Fuzzy Multi-Mode Time–Cost–Quality Trade-Off Optimization in Construction Management of Hydraulic Structure Projects

Abstract

Along with the increased use of water resources, some large water conservancy projects began construction to address power supply shortages and control flooding and drainage. As investment grows, construction cycles lengthen, external environmental impacts become bigger, and civil engineering project management becomes more complex. The real aim of the hydraulic-structure engineering project model is to manage ways of delivering the project on time while maintaining reasonable quality standards and building costs, to optimize project value. We note that the trade-off between conflicting objectives in a water conservancy project in an uncertain environment is a difficult task. To simulate the relationship between a project’s construction quality and its time limit, two new piecewise functions—a double exponential function and a quadratic function—were proposed, and then a fuzzy multi-mode discrete time–cost–quality trade-off concept for water-management projects was established. This model finds the best solution to an NP-hard problem using the particle-swarm optimization algorithm (PSO). A comparison of the calculations to previous studies validates the model and its computational approach. The optimized results of a water conservation project are provided as a conceptual framework for project planning and construction timeframes.

1. Introduction

Water conservation construction projects require a large investment, a long cycle, a complex structure, many technical requirements, and more uncertain factors. As a result, stricter guidelines for managing water conservation construction periods are proposed. Water-structure project construction has three major goals: optimization of time, cost, and quality. Any loss of control over any of these goals will have far-reaching economic and social consequences. Therefore, comprehensive control and management of the three goals is an important part of the construction-period management of a water conservancy project. These three goals influence and constrain each other, and improving one must come at the expense of the others. The question of how to balance the control and management of the three objectives—in order to avoid the occurrence of out-of-control investment, time delays, quality defects, improper decision-making, and other phenomena in the construction of water conservancy projects, and to constantly improve the management level and decision-making level of water conservancy projects—is an urgent problem to be solved in the actual construction of water conservancy projects nowadays. As a result, in order to maximize project value, it is critical to carry out balanced optimization and comprehensive control of the three objectives of construction period, cost, and quality for water conservancy projects.

The optimization of the period–cost–quality balance in water-management project construction research meets the definition of multi-objective engineering project optimization. The three primary goals of project management are to optimize time constraints, cost, and quality. This paper summarizes the current state of research on the balanced optimization of time, cost, and quality in engineering projects.

Since Babu and Suresh [1] first proposed the linear programming model of time-limit–cost–quality equilibrium optimization, the problem of achieving appropriate balance among the three objectives has received a lot of attention, and a lot of research has been conducted. Research into an optimization model of the balance between engineering project duration, cost, and quality is a relevant research direction [2,3,4,5,6], as is research into optimization-model-solving algorithms [7,8], and research into algorithm synthesis [9,10].

Currently, based on the early literature on the problem of optimizing the duration–cost–quality balance of construction projects, the fuzzy objective and decision variables, as well as the situation of multiple operation implementation modes, are not considered. El-rayes and Kandil [11], Mungle et al. [12], and other researchers all contributed to the quality-model study. However, they used expert scoring to determine the quality model factors without factoring in the quantitative coupling relationship between quality, construction period, and cost. Researchers believe that a linear relationship exists between process quality and process duration [1,6], or a quadratic parabolic form, a bell-shaped curve form, a quadratic function relationship [13,14], etc. The preceding assumption tends to ignore the situation of an operation quality of zero, and the quality change rate during the initial stage of operation.

Water conservation projects are large in size, cost a lot of money, and take a long time to develop. Monitoring the project time, cost, and quality during the project’s construction is an important aspect of water conservation strategic planning. Hu [15] considered various objective functions for different stages of hydropower project construction to carry out balanced optimization of the three objectives, and established a genetic algorithm model for comprehensive optimization of hydropower project duration, cost, and quality. Wang et al. [16] investigated the risk of the water conservancy project’s construction schedule, developed a schedule-optimization model that comprehensively considered the cost–quality–completion risk, and solved the satisfying decision scheme using a genetic algorithm. The preceding studies, however, did not take into account the ambiguity of objectives and decision variables, nor the situation of multiple operational implementation modes. Zhang et al. [13] assumed that the relationship between process quality and process duration was a quadratic function, and the aforementioned assumption assessed neither the situation of an operation quality of zero nor the quality change rate in the early stages of operation. Some Chinese experts (Wang Jian (2004 in Chinese) and Yang Yaohong (2006 in Chinese) used utility functions to transform multiple objectives into single objectives to solve the construction-period–cost–quality balance optimization problem, but only provided a set of optimal schemes that could not meet the needs of practical decision-making.

Based on the above situation, this paper considers the fuzzy objective and multiple operation implementation modes caused by uncertain factors such as construction materials, weather, and construction technology; additionally, it carries out comprehensive balanced optimization of the three major objectives of water conservancy project construction period, cost and quality. A fuzzy multi-mode optimization model of construction-period–cost–quality balance is proposed for hydraulic-structure engineering construction projects by using the piecewise function to simulate the relationship between operation quality and operation duration. Meanwhile, three objectives were optimized and solved, and the results are compared with the literature by Mungle et al. [12] to verify the rationality of the model and the effectiveness of the calculation method. Furthermore, the model is applied to a concrete-face rockfill dam project to provide decision-makers with a theoretical foundation.

