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Augment-insert algorithms for the capacitated arc routing problem
0305.0548191 $3.00 + 0.00 Copyright ( 1991 Pergamon Press plc
Compurers Ops Res. Vol. 18. No. 2, pp. 189-198. 1991 Printed in Great Britain. All rights reserved
AUGMENT-INSERT ALGORITHMS FOR THE CAPACITATED ARC ROUTING PROBLEM WEN LEA PEARN* Bell Laboratories, Hoimdel, NJ 07733, U.S.A. (Received
October
1989: in revised ,form April 1990)
Srope and Purpose-The capacitated arc routing problem (CARP), is a capacitated variation of the rural postman problem (RPP), in which there is a capacity constraint associated with each vehicle. The CARP has been shown to be N&hard; therefore, it is difticult to solve the problem exactly. Due to the complexity of the CARP, heuristic solution procedures have been proposed to solve the CARP approximately. This paper introduces two new algorithms to solve the CARP near-optimally, particularly on some classes of networks for which current existing procedures tend to be unsatisfactory. Abstract-The capacitated arc routing problem (CARP) is a generalization of the rural postman problem (RPP), which has many real-world applications. Examples include routing of school buses. mail delivery vehicles, street sweepers and many others. The CARP has been shown to be NP-hard, and, therefore, heuristic procedures are desired to solve the CARP approximately. In this paper, we introduce two new algorithms to solve the CARP near-optimally. Computational results show that the new algorithms outperform the existing procedures on sparse networks with large arc demands. 1. INTRODUCTION
The capacitated arc routing problem (CARP) is a capacitated variation of the arc routing problems, in which there is a capacity constraint associated with each vehicle. The CARP may be stated as follows: given an undirected network G = (V, A), IF’1= n, IAl = m, with nonnegative arc demands and arc costs, find a set of minimal-cost cycles with each cycle passing through the central depot node, which traverses all the demand arcs and the total demand serviced by each vehicle does not exceed the vehicle capacity IV. Thus, the vehicle routes must be designed so that: (1) each positive-demand arc is serviced by exactly one vehicle; (2) each cycle begins and ends at a distinct node called the central depot; (3) the total arc demand serviced by each vehicle does not exceed the vehicle capacity W; and (4) the total routing cost is minimized. Real-world applications of the CARP include: routing of school buses, mail delivery vehicles, street sweepers, snow plows and household refuse collection vehicles; spraying roads with salt, inspecting of electric power lines, or oil or gas pipelines; and reading electric meters. Since the CARP involves capacity constraints, it is difficult to solve the problem exactly even on some approximate, restricted versions of the problem. In fact, it has been shown by Golden and colleagues [l, 2) that the OS-approximate CARP, which is a problem of finding a feasible CARP solution with a cost less than 1.5 times the cost of the optimal CARP solution, on a simple tree network, a simple cycle network and a simple path network, is NP-hard. Pearn et al. [33 presented a network transformation so that the CARP may be fo~ulated as a standard vehicle routing problem (VRP), which allows the CARP to be solved optimally via VRP algorithms through the transformation. However, the resulting number of nodes in the VRP network is (roughly) three times the number of the (positive) demand arcs in the original CARP network. This increase reflects the greater inherent difficulty in solving the CARP, and, therefore, the approach through this network transformation is computationally inefficient. Due to the computational complexity of the problem, many heuristic procedures have been proposed to solve the CARP approximately. Examples include the Construct-Strike algorithm by Christofides [4], the Electric Meter Reader Routing algorithm by Stem and Dror [S], the Path-Scanning and Augment-Merge algorithms by Golden et al. [6], the Modijied Construct-Strike
*Wen Lea Pearn received his Ph.D. from the L’niversity of Maryland. His area of interest includes combinatorial optimization. network analysis and probabilistic modeling. He has contributed articles to variousjournals in these areas.
WEN LEAPEARN
190
and Random Path-Scanning algorithms by Pearn [73 and the Puraflel Insert method by Chapleau et al. [8], Many of these algorithms have been experimented and tested by DeArmon [9], Golden et al. [6] and Pearn [73. These algorithms work well in general and provide near-optimal solutions to the CARP, in particular, the Modified Construct-Strike algorithm by Pearn [7] significantly outperforms many other existing algorithms on dense networks with small arc demands. However, none of these algorithms seem to work well on sparse networks with large arc demands. In this paper, we describe two new algorithms to solve the CARP approximately and provide a near-optimal solution to the CARP, particularly for problems with sparse networks (no greater than 30% complete) and large arc demands.
