Vehicle routing problem: Difference between revisions

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[[File:Illustration of the Vehicle Routing Problem.svg|thumb|An illustration of an instance of the vehicle routing problem in a road network, containing routes for three vehicles to deliver goods from a central depot (D) to 11 locations.]]

The '''vehicle routing problem''' ('''VRP''') is a [[combinatorial optimization]] and [[integer programming]] problem which asks "What is the optimal set of routes for a fleet of vehicles to traverse in order to deliver to a given set of customers?" The problem first appeared, as ''the truck dispatching problem'', in a paper by [[George Dantzig]] and John Ramser in 1959,<ref name=DantzigRamser1959>{{cite journal|last=Dantzig|first=George Bernard|author2=Ramser, John Hubert |title=The Truck Dispatching Problem|journal=Management Science|date=October 1959|volume=6|issue=1|pages=80–91|url=http://andresjaquep.files.wordpress.com/2008/10/2627477-clasico-dantzig.pdf|doi=10.1287/mnsc.6.1.80}}</ref> in which it was applied to petrol deliveries. Often, the context is that of delivering goods located at a central depot to customers who have placed orders for such goods.<ref>{{cite journal |last1=Fisher |first1=Marshall L. |last2=Jaikumar |first2=Ramchandran |title=A generalized assignment heuristic for vehicle routing |journal=Networks |date=June 1981 |volume=11 |issue=2 |pages=109–124 |doi=10.1002/net.3230110205}}</ref> However, variants of the problem consider, e.g, collection of [[Municipal solid waste|solid waste]]<ref>{{cite book |last1=Shuster |first1=Kenneth A. |last2=Schur |first2=Dennis A. |title=Heuristic Routing for Solid Waste Collection Vehicles |date=1974 |publisher=U.S. Environmental Protection Agency}}</ref> and the transport of the elderly and the sick to and from health-care facilities.<ref>{{cite journal |last1=Cordeau |first1=Jean-François |last2=Laporte |first2=Gilbert |title=The dial-a-ride problem: models and algorithms |journal=Annals of Operations Research |date=July 2007 |volume=153 |issue=1 |pages=29-46 |doi=10.1007/s10479-007-0170-8}}</ref><ref>{{cite journal |last1=Cappart |first1=Quentin |last2=Thomas |first2=Charles |last3=Schaus |first3=Pierre |last4=Rousseau |first4=Louis-Martin |title=A Constraint Programming Approach for Solving Patient Transportation Problems |journal=Principles and Practice of Constraint Programming |date=2018 |volume=11008 |pages=490–506 |doi=10.1007/978-3-319-98334-9_32|hdl=2078.1/202079 |hdl-access=free }}</ref> The standard objective of the VRP is to minimise the total route cost.<ref name="toth" />{{rp|5}} Other objectives, such as minimising the number of vehicles used or travelled distance are also considered.<ref>{{cite journal |last1=Gaskell |first1=T. J. |title=Bases for Vehicle Fleet Scheduling |journal=Journal of the Operational Research Society |date=September 1967 |volume=18 |issue=3 |pages=281–295 |doi=10.1057/jors.1967.44}}</ref>

The VRP generalises the [[travelling salesman problem]] (TSP), which is equivalent to requiring a single route to visit all locations. As the TSP is [[NP-hard]], the VRP is also NP-hard.<ref name=toth>{{cite book |editor=Toth, P. |editor2=Vigo, D. |title=The Vehicle Routing Problem |volume=9 |series=Monographs on Discrete Mathematics and Applications |year=2002 |publisher=Society for Industrial and Applied Mathematics |___location=Philadelphia |isbn=0-89871-579-2}}</ref>{{rp|5}}

The '''vehicle routing problem''' ('''VRP''') is a [[combinatorial optimization]] and [[integer programming]] problem which asks "What is the optimal set of routes for a fleet of vehicles to traverse in order to deliver to a given set of customers?" The problem first appeared, as ''the truck dispatching problem'', in a paper by [[George Dantzig]] and John Ramser in 1959,<ref name=DantzigRamser1959>{{cite journal|last=Dantzig|first=George Bernard|author2=Ramser, John Hubert |title=The Truck Dispatching Problem|journal=Management Science|date=October 1959|volume=6|issue=1|pages=80–91|url=http://andresjaquep.files.wordpress.com/2008/10/2627477-clasico-dantzig.pdf|doi=10.1287/mnsc.6.1.80}}</ref> in which it was applied to petrol deliveries. Often, the context is that of delivering goods located at a central depot to customers who have placed orders for such goods. The objective of the VRP is to minimize the total route cost. In 1964, Clarke and Wright improved on Dantzig and Ramser's approach using an effective [[greedy algorithm]] called the savings algorithm.{{Citation needed|date=July 2025}}

