Project Portfolio Selection Considering the Fuzzy Chance Constraint of Water Environmental Restoration

Abstract

The water environment restoration project portfolio (WERP) selection is discussed in this paper. By complying with the analysis of the project’s multidimensional property and operation mode, this paper develops the chance constraint and the management constraint of the WERP from the perspectives of public service and enterprise operation. In addition, the multi-objective mixed integer linear programming model is constructed by combining the expectation method and the fuzzy chance constraint programming method. The results demonstrate that: (1) Our proposed method successfully circumvents the occurrence of local objective optimization within a specific confidence interval, thereby achieving a balance between economic and water environment restoration objectives; (2) including fuzzy chance constraints in our proposed method significantly diminishes the risk of exceeding the WERP capacity, thereby ensuring the effectiveness of water environment restoration by adopting a market-based approach. However, further examination of the impact of various sub-projects in WERP is necessary, along with the integration of novel evolutionary algorithms to enhance the efficiency of our model.

1. Introduction

With the introduction of the concept of the Yangtze River Economic Belt in recent years, the Chinese government has steadily emphasized the ecological environmental conservation of the Yangtze River. The Yangtze River’s self-purification capacity and resource and environment carrying capacity have been overexploited as a result of the Yangtze River’s prior development pattern, which was extensive and less environmentally friendly. The water environment restoration project portfolio (WERP) is crucial for coordinating the development of the Yangtze River Economic Belt and the protection of the Yangtze River. According to estimates, the Chinese government’s policies of WERP will entail nearly thirty billion dollars of investment [1]. The Chinese government’s significant commitment to the water restoration domain creates a vast market for companies. With substantial funding allocated to WERP initiatives, private companies have opportunities to contribute expertise, resources, and innovative solutions to address China’s water environmental challenges and promote sustainable development.

The core of the WERP is typically a project portfolio selection issue, a concept first introduced by Markowz [2]. Traditionally, the primary focus of project portfolio selection is to identify an optimal portfolio of projects from a limited set of alternatives, aiming to maximize the enterprise’s commercial objectives under various resource constraints. Despite the substantial attention given to project portfolio research in commercial, R&D, IT, and other domains, the selection issue of the WERP, as a distinctive project portfolio type, has not yet been extensively explored and investigated.

The WERP is typically comprehensive project portfolio composed of multiple individual projects [3]. Current WERP initiatives encompass a combination of quasi-operational projects (e.g., sewage treatment facilities), nonoperational projects (e.g., lake dredging), and partially operational endeavors (e.g., municipal pipe networks built using BLT). The intricate composition of the WERP sets it apart from traditional project portfolio issues. Moreover, the adoption of the franchise mode in the WERP introduces novel revenue models and operational challenges that differ from those in traditional commercial project portfolios.

The core challenges for WERP’s participants lie in implementing a project portfolio selection that aligns with the commercial objectives, while simultaneously meeting the government’s water restoration requirements [4]. This novel project portfolio issue has achieved limited attention in current research. This novel project portfolio challenge necessitates innovative approaches and decision-making frameworks to ensure that WERP participants can optimize their commercial objectives while effectively contributing to the restoration of water environments as mandated by government regulations [5]. However, this aspect still represents a research gap that requires further investigation and exploration.

The aforementioned studies have demonstrated the WERP is a special project portfolio issue that differs significantly from conventional project portfolios. Thereby, the motivation for this paper is as follows:

• Due to the limited research attention and the absence of a quantitative model, there is a research gap in the WERP selection issue. This paper aims to improve decision-making accuracy for WERP participants by introducing a specialized model. The proposed model provides comprehensive guidance for evaluating projects, prioritizing investments, and allocating resources, thereby enabling participants to optimize the selection of WERP.
• Considering WERP participants face the intricate task of achieving commercial objectives while meeting government water restoration requirements, this paper aims to propose an innovative decision-making approach for selecting project portfolios that effectively address environmental challenges while aligning with commercial goals, realizing a Pareto optimality between profitability and compliance with environmental regulations.

Subsequently, we establish a novel decision-making approach in this paper by integrating fuzzy mathematical and chance constraint techniques into a multi-objective mixed-integer linear programming (MIP) model. Furthermore, the main contributions of this paper can be summarized as follows:

• A novel WERP selection model is developed to address the challenges of selecting a project portfolio in water environment restoration projects. The developed model considers both commercial objectives and government water restoration requirements, effectively balancing profitability and regulatory obligations in the WERP selection issue.
• The proposed method integrates fuzzy mathematical and chance constraint techniques into a MIP model, yielding a robust decision-making approach. By employing fuzzy mathematical methods, the model handles the uncertainty and imprecision inherent in WERP selection, allowing for more robust selection decisions. The integration of chance constraint techniques enhances the reliability of the selected portfolio for providing public services.
• Case studies using publicly available data were conducted to validate the efficacy of the developed WERP selection method. The experimental results confirm the efficiency and flexibility of the proposed approach.

The rest of this paper is organized as follows. Section 2 elaborates on the work related to the proposed method. In Section 3, the construction process of the proposed method is described in detail. In Section 4, the performance of the proposed method is further verified and discussed using an actual WERP case. Finally, the conclusion and future research directions are discussed in Section 5.

2. Related Works

2.1. Project Portfolio Selection

Current project portfolio selection research predominantly focuses on commercial, R&D, and IT projects, with multiple studies exploring strategy orientation, project interaction, and associated risks. Our work builds upon these studies, as the WERP encounters similar constraints such as resource limitations and corporate capabilities. Li et al. [6] studied interruptible projects under resource constraints and constructed a dual-objective project portfolio selection model based on net present value and utility. Tao et al. [7] constructed a robust and adjustable project selection model from the perspective of uncertain revenue interaction and resource interaction. Yang et al. [8] analyzed the impact of organizational task relationships on technical risk events and then constructed a diffusion model of technical risk for the R&D project portfolio. Bai et al. [9] combined QFD theory with portfolio problems, defined the synergy function and fit ratio, and subsequently proposed a portfolio selection model based on strategic proximity. Park et al. [10] divided operational investment projects into three types (sparse, sustainable, and stable projects) and constructed a balanced portfolio model of these three types of projects. However, the existing literature has yet to consider the scenario where the portfolio encompasses a franchise period and bears the responsibility of providing public services. To address this research gap, our study focuses on analyzing the operational approach and unique revenue model of the WERP, aiming to bridge the gap between traditional project portfolio selection research and the specialized needs of WERP.

