# Balancing Project Schedule, Cost, and Value under Uncertainty: A Reinforcement Learning Approach
## 9 Discussion  
During the LPM experiments, we observed that RL generated higher project values, which is noteworthy considering the well-established proficiency of GA, particularly for 10-activity projects, demonstrated in a study with value maximization in a deterministic setting [5]. One potential explanation for this result could be attributed to the RL algorithm's \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **CRL\({}_{1}\)-DRL\({}_{1}\)** & **CRL\({}_{2}\)-DRL\({}_{2}\)** & **CS-DS** \\ \hline Average difference (\%) & \(-6.11\) & \(-4.66\) & \(-2.77\) \\ H\({}_{1}\) & CRL\({}_{1}\) \textless DRL\({}_{1}\) & CRL\({}_{2}\)\textless DRL\({}_{2}\) & CS \textless\textless\textgreater{} DS \\ \(p\)-value & 0.000 & 0.000 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Chance-constrained methods compared with deterministic-constrained counterparts: Difference in project delivery. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **CRL\({}_{1}\)-CS** & **CRL\({}_{1}\)-PR** & **CRL\({}_{1}\)-CRL\({}_{2}\)** \\ \hline Average difference (\%) & \(-3.01\) & \(-21.67\) & \(-4.05\) \\ H\({}_{1}\) & CRL\({}_{1}\) \textless CS & CRL\({}_{1}\) \textless PR & CRL\({}_{1}\)\textless CRL\({}_{2}\) \\ \(p\)-value & 0.000 & 0.000 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: CRL\({}_{1}\) compared with the other chance-constrained methods: Difference in project delivery. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **SA-RL\({}_{1}\)** & **RL\({}_{1}\)-RL\({}_{2}\)** & **RL\({}_{1}\)-TS** \\ \hline Average difference (\%) & 1.70 & 0.93 & 1.87 \\ H\({}_{1}\) & SA \textgreater{} RL\({}_{1}\) & RL\({}_{1}\)\textgreater RL\({}_{2}\) & RL\({}_{1}\)\textgreater TS \\ \(p\)-value & 0.000 & 0.000 & 0.037 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Performance of RL\({}_{1}\), RL\({}_{2}\), SA, and TS. inherent nature, whereby the agent takes immediate actions and receives corresponding rewards, thus continually learning the policy throughout each iteration. In contrast, GA operates in a more randomized manner. Initially, it generates a population of solutions, evaluates each one, and attempts to enhance them through the random mixing of pairs. Discovering optimal solutions through this process takes time for two primary reasons. Firstly, we must await the completion of the entire population's generation and evaluation. Secondly, the fitness value of each solution holds minimal influence on the quality of solutions generated in subsequent iterations, implying that it is not effectively utilized for learning. As stated in the Introduction, LPM aims to create value and minimize waste in the shortest amount of time. One of the main practical implications of our LPM model is the ability it offers project managers to generate alternative solutions and conduct tradeoff analysis, considering different risk levels in terms of time and cost overruns. Each solution generated is an implementable project schedule with the selected mode for each activity, maximizing the project value according to the stakeholders' risk threshold. Our CCBM experiments yielded compelling results, demonstrating that shorter schedules can be obtained by directly solving the chance-constrained model instead of resorting to solving the deterministic model and subsequently incorporating time buffers (Objective 1). This outcome aligns with our expectations. By addressing the chance-constrained problem directly and considering the actual realization of activity durations, we make informed decisions on modes and start times that satisfy the true objective of minimizing the delivery date. In our context, the project duration encompasses the desired on-time probability, including the PB that ensures project completion within the specified timeframe. To the best of our knowledge, no previous work on CCBM scheduling has adopted this outlook. Furthermore, our investigation revealed that the RL approach produces schedules that are competitive when compared with well-established benchmarks (Objective 2).