Quantifying the Decision-Making of PPPs in China by the Entropy-Weighted Pareto Front: A URT Case from Guizhou

: In recent years, value for money (VFM) evaluations using the Public Sector Comparator (PSC) method have gradually been adopted by governments worldwide in the ﬁeld of public investment, and used as decision-making tools for public–private partnership (PPP) projects. However, there has been little research concerned with such emerging decision-making questions with VFM. This paper proposed a quantitative decision-making method of entropy-weighted Pareto front applied in the speciﬁc context of Chinese PPPs, with a case study of an Urban Transit Railway (URT) PPP project from Guizhou province. Factor analysis was used to extract the qualitative indicators of VFM evaluation, associate them with the quantitative characteristics of the project, and thus help public sector decision-makers choose the proper quantitative decision variables.


Introduction
Governments have used public-private partnerships (PPP) contracts worldwide as a tool to develop and manage public infrastructure and services, with the underlying goal of providing the best service at the lowest cost [1]. The World Bank 'PPP Reference Guide 2.0' provides a broad definition of PPP [2]: a long-term contract between a private party and a government entity, for providing a public asset or service, in which the private party bears significant risk and management responsibility, and remuneration is linked to performance. In China, whether or not a PPP project provides value for money (VFM) is measured by a VFM evaluation. The approach and techniques of VFM evaluation contain both qualitative and quantitative aspects to determine the reduction of project life cycle cost, optimize risk allocation, and improve the operational efficiency of a PPP project [3]. How to quantitatively determine the appropriate share of the private sector involvement, the proportion of risk transfer, charge to users, and the project duration becomes an important issue, since the result of decision-making should ensure not only that the public sector pay the lowest PPP project cost, also that the private sector could obtain an expected return.
Since the last decade, research studies have mainly focused on the problems of quantitative decision-making in PPPs. Studies on decision-making objects have included the concession period duration, the capital proportion of the private sector, extra profit allocation, the charge to users, project capacity, operation patterns, and risk transfer. The decision-making methods varied according to these decision objects.

Phase 1-Implement Factor Analysis
All of the steps in Phase 1 will be implemented with SPSS (version 21.0.0.0) software.
Step 1. Identify whether a certain linear relationship exists between the qualitative evaluation indicators. If yes, it will be suitable to extract factors by factor analysis. The common test methods include correlation coefficient matrix, the Bartlett Sphericity (BS) test, and the Kaiser-Meyer-Olkin (KMO) test. The correlation coefficient matrix will be generated at this step.
Step 2. Make a tentative extraction by principal component analysis (PCA) method. According to the correlation coefficient matrix of the original indicators, the initial component analysis will be applied to extract the factors with eigenvalues greater than 1. The initial solution of the factor analysis (the factor-loading matrix) will be obtained at this step.
Step 3. Determine the number of factors to be extracted by observing the scree plot and the result of the total variance interpreted.
Step 4. Implement the orthogonal rotation to the factor-loading matrix by the varimax method, and then output the result by the descending order of first factor loading. The rotated factor-loading matrix will be obtained at this step.

Phase 2-Calculate Entropy Weights
Step 1. Determine the indicators that will be used to calculate the entropy weights of the net cost of the private sector, the public sector, or the users. Suppose that there are n indicators, and that each has m evaluation objects. A m × n matrix will be obtained at this step. a ij (i = 1,2, . . . , m; j = 1,2, . . . n) is supposed to be an element of the matrix as the value of the i-th item in the j-th indicator.
Step 2. Standardize the values of all of the items. a' ij is supposed to be the standardized value of an item. Use: for the indicator on which project has better performance when the value of a ij is larger; or use: for the indicator on which project has the better performance when the value of a ij is smaller.
Step 3. Calculate the proportion y ij of a' ij in the matrix.
Step 4. Calculate the entropy value e j of every indicator.
Step 5. Calculate the difference coefficient d j of every indicator.
Step 6. Calculate the entropy weight of every indicator .

Phase 3-Build the Optimization Model
Step 1. Determine the decision variables of the optimization model based on the factor analysis results in Phase 1.
Step 2. Determine the functions of the optimization model objectives and constraints by the PSC method.
Step 3. Identify whether the optimization model has available solutions when it is required to satisfy the constraints that both the VFM and returns of the private sector are positive.
Step 2. Calculate the entropy-weighted sum of each set in the Pareto front solution sets.
Step 3. Find the minimum weighted sum, and the value set of its corresponding decision variables will be determined as the best quantitative decision of the project.