2. HEURISTICS
EXAMINED
The existing heuristic algorithms, which we wiil examine and compare with our new algorithms, include the Construct-Strike ufgurithm [4-j, the Modi~ed Constrict-Strike algorfthm [7], the Path-Scanning algorithm [6], the Random Path-Scanning algorithm [73 and the Augment-Merge algorithm [6]. In Stem and Dror [SJ, an application of the CARP on the routing electric meter readers problem, was considered. Since working hour time restrictions were imposed in this case, open tours (instead of closed’ tours) may be desired, and tours may start and end at intermediate points of an arc. In the school bus routing system developed by Chapleau et al. [S], besides minimizing the total routing cost incurred, a secondary objective, that of providing “good quality service,” was also considered. These two situations may be viewed as pure CARP applications and thus are not compared with our new algorithm. (a) Construct-Strike algorithm [43 The basic idea behind this algorithm is to construct feasible cycles which, when removed, do not separate the remaining graphs into disconnected components. When the feasible cycles are constructed, we remove them from the graph. This cycle construct-then-remove procedure is repeated until no more cycles can be found. We then move to the second stage of the algorithm. In the second stage, the Minimal Cost l-Matching algorithm is applied to match the odd-degree nodes in the remaining graph so that a Euler cycle may be generated. The algorithm then returns to the first stage and searches for feasible cycles. The feasibfe cycle const~ct-then-remove procedure and the matching procedure are repeated until the whole graph is covered. The complexity of this algorithm is 0(mn3). (b) Modified Construct-Strike algorithm [ 71 Some modifications have been considered to improve the Construct-Strike solutions. In this modified procedure, feasible cycles are constructed and removed repeatedly until no more cycles can be found. The Minimal Spanning Tree algorithm is applied to render the remaining graph connected, and the Matching algorithm is applied so that a Euler cycle may be generated. If feasible cycles can be found in the resulting graph, the algorithm returns to the feasible-cycle Construct-Strike stage. Otherwise, arbitrary arcs are selected to extend the cycle path until the vehicle capacity is exhausted. The Construct-Strike, Minimal Spanning Tree and Matching procedures are repeated until the whole graph is covered. The complexity of this algorithm is 0(mn4). (c) Path-Scanning algorithm [6] The Path-Scanning procedure is based on constructing one cycle at a time using a certain myopic optimization criterion. In forming each cycle, a path is extended by adding the arc that looks most promising until the vehicle capacity is exhausted; then the least-cost return path to the depot is followed to form a complete cycle. In work by Golden et al. [63, five Path-Scanning optimization criteria are considered in the arc selection procedure. Each of these criteria is used to generate a complete solution, and the solution from this approach is the best of these five. The complexity of this algorithm is 0(n3).
Augment-Insert
(d)
Random Path-Scanning
algorithm
algorithms for the CARP
191
[7]
This is a variation of the Path-Scanning algorithm. Instead of using each single criterion to generate a complete solution, one of the five criteria is selected randomly (each criterion receives a probability of 0.2) to use in each step while extending the cycle path, and generate a complete solution. The best of the 30 solutions is then chosen as the solution from this approach. The complexity of this algorithm is 0(n3). (e) Augment-Merge
algorithm
[6]
The basic procedure of this algorithm is outlined as follows (as it appeared in the original paper): ‘(1) Initialize-all demand arcs are serviced by a separate cycle; (2) augment-starting with the longest cycle available, see if a demand arc on a shorter cycle can be serviced on a longer cycle; (3) merge-subject to capacity constraints, evaluate the merging of any two cycles (possibly subject to additional restrictions); then merge the two cycles which yield the largest savings; (4) iterate-repeat Step 3 until finished. Several experiments and modification of the basic AugmentMerge approach are considered in the work by Golden et al. [6]. The number of Augment-Merge program executions was limited to 24, and the solution from this approach was the best of the 24 runs. The complexity of this algorithm is O(n3).
3. THE
AUGMENT-INSERT
ALGORITHMS
Tour-augment methods and insertion procedures have been developed for solving the CARP approximately, such as the Parallel-Insert method by Chapleau et al. [83, the Path-Sunning and Augment-Merge algorithms by Golden et al. [6f and the Random Path-Scanning algorithm by Pearn [7]. In this section, we introduce two new procedures to solve the CARP heuristically, particularly for problems with sparse network and large arc demands. These two new procedures are based on standard ideas used in the design of heuristic algorithms, taking the merits of the Augment-Merge 161 and Parallel-Inert [S] algorithms with some moclihcations. They are very straightforward and are easy to implement. In the new procedures, since the tours servicing demand arcs “far away” from the depot are planned in a more profitable way, problems with sparse networks and large demand should be easier to handle. For convenience, we define the following notations and terms used throughout this paper. Given a CARP network G with depot node 1, define: l d, = cost of the least-cost path between node i and 1; 0 Dij = d,i + dlj; 0 DCYCLEij = the least-cost cycle covers ard (i,i); l QCYCLE, = the least-quantity cycle (with respect to arc demand) covers arc (i, j); l ZNCOST,, = additional cost incurred for inserting arc (u, u) into the current cycle path; 0 S = the set of demand arcs from G that have not been serviced. Augment-Insert
Procedure
Phase-I- augment 1. Let S = G initiaIly. Start with the demand arc (i,j)~S with Dij = maxfu,u~Es(DW)uu}, find the cycle DCYCLEij, then proceed to Step 2. It should be noted that the DCYCLE, searching procedure must be repeated V(U,U)E$ in descending order of D., until the DCYCLEij is found for some demand arc (i, j) E S. If DC YCL Eij does not exist V(i, j) E S, proceed to Phase II.
