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PergamonNonlinear Analysis, The ory, Methods & Applicafionr, Vol. 30. No. 7. pp. 42774288, 1997Proc. 2nd World C ongress of Nonlinear Analysts0 1997 Elsevier Science LtdPII: SO362-546X(97)00127-2
Printed in Great Britain. All rights reserved0362-546X/97 $17.00 +O .O O
ON THE VEHICLE ROUTING PROBLEMN.R Achuthan, L. Caccetta and S.P. Hill
School of Mathematicsnd StatisticsCurtin University of Technology
GPO Box WI987Perth. W estern Australia.
Key Words and Phrases : Branch and Bound, branch and cut , combinatorial optimization, mixed-integer linearprogramming, vehicle routing, vehicle scheduling.1. INTRODUCTION
Vehicle Routing Problemsare concernedwith the delivery of somecommoditiesrom one or moredepots o a numberof customer ocationswith known demand. Such problemsarise in many physicalsystems ealingwith distribution. For example,delivery of commodities uch as mail, food, newspapers,etc. The specificproblemwhich arises s dependent pon he type of constraints ndmanagementbjective.The constraintsof the problem arise from : the vehicle capacity; distance/time estriction; number ofcustomers erviced by a vehicle; and other practical requirements. The management bjectivesusuallyrelates o the minimization of cost/distance r fleet size. In this paper we consider he Vehicle RoutingProblem VRP) having the following features(0 a single ommodity o bedistributed rom a singledepot o customers ithknowndemand.(ii) all customerdemandsre serviced y onevehicle.(iii) eachvehiclehas he same apacityandmakes ne rip.(9 total distance ravelled by eachvehiclecannotexceeda specifiedimit.(4 the objective s o minimize he total distanceravelledby all vehicles.
The resultingproblem s called he Capacity and DistanceRestrictedVehicle Routing Problem CDVRP).Relaxing the distance estriction (iv) gives rise to the so called CapacitatedVehicle Routing ProblemVW.TheseNP-hard [lo] problems ave attractedconsiderable ttentionover the past wo decadesesulting nmany exact and heuristicalgorithms 3,5,6,8]. A good deal of the recent work has beenmotivated by thesuccessof Branch and Cut methods n solving large Travelling SalesmanProblems (TSP) [7,11].Unfortunately the success f this approach n solving he CVRP and CDVRP hasbeen estricted o smallersize problems. In fact, very few CVRPs with over 100 customers ave beensolvedand the situation smuch worse or the hard CDVRP.Recently [1,2,4], we have developedand implemented number of Branch and Cut algorithms hatconsistentlysolveup to 100customer roblems.Our computational uperiorityhasbeenachieved hrough
l improvedsubtourelimination onstraints0 new cutting planes liminating on-optimal olutionshroughadescription f the structureof an optimalsolution.l concurrentuseof equivalent ubtourelimination onstraints.
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The objective of this paper is to brief ly describe this work. We present some mixed integer Linearprogramming models in Section 2. We restrict ourselves to formulations that we have extensively testedcomputationally. Algorithms for the CVRP and the CDVRP arc presented in Section 3 and computationalresults are discussed in Section 4.
2. MODELSWe denote the depot by 1 and the set of customer locations by &= {2,3,...,n}. Let G = (N,E) be the graphrepresenting the vehic le network, with N = { 1,2,...,n} and E = {(ij) : ij E N, i < j}. Further, we adopt the
following notation :q : Demandofcustomerj,j E G. G j : Distance between locations i and j.m : Number of delivery vehicles. Q : Common vehicle capacity.L : Maximum allowable distance of a vehicle route. Li : Length of shortest (l,i)-path in G.
We assume throughout that Q 5. max{q}, L$ i L for every j E& and the distance matrix C = (cij) issymmetric and its elements satisfy the triangle inequali ty. Note that since C is Euclidean, we have L, = q,.For S G & we let 4s) = lower bound on the number of vehicles required to visit all locations of S in anoptimal solution. Note that @) 2 1. We write s for the complement in & of S. For ij E N our decisionvariables are defined as1, f a vehicle travels on a single t rip between i and j,
Xij = 2, if i = 1 and(l,j,l)isaroute,0, otherwise.
