A new method for two-sided matching decision making of PPP projects based on intuitionistic fuzzy choquet integral

Abstract

In the past decade, the public–private partnership (PPP) has been more widely used in worldwide infrastructure construction. Traditional PPP project evaluation methodologies have been incapable of effectively solving the two-sided matching problem between government and enterprises. In this study, we first constructed a bilateral matching satisfaction index system for the PPP project at both the government and enterprise levels, and established the matching satisfaction judgment matrices of the two sides via intuitionistic fuzzy numbers. Considering the influence of multiple attribute decision-making relevance on the attributes of decision, we used the Choquet integral to match the satisfaction evaluation vectors of the two sides, and then obtained the final matching results by constructing a multi-objective decision model. Finally, a case study was implemented to illustrate the practicality and maneuverability of this method. The research results show that the intuitionistic fuzzy Choquet integral data fusion method, which uses a two-sided matching decision-making method for PPP projects, can effectively solve the problem of correlation among attributes in the decision-making process. Consequently, this method can provide a more scientific evaluation of the bilateral matching decision problem between the government and enterprises in PPP projects.

Keywords

Public–private partnership two-sided matching decision-making intuitionistic fuzzy sets Choquet integral

1 Introduction

The accelerated process of urbanization in China has created a situation in which the shortage of funds in infrastructure construction has become a bottleneck; such a situation restricts further development of the economy and urbanization. To alleviate infrastructure funding pressure, an increasing number of governments began to encourage the private sector to participate in public investment projects. Therefore, the public-private partnership (PPP) has become the first choice for the contemporary infrastructure construction by governments [1]. PPP projects often have a longer contract period, which makes the relationship of the participants more complex than those of other procurement systems [2]. The core of the PPP model is the promotion of infrastructure and public service through the cooperation between governments and privateenterprises.

Previous research and practice show that partner selection for PPP projects has changed from a single “lowest price wins” to multi-criteria decision-making mechanism [3 –5], which is mainly concentrated in the project risk sharing mechanism, the operation process, the choice of capital structure, and so on [6 –9]. Moreover, certain scholars have indicated that the choice of the private sector is a key issue to establishing a good PPP [3, 10]. Nevertheless, the establishment of a cooperative relationship should be based on a mutually beneficial situation for both sides rather than the unilateral choice of the private sector. To date, no research has been conducted from the perspective of both the government and private sector; thus, the attainment of a mutually beneficial cooperation has not been clarified.

Multi-attribute decision making is a systematic analysis method used to study the problem of uncertain decision making. The aim is to improve the decision-making process and determine the optimal scheme to meet a certain goal from a series of alternatives [11]. Jean-Luc and Marichal [12] summarized the relevance among the attributes and analyzed the effects of the decision results. Schmeidle and David [13] were the first to apply Choquet integral theory to the decision-making problem. Since then, this theory has since been widely used in data source selection, economic research, performance evaluation, and other fields. Joshi et al. [14] applied the interval valued intuitionistic fuzzy set to solve the multi-criteria group decision making via the Choquet integral operator and the method of ranking the approximate ideal solution (TOPSIS). Zeng [15] developed a new method for intuitionistic fuzzy multiple criteria decision making by integrating induced aggregation operators into the VIKOR method, which can reflect the complex attitudinal character of the decision makers by using order inducing variables; they also provided more complete information for decision-making.

In this paper, a multi-objective decision-making model that can maximize the matching satisfaction of both the government and enterprises is studied; this model considers PPP project characteristics. We introduce intuitionistic fuzzy numbers to reduce the loss of information in the multi-criteria evaluation process of both the government and the enterprises. Furthermore, we use the intuitionistic fuzzy Choquet integration to obtain information integration, and then propose methods to achieve success in public and private matching decision-making recommendations. These results provide a solid basis for the successful implementation of a PPP project.

2 PPP project two-sided matching problem

2.1 Two-sided matching decision problem

Matching problem research originated from the question of marriage and college admission [16]. The two-sided matching problem research has elicited research attention; as a result, studies that have been implemented included those on e-commerce environment, bilateral supply and demand [17, 18], and staff positions in human resources management [19, 20]. Sørensen [21] proposed a structure model based on two-sided matching model construction of risk investment, which can be used to explain the high probability with which companies with more experience in venture capital investment can be listed. Chen et al. [22] proposed a fuzzy method based on axiomatic design (FAD) to identify the matching degree between the demand and supply of knowledge service to promote their continuous improvement and improve efficiency. The two-sided matching diagram can be shown in Fig. 1.