2. Methods and Materials

2.1. Mathematical Basis of Fuzzy Set Theory

2.1.1. Fuzzy Number

There are many types of fuzzy numbers, including triangular fuzzy numbers, trapezoidal fuzzy numbers, six-point fuzzy numbers, and others. A triangular fuzzy number was used in this paper, which is expressed as $\tilde{A}=\left(r_1,r_2,r_3\right)$. The membership function for the deterministic domain is given in Equation (1).

$${\mu }_{\tilde{A}}\left(x\right)=\left\{\begin{array}{c}0,\\ \left(x-r_1\right)/\left(r_2-r_1\right),\\ \left(r_3-x\right)/\left(r_3-r_1\right),\\ 0,\end{array}\right.\begin{array}{c}x<r_1\\ r_1\le x\le r_2\\ r_2\le x\le r_3\\ x>r_3\end{array}$$

(1)

In the formula: x is a variable on the universe of discourse U; and ${\mu }_{\tilde{A}}\left(x\right)$ indicates the degree to which $x\in U$ belongs to $\tilde{A}$, also known as ${\mu }_{\tilde{A}}\left(x\right)$, which is the membership of x to $\tilde{A}$.

2.1.2. Fuzzy-Number Processing

In general, the results of triangular fuzzy-number multiplication and division are no longer triangular fuzzy numbers. However, the results of fuzzy operations are generally regarded as triangular fuzzy numbers, and the rationality of this method is still to be proven. Furthermore, the traditional fuzzy measure of fuzzy events cannot represent decision-makers’ preferences, and the introduction of a fuzzy measure Me embedded with an optimism and pessimism index to describe decision-makers’ attitudes can effectively address the aforementioned shortcomings. Xu and Zhou [17] proposed using the expected-value operation to convert triangular fuzzy numbers into a specific value, and the triangular fuzzy number predictable-value formula is shown in Equation (2).

$$E_{Me}\left[\xi \right]=\left\{\begin{array}{c}\frac{\lambda }{2}{r}_1+\frac{r_2}{2}+\frac{1-\lambda }{2}{r}_3\begin{array}{cc},& r_3\le 0\end{array}\\ \frac{\lambda }{2}\left(r_1+r_2\right)+\frac{\lambda r_3^2-\left(1-\lambda \right)r_2^2}{2\left(r_3-r_2\right)}\begin{array}{cc},& r_2\le 0\le r_3\end{array}\\ \frac{\lambda }{2}\left(r_3+r_2\right)+\frac{\left(1-\lambda \right)r_2^2-\lambda r_1^2}{2\left(r_2-r_1\right)}\begin{array}{cc},& r_1\le 0\le r_2\end{array}\\ \frac{\left(1-\lambda \right)r_1+r_2+\lambda r_3}{2}\begin{array}{cc},& r_1\ge 0\end{array}\end{array}\right.$$

(2)

In the formula, λ is the optimism and pessimism index of decision-makers (0 ≤ λ ≤ 1); the larger λ, the more pessimistic the decision-makers are.

In this paper, the fuzzy variables in the model were all non-negative triangular fuzzy numbers, i.e., $r_1\ge 0$; therefore, $E_{Me}\left[\xi \right]=\frac{\left(1-\lambda \right)r_1+r_2+\lambda r_3}{2}$, e.g., ${\overline{d}}_i\to E\left[{\overline{d}}_i\right]=\frac{\left(1-\lambda \right)d_i{}_1+d_i{}_2+\lambda d_i{}_3}{2}$.

2.2. Fuzzy Multi-Model Construction-Period–Cost–Quality Balance Optimization Model for Hydraulic Structure Project

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. Operation quality level.

Operation Quality Grade	Operation Quality Level	Operation Quality Evaluation
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

.

2.2.2. 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).

$$\begin{array}{c}T={\displaystyle \sum _{i\in cp}{\displaystyle \sum _{k=1}^K{x}_{ik}E\left[{\tilde{d}}_i\right]}}\\ s.t.\left\{\begin{array}{c}{\displaystyle \sum _{k=1}^K{x}_{ik}=1}\\ E{\left[{\tilde{d}}_i\right]}_{\min }\le E\left[{\tilde{d}}_i\right]\le E{\left[{\tilde{d}}_i\right]}_{\max }\\ E\left[{\tilde{t}}_1\right]=0\\ E\left[{\tilde{t}}_i\right]+{\displaystyle \sum _{k=1}^K{x}_{ik}E\left[{\tilde{d}}_i\right]\le E\left[{\tilde{t}}_i\right]}\end{array}\right.\end{array}$$

(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; ${\tilde{d}}_i$ signifies the fuzzy duration of the operation i; j means the tight post-operation of operation i; ${\tilde{t}}_i$ is the fuzzy start time of operation i; ${\tilde{j}}_i$ is the fuzzy start time of job j. The constraint ${\displaystyle \sum _{k=1}^K{x}_{ik}}$ indicates that each operation has only one implementation mode; the constraint $E{\left[{\tilde{d}}_i\right]}_{\min }\le E\left[{\tilde{d}}_i\right]\le E{\left[{\tilde{d}}_i\right]}_{\max }$ indicates that the expected duration of operation i is within the maximum and minimum timeframes; the constraint $E\left[{\tilde{t}}_i\right]+{\displaystyle \sum _{k=1}^K{x}_{ik}E\left[{\tilde{d}}_i\right]\le E\left[{\tilde{t}}_j\right]}$ 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 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).