Note that xi) only needs to be defined for i < j, and qi + q I Q and L, + LI + c,, 5 L. Laporte et al. [9]formulated the problem as follows :Minimize f= CCijXij (2.1)
i
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Second World Congress of Nonlinear Analysts 4279In the above formulation, m can be fixed or variable. Constraints (2.2) and (2.3) are degree constraints onthe depot and the customer locations, respectively. Constraints (2.4) eliminate subtours, that is they prohibit
subtours that are free from the depot, or connected to the depot but violate the capacity or distancerestrictions. In an actual implementation of the above formulation I (S) is only calculated when required.Laporte et al. [6] observed that (2.4) is equivalent to :
CXij + CXij + CXli 2 2r(S) (2.7)i&j& i&,jcS i&ii
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Theorem 2.1 : Let S, Tr,Tz ,..., Tk c &?, be such that :
(4 k22 and c qi > Q, forevery I -< p z q 5 k;i&uTP uTq(b) Ti nTj =+,fori f j;(4 Sr\Ti=+,l
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CXij+C Xii c (&Remark 2.5 : It follows from Theorem2.2 and 2.3 that thereexists an optimal solutionof the CVRP (2.1)to (2.6) with variablem such hat :
if Cqi IQ thenm=landotherwise(i.e. Cqi >Q) m
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g(X) = CCiXij +CCljXlj + M Ccijxii
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3. BRANCH AND CUT ALGORITHMS
h this s&ion we briefly out line our work on the implementat ion of a Branch and Cut algorithm forsolving the CARP and CDVRP uti lizing the cutting planes discussed in Section 2. The basic steps ares~~~eflowdiagramofFigure3.1.In brief, the important features of our method are :
. The init ial relaxed problem does not include the subtour elimination constraints and m isbounded as in (Remark 2.5).. Paessens121Modified SavingsAlgorithm is used o generate n initial upper bound.
l Laporteet al [9] procedures re used o set orced variablesandpurge neffective constraints.o The relaxedLP subproblemsre solvedusingCPLEX [ 131.l We use 6 searchprocedures 2,4] for finding violations of cutting planes 2.4), (2.9), (2.10),(2.12), (2.13), (2.17) and 2.18).l Up to 5 Gomory cuts are usedat the root nodeand the Land Powell rule is used or selectingfrirdional variables or branching.All castmints are retainedwhenbacktracking n the search ree as hey remainvalid.For de& of these rocedures e refer to [ 1,2] for the CVRP and o [4] for the CDVRP.
SZl
Figure 3.1 : Flow Diagram of Algorithm
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4284 Second World Congress of Nonlinear Analysts4. COMPUTATIONAL RESUL TS
We implemented our algorithm in C on a SLJN SPARC II workstation operating a t 28.5 MIPS and ouranalysis represents the most extensive testing ever carried out. For the CVRP, our computational results arebased on 1650 simulated test problems with a variab le number of vehicles m and 24 standard literatureproblems with fixed m. The 1650 test problems are generated as follows :. Number of customers range from 15 to 100 in increments of 5.l The co-ordinates of the customer locations are integers and are randomly generated from asquare of length 1000 units. The qs are then calculated to 2 decimal places.l The customer demands are generated randomly from the integers 1 to 100.l The common vehicle capacity is determined as in Laporte et al [9], that is
where a is a parameter chosen in the interva l [O,l]. We use a-values of 0,0.33, 0.67 and 1.The smaller the u-value the more difficult the problem. For each choice of a andn wegenerate 30 problems.For CDVRP our computational results are based on 8 literature problems plus 4590 test problems ofwhich 1965 are Euclidean. The Euclidean test problems are generated as follows :
l Number of customers range from 15 to 50 in increments of 5.l The parameter a is chosen as : 0, 0.33, 0.67, 1.0. We generate 15 problems for each value of n
and a (a = 0 is only considered for n 5 30).. The maximal allowable distance L for a vehicle route is initia lly specified as1500; 1750; 2000; 2250; and an effective CO 104.
As n increases, we increase the values of L. Each test problem is considered for each choice ofL.l The co-ordinates of the customer locations are integers and are randomly generated from asquare of length 1000 units. The CiIS are then calculated to 2 decimal places. To ensure
feasibility, customer locations are chosen so that c,, 5 750 for every j.l The customer demands q,s are generated randomly from the integers 1 to 100.The Non-Euclidean test problems are generated with the number of customers ranging from 15 to 60, as
for the Euclidean case, except for the following modifications :. The values of L used for all problems are fixed as :1500; 1750; 2000; 2250; to (103.
l The distances c, are generated as follows :(4(b)
For i < j, cI1 s a random real number (rounded to 2 decimal places) chosenfrom the interval (O,lOOO].To increase the likelihood of feasibility, we determine L,, the length of theshortest path from the depot to location i and accept the c,,s only if:(9 cl, I750; or
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SecondWorld Congress of NonlinearAnalysts 4285(ii) for cl, > 750, there exists at least one i # j such that
L, + '21) + Lj 5 1500 and qi + q < Q.Detailed computational results can be found in [2] for the CVRP and in [4] for the CDVRP. Table 4.1
provides a brief comparative analysis for the 1650 simulated CVRPs of Branch and Cut procedures using(2.4) only [l], (2.4) and (2.9) [2] and (2.4), (2.9), (2.10), (2.12), (2.13) and (2.14) [4]. In ourexperimentation we observed that violations of (2.15) were infrequent and consequently it was excluded,For a comparative analysis of the 24 literature problems we refer to [2].Table 4.2 presents our results for the Euclidean CDVRP simulated test problems for 1& 1 = 25 and 45using (2.4), (2.9), (2.10), (2.12), (2.13), (2.14), (2.17) and (2.18). Tab le 4.3 presents similar results for theNon-Euclidean CDVRP simulated test problems for (0 1 = 30 and 60. Full results are in [4].