2. Augment the initial cycle path [consisting of arc (i,j)J by adding demand arcs (u, V)E DC YCLE, in descending order of D*, until the vehicle capacity is exhausted. The least-cost return paths from the endpoints of the extended cycle path to the depot node 1 are followed
to form a complete cycle. Remove this complete cycle from graph G and update S. can be found V(i,j)ES. Proceed to Phase II.
3. Repeat Steps 1 and 2 of Phase I until no more DCYCLE,,
WEN LEA PEARN
192
Pirase ZZ-insertion
Start with the demand arc (i,j)~S with Dij = max,,,,,,,{l),,}, insert the demand arc (u, U)ES into the initial cycle path [consisting of arc (i,j)], provided that INCOST,, < C, where C (to be varied over a range of values) is an upper bound on the insertion cost. Repeat this insertion procedure V(u, u) E S in descending order of 4, until the vehicle capacity is exhausted (or the depot node 1 is reached). The least-cost return paths from the two endpoints of the extended cycle path to the depot node 1 are followed to form a complete cycle. Remove this complete cycle from graph G and update S. Repeat Steps 1 and 2 of Phase II until the whole graph is covered. Choice
of upper
bound C
The upper bound value C for the insertion cost INCOST,,,, should not be set too large or too small. If C is set too large, the procedure attempts to insert demand arcs which are “far away” from the current cycle path, and therefore generates long cycles. On the other hand, if C is set too small, the procedure only inserts demand arcs which are “very close” to the current cycle path; consequently, the cycle path could reach the depot node 1 far before the vehicle capacity is exhausted. Therefore, many additional cycles must be generated to cover the whole graph, We have found no general rules in determining the range of C; however, a reasonable range of C may be set to IX.4maxiev~~~i)l. In computing ZNCOSl;j, we have extended Clarke and Wright’s method [lo] to include both ends of the arc and cycle path; i.e. for a cycle path with endpoints i, j, and a demand arc (u, Y), ~~C~ST~~ = minix,yts&ix + di, -d,,, djx + di, - d,j), where B = ((a, a), (u, u)). Computationalcomplexity The bottleneck of this algorithm is computing INCOST,,, which requires the application of the All-Pair Shortest Path algorithm which is of 0(n3). Therefore, the complexity of this Augment-Insert procedure is O(n3). Two versions of the Augment-Insert procedure are considered and we refer to them as Augment-Insert Algorithm I and Augment-Insert Algorithm II. The former uses the least-cost cycle, DCYCLE,, to augment the initial cycle path, and the latter uses the least-quantity cycle, QCYCLE,. Example
Consider the CARP network depicted in Fig. 1. This problem is taken directly from the 23 test problems mentioned earlier [I, 6, 71 without any modilications. The network has 13 nodes, 23 arcs with density approximately equal to 30%. The vehicle capacity W is set to 35, and the depot
Fig. 1. An example with depot = 1, W = 35 and density = 30%.
Augment-Insert
Algorithm I using C = IO
Table 1. Solution generated by Augment-Insert Cyck
Phase
Starting Arc
1
I
(11.13)
2
i
(11,121
193
algorithms for the CARP
Cycle Path 12-8-9-13-11-10
Total Lord 5 1
1
12 6-12-11-4
Cost
32
94
35
1 a9
Table 2. Performance comparisons of the seven algorithms on the 23 test problems Mod&d Consttuct StlikC
Randan Path Scanning
AugmattInsen
AugmmtInsen
Algorithm1
AlgwirhmIl
9.16
7.59
8.05
12.39
13.95
3.91
2.74
1.61
2.52
4.39
4.83
2
1
2
10
3
1
0
2
2
7
16
6
3
0
34.4
n.5
35.1
521
43.9
48.8
51.5
GmStNP Strike
Avasge 96Above ?he Pmblml LOW?Beund AverageRark Among7bc sevalA@nimmr Numberd Pmbkms Aehievina Lower Baud Numberd FmbIansReceiving
Path Sc8lming
Augment
17.91
11.03
5.13
Mege
Bestsdutions worstsolutionin Terms of % Above llte Lower Bound
node is set to 1. Many (over 43O/o)of the arcs have demand exceeding i/3 of the vehicle capacity; some are far away from the depot. Since f(total arc demand)/(vehicle capacity W) I= [212/35 l= 7, the minimum number of vehicles required in this problem is 7. The solution to this problem, generated by the Augment-Insert Algorithm I with C = 10, is summarized in Table 1. This solution consists of seven cycles and the total cost is 474. Comparing this solution value with that generated by the other live procedures-655 560, 573,645 and 566the improvement is signi~cant. 4. COMPUTATIONAL
RESULTS
It should be noted that Augment-Insert Algorithms I and II are developed specifically for solving the CARP with sparse networks and large arc demands. We do not expect them to work well on dense networks with small arc demands. For such networks, the demand arcs are all very close to the depot node. Therefore, the cycle DCYCLE,, found in Phase I includes only a small number of demand arcs with a relatively small total load (compared with the vehicle capacity W). Consequently, the algorithms create many additional cycles in order to cover the whole graph. Therefore, they may not work well on problems with such characteristics. We experimented with Augment-Insert Algorithms I and II on some test problems using six values of C, which were initially set to C = 0, 5, 10, 1520 and 25 (0 < C G maxi,v{d,il). It appears that the solution obtained on these test problems achieved best problem solutions at C = 0, 5 and 10 for both algorithms, and that solution values increased when the value of C exceeded the range of [O, IO]. Therefore, we limited our testing of the two new algorithms to C = 0, 5 and IO. We first ran through the 23 problems found in the literature that have been tested by DeArmon [93, Golden et al. [6] and Pearn [7]. The sizes of the 23 problems range from ‘7to 27 nodes and from I1 to 55 arcs. Many of these problems are dense networks with small arc demands. Table 2 presents a comparison between the two new algorithms and the live existing solution procedures
194
WEN
LEA
PEARN
Table 3. Solution values generated by the two new algorithms using C = 0. 5 and IO
L * Indicates best solution value.
