1 Introduction
2 Literature Review
2.1 Decision Making Methods for PPP Selection
2.2 Risk Factors and PPP Models
Table 1
The firstlevel risk factors  The secondlevel risk factors  Descriptions 
Constructive risk ${c_{1}}$  Construction cost overrun ${c_{{1_{1}}}}$  The term ‘construction cost overrun’ refers to the possibility that the infrastructure is incapable of delivering within the budget. 
Construction delay ${c_{{2_{1}}}}$  The term ‘construction delay’ refers to the possibility that the officials of the facility are incapable of delivering on time.  
Defective construction ${c_{{3_{1}}}}$  The term ‘defective construction’ refers to the situation in which the equipment, system or facility cannot meet the construction standards and requirements.  
Construction changes ${c_{{4_{1}}}}$  The term ‘construction changes’ refers to the equipment, system or infrastructure that need to be remedied or reworked due to construction defects or design changes.  
Economical risk ${c_{2}}$  Higher level of inflation risk ${c_{{1_{2}}}}$  The term ‘higher level of inflation risk’ refers to the possibility that the actual inflation rate will exceed the projected inflation rate. 
Higher levels of interest rate ${c_{{2_{2}}}}$  The term ‘higher levels of interest rate’ refers to the possibility that the actual interest rate will exceed the projected interest rate, which would lead to the increase of costs required for the construction or operations phase of the project, and would affect the availability and cost of funds.  
Higher levels of exchange rate ${c_{{3_{2}}}}$  The term ‘higher levels of exchange rate’ refers to the possibility that the actual exchange rate will exceed the projected exchange rate, which will lead to the increase of costs required for the construction or operations phase of the project.  
Political interference ${c_{3}}$  Political interference ${c_{3}}$  The term ‘political interference’ refers to the possibility of unforeseeable conduct by the political parties that materially and adversely affect the public decisionmaking process or project implementation. 
Financial risk ${c_{4}}$  Insurance increase ${c_{{1_{4}}}}$  The term ‘insurance increase’ refers to the possibility that the agreed project insurances become insurable or substantially increase in the rates at which insurance premiums are calculated. 
Ownership change ${c_{{2_{4}}}}$  The term ‘ownership change’ refers to the risk that a change in ownership would result in a weakening in its financial standing or support or other detriment to the project.  
Refinancing liabilities ${c_{{3_{4}}}}$  The term ‘refinancing liabilities’ is a postcontracting issue that we cannot model and assess if it would become ‘liability’ risk of the public sector when there is no information on real refinancing structure at the precontracting stage.  
Finance unavailable ${c_{{4_{4}}}}$  The term “finance unavailable” refers to the risk that when debt and/or equity required by the project is not available on the amounts and on the conditions anticipated to perform the project. 
Table 2
PPP models  Descriptions 
Service Contract (SC)  The government outsources several service items of public facilities such as road tolls and cleaning services. However, the government still needs to be responsible for the operation and maintenance of facilities and undertakes the risks of project financing, construction, and operation. Such agreements are usually shorter than five years. 
Management Contract (MC)  The government and the private sector sign an agreement on the operation and maintenance of facilities. Under the agreement, the private sector takes full responsibility for the operation and maintenance, but does not undertake the capital risks. The purpose of this model is to improve the operational efficiency and quality of service of facilities. 
DesignBuild (DB)  The government and the private sector sign an agreement. The private sector is responsible for designing and building the facilities according to the government’s standards and performance requirements. Once the facilities are completed, the government has the ownership and takes charge of the operation and management. Most nonoperating municipal projects, including roads, highways, sewage treatment plants, and other government facilities, can take this model. 
Turnkey Operation (TO)  The government and the private sector sign a turnkey agreement. Under government investment, the private sector is responsible for the design, construction, and operation of the projects for a period of time. The government sets the performance targets and has the ownership of the projects. In this model, the government not only owns the project, but also benefits from private construction and operation. This model is also suitable for the construction and operation of most nonoperating municipal projects. 
LeaseDevelopmentOperation (LDO)  The government and the private sector sign a longterm lease agreement. The private sector leases existing municipal facilities and pays rental fees to the government. The private sector expands existing facilities based on its capital or financing capacity and is responsible for operation and maintenance of the extended municipal facilities for commercial profits. 
BuildTransferOperation (BTO)  The government and the private sector sign an agreement. The construction of the facility is financed by the private sector. When the facility is completed, the private sector transfers the ownership of the facility to the government. Then, the government and the private sector sign a franchise agreement to lease the facilities to partners in the form of longterm leases. During the lease period, the private partner has the opportunity to recover their investments and obtain a reasonable return. 
TransferOperationTransfer (TOT)  The government and the private sector sign a franchise agreement to transfer the completed infrastructure to the franchisor. Based on the future benefits of the project, the government provides onetime funding for the private sector to develop new infrastructure. During the franchise period, the private sector operates projects independently in accordance with state laws, relevant policies and regulations, and government supervision. The private sector recovers cash inflows from projects as a return on investment. At the end of the franchise period, the government withdraws the franchise of projects. In general, the transfer involves only the operation of the project. 
BuildOperationTransfer (BOT)  The government and the private sector sign a franchise agreement that authorizes the private sector to undertake investment, financing, construction, operation, and maintenance of the projects during the franchise period. At the end of the concession period, the government or its affiliates will pay a certain amount of capital (or free of charge) in accordance with the agreement. 
Concession Operation (CO)  The government and the private sector sign a franchise agreement. The private sector has the government franchise and is responsible for financing, construction, operation, maintenance and management of public facilities. Profits are obtained by charging users under the government supervision in a certain period of time. After the expiration of the franchise period, the franchise will be transferred to the government. 
Joint Venture (JV)  The government and the private sector set up a project company to design, finance, build and manage projects jointly. The government and the private sector enjoy rights and responsibilities according to the proportion of equity. 
Equity Transfer (ET)  The government transfers part of the ownership of municipal facilities to the private sector and closely links the interests of the government and the private sector. It not only guarantees the government’s control over municipal facilities, but also acquires the private sector’s technical and managerial experience to a large extent. 
BuildOperation (BBO)  The government sells the original rebuilt and expended municipal infrastructures to the private sector. The private sector is responsible for reconstruction, expansion and permanent ownership of infrastructure. In this model, the government translates all risk into the private sector and only has the regulatory function. 
BuildOwnOperation (BOO)  In this model, there is no need to transfer the ownership of the project to the government. The private sector is responsible for financing, building and ownership of municipal facilities, as well as owning the permanent operation of facilities. The government translates all risk into the private sector and only has the regulatory function. 
Table 3
Contracting out  Component outsourcing Turnkey Lease  Service Contract Management Contract DesignBuild Turnkey Operation LeaseDevelopmentOperation BuildTransferOperation  High public ownership ↓ 
Types of franchise  Partial license (state fee model)  TransferOperationTransfer BuildOperationTransfer  
Concession (private fee model)  Concession Operation  
Types of privatization  Partial privatization  JointVenture Equity Transfer  
Entire privatization  BuyBuildOperation BuildOwnOperation  High privatization 
2.3 Decision Making Methods with SVNHFSs
 (1) All of these methods are based on the assumption that the weighting information is completely known. Thus, none of them can deal with the case where the weighting information is incompletely known.
 (2) Although two references discussed the case where there are interactions among the weights of criteria, they employed the lamdafuzzy measure based Choquet integral. There are two drawbacks of such type of aggregation operators:
 (i) The lamdafuzzy measure can only reflect the complementary, mutual, or independent interactions among the weights of criteria. However, when there are interactions, these three cases may exist simultaneously (Meng and Chen, 2015a);
 (ii) The Choquet integral only considers the interactions between two adjacent coalitions (Meng and Tang, 2013) that cannot globally show the interactions among criteria coalitions.

 (3) The previous distance measures, similarity measures, correlation coefficient of SVNHFSs (Biswas et al., 2016a, Şahin and Liu, 2016, 2017) all need the compared SVNHFSs to have the same length. Otherwise, we need to add extra values into SVNHFSs with the less numbers of elements the hesitant preferred, hesitant indeterminacy and hesitant nonpreferred degree sets. This procedure in fact changes the original information offered by DMs;
 (4) The grey relational analysis method (Biswas et al., 2016b) is based on the offered distance measure that cannot ensure the distance measure between two SVNHFSs to be equal to zero if and only if they are identical.