ALGORlTHMus~(2.4),(2.9~~.10~2.12~ (2.13)md(2.14)
cllat a *Root AVE. Avg. solvedNo. % Time Node out or
ALGORITHMusing (2.4) and (2.9)
Root A vg. Avg. Solved% Tim e Node out or
099.1 251.4 365.1 2599.4 189.5 462.1 29
ALGORITHMusing(2.4)
*Root Avg.% Th?
* Percentage of the original lower bound at the root node in terms of the optimal objective functionvalue.
Table 4.1 : CVRP Comparative Analysis on Simulated Problems
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CPU TIME (SECS) Average No. of CalculationNumber of Problems oflel Solved fD*ra T ROM Avg. Mh Max. EUm+ Chid out of 15 Avg. % %
l MU Node NO . succcsa Exact
%
0 1500 96.8 450.3 13.9 2205.8 50.3 616.3 14 286.5 45.3 32.81750 96.4 1181.9 10.7 3390.2 56.3 1627.2 13 63.7 44.6 39.92000 96.6 984.1 15.8 3473.0 55.7 1212.2 12 9.5 57.9 57.02250 96.6 912.2 15.5 3176.9 56.4 1120.3 12 0.3 75.0 50.0co 96.6 923.5 15.9 2786.8 56.8 1041.8 12 025 l/3 1500 90.8 499.0 10.8 2 130.4 22.4 1330.0 11 2873.2 63.4 27.01750 93.3 690.9 16.3 3065.4 24.4 1881.3 14 1021.4 67.0 31.42000 98.0 107.2 1.0 423.6 24.6 275.5 14 114.9 69.7 27.82250 98.9 48.9 0.8 271.4 24.0 146.8 15 36.7 71.1 45.1
Is0099.5 9.0 0.8 41.8 23.0 14.4 15 0
2r3 87.2 815.9 54.1 2584.9 21.9 2166.9 9 3378.1 72.0 31.81750 91.8 851.9 38.3 2764.8 18.4 4383.4 7 3290.1 83.4 41.62000 95.9 466.0 0.5 1999.5 16.3 2089.6 10 1177.1 87.9 46.32250 97.3 128.0 0.5 560.6 16.5 617.7 13 343.9 89.2 49.4(D 99.6 1.5 0.5 5.6 12.7 2.1 15 0
1 1500 89.2 872.4 11.0 2499.7 20.8 2244.4 10 5945.1 78.8 29.31750 89.1 1077.8 110.3 2779.1 19.0 3668.9 11 3868.5 83.7 36.32000 93.5 530.5 4.8 1615.7 15.4 2334.7 9 1673.1 89.6 52.92250 93.4 303.3 1.3 1350.4 14 .4 1568.7 14 854.6 96.9 59.62Go
100.0 0.7 0.3 1.8 7.7 0.4 15 0l/3 99.0 612.0 15.1 2247.1 34.9 677.0 8 34.6 69.7 52.72750 98.9 832.3 13.1 2811.2 33.0 781.6 11 4.1 37.8 33.33000 98.9 850.6 13.0 2804.2 32.8 830.4 11 0.1 100.0 100.002 98.9 844.2 12.7 2804.0 32.9 821.5 11 0
45 213 2500 99.8 9.8 6.4 13.3 18.5 6.0 2 0.5 0.0 0.02750 98.3 568.8 6.6 2862.9 23.3 1036.4 9 373.3 90.5 48.03000 98.8 455.9 3.8 2891.1 22.7 758.0 10 217.8 88.8 60.72zlo
99.5 158.2 3.3 1135.1 21.7 218.5 15 01 97.6 597.5 169.2 1025.8 21.0 1590.0 2 1051.0 95.4 55.1
2750 98.7 848.3 3.1 3061.1 16.8 1792.0 5 830.8 93.2 51.93000 98.9 408.3 3.2 1423.7 16.9 918.6 7 452.9 95.9 60.0co 99.8 38.7 1.6 493.9 13.8 170.8 15 0
* Percentage of the original lower bound at the root node in terms of the optimal objective functionvalue.
+ The average of the maximum number of cutting plane constraints present at any one stage.+t The average of the number of times g(r*) is calculated (Avg. No.); Percentage of times this
calculation is useful (% success); Percentage of times g(x*) = g(x) (% Exact).Table 4.2 : Euclidean CVDRPs I& 1 = 25,45.