$$\begin{array}{c}C=\left\{\begin{array}{c}{\displaystyle \sum _i{\displaystyle \sum _{k=1}^K{x}_{ik}E\left[{\tilde{C}}_i\right]+T\times IC+\alpha \left(T-D\right)\begin{array}{cc},& T\ge D\end{array}}}\\ {\displaystyle \sum _i{\displaystyle \sum _{k=1}^K{x}_{ik}E\left[{\tilde{C}}_i\right]+T\times IC+\beta \left(T-D\right)\begin{array}{cc},& T<D\end{array}}}\end{array}\right.\\ s.t.\left\{\begin{array}{c}{\displaystyle \sum _{k=1}^K{x}_{ik}=1}\\ E{\left[{\tilde{C}}_i\right]}_{\min }\le E\left[{\tilde{C}}_i\right]\le E{\left[{\tilde{C}}_i\right]}_{\max }\\ \alpha ,\beta ,IC>0\\ C\le B\end{array}\right.\end{array}$$

(4)

In the formula: ${\tilde{C}}_i$ represents the fuzzy direct cost of activity i; T is the actual construction period of the project; IC means the indirect cost coefficient; α denotes the delay penalty coefficient; D stands for the project schedule; β is the early completion reward coefficient; and B is the cost budget. The constraint $E{\left[{\tilde{C}}_i\right]}_{\min }\le E\left[{\tilde{C}}_i\right]\le E{\left[{\tilde{C}}_i\right]}_{\max }$ indicates that the expected direct cost of activity i is within the range of the maximum and minimum cost; Constraint C ≤ 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.

• 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. Zhang [13] and Tran et al. [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).

$$q\left(i\right)=\left\{\begin{array}{c}1-\exp \left(-a{d}_i^b\right)\begin{array}{cc},& d_i<d_{critical}\end{array}\\ 1-\frac{1}{k}{\left(d_i-d_{critical}\right)}^2\begin{array}{cc},& d_i\ge d_{critical}\end{array}\end{array}\right.$$

(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).

$$\begin{gathered} \begin{array}{c}{Q}_1=\left\{\begin{array}{c}{\displaystyle \sum _i}{\displaystyle \sum _{k=1}^K}{x}_{ik}{\omega }_i\left(1-\exp \left(-aE{\left[{\tilde{d}}_i\right]}^b\right)\right)\begin{array}{cc},& E\left[{\tilde{d}}_i\right]\le E\left[{\tilde{d}}_{critical}\right]\end{array}\\ {\displaystyle \sum _i}{\displaystyle \sum _{k=1}^K}{x}_{ik}{\omega }_i{\left(1-\frac{1}{k}\left(E\left[{\tilde{d}}_i\right]-E\left[d_{critical}\right]\right)\right)}^2\begin{array}{cc},& E\left[{\tilde{d}}_i\right]>E\left[{\tilde{d}}_{critical}\right]\end{array}\end{array}\right.\\ \end{array} \\ s.t.\left\{\begin{array}{c}{\displaystyle \sum _{k=1}^K}{x}_{ik}=1\\ {\displaystyle \sum _{i=1}}{\omega }_i=1\\ E\left[{\tilde{d}}_i\right],E\left[{\tilde{d}}_{critical}\right]\ge 0\\ a,b,k>0\end{array}\right. \end{gathered}$$

(6)

In the formula: ω_i stands for the weight of the operation i. The constraint ${\displaystyle \sum _{i=1}{\omega }_i=1}$ 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. El-rayes and Kandil [11], Tareghian and Taheri [18], Mungle [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.

$$Q_1=aT+bC+c{T}^2+d{C}^2+eT\times C+f$$

(7)

In the formula: parameters a, b, c, d, e, and f are constant terms fitted from the original data.

SPSS (Statistical Package for the Social Sciences) statistical analysis software or other graphics and data processing software can be used to analyze the original data for project duration, cost, and quality. The least-squares method is used in SPSS software for nonlinear regression analysis of data to obtain the estimated values of various parameters in the functional relationship, and the quality equation, including construction period and cost, is then obtained. The quality objective function of the construction period of the water conservancy project can then be expressed in the form shown in Equation (8).

