Informatica

As shown in Table 2, there are many PPP models, and the final implementation effect of the project is largely dependent on the selected one. A wide range of studies has been conducted to assist decision making in PPP selection. John and Isr (2003) analysed thirteen PPP projects in North America and Asia and suggested that the project risks, project conditions, and availability of financing are the critical factors in PPP selection. Considering the reference function of similar past projects, Luu et al. (2005) proposed a conceptual framework for case-based PPP selection by integrating the client’s needs, the project characteristics, and external environment. To quantitatively depict the evaluation and selection of PPP projects, Mahdi and Alreshaid (2005) first constructed the index system for evaluating PPP models. Then, the authors proposed a multi-criteria decision-making methodology using the analytic hierarchy process (AHP) to assist the selection of PPP models. According to the degree of legal authorization given by the government to the project company, Ghavamifar and Touran (2008) discussed the selection of PPP, construction management (CM) and design-build (DB) models in transportation projects of all the 50 states in the United States. By taking schedule, cost, owners, project and external environment as input variables and taking cost, schedule, safety, quality and contract as output variables, Chen et al. (2010) constructed the data envelopment analysis (DEA) model to assist owners in selecting PPP models. Combined with modern portfolio theory (MPT) and multi-objective optimization, Weissenböck and Girmscheid (2013) introduced a method for PPP selection for construction firms in selecting highly suitable PPP projects. Dai and Molenaar (2015) presented a risk-based modelling approach to quantify the potential differences in project cost due to the selection of PPP models for highway design and construction. Feng et al. (2018) developed a multi-objective optimization model for balancing public and private interests for PPP models. Furthermore, the authors took Beijing No. 4 Metro Line to show the application of the offered model. Osei-Kyei and Chan (2018b) investigated the differences and similarities on the reasons for implementing PPP in developing and developed economies/countries through Ghana and Hong Kong. Pellegrino et al. (2019) adopted the Monte Carlo simulation as the option-pricing method to test how to ensure the maximum interest rate of private investors in PPP projects.

To deal with the uncertainty in the process of evaluating PPP models, Yuan et al. (2010) researched performance objective attributes in the perspective of different stakeholder groups and combined fuzzy entropy method and fuzzy TOPSIS method to develop an approach for selecting the performance objective levels for PPP models. Shakeri et al. (2015) combined Elimination et Choice Translating Reality (ELECTRE) and Strength, Weakness, Opportunity, Threat (SWOT) methods in fuzzy environment for evaluating private sector for water treatment PPP models in Iran. Valipour et al. (2016) presented a hybrid fuzzy method and a cybernetic analytic network process (CANP) model for identifying sharing risks. Its main principle is to transform linguistic information and expertise into systematic quantitative analysis and use CANP model to address the problems of dependency and feedback between criteria and barriers, as well as the choice of sharing risks. Combining with the local situation, Zhang et al. (2019) developed an indicator system. Then, the authors established an integrated decision-making framework using triangular fuzzy AHP analysis for selecting the suitable PPP model. Based on the theory of intuitionistic fuzzy sets, Wang and Qin (2015) provided an evaluation method to work out the score functions of intuitionistic fuzzy numbers with uncertain weights, which is then used to select appropriate PPP models. On the basis of interval-valued intuitionistic fuzzy set (IVIFS) theory, An et al. (2018) proposed a group decision making method for PPP model selection. Su et al. (2019) introduced the similarity measure with interval neutrosophic information, by which a PPP model selection method with interval neutrosophic set is proposed.

Based on the above literature review, one can verify that although they can deal with the problem of selecting PPP models with fuzzy information, there are some limitations in applications. For example, due to the complexity of PPP model selection, the hesitancy, preferred, indeterminacy and non-preferred information may simultaneously exist. However, none of the previous methods can deal with this case; (2) none of them can cope with the situation where the weights of risk factors are interactive.

