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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis of the MOP under LPVR

#### 2.1. The LPVR Mechanism

#### 2.2. The Interactive Stackelberg Decision-Making Process

## 3. Modeling the Problem by Bi-Level Programing (BLP)

#### 3.1. The Upper Programing (UP) Problem

#### 3.2. The Lower Programing (LP) Problem

#### 3.3. Optimization of Objectives in the BLP Model

## 4. NSGAIII Framework-Based Algorithm for the BLP Model

#### 4.1. Interception Strategy for Reparing Solutions of the UP Problem

#### 4.2. Subsection Approximation Strategy for Optimizing the LP

#### 4.3. Algorithm Design for the Complete BLP Model

## 5. A case Study of the Big Outer Ring Highway PPP Project

#### 5.1. The Big Outer Ring Highway PPP Project

#### 5.2. Evaluation Effects of Different Population Sizes

#### 5.3. Results and Discussion

_{0}) in different indicators. Their definitions are listed in Table A2 of Appendix A. Δ

_{1}~Δ

_{4}show the increments of S

_{1}~S

_{4}when compared to S

_{0}. Changes in $\alpha $, $\epsilon $, ${\mathrm{irr}}^{\mathrm{pri}}$ and $\tau $ are measured in absolute increments as:

_{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.

_{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.

_{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.

_{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%.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Table A1.**Notations and definitions of parameters and variables used in Section 3.1 and Section 3.2.

Notation | Definition |
---|---|

Input Parameters | |

$I$ | Total investment of the project |

$I{}^{\prime}$ | Extra direct investment that can be used to deduct loans, usually arising from a specific finance source such as a public transfer payment from the central government |

$n$ | Construction term of the project |

${q}_{i}$ | Traffic volume of the i-th year in operation period |

${c}_{i}$ | Operation cost of the i-th year in operation period |

${m}_{i}$ | Montage paid of the i-th year in operation period |

${r}_{\mathrm{nt}}$ | The ratio of non-toll income making up to the toll income |

${o}_{i}$ | Ancillary services revenue; i.e., the non-toll incomes of the i-th year in concession term, such as refueling, parking and catering |

${r}^{\mathrm{cap}}$ | The proportion of equity capital funds |

${r}_{j}^{\mathrm{con}}$ | The proportion of the funds invested in the j-th year of construction in the total investment |

${r}^{\mathrm{deb}}$ | Rate of the debt fund |

${r}^{\mathrm{gov}}$ | Discount rate of the public sector |

${r}^{\mathrm{pri}}$ | Discount rate of the private sector |

${e}_{i}$ | The exterior benefits generated from project implementation in the i-th year |

Decision Variables | |

$\alpha $ | Initial investment for construction shared by the public sector. The private sector’s share is notated as $1-\alpha $. |

$\delta $ | Subsidy per vehicle-kilometer |

$\epsilon $ | Ratio of exterior benefits to be collected |

$\rho $ | Toll of per vehicle-kilometer |

$\tau $ | Concession term |

**Table A2.**Notations and definitions of parameters and variables used in Section 5.3. NPV: net present value.

Notation | Definition |
---|---|

${F}_{\mathrm{gov}}^{\mathrm{equity}}$ | NPV of the equity fund shared by the private sector |

${S}_{{r}_{\mathrm{gov}}}$ | NPV of the equity fund shared by the public sector |

${F}_{\mathrm{pri}}^{\mathrm{equity}}$ | NPV of the subsidy paid by the public sector as costs, discounted by ${r}_{\mathrm{gov}}$ |

${\mathrm{irr}}_{\mathrm{pri}}$ | Internal return rate (IRR) of the private sector over the project’s life |

${R}_{\mathrm{sub}}$ | NPV of the subsidy paid by the public sector and received by the private sector as revenue, discounted by ${r}_{\mathrm{pri}}$ |

${R}_{\mathrm{tol}}$ | NPV of toll collected by the private sector |

${R}_{\mathrm{ext}}$ | NPV of the exterior benefits transferred and taken by the private sector as extra revenue |

${C}_{\mathrm{ope}}$ | NPV of the operation costs paid by the private sector |

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**Figure 1.**The technology roadmap of the study in this paper. PPP: public-private partnership; LPVR: least present value of revenue.

