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A generalized exchange heuristic for the capacitatedvehicle routing problemFARID HARCHE a & PARTHASARATHI RAGHAVAN aa Department of Statistics and Operations Research , Stern School of Business, Tisch Hall ,New York University, 40 West 4th St., NY, 10012-1118, U.S.A.Published online: 26 Apr 2007.

To cite this article: FARID HARCHE & PARTHASARATHI RAGHAVAN (1994) A generalized exchange heuristic for the capacitatedvehicle routing problem, International Journal of Systems Science, 25:11, 1911-1920, DOI: 10.1080/00207729408949321

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INT. J. SYSTEMS SCI., 1994, VOL. 25, NO. 11,1911-1920

A generalized exchange heuristic for the capacitated vehicle routingproblem

FA RID HARCHEt and PARTHASARATHI RAGHAVANt

We consider the problem of dispatching the minimum number of vehicles from acentral depot to make deliveries to a set of clients with known demands. Theobjective is to minimize the total distance travelled, subject to vehicle capacityrequirements. We present a new heuristic algorithm for solving this problem. Thealgorithm is based on gener~ized edge-exchangesearch procedures, and relaxationof the capacity .requirements. Computational results, based upon standard testproblems with up to 249customers, indicate that our heuristic compares favourablywith known heuristics in terms of solution quality.

I. IntroductionThe vehicle routing problem (VRP) is a commonly encounlered problem in

distribution management. It has been the subject of extensive research in theliterature; see Bodin et al. (1983) for a comprehensive survey. The importance of thisproblem is reflected by the wide range of its applications, which include school busrouting, waste disposal and mail delivery. In a typical application the dispatcher isfaced with the problem of routing a fleet of vehicles with fixed capacity stationed ata central depot to supply a set of clients with known requirements. The objective isto minimize the total distance travelled.

In practice, there are various restrictions including vehicle capacity constraints,distance constraints, and time windows which complicate the problem. Nevertheless,it is true that in most applications the vehicle capacity constraints are quite important.Here, we study the capacitated vehicle routing problem (CVRP) a variant of theVRP in which only the capacity constraints are considered. The VRP can be describedas follows. One is given a graph G= (V, E) where the node set V, IVI = n + I, consistsof n nodes that represent clients and a special node indexed 0 that designates thedepot, a symmetric cost matrix consisting of costs cij for all i, j pairs of V, a positivedemand d, for each client i E V\ {O}, and a fixed vehicle capacity C; construct acollection of routes which satisfies the following conditions.

(a) Every route starts and ends at the depot.

(b) Every client is served by exactly one route.

(c) The sum of the demands of clients assigned to a route does not exceed thevehicle capacity.

The problem is then to find a collection of routes at minimum total cost. Note thatthe cost is expressed in terms of the distance. Although the CVRP is simple to state,it is, however, hard to solve. In fact any algorithm for the CVR P has to address twoinherently difficult optimization problems: the bin-packing problem, to determine

Received 18 August 1993.t Department of Statistics and Operations Research, Stern School of Business, Tisch Hall,

New York University, 40 West 4th St., NY 10012-1118, U.S.A.

0020-7721/94 $10.00 © 1994 Taylor & Francis Lid.

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the optimal fleet size, and the travelling salesman problem, to determine the optimalroute for the vehicles. Not surprisingly, exact algorithms have met with limited successfor large problems, and heuristics are almost always used for most practical problems.

As a consequence, a considerable amount of research has been directed towardsdeveloping good heuristics for the VRP and its variants. Besides using heuristicsolutions to solve large instances of hard problems, researchers have in recent yearsused heuristic solutions as bounds in search procedures, such as branch-and-boundand branch-and-cut (Padberg and Rinaldi 1989). In fact our main goal for under­taking this work has been the development of a good heuristic that will beembedded in a cutting plane algorithm (Harche and Rinaldi 1995) to solve tooptimality large instances of the CVRP. A number of heuristics have been developedfor the CVRP. Heuristics provided by Clarke and Wright (1964), Gillett and Miller(1974), Christofides and Eilon (1969), Gaskell (1967), Gillett and Johnson (1976),Russell (1977) and Mole and Jameson (1976) can be broadly classified into twocategories: route first, cluster second; and cluster first, route second. Heuristicalgorithms based on mathematical programming techniques include those ofChristofides et al. (1979), Fisher and Jaikumar (1981), Foster and Ryan (1976) andStewart and Golden (1984). In contrast, few attempts have been made to deriveoptimal routes for the CVRP. Exact algorithms are provided by Laporte et al. (1985),Christofides et al. (1981), and Cornuejols and Harche (1994). An excellent review ofexact algorithms was presented by Laporte and Nobert (1987).

