*5.4. Calculation Results*

The designed hybrid algorithm was run in MATLAB(R2018b) on the collected data. After running the program 30 times, an optimal solution was obtained. The total project scheduling time and cost were 48.9 and 8326.54, respectively. The upper planning MRCPSP calculation results are listed in Table 7, showing the start-end time and mode selection of each activity; the corresponding Gantt chart is shown in Figure 6. The calculation result of the MPSCP of the lower planning is shown in Table 8, which defines the transportation volume of the three resources on each transportation route in each time period. The total transportation cost and time were 1144.38 and 13.73, respectively. The convergence iteration is 146 times, and the computation time is 956.3 s.



**Figure 6.** Gantt chart of MRCPSP.

**Table 8.** Resource transportation decision.


#### *5.5. Analysis and Discussion*

#### 5.5.1. Weight Analysis

Different weight settings (i.e.,*μ*<sup>1</sup> and *μ*2) represent different combinations of preferences for decision-makers. To further understand the influence of the weight setting in upper-level planning, a sensitivity analysis was carried out, and the corresponding results are presented in Table 9. Different weight settings led to different results in the upper and lower models, which indicates that the decisions of the two levels are greatly influenced by the upper weight settings and are closely related to each other.


**Table 9.** Weight sensitivity analysis.

#### 5.5.2. Model Comparison

To verify the effectiveness of the model and the superiority of obtaining the optimal and satisfactory solution, the game model was compared with the single-layer model of the MRCPSP and MPSCP, which ignores the conflict.

To establish the corresponding single-layer model, project scheduling and resource supply were combined into a separate optimization problem. The objective function is the duration and cost of project scheduling, *Ft* and *Fc*, the decision variables are also (*v*, *x*), and the constraints include all the constraints in the upper planning. To calculate the comparative rationality of the results, the GADS proposed in the upper planning was also applied to the single-layer model and run in MATLAB(R2018b). Subsequently, the decision results are substituted into *Zc* and *Zt* to calculate the function value, and the objective function value of the single-layer model in the ideal state is obtained.

However, in practice, the lower-level planning MPSCP also has its own optimization objectives and constraints, and there are decision conflicts between the construction department and the resource supplier. Therefore, the ideal optimal solution obtained by the single-level planning model may not be a satisfactory solution for the MPSCP and will usually deviate. Therefore, the results obtained using the ideal single-layer model must be modified as follows:

In the first step, the decision result of the ideal single-layer model was used as the decision result of the upper MRCPSP. In the second step, considering the sequence of decisions, the decision results of the MRCPSP were substituted into the MPSCP to obtain the optimal transportation decision under this situation, namely, the modified solution. In the third step, the result of the transportation decision is substituted into the objective function of the MRCPSP to obtain the objective function value in this case.

In the dynamic game model, considering the hierarchical decision structure and the existence of decision conflicts, the above correction method is repeatedly used to obtain a satisfactory Stackelberg-Nash equilibrium solution. The corresponding calculation results are listed in Table 10, and Table 11 lists the comparison results of the algorithms.



Figure 7 demonstrates the iterative process of the algorithm. The results of algorithm comparison reveal that: <sup>1</sup> All four algorithms can obtain the optimal fitness in 200 iterations, and the hybrid GADS/DIWPSO algorithm has a higher fitness. <sup>2</sup> The computation time and convergence speed of the four algorithms are acceptable, among which GADS/DIWPSO hybrid algorithm is faster than GADS/PSO but slightly slower than GA/DIWPSO and GA/PSO. <sup>3</sup> The GADS/DIWPSO hybrid algorithm has better standard deviation corresponding to fitness, convergence iteration times, and computation time than

other algorithms, showing stable performance, which also reveals that the algorithm can effectively avoid infeasible solutions and reduce the probability of premature convergence. Therefore, the GADS/DIWPSO hybrid algorithm proposed in this study performs better than other algorithms in an acceptable computation time.


**Table 11.** Algorithm comparison results.

**Figure 7.** Algorithm iteration process.

#### *5.6. Management Suggestions*

Through the application of practical cases, some management suggestions are proposed for relevant departments from the perspective of the engineering supply chain:

 <sup>1</sup> When making the project schedule, the decision maker of the engineering project shall ensure that the project schedule and resource supply are within a reasonable range so that the construction schedule based on materials, equipment, and labor force can meet the expected requirements. At the same time, it must be considered that too much or too little resource supply cannot ensure the schedule advancement, because the process sequence and intermittent time in the construction process of the project determine that the actual construction progress cannot violate the internal law of the project. Once the construction progress based on the process is exceeded, quality problems are likely to occur.

