# Balancing Project Schedule, Cost, and Value under Uncertainty: A Reinforcement Learning Approach
## 6 Example  
To provide a concrete illustration of our problem and the RL solution approach, let us consider as an example the development of a radar system, drawn from a real-world project. In Figure 2, we present the project's AON network, while Table 1 provides a list of the project's five activities, each with two available modes. The table includes the optimistic, most likely, and pessimistic durations (O, ML, and P) of the activities, their respective fixed costs (FC), the required resources per period for each mode (engineers, E, and technicians, T), the value parameters, and the income received upon activity completion. It is worth noting that three of the five activities exhibit negative cash flows, i.e., fixed and resource costs, while the remaining two activities yield positive cash flows due to the generated income. This example effectively illustrates how value is defined and measured in practice. The needs and expectations of project stakeholders are translated into value attributes such as range, quality, and reliability (R, Q, and Re, respectively, as shown in Table 1). These attributes are determined by the value parameters associated with each activity mode. Additionally, it is important to consider the resource unit costs per period, which amount to USD 100 for engineers and USD 50 for technicians. In the context of the radar system, we employ the radar equation [101] for computing the radar range [21]. The quality and reliability of the radar system are also considered, as they depend on technical parameters such as transmitter power and antenna gain. These parameters, in turn, are contingent upon the technological alternatives available for each mode. The selection of a mode for the project plan not only determines its value but also has a significant impact on cost and NPV, effectively integrating both components of project value. The equation for calculating radar range \(F_{1}=([TP][RS][AG])^{0.25}\) involves the variables \([TP]\) (transmitter power), \([RS]\) (receiver sensitivity), and \([AG]\) (antenna gain). These \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Activity** & **Mode** & \multicolumn{2}{c}{**Duration**} & \multicolumn{2}{c}{**FC**} & \multicolumn{2}{c}{**Resources**} & \multicolumn{2}{c}{**Value Parameters**} & \multicolumn{1}{c}{**Income**} \\ \cline{3-11}  & & **O** & **ML** & **P** & & **E** & **T** & **R** & **Q** & **Re** & \\ \hline \multirow{3}{*}{Systems engineering} & Small team & 5 & 7 & 10 & 2000 & 1 & 0 & & 0.8 & & \\  & Large team & 3 & 4 & 4 & 4000 & 3 & 1 & & 0.99 & & \\ \hline \multirow{3}{*}{Transmitter design} & Reengineer & 3 & 5 & 8 & 5000 & 2 & 1 & 50 & 0.99 & 0.9 & \\  & New design & 7 & 9 & 11 & 10,000 & 4 & 2 & 100 & 0.95 & 0.8 & \\ \hline \multirow{3}{*}{Receiver design} & Reengineer & 3 & 5 & 9 & 2000 & 2 & 1 & 30 & 0.95 & 0.9 & \\  & New design & 8 & 10 & 11 & 15,000 & 3 & 1 & 200 & 0.8 & 0.99 & \\ \hline \multirow{3}{*}{Antenna design} & Reengineer & 3 & 7 & 9 & 3000 & 3 & 2 & 10 & 0.8 & 0.9 & \\  & New design & 6 & 7 & 9 & 7000 & 5 & 2 & 30 & 0.99 & 0.9 & \\ \hline \multirow{3}{*}{Integration and testing} & In-house & 3 & 4 & 4 & 4000 & 3 & 3 & & 0.99 & 0.9 & \\  & Subcontract & 2 & 2 & 5 & 6000 & 1 & 0 & & 0.9 & 0.99 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of data for radar development. Figure 2: Project network diagram. parameters are extracted from the corresponding activities in Table 1--namely, transmitter design, receiver design, and antenna design. Similarly, the equation for radar quality \(F_{2}=100[SEQ][QT][QR][QA][QI]\) incorporates the impact of various factors denoted by \([SEQ]\), \([QT]\), \([QR]\), \([QA]\), and \([QI]\), which represent systems engineering, transmitter, receiver, antenna, and integration effects on quality, respectively.