2.2 Two-sided matching satisfaction index system of PPP project

The PPP is generally considered a win-win mechanism that represents a collection of resources and efforts of both the public sector and the private sector [23]. The PPP is also seen as an effective way to change the function of government management and to realize the modernization of governance. Bilateral matching of PPP projects involves PPP projects of two-way choice between governments and enterprises, resulting in a long-term and stable partnership, thereby ultimately maximizing satisfaction t of both sides. Therefore, the two-sided matching decision-making process should fully consider the satisfaction rate of both the government and the enterprises.

The PPP is a key issue in the choice of private sector partners. Zhang [10] indicated that suitable business partner selection depends on specific projects to identify and define the appropriate quality standards as well as to develop an effective method for the evaluation. Using a structured questionnaire survey, they established the PPP project evaluation criteria, which comprise four assessment packages as follows: finance, technology, safety and health, and environment. Hwang et al. [24] claimed that PPP projects have always been large and complex, and that the lack of relevant experience will increase the possibility of enterprise project failure. By considering the above research as well as the key factors to the success of the PPP project evaluation of certain provinces and cities in China, this study evaluates companies that bid for PPP project using four aspects: financial strength, technical ability, professional qualifications, and management experience. The results can be helpful to the government in choosing the PPP project.

However, the evaluation indexes of the enterprises to the government are mainly related to the laws and regulations, the return of investment, and the credibility of the government. Laws and regulations are the premise and basis for the matching of the decisions of the government and enterprises. Project appraisal and evaluation, contract performance and change, conflict resolution, and cooperation require clear legal definitions that can promote the implementation of a PPP project [25, 26]. Therefore, to further standardize and encourage enterprises to participate in PPP projects, the government should implement corresponding incentive laws and policies, improve laws and regulations, reform the professional examination and approval process, and streamline the approval process. These procedures can create an effective system environment to promote the implementation of PPP projects.

The investment recovery rate is the most essential criterion in the participation of in a PPP project, and a relatively short time and higher rate of return on investment of the PPP project is more vulnerable to the willingness of enterprises. However, PPP project cycles are generally longer than those of the others, and are accompanied by a series of unpredictable risks [26 –28]. Enterprises face significant economic losses when risk occurs; in the PPP project process, such risk requires the government to provide the corresponding risk-sharing effort. In addition, the government can provide the corresponding price guarantee, interest rate guarantee, compensation guarantee, and restriction of competition to ensure the investment recovery rate of enterprises.

The credibility of the government and the spirit of the contract compose another important factor that influences the enterprise satisfaction of a PPP project [29]. The enterprise is willing to participate in a PPP project as long as open and transparent policy information can be obtained, as well as relevant policies and regulations. Many previous cases have demonstrated that the omission and loss of key information and items will lead to a lack of trust in the government; enterprises will not wish to participate in the project and will raise disputes in the project process. Therefore, the government must exert more efforts to regulate the related process, ensure the disclosure of relevant information, improve the spirit of contract, and fully protect the legitimate rights and interests of enterprises.

2.3 Intuitionistic fuzzy sets and Choquet integrals

In 1965, Zadeh [30] proposed fuzzy set theory, whose core idea is that a value of only 0 or 1 of the characteristic function can be extended to a membership function in the closed unit interval [0, 1] for an arbitrary value.

Definition 1. Let X be a non-empty set; then, $F = {< x, μ_{F} (x) > | x \in X}$ is a fuzzy set, in which μ_F is the membership function of the fuzzy set F, μ_F : X → [0, 1], μ_F (x) is the membership of element x to the set F, and μ_F (x) must be taken a single value in the unit interval [0, 1].

In 1986, Atanassov [31] developed the fuzzy set concept based on the theory of Zadeh, and the definition of intuitionistic fuzzy sets was defined.

