Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Asymmetric Vehicle Routing Problem

Rosa HerreroUniversitat Autònoma de Barcelona

Dr. Alejandro RodríguezUniversitat Politècnica de València

Dr. Angel A. JuanUniversitat Oberta de Catalunya

2012 IN3-HAROSA Workshopfor Junior Researchers

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Outline

1 Asymmetric Capacitated Vehicle Routing Problem

2 On the use of Monte Carlo simulation, cache and splittingtechniques to improve the Clarke and Wright savings heuristics

3 Adaptation to the Asymmetric

4 Results

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Outline

1 Asymmetric Capacitated Vehicle Routing Problem

2 On the use of Monte Carlo simulation, cache and splittingtechniques to improve the Clarke and Wright savings heuristics

3 Adaptation to the Asymmetric

4 Results

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Asymmetric Capacitated Vehicle RoutingProblem

A set of customer demands have to be servedA fleet of homogeneous fleetEach customer is supplied by a single vehicleThe travel cost represents asymmetric distances or travellingtimesMinimizing travel cost

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Example

Solution using minimum distances and real itineraries 5 / 20

Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

References

Laporte, G., Mercure, H., and Nobert, Y. (1986).An exact algorithm for the asymmetrical capacitatedvehicle-routing problem. Networks, 16(1):33–46.Fischetti, M., Toth, P., and Vigo, D. (1994).A branch-and-bound algorithm for the capacitatedvehicle-routing problem on directed-graphs. OperationsResearch, 42(5):846–859.Clarke, G. and Wright, J. (1964).Scheduling of vehicles from central depot to number ofdelivery points. Operations Research, 12(4):568–581.Fisher, M. and Jaikumar, R. (1981).A generalized assignment heuristic for vehicle-routing.Networks, 11(2):109–124.

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

References

Pisinger, D. and Røpke, S. (2007).A general heuristic for vehicle routing problems. Computers& Operations Research, 34(8):2403–2435.Nagata, Y. (2007).Edge assembly crossover for the capacitated vehicle routingproblem. In Cotta, C. and van Hemert, J. I., editors,Evolutionary Computation in Combinatorial Optimization, 7thEuropean Conference, EvoCOP, Lecture Notes in ComputerScience, pages 142–153, Valencia. Springer.

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Outline

1 Asymmetric Capacitated Vehicle Routing Problem

2 On the use of Monte Carlo simulation, cache and splittingtechniques to improve the Clarke and Wright savings heuristics

3 Adaptation to the Asymmetric

4 Results

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Clarke and Wright Savings

Best-known heuristics for solving the VRPGreat simplicity and fast implementationBaseline scenario in which each customer is supplied by aseparate vehicleThe saving supplying to customers i and j for the samevehicle are: Sij = d0i + d0j − dij

It always chooses the edge with the highest savings value ifthe constraints are not violated

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Simulation in Routing via the GeneralizedClarke and Wright Savings heuristic

Author's personal copy

[12], who introduced an entropy function to guide the randomselection of nodes. MCS has also been used in [16,13,24,25] to solvethe CVRP.

3. Designing the SR-GCWS algorithm

In our opinion, recent advances in the development of high-quality pseudo-random number generators [35] have opened newperspectives as regards the use of Monte Carlo simulation (MCS) incombinatorial problems. To test how state-of-the-art randomnumber generators can be used to improve existing heuristics andeven push them to new efficiency levels, we decided to combine aMCS methodology with one of the best-known classical heuristicsfor the CVRP, namely the Clarke and Wright Savings (CWS)method. In particular, we selected the parallel version of thisheuristic as described in [39], since it usually offers better resultsthan the corresponding sequential version [53].

