# A Project Scheduling Game Equilibrium Problem Based on Dynamic Resource Supply
## 1 Introduction  
Managing work in the form of projects has become a common practice to improve work efficiency. Currently, approximately 20% of the world's economic activity is in the form of projects, generating an annual economic value of roughly $12 trillion [1]. Project scheduling refers to the scientific and reasonable arrangement of the beginning and execution times of each activity in a project in order to achieve the established goal [2]. The Resource-Constrained Project Scheduling Problem (RCPSP) is a form of planning based on constraining the resources required by project activities. The Classic RCPSP scheduling decision must satisfy the temporal and resource constraints, and its solution is a scheduling plan that optimizes the management objective under these constraints. [3]. Many scholars have studied the RCPSP, and extension problems have been developed. [PERSON] et al. [4] designed an RCPSP model based on the time window delay from the perspective of owner-contractor interaction. [PERSON] et al. [5] considered the delay penalty on the basis of minimizing the total project time. [PERSON] et al. [6] considered the problem of night shifts in construction projects and minimized the project duration, cost, and utilization of night shifts while meeting the constraints of operational logic and labor availability. In the study of [PERSON] and [PERSON] [7] as well as [PERSON] and [PERSON] [8], an activity can be interrupted after every integer unit of its activity time. [PERSON] and [PERSON] [9] presented a stochastic multiskilled resource scheduling model for RCPSP, which considers the impacts of risk and uncertainty on activity durations. The standard RCPSP assumes that an activity can only be executed in one mode, with a fixed duration and resource requirements. On this basis, [PERSON] [10] proposed a new concept; in practice, management departments can flexibly arrange appropriate execution modes for project activities to achieve corresponding goals, and each mode has different durations and resource demands, i.e., the Multi-mode Resource-Constrained Project Scheduling Problem (MRCPSP). [PERSON] et al. [11] discussed a multi-mode problem without the use of non-renewable resources. [PERSON] et al. [12] considered the MRCPSP with generalized resource constraints. [PERSON] and [PERSON] [13] discussed a special case: in an MRCPSP, each activity requires specific skills, while resources are employees with fixed skills, and employees must be selected according to their skills when arranging activities. MRCPSP is a critical issue in engineering supply chain management, especially in large-scale engineering construction projects. The resource supply is complex and changeable, and the resource transportation policy is updated according to the different ordering schedules of project scheduling [14]. At this point, a Multi-Period Supply Chain Problem (MPSCP) arises, directly affecting both the cost and schedule of the project. If the project schedule is made without considering the constraints of upstream resource supply capacity, the supply delay or interruption of suppliers will delay the construction period and increase both the project cost and risk, among other factors. Similarly, resource supply driven by non-engineering schedule planning will lead to a lower resource utilization rate and a higher inventory cost. In this case, resource constraint is not only a constraint condition of MRCPSP, but also an optimization problem closely related to MRCPSP with the characteristics of a dynamic game. However, in most studies, project scheduling and resource supply are considered as two independent optimization problems, ignoring the interaction and conflict between them, possibly leading to a suboptimal solution for resource supply and project delay. Therefore, it is more realistic to consider project scheduling and resource supply as an integrated system for dynamic game optimization.