2.2. Modeling Method in Project Portfolio Selection

Current project portfolio selection modeling methods commonly rely on linear programming (MIP), integer programming (IP), simulation modeling, heuristics, and metaheuristic algorithms. Bai et al. [11] used a fuzzy Bayesian network to identify key risk factors among projects with associated relationships and constructed a project portfolio model aiming at minimizing associated risks. Chen et al. [12] regarded R&D personnel as a kind of renewable resource and used a discrete Markov chain to describe the personnel updating process. In addition, a project portfolio model with multi-skilled personnel was constructed with the goal of talent cultivation, considering R&D cycles and R&D costs in R&D projects. Li et al. [13] proposed a new type of uncertain organizational risk measurement method, built a new uncertain portfolio optimization model by introducing relevant structural variables, and transformed it into an equivalent dual-standard optimization model for solving problems. Samaneh et al. [14] considered the problem of portfolio selection with flexible project duration under renewable resource constraints and constructed a MIP model. The optimal project portfolio was obtained by solving the model. While existing methods have demonstrated acceptable performance in solving project portfolio issues, they are primarily developed for commercial projects where parameters are typically well-defined due to shorter revenue periods [15,16]. In contrast, our proposed method focuses on addressing the specific uncertainties associated with WERP caused by the long franchise period. By explicitly considering and incorporating this uncertainty into the modeling process, our approach aims to provide more accurate and robust decision-making support.

2.3. Summary of the Related Works

We employ a table to depict visually the differences between the WERP and existing related work; see

Table 1. The related studies in project portfolio selection.

	Type of Objective		Contain Uncertainty Parameter	Project Overall Period (Year)	Considering Operation Period	Modeling Method	Essence of Solution		Project Type
	Single	Multi					Exact	Approx
Our study		✓	✓	5–~20	✓	MIP	✓		Water environment restoration
Li et al. [6]	✓			0–1		MIP		✓	IT
Tao et al. [7]	✓		✓	0–1		MIP	✓		R&D
Yang et al. [8]	✓			0–1		Simulation		✓	Industrial
Bai et al. [9]		✓		1–2	✓	QFD 1		✓	R&D
Park et al. [10]	✓			0–1		Sparse optimize		✓	Commercial
Bai et al. [11]	✓		✓	1–2		Bayesian network		✓	Commercial
Chen et al. [12]		✓		0–1		NSGAII 2		✓	R&D
Li et al. [13]		✓	✓	0–1		IP 3	✓		Finance
Samaneh et al. [14]		✓	✓	0~1.5		MIP		✓	Civil
Feng et al. [16]	✓			0~1		IP 3		✓	Commercial
Rabbani et al. [17]		✓	✓	1–2		MIP		✓	Commercial
Jafarzadeh et al. [18]	✓			0–1		IP 3	✓		Finance
Tofighian et al. [19]		✓		0–1		AOC 4		✓	IT
Huang et al. [20]	✓		✓	0–1	✓	Mean-variance model		✓	R&D
Sefair et al. [21]	✓			0–2		MIP		✓	Industry
Dixit et al. [22]	✓		✓	1–3		MIP	✓		Food
Wu et al. [23]	✓		✓	0–1		NSGAII 2		✓	Energy

Notes: ¹ QFD: quality function deployment; ² NSGAII: non-dominated sorting genetic algorithm; ³ linear programming model; ⁴ AOC: ant colony optimization algorithm.

.

To summarize, none of the available literature addresses project portfolio issues with franchise durations. This study develops a confidence-based multi-objective fuzzy project portfolio selection model for the WERP, including the following characteristics: (1) From the perspective of enterprise operation, multiple types of resource restrictions as well as phased investment plans and organizational constraints have all been taken into account; from the perspective of public services, WERP’s stage objective constraints and overall objective constraints are constructed; (2) the dual objectives of investment cost within the construction period and franchise return and income are considered, and these dual objectives are aggregated and balanced by the satisfaction model; (3) the objective function and constraint conditions with fuzzy parameters are dealt with using the expected value method and the chance constraint approach, respectively.

3. The Proposed Method

3.1. Problem Description

The WERP is a cluster of projects that includes river regulation, greening, roads, pipe networks, and sewage treatment facilities. Compared to commercial projects, the franchise period is a crucial characteristic of WERP and typically extends for 5~20 years. During the franchise period, the corporation that takes over the WERP is responsible for the construction, operation, and maintenance of the water restoration service. The franchise period enables the corporation to recover investments and generate profits while allowing the government to leverage specialist corporations’ expertise and resources for more efficient public service delivery. Moreover, the corporation undertaking the WERP should consider not only the investment returns but also emphasize the water restoration capability of the WERP during the franchise period as another significant objective. Based on the discussion above, the essential considerations in WERP selection can be described as follows:

First, the WERP portfolio selection issue requires additional attention to strike a balance between construction costs and franchise period income. The objective function in most existing portfolio models is net revenue, although this objective is not entirely applicable to WERP. Since the construction investment of WERP is difficult to recover in the short term, a one-sided decision on net income easily leads to a long period of capital occupation and a cash flow crisis. Therefore, it is inevitably necessary to consider comprehensively the dual objectives of construction investment cost and income in franchise periods to solve the problem of WERP portfolio selection.

Second, it is necessary to balance the operating income of the project portfolio and public service provision. The majority of WERP services fall somewhere in the middle between public and private services. As a consequence, WERP portfolio section should begin with a focus on company operations and public services. At the company level, the project portfolio should fulfill a company’s business needs while also being constrained by the company’s available resources and organizational capacity. At the level of public services, the selected portfolio must adhere to established public expectations. In other words, the water restoration services offered by WERP must fulfill the public sector’s minimum requirements, which are generally phased in. This necessitates that the WERP portfolio’s services not only adhere to the overall goal but also take into account the objectives of each stage.

