**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. Liu et al. [4] designed an RCPSP model based on the time window delay from the perspective of owner-contractor interaction. Kim et al. [5] considered the delay penalty on the basis of minimizing the total project time. Cheng 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 Demeulemeester and Herroelen [7] as well as Debels and Vanhoucke [8], an activity can be interrupted after every integer unit of its activity time. Muritala Adebayo

<sup>1</sup> School of Management, Zhejiang University of Technology, Hangzhou 310023, China

Isah and Byung-Soo Kim [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, Elmaghraby [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). Varma et al. [11] discussed a multi-mode problem without the use of non-renewable resources. Zhu et al. [12] considered the MRCPSP with generalized resource constraints. Bellenguez and Emmanuel [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 largescale 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 nonengineering 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.

Relevant research by Sarker [15] demonstrated that the simultaneous optimization of project scheduling and resource supply can improve the efficiency of project scheduling and reduce the overall cost. Xie et al. [16] took the project duration and cost as the optimization objectives, considered the variable resource availability and expressed it by interval variables, and established a dual-objective optimization model of the MRCPSP under the constraint of variable resource availability. Lv et al. [17] further expanded renewable resources into flexible resources with capacity differences, and established a problem model considering capacity differences in which the capacity level affects activity duration. Schwindt and Trautmann [18] considered the time-dependent resource capacity and divided the aggregate demand of intermediate and final products into batches in the batch production mode. Shu-Shun Liu et al. [19] proposed a two-stage optimization model based on constrained programming to address the bridge maintenance scheduling problem.

Many scholars have proposed rich algorithms to solve the integrated system optimization problem of project scheduling and resource supply chains. Asta et al. [20] designed a hybrid algorithm that combines Monte Carlo and hyper-heuristic methods to solve this problem. Xie et al. [21] studied MRCPSP under the condition of uncertain activity duration and designed an approximate dynamic programming algorithm based on the rollout to solve it. Peteghem et al. [22] studied MRCPSP with resource preemption characteristics, introduced an extended serial scheduling generation scheme to improve mode selection, and designed a two-population genetic algorithm. Furthermore, many studies have proven that GA and PSO are more effective and have different advantages in solving such problems [23–27].

GA was first proposed by J. Holland in 1975. It is a random search algorithm that draws on natural selection and genetic mechanisms in the biological world and follows the principle of "survival of the fittest" [28]. Its basic idea is to imitate the natural evolution process through genetic manipulation of individuals with certain structural forms in the population, so as to generate a new population and gradually approach the optimal solution. PSO was proposed by J. Kennedy and R. C. Eberhart in 1995 [29]. It is a random search algorithm based on group cooperation, developed by simulating the foraging behavior of birds. It finds the global optimum by following the currently searched optimum.

The existing research has made important achievements in project scheduling problems and algorithm designs. However, when constructing the model, the interaction between decision makers is ignored. Secondly, the project scheduling problem from the perspective of the engineering supply chain is a multi-objective and multi-stage complex decision problem; previous studies [26,27] have shown that using the bi-level programming method can generate better results.

The innovation of this paper is that the project scheduling problem and resource supply problem are regarded as an integrated system of a dynamic game, involving interactive influences and constraints. Moreover, a two-level multi-objective programming method is adopted, which organizes the whole process of "objective—modeling—algorithm optimization—decision." A large hydropower construction project is taken as an example to prove the scientificity and feasibility of the method.

The rest of the paper is organized as follows: Section 2 gives the key problem statement of MRCPSP-MPSCP integrated system and research methods; Section 3 details the modeling method and hypothesis of establishing the two-level dynamic game model; Section 4 proposes the two-level GADS/DIWPSO hybrid algorithm to solve the established model; Section 5 gives a practical case to emphasize the practicability and effectiveness of the optimization method, and proposes forward management suggestions to related departments; and finally, Section 6 provides conclusions and future research directions.