The Two Types of Optimization Models
There are usually two typical scenarios in the specific context of Chinese PPPs, so two types of optimization models need to be built.
For a newly-built PPP project, if there is no similar project built or under operation within the same city that could provide some guidance in project decision-making, i.e., there is no reference information of the project charge or concession period duration, etc., these pieces of information need to be treated as decision variables in the optimization model. However, the concrete discount rate to calculate the net present value (NPV) of the cost of the public sector should be determined as a parameter with a given value.
Else, if there are similar projects located within the same city, the project charge and concession period duration of the newly-built PPP project probably need to be consistent with the existing projects, i.e., these two parameters, charge and duration, should be treated as parameters with given values in the optimization model. In that scenario, the decision-makers of the public sector need to decide the discount rate within a reasonable range, i.e., the discount rate of the public sector cost could be treated as a decision variable.
In this paper, the optimization model of the first scenario is noted as 'Model 1', while that of the second scenario is noted as 'Model 2'.

A URT PPP Project Case
Guizhou is a province located in southwest China that is well-known for its enriched tourism resources. The capital city is Guiyang city, where two URT lines are to be opened to the public. The case study in this paper is that of Line 2, which is planned to have eight stations: six underground and two on-the-ground. The total length of Line 2 will be 13 km: 9.83 km of underground sections, and 3.17 km of on-the-ground sections. The concession period duration is 29 years: four years of construction and 25 years of operation. The basic public investment benefit rate is 3-8%, the reported discount rate for the public sector is 6.22%, and the expected return rate for the private sector is 6.37%. Table 1 listed the source data extracted from the project report [17], where C ci is the construction cost spent in the i-th year, yuan × 10 4 ; C oi is the operation cost spent in the i-th year, yuan × 10 4 ; C riski is the risk cost occurred in the i-th year, yuan × 10 4 ; Q i is the passenger turnover volume in the i-th year, km × 10 4 ; D i is the total planning operation kilometers of the i-th, km × 10 4 , E i is the project earning of the i-th year, yuan × 10 4 ; and B i is the general public budget of Guiyang city of the i-th year, yuan × 10 8 .