The VRP generalises the [[travelling salesman problem]] (TSP), which is equivalent to requiring a single route to visit all locations. As TSP is [[NP-hard]], the VRP is also NP-hard.<ref name=toth>{{cite book |editor=Toth, P. |editor2=Vigo, D. |title=The Vehicle Routing Problem |volume=9 |series=Monographs on Discrete Mathematics and Applications |year=2002 |publisher=Society for Industrial and Applied Mathematics |___location=Philadelphia |isbn=0-89871-579-2}}</ref>

VRP has many direct applications in industry. Vendors of VRP routing tools often claim that they can offer cost savings of 5%–30%.<ref name="Springer Verlag">{{cite book |url=https://books.google.com/books?id=UMI5GzcNd8wC&q=%22vendors+of+routing+tools%22 |title=Geometric Modelling, Numerical Simulation, and Optimization:: Applied Mathematics at SINTEF |date=2007 |publisher=Springer Verlag |isbn=978-3-540-68783-2 |editor=Geir Hasle |___location=Berlin |pages=397–398 |editor2=Knut-Andreas Lie |editor3=Ewald Quak}}</ref> Commercial solvers tend to use [[Heuristic|heuristics]] due to the size and frequency of real world VRPs they need to solve.{{Citation needed|date=July 2025}}

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** The Capacitated Team Orienteering Problem (CTOP),

** The TOP with Time Windows (TOPTW).

*Vehicle Routing Problem with Backhauling (VRPB): [[Disjoint sets]] of delivery and pickup customers are given. Goods have to be delivered from the depot to the delivery customer and from the pickup customers to the depot.<ref name="deif-backhauling">{{cite journal |last1=Deif |first1=I. |last2=Bodin |first2=L. A. |editor1-last=Kidder |editor1-first=A. |title=Extension of the Clarke and Wright Algorithm for Solving the Vehicle Routing Problem with Backhauling |journal=Proceedings of the Babson College conference on software uses in transportation and logistic management |date=1984 |pages=75-96|___location=Babson Park, MA, US}}</ref> Vehicles may be forbidden from picking up goods from customers until all carried goods have been delivered to delivery customers or allowed interchanging pickups with deliveries at a potential cost.<ref name="deif-backhauling"/><ref>{{cite journal |last1=Golden |first1=B. L. |last2=Baker |first2=E. |last3=Alfaro |first3=J. |last4=Schaffer |first4=J. |editor1-last=Hammesfahr |editor1-first=R. |title=The Vehicle Routing Problem with Backhauling: Two approaches |journal=Proceedings of the Twenty-First Annual Meeting of S.E. TIMS |date=1985 |pages=90-92 |___location=Myrtle Beach, SC, US}}</ref>

*Vehicle Routing Problem with Pickup and Delivery (VRPPD): A number of goods need to be moved from certain pickup locations to other delivery locations. The goal is to find optimal routes for a fleet of vehicles to visit the pickup and drop-off locations.

*Vehicle Routing Problem with [[LIFO (computing)|LIFO]]: Similar to the VRPPD, except an additional restriction is placed on the loading of the vehicles: at any delivery ___location, the item being delivered must be the item most recently picked up. This scheme reduces the loading and unloading times at delivery locations because there is no need to temporarily unload items other than the ones that should be dropped off.

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which imposes that at least {{tmath|r(S)}}arcs leave each customer set <math>S</math>.<ref name=toth />

GCECs and CCCs have an exponential number of constraints so it is practically impossible to solve the linear relaxation. A possible way to solve this is to consider a limited subset of these constraints and add the rest if needed. Identification of the needed constraints is done via a separation procedure. Efficient exact separation methods for such constraints (based on [[Linear programming|mixed integer programming]]) have been developed.<ref>{{Cite journal |last1=Pavlikov |first1=K. |last2=Petersen |first2=N. C. |last3=Sørensen |first3=J. L. |year=2023 |title=Exact Separation of the Rounded Capacity Inequalities for the CapacitatedVehicle Routing Problem |journal=Networks |volume=83 |pages=197–209 |doi=10.1002/net.22183 |s2cid=263321558 |place=Networks|doi-access=free }}</ref>

A different method again is to use a family of constraints which have a polynomial cardinality which are known as the MTZ constraints, they were first proposed for the TSP <ref name=mtz>{{cite journal|last1=Miller|first1=C. E.|last2=Tucker|first2=E. W.|last3=Zemlin|first3=R. A.|title=Integer Programming Formulations and Travelling Salesman Problems|journal=J. ACM|date=1960|volume=7|pages=326–329|doi=10.1145/321043.321046|s2cid=2984845|doi-access=free}}</ref> and subsequently extended by Christofides, Mingozzi and Toth.<ref name=christof>{{cite book|last1=Christofides|first1=N.|last2=Mingozzi|first2=A.|last3=Toth|first3=P.|title=The Vehicle Routing Problem|date=1979|publisher=Wiley|___location=Chichester, UK|pages=315–338}}</ref>