Solving the Asymmetry Multi-Objective Optimization Problem in PPPs under LPVR Mechanism by Bi-Level Programing

: Optimizing the cost and beneﬁt allocation among multiple players in a public-private partnership (PPP) project is recognized to be a multi-objective optimization problem (MOP). When the least present value of revenue (LPVR) mechanism is adopted in the competitive procurement of PPPs, the MOP presents asymmetry in objective levels, control variables and action orders. This paper characterizes this asymmetrical MOP in Stackelberg theory and builds a bi-level programing model to solve it in order to support the decision-making activities of both the public and private sectors in negotiation. An intuitive algorithm based on the non-dominated sorting genetic algorithm III (NSGA III) framework is designed to generate Pareto solutions that allow decision-makers to choose optimal strategies from their own criteria. The e ﬀ ectiveness of the model and algorithm is validated via a real case of a highway PPP project. The results reveal that the PPP project will be ﬁnancially infeasible without the transfer of certain amounts of exterior beneﬁts into supplementary income for the private sector. Besides, the strategy of transferring minimum exterior beneﬁts is more beneﬁcial to the public sector than to users.


Introduction
Public-private partnerships (PPP) have been applied worldwide as a financing instrument to provide public infrastructures and services. In a PPP project, the private sector is usually responsible for the operation of the project and for providing part of the initial investments, with a return on governmental subsidies, user payments or extraneous earnings derived from exterior benefits generated by the project [1]. In other words, when the private sector benefits increase, the costs from the public side are increased.
In the past few decades, the view that cost or benefit allocation among multiple players in a PPP project is actually a multi-objective optimization problem (MOP) has been formed. Koppenjan (2005) investigated the reasons for the stagnation of transport PPP in the Netherlands, finding that the mutually exclusive goals of the public and private sectors affected the sustainability of cooperation [2]. Papajohn (2011) put forward the view that the public is an important stakeholder of PPPs whose interest would be damaged if the government uses PPP to make excessive investments [3]. Garvin (2010) believed that PPP decision-making from the government needs to justify its economic feasibility by

The LPVR Mechanism
The original idea of LPVR is a flexible-term concession; in other words, the concession term is tied to certain return on investment (ROI) indicators for the private sector [17]. When future demand is better than expected, the private sector is allowed to end the contract earlier than estimated. In contrast, when the future demand is not as optimistic as expected, the private sector is allowed to extend the concession term in order to achieve the pre-set ROI. According to Engle et al. (1997,2001), the pre-set ROI will be used at a discounted rate to calculate the private sector's present value of revenue (PVR), which has been taken as the key variable in the tender of the concession contract [15,18]. The private sector with the least PVR, i.e., the LPVR, will be granted the concession contract. Equation (1) presents the general expression of the LPVR [18]: where pri LPVR is the net present value calculated in the base year; 0 C is the initial investment that is paid by the private sector in a PPP project; τ is the estimated concession term; ρ is the unit charge from users; i q is the estimated quantity of service or products provided by the project in the i-th year; and i m is the maintaining and operating cost in i-th year. Theoretically, an LPVR of 0 means that the private sector obtains revenues that are exactly equal to the costs incurred.
In the context of a PPP project, the net present value of revenue of the private sector-the PVR of the private sector-is also affected by decisions from the public sector, which cause variations in both the initial investment section and operation section. Such variations are shown in Equation (2), where (1) Δ is the initial investment shared by the public sector and (2) i Δ contains the operation subsidy and consumed exterior benefits of the i-th year. During the negotiation, the private sector needs to find the least PVR (the LPVR) to win in the bidding process.