on the 23 problems. The performance of the two new algorithms (Augment-Insert Algorithm I and Augment-Insert Algorithm II) on the 23 problems is considered better than the Construct-Strike algorithm, comparable to the Path-Scanning algorithm and is worse than the other three algorithms. The unsatisfactory performance of the new algorithms on the 23 problems, as we mentioned earlier, is because of the characteristics of the networks-dense with small arc demands. For the purpose of testing the two new algorithms, we generated an additional four sets of test problems. These problems are all sparse networks with some large arc demand. Their sizes range from 13 nodes, 23 arcs to 27 nodes, 51 arcs with density ranging from 15 to 30%. These problems are described in the following: 0 Set A: 0 Set B: l Set C: 0 Set D:
six problems; n = 13, m = 23; density z 30%. eight problems; n = 18, m = 30; density ~20%. six problems; n = 22, m = 45; density 5 20%. ten problems; n = 27, m = 51; density z 15%.
Problem generation
These problems were generated from the problems found in the open literature 16, 7, 93 with some modifications, i.e. by arbitrarily: (1) removing demand arcs from the original network (so
Augment-Insert Table 4. Solution
algorithms
values generated
195
for the CARP
by the five existing
algorithms
~
A3
719
611
793
674
589
422
A4
613
560
714
541
530
413
1
634
1
648
1
635
)
646
1
337 1
B8
1
-
Cl
1
889
1
686
1
683
1
715
1
680
1
453
C2
1
807
1
630
1
658
1
643
1
644
1
478 1
C3
1
-
1
686
1
689
1
685
1
659
1
454 1
I
816
I
684
I
686
I
701
I
668
I
450 I
1
719
620
1
449
c4 C5 C6
I
I
I
1
I
1
I
I
640
1 I
I
Dl
I
638 592
1 I
I
I
617
1 I
I
I
1
634
1 I
I
622
641
569
440
444
419
402
432
279
D2
-
447
401
436
428
280
D3
-
519
501
489
479
283
D4
1
-
1
415
1
370
1
367
1
380
1
280 1
D5
1
-
1
5%
1
551
1
598
1
562
1
371 1
D6
1
-
I
426
1
436
1
-
1
412
1
287 I
D7
1
-
1
452
1
413
1
-
1
425
1
280 (
I
I
D8
I
D9I
I
I
DlO
438
-
I
1 1
I
484 I
I
I
484 I
286
482
342
I
485 I
422 I
465
463 1
530
I
443
402 I
I
-
I
486 I
285 I
I
that the density is no greater than 30%); (2) changing depot nodes; (3) changing arc cost; (4) changing arc demand; and (5) reducing the vehicle capacity W, so that the problems generated are significantly different from the original networks. Many of the demand arcs in the new networks are “far away from the depot”, and, therefore, the vehicles must travel along many other arcs before returning to the depot node. Table 3 displays the solution values of the four sets of test problems generated by the two new algorithms using C = 0,s and 10. For example, if we run Augment-Insert Algorithm I on problem 1 with C = 0, 5 and 10, then three solution values-484, 474 and 474- will be generated. Table 4 displays solution values of the four sets of test problems generated by the other five solution procedures. The performance comparisons of the two new algorithms using C = 0, 5 and IO, and the other five algorithms in terms of the average percentage above the problem’s lower bound, the
196
WEN LEA PEARN Table 5. C = 0: performance comparisons of the seven algorithms
56.78
s1.32
130.9
94.8
1
101.5
91.7
1
77.9
1
863
Table 6. C = 5: performance comparisons of the seven algorithms
~
71.95
1 57.91 1 56.78 1 51.32
bwaBotmd Numbaof FmblanaBeDBivin8 Be!uttiam
WmtSOlU!iOOin T-of%Abow TbsLowcrBound
0
1
1
1
13
2
14
I 12lX2 j
101.5 /
130.9 1
94.8
91.1
1
74.4
1
74.s
Table 7. C = 10: performance comparisons of the seven algorithms
LowaBowid Number of
Pmblan#Rec&iq Bat Sdutiav
0
wart solutial in TmnrofZAbove TlwLowa Bound
120.2
I
2
I
11
I 101.5
130.9
94.8
91.7
14 I
74.4
79.5
number of problems receiving the best solution values, and the worst solution in terms of percentage above the lower bound, are displayed in Table 5 (C = 0), Table 6 (C = 5) and Table 7 (C = IO). The performance of the two new algorithms is reasonably good in that: (I) in most (over 80%) of the problems tested, the new algorithms received best problem solutions among all algorithms we examined regardless of which parameter value we chose (in particular, C = 5); (2) the new algorithms outperformed the other five procedures by roughly 26, 12, 11, 12 and 5% of the problem lower bound; (3) the worst solutions in terms of percentage above the problem lower bound obtained by the new algorithms were under 80% (comparing this with that of 91-120% by the other five procedures, the improvement is significant). Further improvement was attempted by taking the best of the solutions generated by the two algorithms using C = 0, 5 and IO. The results of the comparisons are displayed in Table 8. Unfortunately, the improvement is insignificant considering the additional amount of computer time it takes. Problem lower bound
The lower bound used in this paper, for assessing the deviation of the heuristic solutions from optimality, is obtained by using both the Matching Lower Bound [l] and the Node Scanning
Augment-Insert
algorithms for the CARP
Table 8. C = 0, 5 and 10: performance comparisons of the seven algorithms
Table 9. Run time comparisons of the seven algorithms
DIO
27
51
-
10.5
54.9
996.4
40.0
42.3