2.4 Correlation Coefficient
3 Basic Concepts
Definition 1 (See Wang et al., 2010).
(1)
\[ A=\big\{\big\langle x,{T_{A}}(x),{I_{A}}(x),{F_{A}}(x)\big\rangle \hspace{0.1667em}\big\hspace{0.1667em}x\in X\big\},\]Definition 2 (See Torra, 2010).
(2)
\[ E=\big(\big\langle {x_{i}},{h_{E}}({x_{i}})\big\rangle \hspace{0.1667em}\big\hspace{0.1667em}{x_{i}}\in X\big),\]Definition 3 (See Ye, 2015).
(3)
\[ S=\big\{\big\langle {x_{i}},{h_{S}}({x_{i}}),{\iota _{S}}({x_{i}}),{g_{S}}({x_{i}})\big\rangle \hspace{0.1667em}\big\hspace{0.1667em}{x_{i}}\in X\big\},\]Definition 4 (See Şahin and Liu, 2017).
(4)
\[\begin{aligned}{}& {\rho _{\mathrm{SVNHFSs}}}(A,B)=\frac{{C_{\mathrm{SVNHFSs}}}(A,B)}{\sqrt{{C_{\mathrm{SVNHFSs}}}(A,A)}\sqrt{{C_{\mathrm{SVNHFSs}}}(B,B)}}\\ {} & \hspace{1em}=\displaystyle \frac{{\textstyle\sum \limits_{i=1}^{n}}(\frac{1}{{k_{i}}}{\textstyle\sum \limits_{s=1}^{{k_{i}}}}{\gamma _{{A_{\sigma (s)}}}}({x_{i}}){\gamma _{{B_{\sigma (s)}}}}({x_{i}})+\frac{1}{{p_{i}}}{\textstyle\sum \limits_{z=1}^{{p_{i}}}}{\delta _{{A_{\sigma (Z)}}}}({x_{i}}){\delta _{{B_{\sigma (Z)}}}}({x_{i}})+\frac{1}{{l_{i}}}{\textstyle\sum \limits_{t=1}^{{l_{i}}}}{\eta _{{A_{\sigma (t)}}}}({x_{i}}){\eta _{{B_{\sigma (t)}}}}({x_{i}}))}{\sqrt{{\textstyle\sum \limits_{i=1}^{n}}(\frac{1}{{k_{i}}}{\textstyle\sum \limits_{s=1}^{{k_{i}}}}{\gamma _{{A_{\sigma (s)}}}^{2}}({x_{i}})+\frac{1}{{p_{i}}}{\textstyle\sum \limits_{z=1}^{{p_{i}}}}{\delta _{{A_{\sigma (Z)}}}^{2}}({x_{i}})+\frac{1}{{l_{i}}}{\textstyle\sum \limits_{t=1}^{{l_{i}}}}{\eta _{{A_{\sigma (t)}}}^{2}}({x_{i}})}\times \sqrt{{\textstyle\sum \limits_{i=1}^{n}}(\frac{1}{{k_{i}}}{\textstyle\sum \limits_{s=1}^{{k_{i}}}}{\gamma _{{B_{\sigma (s)}}}^{2}}({x_{i}})+\frac{1}{{p_{i}}}{\textstyle\sum \limits_{z=1}^{{p_{i}}}}{\delta _{{B_{\sigma (Z)}}}^{2}}({x_{i}})+\frac{1}{{l_{i}}}{\textstyle\sum \limits_{t=1}^{{l_{i}}}}{\eta _{{B_{\sigma (t)}}}^{2}}({x_{i}})}},\end{aligned}\]4 New SingleValued Neutrosophic Hesitant Fuzzy Correlation Coefficients
4.1 Two New SVNHFCCs in View of Geometric Mean and Maximum
Definition 5.
(5)
\[\begin{aligned}{}& d({\gamma _{i}},{h_{B}})=\underset{{\gamma _{j}}\in {h_{B}}}{\min }{\gamma _{i}}{\gamma _{j}},\end{aligned}\](8)
\[\begin{aligned}{}& {\gamma _{j}^{i}}=\min \big\{{\gamma _{j}}{\gamma _{j}}\in {h_{B}},{\gamma _{i}}{\gamma _{j}}=d({\gamma _{i}},{h_{B}})\big\},\end{aligned}\]Definition 6.
(11)
\[\begin{aligned}{}C(A,B)& ={\sum \limits_{k=1}^{n}}\bigg(\bigg(\frac{1}{l({h_{A}}({x_{k}}))}\sum \limits_{{\gamma _{i}}({x_{k}})\in {h_{A}}({x_{k}})}{\gamma _{i}}({x_{k}}){\gamma _{j}^{i}}({x_{k}})\\ {} & \hspace{1em}+\frac{1}{l({h_{B}}({x_{k}}))}\sum \limits_{{\gamma _{j}}({x_{k}})\in {h_{B}}({x_{k}})}{\gamma _{j}}({x_{k}}){\gamma _{i}^{j}}({x_{k}})\bigg)\\ {} & \hspace{1em}+\bigg(\frac{1}{l({\iota _{A}}({x_{k}}))}\sum \limits_{{\delta _{i}}({x_{k}})\in {\iota _{A}}({x_{k}})}{\delta _{i}}({x_{k}}){\delta _{j}^{i}}\\ {} & \hspace{1em}+\frac{1}{l({\iota _{B}}({x_{k}}))}\sum \limits_{{\delta _{j}}({x_{k}})\in {\iota _{B}}({x_{k}})}{\delta _{j}}({x_{k}}){\delta _{i}^{j}}({x_{k}})\bigg)\\ {} & \hspace{1em}+\bigg(\frac{1}{l({g_{A}}({x_{k}}))}\sum \limits_{{\eta _{i}}({x_{k}})\in {g_{A}}({x_{k}})}{\eta _{i}}({x_{k}}){\eta _{j}^{i}}({x_{k}})\\ {} & \hspace{1em}+\frac{1}{l({g_{B}}({x_{k}}))}\sum \limits_{{\eta _{j}}({x_{k}})\in {g_{B}}({x_{k}})}{\eta _{j}}({x_{k}}){\eta _{i}^{j}}({x_{k}})\bigg)\bigg),\end{aligned}\]Proposition 1.
 (i) $C(A,B)=C(B,A)$;
 (ii) $C(A,A)=2E(A)$, where\[\begin{aligned}{}E(A)& ={\sum \limits_{k=1}^{n}}\bigg(\frac{1}{l({h_{A}}({x_{k}}))}\sum \limits_{\gamma ({x_{k}})\in {h_{A}}({x_{k}})}\gamma {({x_{k}})^{2}}+\frac{1}{l({\iota _{A}}({x_{k}}))}\sum \limits_{\delta ({x_{k}})\in {\iota _{A}}({x_{k}})}\delta {({x_{k}})^{2}}\\ {} & \hspace{1em}+\frac{1}{l({g_{A}}({x_{k}}))}\sum \limits_{\eta ({x_{k}})\in {g_{A}}({x_{k}})}\eta {({x_{k}})^{2}}\bigg).\end{aligned}\]
Definition 7.
Proposition 2.
Proof.
4.2 Two SingleValued Neutrosophic Hesitant Fuzzy 2Additive Shapley Weighted Correlation Coefficients
Definition 8 (See Sugeno, 1974).
Definition 9 (See Grabisch, 1997).
Theorem 1 (See Grabisch, 1997).
(15)
\[ {\phi _{i}}(\mu ,N)=\frac{3n}{2}\mu (i)+\frac{1}{2}\sum \limits_{j\in N\setminus i}\big(\mu (i,j)\mu (j)\big)\]Definition 10.
(16)
\[ C{C_{1}^{\phi }}=\frac{{C_{\phi }}(A,B)}{\sqrt{{E_{\phi }}(A){E_{\phi }}({B^{A}})}+\sqrt{{E_{\phi }}(B){E_{\phi }}({A^{B}})}}.\](17)
\[ C{C_{2}^{\phi }}=\frac{{C_{\phi }}(A,B)}{\max \{{E_{\phi }}(A),{E_{\phi }}({B^{A}})\}+\max \{{E_{\phi }}(B),{E_{\phi }}({A^{B}})\}},\]Proposition 4.