The government and the private sector should assess all potential risk factors throughout the whole life cycle of the project. To show the potential risk factors in the procedure of evaluating PPP models, Jang (2011) offered four primary risk factors and twelve secondary risk factors as shown in Table 1.

On the other hand, since PPP was first introduced by the British government in 1952, various types of PPP models have been proposed. According to the PPP style, Huang (2007) summarized thirteen main PPP models as shown in Table 2.

To indicate their differences clearly, Table 3 shows the features of these thirteen types of PPP models. From the top to the bottom, the degree of privatization is deepening while the private risks are increasing (Huang, 2007).

With the development of fuzzy decision-making theory, many types of generalized fuzzy sets are proposed, such as intuitionistic fuzzy sets (IFSs) (Atanassov, 1986), hesitant fuzzy sets (HFSs) (Torra, 2010), and neutrosophic sets (NSs) (Smarandache, 1999). It is noticeable that NSs are more flexible than IFSs and the latter can be seen as a special case of the former. Considering the advantages of NSs as well as their application in decision making, Wang et al. (2005) introduced the concept of single-valued neutrosophic sets (SVNSs) to express the preferred, indeterminacy and non-preferred information by using three independent variables in $[0,1]$. However, SVNSs still cannot denote the hesitant information. To denote the hesitancy, preferred, indeterminacy and non-preferred information simultaneously, Ye (2015) further presented SVNHFSs that permit the DMs to employ several values in $[0,1]$ to separately denote the hesitant preferred, hesitant indeterminacy and hesitant non-preferred information. All of the above mentioned fuzzy sets can be seen as a special case of SVNHFSs, respectively. Considering its application in decision making, the author defined several basic operations and offered a cosine measure. Then, two single-valued neutrosophic hesitant fuzzy aggregation operators are defined to calculate the comprehensive evaluation values. To show the application of these results, the author studied the selection of the best investment objects. Following the pioneer work of Ye (2015), several aggregation operator based decision making methods with single-valued neutrosophic hesitant fuzzy information are presented. For instance, Liu and Luo (2019) introduced the single-valued neutrosophic hesitant fuzzy ordered weighted averaging (SVNHFOWA) operator and the single-valued neutrosophic hesitant fuzzy hybrid weighted averaging (SVNHFHWA) operator and studied its utilization in software evaluation. However, Akansha and Amit (2019) pointed out that the SVNHFOWA operator does not satisfy monotonicity property and this aggregation operator is unreasonable. Li and Zhang (2018) defined two lamda-fuzzy measure based Choquet integral single-valued neutrosophic hesitant fuzzy aggregation operators to cope with the situation where there are interactive characteristics. Then, the authors discussed its application to the evaluation of emerging technology commercialization. Meanwhile, Pang et al. (2018) proposed a lamda-fuzzy measure based Choquet integral single-valued neutrosophic hesitant fuzzy Heronian mean operator and discussed how to select the best investment country for a Chinese company.

Besides the above aggregation operator based decision making methods, Biswas et al. (2016a) and Şahin and Liu (2016) separately defined two generalized single-valued neutrosophic hesitant weighted distance measure and offered the associated decision making methods. Şahin and Liu (2017) further presented two similarity measures and showed their application in decision making. Ye (2018) used the weighted distance measure offered by Biswas et al. (2016a) and Şahin and Liu (2016) to give a weighted similarity measure. Then, the authors proposed a decision making method. Şahin and Liu (2017) researched correlation and correlation coefficient of SVNHFSs. It is notable that all of these studies employ the example offered by Ye (2015) to show the application of associated measures. Biswas et al. (2016b) offered a grey relational analysis method for decision making with single-valued neutrosophic hesitant fuzzy information based on the offered distance measure and discussed its application in selecting cars. Xu et al. (2019) developed the single-valued neutrosophic hesitant fuzzy TODIM method and discussed its application in venture capital.

