Christensen, Jonas Mark

Røpke, Stefan

The Vehicle Routing Problem with Time Windows (VRPTW) consist of determining a set of feasible vehicle routes to deliver goods to a set of customers using a hierarchical objective; first minimising the number of vehicles used and, second, the total driving distance.A three-stage method is proposed for the VRPTW. The first stage aims at minimising the number of vehicle used, whereas the second and third phase aims at minimising the travel distance. The first stage maintains an ejection pool with temporarily unserved customers, and tries to insert these customer into the existing solution. If a new candidate solution for the minimum number of vehicles is found, a route is removed and the customers from the route is placed in the ejection pool. If the heuristic terminates without finding a new candidate solution where all customers are served, the first stage returns the last found candidate solution that serves all the customers. The second stage usesan Adaptive Large Neighborhood Search (ALNS) algorithm to minimise the travel distance, during the second phase all of the generated routes are considered by solving a set cover problem. The ALNS algorithm uses 4 destroy operators, 2 repair operators and a local search method. The third stage isa Branch-Cut-and-Price (BCP) matheuristic which considers a reduced problem instance, with onlya small subset of the total arcs. This work contributes to the existing literature by integrating a set cover model with ALNS and by using BCP as a post-optimization procedure

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Downloaded from orbit.dtu.dk on: Dec 18, 2017An Adaptive Large Neighborhood Search-based Three-Stage Matheuristic for theVehicle Routing Problem with Time WindowsChristensen, Jonas Mark; Røpke, StefanPublication date:2015Document VersionPeer reviewed versionLink back to DTU OrbitCitation (APA):Christensen, J. M., & Røpke, S. (2015). An Adaptive Large Neighborhood Search-based Three-StageMatheuristic for the Vehicle Routing Problem with Time Windows. Abstract from Odysseus 2015; SixthInternational Workshop on Freight Transportation and Logistics, Ajaccio, Corsika, France.An Adaptive Large Neighborhood Search-basedThree-Stage Matheuristic for the Vehicle RoutingProblem with Time WindowsJonas Christensen1, Stefan Ropke21 Department of Transport ; Technical University of Denmark ; Denmarkjomc@transport.dtu.dk2 Department of Management Engineering ; Technical University of Denmark ; Denmarkropke@dtu.dkKeywords: Vehicle Routing Problem, Large Neighbourhood Search, Matheuristic, Set Cover Prob-lem, Operations Research.1 IntroductionThe Vehicle Routing Problem with Time Windows (VRPTW) consist of determining a set of feasiblevehicle routes to deliver goods to a set of customers using a hierarchical objective; first minimisingthe number of vehicles used and, second, the total driving distance.A three-stage method is proposed for the VRPTW. The first stage aims at minimising the numberof vehicle used, whereas the second and third phase aims at minimising the travel distance. Thefirst stage maintains an ejection pool with temporarily unserved customers, and tries to insert thesecustomer into the existing solution. If a new candidate solution for the minimum number of vehiclesis found, a route is removed and the customers from the route is placed in the ejection pool. If theheuristic terminates without finding a new candidate solution where all customers are served, the firststage returns the last found candidate solution that serves all the customers. The second stage usesan Adaptive Large Neighborhood Search (ALNS) algorithm to minimise the travel distance, duringthe second phase all of the generated routes are considered by solving a set cover problem. The ALNSalgorithm uses 4 destroy operators, 2 repair operators and a local search method. The third stage isa Branch-Cut-and-Price (BCP) matheuristic which considers a reduced problem instance, with onlya small subset of the total arcs. This work contributes to the existing literature by integrating a setcover model with ALNS and by using BCP as a post-optimization procedure.2 Solution MethodEach stage of the method can be seen as separate algorithms that, when combined yields the proposedthree-stage method. The first phase takes as input a feasible solution and tries to minimise thenumber of routes in the solution. The second phase also takes as input a feasible solution andminimises the travel distance while making sure that the number