# An Analytical Model for the Many-to-One Demand Responsive Transit Systems

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review

#### 1.2. Objectives and Contributions

## 2. Methodology

#### 2.1. General Description of the Analytical Model

**A**with area A. Let $R=\{1,2,\dots ,\left|R\right|\}$ denote the set of indices for railway stations in the service region and $j\in R$ be a particular station. We assume that each transit station serves a subregion of $A$, which is known as the “catchment area” [29] that encompasses an area of potential passengers that would be willing to access this station. Each subregion is approximated by a rectangle with length and width, served by a fleet of homogeneous vehicles. Let ${A}_{j}$ denote the subregion served by station $j$ with area ${A}_{j}$. For simplicity, the service region is defined as fully covered by the subregion, that is, ${\sum}_{j\in R}{A}_{j}}=A$. The user demand within the area is assumed to be spatially uniformly distributed within a given time interval of the day with a spatial density ${\lambda}_{j}$ (number of users per unit area). Considering the full flexibility of the DRT service, stops of the DRT vehicle are according to the real-time passenger requests. The total number of stops, ${N}_{j}$, in area ${A}_{j}$ is equal to the number of passengers, that is, ${N}_{j}={\lambda}_{j}{A}_{j}$.

#### 2.2. Agency Cost

#### 2.3. User Cost

_{a},t

_{b}), we have the following equation:

_{l}, is equal to

#### 2.4. Model Formulation

## 3. Numerical Experiments

#### 3.1. Effects of Area of the Service Region and Demand Density Changes

#### 3.2. Effects of Vehicle Occupancy/Fleet Size Changes

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Table 1.**Values of the peddling factor [31].

No. of Stops in ${\mathbf{A}}_{\mathit{j}},\text{}{\mathit{N}}_{\mathit{j}}$ | ${\mathit{K}}_{2}({\mathit{N}}_{\mathit{j}})$ |
---|---|

1 | 0 |

2 | 0.73 |

3 | 0.68 |

4 | 0.63 |

5 | 0.60 |

≥6 | 0.57 |

Symbol | Definition | Baseline Value |
---|---|---|

$A$ | Area of the service region (km^{2}) | - |

b | Sum of pick-up and drop-off times (h) | 0.08 |

$C$ | Vehicle operating cost ($/veh h) | 50 |

$e$ | Scalar factor for earliness delay | 0.5 |

$l$ | Scalar factor for lateness delay | 1.5 |

$N$ | Number of passengers (pax) | - |

${N}_{e}$ | Number of early passengers (pax) | - |

${N}_{l}$ | Number of late passengers (pax) | - |

$Q$ | Vehicle capacity (pax) | 30 |

$v$ | Average vehicle speed (km/h) | 30 |

$w$ | Scalar factor for in-vehicle waiting delay | 1.0 |

β | Cost rate of delay ($/h) | 5 |

${\gamma}_{1}$ | The weight of agency cost | 1 |

${\gamma}_{2}$ | The weight of user cost | 1 |

$\lambda $ | Demand density (pax/km^{2}) | - |

$\mu $ | System service rate (pax/h) | - |

${\phi}_{1j}$ | Distance from the depot to the centroid of ${A}_{j}$ (km) | - |

${\phi}_{2j}$ | Distance from the centroid of ${A}_{j}$ to station $j$ (km) | - |

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**MDPI and ACS Style**

Huang, D.; Tong, W.; Wang, L.; Yang, X. An Analytical Model for the Many-to-One Demand Responsive Transit Systems. *Sustainability* **2020**, *12*, 298.
https://doi.org/10.3390/su12010298

**AMA Style**

Huang D, Tong W, Wang L, Yang X. An Analytical Model for the Many-to-One Demand Responsive Transit Systems. *Sustainability*. 2020; 12(1):298.
https://doi.org/10.3390/su12010298

**Chicago/Turabian Style**

Huang, Di, Weiping Tong, Lumeng Wang, and Xun Yang. 2020. "An Analytical Model for the Many-to-One Demand Responsive Transit Systems" *Sustainability* 12, no. 1: 298.
https://doi.org/10.3390/su12010298