Furthermore, since the payback period of project investment is up to 20 years, it is difficult to predict the method and amount of government financial subsidies for such a lengthy term. Various uncontrollable factors (e.g., nature and society), which are highly unpredictable, have an impact on the project’s performance revenue and WERP’s capacity during a franchise period. Therefore, in this paper, WERP’s capacity, operating performance income, government subsidies, and other parameters are expressed by fuzzy numbers.

The problems to be solved in this paper are as follows. Based on the above issues, the perspectives of enterprise management, multiple resource constraints, investment plan constraints, and organizational capacity constraints are considered. From the perspective of public service, stage objective constraints and overall objective constraints of WERP are considered. By balancing the dual objectives of construction investment cost and franchise return and given the fuzziness of parameters in the long payback period, the selected project portfolio scheme and its execution strategy are finally determined to achieve the optimal decision.

The relevant variables and their meanings are shown in

Table 2. Variables and description.

Variables	Description	Variables	Description
$H^{\left(t\right)}$	Maximum number of simultaneous projects for phase $t$	$O\tilde{P}{C}_j$	Operation maintenance cost of the project $j$
$C{I}^{\left(t\right)}$	Investment in stages during the construction period	$\tilde{S}{I}_j$	Government subsidy proceeds for the project $j$
$C{P}_j$	Construction reserve fee for project $j$	$O\tilde{P}{I}_j$	Operating performance gains of project $j$
$C{C}_j$	Construction investment cost of project $j$	$\gamma$	Discount rate
$y_{jt}$	If $r_{jt}\ne 0$, $y_{jt}=1$, otherwise $y_{jt}=0$	$q_{jt}$	If ${\displaystyle \sum _{t\in T}{r}_{jt}}=1$, $q_{jt}=1$, otherwise $q_{jt}=0$
$z_{jt}$	Operation progress of project $j$ at phase $t$	$r_{jt}$	The construction progress of project $j$ in $t$ phase $r_{jt}\in \left[0,1\right]$
${\widehat{k}}_{jn}$	Class $n$ non-renewable resources for project $j$	${\widehat{K}}_n$	The total amount of resources cannot be updated
$k_{jm}$	Class $m$ renewable resources for project $j$	$K_m^{\left(t\right)}$	Increase of renewable resources in $t$ phase
${\tilde{\theta }}_{js}$	Fuzzy value of WERP capacity in type $s$ of Project $j$	${\xi }_s^{\left(t\right)}$	The minimum value of type $restoration$ requirements in phase $t$
$T$	Set of project portfolio construction cycle, $t\in T$	$P$	Collection of portfolio franchise cycles, $p\in P$
$J$	Collection of alternative projects, $j\in J$	$S$	Set of WERP capacity types, $s\in S$
$M$	Collection of renewable resource types, $m\in M$	$N$	Collection of non-renewable resource types, $n\in N$
$x_j$	Decision variable, representing whether project J is selected over the entire project cycle, $x_j=1$, otherwise $x_j=0$

.

3.2. Model Formulation and Constraints

The model primarily draws upon the actual business content of the WERP. The key to the success of WERP lies in achieving a balance between the operating income and the provision of public services. Therefore, we have developed the model while considering these two crucial aspects. Moreover, in light of the non-linear uncertainty present in the parameters of the WERP, we have innovatively utilized a hybrid modeling method to describe them, thereby enhancing the robustness of the results obtained from our proposed approach.

Specifically, inspired by Wu et al. [24], our objective for the water environment restoration project (WERP) is to maximize operational income while minimizing construction costs. However, the WERP’s heavy reliance on government subsidies, coupled with the significantly lengthy franchise period, introduces uncertainty in operational income. This distinctive characteristic sets the WERP apart from previous business project portfolios. To tackle this issue, we introduce fuzzy mathematical methods to describe this uncertainty. Moreover, the WERP’s primary task is to deliver high-quality water environment restoration services, which have received limited research attention in previous work. To achieve this objective, we adopt a chance constraint modeling approach to ensure the robustness of providing qualified services. In summary, our hybrid modeling approach is better suited for the unique nature of the WERP, allowing the WERP to maximize overall revenue while ensuring high-quality water environment restoration. Our proposed modeling process involves three stages; within the initial stage, the WERP model [M1] and its detailed description are as follows.

$$\left\{\begin{array}{l}\max NP{V}_1={\displaystyle \sum _{p\in P}{\displaystyle \sum _{j\in J}\left(E\left[O\tilde{P}{I}_j\right]+E\left[\tilde{S}{I}_j\right]\right)z_{jt}}{\left(1+\gamma \right)}^{-p}}-{\displaystyle \sum _{p\in P}{\displaystyle \sum _{j\in J}E\left[O\tilde{P}{C}_j\right]z_{jt}}{\left(1+\gamma \right)}^{-p}}\\ \min NP{V}_2={\displaystyle \sum _{t\in T}{\displaystyle \sum _{j\in J}C{C}_j{r}_{jt}{\left(1+\gamma \right)}^{1-t}}}+{\displaystyle \sum _{t\in T}{\displaystyle \sum _{j\in J}C{P}_j{x}_j{\left(1+\gamma \right)}^{1-t}}}\\ s.t. {\displaystyle \sum _{t\in T}{\displaystyle \sum _{j\in J}{r}_{jt}=x_j}}, \forall j,t \end{array}\right.$$

(1)

$${\displaystyle \sum _{j\in J}{y}_{jt}}\le H^{\left(t\right)}, \forall j,t$$

(2)

$$\begin{array}{l}C{I}^{\left(1\right)}-{\displaystyle \sum _{j\in J}C{C}_j{r}_{j1}}-{\displaystyle \sum _{j\in J}C{P}_j{x}_j}\ge 0\\ {\displaystyle \sum _{t\in \tau }C{I}^{\left(t\right)}{\left(1+\gamma \right)}^{1-t}}-{\displaystyle \sum _{t\in T}{\displaystyle \sum _{j\in J}\left(C{C}_j{r}_{j\left(t-1\right)}+C{P}_j{x}_j+O\tilde{P}{C}_j{z}_{j\left(t-1\right)}\right){\left(1+\gamma \right)}^{1-t}}}\\ +{\displaystyle \sum _{t\in \tau }{\displaystyle \sum _{j\in J}\left(O\tilde{P}{I}_j+\tilde{S}{I}_j\right)z_{jt}}{\left(1+\gamma \right)}^{-t}}\ge 0, \forall j,t\end{array}$$

