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Croatian Operational Research Review463CRORR8(2017), 463–469Two models of the capacitated vehicle routing problemZuzana Borˇcinov´a1,*1Faculty of Management Science and Informatics, University ofˇZilina, Univerzitn´a8215/1, 010 026ˇZilina, SlovakiaE-mail:h[email protected]iAbstract.The aim of the Capacitated Vehicle Routing Problem (CVRP) is to find a setof minimum total cost routes for a fleet of capacitated vehicles based at a single depot,to serve a set of customers.There exist various integer linear programming models ofthe CVRP. One of the main differences lies in the way to eliminate sub-tours, i.e. cyclesthat do not go through the depot. In this paper, we describe a well-known flow formulationof CVRP, where sub-tour elimination constraints have a cardinality exponentially growingwith the number of customers. Then we present a mixed linear programming formulationwith polynomial cardinality of sub-tour elimination constraints. Both of the models wereimplemented and compared on several benchmarks.Key words: Capacitated Vehicle Routing Problem, linear programming modelReceived: September 23, 2016; accepted: August 21, 2017; available online: November 30,2017DOI: 10.17535/crorr.2017.00291. Introduction and problem descriptionThe Capacitated Vehicle Routing Problem (CVRP) is one of the fundamental prob-lems in the combinatorial optimization with a number of practical applications intrans- portation, distribution and logistics.The aim of CVRP is to find a set ofminimum total cost routes for a fleet of capacitated vehicles based at a single depot,to serve a set of customers under the following constraints:(1) each route begins and ends at the depot,(2) each customer is visited exactly once,(3) the total demand of each route does not exceed the capacity of the vehicle [8].The first mathematical formulation and algorithm for the solution of the CVRPwas proposed by Dantzig and Ramser [2] in 1959 and five years later, Clarke andWright [1] proposed the first heuristic for this problem.To date, many solutionmethods for the CVRP have been published.General surveys can be found inToth and Vigo [11] and Laporte [7].The CVRP belongs to the category of NPhard problems that can be exactly solved only for small instances of the problem.Therefore, researchers have concentrated on developing heuristic algorithms to solvethis problem (for example [6], [3]).*Corresponding author.c 2017 Croatian Operational Research Society
464Zuzana Borˇcinov´a2. Mathematical formulation od the CVRPLetG= (V, H, c) be a complete directed graph withV={0,1,2, . . . , n}as the setof nodes andH={(i, j) :i, jV, i6=j}as the set of arcs, where node 0 representsthe depot for a fleet ofpvehicles with the same capacityQand remainingnnodesrepresent geographically dispersed customers. Each customeriV- {0}has a cer-tain positive demanddiQ. The non negative travel costcijis associated with eacharc (i, j)H. The cost matrix is symmetric, i.e.cij=cjifor alli, jV, i6=jand

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