## 2 Problem Definition and Literature Review  
In a deterministic and non-preemptive project scheduling, we determine the starting time \(s_{j}\) for each activity from a series of activities numbered from 1 through \(J\). Each activity \(j\) has a processing time \(d_{j}\) as well as precedence constraints with the immediate predecessors \(P_{j}\), which must be finalized before beginning an activity \(j\) as denoted by Equation (1), and resource demand of \(\mathbf{q}_{j}\) units (i.e., \(\mathbf{q}_{j}\in\mathbf{Q}\)) in its active duration as denoted by Equation (2). The vector \(\mathbf{Q}\) contains the different resources required by each activity. All data pointsare assumed to be fixed, known beforehand, and quantified as non-negative integers. Mathematically, this is formulated as [12; 34]:  
\[s_{i}\leq s_{j}-d_{i}\,\ \forall i\in P_{j} \tag{1}\]  
\[\mathbf{q}_{j}\cdot\leq\mathbf{Q}_{rj} \tag{2}\]  
In Equation (2), the vector \(\mathbf{Q}_{rj}\) denotes the available resources to be used by activity \(j\) and \(.\leq\) checks for the availability of all corresponding resources. In Equation (1), an activity \(j\) is made available for execution immediately after all its preceding jobs are completed in a zero time-lag fashion. For RCPSPDC-PEO, a contract receives cash whenever itcompletes the pre-specified activities and at the project completion time \(f_{j+1}\); the objective function is expressed as:  
\[Max.\ Z=I_{f_{j+1}}e^{-df_{j+1}}+\sum_{j=1}^{J}C_{j}e^{-df_{j}}+\sum_{v=1}^{m}I _{v}e^{-df_{v}} \tag{3}\]  
In Equation (3), the completion of activity \(j\) is defined as \(f_{j}=s_{j}+d_{j}\), where project completion time is \(f_{j+1}=\max\bigl{(}\{f_{j}:j=1\ldots\}\bigr{)}\). \(d\), \(C_{j}\) and \(I_{j}\) denote the discount factor, cost, and income for executing job \(j\), respectively. It is worth mentioning that the non-linearity of Equation (3) has further increased the complexity of RCPSPDC-PEO from RCPSP which is already NP-hard. Equation (3) maximizes the sum of various cash flow types discounted by the factor \(d\). There are two mostly used SGS [35; 36; 37; 38] in the literature: These are the Parallel Generation Scheme (PSGS) and the Serial Generation Scheme (SSGS). SGS operates on an activity-incrementation basis, essentially constructing a schedule without a real-time variable. The scheme proceeds through \(\{1,\ldots,J\}\) stages, with one activity being selected for scheduling during each stage from the pool of eligible activities. In this selection process, eligibility is determined primarily by the precedence constraints, while resource constraints are factored in later when determining the activity schedule time. The authors of [19; 22] applied SGSS for stochastic scheduling with ADP in non-deterministic RCPSP. Conversely, PSGS operates using timeline incrementation and is particularly well-suited for tackling hard problems [23]. PSGS excels at generating non-delay schedules [23]. Hence, for the novelty of this study and for its advantages over SSGS, we have employed the PSGS method to generate project schedules. The PSGS with MDP formulation is explained in Section 3.1. RCPSP (RCPSPDC) can be divided into two categories: deterministic and non-deterministic. In deterministic RCPSP (RCPSPDC), details about resource and activity duration are known beforehand. In non-deterministic or uncertain RCPSP (RCPSPDC), this information is only partially or fully unknown. In the literature, the deterministic RCPSPDC has attracted more attention than its non-deterministic counterpart. This is primarily because non-deterministic introduces an additional layer of complexity with its inherent uncertainty, rendering the determination of cash flows associated with the project highly challenging. Consequently, our literature review focuses on the deterministic RCPSPDC, which is also the focus of our study.