## 2 Methods and Materials
### Mathematical Basis of Fuzzy Set Theory
#### 2.2.1 Assumptions  
The following assumptions were made for this problem to simplify the secondary factors in the construction-period-cost-quality balance optimization problem of water conservation projects:  
1. An operation has multiple implementation modes, each with its own operation time, consumption cost, and quality achieved. Besides that, each implementation model's operational time, cost, and quality are ambiguous. 2. Apart from capital constraints, there are no constraints on other resources during project implementation. 3. The quality value in this paper represents the relative quality level, and any real number between 0 and 1 represents the quality of each operation. The overall project quality is the weighted average of the quality of each operation. The operation quality is divided into five levels in this paper, as shown in Table 1.  
2.2 Objective Function of Water Conservancy Project Duration-Cost-Quality Duration Objective Function  
The logical relationship of construction activities is determined by the construction schedule of large-scale water conservation projects. However, due to the influence of many uncertain factors, such as internal collaboration and the external environment, the operation duration the construction process is uncertain, indicating some ambiguity. The critical path in network-planning technology is the path with the longest total work duration, and the total construction period is the sum of the work durations on the critical path. As it is assumed in this paper that the critical path of project construction does not change when the execution modes of each operation differ, the objective function of the construction period of the water conservancy project is shown in Equation (3). \[ \begin{split}& T=\sum\limits_{i\in cp}\sum\limits_{k=1}^{K}x_{ ik}E\left[\widetilde{d_{i}}\right]\\ & s.t.\left\{\begin{aligned} &\sum\limits_{k=1}^{K}x_{ ik}=1\\ & E\left[\widetilde{d_{i}}\right]_{\text{min}}\leq E\left[ \widetilde{d_{i}}\right]\leq E\left[\widetilde{d_{i}}\right]_{\text{max}}\\ & E\left[\widetilde{t_{1}}\right]=0\\ & E\left[\widetilde{t_{i}}\right]+\sum\limits_{k=1}^{K}x_{ik}E \left[\widetilde{d_{i}}\right]\leq E\left[\widetilde{t_{i}}\right]\end{aligned}\right. \end{split}  \tag{3}\]  
In the formula: \(i\) stands for the operation; \(cp\) is the critical path; \(k\) denotes the execution mode; \(K\) is the number of execution modes; \(x_{ik}\) represents the 0-1 binary decision variable; \(\widetilde{d_{i}}\) signifies the fuzzy duration of the operation \(i;j\) means the tight post-operation of operation \(i;\)\(\widetilde{t_{i}}\) is the fuzzy start time of operation \(i;\)\(\widetilde{j_{i}}\) is the fuzzy start time of job \(j\). The constraint \(\sum\limits_{k=1}^{K}x_{ik}\) indicates that each operation has only one implementation mode; the constraint \(E\left[\widetilde{d_{i}}\right]_{\text{min}}\leq E\left[\widetilde{d_{i}} \right]_{\text{}}\leq E\left[\widetilde{d_{i}}\right]_{\text{max}}\) indicates that the expected duration of operation \(i\) is within the maximum and minimum timeframes; the constraint \(E\left[\widetilde{t_{i}}\right]+\sum\limits_{k=1}^{K}x_{ik}E\left[\widetilde{d _{i}}\right]\leq E\left[\widetilde{t_{j}}\right]\) means that the successor cannot be started until the predecessor is completed. Cost Objective Function  
A water conservation project has a lengthy construction period. The total cost of the project is easily affected during the entire construction process by the selection of the construction scheme, the reduction of the construction period, rework and repair, and other factors. As a result, the sole role of the contractor's cost management is to focus on finding the lowest total construction cost. According to this paper, the total project cost decreases