The LPVR Mechanism
The original idea of LPVR is a flexible-term concession; in other words, the concession term is tied to certain return on investment (ROI) indicators for the private sector [17]. When future demand is better than expected, the private sector is allowed to end the contract earlier than estimated. In contrast, when the future demand is not as optimistic as expected, the private sector is allowed to extend the concession term in order to achieve the pre-set ROI. According to Engle et al. (1997,2001), the pre-set ROI will be used at a discounted rate to calculate the private sector's present value of revenue (PVR), which has been taken as the key variable in the tender of the concession contract [15,18]. The private sector with the least PVR, i.e., the LPVR, will be granted the concession contract. Equation (1) presents the general expression of the LPVR [18]: where LPVR pri is the net present value calculated in the base year; C 0 is the initial investment that is paid by the private sector in a PPP project; τ is the estimated concession term; ρ is the unit charge from users; q i is the estimated quantity of service or products provided by the project in the i-th year; and m i is the maintaining and operating cost in i-th year. Theoretically, an LPVR of 0 means that the private sector obtains revenues that are exactly equal to the costs incurred.
In the context of a PPP project, the net present value of revenue of the private sector-the PVR of the private sector-is also affected by decisions from the public sector, which cause variations in both the initial investment section and operation section. Such variations are shown in Equation (2), where ∆ (1) is the initial investment shared by the public sector and ∆ (2) i contains the operation subsidy and consumed exterior benefits of the i-th year. During the negotiation, the private sector needs to find the least PVR (the LPVR) to win in the bidding process.

The Interactive Stackelberg Decision-Making Process
Equation (2) shows that the private sector benefits are affected by the public sector's decisions. Correspondingly, the public sector benefits are also affected by the decisions of the private sector when the control variables ρ and τ of the private sector are used as input parameters for calculating benefits (or costs) at the public side. Supposing ∆ (1) and ∆ (2) i in Equation (2) are determined by strategy u proposed by the public sector, and ρ and τ are the elements of strategy v proposed by the private sector, an interactive decision-making process with the public sector as the leader and the private sector as the follower is shown in Figure 2.

The Interactive Stackelberg Decision-Making Process
Equation (2) shows that the private sector benefits are affected by the public sector's decisions. Correspondingly, the public sector benefits are also affected by the decisions of the private sector when the control variables ρ and τ of the private sector are used as input parameters for calculating benefits (or costs) at the public side. Supposing (1) Δ and (2) i Δ in Equation (2) are determined by strategy u proposed by the public sector, and ρ and τ are the elements of strategy v proposed by the private sector, an interactive decision-making process with the public sector as the leader and the private sector as the follower is shown in Figure 2. The above interactive decision-making process could be characterized by the Stackelberg decision-making problem [19], which has the following characteristics in general: (1) each decisionmaker has specific objectives and control variables, and their strategy will affect the benefits of the others as well as their own; (2) the objectives of different decision-makers are located in different levels, presenting a hierarchical decision-making structure-the upper decision-maker has higher priority in terms of the action order than the lower decision-maker; (3) the decisions of the upper and lower level impact each other, thus forming a relationship of mutual influence and restriction; and (4) the final decision will be a globally feasible solution that satisfies all decision-makers as much as possible.
The Stackelberg decision-making problem could be solved by a bi-level programing (BLP) model [20]. The general form of the BLP model consists of an upper programing (UP) problem and lower programing (LP) problem, which could be expressed as Equations (3)-(6), as shown in [21].
(UP) (5) subject to The above interactive decision-making process could be characterized by the Stackelberg decision-making problem [19], which has the following characteristics in general: (1) each decision-maker has specific objectives and control variables, and their strategy will affect the benefits of the others as well as their own; (2) the objectives of different decision-makers are located in different levels, presenting a hierarchical decision-making structure-the upper decision-maker has higher priority in terms of the action order than the lower decision-maker; (3) the decisions of the upper and lower level impact each other, thus forming a relationship of mutual influence and restriction; and (4) the final decision will be a globally feasible solution that satisfies all decision-makers as much as possible.
The Stackelberg decision-making problem could be solved by a bi-level programing (BLP) model [20]. The general form of the BLP model consists of an upper programing (UP) problem and lower programing (LP) problem, which could be expressed as Equations (3)-(6), as shown in [21].
(UP) min subject to g(u, v) ≤ 0 (6) where F(u, v(u)), u and G(u, v(u)) are the objective function, vector of control variables and constraint of the upper level decision-maker, respectively; f (u, v), v and g(u, v) are those of the lower decision-maker; and u is used as the input parameter when calculating f (u, v) and v(u) is used as the input parameter when calculating F(u, v(u)).