$$\begin{gathered} Q_2=aT+bC+c{T}^2+d{C}^2+eT\times C+f \\ T={\displaystyle \sum _i{\displaystyle \sum _{k=1}^K{x}_{ik}E\left[{\tilde{d}}_i\right]}}\begin{array}{cc},& c=\left\{\begin{array}{c}{\displaystyle \sum _i{\displaystyle \sum _{k=1}^K{x}_{ik}E\left[{\tilde{C}}_i\right]}}+T\times IC+\alpha \left(T-D\right)\begin{array}{cc},& T\ge D\end{array}\\ {\displaystyle \sum _i{\displaystyle \sum _{k=1}^K{x}_{ik}E\left[{\tilde{C}}_i\right]}}+T\times IC+\beta \left(T-D\right)\begin{array}{cc},& T<D\end{array}\end{array}\right.\end{array} \\ s.t.\left\{\begin{array}{c}{\displaystyle \sum _{k=1}^K{x}_{ik}=1}\\ E{\left[{\tilde{d}}_i\right]}_{\min }\le E\left[{\tilde{d}}_i\right]\le E{\left[{\tilde{d}}_i\right]}_{\max }\\ E\left[{\tilde{t}}_1\right]=0\\ \begin{array}{c}E\left[{\tilde{t}}_i\right]+{\displaystyle \sum _{k=1}^K{x}_{ik}E\left[{\tilde{d}}_i\right]\le E\left[{\tilde{t}}_j\right]}\\ E{\left[{\tilde{C}}_i\right]}_{\min }\le E\left[{\tilde{C}}_i\right]\le E{\left[{\tilde{C}}_i\right]}_{\max }\\ \alpha ,\beta ,IC>O\\ C\le B\end{array}\end{array}\right. \end{gathered}$$

(8)

Comprehensive Balanced Optimization Model

Under the premise of meeting engineering construction requirements, a comprehensive equilibrium-optimization model was established to achieve the shortest construction period, the lowest cost, and the highest quality. The two established equilibrium-optimization models are shown in Equations (9) and (10) due to the different quality models.

Balanced optimization Model 1:

$$\begin{gathered} \left\{\begin{array}{c}\min T\\ \min C\\ \max Q_1\end{array}\right. \\ s.t.\left\{\begin{array}{c}E{\left[{\tilde{d}}_i\right]}_{\min }\le E\left[{\tilde{d}}_i\right]\le E{\left[{\tilde{d}}_i\right]}_{\max }\\ E\left[{\tilde{t}}_i\right]+{\displaystyle \sum _{k=1}^K{x}_{ik}E\left[{\tilde{d}}_i\right]\le E\left[{\tilde{t}}_j\right]}\\ E{\left[{\tilde{C}}_i\right]}_{\min }\le E\left[{\tilde{C}}_i\right]\le E{\left[{\tilde{C}}_i\right]}_{\max }\\ \begin{array}{c}C\le B\\ {\displaystyle \sum _{k=1}^K{x}_{ik}=1}\\ {\displaystyle \sum _{i=1}{\omega }_i=1}\\ \begin{array}{c}E\left[{\tilde{t}}_1\right]=0\\ E\left[{\tilde{d}}_i\right],E\left[{\tilde{d}}_{critical}\right]\ge 0\\ \alpha ,\beta ,IC,a,b,k>0\end{array}\end{array}\end{array}\right. \end{gathered}$$

(9)

Balanced optimization Model 2:

$$\begin{gathered} \left\{\begin{array}{c}\min T\\ \min C\\ \max Q_2\end{array}\right. \\ s.t.\left\{\begin{array}{c}{\displaystyle \sum _{k=1}^K{x}_{ik}=1}\\ E{\left[{\tilde{d}}_i\right]}_{\min }\le E\left[{\tilde{d}}_i\right]\le E{\left[{\tilde{d}}_i\right]}_{\max }\\ E\left[{\tilde{t}}_1\right]=0\\ \begin{array}{c}E\left[{\tilde{t}}_i\right]+{\displaystyle \sum _{k=1}^K{x}_{ik}E\left[{\tilde{d}}_i\right]\le E\left[{\tilde{t}}_j\right]}\\ E{\left[{\tilde{C}}_i\right]}_{\min }\le E\left[{\tilde{C}}_i\right]\le E{\left[{\tilde{C}}_i\right]}_{\max }\\ \alpha ,\beta ,IC>0\\ C\le B\end{array}\end{array}\right. \end{gathered}$$

(10)

2.3. Model Solving

Fuzzy multi-mode construction period-cost-quality balance optimization of hydraulic engineering is a multi-objective optimization problem with multiple implementation modes for one operation. As result, the feasible solution space of the problem increases exponentially and becomes an NP-hard problem. Traditional algorithms based on mathematical programming techniques are inefficient, and finding optimal or suboptimal solutions is difficult. Particle-swarm optimization (PSO) has been widely used in the optimization field of construction period, cost, and quality balance due to its advantages of simple principle, strong robustness, easy implementation, and applicability to solving non-convex nonlinear problems [19,20]. PSO first initializes the population before searching n-dimensional space with an m-particles particle swarm. During the search, each particle follows the optimal particle based on its own historical best point and the best points of the other particles in the group. The optimal solution is finally found after many iterations. The ith particle in the population can be expressed as $x_i=\left(x_{i1},x_{i2},\dots ,x_{in}\right)$; the best point of its history is expressed as $p_i=\left(p_{i1},p_{i2},\dots ,p_{in}\right)$; the best point experienced by all particles in the population is expressed as $g=\left(g_1,g_2,\dots ,g_n\right)$; and the particle velocity is expressed as $v_i=\left(v_{i1},v_{i2},\dots ,v_{in}\right)$. The updated formulas of particle velocity and position are shown in Equation (11).

$$v_i^{Gen+1}=\omega v_i^{Gen}+c_1{r}_1\left(p_i-x_i^{Gen}\right)+c_2{r}_2\left(g-x_i^{Gen}\right)x_i^{Gen+1}=x_i^{Gen}+v_i^{Gen+1}$$

(11)

In the formula: Gen is the number of iterations; ω stands for the inertia weight, which determines how much the next generation particle inherits the velocity of the current particle; c₁ and c₂ denote the individual learning factor and the social learning factor, respectively. The learning factor enables the particle to have the ability to learn from the outstanding individuals in the group, to approach its historical optimal point and the historical optimal point within the group; and r₁ and r₂ represent the random numbers [0, 1].

