## 1 Introduction  
In project management, the precise timing for initiating project jobs within the constraints of available resources plays a crucial role in optimizing outcomes across sectors such as fleet management, supply chains, and service delivery. One such problem in this area is the Resource Constraint Project Scheduling with Discounted Cash flows (RCPSPDC), where the objective is to maximize the sum of discounted cash flows from the activities. RCPSPDC is a crucial problem to consider when dealing with long-term projects with significant financial implications [1]. These projects usually involve two parties: the contractor, who is responsible for executing the project, and the client, who outsources the project to the contractor and provides the financing [2]. The details of the financial transactions are usually negotiated when the project's contract is signed [3]. For long-term projects, there are several payment schemes that the two parties can agree upon. Available payment options include Payment at Activities' Completion Times (PAC) [4, 5, 6], ProgressPayments (PP) [7], and Payment at Event Occurrences (PEO) [5; 6]. Although PAC ensures a consistent cash flow for the contractor, it may pose logistical challenges when processing invoices for each individual activity. In PP, payments are made at regular intervals, and the final payment is disbursed upon completion of the project. PP may not be desirable for the client, as it provides little incentive for the contractor to complete the project on time. In contrast, PEO can be seen as an irregular variant of PP. Under PEO, a specific number of project tasks are treated as events, and payments are made to the contractor as soon as these events are completed. During contract negotiations, the number of events and the corresponding payment amounts can be mutually agreed upon to ensure a Pareto-optimal outcome for both the client and the contractor [8; 9; 10; 11]. For seamless project execution, the contractor must maintain a consistent cash inflow of funds to effectively manage expenses such as payroll, equipment costs, and expenditures on raw materials and resources. This study focuses on Resource Constraint Project Scheduling with Discounted Cash flows and Payment at Event Occurrences (RCPSPDC-PEO) from the contractor's perspective. In this scenario, the contractor aims to maximize the Net Present Value (NPV) of the project under the condition that \(m+1\) payments are made throughout the project's duration. These payments are triggered as soon as \(m\) predefined activities are completed, with the final settlement upon project completion. For instance, construction of a new residential complex is one of the practical examples of RCPSPDC-PEO. The contractor is faced with the difficult challenge of sequencing and scheduling a multitude of interdependent and dependent tasks under various resource constraints, ranging from the foundational work to the final utility installations. The allocated budget is usually not paid all at once to prevent unforeseen risks for the client; on the other hand, withholding payment at the completion of the project would also jeopardize the success of the project. Payments are therefore made according to the PEO principle. To optimize the financial outcome of this venture, the contractor aims to maximize the project's NPV, a key variable that determines the project's profitability considering the time value of money. Under the PEO scheme, the company must skillfully manage its limited resources against a backdrop of financial constraints, with cash inflows tied to the achievement of specific project milestones. This requires a judicious planning strategy that aligns task completion with the stipulated payment events to ensure a steady cash flow and, ultimately, the financial success of the project. RCPSPDC is proven to be NP-hard [12]. Both exact (for smaller problems) and approximate solutions (metaheuristics) have been explored in research [13] that could also tackle RCPSPDC-PEO.