Mixed-Integer Linear Programming Models for the Vehicle Routing Problem with Release Times and Reloading at Mobile Satellites

The Vehicle Routing Problem (VRP) is central to last-mile logistics, yet a gap remains when products have late release times and vehicles can be reloaded en route via mobile satellites that rendezvous with reloading vehicles at customer locations. We propose the VRP with Release Times and Reloading at Mobile Satellites (VRP-RT-RMS) and develop two mixed-integer linear programming formulations: a three-index (MILP-3) and a two-index (MILP-2). The objective minimizes total distance subject to capacity, route duration, synchronization, and time constraints. We generated 40 instances from real data (10 per size N∈{10,15,20,25}). En-route reloads simultaneously reduce distance and fleet size and can restore feasibility when the classical VRP is infeasible. To contrast the classical VRP with our VRP-RT-RMS, we analyzed a particular instance with N=10 customers: total distance decreased by 7.26% and the number of vehicles fell from 5 to 3. As instance size grows, MILP-2 shows superior scalability and efficiency compared with MILP-3. Beyond the technical scope, coordinating reloads is pertinent to urban operations with late product releases, lowering kilometers traveled and delivery times.