The Factor Analysis
There were 15 different indicators used in the VFM qualitative evaluation of 19 URT PPP projects [17]: 'Integration' (I1), 'Risk Allocation' (I2), 'Performance' (I3), 'Competitiveness' (I4), 'Government Competency' (I5), 'Bankability' (I6), 'Procurement Potential' (I7), 'Scale' (I8), 'Life span' (I9), 'Capital Diversity' (I10), 'Cost Accuracy' (I11), 'Legal Support' (I12), 'Income Potential' (I13), 'Exemplary' (I14), and 'Acceptance' (15). In the current Chinese PPPs' VFM evaluation system, the first six indicators are stipulated as basic ones that should be used in every PPP case, while the last 10 are specified as optional indicators that could be weighted as 0. In each PPP case, the sum of the weights is 1. Tables A1 and A2 listed the original weights of the 15 indicators among 19 URT PPP projects, in which project 7 did not use the basic indicator 'Bankability'. Table 2 listed the correlation matrix of 15 indicators. Due to the existence of optional indicators, this matrix is not a positive definite, so that the BS test and KMO test could not proceed. Linear correlation could be seen between most indexes, which means these indicators could be used for factor analysis.  I1  I2  I3  I4  I5  I6  I7  I8  I9  I10  I11  I12  I13  I14  In Figure 1, the horizontal axis is the number of the factors that were supposed to be extracted, and the vertical axis is the eigenvalue of the factor. The eigenvalues of the first four factors are high, in which these factors have strong interpretation to 15 indicators. From the fifth factor to the last factor, their eigenvalues are low, i.e., these factors are weak in interpretation of all 15 indicators.   I1  I2  I3  I4  I5  I6  I7  I8  I9  I10  I11  I12  I13  I14  I15  I1 1 In Figure 1, the horizontal axis is the number of the factors that were supposed to be extracted, and the vertical axis is the eigenvalue of the factor. The eigenvalues of the first four factors are high, in which these factors have strong interpretation to 15 indicators. From the fifth factor to the last factor, their eigenvalues are low, i.e., these factors are weak in interpretation of all 15 indicators.  Table 3 is the component matrix of the four extracted factors. The loading of each indicator on the first factor is relatively higher than those of the other three factors. However, the meaning of these four factors is still obscure.   Table 3 is the component matrix of the four extracted factors. The loading of each indicator on the first factor is relatively higher than those of the other three factors. However, the meaning of these four factors is still obscure.  Table 4 displayed the rotated component matrix outputted in the first factor loading descending order. After orthogonal rotation of the original component matrix, the meanings of the four factors become clearer. As could be seen from Table 4, the loadings of 'Procurement Potential', 'Capital Diversity', 'Government Competency' and 'Legal Support' are high on the first factor, in which these indicators are the main components that are interpreted by the first factor. Similarly, the second factor provide interpretation mainly of 'Performance', 'Competitiveness', and 'Scale'. The third factor mainly interpreted 'Integration' and 'Life span'. The fourth factor mainly interpreted 'Acceptance'. From the above paragraph, it seems that the indicators that were mainly interpreted by the first factor on related to government commitments and guarantees; the second factor was associated with the output and performance of the private sector involvement; the third factor was directly related to concession period duration; and the fourth factor reflects the degree of the public service accepted by users who are supposed to be sensitive with price level. Thus, the decision variables could be determined as the benchmark benefit ratio of public investment (r g ), which will be used as the discount rate of the public sector cost, the proportion of the public sector's share in the project company (a), the unit payment to the private sector (b), and the public transport services charge to the users (p).
The degree of total variance interpreted by the extracted factors is shown in Table 5. There are two groups of datasets; each has three columns of data, in turns: the eigenvalue of the factor, the contribution rate of variance interpretation, and the accumulative contribution rate of variance interpretation. In the first group dataset, 78.923% of the total variance of the 15 indicators was interpreted by the four factors before rotation. The second group dataset is the ultimate factor solution after rotation. The accumulative contribution rate of variance interpreted after rotation is not changed. However, after rotation, the factor contributions to the 15 indicators are reallocated, and the meanings of the four factors are less obscure. Model 1 has three objectives: the minimum cost of the private sector, the public sector, or a single user, expressed as equations (7) s.t. and In the above: C PUB = ar cap C c + bD/λ g + aC risky /λ g C PSC = C c + C oy D/λ g + C risky /λ g − pQ(1 + r t )/λ g C VFM = C PSC − C PUB (15) λ p = (1 + r p ) n r p /((1 + r p ) n − 1) λ g = (1 + r g ) n r g /((1 + r g ) n − 1) where [17], a is the project share of the public sector, which is capped at 49% (according to current policy in China); b is the unit payment for the private sector at every year, yuan/km; p is the unit charge to users of the public transportation service provide by the PPP project, yuan/person/km; n is the concession period duration, in years (in the 19 cases collected, the shortest duration is 10 years, and the longest one is 35 years.); λ g and λ p are the coefficients that convert the present value into the unit time (e.g., every year) value; C PRI is the net cost present value of the private sector, including the fund of equity share, the operating and maintenance cost, and risk expenditure (the payments from the public sector and users should be deducted); C PUB is the net cost present value of the public sector, including the share funds, payment for the private sector, and risk expenditure; C PSC is the Public Sector Comparator (PSC), which is the net cost present value of the public sector when implement the URT project in a traditional pattern other than the PPP pattern; C VFM is the difference between C PSC and C PPP that reflects the savings of the public investment for using PPP instead of a traditional pattern; C pri is the unit cost undertaken by the private sector every year, yuan/person/km; C pub is the unit cost undertaken by the public sector every year, yuan/person/km; C user is the unit cost of user, yuan/person/km, which is exactly the p; C c is the total investment of the project, which is also the sum of C ci in Table 1, 793293.91 yuan × 10 4 ; r cap is the capital fund proportion of total investment, 30%; D is the mean of D i in Table 1 Q is the mean of Q i in Table 1, 23533.16 km × person/year; C risky is the mean of C riski in Table 1, 8928.13 yuan × 10 4 /year; r g is the minimum basic public investment benefit rate, 3%, which is also the discount rate for calculating the net cost present value of public sector; r p is the minimum rate of return required by the private sector, 6.37%, which is also the discount rate for calculating the net cost present value of the private sector; r t is the ratio of non-ticket revenue taken up to the total ticket revenue, 15%; E other is the ratio of the external commercial development revenue taken up to the total project investment, which is set at 20% to ensure that the optimization model does have available solutions.