So, our aim was to introduce some nice random behavior withinthe CWS heuristic in order to start an efficient search process insidethe space of feasible solutions. Each of these feasible solutions willconsist of a set of roundtrip routes from the depot that, altogether,satisfy all demands of the nodes by visiting and serving all themexactly once. As stated in Section 2, at each step of the solution-construction process, the CWS algorithm always chooses the edgewith the highest savings value. Our approach, instead, assigns aprobability of selecting each edge in the savings list. Moreover, thisprobability should be coherent with the savings value associatedwith each edge, i.e., edges with higher savings will be more likely tobe selected from the list than those with lower savings. Finally, thisselection process should be done without introducing too manyparameters in the methodology – otherwise, it would be necessaryto perform fine-tuning processes, which tend to be non-trivial andtime-consuming. To reach all those goals, we employ differentgeometric statistical distributions during the CWS solution-construction process: each time a new edge must be selectedfrom the list of available edges, a (quasi-) geometric distribution israndomly selected (details are given in the next section); thisdistribution is then used to assign (quasi-) exponentiallydiminishing probabilities to each eligible edge according to itsposition inside the savings list, which has been previously sortedby its corresponding savings value. That way, edges with highersavings values are always more likely to be selected from the list,but the exact probabilities assigned are variable and they dependupon the concrete distribution selected at each step. By iterating

this methodology, a random but efficient search process is started.Notice that this general approach has similarities with the GreedyRandomized Adaptive Search Procedure (GRASP) [14,15]. GRASP isa typically two-phase approach where in the first phase aconstructive heuristic is randomized. The second phase includesa local search phase. Our proposed approach does without theexpensive local search phase and includes a more detailedrandomized construction step. By doing so we have a moregeneral and less instantiated method as local search needs to beinstantiated for every different problem.

4. A more formal description of the edge selection process

As we have explained in the previous section, during thesolution-construction process, each time a new edge needs to beselected from the savings list, a different quasi-geometricdistribution is chosen. For each edge in the savings list, thisdistribution defines its probability of being selected at the currentedge-selection step of the process. More precisely, each time a newedge must be selected, we choose a real value a, 0 < a < 1, andthen consider the following probability distribution for the randomvariable X = ‘‘node k-th is selected at the current step’’, wherek = 1,2,. . .,s, with s being the current size of the list:

PðX ¼ kÞ ¼ a � ð1� aÞk�1 þ e 8 k ¼ 1;2; . . . ; s

where

e ¼Xþ1

k¼sþ1

a � ð1� aÞk�1 ¼ 1�Xs

k¼1

a � ð1� aÞk�1

Even when the distribution defined here is clearly inspired onthe geometric one, the latter assigns a positive probability to everyvalue in the interval [1, +1) and, therefore does not consider theerror term e. Because of this, we classify the former distribution as aquasi-geometric distribution.

Notice that the list size s diminishes as the process evolves andnew edges are extracted from it. Roughly speaking, if the size of thesavings list in the current step is large enough, the term e is close tozero and, therefore, the parameter a can be interpreted as theprobability of selecting the edge with the highest savings value atthe current step of the solution-construction process. As Fig. 1shows, choosing a relatively low a-value (e.g., a = 0.05) impliesconsidering a large number of edges from the savings list aspotentially eligible, e.g.: assuming s = 100, if we choose a = 0.05then the list of potentially eligible edges will cover about 44 edges

Fig. 1. Effects of the chosen parameter on the geometric distribution.

A.A. Juan et al. / Applied Soft Computing 10 (2010) 215–224 217

Each time a new edge must be selected, a quasi-geometricdistribution assigns a random value at each edge assigningexponentially diminishing probabilities to each eligible edge

according to its position inside the sorted saving list.

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Splitting policies

a b c

d e f

Divide the original set of nodes into disjoint subsets and then tosolve each of these subsets by applying the same methodologydescribed before.