Lower Bound [2]. These two lower bounding procedures have been shown to perform poorly for many instances of problems, in particular, for problems with sparse network and large arc demand. For the four sets of problems we tested, the networks are much more perverse than those tested by DeArmon [9], Golden et al. [6] and Pearn [7]. It might be that which caused such a large deviation of the heuristic solutions from the problem’s lower bounds. Computer
run time
In our testing, the run times for problems with the same size and vehicle capacity are very much the same. However, run times are quite different for problems with the same size but different vehicle capacity. In fact, our testing shows that the run time increases when the vehicle capacity decreases. This is because the vehicle capacity W determines the minimal number of cycles required for the solution and hence determines the number of program iterations. Therefore, instead of showing the average run time, we only display eight test problems (two from each set) in CPU set on the CDC CYBER 720 system (Table 9) as an indication of the relative efficiency of the seven algorithms. Since the solutions generated by the Path-Scanning algorithm, Random Path-Scanning algorithm and Augment-Merge algorithm were obtained by selecting the best among a set of solutions, the run times for these algorithms were calculated by totaling each single execution time. In the Modified Construct-Strike algorithm, however, the first single solution was taken as the solution from this approach. Therefore, only one execution was counted. The four sets of problems we tested are all sparse networks with some large arc demand (or with relatively small vehicle capacity). Each vehicle can only service a small number of demand arcs while it has to travel along many other arcs before returning to the depot. This generates a long cycle. Therefore, the procedures must be repeated as many times as are required to cover all demand arcs. Consequently, it results in taking a great amount of computer time to obtain a complete
198
WEN LEA PEARN
solution. In particular, the Construct-Strike algorithm and the Modified Construct-Strike algorithm involve the Minimal Cost l-Matching applications, which encountered more than 16 odd-degree nodes when the size of the network went up to 27 nodes, 51 arcs; and the matching procedure has to be iteratively repeated so many times that each single execution of the program took longer than 500 CPU sec. It should be noted that both new algorithms are faster than all the existing algorithms (except the Path-Scanning algorithm); in particular, the two new algorithms are IO-20 times faster than the Construct-Strike algorithm, the Modified Construct-Strike algorithm and the Augment-Merge algorithm while the new algorithms outperform all the other live procedures. 5. CONCLUSIONS
In this paper, we have introduced two fast algorithms using tour-augment and insertion ideas. They are very straightforward and are easy to implement. We have tested them on many problems with sparse network and large arc demands, and compared them with live other procedures. The computational results have shown that: (1) both of the two new algorithms significantly outperform the other five algorithms by roughly 26, 12, 11, 12 and 5% of the problem lower bound for all parameter values (in particular, C = 5) we tested; (2) the worst solutions in terms of percentage above the problem’s lower bound obtained by the two new algorithms were all under 80%, as opposed to 91-120% by the other five existing methods; (3) the number of best solutions obtained by the two new algorithms was roughly equal to 14 (46%) and 12 (41%), respectively, as opposed to 1 or 2 (c 7%) by the other five algorithms; (4) the two new algorithms are lo-20 times faster than most existing procedures. While the characteristics of the network may well influence the performance of certain algorithms, we have limited our testing to problems where the underlying networks-are sparse (density is no greater than 30%) with some large arc demands. This is certainly the case in many real-world applications when the operating vehicles have relatively small capacity. We have made no claims that the new algorithms presented in this paper always do better than the other five algorithms on problems with such characteristics. However, we do believe that the relative performance of these seven algorithms on the sample problems in this paper, which were arbitrarily generated from existing test problems (with some modifications) found in the literature, is indicative of their relative performance on the CARP, with such characteristics (sparse network with some large demands), and the new algorithms should be applied along with other existing solution procedures to obtain near-optimal solutions. Acknowledgement-The author would like to thank the anonymous referees for their careful reading of the paper and several valuable suggestions for improvements. REFERENCES 1. B. Golden and R. Wong, Capacitated arc routing problems. Networks 11(3), 305-318 (1981). 2. A. Assad, W. Pearn and B. Golden, Capacitated Chinese postman problem: lower bounds and solvable cases. Am. J. math. Mgmt Sci. 7( I-2), 63-88 (1987). 3. W. Pearn, A. Assad and B. Golden, Transforming arc routing into node routing problem. Computers Ops Res. 14(4). 