of routes used in the input solutionOdysseus 2015 - 515An ALNS-based matheuristic for VRPTWis not exceeded. The third stage takes a subset of the arcs as input and solves this new, smaller,VRPTW instance to optimality using Branch-cut-and-Price. The following subsections explains eachof the phases in more details.2.1 Phase IThe first phase starts by removing a route from the initial solution, and places the customers in anejection pool, which throughout phase I will contain the customers that are unserved in the currentsolution. If no customers remains in the ejection pool, then a new feasible solution has been found,and the number of routes used are decreased by one, in which case the procedure restarts. Phase Iis an adaption of the procedure to minimize the number of routes described in [5].2.2 Phase IIPhase II uses Adaptive Large Neighborhood Search [5] to minimise the distance travelled. In eachiteration the algorithm adaptively chooses between 4 destroy operator, and 2 repair operators, afterwhich the solution is destroyed and repaired using the chosen operators. Hereafter a simple localsearch method is used if the attempted solution is promising. The 4 destroy operators are RandomRemoval, Worst Removal, Related Removal, and Route Removal. The first three are similar to onesdescribed in [2], and Route Removal simply removes a random route. The repair methods are a basicgreedy insertion and a 2-regret insertion heuristic. Both these methods are described in more detailsin [2]. The local search method uses a simple relocate neighborhood, and tries to relocate customersto a cheaper position in the solution. The stopping criteria used in the second phase is based on thenumber of iterations.The second phase maintains a set R with all the routes encountered, to be considered througha set cover problem (SCP). The SCP is solved every t’th iteration with the intend to redirect thesearch towards new unexplored parts of the solution space. The SCP can be stated as followsminIÿrœRc¯rxr|Ax Ø 1, x œ {0, 1}J(1)Where x is a binary decision variable that describes if a route is used in the solution. Here c¯r = M+crwhere the cr is the length of route r. Adding the su ciently large number M puts significance onminimising the number of routes. If a customer is overcovered in the solution it can simply be deletedfrom routes until it is no longer overcovered, thus making the solution feasible again.2.3 Phase IIIThe third phase is used to possibly further improve the solution found at the end of the secondphase. The third phase needs a set of arcs as input for the Branch-Cut-and-Price (BCP) solver. Tofind a small set of ’good’ arcs a set S is maintained throughout the second phase. S contains the nbest unique solution found. The arcs used in the n best solution must be relative ’good’ arcs, andby giving this set of arcs to the BCP-solver it is ensured that the BCP method cannot terminatedwith an optimal solution that is worse than the solution already found.The VRPTW instance defined on the sparse graph is passed to a state-of-the-art BCP solver [4]which is allowed a limited time to find an improved solution.Odysseus 2015 - 516An ALNS-based matheuristic for VRPTWPreliminary results shows that the BCP method is fast for the instances with 100 customers,and also e cient for the instance with 200 customers. For the bigger instances it seldom succeedsin finding an improved solution within the timelimit (600s). A major obstacle is that the pricingproblem in the BCP algorithm becomes too di cult to solve for the larger instances. To alleviatethis we are planning to only solve the pricing problem using heuristics.3 Preliminary ResultsAn algorithm consisting of only phase 2 and 3 of the proposed algorithm, has been tested on the 56Solomon benchmark instances [6]. The Solomon benchmark instances are divided in 2 classes, 1 and2, each with three subclasses C, R, RC. The instances from the class 1 have tight time windows, andthe instances from the class 2 have wide time windows. The customers in the C subclass are clustered,whereas they are placed randomly and random-clustered in the R and RC classes respectively. Thisalgorithm only focus on minimizing the total travel distance. The results are compared with knownoptimal solutions, as reported in [3] and [4]. The algorithm has been executed 10 times for everyinstance. Table 1 shows the preliminary resultsBest found solution Average solution Best found solution Average solutionCTD x¯b CTD x¯ t¯ Opt CTD x¯b CTD x¯ t¯ (s) OptC1 7440.30 0.00% 