Model 2
Model 2 has two objectives: the minimum cost of the private sector and the public sector, as expressed in equations (18) and (19). The cost of single user p is fixed in this scenario. Constraints are the same as with Model 1.
In Model 2, b is capped by 89.49. This value is calculated by: which is stipulated by current PPP policy in China. C oi , C riski , D i , and Q i are listed in Table 1.

The Entropy Weights
The entropy weights of C pri , C pub , and C user in Model 1 are calculated on the variation degrees of E i, B i , and Q i in Table 1. The entropy weights of C PRI and C PPP in Model 2 are calculated by E i, and B i only. Table 6 presents the weights of C pri , C pub , and C user of Model 1, where the weights of C pri and C user are equal, and the weight of C pub is the smallest.  Table 7 presents the weight of C PRI and C PPP of Model 2, where the weight of C PRI is larger than that of C PPP .  Figure 2 depicts the Pareto front of Model 1. Figure 2a-c presents the correlation between each pair of C pri , C pub , and C user , and Figure 2d presents the distribution of the Pareto solutions of the three objectives in a three-dimensional (3D) space. It seems that C pub and C user are linear-correlated. However, it is not easy to determine the relation between C pri and C pub through the displayed complexion. Model 1 and Model 2 used the same configuration of the 'gamultiobj' solver to obtain optimized solutions. The maximum generation is set to 100,000, the initial population size is set to 1000, and the ratio of Pareto front to population size ratio is set to 0.3.

The Entropy Weights
The entropy weights of Cpri, Cpub, and Cuser in Model 1 are calculated on the variation degrees of Ei, Bi, and Qi in Table 1. The entropy weights of CPRI and CPPP in Model 2 are calculated by Ei, and Bi only. Table 6 presents the weights of Cpri, Cpub, and Cuser of Model 1, where the weights of Cpri and Cuser are equal, and the weight of Cpub is the smallest.  Table 7 presents the weight of CPRI and CPPP of Model 2, where the weight of CPRI is larger than that of CPPP.  Figure 2 depicts the Pareto front of Model 1. Figure 2a-c presents the correlation between each pair of Cpri, Cpub, and Cuser, and Figure 2d presents the distribution of the Pareto solutions of the three objectives in a three-dimensional (3D) space. It seems that Cpub and Cuser are linear-correlated. However, it is not easy to determine the relation between Cpri and Cpub through the displayed complexion. Model 1 and Model 2 used the same configuration of the 'gamultiobj' solver to obtain optimized solutions. The maximum generation is set to 100,000, the initial population size is set to 1000, and the ratio of Pareto front to population size ratio is set to 0.3.    Table 8 is the minimum weighted sum and the value of the related decision variables and the value of the three objectives. Cpub is slightly more than Cuser, which means that the user paid less than the public sector to the project. The share ratio of the public sector was close to its upper limit, and the whole concession period duration was much shorter than the longest one of 19 cases (35 years).  (c) C pri and C user ; (d) C pri , C pub and C user . Figure 3 presents the Pareto front of Model 2. It seems that there is a kind of strict linear correlation between C PRI and C PPP . The full version of the Pareto front generated by Model 1 and Model 2 are listed in Table A2.    Table 8 is the minimum weighted sum and the value of the related decision variables and the value of the three objectives. Cpub is slightly more than Cuser, which means that the user paid less than the public sector to the project. The share ratio of the public sector was close to its upper limit, and the whole concession period duration was much shorter than the longest one of 19 cases (35 years).   Table 8 is the minimum weighted sum and the value of the related decision variables and the value of the three objectives. C pub is slightly more than C user , which means that the user paid less than the public sector to the project. The share ratio of the public sector was close to its upper limit, and the whole concession period duration was much shorter than the longest one of 19 cases (35 years). As could be seen from Table 9, each decision variable is close to its own upper limit, and the net return obtained by the private sector through 25 years of operation exceeds the total investment of the project.