Remark 1.
Proposition 5.
5 An Approach to Evaluating PPP Models
5.1 Models for Determining the Optimal Fuzzy Measure
Definition 11.
(20)
\[\begin{aligned}{}& {d_{INH}}(A,B)\\ {} & \hspace{1em}=\frac{1}{3n}{\sum \limits_{i=1}^{n}}\bigg(\frac{1}{{l_{hA}}({x_{i}})+{l_{hB}}({x_{i}})}\bigg(\hspace{0.1667em}\sum \limits_{{\gamma _{{h_{A}}({x_{i}})}}\in {h_{A}}({x_{i}})}\underset{{\gamma _{{h_{B}}({x_{i}})}}\in {h_{B}}({x_{i}})}{\min }{\gamma _{{h_{A}}({x_{i}})}}{\gamma _{{h_{B}}({x_{i}})}}\\ {} & \hspace{2em}+\sum \limits_{{\gamma _{{h_{B}}({x_{i}})}}\in {h_{B}}({x_{i}})}\underset{{\gamma _{{h_{A}}({x_{i}})}}\in {h_{A}}({x_{i}})}{\min }{\gamma _{{h_{B}}({x_{i}})}}{\gamma _{{h_{A}}({x_{i}})}}\bigg)\\ {} & \hspace{2em}+\frac{1}{{l_{\iota A}}({x_{i}})+{l_{\iota B}}({x_{i}})}\bigg(\hspace{0.1667em}\sum \limits_{{\delta _{{h_{A}}({x_{i}})}}\in {\iota _{A}}({x_{i}})}\underset{{\delta _{{h_{B}}({x_{i}})}}\in {\iota _{B}}({x_{i}})}{\min }{\delta _{{\iota _{A}}({x_{i}})}}{\delta _{{\iota _{B}}({x_{i}})}}\\ {} & \hspace{2em}+\sum \limits_{{\delta _{{h_{B}}({x_{i}})}}\in {\iota _{B}}({x_{i}})}\underset{{\delta _{{h_{A}}({x_{i}})}}\in {\iota _{A}}({x_{i}})}{\min }{\delta _{{\iota _{B}}({x_{i}})}}{\delta _{{\iota _{A}}({x_{i}})}}\bigg)\\ {} & \hspace{2em}+\frac{1}{{l_{gA}}({x_{i}})+{l_{gB}}({x_{i}})}\bigg(\hspace{0.1667em}\sum \limits_{{\eta _{{h_{A}}({x_{i}})}}\in {g_{A}}({x_{i}})}\underset{{\eta _{{h_{B}}({x_{i}})}}\in {g_{B}}({x_{i}})}{\min }{\eta _{{h_{A}}({x_{i}})}}{\eta _{{h_{B}}({x_{i}})}}\\ {} & \hspace{2em}+\sum \limits_{{\eta _{{h_{B}}({x_{i}})}}\in {g_{B}}({x_{i}})}\underset{{\eta _{{h_{A}}({x_{i}})}}\in {g_{A}}({x_{i}})}{\min }{\eta _{{g_{B}}({x_{i}})}}{\eta _{{g_{A}}({x_{i}})}}\bigg)\bigg),\end{aligned}\]Proposition 6.
Proof.
(21)
\[\begin{aligned}{}0& \leqslant \sum \limits_{{\gamma _{{h_{A}}({x_{i}})}}\in {h_{A}}({x_{i}})}\underset{{\gamma _{{h_{B}}({x_{i}})}}\in {h_{B}}({x_{i}})}{\min }{\gamma _{{h_{A}}({x_{i}})}}{\gamma _{{h_{B}}({x_{i}})}}\\ {} & \hspace{1em}+\sum \limits_{{\gamma _{{h_{B}}({x_{i}})}}\in {h_{B}}({x_{i}})}\underset{{\gamma _{{h_{A}}({x_{i}})}}\in {h_{A}}({x_{i}})}{\min }{\gamma _{{h_{B}}({x_{i}})}}{\gamma _{{h_{A}}({x_{i}})}}\leqslant {l_{hA}}({x_{i}}){+_{hB}}({x_{i}}).\end{aligned}\](22)
\[\begin{aligned}{}& \frac{1}{{l_{hA}}({x_{i}})+{l_{hB}}({x_{i}})}\bigg(\hspace{0.1667em}\sum \limits_{{\gamma _{{h_{A}}({x_{i}})}}\in {h_{A}}({x_{i}})}\underset{{\gamma _{{h_{B}}({x_{i}})}}\in {h_{B}}({x_{i}})}{\min }{\gamma _{{h_{A}}({x_{i}})}}{\gamma _{{h_{B}}({x_{i}})}}\\ {} & \hspace{1em}+\sum \limits_{{\gamma _{{h_{B}}({x_{i}})}}\in {h_{B}}({x_{i}})}\underset{{\gamma _{{h_{A}}({x_{i}})}}\in {h_{A}}({x_{i}})}{\min }{\gamma _{{h_{B}}({x_{i}})}}{\gamma _{{h_{A}}({x_{i}})}}\bigg)\\ {} & \hspace{1em}+\frac{1}{{l_{hB}}({x_{i}})+{l_{hC}}({x_{i}})}\bigg(\hspace{0.1667em}\sum \limits_{{\gamma _{{h_{B}}({x_{i}})}}\in {h_{B}}({x_{i}})}\underset{{\gamma _{{h_{C}}({x_{i}})}}\in {h_{C}}({x_{i}})}{\min }{\gamma _{{h_{B}}({x_{i}})}}{\gamma _{{h_{C}}({x_{i}})}}\\ {} & \hspace{1em}+\sum \limits_{{\gamma _{{h_{C}}({x_{i}})}}\in {h_{C}}({x_{i}})}\underset{{\gamma _{{h_{B}}({x_{i}})}}\in {h_{B}}({x_{i}})}{\min }{\gamma _{{h_{C}}({x_{i}})}}{\gamma _{{h_{B}}({x_{i}})}}\bigg)\\ {} & \hspace{1em}\frac{1}{{l_{hA}}({x_{i}})+{l_{hC}}({x_{i}})}\bigg(\hspace{0.1667em}\sum \limits_{{\gamma _{{h_{A}}({x_{i}})}}\in {h_{A}}({x_{i}})}\underset{{\gamma _{{h_{C}}({x_{i}})}}\in {h_{C}}({x_{i}})}{\min }{\gamma _{{h_{A}}({x_{i}})}}{\gamma _{{h_{C}}({x_{i}})}}\\ {} & \hspace{1em}+\sum \limits_{{\gamma _{{h_{C}}({x_{i}})}}\in {h_{C}}({x_{i}})}\underset{{\gamma _{{h_{A}}({x_{i}})}}\in {h_{A}}({x_{i}})}{\min }{\gamma _{{h_{C}}({x_{i}})}}{\gamma _{{h_{A}}({x_{i}})}}\bigg)\geqslant 0.\end{aligned}\](23)
\[\begin{aligned}{}{p_{j}^{+}}& ={\bigcup \limits_{i=1}^{m}}{p_{ij}}\\ {} & =\Big\{{\underset{i=1}{\overset{m}{\max }}}\{{\gamma _{ij}}\},{\underset{i=1}{\overset{m}{\min }}}\{{\delta _{ij}}\},{\underset{i=1}{\overset{m}{\min }}}\{{\eta _{ij}}\}\hspace{0.1667em}\big\hspace{0.1667em}{\gamma _{ij}}\in {h_{ij}},{\delta _{ij}}\in {\iota _{ij}},{\eta _{ij}}\in {g_{ij}},i=1,2,\dots ,m\Big\},\end{aligned}\](24)
