## 6 Discussion
### Comparison of the Composite Robustness Criterion and a Single Robustness Criterion  
A total of 200 schedules meeting the logical relationship and resource restrictions of this project are generated randomly and their composite robustness criterion r and three single robustness criteria \(r_{s}\), \(r_{p}\), and \(r_{c}\) are calculated.  
The measured results are divided into five intervals on average. The quantity of schedules that belongs to each interval is shown in Figure 8. The robustness of these schedules measured via \(r\) are mainly within the third and fourth intervals, and a few ones are in the second and fifth intervals, and the schedules in the first interval are the least. This phenomenon complies with the circumstances of the measured results of \(r_{s}\), \(r_{p}\), and \(r_{c}\). Therefore, the robustness measured result of \(r\) is consistent with the results of existing single criteria.  
In order to analyze the representativeness of the proposed composite robustness criterion, which is embodied by the symmetry, concentration, and even dispersion of \(r\) as well as \(r_{s}\), \(r_{p}\), and \(r_{c}\), the skewness, kurtosis and dispersion coefficient of the measured results are calculated. As shown in Figure 9, the kurtosis of \(r_{s}\) and \(r_{c}\) is rather high, which suggests a strong concentration, and thus extreme and invalid evaluations in measuring robustness. The skewness of \(r_{p}\) is very high, which suggests an obvious negative skewness, and its accuracy would be low when measuring the schedule robustness whose fluctuation in structural robustness and quality robustness is small. The dispersion coefficient of \(r_{c}\) is rather large, which suggests poor criterion representativeness, and it would be useless when used individually. From the comparison, it can be drawn that the composite robustness criterion \(r\) has desirable symmetry, concentration, and uniform dispersion, which balances the three single criteria and is representative in robustness measure of the construction schedule.  
To verify the advantage of the proposed composite robustness criterion, the optimization of construction duration and schedule robustness is conducted. Construction duration \(T\) and schedule robustness, which is measured using \(r\), \(r_{s}\), \(r_{p}\), and \(r_{c}\), respectively, are the two optimization objectives. A hybrid grey wolf optimizer with a sine-cosine algorithm is adopted to solve the optimization model [50]. The iteration number is set as 1000. The optimization results in Figure 10 show that when the optimization environment is the same, and optimized duration level is equal, the optimized schedules with the objectives of \(T\)-\(r_{s}\) (or \(T\)-\(r_{p}\), \(T\)-\(r_{c}\)) only have good solution robustness or good quality robustness, but do not have robustness in both aspects. The optimized schedules with the objectives of \(T\)-\(r\) have both good solution robustness and good quality robustness, which ensures the schedules' anti-interference ability from the perspectives of both project execution and project result. Therefore, the composite robustness criterion \(r\) proposed in this paper has the advantages of comprehensively measuring the robustness, in terms of both project execution and project result.  
Figure 8: Distributions of the construction schedules via \(r\), as well as \(r_{s}\), \(r_{p}\), and \(r_{c}\).  
Figure 9: Radar chart of skewness, kurtosis, and dispersion coefficient of \(r\), as well as \(r_{s}\), \(r_{p}\), and \(r_{c}\).