# Layout Optimization for Shared Parking Spaces Considering Shared Parking Walking Time and Parking Fee

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Problem Description

## 4. Morning Commute Equilibrium with Parking

#### 4.1. Travel Pattern When There Are Sufficient Parking Spaces in the Accessorial Parking Lot (Scenario A)

#### 4.2. Travel Patterns Supplemented by Shared Parking When There Is a Shortage of Parking Spaces in the Accessorial Parking Lot (Scenario B)

#### 4.2.1. User Equilibrium Solution for Scenario B

#### 4.2.2. Total Social Cost ($TS{C}_{b}$) and Total Queue Time ($TQ{T}_{b}$) for Scenario B

## 5. Optimal Travel Pattern Solutions under Different Management Objectives

#### 5.1. The Optimal Travel Pattern with the Lowest Total Social Cost

**Lemma**

**1.**

**Proof of Lemma 1.**

**Proposition**

**1.**

**Proof of Proposition 1.**

**Proposition 1(i)**implies that the optimal total social cost in the system is obtained when all commuters park in the accessorial parking lot, illustrating that the total social cost increases owing to the inconvenience of shared parking facilities. In this case, the manager should establish a sufficient number of parking spaces in the accessorial parking lot to meet the parking needs of all commuters and discourage the allocation of shared parking spaces.

**Proposition 1(ii)**shows that it is desirable to reduce the total social cost by implementing shared parking policy to take up part of the parking demand of auto commuters in the road network, provided that the distance between shared parking spaces is sufficiently optimized.

#### 5.2. Optimal Travel Pattern with the Shortest Total Queue Time

**Lemma**

**2.**

**Proof of Lemma 2.**

**Proposition**

**2.**

**Proof of Proposition 2.**

## 6. Numerical Examples

#### 6.1. Comparative Analysis of Total Social Cost for Scenario A and Scenario B

#### 6.1.1. Travel Pattern Analysis for Scenario B with the Lowest Total Social Cost

#### 6.1.2. Comparative Analysis of Total Social Cost for Scenario A and Scenario B

#### 6.2. Comparative Analysis of Total Queue Time in Scenario A and Scenario B

#### 6.2.1. Analysis of the Optimal Travel Mode for Scenario B with the Lowest Total Queue Time

#### 6.2.2. Comparative Analysis of Total Queue Time of Commuters in Scenario A and Scenario B

#### 6.3. Comparative Analysis of Total Social Cost and Total Queue Time under Different Management Objectives

## 7. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Lemma 1

#### Appendix A.2. Proof of Proposition 1

#### Appendix A.3. Proof of Lemma 2

#### Appendix A.4. Proof of Proposition 2

**Table A1.**Optimal travel pattern selection within the capacity range of different accessorial parking lots ${}^{1}$.

${\mathit{n}}_{\mathit{a}}$ | Optimal Travel Pattern for Scenario B | Optimal Travel Pattern under Scenario A and Scenario B Comparison |
---|---|---|

$0<{n}_{a}\le {n}_{a}^{\#}$ | Scenario B(a) or Scenario B(b) | Scenario B(a) or Scenario B(b) |

${n}_{a}^{\#}<{n}_{a}\le {n}_{a}^{*}$ | Scenario B(a) or Scenario B(b) | Scenario A |

${n}_{a}^{*}<{n}_{a}\le M$ | Scenario B(c-3) | Scenario A |

^{1}The optimal travel pattern is chosen with the objective of minimizing the total queue time of commuters in the system.

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**Figure 2.**Travel pattern diagram for scenario A (i.e., sufficient parking spaces in the accessorial parking lot).

**Figure 6.**Comparison of total social cost for scenario A and scenario B with different values of $\omega $.

**Figure 7.**Impact of shared parking fee on total queue time in scenario B with different accessorial parking capacities.