\[\begin{aligned}{}{p_{j}^{}}& ={\bigcup \limits_{i=1}^{m}}{p_{ij}}\\ {} & =\Big\{{\underset{i=1}{\overset{m}{\min }}}\{{\gamma _{ij}}\},{\underset{i=1}{\overset{m}{\max }}}\{{\delta _{ij}}\},{\underset{i=1}{\overset{m}{\max }}}\{{\eta _{ij}}\}\hspace{0.1667em}\big\hspace{0.1667em}{\gamma _{ij}}\in {h_{ij}},{\delta _{ij}}\in {\iota _{ij}},{\eta _{ij}}\in {g_{ij}},i=1,2,\dots ,m\Big\}\end{aligned}\](25)
\[\begin{aligned}{}& \min {\sum \limits_{j=1}^{k}}{\sum \limits_{i=1}^{m}}{\phi _{{c_{j}}}}(\mu ,C)\frac{d({p_{ij}},{p_{j}^{+}})}{d({p_{ij}},{p_{j}^{+}})+d({p_{ij}},{p_{j}^{}})}\\ {} & \text{s.t.}\hspace{2.5pt}\left\{\begin{array}{l@{\hskip4.0pt}l}\mu (C)=1,\hspace{1em}\\ {} \mu (S)\leqslant \mu (T),\hspace{1em}& \forall S,T\subseteq C\hspace{2.5pt}\text{s.t.}\hspace{2.5pt}S\subseteq T,\\ {} \mu ({c_{j}})\geqslant 0,\hspace{1em}& j=1,2,\dots ,n,\end{array}\right.\end{aligned}\](26)
\[\begin{aligned}{}& \min \frac{3n}{2}{\sum \limits_{j=1}^{k}}{\sum \limits_{i=1}^{m}}\frac{d({p_{ij}},{p_{j}^{+}})}{d({p_{ij}},{p_{j}^{+}})+d({p_{ij}},{p_{j}^{}})}\mu ({c_{j}})\\ {} & \hspace{1em}+\frac{1}{2}{\sum \limits_{j=1}^{k}}{\sum \limits_{i=1}^{m}}\frac{d({p_{ij}},{p_{j}^{+}})}{d({p_{ij}},{p_{j}^{+}})+d({p_{ij}},{p_{j}^{}})}(\mu ({c_{j}},{c_{i}})\mu ({c_{i}}))\\ {} & \text{s.t.}\hspace{2.5pt}\left\{\begin{array}{l}\textstyle\sum \limits_{{c_{i}}\subseteq S\setminus {c_{j}}}(\mu ({c_{j}},{c_{i}})\mu ({c_{i}}))\geqslant (s2)\mu ({c_{j}}),\hspace{1em}\forall {c_{j}}\in S\subseteq C,s\geqslant 2,\\ {} \textstyle\sum \limits_{\{{c_{j}},{c_{i}}\}\subseteq C}\mu ({c_{j}},{c_{i}})(c2)\textstyle\sum \limits_{{c_{j}}\in C}\mu ({c_{j}})=1,\\ {} \mu ({c_{j}})\geqslant 0,\hspace{1em}j=1,2,\dots ,k,\end{array}\right.\end{aligned}\](27)
\[\begin{aligned}{}& \min {\sum \limits_{j=1}^{k}}{\sum \limits_{i=1}^{m}}{\phi _{{c_{j}}}}(\mu ,C)\frac{d({p_{ij}},{p_{j}^{+}})}{d({p_{ij}},{p_{j}^{+}})+d({p_{ij}},{p_{j}^{}})}\\ {} & \text{s.t.}\hspace{2.5pt}\left\{\begin{array}{l}\mu (C)=1,\\ {} \mu (S)\leqslant \mu (T),\hspace{1em}\forall S,T\subseteq C\hspace{2.5pt}\text{s.t.}\hspace{2.5pt}S\subseteq T,\\ {} \mu ({c_{j}})\in {W_{j}},\mu ({c_{j}})\geqslant 0,\hspace{1em}j=1,2,\dots ,k,\end{array}\right.\end{aligned}\](28)
\[\begin{aligned}{}& \min \frac{3n}{2}{\sum \limits_{j=1}^{k}}{\sum \limits_{i=1}^{m}}\frac{d({p_{ij}},{p_{j}^{+}})}{d({p_{ij}},{p_{j}^{+}})+d({p_{ij}},{p_{j}^{}})}\mu ({c_{j}})\\ {} & \hspace{1em}+\frac{1}{2}{\sum \limits_{j=1}^{k}}{\sum \limits_{i=1}^{m}}\frac{d({p_{ij}},{p_{j}^{+}})}{d({p_{ij}},{p_{j}^{+}})+d({p_{ij}},{p_{j}^{}})}(\mu ({c_{j}},{c_{i}})\mu ({c_{i}}))\\ {} & \text{s.t.}\hspace{2.5pt}\left\{\begin{array}{l}\textstyle\sum \limits_{{c_{i}}\subseteq S\setminus {c_{j}}}(\mu ({c_{j}},{c_{i}})\mu ({c_{i}}))\geqslant (s2)\mu ({c_{j}}),\hspace{1em}\forall {c_{j}}\in S\subseteq C,\hspace{2.5pt}s\geqslant 2,\\ {} \textstyle\sum \limits_{\{{c_{j}},{c_{i}}\}\subseteq C}\mu ({c_{j}},{c_{i}})(c2)\textstyle\sum \limits_{{c_{j}}\in C}\mu ({c_{j}})=1,\\ {} \mu ({c_{j}})\in {W_{j}},\mu ({c_{j}})\geqslant 0,\hspace{1em}j=1,2,\dots ,k,\end{array}\right.\end{aligned}\]5.2 An Algorithm for Evaluating PPP Models
(29)
\[\begin{aligned}{}{p_{ij}}& =\text{SVNHF2ASWA}({p_{i{1_{j}}}},{p_{i{2_{j}}}},\dots ,{p_{i{q_{j}}}})={\underset{l=1}{\overset{{q_{j}}}{\bigoplus }}}{p_{i{l_{j}}}}{\phi _{{c_{{l_{j}}}}}}\big({\mu ^{j}},{C^{j}}\big)\\ {} & =\Bigg\{\Bigg\{\bigcup \limits_{{\gamma _{i{1_{j}}}}\in {h_{i{1_{j}}}},{\gamma _{i{2_{j}}}}\in {h_{i{2_{j}}}},\dots ,{\gamma _{i{q_{j}}}}\in {h_{i{q_{j}}}}}{\prod \limits_{l=1}^{{q_{j}}}}{\gamma _{i{l_{j}}}}{\phi _{{c_{{l_{j}}}}}}\big({\mu ^{j}},{C^{j}}\big)\Bigg\},\\ {} & \hspace{1em}\Bigg\{\bigcup \limits_{{\delta _{i{1_{j}}}}\in {\iota _{i{1_{j}}}},{\delta _{i{2_{j}}}}\in {\iota _{i{2_{j}}}},\dots ,{\delta _{i{q_{j}}}}\in {\iota _{i{q_{j}}}}}{\prod \limits_{l=1}^{{q_{j}}}}{\delta _{i{l_{j}}}}{\phi _{{c_{{l_{j}}}}}}\big({\mu ^{j}},{C^{j}}\big)\Bigg\},\\ {} & \hspace{1em}\Bigg\{\bigcup \limits_{{\eta _{i{1_{j}}}}\in {g_{i{1_{j}}}},{\eta _{i{2_{j}}}}\in {g_{i{2_{j}}}},\dots ,{\eta _{i{q_{j}}}}\in {g_{i{q_{j}}}}}{\prod \limits_{l=1}^{{q_{j}}}}{\eta _{i{l_{j}}}}{\phi _{{c_{{l_{j}}}}}}\big({\mu ^{j}},{C^{j}}\big)\Bigg\}\Bigg\}\end{aligned}\](30)
\[ {r_{i}^{C{C_{1}^{\phi }}}}=\frac{C{C_{1}^{\phi }}({p_{i}},{p^{+}})}{C{C_{1}^{\phi }}({p_{i}},{p^{+}})+C{C_{1}^{\phi }}({p_{i}},{p^{}})}\](31)
\[ {r_{i}^{C{C_{2}^{\phi }}}}=\frac{C{C_{2}^{\phi }}({p_{i}},{p^{+}})}{C{C_{2}^{\phi }}({p_{i}},{p^{+}})+C{C_{2}^{\phi }}({p_{i}},{p^{}})},\]6 A Case Study
Example 1.