As can be seen from the above formula, the updated particle velocity consists of three parts: the first part represents the particle’s current speed, and the use of the particle’s current state can play a role in balancing the global and local search; the second part represents the particle’s memory ability, which allows the particle to have a global search ability while avoiding falling into the local minimum point; and the third part is the cooperation.

The PSO algorithm’s calculation procedure is as follows:

Step 1: Set the velocity and position of each particle in the population to zero;

Step 2: Determine each particle’s fitness value;

Step 3: Determine the best position for the individual. Each particle’s fitness value is compared to its own best fitness value, and if the current fitness value is high, the current position is updated to the individual’s best position;

Step 4: Determine the best position for the population. Each particle’s fitness value is compared to the best fitness value experienced by all individuals, and if the current fitness value is higher, the current position is updated to the population’s best position;

Step 5: Update particle velocity and position. The velocity and position of the particles in the population are updated according to Equation (11);

Step 6: Calculate the fitness value of the updated particles, compare the selections, and record the individual optimal position and population optimal position of the particle;

Step 7: Determine whether the maximum number of iterations is met; if so, exit the loop and output the optimal solution; otherwise, return to Step 2.

Note: for details about the main model of the PSO algorithm, kindly refer to the relevant literature.

3. Results Analysis and Discussion

3.1. Example Verification

3.1.1. Introduction to Calculation Example

For verification analysis, an engineering project with 18 operations, as defined in the literature by Mungle et al. [12], was selected. The duration and direct cost of the operations were fuzzy. The project’s network plan is depicted in Figure 3, and the relevant data are depicted in

Table 2. Relevant parameters and data of each operation (for the example verification).

Operation Number	Scheme	${\tilde{\mathit{d}}}_{\mathit{i}}/\mathit{D}\mathit{a}\mathit{y}$	${\tilde{\mathit{d}}}_{\mathit{c}\mathit{r}\mathit{i}\mathit{t}\mathit{i}\mathit{c}\mathit{a}\mathit{l}}/\mathit{D}\mathit{a}\mathit{y}$	${\tilde{\mathit{C}}}_{\mathit{i}}/\mathbf{us}\mathit{\$}$	${\mathit{\omega }}_{\mathit{i}}$
1	1	(13, 14, 15)	(13, 14, 15)	(2200, 2400, 2600)	0.03
	2	(13, 15, 17)	(11, 13, 15)	(2000, 2150, 2300)
	3	(14, 16, 18)	(9, 11, 13)	(1800, 1900, 2000)
	4	(18, 21, 24)	(13, 16, 19)	(1400, 1500, 1600)
	5	(22, 24, 26)	(11, 14, 17)	(1100, 1200, 1300)
2	1	(14, 15, 16)	(13, 14, 15)	(2000, 3000, 4000)	0.05
	2	(17, 18, 19)	(14, 16, 18)	(2200, 2400, 2600)
	3	(18, 20, 22)	(13, 16, 19)	(1700, 1800, 1900)
	4	(21, 23, 25)	(15, 17, 19)	(1400, 1500, 1600)
	5	(23, 25, 27)	(13, 15, 17)	(950, 1000, 1050)
3	1	(14, 15, 16)	(13, 15, 17)	(4000, 4500, 5000)	0.08
	2	(20, 22, 24)	(17, 19, 21)	(3000, 4000, 5000)
	3	(30, 33, 36)	(21, 23, 25)	(3000, 3200, 3400)
⁝	⁝	⁝	⁝	⁝	⁝
18	1	(8, 9, 10)	(7, 8, 9)	(2600, 3000, 3400)	0.05
	2	(14, 15, 16)	(9, 11, 13)	(2100, 2400, 2700)
	3	(16, 18, 20)	(7, 10, 13)	(2100, 2200, 2300)

.

3.1.2. Parameter Selection

A comprehensive balanced optimization model (Model 1) was constructed using the method proposed in section “Quality Objective Function”, and the model’s relevant parameters were as follows: λ = 0.5 indicates that decision-makers have no preference; IC = US$50/day; α= US$200/day; β = US$120/day; D = 121days; B = US$180,000 (refer to literature by Mungle et al. [12]); and refer to El-Rayes and Kandil [11] for the weight coefficient. According to the original data, T_min = 104 days; T_max = 169 days; C_min = US$102,900; C_max = US$186,870; Q_min = 0.6193; and Q_max = 0.9876.

The original data were examined and found to conform to the law of operation-quality-decline, allowing the value range of parameter k in quality Model 1 to be determined. From the original data, d₁ = 14, q₁ = 1, d₂ = 15, q₂ = 0.9, and k_min = 10. From d_1′ = 15, q_1′ = 1, d_2′ = 33, and q_2′ = 0.62, k_max = 853 is calculated, so k ∈ [10, 853]. With increasing k, the operation quality q improves. The value of k in all operations is estimated to mostly be around 200, so herein, k is 200.

