# Balancing Project Schedule, Cost, and Value under Uncertainty: A Reinforcement Learning Approach
## 4 Quantitative Models
### Tvnpv  
In line with Section 4.2, we utilize the flow-based formulation introduced by [84] and extend it to encompass multimode projects, NPV, value functions, and chance constraints. The primary objective of the TVNPV model is to maximize the robust project NPV and the project value. To address the chance constraints, we employ SA. We now describe additional parameters and decision variables that are not present in the LPM and CCBM models. Within the problem context, we have \(K\) distinct renewable resources, each associated with a unit cost \(c_{k}\) per period. Additionally, activity \(j\) executed in mode \(m\) incurs a fixed cash inflow or outflow \(c_{jm}\), consisting of fixed costs and payments received. For convenience, we assume that payments are received or made at the end of each activity. To avoid gaps between activities and prevent indefinitely postponing activities with negative cash flows, two main approaches exist in the literature: (1) utilizing a deadline [57] and (2) assuming a sufficiently large payout at the end of the project to offset the gains from postponing activities impacting project completion [66]. In this paper, we adopt the latter approach. When aiming to minimize project duration, a common measure of robustness is the timely project completion probability [44]. In our problem, we adapt this concept and introduce the decision variable rNPV to represent the robust NPV. It signifies the project NPV achieved with a probability of at least \(\gamma\). Thus, instead of evaluating the robustness of a given schedule, we search directly for a schedule with the desired level of robustness. Several parameters are defined within the model. \(NPV^{UP}\) serves as an upper bound for rNPV. \(\widehat{\tau}\) denotes the discount rate, while \(EF_{js}\) and \(LF_{js}\) represent the earliest and latest finish times for activity \(j\) in scenario \(s\), respectively. Additionally, \(T_{\text{max}}\) acts as an upper bound for the project's duration. To represent the finish time of activity \(j\) in scenario \(s\), we introduce the decision variable \(f_{js}\in\left\{EF_{js},\ldots,\,LF_{js}\right\}\). Binary variable \(\widehat{\tau}_{s}\) takes the value \(1\) if the scenario NPV is greater than rNPV. Moreover, decision variable \(\beta_{js}\) represents the discount factor for activity \(j\) in scenario \(s\), and \(\beta^{UP}\) serves as an upper bound for the discount factor. Objective function weights \(w_{1}\) and \(w_{2}\) are included to determine the tradeoff between rNPV and the project value. By solving the MIP for different values of \(w_{1}\) and \(w_{2}\), we can identify the efficient frontier that balances these objectives. To linearize two sets of constraints, we introduce additional variables. Binary variables \(t^{p}_{js}\) are assigned a value of \(0\) for all \(p<f_{js}\) and \(1\) for all \(p\geq f_{js},\ p=0,\ldots,T_{\text{max}}\). Variables \(y_{jms}\) replace the products \(\beta_{js}\delta_{jm}\). The model incorporates constraints (2), (3), (8), (20)-(22), and (25)-(28) from the LPM and CCBM models. We now present the model, providing an explanation of the objective function, followed by an overview of the remaining constraints. Subsequently, we will discuss the linearization of the nonlinear constraints.