# Deling with Uncertainty in the MRCPSP/Max Using Discrete Differential Evolution and Entropy
## 5 Conclusions  
Uncertainty greatly impacts the dynamic behavior of real-world environments. Adaptive capacity helps to improve preparedness in dynamic environments, and managers must respond effectively to changes in environmental conditions. Researchers and practitioners have sought to optimize schedules and quality in many studies, and improving schedules remains a pressing concern. In this paper, we tried to resolve the multi-mode resource-constrained project scheduling problem (MRCPSP/max), which is not a common domain for the DDE algorithm, and we specifically considered entropy, which helped to deal with uncertainty.  
We focus on three main contributions in this paper. Initially, we explored the characterization of MRCPSP/max under uncertainty conditions and confirmed the need for managers to recognize and positively respond to disruptive events. Using entropy to determine disruptive events and response intervals in scheduling, we demonstrated this proposition. Then, we formulated the robustness attribute as a scheduling adaptability maximization problem and a three-stage schedule generation framework to enhance resilience by absorbing the impact of unexpected events, while rescheduling during execution. Our final contribution was a discrete framework for the differential evolution algorithm. In our application of DDE, the encoding of the DDE algorithm consisted of two vectors representing the task sequence and the execution modes for each activity. Several potential resource conflicts and priority constraints were considered in our design. The proposed DDE was evaluated by solving test instances of benchmark sets by comparing its performance to the best known optima (BKO) and the previous application based on the artificial bee colony (ABC) approach.  
The findings indicated that, for all algorithms, the problem's complexity influences the number of optima found and the average run time. Overall, a more effective algorithm is the DDE algorithm, as it offers more optimal solutions and a higher number of them. Additionally, we were able to determine when practitioners need to add response time intervals at certain range levels, such as 8% to 15% in this case, to benefit from schedule robustness. Finally, compared to the BKO for every stage, the stability of the solutions  
Figure 8: Robustness measure CI for each algorithm and stage for MM100 instances.  
provided by the DDE demonstrated its algorithmic advantage in terms of resilience. Unfortunately, the more realistic the academic model is, the more difficult it is to solve the problem; the MRCPSP/max is simply one of the very difficult problems.  
Nonetheless, the encouraging computational results may lead to future implications along other lines. First of all, it is interesting to study which features make instances of MRCPSP/max difficult or easy to solve. In this sense, future studies may further enhance scheduling efficiency by examining various criteria for activity prioritization and mode selection. Another interesting line of research is investigating other potential encoding scheme frameworks in order to capture problem-specific aspects. Furthermore, efforts to examine other approaches to dealing with uncertainty in project scheduling and the verification of their performance using real-world data are also necessary.  
Conceptualization, A.H.-L.C., Y.-C.L. and J.D.P.; methodology, Y.-C.L. and J.D.P.; software, J.D.P.; validation, Y.-C.L. and J.D.P.; formal analysis, J.D.P.; writing--original draft preparation, A.H.-L.C., Y.-C.L. and J.D.P.; writing--review and editing, A.H.-L.C. and Y.-C.L.; visualization, Y.-C.L. and J.D.P.; supervision, A.H.-L.C.; project administration, A.H.-L.C.; funding acquisition, A.H.-L.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Ministry of Science and Technology, Taiwan, ROC, grant numbers: MOST 103-2221-E-253-005, and MOST 104-2221-E-253-002.  
The authors declare no conflict of interest.