(3)

$${\displaystyle \sum _{j\in J}{\widehat{k}}_{jn}}{r}_{jt}\le {\widehat{K}}_n, \forall j,t,n$$

(4)

$${\displaystyle \sum _{t\in T}{K}_m^{\left(t\right)}\ge {\displaystyle \sum _{t\in T}{\displaystyle \sum _{j\in J}{k}_{jm}}{r}_{jt}}}, \forall j,t,m$$

(5)

$$C_r\left\{{\displaystyle \sum _{t\in T}{\displaystyle \sum _{j\in J}{q}_{jt}{\tilde{\theta }}_{js}}}\ge {\xi }_s^{\left(t\right)}\right\}, \forall j,t$$

(6)

$$x_j,y_{jt},r_{jt},q_{jt},z_{jt}\in \left\{0,1\right\}, \forall j,t$$

(7)

The objective of the WERP model [M1] is to maximize income over the franchise period while minimizing total investment during the construction phase. To achieve this, we establish two objective functions, detailed as follows:

Objective function 1 aims to maximize income during the franchise period. This objective differs from previous studies due to the WERP’s inability to recover its investment in the short term. Consequently, we devise a novel objective to evaluate the revenue generated by the WERP, in comparison with Kannimuthu et al. [25]. First, the government provides subsidies $\tilde{S}{I}_j$ to support the project’s operations throughout the franchise period. This is followed by the operating performance income $O\tilde{P}{I}_j$, which is directly related to the performance appraisal results, including the availability service fee, sewage treatment fee, water price revenue sharing, and performance appraisal payment. The last component is the operation maintenance cost $O\tilde{P}{C}_j$, which is the necessary expenditure to maintain the project’s operations.

Objective function 2 aims to minimize the total investment during the construction period, similar to previous studies [26], and comprises two components. Specifically, the first component is the project construction investment $C{C}_j$, which is the project price required by the construction project. The other is the reserve $C{P}_j$ at the initial stage of project construction, which represents the organizational cost to be paid during the initial preparation period of project construction.

The construction and franchise phases of the WERP involve long investment periods, significant investments, and various uncertain factors. Therefore, WERP enterprises often opt for rolling the WERP with smaller-scale investment cycles, aiming to control investment risks effectively. This is similar to the development strategy of some large commercial projects [27], however, the key distinction lies in the fact that the WERP has both stage-specific and overall public service objectives to fulfill. Subsequently, we provide an elaborate exposition on the construction of the constraints, referring to the aforementioned similarities and differences.

To highlight our contributions, we begin by introducing the novel constraints proposed in this paper.

In contrast to previous work [28], we employ a more refined stage investment constraint by innovatively incorporating a recoverable investment strategy, which is more in line with the actual investment situation of WERP. Specifically, we introduce a novel variable, the recoverable working capital $R{C}^{\left(t\right)}$, utilizing a recursive method to construct the stage investment constraint, the recursive process of the refined stage investment constraint can be detailed as follows:

$$C{I}^{\left(1\right)}-{\displaystyle \sum _{j\in J}C{C}_j{r}_{j1}}-{\displaystyle \sum _{j\in J}C{P}_j{x}_j}\ge 0$$

(8)

$$\begin{array}{l}R{C}^{\left(t\right)}=C{I}^{\left(t\right)}+\left[R{C}^{\left(t-1\right)}-{\displaystyle \sum _{j\in J}\left(C{C}_j{r}_{j\left(t-1\right)}+C{P}_j{x}_j+O\tilde{P}{C}_j{z}_{j\left(t-1\right)}\right)}\right]\left(1+\gamma \right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{j\in J}\left(O\tilde{P}{I}_j+\tilde{S}{I}_j\right)z_{j\left(t-1\right)}}\left(1+\gamma \right)\end{array}$$

(9)

$$R{C}^{\left(t\right)}-{\displaystyle \sum _{j\in J}\left(C{C}_j{r}_{j\left(t-1\right)}+C{P}_j{x}_j+O\tilde{P}{C}_j{z}_{j\left(t-1\right)}\right)}\ge 0,\forall t,j$$

(10)

where $C{I}^{\left(1\right)}$ denotes the initial construction investment, calculated by net present value, and Equation (8) represents the relationship between $C{I}^{\left(1\right)}$, $C{C}_j$, and $C{P}_j$ in the initial investment stage. Equation (9) indicates that $R{C}^{\left(t\right)}$ in stage $t$ is derived from the investments and income of the preceding investment stage (in stage $t-1$). Equation (11) denotes the allocation of a sufficient fee to meet both the construction expenses for the projects to be built and the operating expenses for the projects already completed at each stage. By combining Equations (8)–(10), the stage investment constraint Equation (11) can be constructed through recursive reasoning:

$$\begin{array}{l}{\displaystyle \sum _{t\in \tau }C{I}^{\left(t\right)}{\left(1+\gamma \right)}^{1-t}}-{\displaystyle \sum _{t\in T}{\displaystyle \sum _{j\in J}\left(C{C}_j{r}_{j\left(t-1\right)}+C{P}_j{x}_j+O\tilde{P}{C}_j{z}_{j\left(t-1\right)}\right){\left(1+\gamma \right)}^{1-t}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{t\in \tau }{\displaystyle \sum _{j\in J}\left(O\tilde{P}{I}_j+\tilde{S}{I}_j\right)z_{jt}}{\left(1+\gamma \right)}^{-t}}\ge 0\end{array}$$

(11)

Considering that the WERP has both stage-specific and overall public service objectives to fulfill, we propose a novel chance constraint by introducing new variables ${\tilde{\theta }}_{js}$ and ${\xi }_s^{\left(t\right)}$, to evaluate the water restoration capability of WERP, depicted as follows:

$$C_r\left\{{\displaystyle \sum _{t\in T}{\displaystyle \sum _{j\in J}{q}_{jt}{\tilde{\theta }}_{js}}}\ge {\xi }_s^{\left(t\right)}\right\}, \forall j,t$$

(12)

where Equation (12) indicates that the WERP should meet the water environment restoration requirement in each investment stage within a specified confidence level ${\xi }_s^{\left(t\right)}$. As the detailed derivation of this constraint necessitates a fuzzy form, we provide a comprehensive discussion of it in the subsequent section.