7440.30 0.00% 31 90/90 C2 4699.00 0.00% 4699.00 0.00% 52 80/80R1 14085.20 0.01% 14094.02 0.09% 150 82/120 R2 9602.40 0.05% 9624.85 0.30% 114 35/110RC1 10676.70 0.01% 10683.41 0.08% 239 46/90 RC2 8005.80 0.01% 8016.08 0.14% 62 54/80All 54509.40 0.01% 54557.66 0.11% 110 387/560Table 1: Overall solution statistics. CTD is the cumulative travel distance. x¯b is the gap for the bestsolution, and x¯ is the gap for the average solution. t¯ is the average time spent in seconds, and Opt denotesout of how many optimal solutions were found for each of the instance classes. Distances are truncated toone decimal as it is common in the literature on exact methods for the VRPTW.The preliminary results shows that the algorithm is very good at minimizing the travel distance, andon average finds solutions that are 0.11% from the known optimal solution. Moreover the algorithmfinds the optimal solution for 49 out of 56 instances. Table 2 shows how the di erent parts of thealgorithm contributes to the overall results. Here 3 di erent algorithm have been tested;1. ALNS: Only ALNS, the set cover problems are not solved.2. ALNS SCP: ALNS with the inclusion of the set cover problems. This algorithm correspondsto phase II as described in section 2.2.3. ALNS SCP BCP: This algorithm uses both the second and third phase.Table 2 shows the benefit of consider all the routes through a set cover model, as this greatlyincrease the quality of the solutions while incurring a reasonable increase in the time used. The BCPmethod increases the quality of the best solutions, and increases the number of optimal solutionsfound.Figure 1 shows how the second phase converges. For this test, the stopping criteria was 25000iterations and the set cover model was solved every 2500’th iteration. The figure shows that solvingthe SCP greatly helps the convergence in the beginning of the algorithm. In the end of the algorithmthe e ect is not as huge, but there is still a noticeable change.Odysseus 2015 - 517An ALNS-based matheuristic for VRPTWBest Solution Average Solution Optimalst¯ x¯ worst x¯ worst Total InstancesALNS 59.01 0.13% 1.64% 0.43% 2.45% 264/560 35/56ALNS SCP 90.55 0.03% 0.54% 0.13% 0.74% 353/560 42/56ALNS SCP BCP 109.87 0.01% 0.54% 0.11% 0.70% 387/560 49/56Table 2: Performance of the algorithms for the Solomon instances. t¯ is the average time used for thealgorithm, x¯ denotes the average gap and worst is the worst achieved solution both when considering the bestsolution, or the average solution. The Optimal columns denotes how many optimal solutions were found, andfor how many instances the algorithm found an optimal solution. The number after the ’/’ is the number ofpossible optimal solutions. Each instance were solved 10 times for each of the algorithms0 2500 5000 7500 10000 12500 15000 17500 20000 22500 250000%1%2%3%4%5%IterationGapGapFigure 1: Convergence of phase IICurrently, work on finalizing and fine-tuning the vehicle minimization phase is going on. Resultsfrom this phase will be available at the the time of the presentation and the algorithm will be testedon the larger instances from Gehring and Homberger’s benchmark set [1].References[1] Hermann Gehring and Jo¨rg Homberger. “Parallelization of a Two-Phase Metaheuristic for Rout-ing Problems with Time Windows”. English. In: Journal of Heuristics 8.3 (2002), pp. 251–276.issn: 1381-1231.[2] David Pisinger and Stefan Ropke. “A general heuristic for vehicle routing problems”. In: Com-puters and Operations Research 34 (2007), pp. 2403–2435. issn: 03050548.[3] Roberto Roberti. “Exact Algorithms for Di erent Classes of Vehicle Routing Problems”. PhDthesis. University of Bologna, 2012.[4] Stefan Ropke. “Branching decisions in branch- and-cut-and-price algorithms for vehicle routingproblems”. In: Column Generation. 2012.[5] Stefan Ropke and David Pisinger. “An Adaptive Large Neighborhood Search Heuristic for thePickup and Delivery Problem with Time Windows”. In: Transportation Science 40 (2006),pp. 455–472. issn: 0041-1655.[6] Marius M. Solomon. “Algorithms for the vehicle routing and scheduling problems with timewindow constraints”. In: Operations research 35 (1987), pp. 254–265. issn: 0030-364X.Odysseus 2015 - 518

An Adaptive Large Neighborhood Search-based Three-Stage Matheuristic for the Vehicle Routing Problem with Time Windows

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An Adaptive Large Neighborhood Search-based Three-Stage Matheuristic for the Vehicle Routing Problem with Time Windows

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