Modeling the Problem by Bi-Level Programing (BLP)
There are two prerequisites for both parties to participate in the PPP project: one is that the public sector should accomplish "value for money" (VFM)-the lifecycle costs paid in the PPP mode should be no more than that in the traditional investment mode (without private funding)-while the other one is that the private sector can obtain an actual return level that is no less than the pre-set rate of return. In other words, the values of VFM and LPVR are required to be positive to satisfy the two prerequisites.
Definitions of parameters and variables in the formulas of the model are listed in Table A1 of Appendix A.

The Upper Programing (UP) Problem
The UP problem is a three-objective MOP that is used to minimize the lifecycle cost of the public sector, the payment level of users and the consumption of exterior benefits. The constraint for the public sector to participate in a PPP project is to realize "value for money", i.e., the value of VFM should be non-negative. The objectives and constraints of the UP problem are formally expressed as follows: where G α, δ, ν (α,δ,ε) (ρ, τ) is the net present value of the lifecycle cost paid by the public sector under the PPP mode; ρ is the unit toll per vehicle-kilometer, representing the payment level of users; and ε is the ratio of collected exterior benefits used as supplementary income for the project. ν (α,δ,ε) (ρ, τ) is the input variable combination of ρ * , τ * , which is solved by the lower programing (LP) model upon the upper decision {α, δ, ε}. The full expression of the UP objective is: which consists of two terms. The first term of Equation (11) is the net present value (NPV) of equity funds shared by the public sector, and the second term is the NPV of subsidies that the public sector expends during the whole operation period. The value of VFM is defined as the difference between the lifecycle cost under traditional investment mode, called the public sector comparator (PSC) value, and that under the PPP mode expressed by G α, δ, ν (α,δ,ε) (ρ, τ) . Under the traditional investment mode, the equity funds of construction investment and operation costs are all afforded by the public sector. The value of PSC can be expressed as: Ir cap r con (1−α)Ir cap r con

The Lower Programing (LP) Problem
In the LP problem, the private sector company reduces their benefits due not to an original intention but to improve the chance of being selected as the partner, especially in a competitive bidding scenario under the LPVR mechanism. The objective function of the private sector company is to minimize their PVR, which is constrained by their own non-negative requirement. The model of the LP problem is formally expressed as follows: where P(α , δ , ε , ρ, τ) is the net present value of the private sector benefit, and {α , δ , ε } is given by the upper decision-maker (the public sector). The full expression of P(α , δ , ε , ρ, τ) is:

Optimization of Objectives in the BLP Model
As is known, it is difficult to find an absolute optimal solution in a MOP with conflicted objectives in which a series of comparatively optimal solutions, called Pareto solutions, could be obtained. Each Pareto solution performs better for at least one objective and worse for at least one objective than other Pareto solutions [22]. Pareto solutions can leave the decision-makers with adequate space for strategies without any requirements of weighting the multiple objectives as one general objective in advance.
In this paper, Pareto solutions will be solved and then compared to the original case in a case study to determine which objectives are present and the degree to which they are optimized. Figure 3 shows the schematic of how the Pareto solutions are demonstrated in the final results. For instance, in Figure 3, Pareto solution 1 reduces the payment of the public sector and users but increases the consumption of exterior benefits (EB) and the LPVR of the private sector; Pareto solution 2 reduces the payment of the public sector and users, keeps the consumption of EB the same, and lifts the LPVR; and Pareto solution 3 reduces the public sector costs, keeps the user payment level the same, and increases the consumption of EB and LPVR. It is noticeable that when the value of LPVR is more than zero, this means that the private sector could obtain more revenue than expected.

