## 2 Methods and Materials
### Mathematical Basis of Fuzzy Set Theory
#### 2.2.1 Assumptions
[14] believed that the relationship between process quality and process duration was a quadratic function, and the above assumption considered neither the situation of an operation quality of zero nor the quality change rate at the initial stage of operation. Because of the complexities of water conservancy engineering-process construction, the following assumptions were made about the process's operation quality: when the operation has not begun, the working quality is zero; in the early stages of operation, the quality of operation is greatly improved with a massive input of materials, equipment, personnel, and other resources; when the duration of operation reaches the optimal time, the quality level of extended operation time increases less; and when the operation duration reaches the critical time, the quality level will decrease if the operation duration is prolonged. Figure 2 depicts the relationship curve between operation quality and operation duration. Fitting the piecewise function curve with the double exponential function and quadratic function yields the function relation, as shown in Equation (5). \[q(i)= \begin{cases}\begin{array}{c}1-\exp\left(-ad_{i}^{\Phi}\right),\quad d_ {i}<d_{critical}\\ 1-\frac{1}{k}\left(d_{i}-d_{critical}\right)^{2},\quad d_{i}\geq d_{critical} \end{array}\end{cases}  \tag{5}\]  
In the formula: \(d_{i}\) is the duration of operation \(i\); \(d_{critical}\) is the critical time of operation \(i\); \(a\), \(b\), and \(k\) are normal terms. The quality of the entire project was recorded as the weighted average of the quality of each operation, and the quality objective function of the water conservancy project construction period is shown in Equation (6). \[Q_{1}= \begin{cases}\begin{array}{c}\sum\limits_{i}\sum\limits_{k=1}^{K}x_{ ik}\omega_{i}\Big{(}1-\exp\left(-aE\left[\widetilde{d}_{i}\right]^{k}\right) \Big{)},\end{array}&E\left[\widetilde{d}_{i}\right]\leq E\Big{[}\widetilde{d} _{critical}\Big{]}\\ \sum\limits_{i}\sum\limits_{k=1}^{K}x_{ik}\omega_{i}\Big{(}1-\frac{1}{k} \Big{(}E\left[\widetilde{d}_{i}\right]-E\left[d_{critical}\right]\Big{)}\Big{)} ^{2},\end{array}&E\left[\widetilde{d}_{i}\right]>E\Big{[}\widetilde{d}_{critical }\Big{]}\end{cases}  \tag{6}\]  
Figure 2: The relationship between operation quality and operation duration. In the formula: \(\omega_{i}\) stands for the weight of the operation \(i\). The constraint \(\sum\limits_{i=1}\omega_{i}=1\) means that the sum of the weights of each operation is 1. 2. Quality-Period-Cost Model  
The preceding section discusses the relationship between quality and construction period, makes assumptions about the relationship based on previous research results, and establishes the quality-construction-period relationship model. However, project quality is closely related to cost as well as construction duration. [PERSON] and [PERSON] [11], [PERSON] and [PERSON] [18], [PERSON] [12], and others used expert scoring to determine the factors in the quality model without taking into account the quantitative relationship between construction period, quality, and cost. This section focuses on the relationship between quality, duration, and cost. Engineering quality quantification has long been a hot research topic in the industry. It is difficult to quantify the relationship between quality, duration, and cost. This research examines the quantification of engineering quality through the processing of engineering data. When obtaining relevant information and data for new projects is difficult, existing data from similar projects of the same scale can be used for analysis, and if necessary, appropriate corrections can be made based on the actual situation of the new project. The functional relationship between project quality, construction period, and the cost is first established. \[Q_{1}=aT+bC+cT^{2}+dC^{2}+eT\times C+f \tag{7}\]  
In the formula: parameters \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\) are constant terms fitted from the original data.