Although there are many studies about decision making with single-valued neutrosophic hesitant fuzzy information, there are still several restrictions.

From the above analysis for the new method, one can verify that it avoids all of the above listed limitations.

Correlation coefficient is an important measure for decision making, which has been widely researched and used in decision making with different types of fuzzy sets. For example, fuzzy correlation coefficient (Murthy et al., 1985), intuitionistic fuzzy correlation coefficients (Szmidt and Kacprzyk, 2010), hesitant fuzzy correlation coefficients (Meng and Chen, 2015a), and neutrosophic correlation coefficients (Ye, 2013, 2014b). As for correlation coefficient of SVNHFSs, we only find one reference (Şahin and Liu, 2017). However, the rationality of this correlation coefficient needs to be further discussed. For example, it can only deal with the situation where the compared SVNHFSs have the same length. Otherwise, we need to add extra values into the shorter SVNHFSs. However, this process in fact distorts the original information. Furthermore, Şahin and Liu’s method assumes that the criteria are independent and the weights are completely known. All of these aspects restrict its application.

Considering the limitations of previous research about selecting PPP models, decision making methods with single-valued neutrosophic hesitant fuzzy information and single-valued neutrosophic hesitant fuzzy correlation coefficient, this paper introduces a new correlation coefficient based method for selecting PPP models in the setting of SVNHFSs.

Based on Meng and Chen’s hesitant fuzzy correlation coefficients (Meng and Chen, 2015b), this subsection proposes two new correlation coefficients of SVNHFSs, which have two advantages: (i) the lengths of compared SVNHFEs can be different; (ii) they are not limited to the ordered elements to define the correlation between SVNHFSs. Before offering the definition of new correlation coefficients, we first propose the following distances.

If there is more than one value in ${h_{B}}$, ${\iota _{B}}$, and ${g_{B}}$ that satisfy (5)–(7), respectively, let

Based on the above defined distance measures, we next consider the correlation between SVNHFSs.

On the basis of the defined correlation between SVNHFSs, we further define correlation coefficients of SVNHFSs as follows:

Correlation coefficients (12) and (13) neither consider the lengths of SVNHFEs nor arrange their possible values in an increasing order. Generally speaking, the optimistic DMs can apply correlation coefficient (12), while the pessimistic DMs could employ (13).

When correlation coefficients in Definition 7 are used to calculate the example in Section 3, one can obtain that $C{C_{1}}(A,B)=0.631$ and $C{C_{2}}(A,B)=0.377$.

To show the rationality of correlation coefficients offered in Definition 7, we consider the following several desirable properties:

For the geometric mean based SVNHFCC and the maximum based SVNHFCC shown in Definition 7, we have the following relationship:

Correlation coefficients offered in Section 4.1 are based on the assumption that all SVNHFEs have the same importance. When the finite set $X=\{{x_{1}},{x_{2}},\dots ,{x_{n}}\}$ denotes criteria in decision making, this hypothesis is unreasonable. To indicate their differences in importance, this section further introduces two weighted correlation coefficients.

Some scholars noted that the weights of elements in a set may be interactive (Meng et al., 2018; Xu, 2010). To cope with the situation, fuzzy measure (Sugeno, 1974) is an effective tool.

In multi-criteria decision making, $\mu (A)$ can be viewed as the importance of the criteria set A. In addition to the usual weights of the criteria set taken separately, the weight of any combination of criteria is also defined.

Considering the fact that fuzzy measures are too complex in practical application, 2-additive measures introduced by Grabisch (1997) are a good choice that can reduce the complexity of fuzzy measures.

Based on Definition 9, Grabisch (1997) proposed the following theorem to determine a 2-additive measure.

Although 2-additive measure is powerful to reflect the interactions among the weights of criteria, it cannot be used as the weights directly. Considering this issue, the Shapley function (Shapley, 1953) is a useful tool. With respect to the 2-additive measure μ, the Shapley function can be expressed as in Meng and Tang (2013): for any $i\in N$.

