X Wang B Golden and E Wasil INFORMS San Francisco November 2014 Introduction Minmax objective In the MultiDepot VRP the objective is to minimize the total distance traveled by all vehicles ID: 261471
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The Min-Max Split Delivery Multi-Depot Vehicle Routing Problem with Minimum Delivery Amounts
X. Wang, B. Golden, and E. WasilINFORMSSan FranciscoNovember 2014 Slide2
Introduction: Min-max objective
In the Multi-Depot VRP, the objective is to minimize the total distance traveled by all vehiclesIn the min-max MDVRP, the objective is to minimize the maximum distance traveled by a vehicle (Carlsson et al. 2009)1Slide3
IntroductionWhy is the min-max objective important?
Applications Disaster relief effortsServe all victims as soon as possible
Computer networks
Minimize maximum latency between a server and a client
Workload balance
Balance workload among drivers or across time horizon
2Slide4
Introduction: Split serviceYakici and
Karasakal (2013) studied a min-max service VRP with split delivery and heterogeneous demandDuration of a route = travel time + service timeService times can significantly change the optimal routing plan of the min-max VRP (Bertazzi et al. 2014)3Slide5
Introduction: Minimum deliverySplit delivery may inconvenience the customers
Gulczynski et al. (2010) introduced a split delivery VRP with minimum delivery amounts4Slide6
Introduction: Min-max SDMDVRP-MDAWe want to develop an algorithm for a problem with
Min-max objectiveMultiple depotsService timesSplit deliveriesMinimum delivery amounts5Slide7
Structural Propertiesk-
split cycle (Dror and Trudeau, 1970)Any min-max SDMDVRP (no minimum delivery requirement) has an optimal solution in which there is no k-split cycle6Slide8
Structural properties: ClustersConsider an auxiliary graph
Vertices: routesEdges: customers with split serviceA cluster of routes is a set of routes with the corresponding vertices in a connected component of the auxiliary graph7Slide9
Structural properties: Clusters
8Slide10
Structural propertiesAny min-max SDMDVRP has an optimal solution such that any two routes that split a customer have the same duration
Any min-max SDMDVRP has an optimal solution with all routes in the same cluster having the same duration (balanced clusters)9Slide11
Algorithm: Cluster balance subroutineThe balanced structure is frequently disrupted during the local search procedure
We developed a cluster balance subroutine using a network model toRestore balance if possibleBreak up clusters if balance cannot be restored 10Slide12
Algorithm: Cluster balance subroutineCluster
Auxiliary graph
11Slide13
Algorithm: Cluster balance subroutine
Compute the target durationDetermine the flows that minimize the maximum deviation of the route durations from the target durationIf the maximum deviation is zero, balance is restored; otherwise, break the cluster12Slide14
AlgorithmInitialization
We modified MD (Wang et al. 2014) to initialize a feasible solution with no split deliveriesLocal search (ignoring minimum delivery amounts)Step 1. From the cluster with the longest route duration, identify a customer to split, starting from the end customers13Slide15
AlgorithmLocal search
Step 2. Locate a position in another cluster to insert the customer (cheapest insertion)Step 3. Merge the two clustersStep 4. Restore balance in the merged clusterImproved – go back to Step 1Not improved – try splitting another customerStep 5. Stop if we have tried to split every customer in the cluster14Slide16
Algorithm: Local searchBefore merge
Merged cluster15Slide17
Algorithm: Local searchMerged cluster
Balanced cluster16Slide18
Algorithm
PerturbationStep 1. Perturb the locations of the depots
17Slide19
Algorithm
PerturbationStep 2. Solve the new problemStep 3. Set the depots back to the their original positionsStep 4. Solve the problem and update the solutionStep 5. Repeat the process until there is no improvement for five consecutive perturbations18Slide20
Algorithm: Satisfy the minimum delivery amounts
Apply the cluster balance subroutine with additional constraintsIf delivered service <= minimum amount/2Remove all serviceIf delivered service > minimum amount/2Increase the service delivered to the minimum amount19Slide21
Computational resultsGenerated 258 test instances from the 43 instances in Wang et al. (2014)
Service timeShort service [1 – 10)Medium service [10 – 100)Long service [100 – 1000)Customer-to-vehicle ratioShort route (less than 20)Medium route(between 20 and 50)Long route (between 50 and 100)20Slide22
Computational results
(%)Short route
Medium
route
Long route
Average
Short service
2.64
0.68
0.28
1.24
Medium
service
4.60
1.08
0.33
2.08
Long service
7.80
1.61
0.55
3.45
Average
5.01
1.12
0.39
2.26
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Table 1: Savings from non-split solutionsSlide23
Computational results:
MDA fraction0
0.1
0.2
0.3
0.4
Short service
1.24
1.21
1.11
1.00
0.84
Medium
service
2.08
1.89
1.53
1.23
0.76
Long service
3.45
3.19
2.83
2.39
1.73
Short route
5.01
4.71
4.16
3.49
2.56
Medium
route
1.12
1.01
0.84
0.73
0.51
Long route
0.39
0.34
0.27
0.22
0.14
Average
2.26
2.10
1.83
1.54
1.11
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Table 2: Average savings from splitting with various minimum delivery fractionsSlide24
ConclusionsWe developed a heuristic that solved the min-max
SDMDVRP – MDA in four stagesIn future work, we want to improve the algorithm further and compare its performance to other possible approaches23Slide25
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Q & Awangxy@umd.edu