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...

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Bibliographic Details
Published in:Mathematics (Basel) Vol. 13; no. 22; p. 3638
Main Authors: Soto-Concha, Raúl, Morillo-Torres, Daniel, Escobar, John Willmer, Mena-Reyes, Jorge Félix, Linfati, Rodrigo
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.11.2025
Subjects:
Collaboration
Costs
Customers
Integer programming
Inventory
last-mile logistics
Linear programming
Logistics
Mixed integer
Mobile communication systems
mobile satellites
Orbital rendezvous
release times
route reloading
Satellites
Synchronism
Vehicle routing
vehicle routing problem
Vehicles
Wireless communication systems
Chile
ISSN:2227-7390, 2227-7390
Online Access:Get full text
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Summary: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.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math13223638