Definition 2. An intuitionistic fuzzy set (IFS) over X is an object with the form $\begin{matrix} A = {< x, u_{A} (x), v_{A} (x) > | x \in X} \\ u_{A} : X \to [0, 1], x \in X \to u_{A} (x) \in [0, 1], \\ v_{A} : X \to [0, 1], x \in X \to v_{A} (x) \in [0, 1], \end{matrix}$ where the functions u_A (x) and v_A (x) define the degree of membership and the degree of non-membership of the element, and for every x ∈ X : 0 ≤ u_A (x) + v_A (x) ≤1.

In addition, π_A (x) =1 - u_A (x) + v_A (x) is called the intuitionistic fuzzy index, which indicates the degree of hesitation of x to the intuitionistic fuzzy set X.

In 2006, Xu and Yager [32] completed the definition of intuitionistic fuzzy number comparison and sorting method, which was based on the scoring function [33] and the accuracy function [34].

Definition 3. Let α = (u_α, v_α) and β = (u_β, v_β) be two intuitionistic fuzzy values, the s (α) = u_α - v_α and s (β) = u_β - v_β be the scores of α and β, h (α) = u_α + v_α and h (β) = u_β + v_β be the degrees of accuracy of α and β. We can compare them as follows:

If s (α) < s (β), then α < β;

If s (α) < s (β), then

If h (α) < h (β), then α < β;

If h (α) < h (β), then α = β.

Definition 4. A fuzzy measure on X is a set function u : X → [0, 1] such that

μ (∅) =0, μ (X) =1;

A, B ⊆ X, A ⊆ B implies μ (A) ≤ μ (B).

Definition 5. Let f be a real-valued function on X; the Choquet integral of f with respect to a fuzzy measure μ on X is defined as: $C_{μ} (f) = \sum_{i = 1}^{n} f_{(i)} [μ (A_{(i)}) - μ (A_{(i + 1)})]$ where the (·) used for indices represent a permutation on X such that f₍₁₎ ≤ f₍₂₎ ≤ ⋯ ≤ f_(n), A_(i) = {i, …, n} and A_(n+1) =∅.

Definition 6. Let ${\tilde{α}}_{i} = (u_{{\tilde{α}}_{i}}, v_{α_{i}}), (i = 1, 2, \dots, n)$ be a collection of X intuitionistic fuzzy values, where μ is a fuzzy measure on X. Then, ${\tilde{α}}_{i}$ is a discrete intuitionistic fuzzy Choquet integral operator with respect to μ being defined as [2]: $\begin{matrix} {IFC}_{μ} ({\tilde{α}}_{1}, \dots, {\tilde{α}}_{n}) \\ = {\tilde{α}}_{(1)} [μ (A_{(1)}) - μ (A_{(2)})] \oplus {\tilde{α}}_{(2)} [μ (A_{(2)}) - μ (A_{(3)})] \\ \oplus \dots \oplus {\tilde{α}}_{(n)} [μ (A_{(n)}) - μ (A_{(n + 1)})] \\ = \sum_{i = 1}^{n} \oplus {\tilde{α}}_{(i)} [μ (A_{(i)}) - μ (A_{(i + 1)})] \\ = (1 - \prod_{i = 1}^{n} {(1 - u_{{\tilde{α}}_{(i)}})}^{μ (A_{(i)}) - μ (A_{(i + 1)})}, \\ \prod_{j = 1}^{n} {(v_{{\tilde{α}}_{(i)}})}^{μ (A_{(i)}) - μ (A_{(i - 1)})}), \end{matrix}$ where the (·) used for indices represent a permutation on X such that ${\tilde{α}}_{(1)} \leq {\tilde{α}}_{(2)} \leq \dots \leq {\tilde{α}}_{(n)}$ , A_(i) = {i, …, n} and A_(n+1) =∅.

3 Two-sided matching decision-making model

3.1 Two-sided matching model based on Choquet integral

Considering the problems in the process of two-sided matching decision, let both subject sides involved in the matching be D = {D₁, D₂, ⋯ , D_n} and P = {P₁, P₂, ⋯ , P_n} respectively.

