# Balancing Project Schedule, Cost, and Value under Uncertainty: A Reinforcement Learning Approach
## 4 Quantitative Models
### Ccbm  
Traditionally, the CCBM scheduling approach begins by creating a baseline schedule that minimizes the project duration based on deterministic estimates of activity durationssuch as the median or mean values [82]. This involves solving the resource-constrained project scheduling problem (RCPSP) or its multimode extension. Due to the NP-hard nature of the problem [83], heuristic methods are commonly used, especially for larger projects. Once a baseline schedule is established, the PBs and FBs are inserted using a buffer-sizing technique such as the methods discussed in Section 2.2. The aim is to create a stable schedule that can be evaluated using robustness measures described in relevant literature, such as the standard deviation of project length, stability cost, and timely project completion probability [44]. The on-time probability not only serves as an indicator of schedule robustness but also plays a role in buffer calculation. [PERSON] and [PERSON] [26] proposed the use of Monte Carlo simulation to determine the cumulative distribution function (CDF) for project completion time, thereby determining the size of the PB. For example, if we aim for a 95% probability of completing the project on schedule, the PB would be the difference between the project duration at the 95 th percentile and the duration of the baseline schedule. Let us go one step further. If we directly search for the shortest time-buffered schedule that meets the desired on-time probability, we can identify a schedule with the same probability but a shorter duration. This leads us to the chance-constrained CCBM problem, where we consider multimode projects. In this problem formulation, we not only search for the schedule but also determine the activity modes that result in the shortest project duration while meeting the desired on-time probability. This duration encompasses the baseline schedule, nominal (deterministic) activity durations, and the PB. In this paper, we define project delivery as this time-buffered project duration, which represents the deadline we can meet (i.e., deliver the project) with the desired on-time probability. To model our chance-constrained CCBM problem, we adopt a MILP approach using the flow-based formulation described in [84], extending it to accommodate multimode projects. We handle the chance constraints using SA as in the LPM model but require additional parameters and decision variables that were not present there. There are \(K\) distinct renewable resources available, each having a total availability of \(\mathcal{R}^{k}\) units. When activity \(j\) is executed in mode \(m\), it requires \(r_{jm}^{k}\) units of resource \(k\). For a given scenario \(s\), the earliest start time for activity \(j\) is denoted as \(ES_{js}\), while \(LS_{js}\) represents the latest start time. The project delivery is represented by decision variable \(D_{p}\). Project milestones 0 and \(J+1\) have a singular mode, zero duration, and no resource requirements. Binary decision variable \(z_{ij}\) is employed to indicate if activity \(j\) commences after the completion of activity \(i\), taking a value of 1 in such cases. The flow variable \(\phi_{ij}^{k}\) models the amount of resource \(k\) transferred from activity \(i\) to activity \(j\). Our chance-constrained CCBM model incorporates constraints (3), (8), (12), and (16) from the LPM model. Moving forward, we will present the model, followed by an explanation of the objective function and the remaining constraints. \[\text{Min }D_{p}, \tag{18}\]  
subject to:  
\[t_{J+1,s}-\theta(1-\tau_{s})\leq D_{p},\ \forall s=1,\ldots,S, \tag{19}\]  
\[z_{ij}+z_{ji}\leq 1,\ \forall i=0,\ldots,J,\ \forall j=1,\ldots,J+1,\ \forall i<j, \tag{20}\]  
\[z_{ij}+z_{jh}-z_{ih}\leq 1,\ \forall i,j,h=0,\ldots,J+1,\ \forall i\
eq j\
eq h, \tag{21}\]  
\[z_{ij}=1,\ \forall i\in\mathcal{P}(j),\ \forall j=1,\ldots,J+1, \tag{22}\]  
\[t_{js}-t_{is}-Mz_{ij}\geq\sum_{m=1}^{M_{i}}\delta_{im}d_{ims}-M,\ \forall i,j=0,\ldots,J+1,\ \forall i\
eq j,\ \forall s=1,\ldots S, \tag{23}\]\[