 <sup>2</sup> The engineering supply chain generally involves multiple stakeholders such as owners, contractors, resource suppliers, and transportation agents. Different stakeholders are responsible for various professional tasks. These tasks are often interrelated, and if considered separately and while ignoring the conflicts of various stakeholders, they can lead to suboptimal solutions, which in turn can cause economic losses, construction delays, and other problems. Therefore, in the actual implementation of engineering projects, inherent conflicts and complex interactions must be identified and resolved.

 <sup>3</sup> In engineering practice, project managers must consider all kinds of resources, such as the labor force, materials, and equipment as a whole. The disharmony between any type of resource and other resources may cause resource redundancy or project stagnation at a certain link in an engineering project.

 <sup>4</sup> Modeling the decision-making process helps to understand the complexity and conflicts involved in the supply chain and then conducts quantitative analysis to determine a satisfactory equilibrium strategy. For example, the new Stackelberg dynamic game model proposed for the MRCPSP-MPSCP integrated system is more suitable than the corresponding single-layer model. In addition, the preference setting of the multi-objective function is important, and different preference combinations lead to different results.

#### **6. Conclusions and Future Research**

This study investigated the integration of multimode project scheduling and resource supply in an engineering supply chain. Resource constraint is not only a constraint condition of the engineering supply chain, but is often a separate optimization problem. Therefore, integrating resource supply into project scheduling is an MRCPSP-MPSCP integrated system with multi-agent decision-making characteristics and a hierarchical decision-making structure. Resolving conflicts in this integrated system helps ensure that the project runs successfully at an acceptable cost and is completed on time. On this basis, a Stackelberg dynamic game model was established, and a two-level multi-objective programming method was designed to further solve internal conflicts. Subsequently, a two-layer GADS/DIWPSO hybrid algorithm with an interactive evolution mechanism was proposed to solve the new Stackelberg model, and a satisfactory Stackelberg-Nash equilibrium solution was determined through a repeated dynamic interaction process. This provides theoretical significance for solving related problems of engineering supply chain.

In the context of the global impact of COVID-19, coordinated optimization and sustainable operation of the engineering supply chain play an important role in the recovery of the industrial economy. This study provides a theoretical basis and algorithm support for how engineering and construction departments and resource suppliers in the supply chain promote the optimization of overall benefits. For the engineering construction department, considering the limitation of resource supply, more thought is devoted to the project scheduling problem to ensure the overall operation of the project. For resource suppliers, considering the characteristics of master-slave decision-making, this study provides a reference for the formulation of a resource transportation strategy, and finally promotes mutual benefit on both sides to achieve better cooperation results.

After discussion and analysis, it can be discovered that in the engineering supply chain, the multi-period resource supply problem does have an impact on the project scheduling. Therefore, the dynamic game model for the MRCPSP-MPSCP integrated system is more realistic, and the proposed two-level multi-objective programming method and GADS/DIWPSO hybrid algorithm can solve the conflicts between stakeholders, and finally realize the Stackelberg-Nash equilibrium strategy. In conclusion, when solving similar problems, researchers should start from reality, fully consider the conflicts of interest among participants, and make reasonable assumptions. Only in this way can a better decision plan be generated.

However, there are still some limitations in this study: <sup>1</sup> he scheduling problem of multiple projects is not considered; <sup>2</sup> the mixed transportation of multi-type vehicles is not considered in terms of resource transportation; and <sup>3</sup> more participants can be considered in a large engineering supply chain, such as material manufacturers and transportation agents. These limitations will form the basis for future research.

**Author Contributions:** Conceptualization, C.F.; methodology, C.F. and S.H.; data analysis, C.F., S.H. and Y.M.; writing—original draft preparation, C.F. and S.H.; writing—review and editing, Z.L.; supervision, Z.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the National Natural Science Foundation of China (No. 71702167 and No. 72202056).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All processed data used in the study have been shown in the article.

**Acknowledgments:** The authors gratefully acknowledge the funding and support provided by the National Natural Science Foundation of China (No. 71702167 and No. 72202056) and are grateful to the editors and the reviewers for their insightful comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