Table 4
${c_{11}}$: Construction cost overrun  ${c_{21}}$: Construction delay  ${c_{31}}$: Defective construction  ${c_{41}}$: Construction changes  
${a_{1}}$ (SC)  $\{\{0.3,0.5\},\{0.1,0.2\},\{0.3,0.4\}\}$  $\{\{0.5,0.6\},\{0.2,0.3\},\{0.3,0.4\}\}$  $\{\{0.2,0.3\},\{0.1,0.2\},\{0.5,0.6\}\}$  $\{\{0.6,0.7\},\{0.1,0.2\},\{0.2,0.3\}\}$ 
${a_{2}}$ (MC)  $\{\{0.6,0.7\},\{0.1\},\{0.2\}\}$  $\{\{0.6,0.7\},\{0.1,0.2\},\{0.1,0.2\}\}$  $\{\{0.5,0.6\},\{0.4\},\{0.2,0.3\}\}$  $\{\{0.5,0.7\},\{0.2\},\{0.3\}\}$ 
${a_{3}}$ (DB)  $\{\{0.4,0.6\},\{0.1\},\{0.3\}\}$  $\{\{0.7,0.8\},\{0.1\},\{0.2\}\}$  $\{\{0.4,0.6\},\{0.2,0.3\},\{0.4\}\}$  $\{\{0.3,0.5\},\{0.2\},\{0.1,0.2,0.3\}\}$ 
${a_{4}}$ (TO)  $\{\{0.1,0.4\},\{0.5\},\{0.6\}\}$  $\{\{0.8\},\{0.1\},\{0.2\}\}$  $\{\{0.4,0.5\},\{0.2,0.3\},\{0.1\}\}$  $\{\{0.4,0.6\},\{0.2\},\{0.3\}\}$ 
${a_{5}}$ (LDO)  $\{\{0.7\},\{0.2,0.3\},\{0.1\}\}$  $\{\{0.5,0.6\},\{0.3,0.4\},\{0.2\}\}$  $\{\{0.7,0.8\},\{0.1,0.2\},\{0.1\}\}$  $\{\{0.3,0.5\},\{0.2,0.3\},\{0.1\}\}$ 
${a_{6}}$ (BTO)  $\{\{0.6\},\{0.3,0.4\},\{0.1\}\}$  $\{\{0.8\},\{0.2\},\{0.1\}\}$  $\{\{0.5,0.7\},\{0.3\},\{0.1,0.2\}\}$  $\{\{0.4,0.6\},\{0.2,0.4\},\{0.1\}\}$ 
${a_{7}}$ (TDT)  $\{\{0.4,0.7\},\{0.3\},\{0.1\}\}$  $\{\{0.6\},\{0.4\},\{0.1,0.3\}\}$  $\{\{0.5\},\{0.3,0.4\},\{0.1,0.2\}\}$  $\{\{0.1\},\{0.2\},\{0.6,0.8\}\}$ 
${a_{8}}$ (BOT)  $\{\{0.5,0.8\},\{0.1,0.2\},\{0.1\}\}$  $\{\{0.4,0.7\},\{0.3\},\{0.1\}\}$  $\{\{0.2,0.3\},\{0.4\},\{0.5,0.6\}\}$  $\{\{0.7\},\{0.2,0.3\},\{0.1,0.2\}\}$ 
${a_{9}}$ (CO)  $\{\{0.4,0.5,0.7\},\{0.2,0.3\},\{0.1\}\}$  $\{\{0.4,0.5\},\{0.3,0.4\},\{0.1\}\}$  $\{\{0.6,0.8\},\{0.1,0.2\},\{0.1\}\}$  $\{\{0.5,0.8\},\{0.2\},\{0.1,0.2\}\}$ 
${a_{10}}$ (JV)  $\{\{0.6\},\{0.4\},\{0.1,0.3\}\}$  $\{\{0.3,0.5\},\{0.4\},\{0.1,0.2\}\}$  $\{\{0.7\},\{0.3\},\{0.1\}\}$  $\{\{0.5\},\{0.3,0.5\},\{0.2\}\}$ 
${a_{11}}$ (ET)  $\{\{0.4\},\{0.2,0.3\},\{0.1,0.2\}\}$  $\{\{0.1,0.2\},\{0.2,0.3,0.4\},\{0.6\}\}$  $\{\{0.5\},\{0.2\},\{0.1\}\}$  $\{\{0.4,0.7\},\{0.3\},\{0.1\}\}$ 
${a_{12}}$ (BBO)  $\{\{0.1\},\{0.1,0.2\},\{0.6,0.8\}\}$  $\{\{0.7\},\{0.1,0.2,0.3\},\{0.1\}\}$  $\{\{0.6\},\{0.2,0.4\},\{0.1\}\}$  $\{\{0.3,0.5\},\{0.2\},\{0.1\}\}$ 
${a_{13}}$ (BOO)  $\{\{0.4\},\{0.3\},\{0.2,0.1\}\}$  $\{\{0.6\},\{0.2,0.4\},\{0.1,0.2\}\}$  $\{\{0.1,0.2\},\{0.2,0.3\},\{0.5,0.7\}\}$  $\{\{0.7\},\{0.1,0.3\},\{0.2\}\}$ 
Table 5
${c_{12}}$: Higher level of inflation rate  ${c_{22}}$: Higher level of interest rate  ${c_{32}}$: Volatility of exchange rate  
${a_{1}}$ (SC)  $\{\{0.4,0.7\},\{0.2,0.3\},\{0.1,0.3\}\}$  $\{\{0.1,0.3\},\{0.2\},\{0.6,0.7\}\}$  $\{\{0.5,0.7\},\{0.3\},\{0.1,0.3\}\}$ 
${a_{2}}$ (MC)  $\{\{0.4,0.6\},\{0.2,0.3,0.4\},\{0.1,0.2\}\}$  $\{\{0.4,0.6\},\{0.3\},\{0.1,0.2\}\}$  $\{\{0.6,0.7\},\{0.3\},\{0.1\}\}$ 
${a_{3}}$ (DB)  $\{\{0.1\},\{0.1\},\{0.7,0.8,0.9\}\}$  $\{\{0.4,0.7\},\{0.2\},\{0.1,0.3\}\}$  $\{\{0.3,0.5\},\{0.4\},\{0.2,0.3\}\}$ 
${a_{4}}$ (TO)  $\{\{0.5,0.7,0.8\},\{0.1\},\{0.2\}\}$  $\{\{0.4,0.6\},\{0.2,0.4\},\{0.1\}\}$  $\{\{0.6\},\{0.3,0.4\},\{0.1\}\}$ 
${a_{5}}$ (LDO)  $\{\{0.5\},\{0.1,0.4\},\{0.3\}\}$  $\{\{0.1\},\{0.1,0.3\},\{0.5,0.7\}\}$  $\{\{0.4,0.7\},\{0.1,0.3\},\{0.2\}\}$ 
${a_{6}}$ (BTO)  $\{\{0.8\},\{0.1,0.2\},\{0.1\}\}$  $\{\{0.6,0.7\},\{0.3\},\{0.1,0.2\}\}$  $\{\{0.4,0.6\},\{0.3,0.4\},\{0.1,0.2\}\}$ 
${a_{7}}$ (TDT)  $\{\{0.3,0.6\},\{0.4\},\{0.2,0.3\}\}$  $\{\{0.5,0.6\},\{0.3,0.4\},\{0.1,0.2\}\}$  $\{\{0.6,0.8\},\{0.1,0.2\},\{0.1\}\}$ 
${a_{8}}$ (BOT)  $\{\{0.1\},\{0.3\},\{0.4,0.7\}\}$  $\{\{0.2,0.3\},\{0.4\},\{0.4,0.5\}\}$  $\{\{0.4,0.7\},\{0.3\},\{0.2\}\}$ 
${a_{9}}$ (CO)  $\{\{0.4,0.5\},\{0.3,0.5\},\{0.2\}\}$  $\{\{0.5,0.6\},\{0.1,0.3\},\{0.2\}\}$  $\{\{0.6\},\{0.3,0.4\},\{0.1,0.2\}\}$ 
${a_{10}}$ (JV)  $\{\{0.3,0.6\},\{0.4\},\{0.2,0.4\}\}$  $\{\{0.6,0.7\},\{0.2,0.3\},\{0.1\}\}$  $\{\{0.5\},\{0.3,0.4\},\{0.1,0.3\}\}$ 
${a_{11}}$ (ET)  $\{\{0.5\},\{0.3,0.4\},\{0.2\}\}$  $\{\{0.4,0.6\},\{0.3,0.4\},\{0.1,0.3\}\}$  $\{\{0.3,0.6\},\{0.4,0.5\},\{0.2\}\}$ 
${a_{12}}$ (BBO)  $\{\{0.1\},\{0.1,0.2\},\{0.7,0.8\}\}$  $\{\{0.1\},\{0.2,0.4\},\{0.4,0.6\}\}$  $\{\{0.1,0.5\},\{0.2,0.4\},\{0.3,0.5\}\}$ 