**Figure 2.**An interactive decision-making process in a PPP project context initialized by the public sector as the leader.

**Figure 3.**How the objectives of the upper programing (UP) problem and the lower programing (LP) problem are optimized.

**Figure 4.**Surface diagram with contours of the functional relationships between ${F}_{\mathrm{LP}}$, $\rho $ and $\tau $. ${r}^{\mathrm{pri}}=0.08$ and ${\tau}^{\mathrm{max}}=25$.

**Figure 5.**Process of the entire algorithm used to solve the complete bi-level programing (BLP) model. NSGAIII: non-dominated sorting genetic algorithm III.

**Figure 7.**The distribution of the LP problem’s objective values, calculated by two sizes of population under the NSGAIII algorithm. Three objectives of the UP problem are shown, which are the lifecycle cost of the public sector (${C}_{\mathrm{gov}}$), user payment (${C}_{\mathrm{user}}$), and the consumption of exterior benefit (${C}_{\mathrm{ext}}$).

**Figure 8.**The convergence of the LP problem’s objective values calculated by two sizes of population under the NSGAIII algorithm. (

**a**) shows convergence of all 50 generations, and (

**b**) shows convergence of the 3rd–50th generations.

**Figure 9.**The convergence of the LP problem’s objective values calculated by two sizes of population under NSGAIII algorithm. (

**a**) shows the stacked plot of all 120 solutions with a dashed box outlining the general locations of those selected optimal solutions. (

**b**) displays contents in the dashed box of (

**a**), and each dashed line represents one selected optimal solution.

Year | Operation Subsidy in Original Case ($ Million) | $\mathbf{Traffic}\text{}{\mathit{q}}_{\mathit{i}}\text{}(\mathbf{Year},\text{}\mathbf{Million}-\mathbf{Vehicle})$ | Unit Traffic Subsidy ($ Vehicle-km) | $\mathbf{Operation}\text{}\mathbf{Cos}\mathbf{t}\text{}{\mathit{c}}_{\mathit{i}}\text{}(\$\text{}\mathbf{Million})$ | $\mathbf{Exterior}\text{}\mathbf{Benefit}\text{}\left(\mathbf{EB}\right){\mathit{e}}_{\mathit{i}}\text{}(\$\text{}\mathbf{Million})$ |
---|---|---|---|---|---|