The heuristic developed here begins by solving the underlying bin-packingproblem to determine the minimum number of vehicles required to meet the demandsof all customers. Then the heuristic alternates betweerrtwo phases. In the first,infeasible solutions with respect to the vehicle capacity are generated, while the secondphase attempts to render feasible the infeasible solutions found in the first phase. Edgeexchanges are applied to improve solutions and establish feasibility. Section 2 outlinesseveral improvement procedures that are based on edge exchanges. A detaileddescription of the algorithm is given in § 3. In § 4 computational results based on aset of standard problems are reported. Finally, § 5 briefly summarizes our work.

2. Edge exchangesEdge exchange search techniques have played an important role in the develop­

ment of efficient heuristics for routing problems. In particular, they have beensuccessfully applied to the travelling salesman problem (TSP) (see Lin and Kernighan(1973). Moreover, these techniques have been extended to vehicle routing problems(Christofides and Eilon 1969, Russell 1977, Stewart and Golden 1984). The rationalefor these methods is to improve the existing solution by replacing a set of edges byanother set of profitable edges in such a way so as to preserve the feasibility of theresulting solution. Usually the process ends when no further improvements can bemade. In our setting we distinguish between two distinct classes of edge exchangetechniques. The first, called intra-route edge exchanges (also known as Lin-Kernighanr-opt procedures), consists of performing edge exchanges within each route of a setof routes. Therefore they affect the order in which clients are visited in a route.Clearly, the intra-route procedures only require information about the route whichis subjected to exchanges. Edge exchange procedures of this class, in our setting, donot affect the assignment of clients to routes. In contrast, the second class, called theinter-route exchanges which involve exchanging edges and swapping of nodes

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Generalized exchange heuristicfor CVRP 1913

between several routes, affects not only the order in which the nodes are visited butalso the assignment of nodes (clients) to routes. They require information aboutseveral routes, possibly all the routes. This is important because of the inherentbin-packing features of the problem considered here. Two simple tests, defined below,are required to guarantee that the new solution obtained after inter-route exchangesis a set of routes which satisfies capacity constraints. It is possible to find the optimalroutes by enumerating all possible inter-route exchanges. However, the number ofsuch exchanges is prohibitive. Thus the choice of the inter-route edge exchanges wasbased on the ease of implementation as well as the computational effort needed.Let us now outline the inter-route edge exchanges used and implemented inour procedure. Denote the routes by T;, i = I, ... , t. Let D, be the sum ofthe demands of all clients assigned to route T;. Define the slack 5, = C - D"i = I, ... , t, where C is the vehicle capacity. The following exchanges areperformed.

An insertion of a node from one route into another route. This process involvesthe replacement of three edges by three new edges. As shown in Fig. I, node k fromT2 is inserted into route T1. The edges (0, k), (k, I) and (i, j) are deleted and replacedby (i, k), (j, k) and (0, I). This exchange is worthwhile if the following conditions aresatisfied: d. ~ 5,; and cij + co. + Ck/ > Ci. + Ckj + Cot·

A swap of multiple node sets between a pair of routes. From Fig. 2 it is seen thatnodes}, k and I of T1 are swapped with nodes q and p of T2 • Four edges are deletedand four new edges are created. Here we need to check that

5, = (dj + d. + d,) ;::; dq + d; and 52 + (dq + dp) ;::; d, + d. + d,

c'p + cj h + C'U+ C,q < cij + chp + cqu + c"

A circular swapping (relocation) of three single nodes belonging to three differentroutes. As is illustrated in Fig. 3, node i in route T, goes to route T2 , displacing nodeI in route T2 to route T3 , which in place displaces node q in route T3 to route TI •

As a result, six edges are replaced by six new edges. In order to perform this exchange,we need to check that

51 - d, ;::; dq , 52 - d, ;::; d, and 53 - dq ;::; d,

All the exchanges described above have been implemented in the heuristic wepresent in the next section. Moreover, extensions of the first and third exchanges, inwhich node sets involving at least two nodes are swapped, have also been implemented.The exchanges described above are distinct from one another. That is, none of theexchanges is implied by any of the others.

(a) (b)

Figure I. (a) Before exchange; (b) after exchange.

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F. Harche and P. Raghavan

(b)

Figure 2. (a) Before exchange; (b) after exchange.

