*2.1. Problem Description*

Project scheduling has always been considered the core of engineering supply chains, as the construction and operation of the supply chain are driven by the development of the project schedule. The engineering construction department first defines the resource demand of each demand point in each time period by forming a project schedule plan; next, the resource supplier attempts to meet the resource demand. However, as the resource supplier is also a decision-making subject with its own constraints, it optimizes its cost and time goals by formulating a resource transportation strategy and sends this information back to the engineering construction department, thus affecting the formulation of the project's schedule. Conflict and cooperation coexist in the engineering supply chain. The engineering construction department has higher decision-making power (i.e., the leader), whereas the resource supplier is subordinate (i.e., the follower). This "leader–follower" behavior is, in its essence, a Stackelberg game, with the characteristics of multi-periodicity in practice. Therefore, project scheduling and resource supply comprise an inseparable integrated system, which is the game analysis and dynamic coordination problem of the integrated system of "project scheduling–resource supply" from the perspective of the engineering supply chain. The successful operation of this system helps reduce project costs, shorten construction periods, and improve project quality and resource utilization.

The research object of this project is a large hydropower construction project located in southeast China. A concrete double-curvature arch dam is the main project, with many construction activities with priority relationships and shared resources; each activity has several alternative modes, and each mode has a certain duration and resource demand. To meet the requirements of shared resources, it is necessary to specify the ordering time and quantity in each time period when making the project scheduling scheme, and the resource supplier further formulates the resource transportation strategy. These constitute the dynamic game decision-making system of the MRCPSP-MPSCP integrated system, and the structural model is shown in Figure 1.

**Figure 1.** Structural model of the MRCPSP-MPSCP integrated system.

#### *2.2. Research Methodology*

In this paper, we adopt a two-level multi-objective mode of programming which informs the whole process of "objective—modeling—algorithm—optimization—decision." According to the characteristics of the dynamic game of this problem in the engineering supply chain, we adopt a two-level modeling method to express the interaction between MRCPSP and MPSCP. To determine the optimal equilibrium strategy of the model, a twolayer hybrid algorithm, composed of a GA with double strings and an improved PSO, is proposed. Considering the existence of many uncertainties in the engineering supply chain, for example, the project activity time is a typical uncertain variable; Bidot et al. [30] considered a project schedule with a random activity duration. In addition, factors such as weather conditions, labor efficiency, and transportation environment make the decisionmaking process more complicated. Therefore, random variables are used in this study to describe various variables in an uncertain environment. Finally, the applicability and effectiveness of the proposed optimization method are evaluated through a case of a large hydropower construction project.

#### **3. Model Establishment**

To properly express the dynamic game characteristics of the MRCPSP-MPSCP integrated system, a two-level multi-objective programming model is established, which includes the upper and lower models.

#### *3.1. Symbols and Assumptions*

3.1.1. Indicators


*i*: Activity mode index, *i* ∈ *I* = {1, 2, . . . , *I*} *s*: Demand point index, *s* ∈ *S* = {1, 2, . . . , *S*}

### 3.1.2. Parameters Related to Project Scheduling

*B*: Total available budget

*D*: Project planning cycle

*ICs*: Inventory capacity at the demand point *s*

*Pj*: Set of predecessors of activity *j*

*cjm*: Direct cost of activity *j* in mode *m*

*djm*: Operation time of activity *j* in *m* mode

*rjmk*: The demand of *m* mode of activity *j* for resource *k* in each time period

*rk*: Maximum supply capacity of resource *k* in each time period

*c*0: Overhead cost per time period

*EFj*: The earliest completion time of activity *j*

*LFj*: The latest completion time of activity *j*

*Rs*: The set of activities for which demand point *s* is responsible

*Pck*: Unit purchase cost of resource *k*

*Ock*: Each order cost of resource *k*

*Ick*: Storage cost of resource *k* in each time period

3.1.3. Parameters Related to Resource Supply

*T*(*t*): Delivery date of time period *t*

*Rk*: Maximum amount of resource *k* transported each time

*Pik*: Supply capacity of resource *k* at supply point *i*

*cisk*: Unit transportation cost of resource *k* on the transportation path (*i*,*s*)

*tisk*: Unit transportation time of resource *k* on the transportation path (*i*,*s*)

3.1.4. Decision Variables

*visk*(*t*): The allocation of resource *k* on transportation path (*i*,*s*) in time period *t*

*xjmt* = 1, If activity *j* executes mode *m* in time period *t* 0, Otherwise , represents the mode

selection of activity *j*

*zkt* = 1, If resource *k* transported at the beginning of time period *t* 0, Otherwise , represents

whether resource *k* is transported during time period *t*
