## 1 Introduction
The metaheuristics approach takes a set of feasible solutions (i.e., scheduled projects) called population and employs various exploration and exploitation techniques to find quasi-optimal solutions in a reasonable time [14]. Studies have demonstrated that hybrid approaches, intelligently combining different methods (metaheuristics or other exact methods), outperform standalone metaheuristics by leveraging their collective strengths and mitigating individual limitations [14; 15]. This preference for hybridization [15; 16] resonates with the No Free Lunch Theorem (NFLT) [17], which argues against the feasibility of a universal standalone metaheuristic. Alternatively, in combinatorial problems like RCPSP and RCPSPDC, the scheduling scheme can be viewed as a Markov process [18]. That is, at any decision point in a scheduling process, a job is scheduled using a certain policy derived from the information available at the current state. In this sense, a determination of scheduling policy, also known as closed-loop policy [19], can be seen as a Markov decision process (MDP) [18; 20]. However, computing the optimal closed-loop policy in this setting as Dynamic Programming (DP) suffers from the curse of dimensionality [19]. The approximation of the cost-to-go or the benefit-to-go function [21] to approximate the decision policy [22] to resolve this intractability issue results in an approximate dynamic programming (ADP). To solve a deterministic RCPSPDC-PEO, ADP can be embedded into the schedule generation scheme (SGS) [1; 23; 24]. SGS then decides based on a derived scheduling policy [22] when there are multiple activities to be executed, but the quality of the final schedule is highly dependent on the policy it utilized. Motivated by these facts, this study proposes a novel architecture: a hybridization of approximate dynamic programming (ADP) [19; 22; 25] with the Discrete Cuckoo Search (DCS) [26; 27; 28], Estimation of Distribution Algorithm (EDA) [29; 30], and Ant Colony Optimization (ACO) [31; 32] to solve the RCPSPDC-PEO. Applications of ADP to solve RCP-SPDC or RCPSPDC-PEO have not been explored so far, especially in combination with metaheuristic approaches [11]. In addition, this proposal is driven by several compelling reasons. First, ADP's dynamic and Markov properties enables us to progressively explore solutions, markedly reducing computation time by eliminating the need to revisit previously executed jobs, a common occurrence in metaheuristic approaches. Second, ADP's policy approximation through Monte Carlo (MC) simulations is inherently parallelizable, leveraging the capabilities of multi-core CPU computers to our advantage. Third, the historical data derived from MC simulations can be efficiently integrated into powerful distribution-based metaheuristics such as EDA and ACO, allowing for the reuse of information. Fourth, unlike the often-challenging parameter management in metaheuristics hybrid systems, ADP offers the convenience of independent parameter control. Finally, the refined solutions produced by ADP are ideal for application in DCS, which significantly benefits from the high-quality solutions it employs. Thus, this study makes the following contributions:  
1. Review and proposal of 13 different heuristics for RCPSPDC-PEO;
2. Proposal of ADP scheduling based on MC and MDP for RCPSPDC-PEO, which can be implemented on multi-core CPU computers;
3. Proposal of DCS as a solution improvement to ADP for RCPSPDC-PEO; and
4. Proposal of path estimation methods based on EDA and ACO. This endeavor results in six models based on ADP and ADP with DCS. To compare their performance, we first customized the j30, j60, and j90 project instances of PSPLIB [33]. Our experimental analysis performed on these customized datasets designed to investigate the effectiveness of the six different models showed that ADP-DCS provided better results than ADP alone. Using EDA and ACO as priorities for MC simulations further improved the solutions.