3. The algorithmIn this section we describe an algorithm which proceeds by performing edge

exchanges through a series of feasible and infeasible routes. The algorithm terminateswhen no further improvement can be made. The method can be viewed as a variationof the one which has been used by Russell (1977). A distinct and practical feature ofthe algorithm is that it accepts infeasible routes; see also Stewart and Golden (1984).This strategy has proven particularly useful, as it creates several set of routes andtherefore may increase the likelihood of finding better feasible solutions. We nowdiscuss various features of the algorithm.

Unlike most of the known heuristics in the literature for the problem consideredhere, the present heuristic addresses the problem of finding the minimum number ofvehicles required explicitly. The only other such attempt appears in Fisher andJaikumar (1981). The proposed procedure consists of two distinct parts. The firstpart (Step 0 below) is used to determine the smallest feasible number k of vehiclesby solving the underlying bin-packing problem using the first-fit decreasing (FFD)algorithm (Johnson (1974). The second part (Steps 1-6 below) is concerned withfinding the shortest set of routes using k vehicles. It should be clear that the value kused in the second part of the procedure is identical to the value obtained in the firstpart. The second part of the algorithm is based on generalized edge exchange searchprocedures and relaxation of the capacity requirements.

(a)

T,b ---<!}--%-

T,k ---0----G--

T,- --G-- --G)---=- (b)

}Q\ I ,\ I ,\ I ")' , ,\ I, T,

k\>Q,'@--'I 'I .-

), 1',/ I,

I '~- "0-2-Figure 3. (a) Before exchange; (b) after exchange.

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Generalized exchange heuristicfor C VRP

Generalized exchange algorithm

1915

Step O. Find the minimum number k of vehicles required using the FFD bin packingalgorithm (Johnson 1974).

Step I. Choose a value of the perturbation parameter ,1.C and relax vehicle capacityby setting C= C + ,1.c,

Step 2. Construct an initial set of k routes using a nearest-neighbour heuristic (thisset of routes is only feasible with respect to the relaxed capacity C).

Step 3. (Improvement step): apply both intra-route and inter-route edge exchanges.

Step 4. If the current set of routes is infeasible with respect to the true capacity, C,go to Step 6. Else, decrement the perturbation by a constant ratio,1.C +- 0·75 * ,1.c, and set C= C + ,1.c,

Step 5. If C= C, write the best feasible solution. Stop. Else go to Step 2.

Step 6. (Make-feasible step): perform inter-route edge exchanges to establish feasi­bility with a minimal increase (if possible) in total distance. Record the bestfeasible solution. Go to Step 3.

We now discuss the essential steps of the algorithm. The value of the change incapacity ,1.C was established empirically. For the test problems reported in this paper,we ran the algorithm for every integer in the interval [5,20] until a good solutionresulted.

The second part of the algorithm is initiated with a perturbed capacity C=C + ,1.c, Several factors must be considered in selecting ,1.c, First, the ability torestore feasibility is a critical factor. If ,1.C is large the process of restoring feasibilitymay fail (by choosing wrong sequence of exchanges), which prematurely terminatesthe algorithm. Further, when the perturbed capacity ,1.C is large, the algorithm takeslonger to re-establish feasibility, since the capacity is reduced by a constant ratio andthus needs to iterate over a large number of perturbed capacities. To summarize, itis imperative that the change in capacity be small so that 'slightly' infeasible solutionscan be generated and feasibility quickly achieved.

The initial (infeasible) set of routes is constructed, using k and C, as follows. First,k clusters (bins) are formed by solving the bin-packing problem. Note that by relaxingthe capacity C, the bin-packing solution may yield k - [ clusters. In this case somearbitrary client, already assigned to some cluster, will be reassigned to the kth cluster.Once k clusters have been formed then we generate a travelling salesman (denotedTSP) route of all nodes, including the depot, within each cluster. This is accomplishedby using the so-called 'nearest-neighbour' approach for theTSP (Golden and Stewart1985).

Following the generation of routes, the algorithm consists of two phases. In thefirst phase, called the improvement phase, edge exchanges are performed to improvethe solution; the routes in this phase are, however, infeasible with respect to the truevehicle capacity. We begin by applying intra-route edge exchanges (or Lin-Kernighanr-opt procedures) on each route. We perform inter-route and intra-route exchangesrepeatedly until no further improvement is possible. At this point we enter the secondphase, called the make-feasible phase, which is a systematic procedure to restorefeasibility. It is used to impose the original capacity constraints, thereby producing

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a feasible set of routes. The procedure starts with an infeasible set of routes whichis then subjected to inter-route exchanges to remove infeasibilities in the routes. Inthis routine we focus our attention only on routes with demands which exceed thetrue capacity C. For every such route we do the following: among all the exchangesthat would reduce the total demand of a route, we choose the one that minimizesthe increase in the tour length. It is appropriate to observe that the inter-route edgeexchanges are essential to achieve feasibility. Having obtained a feasible solution, wethen perform both intra-route and inter-route edge exchanges which lead to potentialimprovement of the route length while preserving feasibility. The resulting improvedfeasible solution is then stored. Next, the perturbation value ~C is suitably decreasedby a constant ratio and the process is repeated. The algorithm terminates when therelaxed capacity C is equal to the true capacity C and no further improvement canbe achieved. The reason for iterating over different values of the relaxed capacity isto discover in our search high-quality solutions that may not be found otherwise.

