# The Optimization of Bus Departure Time Based on Uncertainty Theory—Taking No. 207 Bus Line of Nanchang City, China, as an Example

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## Abstract

**:**

## 1. Introduction

## 2. Uncertainty Theory

#### 2.1. Uncertain Measure and Uncertain Space

**Definition**

**1.**

- Axiom 1
- (normality): For the universal set $\mathsf{\Gamma}$, ${\rm M}\left\{\mathsf{\Gamma}\right\}=1$
- Axiom 2
- (self-duality): $\mathsf{{\rm M}}\left\{\mathsf{\Lambda}\right\}+\mathsf{{\rm M}}\left\{{\mathsf{\Lambda}}^{C}\right\}=1$ for any event $\mathsf{\Lambda}$. ${\mathsf{\Lambda}}^{C}$ is the complement of $\mathsf{\Lambda}$;
- Axiom 3
- (sub-additivity): For every countable sequence of events ${\mathsf{\Lambda}}_{1},{\mathsf{\Lambda}}_{2},\cdots ,$ we have $\mathsf{{\rm M}}\left\{{\displaystyle \underset{i=1}{\overset{\infty}{\cup}}\mathsf{\Lambda}}\right\}\le {\displaystyle \sum _{i=1}^{\infty}\mathsf{{\rm M}}\left\{{\mathsf{\Lambda}}_{i}\right\}}$.

**Definition**

**2.**

#### 2.2. Uncertain Variable and Uncertain Distribution

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

## 3. Uncertainty Bi-level Programming Model for Departure Frequency of a Bus Line

#### 3.1. Model Construction

- (i).
- The bus vehicle configuration is performed after the bus line is determined;
- (ii).
- The bus travel speed is obtained by averaging the historical data;
- (iii).
- The boarding rule is first-come-first-service;
- (iv).
- Considering single-type bus vehicles, the bus departs according to the timetable specification, without considering emergencies;
- (v).
- Failure to board a bus is counted as a retained guest, which affects the satisfaction of bus services. Additionally, the passenger’s arrival follows a linear uncertain distribution (the chi-squared test process to verify the distribution is given in the case analysis).

#### 3.1.1. Upper-Level Objective Function Considering Minimization of Passengers’ Waiting Time Cost

#### 3.1.2. Lower-Level Objective Function

^{2}); S is the parking area of bus vehicles, m

^{2}; C is the average purchase cost of present vehicle, CNY; d is the average operating life of the bus vehicle, day; Cs is the operating cost of the bus line per day; ${f}_{i}$ has the same meaning as formula (3); w is the operation maintenance and fuel cost per bus per shift, which can be expressed as follows:

#### 3.1.3. Constraints

#### 3.2. Optimization Model of Non-Uniform Departure Interval

_{1}= 0 ), hour; ${t}_{ij}$ is the time area not covered by the starting time of the departure interval of the current period (T

_{1}= 0), hour; ${q}_{TIJ}$ is the number of passengers arriving at the bus stop j during the last departure interval TI of period i, person; ${q}_{tij}$ is the passenger flow at the bus stop j during the starting departure interval ti in the period i, person.

#### 3.3. Model Solution

#### 3.3.1. Lower-Level Programming Model Solution

#### 3.3.2. Upper-Level Programming Model Solution

## 4. Case Analysis

#### 4.1. Case Introduction

_{i}presents the actual observation value, F

_{i}means the model theoretical value, and the sum of the values in the last column of Table 4 is the chi-squared value, which indicates the degree of deviation between the actual observation values and the model theoretical values. Since the number of sample groups $g=7$, there are two parameters $a,b$ in the linear uncertain distribution, the number of parameters $l=2$, so the degree of freedom of the chi-squared statistic is $DF=g-1-l=7-1-2=4$. In checking the chi-squared distribution quantile table, we receive ${\chi}_{0.05}^{2}=9.488>4.97$, so there is a 95% probability that the number of passengers obeys a linear uncertain distribution. The chi-squared test for the number of bus passengers at each station obeying a linear uncertain distribution is similar.