The establishment of other constraints extensively relies on previous studies, acting as fundamental components in a typical project portfolio issue. However, these constraints are not the main focus of this research, which can be briefly described as follows:

Equation (1) is the project completion constraint, indicating that if a project is selected, it must be completed within the entire investment period [8,12]; Equation (2) is the constraint on the number of executable projects in the entire investment period, and each period is determined by the organizational capacity of the enterprise [10]; Equation (4) is the nonrenewable resource constraint [13]; Equation (5) is the renewable resource constraint; Equation (7) is the 0–1 constraint of the corresponding decision variables [16].

3.3. Model Transformation

In this study, we employ fuzzy mathematical methods to handle effectively the inherent uncertainty and vagueness prevalent in the selection process of the water environment restoration project (WERP). The rationale for utilizing these methods can be elaborated and explained as follows:

• Fuzzy mathematics provides a robust framework for modeling and reasoning under conditions of uncertainty. It enables the WERP model to address incomplete or ambiguous information, such as fluctuating government subsidies, which hold significance for the WERP due to its longer franchise period, potentially leading to increased uncertainty compared with other commercial projects.
• Fuzzy mathematics improves decision-making consistency by accommodating diverse data types and gradual transitions among alternatives in project portfolio issues. It can effectively capture the interdependencies and nuances among different water restoration capabilities in the WERP, including river regulation, road construction, pipe networks, and sewage treatment facilities.

Employing fuzzy sets allows the proposed model to capture and represent the imprecise nature of these variables, thereby facilitating more robust modelling and decision-making processes. Specifically, we set two kinds of fuzzy parameters: one is the parameters of the franchise period, including government subsidy income $\tilde{S}{I}_j$, operating performance income $O\tilde{P}{I}_j$ and operating maintenance cost $O\tilde{P}{C}_j$; the other is WERP’s water restoration capacity ${\tilde{\theta }}_{js}$. The ambiguity of the two kinds of parameters is not consistent. Comparatively, $\tilde{S}{I}_j$, $O\tilde{P}{I}_j$, and $O\tilde{P}{C}_j$ have higher volatility and ambiguity in a long franchise period. However, ${\tilde{\theta }}_{js}$ can give relatively accurate predictions based on historical experience and expert data. Based on this, this paper utilizes the expected value method and the chance constraint programming method to transform the variables into clear and equivalent forms to realize calculations [29].

Assuming that fuzzy variables $O\tilde{P}{I}_j$, $\tilde{S}{I}_j$, $O\tilde{P}{C}_j$, and ${\tilde{\theta }}_{js}$ are trapezoidal fuzzy variables, their trapezoidal structural elements can be written as: $O\tilde{P}{I}_j=\left[{\eta }_1^j,{\eta }_2^j,{\eta }_3^j,{\eta }_4^j\right]$, $\tilde{S}{I}_j=\left[{\beta }_1^{js},{\beta }_2^{js},{\beta }_3^{js},{\beta }_4^{js}\right]$, $O\tilde{P}{C}_j=\left[{\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j,{\alpha }_4^j\right]$, ${\tilde{\theta }}_{js}=\left[{\theta }_1^{js},{\theta }_2^{js},{\theta }_3^{js},{\theta }_4^{js}\right]$, and ${\beta }_1^j\le {\beta }_2^j\le {\beta }_3^j\le {\beta }_4^j$, ${\eta }_1^j\le {\eta }_2^j\le {\eta }_3^j\le {\eta }_4^j$, ${\alpha }_1^j\le {\alpha }_2^j\le {\alpha }_3^j\le {\alpha }_4^j$, ${\theta }_1^{js}\le {\theta }_2^{js}\le {\theta }_3^{js}\le {\theta }_4^{js}$.

According to the literature [30,31,32], the fuzzy variable $O\tilde{P}{I}_j$ can be expressed as follows based on the expected value distribution:

$$E\left[O\tilde{P}{I}_j\right]={\displaystyle {\int }_0^{+\infty }{C}_r\left\{O\tilde{P}{I}_j\ge \beta \right\}d}\beta -{\displaystyle {\int }_{-\infty }^0{C}_r\left\{O\tilde{P}{I}_j\le \beta \right\}d}\beta$$

(13)

Thus, the fuzzy expectation of $O\tilde{P}{I}_j$ is $E\left[O\tilde{P}{I}_j\right]=\left({\eta }_1^j+{\eta }_2^j+{\eta }_3^j+{\eta }_4^j\right)/4$, $E\left[\tilde{S}{I}_j\right]$, and $E\left[O\tilde{P}{C}_j\right]$ can be obtained in the same way.

Considering that ${\tilde{\theta }}_{js}$ is a fuzzy variable and its nonnegative linear combination ${\displaystyle \sum _{t\in T}{\displaystyle \sum _{j\in J}{q}_{jt}{\tilde{\theta }}_{js}}}$ is still a fuzzy variable, denoted as $f\left[{\tilde{\theta }}_{js}\right]$ for convenience of expression, the credibility distribution of the water environment restoration constraints of the WERP portfolio can be described as follows:

$$Cr\left\{f\left[{\tilde{\theta }}_{js}\right]\ge {\xi }_s^{\left(t\right)}\right\}=\frac{1}{2}\left(Pos\left\{f\left[{\tilde{\theta }}_{js}\right]\ge {\xi }_s^{\left(t\right)}\right\}+1-Pos\left\{f\left[{\tilde{\theta }}_{js}\right]<{\xi }_s^{\left(t\right)}\right\}\right)$$

(14)

For given confidence levels ${\delta }_{js}\in \left(0.5,1\right]$, the above equation can be expanded as follows (see Appendix A for the detailed process):

$$Cr\left\{f\left[{\tilde{\theta }}_{js}\right]\ge {\xi }_s^{\left(t\right)}\right\}\ge {\delta }_{js}\iff {\displaystyle \sum _{t\in T}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{s\in S}\left\{\left(2{\delta }_{js}-1\right)q_{jt}{\theta }_4^{js}+2\left(1-{\delta }_{js}\right)q_{jt}{\theta }_3^{js}\right\}}}}\ge {\xi }_s^{\left(t\right)}$$