Definition 7. Suppose Φ : D ∪ P → D ∪ P is a 1-1 map, ∀D_i ∈ D, ∀P_j ∈ P, subject to: 1) Φ (D_i) ∈ P; 2) Φ (P_j) ∈ D ∪ {P_j}; 3) Φ (D_i) = P_j if and only if Φ (P_j) = D_i. Then, Φ is the two-sided matching between set D and set P. Specifically, when Φ (D_i) = P_j, then D_i and P_j are matching in set Φ, and the (D_i, P_j) is defined as a matching pair. When Φ (P_j) = P_j, then P_j cannot be matching in set Φ.

Let $C^{1} = {C_{1}^{1}, C_{2}^{1}, \dots, C_{s}^{1}}$ be the matching satisfaction evaluating index set, and E_σij = {E_1ij, E_2ij, ⋯ , E_sij} be the evaluation of subject D_i to subject P_j based on $C_{σ}^{1} (σ = 1, 2, \dots, s)$ , which indicates the σth evaluation index. In addition, let μ¹ = {μ¹ (A₍₁₎) , μ¹ (A₍₂₎) , ⋯ , μ¹ (A_(s))} be the fuzzy measure vector of the index set given by the experts, where $A_{(j)} = {(C_{j}^{1}), \dots (C_{s}^{1})}$ and μ¹ (A_(j)) indicate the measurement information (importance degree) determined by considering the correlation information between the index set ${C_{j}^{1}, C_{j + 1}^{1}, \dots, C_{s}^{1}}$ .

By contrast, let $C^{2} = {C_{1}^{2}, C_{2}^{2}, \dots, C_{t}^{2}}$ be the matching satisfaction evaluating index set and F_τij = {F_1ij, F_2ij, ⋯ , F_tij} be the evaluation of subject P_j to subject D_i based on $C_{τ}^{2} (τ = 1, 2, \dots, t)$ , which indicates the τth evaluation index. In addition, let μ² = {μ² (B₍₁₎) , μ² (B₍₂₎) , ⋯ , μ² (B_(t))} be the fuzzy measure vector of the index set given by the experts, where $B_{(j)} = {(C_{j}^{2}), \dots, (C_{t}^{2})}$ and μ² (B_(j)) indicate the measurement information (importance degree) determined by considering the correlation information between the index set ${C_{j}^{2}, C_{j + 1}^{2}, \dots, C_{t}^{2}}$ .

All the evaluation information in the matching process is the intuitionistic fuzzy number, and each satisfaction evaluation index cannot be a simple linear summation because of non-independence. However, the index can be used to reduce the loss of fuzzy evaluation information and to further improve the accuracy of the evaluation by using the intuitionistic fuzzy Choquet integral. Based on the intuitionistic fuzzy Choquet integral evaluation model mentioned above, the model is obtained by set {D_i} and {P_j} to match the satisfaction of the vector α_ij and β_ij; such a match establishes a two-sided matching decision model of satisfaction to the maximum.

Suppose x_ij represents a 0 - 1 variable, when x_ij = 0 means that D_i and P_j cannot be matching in the set Φ, or μ (D_i) ≠ P_j; x_ij = 1 means that D_i and P_j have the matching pair (D_i, P_j) in the set Φ, or μ (D_i) = P_j. Then, a multiple objective optimization model can be set up as follows:

$\begin{matrix} max Z_{D} = \sum_{i = 1}^{n} \sum_{j = 1}^{m} α_{ij} x_{ij} (a) \\ max Z_{P} = \sum_{i = 1}^{n} \sum_{j = 1}^{m} β_{ij} x_{ij} (b) \\ s . t . & \sum_{j = 1}^{m} x_{ij} \leq 1, i = 1, 2, \dots, n; (c) \end{matrix}$ (1) $\begin{matrix} \sum_{i = 1}^{n} x_{ij} \leq 1, j = 1, 2, \dots, m; (d) \\ x_{ij} = 0 or 1, i = 1, 2, \dots, n; \\ j = 1, 2, \dots, m . (e) \end{matrix}$

In the model above, formulas (a) and (b) are the objective functions and formulas (c) and (d) are the constraint conditions. The aim of formula (a) is to maximize the satisfaction of all the D_i to P_j; the aim of formula (b) is to maximize the satisfaction of all the P_j to D_i. Formula (c) indicates that each subject D_i can only have no more than one matching pair of P_j; formula (d) indicates that each subject P_j can only have no more than one matching pair of D_i.