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

SR-GCWS-CS

Splitting

Baseline scenario

SR-GCWS

Improve using Cache

Baseline scenario

SR-GCWS

Improve using Cache

Current solution ≤CWS sol

Update Best Sol

Update Best Sol

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Outline

1 Asymmetric Capacitated Vehicle Routing Problem

2 On the use of Monte Carlo simulation, cache and splittingtechniques to improve the Clarke and Wright savings heuristics

3 Adaptation to the Asymmetric

4 Results

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Savings

The saving supplying to customers i and j for thesame vehicle are: Sij = d̂0i + d̂0j − d̂ij

d̂0i =d0i+di0

2

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Savings

The saving supplying to customers i and j for thesame vehicle are: Sij = d̂0i + d̂0j − d̂ij

d̂0i =d0i+di0

2

ij

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Savings

The saving supplying to customers i and j for thesame vehicle are: Sij = d̂0i + d̂0j − d̂ij

d̂0i =d0i+di0

2

ij

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Improve using Cache

Improve nodes order trying to sort its nodes in a moreefficient way by eliminating possible knots in the currentroute.Compare with its reverse route.Check if it exists already a cached route covering the sameset of nodes.

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Outline

1 Asymmetric Capacitated Vehicle Routing Problem

2 On the use of Monte Carlo simulation, cache and splittingtechniques to improve the Clarke and Wright savings heuristics

3 Adaptation to the Asymmetric

4 Results

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Results

20 asymmetric CVRP instances have been usedWith 50 or 100 customersA fleet from 2 to 7 vehicles10 random seeds1 minute of Elapsed TimeProcessor: Intel Pentium Dual CPU 2.40 GHz with 3.25 GBRam

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Results

Problem nCustomers nVehicles Gap CWS Gap Avg10 Gap Best10G-C-CAA0501NA 50 2 5.77 % 1.56 % 1.45 %G-A-CAA0501NA 50 2 19.36 % 0.56 % 0.08 %G-C-CAA0502NA 50 3 7.93 % 0.00 % 0.00 %G-A-CAA0502NA 50 3 7.47 % 0.00 % 0.00 %G-C-CAA0503NA 50 4 4.66 % 0.00 % 0.00 %G-A-CAA0503NA 50 4 9.34 % 0.00 % 0.00 %G-C-CAA0504NA 50 2 4.14 % 0.00 % 0.00 %G-A-CAA0504NA 50 2 10.40 % 0.95 % 0.95 %G-C-CAA0505NA 50 3 8.52 % 0.00 % 0.00 %G-A-CAA0505NA 50 3 14.71 % 1.99 % 0.68 %G-C-CAA1001NA 100 5 16.10 % 1.43 % 1.27 %G-A-CAA1001NA 100 5 11.86 % 3.14 % 2.37 %G-C-CAA1002NA 100 5 10.61 % 2.03 % 1.67 %G-A-CAA1002NA 100 5 11.59 % 1.24 % 1.03 %G-C-CAA1003NA 100 5 8.66 % 1.44 % 1.36 %G-A-CAA1003NA 100 5 13.80 % 2.19 % 1.59 %G-C-CAA1004NA 100 6 9.54 % 0.57 % 0.40 %G-A-CAA1004NA 100 6 12.31 % 0.75 % 0.74 %G-C-CAA1005NA 100 7 11.33 % 0.06 % 0.05 %G-A-CAA1005NA 100 7 9.26 % 1.81 % 1.17 %

AVG 10.37 % 0.99 % 0.74 %MAX 19.36 % 3.14 % 2.37 %

Gap respect to best-known solution [Nagata, Y. (2007). ]18 / 20

Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

G-C-CAA1005NA v.s. G-A-CAA1005NA

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Outline

ACVRP

SR-GCWS-CS

Adaptation tothe Asymmetric

Results

Asymmetric Vehicle Routing Problem

Rosa HerreroUniversitat Autònoma de Barcelona

Dr. Alejandro RodríguezUniversitat Politècnica de València

Dr. Angel A. JuanUniversitat Oberta de Catalunya

2012 IN3-HAROSA Workshopfor Junior Researchers

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