${a_{13}}$ (BOO)  $\{\{0.3\},\{0.2,0.4\},\{0.3,0.5\}\}$  $\{\{0.1,0.3\},\{0.2,0.4\},\{0.6,0.7\}\}$  $\{\{0.5,0.6\},\{0.1,0.4\},\{0.2\}\}$ 
Table 6
${a_{1}}$ (SC)  ${a_{2}}$ (MC)  ${a_{3}}$ (DB)  ${a_{4}}$ (TO)  ${a_{5}}$ (LDO)  ${a_{6}}$ (BTO)  ${a_{7}}$ (TDT)  
${c_{3}}$: Politics risk  $\{\{0.2,0.4\},\{0.6\},\{0.5\}\}$  $\{\{0.6,0.7\},\{0.3\},\{0.1,0.2\}\}$  $\{\{0.6\},\{0.3,0.4\},\{0.2\}\}$  $\{\{0.1,0.2\},\{0.7\},\{0.3\}\}$  $\{\{0.5,0.7\},\{0.3\},\{0.1,0.2\}\}$  $\{\{0.2,0.3\},\{0.5\},\{0.7\}\}$  $\{\{0.3\},\{0.5,0.6\},\{0.4\}\}$ 
${a_{8}}$ (BOT)  ${a_{9}}$ (CO)  ${a_{10}}$ (JV)  ${a_{11}}$ (ET)  ${a_{12}}$ (BBO)  ${a_{13}}$ (BOO)  
${c_{3}}$: Politics risk  $\{\{0.7\},\{0.3\},\{0.1,0.2\}\}$  $\{\{0.6,0.8\},\{0.2\},\{0.1\}\}$  $\{\{0.3\},\{0.5,0.6\},\{0.4\}\}$  $\{\{0.2\},\{0.5,0.6\},\{0.4\}\}$  $\{\{0.1\},\{0.4\},\{0.5,0.6\}\}$  $\{\{0.8\},\{0.1,0.2\},\{0.2\}\}$ 
Table 7
${c_{14}}$: Construction cost overrun  ${c_{24}}$: Construction delay  ${c_{34}}$: Defective construction  ${c_{44}}$: Construction changes  
${a_{1}}$ (SC)  $\{\{0.1,0.2\},\{0.4\},\{0.2,0.6\}\}$  $\{\{0.5,0.7\},\{0.2,0.3\},\{0.1\}\}$  $\{\{0.1\},\{0.2,0.3\},\{0.5,0.7\}\}$  $\{\{0.7\},\{0.1,0.2\},\{0.2,0.3\}\}$ 
${a_{2}}$ (MC)  $\{\{0.5,0.7\},\{0.1\},\{0.3\}\}$  $\{\{0.4,0.7\},\{0.2,0.3\},\{0.1,0.2\}\}$  $\{\{0.5,0.6\},\{0.4\},\{0.2\}\}$  $\{\{0.5,0.7\},\{0.2,0.3\},\{0.1\}\}$ 
${a_{3}}$ (DB)  $\{\{0.4,0.6\},\{0.1,0.3\},\{0.2\}\}$  $\{\{0.1\},\{0.3\},\{0.7,0.9\}\}$  $\{\{0.4,0.5\},\{0.4\},\{0.1,0.3\}\}$  $\{\{0.3,0.5\},\{0.1\},\{0.2,0.3\}\}$ 
${a_{4}}$ (TO)  $\{\{0.1,0.4\},\{0.5\},\{0.6\}\}$  $\{\{0.7,0.8\},\{0.1\},\{0.2\}\}$  $\{\{0.4,0.6\},\{0.2,0.3\},\{0.1,0.2\}\}$  $\{\{0.6\},\{0.2,0.4\},\{0.3\}\}$ 
${a_{5}}$ (LDO)  $\{\{0.7\},\{0.1,0.3\},\{0.1\}\}$  $\{\{0.5\},\{0.3,0.5\},\{0.2\}\}$  $\{\{0.1\},\{0.1,0.2\},\{0.5,0.8\}\}$  $\{\{0.3,0.4\},\{0.2,0.3\},\{0.1\}\}$ 
${a_{6}}$ (BTO)  $\{\{0.6\},\{0.3,0.4\},\{0.1,0.2\}\}$  $\{\{0.9\},\{0.1\},\{0.4\}\}$  $\{\{0.1\},\{0.3\},\{0.5,0.6\}\}$  $\{\{0.4,0.6\},\{0.2,0.3\},\{0.1\}\}$ 
${a_{7}}$ (TDT)  $\{\{0.4,0.7\},\{0.2,0.3\},\{0.1\}\}$  $\{\{0.4,0.6\},\{0.3\},\{0.1,0.2\}\}$  $\{\{0.5\},\{0.3,0.5\},\{0.1,0.2\}\}$  $\{\{0.6,0.8\},\{0.2\},\{0.1\}\}$ 
${a_{8}}$ (BOT)  $\{\{0.1\},\{0.1,0.2\},\{0.6,0.8\}\}$  $\{\{0.4,0.7\},\{0.2,0.3\},\{0.1\}\}$  $\{\{0.5,0.6\},\{0.4\},\{0.2,0.3\}\}$  $\{\{0.4,0.7\},\{0.3\},\{0.1,0.2\}\}$ 
${a_{9}}$ (CO)  $\{\{0.6,0.7\},\{0.1,0.3\},\{0.1\}\}$  $\{\{0.4,0.5\},\{0.3\},\{0.1,0.2\}\}$  $\{\{0.1\},\{0.1,0.2\},\{0.5,0.8\}\}$  $\{\{0.1,0.2\},\{0.3\},\{0.4,0.6\}\}$ 
${a_{10}}$ (JV)  $\{\{0.6\},\{0.3,0.4\},\{0.1,0.3\}\}$  $\{\{0.3,0.6\},\{0.4\},\{0.2,0.3\}\}$  $\{\{0.7\},\{0.2,0.3\},\{0.1\}\}$  $\{\{0.5\},\{0.3,0.4\},\{0.1\}\}$ 
${a_{11}}$ (ET)  $\{\{0.4,0.7\},\{0.2,0.3\},\{0.1,0.2\}\}$  $\{\{0.1,0.2\},\{0.2,0.4\},\{0.6\}\}$  $\{\{0.5\},\{0.2,0.4\},\{0.1\}\}$  $\{\{0.1\},\{0.4\},\{0.3,0.6\}\}$ 
${a_{12}}$ (BBO)  $\{\{0.1\},\{0.1,0.2\},\{0.7,0.8\}\}$  $\{\{0.7\},\{0.2,0.3\},\{0.1,0.2\}\}$  $\{\{0.5,0.6\},\{0.2,0.4\},\{0.1\}\}$  $\{\{0.3,0.5\},\{0.2,0.4\},\{0.1\}\}$ 
${a_{13}}$ (BOO)  $\{\{0.5\},\{0.3\},\{0.1,0.2\}\}$  $\{\{0.6\},\{0.2,0.4\},\{0.1,0.3\}\}$  $\{\{0.1,0.2\},\{0.2,0.4\},\{0.6,0.7\}\}$  $\{\{0.5\},\{0.1,0.4\},\{0.2\}\}$ 
Table 8
The firstlevel risk factor  The range of known weighting information  The secondlevel risk factor  The range of known weighting information 
${c_{1}}$: Construction risk  $[0.1,0.2]$  ${c_{{1_{1}}}}$: Construction cost overrun  $[0.1,0.2]$ 
${c_{{2_{1}}}}$: Construction delay  $[0.4,0.45]$  
${c_{{3_{1}}}}$: Defective construction  $[0.2,0.3]$  
${c_{{4_{1}}}}$: Construction changes  $[0.3,0.4]$  
${c_{2}}$: Economy risk  $[0.25,0.4]$  ${c_{{1_{2}}}}$: Higher level of inflation rate  $[0.15,0.4]$ 
${c_{{2_{2}}}}$: Higher level of interest rate  $[0.25,0.4]$  
${c_{{3_{2}}}}$: Volatility of exchange rate  $[0.3,0.5]$  
${c_{3}}$: Politics risk  $[0.35,0.45]$  ${c_{3}}$: Politics risk  $[0.35,0.5]$ 
${c_{4}}$: Finance risk  $[0.15,0.3]$  ${c_{{1_{4}}}}$: Insurance increases  $[0.1,0.25]$ 
