# Balancing Project Schedule, Cost, and Value under Uncertainty: A Reinforcement Learning Approach
## 2 Literature Review
### Max-NPV Problem  
Extensive research has been conducted regarding the max-NPV problem. [PERSON] [56] conducted an investigation into the deterministic problem, employing a linearization technique that approximated the objective function through the first terms of the Taylor expansion. Subsequently, a wealth of additional studies have contributed to the existing body of knowledge on the max-NPV problem. Notably, ref. [57] demonstrated its NP-hardness, while [58] proposed a precise solution approach tailored to smaller projects as well as a Lagrangian relaxation method coupled with a decomposition strategy for more extensive problems. Additionally, ref. [57] devised a methodology that involved grouping activities together, and in a subsequent publication [59], they further expanded their work to encompass capital constraints and various cash outflow models. [PERSON] [60] explored projects characterized by payment milestones and investigated diverse scheduling techniques, including activity right-shift, backward scheduling, and left-right justification. The deterministic max-NPV problem, when extended to accommodate multiple modes, presents a multimode variant of the original problem. [PERSON] et al. [61] successfully achieved optimal solutions for projects containing up to 30 activities and three modes by utilizing a network flow model. Building upon the scheduling technique mentioned earlier in [57], ref. [62] further expanded their approach to incorporate multimode projects and diverse payment models for cash inflows. The examination of these payment models continued in the context of the max-NPV discrete time/cost tradeoff problem. In this regard, ref. [63] compared the impact of three distinct solution representations by integrating them into an iterated local search algorithm. Additionally, ref. [64] addressed a bi-objective optimization problem, aiming to balance the NPV between the contractor and the client. The stochastic max-NPV problem serves as another extension to the original deterministic max-NPV problem, introducing random variables for activity durations and cash flows. [PERSON] and [PERSON] [65] provided an in-depth review of the early literature on this subject. In their study, [PERSON] et al. [66] focused on maximizing the expected value of NPV (eNPV) while considering variable activity durations, the risk associated with activity failure, and different approaches or modules to mitigate this risk. Resource constraints, however, are not taken into account in their analysis. Similarly, ref. [6] explored the notion of a general project failure risk that diminishes as project progress is made. They also considered activity-specific risks. It is worth noting that completing earlier activities sooner not only eliminates the risk of failure, thereby improving the eNPV, but also potentially accelerates costs, consequently worsening the eNPV. Incorporating weather condition modeling into stochastic durations, ref. [67] introduced decision variables in the form of gates. These gates dictate when resources become available for specific activities, allowing for a more comprehensive analysis of the problem at hand. In the study conducted by [68], optimal solutions on a global scale were identified for the stochastic NPV problem. Specifically, the focus was on activity durations that followed a phase-type distribution, deterministic cash flows, and the absence of resource constraints. Expanding on these findings, the authors further applied them to determine the optimal sequence of stages in multistage sequential projects characterized by stochastic stage durations. In doing so, exact expressions in closed form were derived for the moments of NPV, with the utilization of a three-parameter lognormal distribution to accurately approximate the distributions of NPV [69]. Additionally, it was demonstrated that this problem is equivalent to the least-cost fault detection problem, which was established by [69] and [70]. [PERSON] and [PERSON] [71] contributed to the field by presenting a novel and efficient algorithm.