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l&l
30
60
a
0
l/3
213
1
l/3
213
1
T1500175020002250
001500175020002250
m1500175020002250
al1500175020002250Is0175020002250
m1500175020002250
cm1500175020002250
m -
Root*%92.091.593.093.393.298.698.798.998.998.999.799.599.699.699.697.398.299.299.3
100.098.898.398.598.598.599.899.899.999.999.999.099.499.899.799.8
CPU TIME (SECS)
Avg. Min. T aX.3054.7 3054.7 3054.71125.8 726.1 1843.11133.9 780.2 1957.51038.4 550.7 1549.81187.1 531.3 2246.4
19.6 0.6 71.816.8 0.6 61.730.0 0.6 114.729.3 0.6 102.729.2 0.6 102.2
1.4 0.4 4.41.4 0.4 4.81.1 0.4 3.01.1 0.4 3.01.1 0.4 3.0
133.3 0.6 710.965.2 0.4 488.341.7 0.3 276.6
209.4 0.3 2393.00.6 0.3 1.5
1614.2 14.8 3084.31759.4 35.6 2868.41668.5 6.3 3181.11753.1 6.3 3159.91777.8 6.2 3161.0
24.5 1.9 210.19.2 1.9 37.4
10.8 1.9 33.210.8 1.9 33.210.8 1.9 33.2
497.0 5.3 2013.6446.7 2.6 1857.463.8 2.2 538.3
6.6 1.8 23.06.8 1.8 34.5
AverageNumber ofElim 1 ChildMPL Nodes
58.0 3444.052.3 1027.351.6 978.051.2 896.4
13.8 49.514.5 102.814.5 101.1
4.02.12.12.1
932.8556.5351.7
1873.1
17.5 1862.016.9 1693.817.0 1826.9
7.9 6.98.1 10.78.1 10.78.1 10.7
f
6.9 1109.66.3 990.05.1 122.74.7 7.34.7 7.3
T No. ofProblemsSolvedout of 15(M)(3,3)
555
151515151515151515151515151515101112131315151515151069
1215
* Percentage of the original lower bound at the root node in terms of the optimal objective functionvalue.+ The average of the maximum number of cutting plane constraints present at any one stage.
i+ First value in brackets indicates the number of problems solved optimally; the second figuredenotes the number of infeasibilities detected. Other values are the number of problems solvedoptimally.
Table 4.3 : Non-Euclidean CVDRPs I# 1 = 30,60.
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4288 Second World Congress of Nonlinear AnalystsREFERENCES
1.2.3.4.5.6.7.
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ACHUTHAN, N.R., CACCETTA, L. and HILL, S.P., A new subtour elimination constraint for thevehicle routing problem, EJ0.R. 91,573-586 (1996).ACHUTHAN, N.R., CACCETTA, L. and HILL, S.P., An improved branch and cut algor ithm forthe capacitated vehicle routing problem. (submitted).ACHUTHAN, N.R., CACCETTA, L., CARLTON, M. and SARWADI, Comparison of vehic lerouting heuristics, (submitted).ACHUTHAN, N.R., CACCETTA, L. and HILL, S.P., The vehic le routing problem with capacityand distance restrictions (submitted).BODIN, L.D., GOLDEN, B.L., ASSAD, A. and BALL, M., Routing and scheduling of vehicles andcrews : the state of the art. Comp and Oper. Rex. 10,69-2 11 (1983).CHRISTOFIDES, N., Vehicle routing, in The Travelling Salesman Problem (E. Lawler et al,Editors), 43 l-448 (1985).CROWDER H. and PADBERG, M., Solving large-scale symmetric travelhng salesman problems tooptimality, Management Science 26,495-509 (1980).LAPORTE, G., The vehic le routing problem : An overview of exact and approximate algorithms,EJ0.R. 59,213-247 (1992).LAPORTE, G., NORBERT, Y. and DESROCHERS, M., Optimal routing under capacity anddistance restrictions, Operations Research 33, 1050-1073 (1985).LENSTRA, J.K. and RINNOOY KAN, A.H.G., Complexity of vehic le routing and schedulingprob1enqNetwork.r 11,221-227 (1981).PADBERG, M. and RINALDI, G., A branch and cut algor ithm for the resolution of large scalesymmetric travel ling salesman problems, Si%4 Review 33, 60-l 00 (199 1).PAESSENS, H., The savings algorithm for the vehic le routing problem, E.J. 0. R. 34, 336-344(1988).Using the CPLEX Library and CPLEX Mixed Integer Library, CPLEX Optimization Inc., 930Tahoe Blvd # 802-297, Incline Village, NV 89451, U.S.A., (1993).