285-288 (1987). 4. N. Christofides, The optimal traversal of a graph. Omega 1, 719-732 (1973). 5. H. Stem and M. Dror, Routing electric meter readers. Computers Ops Res. 6, 209-223 (1979). 6. B. Golden, J. DeArmon and E. Baker, Computational experiments with algorithms for a class of routing problems. Compurers Ops Res. 10, 47-69 (1983). 7. W. Pearn, Approximate solutions for the capacitated arc routing problem. Compurers Ops Res. 16, 589-600 (1989). 8. L. Chapleau, J. Ferland, G. Lapalme and J. Rousseau, A parallel insert method for the capacitated arc routing problem. Ops Res. Lerr. 3, 95-99 (1984). 9. J. DeArmon, A comparison of heuristics for the capacitated Chinese postman problem. Master’s Thesis, Univ. of Maryland at College Park (1981). IO. C. Clarke and J. White, Scheduling of vehicles from a central depot to a number of delivery points. Ops Res. 12(4), 568-58 I (1964). Il. S. Roy and J. Rousseau, The capacitated Canadian postman problem. INFOR 27(l), 58-73 (1989).
Compurers Ops Res. Vol. 18. No. 2, pp. 189-198. 1991 Printed in Great Britain. All rights reserved
AUGMENT-INSERT ALGORITHMS FOR THE CAPACITATED ARC ROUTING PROBLEM WEN LEA PEARN* Bell Laboratories, Hoimdel, NJ 07733, U.S.A. (Received
October
1989: in revised ,form April 1990)
Srope and Purpose-The capacitated arc routing problem (CARP), is a capacitated variation of the rural postman problem (RPP), in which there is a capacity constraint associated with each vehicle. The CARP has been shown to be N&hard; therefore, it is difticult to solve the problem exactly. Due to the complexity of the CARP, heuristic solution procedures have been proposed to solve the CARP approximately. This paper introduces two new algorithms to solve the CARP near-optimally, particularly on some classes of networks for which current existing procedures tend to be unsatisfactory. Abstract-The capacitated arc routing problem (CARP) is a generalization of the rural postman problem (RPP), which has many real-world applications. Examples include routing of school buses. mail delivery vehicles, street sweepers and many others. The CARP has been shown to be NP-hard, and, therefore, heuristic procedures are desired to solve the CARP approximately. In this paper, we introduce two new algorithms to solve the CARP near-optimally. Computational results show that the new algorithms outperform the existing procedures on sparse networks with large arc demands. 1. INTRODUCTION
The capacitated arc routing problem (CARP) is a capacitated variation of the arc routing problems, in which there is a capacity constraint associated with each vehicle. The CARP may be stated as follows: given an undirected network G = (V, A), IF’1= n, IAl = m, with nonnegative arc demands and arc costs, find a set of minimal-cost cycles with each cycle passing through the central depot node, which traverses all the demand arcs and the total demand serviced by each vehicle does not exceed the vehicle capacity IV. Thus, the vehicle routes must be designed so that: (1) each positive-demand arc is serviced by exactly one vehicle; (2) each cycle begins and ends at a distinct node called the central depot; (3) the total arc demand serviced by each vehicle does not exceed the vehicle capacity W; and (4) the total routing cost is minimized. Real-world applications of the CARP include: routing of school buses, mail delivery vehicles, street sweepers, snow plows and household refuse collection vehicles; spraying roads with salt, inspecting of electric power lines, or oil or gas pipelines; and reading electric meters. Since the CARP involves capacity constraints, it is difficult to solve the problem exactly even on some approximate, restricted versions of the problem. In fact, it has been shown by Golden and colleagues [l, 2) that the OS-approximate CARP, which is a problem of finding a feasible CARP solution with a cost less than 1.5 times the cost of the optimal CARP solution, on a simple tree network, a simple cycle network and a simple path network, is NP-hard. Pearn et al. [33 presented a network transformation so that the CARP may be fo~ulated as a standard vehicle routing problem (VRP), which allows the CARP to be solved optimally via VRP algorithms through the transformation. However, the resulting number of nodes in the VRP network is (roughly) three times the number of the (positive) demand arcs in the original CARP network. This increase reflects the greater inherent difficulty in solving the CARP, and, therefore, the approach through this network transformation is computationally inefficient. Due to the computational complexity of the problem, many heuristic procedures have been proposed to solve the CARP approximately. Examples include the Construct-Strike algorithm by Christofides [4], the Electric Meter Reader Routing algorithm by Stem and Dror [S], the Path-Scanning and Augment-Merge algorithms by Golden et al. [6], the Modijied Construct-Strike
*Wen Lea Pearn received his Ph.D. from the L’niversity of Maryland. His area of interest includes combinatorial optimization. network analysis and probabilistic modeling. He has contributed articles to variousjournals in these areas.