${c_{{2_{4}}}}$: Ownership change  $[0.3,0.45]$  
${c_{{3_{4}}}}$: Refinancing liabilities  $[0.2,0.3]$  
${c_{{4_{4}}}}$: Finance unavailable  $[0.4,0.6]$ 
(32)
\[\begin{aligned}{}& \min 8.443(\mu ({c_{{1_{1}}}})+\mu ({c_{{2_{1}}}})+\mu ({c_{{3_{1}}}})+\mu ({c_{{4_{1}}}}))+4.327\mu ({c_{{1_{1}}}},{c_{{2_{1}}}})\\ {} & \hspace{1em}+4.116\mu ({c_{{3_{1}}}},{c_{{4_{1}}}}))+4.633\mu ({c_{{1_{1}}}},{c_{{3_{1}}}})+3.81\mu ({c_{{2_{1}}}},{c_{{4_{1}}}})\\ {} & \hspace{1em}+3.773\mu ({c_{{1_{1}}}},{c_{{4_{1}}}})+4.67\mu ({c_{{2_{1}}}},{c_{{3_{1}}}})\\ {} & \text{s.t.}\hspace{2.5pt}\left\{\begin{array}{l}\mu ({c_{{1_{1}}}})+\mu ({c_{{2_{1}}}})\mu ({c_{{1_{1}}}},{c_{{2_{1}}}})\leqslant 0,\mu ({c_{{1_{1}}}})+\mu ({c_{{3_{1}}}})\mu ({c_{{1_{1}}}},{c_{{3_{1}}}})\leqslant 0\\ {} \mu ({c_{{1_{1}}}})+\mu ({c_{14}})\mu ({c_{{1_{1}}}},{c_{14}})\leqslant 0,\mu ({c_{{2_{1}}}})+\mu ({c_{{3_{1}}}})\mu ({c_{{2_{1}}}},{c_{{3_{1}}}})\leqslant 0\\ {} \mu ({c_{{2_{1}}}})+\mu ({c_{{4_{1}}}})\mu ({c_{{2_{1}}}},{c_{{4_{1}}}})\leqslant 0,\mu ({c_{{3_{1}}}})+\mu ({c_{{4_{1}}}})\mu ({c_{{3_{1}}}},{c_{{4_{1}}}})\leqslant 0\\ {} \mu ({c_{{1_{1}}}})+\mu ({c_{{2_{1}}}})+\mu ({c_{{3_{1}}}})\mu ({c_{{1_{1}}}},{c_{{2_{1}}}})\mu ({c_{{1_{1}}}},{c_{{3_{1}}}})\leqslant 0\\ {} \mu ({c_{{1_{1}}}})+\mu ({c_{{2_{1}}}})+\mu ({c_{{3_{1}}}})\mu ({c_{{1_{1}}}},{c_{{2_{1}}}})\mu ({c_{{2_{1}}}},{c_{{3_{1}}}})\leqslant 0\\ {} \mu ({c_{{1_{1}}}})+\mu ({c_{{2_{1}}}})+\mu ({c_{{3_{1}}}})\mu ({c_{{1_{1}}}},{c_{{3_{1}}}})\mu ({c_{{2_{1}}}},{c_{{3_{1}}}})\leqslant 0\\ {} \mu ({c_{{1_{1}}}})+\mu ({c_{{2_{1}}}})+\mu ({c_{{4_{1}}}})\mu ({c_{{1_{1}}}},{c_{{2_{1}}}})\mu ({c_{{1_{1}}}},{c_{{4_{1}}}})\leqslant 0\\ {} \mu ({c_{{1_{1}}}})+\mu ({c_{{2_{1}}}})+\mu ({c_{{4_{1}}}})\mu ({c_{{1_{1}}}},{c_{{2_{1}}}})\mu ({c_{{2_{1}}}},{c_{{4_{1}}}})\leqslant 0\\ {} \mu ({c_{{1_{1}}}})+\mu ({c_{{2_{1}}}})+\mu ({c_{{4_{1}}}})\mu ({c_{{1_{1}}}},{c_{{4_{1}}}})\mu ({c_{{2_{1}}}},{c_{{4_{1}}}})\leqslant 0\\ {} \mu ({c_{{1_{1}}}})+\mu ({c_{{3_{1}}}})+\mu ({c_{{4_{1}}}})\mu ({c_{{1_{1}}}},{c_{{3_{1}}}})\mu ({c_{{1_{1}}}},{c_{{4_{1}}}})\leqslant 0\\ {} \mu ({c_{{1_{1}}}})+\mu ({c_{{3_{1}}}})+\mu ({c_{{4_{1}}}})\mu ({c_{{1_{1}}}},{c_{{3_{1}}}})\mu ({c_{{3_{1}}}},{c_{{4_{1}}}})\leqslant 0\\ {} \mu ({c_{{1_{1}}}})+\mu ({c_{{3_{1}}}})+\mu ({c_{{4_{1}}}})\mu ({c_{{1_{1}}}},{c_{{4_{1}}}})\mu ({c_{{3_{1}}}},{c_{{4_{1}}}})\leqslant 0\\ {} \mu ({c_{{2_{1}}}})+\mu ({c_{{3_{1}}}})+\mu ({c_{{4_{1}}}})\mu ({c_{{2_{1}}}},{c_{{3_{1}}}})\mu ({c_{{2_{1}}}},{c_{{4_{1}}}})\leqslant 0\\ {} \mu ({c_{{2_{1}}}})+\mu ({c_{{3_{1}}}})+\mu ({c_{{4_{1}}}})\mu ({c_{{2_{1}}}},{c_{{3_{1}}}})\mu ({c_{{3_{1}}}},{c_{{4_{1}}}})\leqslant 0\\ {} \mu ({c_{{2_{1}}}})+\mu ({c_{{3_{1}}}})+\mu ({c_{{4_{1}}}})\mu ({c_{{2_{1}}}},{c_{{4_{1}}}})\mu ({c_{{3_{1}}}},{c_{{4_{1}}}})\leqslant 0\\ {} \mu ({c_{{1_{1}}}},{c_{{2_{1}}}})+\mu ({c_{{1_{1}}}},{c_{{3_{1}}}})+\mu ({c_{{1_{1}}}},{c_{{4_{1}}}})+\mu ({c_{{2_{1}}}},{c_{{3_{1}}}})+\mu ({c_{{2_{1}}}},{c_{{4_{1}}}})\\ {} \hspace{1em}+\mu ({c_{{3_{1}}}},{c_{{4_{1}}}})2(\mu ({c_{{1_{1}}}})+\mu ({c_{{2_{1}}}})+\mu ({c_{{3_{1}}}})+\mu ({c_{{4_{1}}}}))=1\\ {} \mu ({c_{{1_{1}}}})\in [0.1,0.2],\hspace{1em}\mu ({c_{{2_{1}}}})\in [0.4,0.45],\\ {} \mu ({c_{{3_{1}}}})\in [0.2,0.3],\hspace{1em}\mu ({c_{{4_{1}}}})\in [0.3,0.4].\end{array}\right.\end{aligned}\](33)
\[\begin{aligned}{}& \min 8.443(\mu ({c_{1}})+\mu ({c_{2}})+\mu ({c_{3}})+\mu ({c_{4}}))+4.327\mu ({c_{1}},{c_{2}})\\ {} & \hspace{1em}+4.116\mu ({c_{3}},{c_{4}}))+4.633\mu ({c_{1}},{c_{3}})+3.81\mu ({c_{2}},{c_{4}})+3.773\mu ({c_{1}},{c_{4}})\\ {} & \hspace{1em}+4.67\mu ({c_{2}},{c_{3}})\\ {} & \text{s.t.