**Figure 8.**Comparison of total queue time for scenario A and scenario B with different values of $\omega $.

**Figure 9.**Relationship between total social cost and total queue time with shared parking fee in scenario B.

**Figure 10.**Relationship between total social cost and total queue time for commuters with the discrete degree $\omega $ of space layout of shared parking spaces in scenario B.

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

N | Total number of auto commuters. | ${n}_{a}$ | Number of accessorial parking spaces. |

s | Capacity of the road bottleneck. | ${t}^{*}$ | Desired arrival time. |

$\alpha $ | Value of unit travel time. | t | Departure time for commuters. |

$\beta $ | Value of unit time for early arrival. | n | Serial number of shared parking space. |

$\gamma $ | Value of unit time for late arrival. | $\omega $ | Walking time through a parking space. |

$\lambda $ | Walking time cost for commuters. | ${\tau}_{a}$ | Accessorial parking fee. |

${N}_{a}$ | Number of commuters choosing accessorial parking. | ${\tau}_{b}$ | Shared parking fee. |

${N}_{b}$ | Number of commuters choosing shared parking. | $\Delta \tau $ | Fee margin between two types of parking lots |

**Table 2.**Boundary conditions of different travel patterns under scenario B and the range of parking fee margin $\Delta \tau $ and accessorial parking capacity ${n}_{a}$ for the two types of parking lots.

Scenario | Boundary Conditions | $\mathbf{\Delta}\mathit{\tau}$ | ${\mathit{n}}_{\mathit{a}}$ | |
---|---|---|---|---|

B(a) | ${t}_{2}<{t}_{3}$ | $\Delta \tau >\frac{\beta \xb7{n}_{a}}{s}$ | ${n}_{a}\in [0,N)$ | |

B(b) | ${t}_{2}={t}_{3}$ | $\Delta \tau =\frac{\beta \xb7{n}_{a}}{s}$ | ${n}_{a}\in [0,N)$ | |

B(c) | B(c-1) | ${t}_{2}>{t}_{3},{t}_{2}={t}_{4}<{t}^{*}$ | $\frac{{n}_{a}(\beta +\gamma )-N\gamma}{s}-\omega (N-{n}_{a})(\gamma +\lambda )<\Delta \tau <\frac{\beta \xb7{n}_{a}}{s}$ | ${n}_{a}\in [0,N)$ |

B(c-2) | ${t}_{2}>{t}_{3},{t}_{2}={t}_{4}={t}^{*}$ | $\Delta \tau =\frac{{n}_{a}(\beta +\gamma )-N\gamma}{S}-\omega (N-{n}_{a})(\gamma +\lambda )$ | ${n}_{a}\in [\frac{N(\gamma +\omega s(\lambda +\gamma \left)\right)}{\left(\beta +\gamma \right)+\omega s\left(\lambda +\gamma \right)},N)$ | |

B(c-3) | ${t}_{2}>{t}_{3},{t}_{2}={t}_{4}>{t}^{*}$ | $0\le \Delta \tau <\frac{{n}_{a}(\beta +\gamma )-N\gamma}{S}-\omega (N-{n}_{a})(\gamma +\lambda )$ | ${n}_{a}\in [\frac{N(\gamma +\omega s(\lambda +\gamma \left)\right)}{\left(\beta +\gamma \right)+\omega s\left(\lambda +\gamma \right)},N)$ |

**Table 3.**Optimal parking fee and travel patterns corresponding to different accessorial parking capacities under scenario B at the lowest total social cost.