3.2 Model solution

To solve the multi-objective model, taking into account the membership vector S ∈ [0, 1], compute the linear weighted sum of formulas (a) and (b). If w_D and w_P are the weights of the target Z_D and Z_P, w_D, w_P ∈ (0, 1), w_D + w_P = 1, then the multi-objective model can be transformed into a single objective linear programming model: $\begin{matrix} max Z_{D, P} & = & w_{D} \sum_{i = 1}^{n} \sum_{j = 1}^{m} α_{ij} x_{ij} + w_{P} \sum_{i = 1}^{n} \sum_{j = 1}^{m} β_{ij} x_{ij} \\ = & \sum_{i = 1}^{n} \sum_{j = 1}^{m} γ_{ij} x_{ij} \\ s . t . & \sum_{j = 1}^{m} x_{ij} \leq 1, i = 1, 2, \dots, n; \end{matrix}$ (2) $\begin{matrix} \sum_{i = 1}^{n} x_{ij} \leq 1, j = 1, 2, \dots, m; \\ x_{ij} = 0 or 1, i = 1, 2, \dots, n; \\ j = 1, 2, \dots, m . \end{matrix}$ where γ_ij = w_Dα_ij + w_Pβ_ij and w_D and w_P reflect the importance relative to the total target Z_D,P in the actual two-sided matching decision making. If we wish to maintain equal opportunities for both sides, we can let w_D = w_P = 0.5. When the status of the two sides cannot be equal (w_D ≠ w_P), which means that one party is more important than the other, the following comparison objectives can be used to determine their respective weights by Z_D and Z_P. For example, if the goal Z_D is more important than Z_P, and the importance degree is λ, λ > 1, then, $w_{D} = \frac{λ}{1 + λ}, w_{P} = \frac{λ}{1 + λ} .$

The targets and constraints in model (2) are all linear functions, which is a 0 - 1 integer programming with at least m · n variables and 2^mn feasible solutions. Therefore, based on linear programming theory, model (2) has the optimal solution, and the optimal solution is the effective solution of model (1). MATLAB software is used to solve the problem model (2). The above modeling and solving process can be summarized as shown in Fig. 2.

4 A real PPP project matching example and comparison analyses

A local Chinese government procurement agency is responsible for the PPP projects in the region. This agency provides information consulting and intermediary services. To improve the matching efficiency and success rate of the government and enterprises in the PPP project, as well as to maximize the benefits from cooperation and reduce the risk of cooperation, the agency collected three government PPP projects (D₁, D₂, D₃) and six bidding enterprises (P₁, P₂, P₆). The agency conducted a comprehensive survey of the actual operation of the situation and information summary during a certain period. The agency hired experts to conduct multi-index evaluation to form a reasonable match and help the bidding companies in terms of classified guidance and counseling services. Considering the financial pressures and the risk of the project, an enterprise can successfully invest in only one project within a period of time. To solve the two-sided matching problem of the PPP project, the calculation process of the method should be explained in detail.

Step 1.

The intuitionistic fuzzy values decision matrix of the bidding companies P_j is developed based on the four evaluating criteria, namely, financial strength, technical ability, professional qualifications, and management experience. Eight experts in the government procurement expert database are invited to evaluate the core competencies of six candidates. The intuitionistic fuzzy decision matrix of the candidates is obtained and shown in Table 1. However, the bidding enterprises evaluated the government PPP projects from three core competencies, the laws and regulations that should be protected, rate of return on investment, and credibility of the government, as shown in Table 2.

Step 2.

Tables 1 and 2 indicate that according to Definition 3, we can reorder the partial evaluation of the candidates α_ij and β_ij such that ${\tilde{α}}_{ij (k)} \leq {\tilde{α}}_{ij (k + 1)}$ , i = 1, 2, ⋯ , 6, and ${\tilde{β}}_{ij (k)} \leq {\tilde{β}}_{ij (k + 1)}$ , j = 1, 2, 3.