Since the original data do not show the law of the rising stage of operation quality, this paper intends to derive the value ranges of a and b from the total quality value range.

The double exponential function’s parameters are a and b, and the values of a and b affect the value of engineering quality Q₁. Figure 4 depicts the specific relationship.

After determining a and b, the corresponding Q₁ value is an interval range. The average value of the interval is used as the point to plot the relationship between the three in this section.

In the original example, 0.6193 ≤ Q ≤ 0.9876, so 0.02 ≤ a ≤ 0.5, 0.7 ≤ b ≤ 1.7. This paper considers a = 0.05 and b = 1.5.

A comprehensive equilibrium-optimization model (Model 2) was built using the method proposed in section “Quality Objective Function”. The model’s relevant parameters included λ = 0.5; IC = US$50/day; α = US$200/day; β = US$120/day; D = 121 days; and B = US$180,000. The SPSS21.0 software was used to perform nonlinear regression analysis on the original data from Mungle et al. [12], and the estimated values of the parameters were a = −3.963, b = −0.001, c = 0.054, d = e = 0, and f = 136.21.

The MATLAB program and PSO were used to solve Models 1 and 2. The parameters of PSO were: population size m = 20; maximum evolutionary generation Gen = 100; inertia weight ω was linearly reduced from 0.9 to 0.4; c₁ = c₂ = 2; v_max = 0.5; and the optimization results were obtained after running for 7.5 min. Mungle et al. [12] calculated the optimization result in 18.2 min using a fuzzy clustering genetic algorithm. Figure 5 and

Table 3. Optimization results of literature by Mungle et al. [12] and optimization results of Models 1 and 2.

Reference by Mungle et al. [12] Optimization Results	Duration/Day	Cost/US$	Quality	Model 1—Optimized Results	Duration/Day	Cost/US$	Quality	Model 2—Optimized Results	Duration/Day	Cost/US$	Quality
1	104	168,480	0.85667	1	104	164,168	0.9319	1	104	164,015	0.9245
2	104	160,860	0.83167	2	104	157,410	0.9421	2	105	158,328	0.9028
3	104	153,968	0.79600	3	105	162,180	0.9502	3	105	152,275	0.8902
4	105	143,345	0.78634	4	106	152,105	0.9487	4	106	160,478	0.9103
5	106	153,120	0.83233	5	106	154,830	0.9505	5	107	155,320	0.8911
6	106	136,858	0.75267	6	106	155,230	0.9592	6	108	146,450	0.8709
7	107	155,130	0.84534	7	108	141,518	0.9438	7	109	141,870	0.8585
8	108	147,460	0.80800	8	108	144,990	0.9644	8	110	138,520	0.8484
9	108	124,110	0.70734	9	108	146,760	0.9652	9	110	134,335	0.8436
10	109	136,975	0.77100	10	108	149,528	0.9693	10	111	152,820	0.9214
11	110	127,688	0.74067	11	109	139,760	0.9716	11	112	145,227	0.9005
12	111	158,415	0.86234	12	111	137,520	0.9741	12	113	131,310	0.8663
13	111	142,308	0.79267	13	111	136,075	0.9729	13	114	128,900	0.8480
14	112	149,030	0.83300	14	112	135,210	0.9778	14	116	122,175	0.8325
15	114	131,568	0.77434	15	113	134,360	0.9809	15	118	135,508	0.8767
16	114	116,618	0.74734	16	114	129,430	0.9772	16	119	114,070	0.8228
17	114	113,118	0.71967	17	115	125,508	0.9803	17	120	125,805	0.8404
18	116	148,870	0.84134	18	116	122,310	0.9797	18	121	117,960	0.8117
19	116	140,870	0.80500	19	117	119,430	0.9771	19	122	133,520	0.8520
20	118	123,050	0.75500	20	119	115,250	0.9726	20	122	120,933	0.8383
21	119	116,340	0.74567	21	120	113,350	0.9738	21	123	109,268	0.8003
22	121	140,815	0.79600	22	120	110,270	0.9712
23	123	126,060	0.77334	23	123	109,550	0.9603
24	123	132,000	0.77534
25	123	111,355	0.69600

show both results.

Model 1 optimization obtained 23 Pareto optimal solutions; the 3D optimized results are shown in Figure 6a. The two-dimensional diagrams of period–cost, period–quality, and cost–quality relationships are shown in Figure 6b–d.

Model 2 optimization obtained 21 Pareto optimal solutions; the 3D optimized results are shown in Figure 6a. The two-dimensional diagrams of period–cost, period–quality, and cost–quality relationships are shown in Figure 7b–d.

The optimization results in this paper are consistent with the literature by Mungle et al. [12] in terms of Pareto solution number, duration optimization interval, and cost optimization interval, as shown in Figure 5; however, the optimization quality in this paper is relatively high. This is because the quality model adopted in literature by Mungle et al. [12] is $Q=\alpha Q_{\min }+\left(1-\alpha \right)Q_{average}$, and the quality optimization interval is between the minimum quality and the average quality, that is, $0.6193\le Q\le 0.8945$.

