# Balancing Project Schedule, Cost, and Value under Uncertainty: A Reinforcement Learning Approach
## 1 Introduction
Decision makers can use our approach to plot an efficient frontier with the optimal project plans for given probabilities of meeting the schedule and budget. The second challenge that we explore in this paper is a new formulation of Critical Chain Buffer Management (CCBM), which is a well-known framework that addresses uncertainty and the issue of project overruns. The usual procedure for generating a schedule within this framework is to use an appropriate scheduling method to find a baseline schedule that is optimal or near optimal for the problem with fixed activity durations and then apply a buffer sizing technique to add time buffers--namely, a project buffer (PB) and feeding buffers (FBs). In this paper, we focus on solving the chance-constrained CCBM problem. We present a mixed-integer linear programming (MILP) model for the multimode problem and propose an RL-based algorithm to solve it. We demonstrate that solving the chance-constrained CCBM problem produces shorter project durations than solving the deterministic-constrained problem and then adding time buffers. We also prove that our RL-based method is effective in creating CCBM schedules compared with established benchmarks. The third challenge that we address in this paper is building a novel model that integrates uncertainty and chance constraints with two of the most important objectives in project management: the maximization of the project's net present value (NPV) and project value. The maximization of the project NPV (max-NPV) problem is highly relevant in the current context. Decision makers need to compare different project alternatives, make go/no go decisions, and decide which projects will be in their project portfolio [6]. Nevertheless, it is well known that the evaluation of a project should not only depend on financial factors; a project can have a high NPV but fail to deliver the expected value to customers and other stakeholders. As mentioned earlier, project value is becoming an essential factor in project management. Most studies have investigated the max-NPV problem and project value separately instead of integrating them. We argue that considering both goals together provides a more comprehensive assessment of a project when evaluating different project options. We propose a new formulation of the optimization problem that combines both NPV and project value, which we call the tradeoff between project value and NPV (TVNPV). We design RL-based algorithms to solve the TVNPV problem and explore the tradeoff between attaining both objectives. Each of the three challenges is critical because they help to solve different aspects of the situation pointed out above: the high percentage of time and cost overruns, caused in part by project uncertainty, and the failure to deliver value to stakeholders. The LPM challenge tackles value and adopts due date and budget chance constraints to avoid overruns. CCBM focuses directly on the robust minimum duration project plans to avoid delays. TVNPV takes a more holistic approach and deals both with the financial objective in the context of risk by maximizing a robust formulation of the NPV and satisfying stakeholders' needs and expectations by maximizing value. Our study presents several novel contributions. Firstly, in terms of problem formulation, we introduce three new models: an LPM one, a CCBM one, and a TVNPV one. Our LPM is the first model to maximize a value function with chance constraints. Our CCBM model is novel since it tackles the chance-constrained problem directly and addresses multimode problems, which is rare in the literature. The TVNPV model considers project value and NPV in tandem for the first time and introduces the concept of robust NPV. Secondly, in terms of solution methods, we apply RL to the three problems. This heuristic approach is seldom employed within the realm of project management and has yet to be applied to similar problems of this nature. The rest of the paper is organized as follows: Section 2 provides a literature review of the relevant research in the field.