Autonomous and conventional bus fleet optimization for fixed-route operations considering demand uncertainty

Abstract

The emerging technology of autonomous vehicles has been widely recognized as a promising urban mobility solution in the future. This paper considers the integration of autonomous vehicles into bus transit systems and proposes a modeling framework to determine the optimal bus fleet size and its assignment onto multiple bus lines in a bus service network considering uncertain demand. The mixed-integer stochastic programming approach is applied to formulate the problem. We apply the sample average approximation (SAA) method to solve the formulated stochastic programming problem. To tackle the nonconvexity of the SAA problem, we first present a reformulation method that transforms the problem into a mixed-integer conic quadratic program (MICQP), which can be solved to its global optimal solution by using some existing solution methods. However, this MICQP based approach can only handle the small-size problems. For the cases with large problem size, we apply the approach of quadratic transform with linear alternating algorithm, which allows for efficient solution to large-scale instances with up to thousands of scenarios in a reasonable computational time. Numerical results demonstrate the benefits of introducing autonomous buses as they are flexible to be assigned across different bus service lines, especially when demand uncertainty is more significant. The introduction of autonomous buses would enable further reduction of the required fleets and total cost. The model formulation and solution methods proposed in this study can be used to provide bus transit operators with operational guidance on including autonomous buses into bus services, especially on the autonomous and conventional bus fleets composition and allocation.

Appendix 1

Minimizing the objective function is equivalent to maximizing the following function:

$$\begin{aligned} \max \quad&\frac{\delta w_t |R||L| }{|R|} - Z(P2) \nonumber \\&= -\alpha ^a M - \sum _{l} \alpha ^c N_{l} - \sum _{l} c_{l}^c f_{l}^c - \frac{1}{|R|}\left[ \sum _{r} \sum _{l} c_l^a f_{l,r}^a +\sum _{r} \sum _{l} c_{p} \cdot u_{l, r}\right] \nonumber \\&+ \frac{w_t}{|R|} \sum _{r} \sum _{l} \frac{\delta (f_l^c + f_{l,r}^a) - k s_{l,r}}{f_l^c + f_{l,r}^a} \end{aligned}$$

(40)

where Z(P2) is the objective function of P2. Define a variable y such that $y_{l,r} = \frac{\sqrt{\delta (f_l^c + f_{l,r}^a) - k s_{l,r}}}{f_l^c + f_{l,r}^a}$. (40) is equivalent to

$$\begin{aligned} \max \quad&\frac{w_t}{|R|}\left( \sum _l \sum _r 2 y_{l,r} \sqrt{\delta (f_l^c + f_{l,r}^a) - k s_{l,r}}-\sum _l \sum _r y_{l,r}^2(f_l^c + f_{l,r}^a)\right) \nonumber \\&-\alpha ^a M - \sum _{l} \alpha ^c N_{l} - \sum _{l} c_{l}^c f_{l}^c - \frac{1}{|R|}\left[ \sum _{r} \sum _{l} c_l^a f_{l,r}^a +\sum _{r} \sum _{l} c_{p} \cdot u_{l, r}\right] \end{aligned}$$

(41)

since plugging $y_{l,r}$ into (41) will lead to (40). Finally, define function $h(f_l^c, f_{l,r}^a,s_{l,r}) = \sqrt{\delta (f_l^c + f_{l,r}^a) - k s_{l,r}}$, we can rewrite the formulation in the hypograph form of h, i.e., $\beta \le h$, leading to the formulation in P4. Note that, to ensure the square root function is well-defined, $\delta$ should be chosen such that $\delta (f_l^c + f_{l,r}^a) - k s_{l,r} > 0, \forall l \in I, r \in R$.