Category (1) | ${\mathit{n}}_{\mathit{a}}$ (2) | $\mathit{TSC}$ (3) | $\mathbf{\Delta}\mathit{\tau}$ (4) | Optimal Travel Pattern (5) | |
---|---|---|---|---|---|

1 | $0<{n}_{a}<\frac{N\gamma}{\beta +\gamma}$ | $TS{C}_{b1}$ | $\Delta {\tau}_{opt}=\frac{\beta \xb7{n}_{a}}{s}$ | B(b) | |

2 | ${n}_{a}=\frac{N\gamma}{\beta +\gamma}$ | $TS{C}_{b2}$ | $0\le \Delta {\tau}_{opt}\le \frac{N\beta \gamma}{s(\beta +\gamma )}$ | B(b), B(c-1) | |

3 | $\frac{N\gamma}{\beta +\gamma}<{n}_{a}<N$ | $\frac{N\gamma}{\beta +\gamma}<{n}_{a}<\frac{N(\gamma +\omega s(\lambda +\gamma \left)\right)}{(\beta +\gamma )+\omega s(\lambda +\gamma )}$ | $TS{C}_{b3}$ | $\Delta {\tau}_{opt}=0$ | B(c-1) |

${n}_{a}=\frac{N(\gamma +\omega s(\lambda +\gamma \left)\right)}{(\beta +\gamma )+\omega s(\lambda +\gamma )}$ | B(c-2) | ||||

$\frac{N(\gamma +\omega s(\lambda +\gamma \left)\right)}{(\beta +\gamma )+\omega s(\lambda +\gamma )}<{n}_{a}<N$ | B(c-3) |

_{b2}= $\frac{\beta \gamma}{\beta +\gamma}\frac{{N}^{2}}{s}+\frac{{N}^{2}\beta 2\left(\gamma +\lambda \right)\omega}{{\left(\beta +\gamma \right)}^{2}}$; 3 TSC

_{b3}= $\frac{\beta \gamma}{\beta +\gamma}\frac{{N}^{2}}{s}+\frac{\beta \left(\gamma +\lambda \right)\omega N\left(N-{n}_{a}\right)}{\beta +\gamma}$.

**Table 4.**Optimal travel patterns for different accessorial parking capacities ${n}_{a}$ under scenario B (with the lowest total social cost as the management objective).

Category | ${\mathit{n}}_{\mathit{a}}$ (Vehicles) | ${\mathit{\tau}}_{\mathit{b}}^{*}$ (Yuan) | TSC (Yuan) | Optimal Travel Pattern |
---|---|---|---|---|

Category 1 | 120 | 9 | 1496 | Scenario B(b) |

Category 2 | 200 | [5, 11.7] | 1672 | Scenario B(b), Scenario B(c) |

Category 3 | 226 | 5 | 1620 | Scenario B(c) |

**Table 5.**Optimal travel patterns for different accessorial parking capacities ${n}_{a}$ under scenario B (with the shortest total queue time for commuters as the management objective).

Category | ${\mathit{n}}_{\mathit{a}}$ (Vehicles) | ${\mathit{\tau}}_{\mathit{b}}^{*}$ (Yuan) | TQT (hour) | Optimal Travel Pattern |
---|---|---|---|---|

Category 1 | 120 | [9, ∞) | 39.71 | Scenario B(a), Scenario B(b) |

Category 2 | 200 | [11.7, ∞) | 68.41 | Scenario B(a), Scenario B(b) |

Category 3 | 226 | 5 | 81.52 | Scenario B(c) |

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## Share and Cite

**MDPI and ACS Style**

Ji, Y.; Lu, X.; Jiang, H.; Zhu, X.; Wang, J.
Layout Optimization for Shared Parking Spaces Considering Shared Parking Walking Time and Parking Fee. *Sustainability* **2022**, *14*, 5635.
https://doi.org/10.3390/su14095635

**AMA Style**

Ji Y, Lu X, Jiang H, Zhu X, Wang J.
Layout Optimization for Shared Parking Spaces Considering Shared Parking Walking Time and Parking Fee. *Sustainability*. 2022; 14(9):5635.
https://doi.org/10.3390/su14095635

**Chicago/Turabian Style**

Ji, Yangbeibei, Xueqing Lu, Hanwan Jiang, Xinyang Zhu, and Jiao Wang.
2022. "Layout Optimization for Shared Parking Spaces Considering Shared Parking Walking Time and Parking Fee" *Sustainability* 14, no. 9: 5635.
https://doi.org/10.3390/su14095635