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The Dial-A-Ride Problem considering the in-vehicle crowding inconvenience due to COVID-19

The Dial-A-Ride Problem (DARP) has received significant attention during the COVID-19 pandemic. During the pandemic's peak, public transport ridership was reduced up to 90% in several countries and many public transport users had to seek less crowded alternatives in DARP services. Such alternatives are flexible modes that do not operate on fixed lines (i.e., on-demand minibuses, shared vehicles). However, the standard Dial-A-Ride Problem (DARP) does not consider the in-vehicle crowding as long as the capacity of the vehicle is not exceeded. To rectify this, this study proposes a new formulation of the DARP that considers also the inconvenience of passengers due to the in-vehicle crowding levels in the objective function of the problem. In our formulation, we consider a progressive penalization of the increase of in-vehicle crowding to account for social distancing. This is modeled with piecewise linear functions that map the inconvenience of passengers to the in-vehicle crowding levels. The proposed model is a MINLP and it is reformulated as a MILP that can be solved with branch-and-bound and linear programming. This model is implemented in numerical experiments with benchmark DARP datasets to investigate the increases of the vehicle route costs when seeking to reduce the in-vehicle crowdedness.