2019 | 24.22 | 231.49 | 0.10462 | 27.19 | 111.57 |

2020 | 30.65 | 292.94 | 0.10462 | 28.28 | 220.54 |

2021 | 38.78 | 370.70 | 0.10462 | 29.42 | 265.32 |

2022 | 49.08 | 469.10 | 0.10462 | 30.61 | 301.64 |

2023 | 62.11 | 593.63 | 0.10462 | 31.85 | 332.97 |

2024 | 66.79 | 634.72 | 0.10523 | 33.15 | 385.35 |

2025 | 71.42 | 678.65 | 0.10523 | 34.51 | 439.89 |

2026 | 76.36 | 725.63 | 0.10523 | 35.93 | 496.65 |

2027 | 81.65 | 775.85 | 0.10524 | 37.42 | 555.69 |

2028 | 87.30 | 829.55 | 0.10523 | 38.97 | 617.05 |

2029 | 87.62 | 845.48 | 0.10363 | 40.60 | 638.83 |

2030 | 89.30 | 861.71 | 0.10363 | 42.30 | 661.38 |

2031 | 91.01 | 878.26 | 0.10363 | 44.08 | 684.73 |

2032 | 92.76 | 895.12 | 0.10363 | 45.94 | 708.90 |

2033 | 94.54 | 912.30 | 0.10363 | 47.89 | 733.94 |

2034 | 94.47 | 920.81 | 0.10259 | 49.93 | 759.88 |

2035 | 95.35 | 929.39 | 0.10259 | 52.06 | 786.76 |

2036 | 96.23 | 938.05 | 0.10259 | 54.29 | 814.61 |

2037 | 97.13 | 946.80 | 0.10259 | 56.63 | 842.78 |

2038 | 98.04 | 955.62 | 0.10259 | 59.08 | 871.33 |

2039 | 97.41 | 964.53 | 0.10100 | 61.64 | 876.72 |

2040 | 98.32 | 973.52 | 0.10100 | 64.32 | 899.72 |

2041 | 99.24 | 982.60 | 0.10099 | 67.12 | 923.68 |

2042 | 100.16 | 991.75 | 0.10100 | 70.06 | 948.55 |

2043 | 101.09 | 1001.01 | 0.10099 | 73.14 | 974.45 |

$\mathbf{Share}\text{}\mathbf{of}\text{}\mathbf{Equity}\text{}\mathbf{Fund}\text{}\mathit{\alpha}$ (%) | $\mathbf{Unit}\text{}\mathbf{Subsidy}\text{}\mathit{\delta}$ ($ Vehicle-km) | $\mathbf{Ratio}\text{}\mathbf{of}\text{}\mathbf{EB}\text{}\mathbf{Transferred}\text{}\mathit{\epsilon}$ (%) | $\mathbf{Unit}\text{}\mathbf{Toll}\text{}\mathit{\rho}$ ($ Vehicle-km) | $\mathbf{Concession}\text{}\mathbf{Term}\text{}\mathit{\tau}$ (Years) | |
---|---|---|---|---|---|

Upper limit | ${\alpha}^{\mathrm{max}}=49\%$ | ${\delta}^{\mathrm{max}}=0.10524$ | $\epsilon =100\%$ | $\rho =0.09928$ | $\tau =25$ |

Lower limit | ${\alpha}^{\mathrm{min}}=0\%$ | ${\delta}^{\mathrm{min}}=0$ | $\epsilon =0\%$ | $\rho =0$ | $\tau =15$ |

$\mathbf{Share}\text{}\mathbf{of}\text{}\mathbf{Equity}\text{}\mathbf{Fund}\text{}\mathit{\alpha}$ (%) | $\mathbf{Unit}\text{}\mathbf{Subsidy}\text{}\mathit{\delta}$ ($ Vehicle-km) | $\mathbf{Ratio}\text{}\mathbf{of}\text{}\mathbf{EB}\text{}\mathbf{Transferred}\text{}\mathit{\epsilon}$ (%) | $\mathbf{Unit}\text{}\mathbf{Toll}\text{}\mathit{\rho}$ ($ Vehicle-km) | $\mathbf{Concession}\text{}\mathbf{Term}\text{}\mathit{\tau}$ (Years) | |
---|---|---|---|---|---|

S_{0} | 49.00% | 0.1034 | 0.00% | 0.0976 | 25 |

S_{1} | 0.00% | 0.0000 | 34.71% | 0.0820 | 15 |

Δ_{1} | −49.00% | −100.00% | +34.71% | −15.94% | −40.00% |

S_{2} | 48.22% | 0.0726 | 29.43% | 0.0000 | 15 |

Δ_{2} | −0.78% | −29.84% | +29.43% | −100.00% | −40.00% |

S_{3} | 47.30% | 0.1035 | 3.55% | 0.0993 | 20 |

Δ_{3} | −1.70% | +0.05% | +3.55% | +1.75% | −20.00% |

S_{4} | 45.05% | 0.0344 | 30.93% | 0.0302 | 15 |

Δ_{4} | −3.95% | −66.69% | +30.93% | −69.03% | −40.00% |

**Table 4.**Changes in indicators reflecting the benefits and costs of the public sector. PSC: public sector comparator. VFM: value for money.