(15)

In summary, the multi-objective fuzzy chance-constrained programming model [M1] can be transformed into a clear equivalent form, denoted as model [M2], which is expressed as follows:

$$\left\{\begin{array}{l}\max NP{V}_1={\displaystyle \sum _{p\in P}{\displaystyle \sum _{j\in J}\left[\left({\beta }_1^j+{\beta }_2^j+{\beta }_3^j+{\beta }_4^j\right)/4+\left({\eta }_1^j+{\eta }_2^j+{\eta }_3^j+{\eta }_4^j\right)/4\right]z_{jt}}{\left(1+\gamma \right)}^{-p}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\displaystyle \sum _{p\in P}{\displaystyle \sum _{j\in J}\left[\left({\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j+{\alpha }_4^j\right)/4\right]z_{jt}}{\left(1+\gamma \right)}^{-p}}\\ \max NP{V}_2={\displaystyle \sum _{t\in T}{\displaystyle \sum _{j\in J}C{C}_j{r}_{jt}{\left(1+\gamma \right)}^{1-t}}}+{\displaystyle \sum _{t\in T}{\displaystyle \sum _{j\in J}C{P}_j{x}_j{\left(1+\gamma \right)}^{1-t}}}\\ s.t.\hspace{1em}\hspace{1em}\left(1\right)~\left(5\right),\hspace{1em}\hspace{1em}\left(7\right),\hspace{1em}\hspace{1em}\left(14\right)\end{array}\right.$$

3.4. Compromise Satisfaction Model of Dual Objectives

The clear equivalence model [M2] is a multi-objective MIP model. The general solution to the multi-objective optimization problem is to transform the multi-objective optimization problem into a single-objective optimization problem. The main methods are the ideal point method, the weighted average method, the grade weight method, and the risk preference coefficient method. These methods are simple and effective, but it is difficult to describe managers’ decision-making psychology in detail. To this end, this paper adopts the Zimmerman model [33,34]. To solve the problem that Zimmerman’s model is prone to fall into when local goal optimization and goals conflict, a psychological attention index is added, which can measure and gradually adjust the satisfaction level of each objective function according to the preferences of decision makers and ensure that the algorithm only searches for satisfactory solutions, thereby effectively reducing the search time required for understanding. Therefore, this paper rewrites the [M2] model into the [M3] model through the improved Zimmerman model to further solve the WERP portfolio selection problem. The specific solution process is as follows:

For each target, find the optimal solution of the single target, recorded as the upper bound target value and the upper bound solution, denoted as $NP{V}_i\left(NP{V}_i^{\delta ,U},x_i^{\delta ,U}\right),i=1,2$. Then, the corresponding lower bound target value can be obtained by the following method:

$$NP{V}_i^{\delta ,L}=NPV\left(x_j^{\delta ,U}\right),NP{V}_j^{\delta ,L}=NPV\left(x_i^{\delta ,U}\right),i\ne j$$

(16)

Take the upper and lower bound solutions of the target value as the left and right boundaries of the satisfactory solution interval. Then, the proximity ${\mu }_i\left(x\right)$ of the target value corresponding to the feasible solution and the upper bound target value can be expressed as:

$${\mu }_i\left(x\right)=\left(NP{V}_i\left(x\right)-NP{V}_i^{\delta ,L}\right)/\left(NP{V}_i^{\delta ,U}-NP{V}_i^{\delta ,L}\right)$$

(17)

Use the improved Zimmerman model to aggregate the upper bound proximity of feasible solutions with dual objectives and obtain the compromise satisfaction model based on global objective optimization [M3]:

$$\left\{\begin{array}{l}\max \nu ={\displaystyle \sum _{i\in I}{\omega }_i{\left(1-{\phi }_i\right)}^{{\lambda }_i}}\\ s.t.\hspace{1em} 1-{\mu }_i\left(x\right)\le {\phi }_i, \forall i\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(18\right)\\ \hspace{1em}\hspace{1em} x\in W,{\omega }_i,{\lambda }_i\in \left[0,1\right], \forall i\hspace{1em}\hspace{1em}\hspace{1em}\left(19\right)\end{array}\right.$$

where $W$ is the feasible region solution constituted by constraints of model [M2]; ${\lambda }_i$ stands for the psychological concern index; ${\mu }_i\left(x\right)$ represents the closeness between the target value corresponding to the feasible solution and the upper bound target value; and $\nu$ represents the compromise satisfaction. ${\phi }_i$ is the decision maker’s dissatisfaction with the target value; ${\lambda }_i$ is the psychological concern index, and the size of ${\lambda }_i$ can reflect the attentiveness of decision makers. The closer ${\lambda }_i$ is to 1, the more attention is given to the target value; otherwise, higher attention is given.

Subsequently, our work outlines the process by which the decision maker (DM) can select the optimal WERP using our proposed method:

Step 1: Set the initial value of the parameters: ${\phi }_i=0.5$ and ${\xi }_s^{}=0.55$.

Step 2: Based on the set parameters, optimize model [M3] to obtain the corresponding optimal values ($NP{V}_i$ and $NP{V}_i^{\delta ,U}$) in [M2].

Step 3: DM adjusts the RATIO based on the distance from $NP{V}_i$ to $NP{V}_i^{\delta ,U}$.

Step 4: DM adjusts ${\xi }_s^{}$.

Step 5: Repeat steps 2 to 4 until the DM obtains the most satisfactory $NP{V}_i$ and the corresponding WERP.

It is essential to highlight that the overall process heavily relies on the decision preferences of the DM concerning construction investment, operating revenue, and water environment services. Our method respects the DM’s autonomy in decision making to the utmost extent, aiming to serve as an efficient tool to facilitate the decision-making process rather than replacing the DM. Furthermore, our proposed method offers flexibility and provides additional reference information to assist in decision making, which is further elaborated in Section 4.