After determining the relevant parameters, the optimization interval of the quality model in this paper is between [0.9233 and 0.9876], indicating that the quality level of the model optimization in this paper is relatively high. The PSO algorithm adopted in this paper has a fast convergence speed and short solving time, and can quickly find the high-quality interval and search for optimization. The results of the calculations show that the model in this paper can achieve a better balance of the three objectives of optimizing construction period, cost, and quality.

As can be seen from Table 3, the optimized construction period interval in the literature by Mungle et al. [12] is [104–123], the cost (US$) interval is [111,355–168,480], and the quality interval is [0.69600~0.86234]; the optimized period interval of Model 1 is [104~123], the cost (US$) interval is [109,550~164,168], and the quality interval is [0.9319~0.9809]. In terms of the number of Pareto solutions, the optimized interval of the construction period, and the optimized interval of cost, the optimized results of Model 1 are consistent with the literature by Mungle et al. [12]; however, the optimized quality level of Model 1 is higher. This is because the quality and quantity mode adopted in the literature by Mungle et al. [12] is $Q=\alpha Q_{\min }+\left(1-\alpha \right)Q_{average}$, and the optimized quality interval is between the minimum quality and the average quality, i.e., 0.6193 ≤ Q ≤ 0.8945. In Model 1, the optimized quality interval objective function after determining the relevant parameters is between [0.9233 and 0.9876], so the optimized quality level of Model 1 is relatively high.

Still from Table 3, it is found that the optimized period interval of Model 2 is [104–123], the optimized cost (US$) interval is [109,268–164,015], and the optimized quality interval is [0.8003–0.9245]. The optimized results of Model 2 are consistent with the literature by Mungle et al. [12] on Pareto solution number, optimized period interval, and optimized cost interval. Model 2’s optimized quality level, on the other hand, is greater than 0.8, which is a good level, and higher than most optimizations in the literature by Mungle et al. [12]. This is because Mungle et al. [12] ’s quality model in the literature is conservative.

The optimized quality does not exceed the project’s average quality. Model 2’s quality objective function reflects the relationship between construction period and cost, which is consistent with engineering practice. The optimized construction-period–cost–quality balance can meet the project’s actual needs.

When the optimized results of Models 1 and 2 are compared, the optimized quality of Model 2 is lower than that of Model 1, but its optimized cost is higher than that of Model 1. When the optimized quality results of Model 1 and 2 are compared, the optimized quality differentiation of Model 1 is smaller, while that of Model 2 is larger, making it easier for decision-makers to make subjective decisions.

The calculation results show that Models 1 and 2 can achieve a better balance of the three objectives of the construction period, cost, and quality. Decision-makers can select an appropriate model based on the project’s actual situation to comprehensively balance and optimize the project’s construction period, cost, and quality; formulate the project resource-allocation scheme and project-scheduling scheme; and effectively control the three project objectives.

3.1.3. Engineering Applications

Since Model 2 considers the coupling relationship between the three objectives of optimizing construction-period–cost–quality, which is relatively close to the actual situation of the project, Model 2 was applied to a single project concerning a concrete-face rockfill dam in a water conservancy project for application analysis. The system design of the concrete-face rockfill dam project is shown in Figure 8.

This single project included 11 operations, and the relevant parameters and data can be found in

Table 4. Relevant parameters and data of each operation (for the engineering application of the concrete-face rockfill dam project).

Operation Number	Scheme	${\tilde{\mathit{d}}}_{\mathit{i}}/\mathit{m}\mathit{o}\mathit{i}\mathit{s}$	${\tilde{\mathit{C}}}_{\mathit{i}}/\mathit{u}\mathit{s}\mathit{\$}$	$\tilde{\mathit{q}}$
1	1	(1, 2, 3)	(7,575,757; 8,333,333; 9,090,909)	(0.90, 0.92, 0.94)
	2	(2, 3, 4)	(6,666,666; 7,272,727; 7,878,787)	(0.88, 0.89, 0.90)
2	1	(3, 4, 5)	(9,242,424; 10,151,515; 11,060,606)	(0.83, 0.86, 0.89)
	2	(4, 5, 6)	(8,636,363; 9,393,939; 10,151,515)	(0.74, 0.76, 0.78)
3	1	(2, 3, 4)	(8,484,848; 8,939,393; 9,393,939)	(0.70, 0.72, 0.74)
	2	(3, 4, 5)	(7,878,787; 8,333,333; 8,787,878)	(0.81, 0.82, 0.83)
	3	(4, 5, 6)	(7,272,727; 7,727,272; 8,181,818)	(0.78, 0.80, 0.82)
4	1	(2, 3, 4)	(3,484,848; 3,787,878; 4,090,909)	(0.85, 0.86, 0.87)
	2	(3, 4, 5)	(2,878,787; 3,181,818; 3,484,848)	(0.80, 0.83, 0.86)
5	1	(4, 5, 6)	(6,363,636; 6,818,181; 7,272,727)	(0.92, 0.94, 0.96)
	2	(5, 6, 7)	(5,606,060; 6,060,606; 6,515,151)	(0.87, 0.89, 0.91)
6	1	(1, 2, 3)	(5,454,545; 5,757,575; 6,060,606)	(0.75, 0.77, 0.79)
	2	(2, 3, 4)	(4,924,242; 5,303,030; 5,681,818)	(0.96, 0.97, 0.98)
	3	(3, 4, 5)	(4,621,212; 4,848,484; 5,075,757)	(0.91, 0.92, 0.93)
7	1	(7, 8, 9)	(8,181,818; 8,484,848; 8,787,878)	(0.78, 0.79, 0.80)
	2	(9, 10, 11)	(7,500,000; 7,727,272; 7,954,545)	(0.76, 0.77, 0.78)
	3	(10, 12, 14)	(6,818,181; 6,969,696; 7,121,212)	(0.70, 0.73, 0.76)
8	1	(1, 2, 3)	(5,303,030; 5,454,545; 5,606,060)	(0.86, 0.87, 0.88)
	2	(2, 3, 4)	(4,848,484; 5,000,000; 5,151,515)	(0.84, 0.85, 0.86)
	3	(3, 4, 5)	(3,636,363; 4,545,454; 5,454,545)	(0.80, 0.82, 0.84)
9	1	(1, 2, 3)	(2,348,484; 2,878,787; 3,409,090)	(0.97, 0.98, 0.99)
	2	(2, 3, 4)	(1,742,424; 2,121,212; 2,500,000)	(0.94, 0.96, 0.98)
10	1	(1, 2, 3)	(3,106,060; 3,787,878; 4,469,696)	(0.88, 0.90, 0.92)
	2	(2, 3, 4)	(2,954,545; 3,333,333; 3,863,636)	(0.86, 0.88, 0.90)
11	1	(1, 2, 3)	(757,575; 909,090; 1,060,606)	(0.91, 0.94, 0.97)
	2	(2, 3, 4)	(454,545; 606,060; 757,575)	(0.79, 0.83, 0.87)