WEN LEAPEARN
190
and Random Path-Scanning algorithms by Pearn [73 and the Puraflel Insert method by Chapleau et al. [8], Many of these algorithms have been experimented and tested by DeArmon [9], Golden et al. [6] and Pearn [73. These algorithms work well in general and provide near-optimal solutions to the CARP, in particular, the Modified Construct-Strike algorithm by Pearn [7] significantly outperforms many other existing algorithms on dense networks with small arc demands. However, none of these algorithms seem to work well on sparse networks with large arc demands. In this paper, we describe two new algorithms to solve the CARP approximately and provide a near-optimal solution to the CARP, particularly for problems with sparse networks (no greater than 30% complete) and large arc demands.
2. HEURISTICS
EXAMINED
The existing heuristic algorithms, which we wiil examine and compare with our new algorithms, include the Construct-Strike ufgurithm [4-j, the Modi~ed Constrict-Strike algorfthm [7], the Path-Scanning algorithm [6], the Random Path-Scanning algorithm [73 and the Augment-Merge algorithm [6]. In Stem and Dror [SJ, an application of the CARP on the routing electric meter readers problem, was considered. Since working hour time restrictions were imposed in this case, open tours (instead of closed’ tours) may be desired, and tours may start and end at intermediate points of an arc. In the school bus routing system developed by Chapleau et al. [S], besides minimizing the total routing cost incurred, a secondary objective, that of providing “good quality service,” was also considered. These two situations may be viewed as pure CARP applications and thus are not compared with our new algorithm. (a) Construct-Strike algorithm [43 The basic idea behind this algorithm is to construct feasible cycles which, when removed, do not separate the remaining graphs into disconnected components. When the feasible cycles are constructed, we remove them from the graph. This cycle construct-then-remove procedure is repeated until no more cycles can be found. We then move to the second stage of the algorithm. In the second stage, the Minimal Cost l-Matching algorithm is applied to match the odd-degree nodes in the remaining graph so that a Euler cycle may be generated. The algorithm then returns to the first stage and searches for feasible cycles. The feasibfe cycle const~ct-then-remove procedure and the matching procedure are repeated until the whole graph is covered. The complexity of this algorithm is 0(mn3). (b) Modified Construct-Strike algorithm [ 71 Some modifications have been considered to improve the Construct-Strike solutions. In this modified procedure, feasible cycles are constructed and removed repeatedly until no more cycles can be found. The Minimal Spanning Tree algorithm is applied to render the remaining graph connected, and the Matching algorithm is applied so that a Euler cycle may be generated. If feasible cycles can be found in the resulting graph, the algorithm returns to the feasible-cycle Construct-Strike stage. Otherwise, arbitrary arcs are selected to extend the cycle path until the vehicle capacity is exhausted. The Construct-Strike, Minimal Spanning Tree and Matching procedures are repeated until the whole graph is covered. The complexity of this algorithm is 0(mn4). (c) Path-Scanning algorithm [6] The Path-Scanning procedure is based on constructing one cycle at a time using a certain myopic optimization criterion. In forming each cycle, a path is extended by adding the arc that looks most promising until the vehicle capacity is exhausted; then the least-cost return path to the depot is followed to form a complete cycle. In work by Golden et al. [63, five Path-Scanning optimization criteria are considered in the arc selection procedure. Each of these criteria is used to generate a complete solution, and the solution from this approach is the best of these five. The complexity of this algorithm is 0(n3).
Augment-Insert
(d)
Random Path-Scanning
algorithm
algorithms for the CARP
191
[7]
This is a variation of the Path-Scanning algorithm. Instead of using each single criterion to generate a complete solution, one of the five criteria is selected randomly (each criterion receives a probability of 0.2) to use in each step while extending the cycle path, and generate a complete solution. The best of the 30 solutions is then chosen as the solution from this approach. The complexity of this algorithm is 0(n3). (e) Augment-Merge
algorithm
[6]
The basic procedure of this algorithm is outlined as follows (as it appeared in the original paper): ‘(1) Initialize-all demand arcs are serviced by a separate cycle; (2) augment-starting with the longest cycle available, see if a demand arc on a shorter cycle can be serviced on a longer cycle; (3) merge-subject to capacity constraints, evaluate the merging of any two cycles (possibly subject to additional restrictions); then merge the two cycles which yield the largest savings; (4) iterate-repeat Step 3 until finished. Several experiments and modification of the basic AugmentMerge approach are considered in the work by Golden et al. [6]. The number of Augment-Merge program executions was limited to 24, and the solution from this approach was the best of the 24 runs. The complexity of this algorithm is O(n3).