4. Computational resultsIn this section we will investigate the computational efficiency of the generalized

exchange heuristic. The analysis is made by considering only the quality of thesolutions produced. The heuristic algorithm described in this paper was coded inFO RTRA N and run on a VAX 8700 for a set of 15 test problems. These problemshave long been used as standard problems in the vehicle routing literature. Table Iprovides information on these problems. Note that problem 15 is a single depotversion of the problem from Gillet and Johnson (1976). For each problem Table Igives the number of customers, the vehicle capacity, the minimum number of vehiclesused, the respective source, the value of the perturbation parameter ~C, and thevehicle utilization (percent), which is expressed as

vehicle utilization = Liev di

100(k x C)

where Liev d, is the total demand, k is the number of vehicles used in the solution,and C is the vehicle capacity.

The vehicle utilization reflects the percentage of vehicle capacity used, and thusthc degree of difficulty of the test problems. Table 1 reveals a high degree of vehicleutilization. It will be useful to make some observations regarding the perturbationparameter ~c. First, we report only the best initial value of ~C (initialized in Step1 of the second part of the algorithm) that produced the best suboptimal solution.

Second, no rigorous justification can be given with respect to the variations ofthe perturbation values for our test problems. This provides evidence that thealgorithm is fairly sensitive to problem data and parameters. Indeed, different initialperturbation values may lead to different heuristic solutions. However, as pointedout in the preceding section, we suggest that ~C be chosen empirically in the interval[5,20] for our test problems. In general, however, the choice. of the perturbationvalue ~C depends on the problem data (that is, demand distribution, vehicle capacityand number of vehicles used) and can only be determined after trying different values.

Third, it was found that for problems I, 2 and 4 in Table 1, any initial values of~c in the range [5,20] yield the same best heuristic solutions reported. It is temptingto argue that for these problems, the small change in capacity (that is, 0·11% and0'12% of the vehicle capacity for problems 1, 2 and 4, respectively) had no significant

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Generalized exchange heuristic for C VRP 1917

Number Vehicleof Number of utilization

Problem clients Capacity vehiclest Reference AC (%)

I 22 4500 3 Gaskell 1967 5 75-472 29 4500 3 Gaskell 1967 5 94·443 30 140 7 Clarke and Wright 1964 5 92·144 32 8000 4 Gaskell 1967 10 91·785 50 160 5 Christofides and Eilon

1969 5 97·126 75 140 10 Christofides and Eilon

1969 15 97-437 75 100 14 Gillett and Miller 1974 10 97-438 75 180 8 Gillett and Miller 1974 20 94·729 75 220 7 Gillett and Miller 1974 10 88·57

10 100 200 8 Christofides and Eilon1969 8 91·12

11 100 112 14 Gillett and Miller 1974 6 92-9812 120 200 7 Christofides et al. 1979 10 98·2113 150 200 12 Christofides et al. 1979 IS 93·1314 199 200 17 Christofides et al. 1979 15 93·5615 249 500 25 Gillett and Johnson 1976 10 96·85

t Minimum number of vehicles provided by the bin-packing solution.

Table 1. Test problem characteristics.

effect on the solution quality. Finally, the number of decrements of AC (that is, thenumber of times Step 4 of the algorithm is invoked) is determined by both the initialvalue of AC and the decrementing ratio 0·75 (this ratio has been empirically chosen).For example, when AC = 20 the number of decrements is 9. For all test problems,this number ranges between 9 (when AC = 20) and 4 (when AC = 5).

A comparative computational performance summary of various competingalgorithms is documented in Table 2. The solution quality generated by thegeneralized exchange heuristic, as well as those of the heuristic algorithms listed inthe table, are given. For all test problems, the following procedure was used toevaluate the distance between any two points (x , Yl) and (x z, Yz):

INT (SQRT «Xl - xz)Z + (Yl - Yz)Z) + 0·5)

One can observe from Table 2 that solutions for the problems 1,2 and 4 have beensignificantly improved. However, these results should be viewed conservatively forthe following reason. Though these problems have both capacity and distanceconstraints, only capacity constraints have been considered in our computation.Table 2 reveals that a large portion of improved solutions were obtained by theproposed heuristic. For seven out of fifteen problems, a better solution than the bestknown was produced. Moreover, in six other problems the new solutions were eithercomparable or identical to the best known solutions. In all cases our heuristicproduced routes which require the minimum number of vehicles.