.

The concrete-face rockfill dam project’s construction-period-cost-quality comprehensive balanced optimization model was constructed by using the second method proposed in section “Quality Objective Function”. The relevant parameters in the model were: λ = 0.5 indicates that the decision-makers have no preference; the penalty coefficient α for the delay was set at US$1,212,121/month; the reward coefficient β for early completion was set as US$606,060/month; the indirect cost coefficient IC was set at US$757,575/month; the planned construction period D was 42 months; and the cost budget B was set at US$90,909,090. The project’s quality had to be above good. The SPSS21.0 software was used for nonlinear regression data analysis, and the estimated values of the parameters were a = −0.783, b = 0.001, c = −0.146, d = e = 0, and f = 91.917. The model was solved through the MATLAB program and PSO. The parameters of PSO were: the population size m = 20; maximum evolutionary generation Gen = 100; inertia weight ω decreases linearly from 0.9 to 0.4; c₁ = c₂ = 2; v_max = 0.5. The solution results are shown in

Table 5. Optimization result data of concrete-face rockfill dam project.

Solution Number	Duration/Month	Cost/US$	Quality
1	43	82,575,757	0.8032
2	41	83,787,878	0.8145
3	40	84,545,454	0.8280
4	38	85,606,060	0.8316
5	37	86,666,666	0.8457
6	36	87,727,272	0.8621
7	34	88,484,848	0.8658
8	32	89,696,969	0.8796

.

The concrete-face rockfill dam project was made up of 11 operations, with an average of 2.4 execution modes for each operation. Therefore, the feasible solution space was 2.4¹¹, that is, there were 15217 combination schemes with different values of construction-period–cost–quality. PSO was used to find eight non-dominant solutions in a large search space, allowing decision-makers to choose from a narrower range of options.

The optimized construction period of the concrete-face rockfill dam project, as shown in Table 5, is between [32–43] months, which meets the control requirements of the planned construction period of 42 months. The optimized cost was between [US$82,469,400–89,581,440], which is less than the budgeted cost of US$90,909,090, and meets the cost control requirements. The optimized quality level is between [0.8032–0.8796]. Referring to the operation quality standard in Table 1, the optimized quality in this example reaches a good level, which meets the requirements of the engineering quality standards.

As a result, the optimized construction-period–cost–quality of the concrete face rockfill dam project meets the project’s actual requirements. The scheme can be selected by the decision-makers based on the specific requirements of the project construction, to serve as a reference basis for the decision-makers to carry out project planning and scheduling.

4. Conclusions

This paper aimed to investigate time–cost–quality trade-off optimization in water conservancy project construction management, and introduced a model for the time–cost–quality trade-off problem. The following are the paper’s main research insights: the piecewise function between quality and time, as well as the fitting function between and among quality, time, and cost, were developed for water conservancy projects; the fuzzy multi-mode discrete time–cost–quality trade-off models for water project construction were proposed based on the quality models; the models can solve the time–cost–quality trade-off problem in a fuzzy, uncertain environment with multiple operating modes for one operation; compared with the previous quality model, the suggested models overcome the subjectivity of expert scoring; they consider the situation of an operation quality of zero and the problem of quality change rate at the initial stage of operation, and are more in line with the objective reality of the water conservancy project construction process. The particle-swarm algorithm was used to efficiently solve the equilibrium-optimization model. The comparison of the optimization results to the relevant literature reveals that the proposed model’s optimization quality is high, confirming the model’s rationality and the calculation method’s validity. Finally, model 2 presented in this paper was applied to a water-management project as a reference for decision-makers to optimize and control the target, which plays an important guiding role in engineering construction.