NSGAIII Framework-Based Algorithm for the BLP Model
BLP models are difficult to solve directly because of their status as NP-hard problems. The model built in Section 3 adds even difficulty since the UP problem is a three-objective nonlinear MOP and the LP problem is a mixed-integer nonlinear problem. Interactions between the two levels are hardly predicted; thus, the widely used Karush-Kuhn-Tucker (KKT) conditions [23] or sensitivity analysisbased methods [24] would not work for this problem.
In studies [25][26][27][28] aimed at solving such complex BLP models in recent years, embeddedcomputational approaches are applied. In general, the outer layer of their framework contains a heuristic algorithm to solve the UP problem, such as the genetic algorithm (GA), simulated annealing (SA) and particle swarm optimization (PSO). Then, a specific algorithm is applied to provide the optimal solutions to the LP problem, which are used as input parameters to compute the UP problem.
In this paper, the general idea used to solve the complete BLP model is to create an embeddedcomputational framework. The algorithm of non-dominating GAs-known as NSGAIII-is used to solve the three-dimensional MOP in the upper level. Additionally, the algorithm based on variable discretization is taken to quickly produce optimal solutions to the LP problem. NSGAIII is an algorithm developed on the GA framework to solve many-objective problems (MaOPs) [29] and has a good reputation for improving the evenness and diversity of solutions via the elite selection strategy of "reference points" [30].
In the built BLP model, problems α δ ε is an optimal solution of the upper problem. There are two main targets of the algorithm design of the BLP model: (1) to ensure that the LP problem produces feasible solutions upon each decision from the UP; and (2) to create a fast searching technique, which is required to solve the LP problem to avoid embedded iterations that would consume too much computing time, since the algorithm of the UP problem is evolution-based.

Interception Strategy for Reparing Solutions of the UP Problem
The main idea of this strategy is to intercept the UP solution{ } , , α δ ε that causes the LP not to be able to find a feasible solution under its own constraints. The interception strategy process includes the following steps.

NSGAIII Framework-Based Algorithm for the BLP Model
BLP models are difficult to solve directly because of their status as NP-hard problems. The model built in Section 3 adds even difficulty since the UP problem is a three-objective nonlinear MOP and the LP problem is a mixed-integer nonlinear problem. Interactions between the two levels are hardly predicted; thus, the widely used Karush-Kuhn-Tucker (KKT) conditions [23] or sensitivity analysis-based methods [24] would not work for this problem.
In studies [25][26][27][28] aimed at solving such complex BLP models in recent years, embedded-computational approaches are applied. In general, the outer layer of their framework contains a heuristic algorithm to solve the UP problem, such as the genetic algorithm (GA), simulated annealing (SA) and particle swarm optimization (PSO). Then, a specific algorithm is applied to provide the optimal solutions to the LP problem, which are used as input parameters to compute the UP problem.
In this paper, the general idea used to solve the complete BLP model is to create an embedded-computational framework. The algorithm of non-dominating GAs-known as NSGAIII-is used to solve the three-dimensional MOP in the upper level. Additionally, the algorithm based on variable discretization is taken to quickly produce optimal solutions to the LP problem. NSGAIII is an algorithm developed on the GA framework to solve many-objective problems (MaOPs) [29] and has a good reputation for improving the evenness and diversity of solutions via the elite selection strategy of "reference points" [30].
In the built BLP model, problems of both levels have independent constraints. Every individual in the final Pareto solution set of α, δ, ε, ρ, τ should be globally feasible, i.e., satisfying the constraints of both the UP problem and the LP problem. In each part of the combination of α, δ, ε, ρ, τ , ρ, τ is solved based on {α, δ, ε}, and {α, δ, ε} is an optimal solution of the upper problem. There are two main targets of the algorithm design of the BLP model: (1) to ensure that the LP problem produces feasible solutions upon each decision from the UP; and (2) to create a fast searching technique, which is required to solve the LP problem to avoid embedded iterations that would consume too much computing time, since the algorithm of the UP problem is evolution-based.

Interception Strategy for Reparing Solutions of the UP Problem
The main idea of this strategy is to intercept the UP solution {α, δ, ε} that causes the LP not to be able to find a feasible solution under its own constraints. The interception strategy process includes the following steps. Step 1: Generate N individuals of α, δ, ε, ρ max , τ max by the GA operator. ρ max and τ max are the upper limits of ρ and τ, respectively. N is the size of the population.
Step 2: Compute the value of the LP constraint function with N individuals of α, δ, ε, ρ max , τ max . Count the number of individuals which violate the LP constraint, and notate it as n.