It is difficult to consider computational performance in terms of solution times,since most of the studies in the vehicle routing literature were performed on differentcomputers. The solution times obtained with the proposed heuristic are presented inTable 3. All times reported in Table 3 are exclusive of both input/output times andthe bin-packing solution times (Step 0 of the algorithm). Moreover, the solution times

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'"00

Clarke Gillett Foster Fisher Stewartand and and and Christofides and

Problem Wright Miller Russell Ryan Kaikumar et a/. Golden This paper

1 955 (5) 956 (5) 956 (5) 953 (5) ~ 569U2 963 (5) 875 (4) 875 (4) 873 (4) - 534U3 1427 (8) 1214 1377 (8) - 1212U 1216 ~

4 839 (5) 810 810 809 492U ::J:::5 585 546 524 521t 524 534 521t 521U '".,"6 900 865 854 852 857 857 (11) 847t 847t ;:,-

'"7 1127 (15) 1061 1081 1058 (15) I042t '"8 754 745t 760 751 751 ;:,;:,.

9 - 715 697 692t - 692t 695 :-010 887 862 833 825t 833 851 829 825t :::<:l11 1176 1114 1116 - 1117 1113t '"""12 1066 I045t ;:,-

'"13 1204 1079 - 1014t 1064 1070 "'"14 1540 1389" 1420 1418 1397 ;:,

15 5918t

t Best known solution.t Optimal solutions (Harche and Rinaldi 1994). For problem 5 see also Cornuejols and Harche (1994).

Table 2. Comparison of algorithms based on solution values. (The numbers of routes used that are different from the number of vehiclesshown in Table I are given in parentheses.),

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Problem 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Time (s) 2·0 3'5 4·1 5·9 47-6 53-9 36·5 76·7 74·5 167-9 189·2 201·3 256·7 399·7 569·8

Table 3. Solution time.

are given for one initial value of ~C and all its subsequent decrements (Step 4 of thealgorithm). It can be seen from Table 3 that the performance of the heuristic in termsof computational effort is reasonable.

5. ConclusionsA new heuristic that is based on generalized edge exchange techniques has been

proposed. A useful feature of the heuristic is that it does not maintain feasibilitythroughout. Moreover, it yields the best solution with the minimum number ofvehicles required. Further, the solutions produced by the heuristic were found to be,in most cases, of high quality. Finally, the heuristic can be easily extended to handlegeneral graphs as well as a number of constraints arising in vehicle routing problems.

ACKNOWLEDGMENTS

The authors would like to thank Professor G. Rinaldi for several stimulatingdiscussions that motivated the work done in this paper, and for providing us withan implementation of the Lin-Kernighan procedure, written by Professors M. W.Pad berg and G. Rinaldi (1989). The authors are also grateful to two anonymousreferees for their constructive suggestions.

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(New York: Wiley), Chap. 11.CHRISTOFIDES, N., MINGOZZI, A., and TOTH, P., 1981, Math. Proqram., 20, 255.CLARKE, G., and WRIGHT, J. W.. 1964, Oper. Res., 12,568.CORNUEJOLS, G., and HARCHE, F., 1994, Math. Proqram., 60, 21-52.FISHER, M., and JAIKUMAR, R., \981, Networks, II, 109.FOSTER, B. A., and RYAN, D. M., 1976, Oper. Res. Q., 27, 367.GASKELL, T. J., 1967, Oper. Res. Q., 18, 281.GILLETT, B., and JOHNSON, J., 1976, Omega, 4, 711.GILLETT, B., and MILLER, L., 1974, Oper. Res., 22, 340.GOLDEN, B. L., and STEWART, W. R., 1985, Empirical analysis of heuristics. edited by E. L.

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JOHNSON, D. S., 1974, J. Comput. System Sci., 8, 272.LAPORTE, G., and NORBERT, Y., \987, AIIII. Discrete Math., 31, 147.LAPORTE, G., NORBERT, Y., and DESROCHERS, M.. 1985, Oper. Res .. 33, 1050.LIN, S., and KERNIGHAN, B., 1973, Oper. Res., 21, 498.MOLE, R.. and JAMESON, S., 1976, Oper. Res. Q., 27, 503.

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