Based on the 2-additive Shapley function and correlation coefficients offered in Section 4.1, single-valued neutrosophic hesitant fuzzy 2-additive Shapley weighted correlation coefficients (SVNHF-2ASWCCs) are defined as follows:

Furthermore, when we have ${\omega _{{x_{k}}}}=1/n$ for all $k=1,2,\dots ,n$, correlation coefficients in Definition 10 reduce to those in Definition 7.

Similar to correlation coefficients offered in Section 4.1, we obtain the following property:

Noticeably, the pessimists can choose (12) or (14), while the optimists could adopt (11) and (13).

Before introducing models for determining the optimal fuzzy measure on the attribute set, we define the following improvement normalized Hamming distance measure for SVNHFSs:

To show the rationality of the Hamming distance measure listed in Definition 11, we offer the following property:

Considering a decision-making problem, let $A=\{{a_{1}},{a_{2}},\dots ,{a_{m}}\}$ be a set of alternatives, and $C=\{{c_{1}},{c_{2}},\dots ,{c_{k}}\}$ be a set of criteria. Suppose that ${p_{ij}}=({h_{ij}},{\iota _{ij}},{g_{ij}})$ is the SVNHFE of the alternative ${a_{i}}$ for the criterion ${c_{j}}$, $i=1,2,\dots ,m$; $j=1,2,\dots ,k$. By $D={({p_{ij}})_{m\times k}}$, we denote the associated single-valued neutrosophic hesitant fuzzy decision matrix. Let ${p_{j}}$ be the jth column of D, and let for each $j=1,2,\dots ,k$.

Let ${p^{+}}=\{{p_{1}^{+}},{p_{2}^{+}},\dots ,{p_{k}^{+}}\}$ and ${p^{-}}=\{{p_{1}^{-}},{p_{2}^{-}},\dots ,{p_{k}^{-}}\}$. When the fuzzy measure μ on the criteria set C is completely unknown, we establish the following model: where ${\phi _{{c_{j}}}}(\mu ,C)$ is the Shapley value of the criterion ${c_{j}}$ with respect to the fuzzy measure μ, $d({p_{ij}},{p_{j}^{+}})$ is the improvement normalized Hamming distance measure between ${p_{ij}}$ and ${p_{j}^{+}}$, and $d({p_{ij}},{p_{j}^{-}})$ is the improvement normalized Hamming distance measure between ${p_{ij}}$ and ${p_{j}^{-}}$.

When μ is a 2-additive measure, we get where the notations are as shown in model (25).

When the fuzzy measure μ on the criteria set C is incompletely known, model for ascertaining the optimal fuzzy measure μ is built as follows: where ${W_{j}}$ is the known weight information, and other notations are as shown in model (25).

Similarly, we derive the following model for the optimal 2-additive measure μ on C: where other notations are as shown in models (26) and (27).

Models (25) to (28) degenerate to corresponding models for the optimal additive measure on the criteria set C if there are no interactions.

This subsection introduces an algorithm for evaluating PPP models under single-valued neutrosophic hesitant fuzzy environment, which can deal with the situation where the weighting information with interactive characteristics is incompletely known. Based on the established models and the defined correlation coefficients, the main procedure is given as follows:

Step 1: With respect to a PPP decision-making problem, DMs evaluate the PPP model ${a_{i}}$ with respect to the risk factor ${c_{{l_{j}}}}$, which is denoted by the SVNHFE ${p_{i{l_{j}}}}=({h_{i{l_{j}}}},{\iota _{i{l_{j}}}},{g_{i{l_{j}}}})$ for all $i=1,2,\dots ,m$, where j is the first-level risk factor, $j=1,2,\dots ,k$, and ${l_{j}}$ is the second-level risk factor, ${l_{j}}={1_{j}},{2_{j}},\dots ,{q_{j}}$. The associated single-valued neutrosophic hesitant fuzzy matrices are defined as ${P^{j}}={({p_{i{l_{j}}}})_{m\times {q_{j}}}}$, $j=1,2,\dots ,k$;