#### 4.2. Model Solution and Analysis

#### 4.2.1. Initial Values

#### 4.2.2. Model Solution

#### 4.2.3. Analysis of Model Results

- (1)
- Passenger information acquisition index [24]

- (2)
- Passenger waiting time index [24]

- (3)
- Time index of unsatisfied demand [24]

## 5. Discussions and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Relationship between departure frequency and objective function value in lower-level function.

**Table 1.**The index of symbols used in Section 2.

Symbol | Description |

L | a $\sigma -$ algebra on a non-empty set $\mathsf{\Gamma}$ |

$\mathsf{\Gamma}$ | a non-empty set |

$\mathsf{{\rm M}}:L\to [0,1]$ | set function from L to interval [0.1] |

$\mathsf{\Lambda}$ | an event |

${\mathsf{\Lambda}}^{C}$ | the complement of $\mathsf{\Lambda}$; |

$(\mathsf{\Gamma},L,\mathsf{{\rm M}})$ | uncertainty space, composed of $\mathsf{\Gamma}$, L and the uncertainty measure M |

$\Re $ | the real number set |

$\xi $ | an uncertain variable, it is a measurable function from the uncertain space $(\mathsf{\Gamma},L,\mathsf{{\rm M}})$ to the real number set $\Re $ |

$B$ | any Borel set |

$\mathsf{\Phi}(x)$ | uncertain distribution |

$L(a,b)$ | linear uncertainty distribution with a, b as upper and lower bounds |

$N(e,\text{}\sigma )$ | normal uncertainty distribution with parameters $e\in \Re \text{},\text{}\sigma \in {\Re}^{+}$ |

**Table 2.**The cards volume and number of vehicles in 7:00–8:00 am from 25–31 March 2019 for bus line 207.

Date | March 25 | March 26 | March 27 | March 28 | March 29 | March 30 | March 31 |

Swipe mount | 1085 | 1215 | 1100 | 1180 | 1128 | 956 | 588 |

Departures | 13 | 13 | 12 | 11 | 12 | 12 | 13 |

**Table 3.**Number of passengers getting on and off bus stop during 7:00–8:00 am on 29 March 2019 for bus 207.

Stop | Boarding | Getting Off | Stop | Boarding | Getting Off | Stop | Boarding | Getting Off |

1 | 79 | 0 | 8 | 138 | 64 | 15 | 13 | 87 |

2 | 47 | 0 | 9 | 119 | 167 | 16 | 26 | 157 |

3 | 40 | 0 | 10 | 79 | 81 | 17 | 36 | 85 |

4 | 93 | 11 | 11 | 13 | 63 | 18 | 13 | 60 |

5 | 106 | 14 | 12 | 27 | 80 | 19 | 32 | 80 |

6 | 119 | 35 | 13 | 79 | 47 | |||

7 | 158 | 50 | 14 | 40 | 114 |

**Table 4.**Chi-squared test statistics for passenger numbers satisfying linear uncertainty distributions.

Date | ${f}_{i}$ | ${P}_{i}$ | ${F}_{i}$ | ${f}_{i}-{F}_{i}$ | ${({f}_{i}-{F}_{i})}^{2}$ | ${({f}_{i}-{F}_{i})}^{2}/{F}_{i}$ |