}\hspace{2.5pt}\left\{\begin{array}{l}\mu ({c_{1}})+\mu ({c_{2}})\mu ({c_{1}},{c_{2}})\leqslant 0,\mu ({c_{1}})+\mu ({c_{3}})\mu ({c_{1}},{c_{3}})\leqslant 0\\ {} \mu ({c_{1}})+\mu ({c_{4}})\mu ({c_{1}},{c_{4}})\leqslant 0,\mu ({c_{2}})+\mu ({c_{3}})\mu ({c_{2}},{c_{3}})\leqslant 0\\ {} \mu ({c_{2}})+\mu ({c_{4}})\mu ({c_{2}},{c_{4}})\leqslant 0,\mu ({c_{3}})+\mu ({c_{4}})\mu ({c_{3}},{c_{4}})\leqslant 0\\ {} \mu ({c_{1}})+\mu ({c_{2}})+\mu ({c_{3}})\mu ({c_{1}},{c_{2}})\mu ({c_{1}},{c_{3}})\leqslant 0\\ {} \mu ({c_{1}})+\mu ({c_{2}})+\mu ({c_{3}})\mu ({c_{1}},{c_{2}})\mu ({c_{2}},{c_{3}})\leqslant 0\\ {} \mu ({c_{1}})+\mu ({c_{2}})+\mu ({c_{3}})\mu ({c_{1}},{c_{3}})\mu ({c_{2}},{c_{3}})\leqslant 0\\ {} \mu ({c_{1}})+\mu ({c_{2}})+\mu ({c_{4}})\mu ({c_{1}},{c_{2}})\mu ({c_{1}},{c_{4}})\leqslant 0\\ {} \mu ({c_{1}})+\mu ({c_{2}})+\mu ({c_{4}})\mu ({c_{1}},{c_{2}})\mu ({c_{2}},{c_{4}})\leqslant 0\\ {} \mu ({c_{1}})+\mu ({c_{2}})+\mu ({c_{4}})\mu ({c_{1}},{c_{4}})\mu ({c_{2}},{c_{4}})\leqslant 0\\ {} \mu ({c_{1}})+\mu ({c_{3}})+\mu ({c_{4}})\mu ({c_{1}},{c_{3}})\mu ({c_{1}},{c_{4}})\leqslant 0\\ {} \mu ({c_{1}})+\mu ({c_{3}})+\mu ({c_{4}})\mu ({c_{1}},{c_{3}})\mu ({c_{3}},{c_{4}})\leqslant 0\\ {} \mu ({c_{1}})+\mu ({c_{3}})+\mu ({c_{4}})\mu ({c_{1}},{c_{4}})\mu ({c_{3}},{c_{4}})\leqslant 0\\ {} \mu ({c_{2}})+\mu ({c_{3}})+\mu ({c_{4}})\mu ({c_{2}},{c_{3}})\mu ({c_{2}},{c_{4}})\leqslant 0\\ {} \mu ({c_{2}})+\mu ({c_{3}})+\mu ({c_{4}})\mu ({c_{2}},{c_{3}})\mu ({c_{3}},{c_{4}})\leqslant 0\\ {} \mu ({c_{2}})+\mu ({c_{3}})+\mu ({c_{4}})\mu ({c_{2}},{c_{4}})\mu ({c_{3}},{c_{4}})\leqslant 0\\ {} \mu ({c_{1}},{c_{2}})+\mu ({c_{1}},{c_{3}})+\mu ({c_{1}},{c_{4}})+\mu ({c_{2}},{c_{3}})+\mu ({c_{2}},{c_{4}})\\ {} \hspace{1em}+\mu ({c_{3}},{c_{4}})2(\mu ({c_{1}})+\mu ({c_{2}})+\mu ({c_{3}})+\mu ({c_{4}}))=1\\ {} \mu ({c_{1}})\in [0.1,0.2],\hspace{1em}\mu ({c_{2}})\in [0.4,0.45],\\ {} \mu ({c_{3}})\in [0.2,0.3],\hspace{1em}\mu ({c_{4}})\in [0.3,0.4].\end{array}\right.\end{aligned}\]$C{C_{1}^{\phi }}({p_{i}},{p^{+}})$  $C{C_{1}^{\phi }}({p_{i}},{p^{}})$  $C{C_{2}^{\phi }}({p_{i}},{p^{+}})$  $C{C_{2}^{\phi }}({p_{i}},{p^{}})$  
${a_{1}}$  0.77  0.9  0.74  0.80 
${a_{2}}$  0.96  0.66  0.86  0.53 
${a_{3}}$  0.91  0.74  0.77  0.59 
${a_{4}}$  0.71  0.82  0.67  0.69 
${a_{5}}$  0.94  0.71  0.79  0.55 
${a_{6}}$  0.76  0.81  0.72  0.80 
${a_{7}}$  0.78  0.86  0.67  0.70 
${a_{8}}$  0.93  0.65  0.85  0.58 
${a_{9}}$  0.97  0.57  0.94  0.49 
${a_{1}}0$  0.79  0.83  0.72  0.73 
${a_{1}}1$  0.68  0.88  0.54  0.69 
${a_{1}}2$  0.63  0.92  0.52  0.76 
${a_{1}}3$  0.94  0.63  0.85  0.57 
Table 10
Geometric mean based SVNHF2ASWCC  Maximum based SVNHF2ASWCC  Şahin and Liu’s correlation coefficient (Şahin and Liu, 2017)  
Ranking values of ${a_{1}}$  0.461  0.481  0.617 
Ranking values of ${a_{2}}$  0.593  0.619  0.875 
Ranking values of ${a_{3}}$  0.553  0.567  0.736 
Ranking values of ${a_{4}}$  0.464  0.493  0.594 
Ranking values of ${a_{5}}$  0.57  0.59  0.826 
Ranking values of ${a_{6}}$  0.484  0.475  0.651 
Ranking values of ${a_{7}}$  0.476  0.489  0.639 
Ranking values of ${a_{8}}$  0.59  0.596  0.804 
Ranking values of ${a_{9}}$  0.63  0.657  0.892 
Ranking values of ${a_{10}}$  0.488  0.494  0.658 
Ranking values of ${a_{11}}$  0.436  0.439  0.509 
Ranking values of ${a_{12}}$  0.408  0.407  0.468 
Ranking values of ${a_{13}}$  0.6  0.6  0.834 
Ranking using geometric mean based SVNHF2ASWCC  ${a_{9}}\succ {a_{13}}\succ {a_{2}}\succ {a_{8}}\succ {a_{5}}\succ {a_{3}}\succ {a_{10}}\succ {a_{6}}\succ {a_{7}}\succ {a_{4}}\succ {a_{1}}\succ {a_{11}}\succ {a_{12}}$  
Ranking using maximum based SVNHF2ASWCC  ${a_{9}}\succ {a_{2}}\succ {a_{13}}\succ {a_{8}}\succ {a_{5}}\succ {a_{3}}\succ {a_{10}}\succ {a_{4}}\succ {a_{1}}\succ {a_{7}}\succ {a_{6}}\succ {a_{11}}\succ {a_{12}}$  
Ranking using Şahin and Liu’s correlation coefficient  ${a_{9}}\succ {a_{2}}\succ {a_{13}}\succ {a_{5}}\succ {a_{8}}\succ {a_{3}}\succ {a_{10}}\succ {a_{6}}\succ {a_{7}}\succ {a_{1}}\succ {a_{4}}\succ {a_{11}}\succ {a_{12}}$ 
Table 11
Methods  Does it change the original decision making information?  Does it need the length of the compared SVNHFSs to be equal?  Does it consider the situation where the weighting information is incompletely known?  Can it deal with the case where there are interactive characteristics? 
Şahin and Liu’s method (Şahin and Liu, 2017)  YES  YES  NO  NO 
New method  NO  NO  YES  YES 