Step 2: Model (26) or (28) is used to determine the optimal 2-additive measure ${\mu ^{j}}$ on the second-level risk factor set ${C^{j}}=\{{c_{{1_{j}}}},{c_{{2_{j}}}},\dots ,{c_{{q_{j}}}}\}$, $j=1,2,\dots ,k$. Then, (15) is adopted to calculate the Shapley values ${\phi _{{c_{{l_{j}}}}}}({\mu ^{j}},{C^{j}})$, ${l_{j}}={1_{j}},{2_{j}},\dots ,{q_{j}}$, and $j=1,2,\dots ,k$;

Step 3: Using the single-valued neutrosophic hesitant fuzzy 2-additive Shapley weighted aggregation (SVNHF-2ASWA) operator to calculate the comprehensive SVNHFEs for first-level risk factor ${c_{j}}$, where for all $i=1,2,\dots ,m$ and all $j=1,2,\dots ,k$;

Step 4: Again using model (26) or (28) to obtain the optimal 2-additive measure μ on the first-level risk factor set $C=\{{c_{1}},{c_{2}},\dots ,{c_{k}}\}$. Meanwhile, (15) is adopted to calculate the Shapley values ${\phi _{{c_{j}}}}(\mu ,C)$, $j=1,2,\dots ,k$;

Step 5: Let ${p_{i}}=({p_{i1}},{p_{i2}},\dots ,{p_{ik}})$ with ${p_{ij}}=\{{h_{ij}},{\iota _{ij}},{g_{ij}}\}$ for all $i=1,2,\dots ,m$, and all $j=1,2,\dots ,k$. Using the SVNHF-2ASWCCs shown in Definition 10 to calculate the correlation coefficients $\left\{\begin{array}{l}C{C_{1}^{\phi }}({p_{i}},{p^{+}})\\ {} C{C_{1}^{\phi }}({p_{i}},{p^{-}})\end{array}\right.$ or $\left\{\begin{array}{l}C{C_{2}^{\phi }}({p_{i}},{p^{+}})\\ {} C{C_{2}^{\phi }}({p_{i}},{p^{-}})\end{array}\right.$, where ${p^{+}}={\textstyle\bigcup _{j=1}^{k}}{p_{j}}=\{{\max _{i=1}^{m}}\{{\gamma _{i}}\},{\min _{i=1}^{m}}\{{\delta _{i}}\},{\min _{i=1}^{m}}\{{\eta _{i}}\}|{\gamma _{i}}\in {h_{i}},{\delta _{i}}\in {\iota _{i}},{\eta _{i}}\in {g_{i}},i=1,2,\dots ,m\}$, ${p^{-}}={\textstyle\bigcup _{j=1}^{k}}{p_{j}}=\{{\min _{i=1}^{m}}\{{\gamma _{i}}\},$ ${\max _{i=1}^{m}}\{{\delta _{i}}\},{\max _{i=1}^{m}}\{{\eta _{i}}\}|{\gamma _{i}}\in {h_{i}},{\delta _{i}}\in {\iota _{i}},{\eta _{i}}\in {g_{i}},i=1,2,\dots ,m\}$;

Step 6: Based on the SVNHF-2ASWCCs, we employ the following equation to calculate the final ranking values or where $i=1,2,\dots ,m$;

Step 7: According to the final ranking values ${r_{i}^{C{C_{1}^{\phi }}}}$ or ${r_{i}^{C{C_{2}^{\phi }}}}$, $i=1,2,\dots ,m$, we can obtain the ranking of PPP models and then select the best one;

Step 8: End.

To show the procedure of the above algorithm clearly, please see Fig. 1.