Timetable optimization for single bus line involving fuzzy travel time

Abstract

Timetable optimization is an important step for bus operations management, which essentially aims to effectively link up bus carriers and passengers. Generally speaking, bus carriers attempt to minimize the total travel time to reduce its operation cost, while the passengers attempt to minimize their waiting time at stops. In this study, we focus on the timetable optimization problem for a single bus line from both bus carriers’ perspectives and passengers’ perspectives. A bi-objective optimization model is established to minimize the total travel time for all trips along the line and the total waiting time for all passengers at all stops, in which the bus travel times are considered as fuzzy variables due to a variety of disturbances such as weather conditions and traffic conditions. A genetic algorithm with variable-length chromosomes is devised to solve the proposed model. In addition, we present a case study that utilizes real-life bus transit data to illustrate the efficacy of the proposed model and solution algorithm. Compared with the timetable currently being used, the optimal bus timetable produced from this study is able to reduce the total travel time by 26.75% and the total waiting time by 9.96%. The results demonstrate that the established model is effective and useful to seek a practical balance between the bus carriers’ interest and passengers’ interest.

Appendix A

In this appendix, some basic concepts and definitions about credibility theory are introduced.

Let ${\Theta }$ be a nonempty set, and let $\mathcal {A}$ be its power set. Each element of ${\mathcal {A}}$ is called an event. Credibility measure is a set function from ${\mathcal {A}}$ to [0, 1]. For each event, its credibility indicates the chance that the event will occur. In order to ensure that the set function has certain mathematical properties, Li and Liu (2006) provided the following four axioms:

Axiom 1

(Normality) Cr $\{{\Theta }\}=1$.

Axiom 2

(Monotonicity) Cr $\{A\}\le$ Cr $\{B\}$ for any events $A\subseteq B$.

Axiom 3

(Duality) Cr $\{A\}+$ Cr $\{A^c\}=1$ for any event A.

Axiom 4

(Maximality) Cr $\{\cup _i A_i\}= \sup _i$ Cr $\{A_i\}$ for any collection of events $\{A_i\}$ with $\sup _i$ Cr $\{A_i\}<0.5$.

Let ${\Theta }$ be a nonempty set, ${\mathcal {A}}$ the power set, and $\mathbf{Cr }$ a credibility measure. Then the triplet $({\Theta },{\mathcal {A}},$ Cr) is called a credibility space. Let $\xi$ be a fuzzy variable on the credibility space $({\Theta },\mathcal {A},$ Cr). Then, its expected value (Liu and Liu 2002) is defined as

$$\begin{aligned} E[\xi ]=\int _0^{+\infty } \mathbf{Cr }\{\xi \ge r\}dr - \int _{-\infty }^{0}\mathbf{Cr }\{\xi \le r\}dr, \end{aligned}$$

provided that at least one of the two integrals is finite.

Example 1

Assume that $\xi$ is a simple fuzzy variable taking district values in $\{x_1,x_2,\ldots ,x_\mathrm{m}\}$. If $\xi$ has the following credibility function

$$\begin{aligned} v(x) = \left\{ \begin{array}{ll} v_1, &{} \quad \text{ if } \ x=x_1\\ v_2, &{} \quad \text{ if } \ x=x_2\\ \cdots &{} \quad \cdots \\ v_\mathrm{m}, &{} \quad \text{ if } \ x=x_\mathrm{m},\\ \end{array} \right. \end{aligned}$$

then it has expected value

$$\begin{aligned} E[\xi ]=\sum \limits _{i=1}^m w_i x_i, \end{aligned}$$

(11)

where for each $1 \le i \le m$, the weight $w_i$ is given by

$$\begin{aligned} w_i= & {} \max \limits _{x_j \le x_i} v_j \wedge 0.5 - \max \limits _{x_j < x_i} v_j \wedge 0.5 \\&+\max \limits _{x_j \ge x_i} v_j \wedge 0.5 - \max \limits _{x_j> x_i}v_j \wedge 0.5. \end{aligned}$$