Subsection Approximation Strategy for Optimizing the LP
When solving the LP, {α, δ, ε} is taken as the given parameter set. Besides this, τ is a set with a limited number of integer elements indicating the year of concession term, as τ= {1, 2, · · · , τ max }. Thus, Equation (16) can be equivalently transformed into Equation (17): Figure 4 shows the three-dimensional surface diagram with contours of the functional relationships between F LP , ρ and τ, where ρ is taken as a discretized variable. The F LP value remains the same at different positions on the same contour line. It can be seen from Figure 2 that larger values of ρ and τ produce larger values of F LP . Besides this, under the same value of F LP , ρ and τ present a kind of inverse correlation. Assuming that the private sector is more averse to long-term concessions than charging lower prices, the process of the subsection approximation strategy includes the following steps.
Symmetry 2020, 12, x FOR PEER REVIEW 8 of 19 Step  Thus, Equation (16) can be equivalently transformed into Equation (17): Figure 4 shows the three-dimensional surface diagram with contours of the functional relationships between LP F , ρ and τ , where ρ is taken as a discretized variable. The LP , ρ and τ present a kind of inverse correlation. Assuming that the private sector is more averse to long-term concessions than charging lower prices, the process of the subsection approximation strategy includes the following steps. Step 1: Discretize the continuous decision variable Step 2: According to the order that  Step 1: Discretize the continuous decision variable ρ ∈ [ρ min , ρ max ] according to some acceptable precision. For example, ρ min = 0 and ρ max = 0.69 at a precision of 0.01; then, ρ can be segmented into 70 pieces as ρ = {0.00, 0.01, 0.02, · · · , 0.69}.
Step 4: The finally obtained α, δ, ε, p , τ is taken as the approximate optimal solution of the LP.

Algorithm Design for the Complete BLP Model
The entire process of the proposed algorithm for the built BLP model is implemented with the 'PlatEMO' [31] source code, including the following steps ( Figure 5):   Step 0 Initialize the parameters of the NSGAIII algorithm. These parameters are population size N, the number of objective functions M and the maximum generation times gen. The actual size of population N will be determined by the inherent reference point generation method of NSGAIII [32].
Step 1: Generate a population of N individuals of {α, δ, ε} by real coding. In other words, generate N random numbers between the upper limit and lower limit of each variable (α, δ or ε), then rearrange them as an N × 3 matrix. Then, perform the interception strategy listed in Section 4.1 to obtain N globally feasible solutions, where each of them is a combination of α, δ, ε, p, τ .
Step 2: Calculate values of the UP problem's objective functions with N α, δ, ε, p, τ and find the minimum value for each objective to form the coordinate set Z min of the "ideal point". For example, each of z Step 3: Generate N parent individuals of {α, δ, ε} by the tournament selection strategy [33], then perform the Simulate Binary Crossover (SBX) [34] and Polynomial Mutation (PM) [35] operations to obtain the child population for the current epoch. Repeat the strategy listed in Section 4.1 with the tournament, SBX and PM operations as needed to make all child individuals available to solve the solutions of the LP.
Step 4: Combine the parent population and child population together to obtain 2N individuals of α, δ, ε, p, τ ; then, calculate their objective functions as fitness.
Step 5: Perform the Effective Non-Dominating Sorting (ENS) strategy [36] on the 2N individuals and allocate a non-dominating serial number to each individual. The elite selection strategy based on reference points [37] should be carried out for those individuals with the largest serial number.
Step 6: Repeat Step 3 to Step 5 until the maximum generation time gen is reached. The final population of individuals is taken as the Pareto solution set of the entire BLP.