${\mathit{C}}_{\mathbf{gov}}$ ($ Million) | PSC ($ Million) | VFM ($ Million) | ${\mathit{F}}_{\mathbf{gov}}^{\mathbf{equity}}$ ($ Million) | ${\mathit{S}}_{{\mathit{r}}_{\mathbf{gov}}}$ ($ Million) | |
---|---|---|---|---|---|

S_{0} | 1267.02 | 1071.83 | −195.19 | 238.00 | 1043.38 |

S_{1} | 0.00 | 1289.00 | 1289.00 | 0.00 | 0.00 |

Δ_{1} | −100.00% | +20.26% | +760.38% | −100.00% | −100.00% |

S_{2} | 677.91 | 1816.86 | 1138.95 | 234.21 | 457.83 |

Δ_{2} | −46.50% | +69.51% | +683.50% | −1.59% | −56.12% |

S_{3} | 1080.09 | 1100.84 | 20.75 | 229.76 | 864.19 |

Δ_{3} | −14.75% | +2.71% | +110.63% | −3.46% | −17.17% |

S_{4} | 422.94 | 1622.39 | 1199.44 | 218.81 | 217.33 |

Δ_{4} | −66.62% | +51.37% | +714.50% | −8.06% | −79.17% |

LPVR ($ Million) | ${\mathbf{irr}}_{\mathbf{pri}}$ (%) | ${\mathit{R}}_{\mathbf{sub}}$ ($ Million) | ${\mathit{R}}_{\mathbf{tol}}$ ($ Million) | ${\mathit{R}}_{\mathbf{ext}}$ ($ Million) | ${\mathit{F}}_{\mathbf{pri}}^{\mathbf{equity}}$ ($ Million) | ${\mathit{C}}_{\mathbf{ope}}$ ($ Million) | |
---|---|---|---|---|---|---|---|

S_{0} | −15.82 | 7.60% | 630.23 | 606.52 | 0.00 | 247.71 | 360.58 |

S_{1} | 0.20 | 8.00% | 0.00 | 364.98 | 1054.59 | 485.71 | 253.18 |

Δ_{1} | +101.26% | +0.40% | −100.00% | −39.82% | - | +96.08% | −29.78% |

S_{2} | 0.29 | 8.01% | 316.56 | 0.00 | 894.28 | 251.50 | 253.18 |

Δ_{2} | +101.82% | +0.41% | −49.77% | −100.00% | - | +1.53% | −29.78% |

S_{3} | 0.20 | 8.01% | 556.02 | 544.19 | 138.90 | 255.95 | 311.51 |

Δ_{3} | +101.29% | +0.41% | −11.78% | −10.28% | - | +3.32% | −13.61% |

S_{4} | 0.13 | 8.00% | 150.27 | 134.47 | 939.67 | 266.91 | 253.18 |

Δ_{4} | +100.83% | +0.40% | −76.16% | −77.83% | - | +7.75% | −29.78% |

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## Share and Cite

**MDPI and ACS Style**

Liu, F.; Liu, J.; Yan, X.
Solving the Asymmetry Multi-Objective Optimization Problem in PPPs under LPVR Mechanism by Bi-Level Programing. *Symmetry* **2020**, *12*, 1667.
https://doi.org/10.3390/sym12101667

**AMA Style**

Liu F, Liu J, Yan X.
Solving the Asymmetry Multi-Objective Optimization Problem in PPPs under LPVR Mechanism by Bi-Level Programing. *Symmetry*. 2020; 12(10):1667.
https://doi.org/10.3390/sym12101667

**Chicago/Turabian Style**

Liu, Feiran, Jun Liu, and Xuedong Yan.
2020. "Solving the Asymmetry Multi-Objective Optimization Problem in PPPs under LPVR Mechanism by Bi-Level Programing" *Symmetry* 12, no. 10: 1667.
https://doi.org/10.3390/sym12101667