6 and 7 show the forecasting accuracy for the labour and material cost objects obtained with five different fitness functions for the four populations. The fitness function values of each population are the minimum values obtained through testing five different generations from the first level. Figure 6 shows the fitness values for the labour cost object. The figure shows five different curves for five different fitness functions. Figure 5: Material cost data forecasting using the multi-objective GA based on the ARIMA model. Concerning the first population, \(fitness(MAPE)\) has the highest error rate, 3.5, while \(fitness(MdAPE)\) and \(fitness(RMSPE)\) have the same value, 2.01. \(fitness(RMdSPE)\) has a value of 2.15 and \(fitness(MASE)\) has the lowest value, 0.89. Concerning the second population, \(fitness(MdAPE)\) and \(fitness(RMSPE)\) show the same value, 0.02, which is the lowest fitness value for this population. The value for \(fitness(MASE)\) is 0.12, while the values for \(fitness(RMdSPE)\) and \(fitness(MAPE)\) are 0.58 and 0.59, respectively. Concerning the third population, \(fitness(MdAPE)\) and \(fitness(RMSPE)\) show the same value, 0.13, which is the lowest fitness value for this population. The value for \(fitness(RMdSPE)\) is 0.14, which is higher than that for \(fitness(MdAPE)\) and \(fitness(RMSPE)\), while the values for \(fitness(MASE)\) and \(fitness(MAPE)\) are 0.47 and 1.7, respectively. Finally, concerning the fourth population, \(fitness(MdAPE)\) and \(fitness(RMSPE)\) show the same value, 0.08, which is the lowest  
Figure 6: Labour cost error rate. Figure 7: Material cost error rate. fitness value for this population. \(fitness(MASE)\) has a value of 0.19, while \(fitness(MAPE)\) and \(fitness(RMdSPE)\) have higher values, 0.43 and 0.52, respectively. Overall, the fitness functions are variants. \(fitness(MAPE)\) has the highest of all the curves of the four populations because it involves small data counts. \(fitness(MdAPE)\) and \(fitness(RMSPE)\) have almost equal values over the four populations, while their values can be regarded as almost close when one takes all the 15 generations for the four populations into account. The \(fitness(RMdSPE)\) and \(fitness(MASE)\) curves show a sensitivity to the data caused by the population randomization. Selecting a proper population for the labour cost data is quite difficult due to the variety of fitness values. In this study, we considered the population that was selected by more than two of the fitness functions to have a low forecasting error rate. The thirteenth generation for the second population was selected as having the lowest forecasting error rate with a suitable selection of input data. The labour cost data were selected using three fitness functions, \(fitness(MdAPE)\), \(fitness(RMdSPE)\) and \(fitness(MASE)\), with values of 0.02, 0.02 and 0.12, respectively. Figure 7 shows the fitness values for the material cost object. The figure shows that, concerning the first population, \(fitness(MAPE)\) has the highest error rate, 3.54, while \(fitness(MdAPE)\) and \(fitness(RMdSPE)\) have the same value, 1.97. \(fitness(RMSPE)\) has a value of 2.54 and \(fitness(MASE)\) has the lowest value, 0.79. With regard to the second population, \(fitness(MdAPE)\) and \(fitness(RMdSPE)\) show the same value, 1.35, which is the lowest fitness value for this population.