The Big Outer Ring Highway PPP Project
According to the PPP implementation plan of the Big Outer Ring Highway (Figure 6), the total investment (I) is about 13.5 billion Yuan (1.94 billion USD at an exchange rate of 1:6.9501), the construction period is 2 years (2017-2018) and the operation period is 25 years (2019-2043). The capital fund ratio (r cap ) is 25% of the total investment, and the project has been granted a sum of extra funding (I ) from the central government equal to the amount of the capital funding. The interest rate of long-term loans for more than five years (r deb ) is 4.9%. The expected ratio of non-toll income to toll income (r nt ) is 2%. The least expected ROI of the private sector (r pri ) is 8%, and the ROI of the public sector (r gov ) is 4.25%. Other input parameters of the operation period are listed in Table 1.
The upper limit and lower limit of the decision variables are shown in Table 2. The upper limit of variable δ is determined according to the maximum unit traffic subsidy (shown in Table 2, column 4), while other values are extracted from the project plan report.
1 Figure 6. Sketch of the location of the Big Outer Ring Highways on a plain map.   Table 2. Values of the upper limit and lower limit of decision variables.

Concession Term τ (Years)
Upper limit

Evaluation Effects of Different Population Sizes
Population size is one of the key parameters of the non-dominating sorting GA algorithms. The effects of two population sizes N = 60 and N = 120 are compared with the NSGAIII algorithm. Figure 7 shows the distribution of the Pareto solutions of the two population sizes from different views in the three-dimensional space. Arrows point to the convergence direction. It can be seen from Figure 7 that the group of N = 120 performs better in terms of the convergence and evenness of solutions. There are also some solutions of group N = 60 that are relatively far away from the front edge (in the convergence direction) of the solution set, which means that these solutions are inferior to those that are right on or close to the front edge.
It can be seen from Figure 8a that a larger population creates better convergence, even though there is little difference between the two sizes of population in general. Values of the LP objective functions in the first two iterations are much higher than those in the subsequent iterations; thus, the convergence curve starts from the third to the 50th iteration, as shown independently in Figure 8b. It can be seen in Figure 8b that a larger population size leads to a better convergence than a smaller population size after the 17th iteration. Results of the group are kept for analysis in later sections of this paper.

Results and Discussion
gov C , user C , ε , VFM, LPVR and the private sector's IRR for all the 120 Pareto solutions, arranged in the order of increasing LPVR values, are plotted as stacked lines in Figure 9a. From the 120 Pareto solutions, the optimal strategies for minimizing the costs of the public sector (S1), minimizing user payment (S2), minimizing the transferred exterior benefits (S3) and minimizing the LPVR of the private sector (S4) are drawn out separately in Figure 9b. For each type of optimal strategy, only the strategies which generated the lowest LPVR are chosen. In that case, all four optimal strategies are located within the range of the first 10 solutions.

Results and Discussion
C gov , C user , ε, VFM, LPVR and the private sector's IRR for all the 120 Pareto solutions, arranged in the order of increasing LPVR values, are plotted as stacked lines in Figure 9a. From the 120 Pareto solutions, the optimal strategies for minimizing the costs of the public sector (S1), minimizing user payment (S2), minimizing the transferred exterior benefits (S3) and minimizing the LPVR of the private sector (S4) are drawn out separately in Figure 9b. For each type of optimal strategy, only the strategies which generated the lowest LPVR are chosen. In that case, all four optimal strategies are located within the range of the first 10 solutions.  It can be seen from Figure 9a that VFM and LPVR are both less than zero in the original case (the red dot lines), illustrating that strategies of the original case are not feasible for implementing the project via PPP. The ratio of consumed exterior benefits in all Pareto solutions is above that in the original case, which means the problem might be infeasible without transferring certain amounts of exterior benefits as supplementary income into the project. Variation of C gov and of VFM are not linearly correlated, revealing their sum, i.e., the PSC value is not stable. Therefore, it is not appropriate to use the VFM value to evaluate different PPP strategies.
Tables 3-5 separately show the changes when each optimal solution is compared with the original case solution (S 0 ) in different indicators. Their definitions are listed in Table A2 of Appendix A. ∆ 1~∆4 show the increments of S 1~S4 when compared to S 0 . Changes in α, ε, irr pri and τ are measured in absolute increments as: Changes in δ, ρ, C gov , PSC, VFM, F equity gov , S r gov , and B pri ,R sub ,R tol , F equity pri and C ope are measured in relative increments as: The following discussion is based on the results shown in Tables 3-5. S 1 offers a scheme of nearly complete privatization, which is without any capital fund or subsidy paid by the public sector. The results of S 1 reveal that the public sector can achieve maximum benefit when delivering the project in a fully market-based schema.
S 2 provides a schema equivalent to opening up the project to the public for free. Besides, adding 29.43% of exterior benefits realizes a 46.5% reduction in the public sector cost.
S 3 shows that at least 3.55% of exterior benefits need to be transferred into the project to meet the two prerequisites of PPP. In S3, the unit payment of users increases by 1.75% while the public sector cost reduces by 14.75%, which indicates that the strategy of minimum consumption of exterior benefits is more beneficial to the public sector than to users.
In S 4 , the actual IRR of the private sector is almost the same as the other three optimal strategies. Compared with S 1 , S 2 and S 3 , S 4 provides a more balanced schema, in which the consumption of exterior benefits increases by 30.93%, the cost of the public sector reduces by 66.62% and the unit payment of users decreases by 69.03%.
It can be seen in all four optimal strategies that shortening the concession term plays the most critical role in cutting down the cost of the public sector and increasing the benefit of the private sector. Both parties benefit from the reduction of their own costs in the operation period.

