## 3 The MOFOA-HS for the Multi-Objective F-MSRCPSP
### Population Initialization
As the \(t\)-test results show, there are only six tests that fail to accept the null hypothesis for 22, and all tests accept the null hypothesis for 22. In general, based on the above experimental results, the MOFOA-HS algorithm is more effective than NSGAII, MOFOA, and MOEA/D in solving the proposed F-MSRCPSP. ## 5 Conclusions and Future Works

This study proposes a novel F-MSRCPSP, where the execution of an activity requires a combination of multiple resources. In the F-MSRCPSP, the resource combination may vary over a time period, which results in changes in the duration and cost of the activity. To solve the concerned problem, a hybrid multi-objective fruit fly optimization algorithm with heuristic strategies is proposed. A novel solution representation based on an activity sequence list and a resource pattern list is designed. Two scheduling generation schemes, i.e., the FPSGS and the FSSGS, are introduced. Two heuristic strategies, i.e., the GSS and the NGSS, are adopted for specific resource allocation. Then, an initialization method is designed to balance the uniform distribution of the population. Additionally, a series of operators are designed to accelerate the convergence speed and strengthen the exploration of the algorithm, such as a rejoining operator, a contraposition selection operator, an empirical rearrangement and reselection operator, a local and global balanced operator, and so on. We have compared the MOFOA-HS to other excellent algorithms, i.e., NSGAII, MOFOA, and MOEA/D. The results of the comparative experiment and \(t\)-test statistical analysis show that the proposed MOFOA-HS algorithm obtains the best performance among the compared algorithms. In future work, it is important to consider the preemptive F-MSRCPSP. In real-world projects, we often encounter a complex and uncertain environment, which can result in a reduction in available resources and the need to temporarily add external activities. Furthermore, the integration of solutions derived from heuristic algorithms should be explored as an initial solution to accelerate the exact solution of the mathematical model. Methodology, X.L. ([PERSON]) and B.D. ; Software, X.L. ([PERSON]); Validation, J.G. ; Investigation, J.G. ; Writing--original draft, X.L. ([PERSON]) and X.L. ([PERSON]); Supervision, S.G. and B.D. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The data presented in this study are available on request from the corresponding author. The authors declare no conflict of interest. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{4}{c}{**HV**} & \multicolumn{4}{c}{**IGD**} \\ \cline{3-8}  & & **NSGA-II** & **MOFOA** & **MOEA/D** & **NSGA-II** & **MOFOA** & **MOEA/D** \\ \hline
100\_15\_65\_9 & \(p\)-value & 0.67 & 0.03 & 0.58 & 0.74 & 1.00 & 0.77 \\  & Null hypothesis & Y & N & Y & Y & Y & Y \\
200\_10\_50\_9 & \(p\)-value & 0.45 & 0.40 & 0.88 & 0.79 & 0.99 & 0.83 \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\
200\_15\_54\_15 & \(p\)-value & 0.11 & 0.00 & 0.01 & 0.96 & 1.00 & 0.96 \\  & Null hypothesis & Y & N & N & Y & Y & Y \\
200\_20\_145\_15 & \(p\)-value & 0.87 & 0.13 & 0.55 & 0.08 & 1.00 & 0.97 \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\ \hline \hline \end{tabular}
\end{table} 
Table 10: _Cont._

## References

* [PERSON] and [PERSON] (2014) [PERSON]; [PERSON] Critical chain and evidence reasoning applied to multi-project resource schedule in automobile R&D process. _Int. J. Proj. Manag._**2014**, _32_, 166-177. * [PERSON] and [PERSON] (2021) [PERSON]; [PERSON] Integrating Schedule Risk Analysis with Multi-S skilled Resource Scheduling to Improve Resource-Constrained Project Scheduling Problems. _Appl. Sci._**2021**, _11_, 650. [CrossRef]
* [PERSON] et al. (2017) [PERSON]; [PERSON]; [PERSON] Scheduling deadline constrained scientific workflows on dynamically provisioned cloud resources. _Future Gener. Comput. Syst._**2017**, _75_, 348-364. [CrossRef]
* [PERSON] et al.