The evaluation for the reordering of D₁ → P₁ is as follows: $\begin{matrix} {\tilde{α}}_{11 (1)} & = & (0.3, 0.4), {\tilde{α}}_{11 (2)} = (0.4, 0.3), \\ {\tilde{α}}_{11 (3)} & = & (0.6, 0.3), {\tilde{α}}_{11 (4)} = (0.7, 0.1) . \end{matrix}$

The evaluation for the reordering of P₁ → D₁ is as follows: $\begin{matrix} {\tilde{β}}_{11 (1)} & = & (0.4, 0.5), {\tilde{β}}_{11 (2)} = (0.3, 0.1), \\ {\tilde{β}}_{11 (3)} & = & (0.5, 0.3) . \end{matrix}$

Similarly, we can draw the reordering of all.

Step 3.

The matching sides of government PPP projects among the satisfaction evaluation indicators are not completely independent; these indicators include the technology capability $C_{2}^{1}$ and the professional qualification $C_{3}^{1}$ , laws and regulations $C_{1}^{2}$ and the government credibility $C_{3}^{2}$ , and so on. Tan [35] indicated that the following can be assumed: $\begin{matrix} μ (C_{1}^{1}) = 0.36, μ (C_{2}^{1}) = 0.27, μ (C_{3}^{1}) = 0.25, \\ μ (C_{4}^{1}) = 0.27, μ (C_{1}^{1}, C_{2}^{1}) = 0.74, \\ μ (C_{1}^{1}, C_{3}^{1}) = 0.67, μ (C_{1}^{1}, C_{4}^{1}) = 0.52, \\ μ (C_{2}^{1}, C_{3}^{1}) = 0.45, μ (C_{2}^{1}, C_{4}^{1}) = 0.43, \\ μ (C_{3}^{1}, C_{4}^{1}) = 0.37, μ (C_{1}^{1}, C_{2}^{1}, C_{3}^{1}) = 0.78, \\ μ (C_{1}^{1}, C_{2}^{1}, C_{4}^{1}) = 0.72, μ (C_{1}^{1}, C_{3}^{1}, C_{4}^{1}) = 0.67, \\ μ (C_{2}^{1}, C_{3}^{1}, C_{4}^{1}) = 0.60, μ (C_{1}^{1}, C_{2}^{1}, C_{3}^{1}, C_{4}^{1}) = 1.0 . \end{matrix}$

In addition, we can suppose that: $\begin{matrix} μ (C_{1}^{2}) = 0.55, μ (C_{2}^{2}) = 0.43, μ (C_{3}^{2}) = 0.30, \\ μ (C_{1}^{2}, C_{2}^{2}) = 0.86, μ (C_{1}^{2}, C_{3}^{2}) = 0.78, \\ μ (C_{2}^{2}, C_{3}^{2}) = 0.69, μ (C_{1}^{2}, C_{2}^{2}, C_{3}^{2}) = 1.0 . \end{matrix}$

Step 4.

By using the intuitionistic fuzzy Choquet integral operator (IFC), the following can be defined: $\begin{matrix} {\tilde{α}}_{ij} & = & {IFC}_{μ} ({\tilde{α}}_{ij 1}, \dots, {\tilde{α}}_{ijn}) \\ = & (1 - \prod_{k = 1}^{n} {(1 - u_{\tilde{α} ij (k)})}^{μ (A_{(k)}) - μ (A_{(k + 1)})}, \\ \prod_{k = 1}^{n} {(v_{\tilde{α} ij (k)})}^{μ (A_{(k)}) - μ (A_{(k + 1)})}) \end{matrix}$

For example: $\begin{matrix} {\tilde{α}}_{11} & = & {IFC}_{μ} ({\tilde{α}}_{111}, \dots, {\tilde{α}}_{114}) \\ = & (1 - \prod_{k = 1}^{4} {(1 - u_{\tilde{α} ij (k)})}^{μ (A_{(k)}) - μ (A_{(k + 1)})}, \\ \prod_{k = 1}^{4} {(v_{\tilde{α} ij (k)})}^{μ (A_{(k)}) - μ (A_{(k + 1)})}) \\ = & (1 - (1 - 0.3)^{1 - 0.82} (1 - 0.4)^{0.82 - 0.67} \\ (1 - 0.3)^{0.67 - 0.25} (1 - 0.3)^{0.25}, (0.4)^{1 - 0.82} \\ (0.3)^{0.82 - 0.67} (0.3)^{0.67 - 0.25} (0.1)^{0.25}) \\ = & (0.56, 0.24) \end{matrix}$

