The crowd-shipping with penalty cost function and uncertain travel times

The vehicle routing problem with time windows and occasional drivers (VRPODTW) is an extension of the vehicle routing problem with time windows, where ordinary people may perform deliveries support- ing company drivers to serve a set of customers. We consider a VRPODTW variant with uncertain travel times and a penalty for each missed delivery, i.e., when either company drivers or occasional drivers ar- rive after the ending of the time window and the delivery does not take place. We formulate the problem with a chance-constrained stochastic model imposing a probability on the maximum number of missing deliveries. Then, we propose an equivalent adjustable robust formulation via uncertain polytope whose feasibility guarantees the probability constraint. We define two optimal solution approaches based on Benders’ decomposition and column-and-row generation. For the former, we consider logic and optimal- ity cuts. The column-and-row generation relies on a relaxation of the uncertain polytope where mean- ingful realizations of the uncertain travel times are included on the fly. Numerical results are collected on benchmarks for the VRPODTW, properly modified to take into account the uncertainty. We analyze the behavior of the proposed optimal strategies, and discuss the benefit of addressing the uncertain prob- lem showing better resource allocation with the robust solutions compared with the nominal ones, via a sampling analysis.