## 3 Proposed Methods
### PSGS and MDP Formulation  
For deterministic scheduling, the MDP of RCPSPDC-PEO must be embedded into the SGS; in our case, we employ PSGS. Our MDP has the following five components:  
1. Stages: We denote the stages by \(v\in[0,\dots,\mathrm{F}]\); \(v\) gets incremented whenever a job is scheduled. In other words, whenever the number of unscheduled jobs decreases, \(v\) gets incremented. \(v=0\) corresponds to no job being scheduled and \(v=F\) corresponds to the stage where all the jobs are being scheduled. The number of stages therefore corresponds to the size of the project. 2. States: The state of stage \(v\) is described as \(S_{v}=\left\{\mathbf{A}_{v},\mathbf{C}_{v},\mathbf{Q}_{rv},\mathbf{A}_{v},t_{v}\right\}\). In \(S_{v}\), the elements \(\mathbf{A}_{v},\mathbf{C}_{v},\mathbf{Q}_{rv}\) are the vectors containing the active jobs at time \(v\), the completed jobs at \(v\), and the resources available at \(v\), respectively. The vector \(\mathbf{A}_{v_{t}}\) contains the completion time of the active jobs, and \(t_{v}\) denotes the completion time of the jobs that are completed immediately before the state \(S_{v}\). The initial and final states are given by \(S_{0}=\left\{\emptyset,\emptyset,\emptyset,\mathbf{Q}_{rv},\emptyset,0\right\}\) and \(S_{F}=\left\{\emptyset,\left\{1,\dots\right\},\mathbf{Q}_{rv},\emptyset,f_{f+1}\right\}\), where \(\mathbf{Q}_{rv}\) = \(\mathbf{Q}_{rF}\) - \(\mathbf{Q}\); \(\mathbf{Q}\) is the vector containing the total resources available when no job is being executed and. = denotes the element wise comparison in the vectors. 3. Decision sets: Suppose \(E_{v}:=\chi(S_{v})\) consists of eligible activities, that is, jobs that fulfill Equations (1) and (2), then the decision space \(x_{v}\subseteq\chi(S_{v})\) contains jobs that are feasible for execution. 4. Transition process: The transition from stage \(v\) to \(v+1\) takes place with \(S_{v+1}=S^{M}(S_{v},x_{v})\) using the transition function \(S^{M}(\cdot)\). The transitions of different components are presented in Equations (4a) to (4k): \[\text{if }\mathbf{q}_{j}\cdot\geq\mathbf{Q}_{rj},\forall j\in E_{v}\text{ or }E_{v}=\emptyset\] (4a) \[t_{v+1}=\min\Bigl{(}\left\{f_{j}:j\in\mathbf{A}_{v},f_{j}\in\mathbf{A}_{v}\right\} \Bigr{)}\] (4b) \[\mathbf{C}_{v+1}=\mathbf{C}_{v}\cup\left\{j:f_{j}\leq t_{v+1},\ \forall j\in\mathbf{A}_{v},f_{j}\in\mathbf{A}_{v_{t}}\right\}\] (4c) \[\mathbf{Q}_{rv+1}=\mathbf{Q}\cdot-\sum_{k=1}^{|\mathbf{q}|}\Bigl{[}\mathbf{q}_{1}\dots\mathbf{q}_ {j}\dots\mathbf{q}_{l}\Bigr{]}^{-1},\,j=[1,\dots,l]\in\mathbf{A}_{v}\] (4d) \[\mathbf{A}_{v+1}=\left\{j:f_{j}>t_{v+1},\ \forall j\in\mathbf{A}_{v},f_{j}\in\mathbf{A}_{v_{ t}}\cup x_{v0}\right\}\] (4e) \[\text{if }\mathbf{q}_{j}\cdot\leq\mathbf{Q}_{rj}\,\ \exists j\in E_{v}\text{ or }\mathbf{A}_{v}=\emptyset\] (4f) \[t_{v+1}=t_{v}\] (4g) \[\mathbf{C}_{v+1}=\mathbf{C}_{v}\] (4h) \[\mathbf{A}_{v+1}=\mathbf{A}_{v}\cup x_{v}\] (4i) \[\mathbf{Q}_{rv+1}=\mathbf{Q}\cdot-\sum_{k=1}^{|\mathbf{q}|}\Bigl{[}\mathbf{q}_{1}\dots\ \mathbf{q}_{j}\dots\mathbf{q}_{l}\Bigr{]}^{-1},\,j=[1,\dots,l]\in\mathbf{A}_{v}\] (4j) \[t_{v+1}=\min\Bigl{(}\left\{f_{j}:j\in\mathbf{A}_{v},f_{j}\in\mathbf{A}_{v}\right\} \Bigr{)}\]0 (4k)  
In this transition, the process must follow a logical control flow. That is, if Equation (4a) is true, Equations (4b)-(4e) are executed, followed by Equation (4k). Equation (4a)checks whether \(E_{v}\), an eligible set at state \(S_{v}\), is empty or whether the resources required by each activity of \(E_{v}\) are less than the balance resources if \(E_{v}\) is not empty. Equation (4a) assesses whether the set of eligible activities at state \(S_{v}\), denoted as \(E_{v}\), is empty. If \(E_{v}\) is not empty, the equation further evaluates whether the resources required for each activity within \(E_{v}\) exceed the available balance of resources. Equation (4b) determines the times associated with state \(S_{v+1}\) by taking the minimum of the finish times of the active jobs from the state \(S_{v}\).