The experts hired by the government procurement agency evaluate the bidding companies and the PPP projects with the index sets ${C_{1}^{1}, C_{2}^{1}, C_{3}^{1}, C_{4}^{1}}$ and ${C_{1}^{2}, C_{2}^{2}, C_{3}^{2}}$ as follows: $\begin{matrix} {[α (D_{ij})]}_{3 \times 6} & = & [\begin{matrix} (0.56, 0.24) & (0.65, 0.20) & (0.35, 0.30) & (0.47, 0.31) & (0.51, 0.28) & (0.34, 0.48) \\ (0.67, 0.18) & (0.37, 0.50) & (0.55, 0.22) & (0.34, 0.37) & (0.50, 0.35) & (0.61, 0.17) \\ (0.40, 0.17) & (0.58, 0.22) & (0.43, 0.36) & (0.33, 0.31) & (0.65, 0.18) & (0.58, 0.22) \end{matrix}] \\ {[β (P_{ij})]}_{3 \times 6} & = & [\begin{matrix} (0.43, 0.23) & (0.52, 0.25) & (0.60, 0.30) & (0.63, 0.16) & (0.74, 0.19) & (0.30, 0.25) \\ (0.71, 0.15) & (0.65, 0.14) & (0.45, 0.25) & (0.58, 0.18) & (0.40, 0.36) & (0.83, 0.14) \\ (0.54, 0.21) & (0.65, 0.16) & (0.49, 0.30) & (0.41, 0.42) & (0.61, 0.26) & (0.64, 0.27) \end{matrix}] \end{matrix}$

Definition 3 indicates that the intuitionistic fuzzy evaluation matrix can be obtained as follows: $\begin{matrix} {[α (D_{ij})]}_{3 \times 6} & = & [\begin{matrix} 0.32 & 0.45 & 0.05 & 0.16 & 0.23 & - 0.14 \\ 0.49 & - 0.13 & 0.33 & - 0.03 & 0.15 & 0.44 \\ 0.23 & 0.36 & 0.07 & 0.02 & 0.47 & 0.23 \end{matrix}] \\ {[β (P_{ij})]}_{3 \times 6} & = & [\begin{matrix} 0.20 & 0.27 & 0.30 & 0.47 & 0.55 & 0.05 \\ 0.56 & 0.51 & 0.20 & 0.40 & 0.04 & 0.69 \\ 0.33 & 0.49 & 0.19 & - 0.01 & 0.35 & 0.37 \end{matrix}] \end{matrix}$

Step 5.

By using the two-sided matching model (2) and the optimization toolbox of MATLAB, we can obtain the optimal solution as x₁₂ = x₂₆ = x₃₅ = 1, and the others as x_ij = 0. Therefore, the optimal matching result is: D₁ ↔ P₂, D₂ ↔ P₆, D₃ ↔ P₅, which shows that the government PPP project D₁ matches with bidding company P₂, project D₂ matches with company P₆, and project D₃ matches with company P₅.

To determine the effects of the different values of ω₁ and ω₂, we conduct sensitivity analysis of the model, and the results are shown in Table 3.

5 Conclusion

The governments on all levels in developing countries have established corresponding basic public procurement platforms and policies for promoting innovation of the PPP projects [36, 37]. The two-way selection of private and public partners is the basis for the implementation of a PPP project. Considering the correlation index in the satisfaction evaluation characteristics, we combine the intuitionistic fuzzy Choquet integral and multi-objective decision problem method to obtain a satisfactory evaluation of government and enterprise PPP projects. We use intuition fuzzy Choquet integral aggregation and calculate the objective decision data and expert subjective preference information of both the government and enterprises. After forming a matching satisfaction judgment matrix, the final evaluation results are obtained by a multi-objective decision model. Intuitionistic fuzzy set theory is adopted in the paper; the theory reduces the loss of any information in the process of aggregation and considers the interaction among the information. Finally, a matching case computation and analyses show the clear logic of the method proposed in this paper; this method not only has better theoretical support, but also considers practicality and operability. This method can effectively improve the decision-making efficiency in relation to practical problems and provide a new way to solve the two-sided matching of a PPPproject.

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