Last mile deliveries with lockers: formulations and algorithms

Last‐mile delivery with drone and lockers

“…We first propose a formulation inspired by Roberti and Ruthmair's [69] TSP-D model and Buzzega and Novellani's [12] TSPL model, which uses the following sets of variables:…”

“…Constraints (11) impose that if a locker is used, then one arc entering the locker must take value one. Constraints (12) impose a maximum capacity on the number of deliveries assigned to each locker.…”

“…The formulation described in this section, Formulation Three, reduces the number of needed binary variables and constraints with respect to Formulation Two. It is inspired by those proposed in Dell'Amico et al [22] for the FSTSP (compared to which we reduced the number of linear variables and big-M constraints in the drone component of the model) and in Buzzega and Novellani [12] for the TSPL, and makes use of the already defined variables x T ij , x L ij , x R ij , y ib , t i and the following new ones:…”

“…The MILP formulation models hard time windows but the authors solve the case with soft time windows, to increase efficiency. Buzzega and Novellani [12] propose improved MILP formulations and branch‐and‐cut (B&C) algorithms for a similar problem that they call the vehicle routing problem (VRP) with lockers , which may or may not consider (hard) time windows and multiple vehicles. The cost of servicing a customer via locker is included in the objective function as a penalty, together with the traveling costs, that must be minimized.…”

“…We first propose a formulation inspired by Roberti and Ruthmair's [69] TSP‐D model and Buzzega and Novellani's [12] TSPL model, which uses the following sets of variables: $$$ {x}_{ij}^T $$$, a binary variable that equals 1 if the truck travels arc $$$ \left(i,j\right)\in A $$$, 0 otherwise. $$$ {x}_{ij}^D $$$, a binary variable that takes value 1 if the drone travels the arc $$$ \left(i,j\right)\in A $$$; 0 otherwise. Note that the variable $$$ {x}_{ij}^D $$$ may take value 1 also if the drone travels the arc $$$ \left(i,j\right) $$$ on the truck.…”