4. Application of Proposed Method

Taking as an example the ecological projects invested by the Y Ecology and Environment Co., Ltd., a large state-owned enterprise, the enterprise plans to select a WERP from 12 water environment restoration projects (P1~P12) with a construction investment period of 30 months, taking the first quarter as the unit cycle. After investigation, the determined parameters and fuzzy parameters of the project are listed in

Table 3. Project deterministic parameters.

	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12
$CC$	72.25	80.75	93.5	72.25	156	155	150	157	213	175	169	218
$CP$	1.50	1.68	1.94	1.50	5.05	3.22	4.99	3.22	5.46	3.63	3.51	4.46
$T$/year	0.49	0.49	0.46	0.45	0.74	0.735	0.735	0.725	0.965	0.985	0.98	1.2
$P$/year	15	15	15	15	15	15	20	20	20	15	15	15

and Appendix A 

Table A1. Project deterministic parameters.

	A1	A2	A3	A4	A5	$\mathit{S}\mathit{I}$	$\mathit{O}\mathit{P}\mathit{I}$	$\mathit{O}\mathit{P}\mathit{C}$
P1	3.19, 3.99, 4.98, 5.23	0.42, 0.52, 0.65, 0.68	0.63, 0.79, 0.99, 1.04	0.16, 0.20, 0.25, 0.26	1.96, 2.45, 3.06, 3.22	9.82, 15.70, 23.56, 29.40	34.32, 54.91, 82.36, 102.96	3.57, 5.70, 8.56, 10.70
P2	4.01, 5.02, 6.27, 6.58	0.08, 0.10, 0.13, 0.14	0.99, 1.24, 1.55, 1.63	0.20, 0.25, 0.31, 0.33	3.04, 3.80, 4.75, 4.99	15.52, 24.83, 37.25, 46.56	33.32, 53.30, 79.96, 99.95	3.55, 5.68, 8.52, 10.65
P3	1.28, 1.60, 2.00, 2.10	0.26, 0.32, 0.41, 0.43	3.41, 4.26, 5.32, 5.59	0.33, 0.42, 0.52, 0.55	0.32, 0.40, 0.50, 0.52	15.42, 24.67, 37.01, 46.26	40.70, 65.13, 97.69, 122.11	3.57, 5.70, 8.56, 10.70
P4	2.49, 3.11, 3.89, 4.09	0.27, 0.33, 0.42, 0.44	3.82, 4.77, 5.97, 6.27	0.06, 0.07, 0.09, 0.10	1.51, 1.89, 2.36, 2.48	9.47, 15.16, 22.74, 28.42	34.64, 55.42, 83.14, 103.92	3.55, 5.67, 8.51, 10.64
P5	6.18, 7.72, 9.65, 10.13	0.90, 1.13, 1.41, 1.48	1.42, 1.77, 2.22, 2.33	0.36, 0.45, 0.57, 0.59	3.86, 4.82, 6.03, 6.33	23.68, 37.89, 56.84, 71.05	65.95, 105.51, 158.27, 197.84	3.04, 4.86, 7.30, 9.12
P6	8.48, 10.60, 13.25, 13.91	0.19, 0.24, 0.30, 0.32	2.15, 2.69, 3.36, 3.53	0.44, 0.55, 0.68, 0.72	6.73, 8.41, 10.51, 11.04	21.21, 33.94, 50.91, 63.63	68.28, 109.25, 163.88, 204.84	3.80, 6.08, 9.12, 11.40
P7	2.87, 3.59, 4.49, 4.71	0.53, 0.66, 0.83, 0.87	7.11, 8.89, 11.11, 11.66	0.76, 0.95, 1.19, 1.25	0.69, 0.86, 1.08, 1.14	20.32, 32.51, 48.76, 60.95	66.54, 106.47, 159.70, 199.62	3.51, 5.61, 8.41, 10.52
P8	5.28, 6.60, 8.25, 8.67	0.60, 0.76, 0.94, 0.99	8.34, 10.42, 13.03, 13.68	0.11, 0.14, 0.17, 0.18	3.36, 4.20, 5.25, 5.51	29.52, 47.24, 70.85, 88.57	59.87, 95.79, 143.69, 179.61	3.85, 6.16, 9.24, 11.55
P9	9.07, 11.34, 14.18, 14.88	1.32, 1.65, 2.06, 2.16	2.18, 2.73, 3.41, 3.58	0.59, 0.74, 0.92, 0.97	5.93, 7.41, 9.26, 9.72	39.39, 63.03, 94.54, 118.18	80.28, 128.45, 192.68, 240.85	4.40, 7.03, 10.55, 13.19
P10	4.41, 5.52, 6.90, 7.24	0.78, 0.98, 1.22, 1.28	11.38, 14.23, 17.79, 18.67	1.19, 1.49, 1.87, 1.96	1.12, 1.40, 1.74, 1.83	39.11, 62.58, 93.87, 117.34	63.60, 101.76, 152.65, 190.81	4.36, 6.97, 10.45, 13.07
P11	8.16, 10.21, 12.76, 13.39	0.93, 1.16, 1.45, 1.52	11.93, 14.91, 18.64, 19.57	0.17, 0.21, 0.27, 0.28	5.02, 6.27, 7.84, 8.23	37.22, 59.54, 89.32, 111.65	61.16, 97.86, 146.80, 183.49	4.35, 6.96, 10.44, 13.05
P12	16.79, 20.9, 26.23, 27.55	0.37, 0.47, 0.59, 0.61	4.41, 5.51, 6.89, 7.23	0.94, 1.18, 1.47, 1.54	12.90, 16.13, 20.16, 21.17	50.32, 80.51, 120.76, 150.9	74.82, 119.71, 179.56, 224.45	4.39, 7.02, 10.52, 13.16

, respectively. The main environmental WERP capacity indicators required by the environmental protection department are as follows (

Table 4. WERP capacity indicators.

	T1	T2	T3	T4	T5	T6	T7	T8	T9	T10
A1	/	/	/	13.5	13.5	13.5	17.25	36.75	36.75	45.0
A2	/	/	/	1.8	1.8	1.8	1.8	3.2	3.2	4.4
A3	/	/	/	15.0	25.0	25.0	39.0	52.0	52.0	56.0
A4	/	/	/	0.5	1.0	1.0	1.2	2.5	2.5	3.0
A5	/	/	/	/	8.25	8.25	9.0	24.0	24.0	31.5

Notes: “/” denotes the absence of any requirements in this stage.