Conclusions
This paper studies the problem of Stackelberg game optimization under an LPVR mechanism when negotiating a transport PPP project between the public sector and the private sector. The decision-making scenario and the characteristics of the Stackelberg game during procurement are described and analyzed. Based on that analysis, a BLP model is constructed. The objective of the UP problem is to minimize the costs, including the public sector cost and the user payment. The constraint of the UP is to realize value for money for the public sector. The objective of the LP problem is to minimize the difference between the actual returns and the expected returns of the private sector, and its constraint is that the actual returns must be no less than the expected level.
An intuitive algorithm based on the NSGAIII framework is designed to solve the BLP model. In order to generate global feasible solutions that satisfy both the UP and LP constraints, an interception strategy is carried out to avoid infeasible solutions from UP to LP. Based on the analysis of the objective function of the LP, a subsection approximating algorithm to solve the LP by discretizing variables of the concession term and toll is proposed.
The effectiveness of the model and algorithm is validated by the PPP project of the Beijing Big Outer Ring highway. Solutions maximizing each single objective are selected from the Pareto solution set. Costs and benefits for both the public sector and the private sector under different solutions are compared and discussed. Main findings include that (1) the PPP project will be financially infeasible without transferring certain amounts of exterior benefits into supplementary income for the private sector; (2) the strategy of transferring the minimum exterior benefits is more beneficial to the public sector than to users; (3) the public sector can achieve maximum benefit when delivering the project in a fully market-based schema; and (4) the value of VFM is not suitable for use as the evaluation criteria for PPP investment schemas since its computation base, the PSC, is not static.
It is necessary to know that results in this study are case-specific, as it has not been ensured that each project generates the same level of returns. This paper is models the Stackelberg decision-making problem of two parties only. In future research, the public, local residents, tax payers, pure investors and other stakeholders in PPPs could be involved, and a multi-layered Stackelberg MOP could be built to further simulate more complex decision-making scenarios.
Author Contributions: F.L. drafted the original manuscript; J.L. and X.Y. supervised the entire study work and writing of the paper. All authors have read and agreed to the published version of the manuscript. The proportion of the funds invested in the j-th year of construction in the total investment r deb Rate of the debt fund r gov Discount rate of the public sector r pri Discount rate of the private sector e i The exterior benefits generated from project implementation in the i-th year Decision Variables α Initial investment for construction shared by the public sector. The private sector's share is notated as 1 − α. δ Subsidy per vehicle-kilometer ε Ratio of exterior benefits to be collected ρ Toll of per vehicle-kilometer τ Concession term In Table A1, o i and m i are expressed separately as follows.  NPV of the equity fund shared by the private sector S r gov NPV of the equity fund shared by the public sector F equity pri NPV of the subsidy paid by the public sector as costs, discounted by r gov irr pri Internal return rate (IRR) of the private sector over the project's life R sub NPV of the subsidy paid by the public sector and received by the private sector as revenue, discounted by r pri R tol NPV of toll collected by the private sector R ext NPV of the exterior benefits transferred and taken by the private sector as extra revenue C ope NPV of the operation costs paid by the private sector