Solving the Asymmetry Multi-Objective Optimization Problem in PPPs under LPVR Mechanism by Bi-Level Programing
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
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
Notation | Definition |
---|---|
Input Parameters | |
Total investment of the project | |
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 | |
Construction term of the project | |
Traffic volume of the i-th year in operation period | |
Operation cost of the i-th year in operation period | |
Montage paid of the i-th year in operation period | |
The ratio of non-toll income making up to the toll income | |
Ancillary services revenue; i.e., the non-toll incomes of the i-th year in concession term, such as refueling, parking and catering | |
The proportion of equity capital funds | |
The proportion of the funds invested in the j-th year of construction in the total investment | |
Rate of the debt fund | |
Discount rate of the public sector | |
Discount rate of the private sector | |
The exterior benefits generated from project implementation in the i-th year | |
Decision Variables | |
Initial investment for construction shared by the public sector. The private sector’s share is notated as . | |
Subsidy per vehicle-kilometer | |
Ratio of exterior benefits to be collected | |
Toll of per vehicle-kilometer | |
Concession term |
Notation | Definition |
---|---|
NPV of the equity fund shared by the private sector | |
NPV of the equity fund shared by the public sector | |
NPV of the subsidy paid by the public sector as costs, discounted by | |
Internal return rate (IRR) of the private sector over the project’s life | |
NPV of the subsidy paid by the public sector and received by the private sector as revenue, discounted by | |
NPV of toll collected by the private sector | |
NPV of the exterior benefits transferred and taken by the private sector as extra revenue | |
NPV of the operation costs paid by the private sector |
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Year | Operation Subsidy in Original Case ($ Million) | Unit Traffic Subsidy ($ Vehicle-km) | |||
---|---|---|---|---|---|
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 |
(%) | ($ Vehicle-km) | (%) | ($ Vehicle-km) | (Years) | |
---|---|---|---|---|---|
Upper limit | |||||
Lower limit |
(%) | ($ Vehicle-km) | (%) | ($ Vehicle-km) | (Years) | |
---|---|---|---|---|---|
S0 | 49.00% | 0.1034 | 0.00% | 0.0976 | 25 |
S1 | 0.00% | 0.0000 | 34.71% | 0.0820 | 15 |
Δ1 | −49.00% | −100.00% | +34.71% | −15.94% | −40.00% |
S2 | 48.22% | 0.0726 | 29.43% | 0.0000 | 15 |
Δ2 | −0.78% | −29.84% | +29.43% | −100.00% | −40.00% |
S3 | 47.30% | 0.1035 | 3.55% | 0.0993 | 20 |
Δ3 | −1.70% | +0.05% | +3.55% | +1.75% | −20.00% |
S4 | 45.05% | 0.0344 | 30.93% | 0.0302 | 15 |
Δ4 | −3.95% | −66.69% | +30.93% | −69.03% | −40.00% |
($ Million) | PSC ($ Million) | VFM ($ Million) | ($ Million) | ($ Million) | |
---|---|---|---|---|---|
S0 | 1267.02 | 1071.83 | −195.19 | 238.00 | 1043.38 |
S1 | 0.00 | 1289.00 | 1289.00 | 0.00 | 0.00 |
Δ1 | −100.00% | +20.26% | +760.38% | −100.00% | −100.00% |
S2 | 677.91 | 1816.86 | 1138.95 | 234.21 | 457.83 |
Δ2 | −46.50% | +69.51% | +683.50% | −1.59% | −56.12% |
S3 | 1080.09 | 1100.84 | 20.75 | 229.76 | 864.19 |
Δ3 | −14.75% | +2.71% | +110.63% | −3.46% | −17.17% |
S4 | 422.94 | 1622.39 | 1199.44 | 218.81 | 217.33 |
Δ4 | −66.62% | +51.37% | +714.50% | −8.06% | −79.17% |
LPVR ($ Million) | (%) | ($ Million) | ($ Million) | ($ Million) | ($ Million) | ($ Million) | |
---|---|---|---|---|---|---|---|
S0 | −15.82 | 7.60% | 630.23 | 606.52 | 0.00 | 247.71 | 360.58 |
S1 | 0.20 | 8.00% | 0.00 | 364.98 | 1054.59 | 485.71 | 253.18 |
Δ1 | +101.26% | +0.40% | −100.00% | −39.82% | - | +96.08% | −29.78% |
S2 | 0.29 | 8.01% | 316.56 | 0.00 | 894.28 | 251.50 | 253.18 |
Δ2 | +101.82% | +0.41% | −49.77% | −100.00% | - | +1.53% | −29.78% |
S3 | 0.20 | 8.01% | 556.02 | 544.19 | 138.90 | 255.95 | 311.51 |
Δ3 | +101.29% | +0.41% | −11.78% | −10.28% | - | +3.32% | −13.61% |
S4 | 0.13 | 8.00% | 150.27 | 134.47 | 939.67 | 266.91 | 253.18 |
Δ4 | +100.83% | +0.40% | −76.16% | −77.83% | - | +7.75% | −29.78% |
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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
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 StyleLiu, 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