): ① A1: urban sewage treatment capacity (ten thousand $t/d$); ② A2: area of regulation around the lake (ten thousand $m^2$); ③ A3: total removal of silt from polluted river (ten thousand $m^2$); ④ A4: storage capacity (ten thousand $m^2$); ⑤ A5: greening project area (ten thousand $m^2$). All the computational experiments in this study were carried out on a computer with a CPU frequency of 2.30 GHz, a memory of 8 G, and a 64-bit operating system configuration. The optimization models were compiled in MATLAB R2020b and solved by the YALMIP/CPLEX programming solver.

According to the opinions of the expert group, we set ${\delta }_{js}=0.85,j=1,\cdots ,5$ and applied YALMIP/CPLEX to solve the model. The optimal decision scheme is {1,2,4,5,7,8,12}. The compromise satisfaction of the optimal combination is 0.90. The project portfolio and its implementation plan are shown in Figure 1. To further expand the decision-making space, based on the optimal solution, we further analyzed the extra investment required to achieve the single-goal optimal franchise return so that the decision makers can balance the relationship between the two goals. The additional investment rate was set as $PL$, and the calculation formula is $PL=\left(F_1^{*}-F_1\right)/F_1$, where $F_1$ is the optimal solution of construction investment and $F_1^{*}$ is the lower bound function value of $F_1$ obtained by maximizing $F_2$. For example, at the point of satisfaction of 0.83, the expected variation range of construction investment is [0, 10.5%]. This indicates that at this time, if the decision maker wants to maximize the franchise return, he or she needs to make an additional investment of 10.5% based on the current optimal solution of construction investment.

When the decision maker sets the confidence level at different values, the influence of the change in confidence level on the optimal solution can be further analyzed. Figure 2a shows the comparison between the WERP capacity and the minimum limit plane of restoration demand under different types of constraints. For easy observation, a section view of A3 type is shown (Figure 2b).

Under certain constraints, each WERP capacity is close to the restoration demand limit. Once the WERP capacity is disturbed, the expected WERP capacity of the project exceeds the limit, which makes it unlikely that the actual restoration demand will be met. However, chance constraints can significantly reduce the out-of-limit risk of WERP capacity, and the confidence level is higher. The larger the margin of the capacity value, the smaller the risk of exceeding the limit is. It can be seen that the chance constraint model is helpful to improve the WERP capacity of the project portfolio and ensure the quality of public service of enterprises.

According to Figure 3, as the specified confidence level increases, satisfaction with the solution decreases gradually. This is because, in order to reduce the risk at a higher confidence level and meet the demand for WERP, some projects with high returns and low investment need to be abandoned, and projects with high WERP capacity need to be selected, thus reducing the proximity between the target value corresponding to the feasible solution and the upper target value, which causes a declining trend in satisfaction.

Furthermore, when the confidence level specified by the decision maker is not a definite value but an acceptable confidence interval, the stability of different optimal solutions within the confidence interval can be obtained. The confidence interval of the project’s WERP capacity was set as [0.75~0.9], and the step size was 0.05. A total of 1024 optimal combinations were obtained through computational experiments, and the statistical results were analyzed (Figure 4).

Accordingly, in certain scenarios (marked by red dotted line in Figure 4), optimization of construction investment will lead to a reduction of franchise returns, and vice versa, indicating that there is a certain conflict relationship between dual objectives in certain project portfolios under certain confidence intervals. Optimization of one target will result in the loss of another target, but the overall fluctuation of the final optimization result is slight. It is therefore indicated that the solution method presented in this paper can effectively avoid an outcome in which the model falls into local objective optimization and can balance dual objectives effectively.

As shown in Figure 5, a total of five groups of optimal project portfolio schemes were obtained within the specified confidence interval (W1:{1,2,4,6,7,8,9}, W2:{1,2,3,4,7,9,12}, W3:{2,4,7,9,10,12}, W4:{1,2,4,5,7,8,12}, and W5:{1,2,3,5,6,10,11}). Specifically, portfolio scheme W4 was the optimal scheme 384 times, with a frequency of 37.5%, indicating that the solution has high stability. By continuing the investigation of the selection of individual projects, we revealed that Project 2 is the required project for each optimal combination. In addition, decision makers can identify key project sets based on the frequency of project selection, which should be considered in future portfolio selections.

5. Conclusions

A type of project portfolio selection with franchise periods, termed WERP, was investigated in this study. Taking into account the operational aspects of enterprises and the provision of public service, the method incorporates internal constraints related to enterprise operations and water environment restoration requirements. Furthermore, a dual-objective function was designed that considers construction investment cost and franchise return income based on the payment and operation modes specific to such projects. Taking into account the long franchise period and the fuzziness of individual metrics, a suitable multi-objective fuzzy programming model was established by combining the expected value method and the chance constraint method. The influence of the confidence level on the optimal solution was analyzed, and computational experiments provided insights into the selection frequency and stability of the project portfolio.

• The project portfolio selection method proposed in this paper effectively prevents the model from becoming stuck in local objective optimization within a specific confidence interval. By ensuring high satisfaction in the results, it achieves an effective balance. Additionally, the calculation of the extra investment rate expands the decision-making space for decision makers based on the optimal solution.
• The introduction of the proposed fuzzy chance constraint effectively reduces the risk of WERP capacity exceeding limits. A higher confidence level provides a larger margin for the distance limit of the WERP capacity value in the optimal portfolio, resulting in a smaller out-of-limit risk and higher quality of public service. However, if satisfaction with the solution decreases, the decision maker can adjust the relevant parameters according to their preferences.
• This paper provides valuable insights for enterprise managers in the context of WERP project portfolio selection. Additionally, it offers recommendations to governments on setting rational water environment restoration goals and designing incentive policies, such as subsidies or tax cuts, to facilitate mutually beneficial collaborations between enterprises and public entities, fostering a win–win outcome for all stakeholders.

The proposed method in our work still has some limitations, especially the influence of various sub-projects within the integrated water environment management project on the optimal solution, and the combination of such problems with evolutionary algorithms is worthy of further study.