Bi-objective autonomous vehicle repositioning problem with travel time uncertainty

Abstract

We study the problem of repositioning autonomous vehicles in a shared mobility system in order to simultaneously minimize the unsatisfied demand and the total operating cost. We first present a mixed integer linear programming formulation for the deterministic version of the problem. We extend this formulation to make it easier to work with in the non-deterministic setting. We then show how the travel time uncertainty can be incorporated into this extended deterministic formulation using chance-constraint programming. Finally, two new reformulations for the proposed chance-constraint program are developed. We show a critical result that the size of one of the reformulations (in terms of the number of variables and constraints) does not depend on the number of scenarios, and so it outperforms the other reformulation. Both reformulations are bi-objective mixed integer linear programs with a finite number of nondominated points and so they can be solved directly by algorithms such as the balanced box method (Boland et al. in INFORMS J Comput 27(4):735–754, 2015). A computational study demonstrates the efficacy of the proposed reformulations.

Appendix

In this section, we extend the results of Sect. 4.1. The imposed time limit is 3600 s. We solved each of the 20 small instances presented in Sect. 4.1 under three different settings including: 30, 60, and 90 scenarios denoted by S30, S60 and S90, respectively. The results are shown in Tables 5, 6 and 7 in which the maximum probability allowed for violating the uncertain constraints are different in these tables. Specifically, in Table 5, we set $\alpha _{it}^k=0.1$ for each $i\in \mathcal {N}$, $t\in \mathcal {T}\backslash \{1\}$ and $k\in \mathcal {M}$. In Table 6, we set $\alpha _{it}^k=0.01$ for each $i\in \mathcal {N}$, $t\in \mathcal {T}\backslash \{1\}$ and $k\in \mathcal {M}$. Finally, in Table 7, we set $\alpha _{it}^k=0.001$ for each $i\in \mathcal {N}$, $t\in \mathcal {T}\backslash \{1\}$ and $k\in \mathcal {M}$. Overall, we observe that for smaller maximum probability allowed for violating the uncertain constraints, both P5 and P6 can be solved faster but the number of nondominated points does not change.

Table 5 Numerical results obtained by running the BBM to solve P5 and P6 when $\alpha _{it}^k=0.1$ for each $i\in \mathcal {N}$, $t\in \mathcal {T}\backslash \{1\}$ and $k\in \mathcal {M}$

Table 6 Numerical results obtained by running the BBM to solve P5 and P6 when $\alpha _{it}^k=0.01$ for each $i\in \mathcal {N}$, $t\in \mathcal {T}\backslash \{1\}$ and $k\in \mathcal {M}$

Table 7 Numerical results obtained by running the BBM to solve P5 and P6 when $\alpha _{it}^k=0.001$ for each $i\in \mathcal {N}$, $t\in \mathcal {T}\backslash \{1\}$ and $k\in \mathcal {M}$