Discussion
From the collected 19 URT PPP project cases, it is understandable that to certain extent, the weight distribution among the 15 indicators in VFM qualitative evaluation reflected some of the quantifiable characteristics of the project that were valued by the experts who graded on the mark sheet. However, it may be difficult to directly obtain the correlation between the weight of the qualitative indicators and quantifiable characteristics. Factor analysis, as a common method of dimensionality reduction, could be used to explore such correlations by analyzing the data weights, and to find out the most important aspects and which qualitative evaluation indicators contributed to them substantially. Due to the application of the optional indicators in Chinese PPP practice, not all 15 indicators could be given non-zero weights, which then means that the correlation matrix is not positive definite, and the application of factor analysis tests are restricted. The unifying indicators in each evaluation may help in determining the quantitative decision variables with a more explanatory result factor analysis.
The entropy weight method is one kind of objective weighting that could avoid the bias caused by the expert preference that may occur in the subjective weighting methods. Under the same evaluation indicator, the greater the difference of the observed values, the smaller the entropy value of the corresponding indicator, and the greater the entropy weight. Initially, for Model 1, the author used the Pareto solutions directly as evaluation items under all three objectives-C pri , C pub , and C user -and found out that no matter how many times the experiment was conducted, the weights were equally distributed among the three evaluation indicators. The possible reason for that is the variance of observed values under each indicator produced by the solver equally. Thus, in order to reflect the difference degree of C pri , C pub , and C user , substituted indicators are needed.
This paper used E i , Bi, and Q i as the entropy-weighting indicators for Model 1, because the operation cost in every year is fixed by C oy and Q. To be consistent with Model 1 in entropy-weighting, Model 2 selected E i and Bi for calculating the weights of C PRI and C PUB . As seen from the results, in Model 1, the private sector and the users are equally weighted, and the public sector is slightly lower. In Model 2, the private sector has a slightly higher weight than the public sector. These results are basically consistent with the reality of Chinese PPPs practice, in which the public sector tent gives more benefits to the private sector and the users to enhance their willingness of participation, with non-negative VFM as the prerequisite of the project.
As seen in the Pareto front results of both Model 1 and Model 2, the private sector achieved profits based on the payment from the public sector and the users. In Model 1, b and p had the upper limit as positive infinity, while in Model 2, b had the upper limit and p was a constant parameter. However, b was much lower in Model 1 than in Model 2; since the users were burdened with a higher unit cost in Model 1, that unit price may be too expensive for the local citizen. In Model 2, in which the private sector yielded at a high level, the possible reason was that the decision variables a, b, and r g could not keep the private sector from over-profiting, reflecting that the current policies may not be reasonable. Figure 4 displays the variance (%) of C PRI and C PUB calculated by the decision picked out according to the minimum weighted sum in both models when some parameter added its own −20%, −10%, 0, +10%, +20%, respectively. Figure 4a shows that C PRI is very sensitive to most of the six parameters, r g and C riksy not included. Table 10 shows the actual values of C PRI when these parameters changed; the positive values are underlined, revealing that the private sector had a net loss, which failed to satisfy the prerequisites of a PPP project. Figure 4c presented that the C PRI value in Model 2 was insensitive to the parameters, except for r p . Figure 4b-d showed that C PUB was insensitive to the parameter changes in both models.
Sustainability 2018, 10, x FOR PEER REVIEW 13 of 16 was insensitive to the parameters, except for rp. Figure 4b-d showed that CPUB was insensitive to the parameter changes in both models.

Conclusions
In this paper, factor analysis was carried out on the qualitative evaluation weights of the 19 URT PPP cases, and the results were used to determine the quantitative decision variables of the

Conclusions
In this paper, factor analysis was carried out on the qualitative evaluation weights of the 19 URT PPP cases, and the results were used to determine the quantitative decision variables of the optimization models. Two types of nonlinear multi-objective models were established to solve the Pareto front solution sets, with constraints expressed as non-negative VFM values, and the return of the private sector upon the PSC method. The entropy-weights method was applied to evaluate the Pareto solutions, and the one with the minimum weighted sum was picked up. The calculated entropy weights showed consistency with the reality of Chinese PPPs practice. Finally, the sensitivity analysis of the present value of the public sector and private sector cost calculated by the determined decision variable value is obtained.
This paper provided a basic model under the scenario of static decision-making. The following two were not discussed throughout this article, but deserve further study: One is the financing decision model of a cost-recovery URT PPP project based on add-valuecapture (LVC) theory as the extension of the basic model in this paper. The other is the multi-stage dynamic and flexible decision-making problems over a whole life cycle of the URT PPP project, which can be divided into different stages of financing, bidding, construction, operation, maintaining, retrofitting, capital transferring, etc.