1 | 1085 | 0.15 | 1090 | −5.24 | 27.41 | 0.03 |

2 | 1215 | 0.17 | 1234 | −19.12 | 365.72 | 0.30 |

3 | 1100 | 0.15 | 1107 | −6.84 | 46.75 | 0.04 |

4 | 1180 | 0.16 | 1195 | −15.38 | 236.69 | 0.20 |

5 | 1128 | 0.16 | 1138 | −9.83 | 96.61 | 0.08 |

6 | 956 | 0.13 | 947 | 8.55 | 73.05 | 0.08 |

7 | 588 | 0.07 | 540 | 47.86 | 2290.89 | 4.24 |

sum | 7252 | 1.00 | 7252 | — | — | ${\chi}^{2}=$ 4.97 |

**Table 5.**Chi-squared test of normal uncertainty distribution for running time at peak hours (25–31 March 2019).

Running Time, Minute | $\overline{{f}_{i}}$ | $\overline{{P}_{i}}$ | $\overline{{F}_{i}}$ | $\overline{{f}_{i}}-\overline{{F}_{i}}$ | ${(\overline{{f}_{i}}-\overline{{F}_{i}})}^{2}$ | ${(\overline{{f}_{i}}-\overline{{F}_{i}})}^{2}/\overline{{F}_{i}}$ |

[20, 22] | 5 | 0.05 | 8 | −2.78 | 7.74 | 0.99 |

[22, 24] | 10 | 0.07 | 11 | −0.79 | 0.62 | 0.06 |

[24, 26] | 16 | 0.10 | 15 | 1.41 | 1.99 | 0.14 |

[26, 28] | 23 | 0.13 | 19 | 3.87 | 14.98 | 0.78 |

[28, 30] | 32 | 0.16 | 24 | 7.79 | 60.69 | 2.51 |

[30, 32] | 22 | 0.16 | 24 | −2.21 | 4.88 | 0.20 |

[32, 34] | 17 | 0.13 | 19 | −2.13 | 4.54 | 0.24 |

[34, 36] | 12 | 0.10 | 15 | −2.59 | 6.70 | 0.46 |

[36, 38] | 10 | 0.07 | 11 | −0.79 | 0.62 | 0.06 |

[38, 40] | 6 | 0.05 | 8 | −1.78 | 3.18 | 0.41 |

Sum | 153 | 1 | 153 | — | — | ${\chi}^{2}=$ 5.84 |

Index | Values | |||||||
---|---|---|---|---|---|---|---|---|

Departure Frequency | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |

Operation cost, CNY | 4546 | 3877 | 3375 | 2984 | 2671 | 2416 | 2203 | 2322 |

Difference from the minimum value | 2343 | 1674 | 1172 | 781 | 468 | 213 | 0 | 119 |

Percentage of difference to minimum | 106.4% | 76.0% | 53.2% | 35.5% | 21.2% | 9.7% | 0.0% | 5.4% |

Index | Degree of Passenger Information Acquisition | ||||

Evaluation Criterion | Excellent | Good | Average | Poor | Extremely Poor |

value | 1 | 0.75 | 0.5 | 0.25 | 0 |

**Table 8.**Comparison of indicators before and after vehicle allocation optimization for the bus line.

Index | Non-Uniform Scheduling | Uniform Scheduling | Index Weight |

Passenger-related information acquisition | 0.75 | 1 | 0.2 |

Passenger waiting time, B1 | 0.8 | 0.8 | 0.4 |

Unsatisfied demand time, B2 | 0.6 | 0.33 | 0.4 |

Overall index value | 0.71 | 0.65 |

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## Share and Cite

**MDPI and ACS Style**

Xue, Y.; Cheng, L.; Jiang, H.; Guo, J.; Guan, H.
The Optimization of Bus Departure Time Based on Uncertainty Theory—Taking No. 207 Bus Line of Nanchang City, China, as an Example. *Sustainability* **2023**, *15*, 7005.
https://doi.org/10.3390/su15087005

**AMA Style**

Xue Y, Cheng L, Jiang H, Guo J, Guan H.
The Optimization of Bus Departure Time Based on Uncertainty Theory—Taking No. 207 Bus Line of Nanchang City, China, as an Example. *Sustainability*. 2023; 15(8):7005.
https://doi.org/10.3390/su15087005

**Chicago/Turabian Style**

Xue, Yunqiang, Lin Cheng, Haoran Jiang, Jun Guo, and Hongzhi Guan.
2023. "The Optimization of Bus Departure Time Based on Uncertainty Theory—Taking No. 207 Bus Line of Nanchang City, China, as an Example" *Sustainability* 15, no. 8: 7005.
https://doi.org/10.3390/su15087005