## 2 Methods and Materials
### Mathematical Basis of Fuzzy Set Theory
#### 2.2.1 Assumptions
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Operation Quality Grade** & **Operation Quality Level** & **Operation Quality Evaluation** \\ \hline Level 1 & 0.9–1.0 & Excellent \\ Level 2 & 0.8–0.9 & Good \\ Level 3 & 0.7–0.8 & Qualified \\ Level 4 & 0.6–0.7 & Basic qualification \\ Level 5 & 0.6 or less & Failed \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Operation quality level. gradually as the actual construction period evolves; when the critical period is reached, the total project cost is the lowest and when the actual time limit exceeds the critical time limit, the total project cost increases. Figure 1 depicts a schematic diagram of the relationship between project construction costs and the construction period. This paper considers the total cost of the water-management project construction period, which includes direct costs, indirect costs, penalty costs for delays, and reward costs for early completion, without taking into account the time value of capital. The direct cost of the project is the sum of the direct costs of each operation in the project; the indirect cost refers to project review and management expenses, which are assumed to have a linear relationship with the construction period; the penalty cost of delay and the reward cost of early completion are the costs incurred by using additional economic means to ensure the project's completion on time. The cost objective function for the water conservation project during the construction period is shown in Equation (4). \[C= \begin{cases}\sum\limits_{i}\sum\limits_{k=1}^{K}x_{ik}E\Big{[}\widetilde{ C}_{i}\Big{]}+T\times IC+\alpha(T-D),&T\geq D\\ \sum\limits_{i}\sum\limits_{k=1}^{K}x_{ik}E\Big{[}\widetilde{C}_{i}\Big{]}+T \times IC+\beta(T-D),&T<D\\ \end{cases}  \tag{4}\]  
\[s.t. \begin{cases}\sum\limits_{k=1}^{K}x_{ik}=1\\ E\Big{[}\widetilde{C}_{i}\Big{]}_{\min}\leq E\Big{[}\widetilde{C}_{i}\Big{]} \leq E\Big{[}\widetilde{C}_{i}\Big{]}_{\max}\\ \alpha,\beta,IC>0\\ C\leq B\\ \end{cases}  \tag{5}\]  
In the formula: \(\widetilde{C}_{i}\) represents the fuzzy direct cost of activity \(i\); \(T\) is the actual construction period of the project; \(IC\) means the indirect cost coefficient, \(\alpha\) denotes the delay penalty coefficient; \(D\) stands for the project schedule; \(\beta\) is the early completion reward coefficient; and \(B\) is the cost budget. The constraint \(E\Big{[}\widetilde{C}_{i}\Big{]}_{\min}\leq E\Big{[}\widetilde{C}_{i}\Big{]} \leq E\Big{[}\widetilde{C}_{i}\Big{]}_{\max}\) indicates that the expected direct cost of activity \(i\) is within the range of the maximum and minimum cost; Constraint \(C\leq B\) means that the total cost of the project cannot exceed the budget. Quality Objective Function  
The safety of downstream people's lives and properties is linked to the construction of water conservation projects, particularly the construction of water-retaining structures on rivers. The project's construction quality will affect not only the building's service life and benefits, but also the cost of reconstruction and maintenance. What is worse, once the accident occurs, it will result in irreparable losses to the national economy, life, and property. Figure 1: Diagram depicting the relationship between construction cost and construction period. Quality-Duration Model  
In theory, the project quality standard does not change or decrease as a result of a compressed construction period, but in practice, the project quality is usually affected by a truncated construction period. Experts assumed a linear relationship between activity quality and time in the study of the quality model, used a bell curve to simulate the duration-quality relationship, and assumed that the operation time and quality are in the form of a quadratic parabola. [PERSON] [13] and [PERSON] et al.