3. THE
AUGMENT-INSERT
ALGORITHMS
Tour-augment methods and insertion procedures have been developed for solving the CARP approximately, such as the Parallel-Insert method by Chapleau et al. [83, the Path-Sunning and Augment-Merge algorithms by Golden et al. [6f and the Random Path-Scanning algorithm by Pearn [7]. In this section, we introduce two new procedures to solve the CARP heuristically, particularly for problems with sparse network and large arc demands. These two new procedures are based on standard ideas used in the design of heuristic algorithms, taking the merits of the Augment-Merge 161 and Parallel-Inert [S] algorithms with some moclihcations. They are very straightforward and are easy to implement. In the new procedures, since the tours servicing demand arcs “far away” from the depot are planned in a more profitable way, problems with sparse networks and large demand should be easier to handle. For convenience, we define the following notations and terms used throughout this paper. Given a CARP network G with depot node 1, define: l d, = cost of the least-cost path between node i and 1; 0 Dij = d,i + dlj; 0 DCYCLEij = the least-cost cycle covers ard (i,i); l QCYCLE, = the least-quantity cycle (with respect to arc demand) covers arc (i, j); l ZNCOST,, = additional cost incurred for inserting arc (u, u) into the current cycle path; 0 S = the set of demand arcs from G that have not been serviced. Augment-Insert
Procedure
Phase-I- augment 1. Let S = G initiaIly. Start with the demand arc (i,j)~S with Dij = maxfu,u~Es(DW)uu}, find the cycle DCYCLEij, then proceed to Step 2. It should be noted that the DCYCLE, searching procedure must be repeated V(U,U)E$ in descending order of D., until the DCYCLEij is found for some demand arc (i, j) E S. If DC YCL Eij does not exist V(i, j) E S, proceed to Phase II.
2. Augment the initial cycle path [consisting of arc (i,j)J by adding demand arcs (u, V)E DC YCLE, in descending order of D*, until the vehicle capacity is exhausted. The least-cost return paths from the endpoints of the extended cycle path to the depot node 1 are followed
to form a complete cycle. Remove this complete cycle from graph G and update S. can be found V(i,j)ES. Proceed to Phase II.
3. Repeat Steps 1 and 2 of Phase I until no more DCYCLE,,
WEN LEA PEARN
192
Pirase ZZ-insertion
Start with the demand arc (i,j)~S with Dij = max,,,,,,,{l),,}, insert the demand arc (u, U)ES into the initial cycle path [consisting of arc (i,j)], provided that INCOST,, < C, where C (to be varied over a range of values) is an upper bound on the insertion cost. Repeat this insertion procedure V(u, u) E S in descending order of 4, until the vehicle capacity is exhausted (or the depot node 1 is reached). The least-cost return paths from the two endpoints of the extended cycle path to the depot node 1 are followed to form a complete cycle. Remove this complete cycle from graph G and update S. Repeat Steps 1 and 2 of Phase II until the whole graph is covered. Choice
of upper
bound C
The upper bound value C for the insertion cost INCOST,,,, should not be set too large or too small. If C is set too large, the procedure attempts to insert demand arcs which are “far away” from the current cycle path, and therefore generates long cycles. On the other hand, if C is set too small, the procedure only inserts demand arcs which are “very close” to the current cycle path; consequently, the cycle path could reach the depot node 1 far before the vehicle capacity is exhausted. Therefore, many additional cycles must be generated to cover the whole graph, We have found no general rules in determining the range of C; however, a reasonable range of C may be set to IX.4maxiev~~~i)l. In computing ZNCOSl;j, we have extended Clarke and Wright’s method [lo] to include both ends of the arc and cycle path; i.e. for a cycle path with endpoints i, j, and a demand arc (u, Y), ~~C~ST~~ = minix,yts&ix + di, -d,,, djx + di, - d,j), where B = ((a, a), (u, u)). Computationalcomplexity The bottleneck of this algorithm is computing INCOST,,, which requires the application of the All-Pair Shortest Path algorithm which is of 0(n3). Therefore, the complexity of this Augment-Insert procedure is O(n3). Two versions of the Augment-Insert procedure are considered and we refer to them as Augment-Insert Algorithm I and Augment-Insert Algorithm II. The former uses the least-cost cycle, DCYCLE,, to augment the initial cycle path, and the latter uses the least-quantity cycle, QCYCLE,. Example
Consider the CARP network depicted in Fig. 1. This problem is taken directly from the 23 test problems mentioned earlier [I, 6, 71 without any modilications. The network has 13 nodes, 23 arcs with density approximately equal to 30%. The vehicle capacity W is set to 35, and the depot
Fig. 1. An example with depot = 1, W = 35 and density = 30%.
Augment-Insert
Algorithm I using C = IO
Table 1. Solution generated by Augment-Insert Cyck
Phase
Starting Arc
1
I
(11.13)
2
i
(11,121
193
algorithms for the CARP
Cycle Path 12-8-9-13-11-10
Total Lord 5 1
1
12 6-12-11-4
Cost
32
94
35
1 a9
Table 2. Performance comparisons of the seven algorithms on the 23 test problems Mod&d Consttuct StlikC
Randan Path Scanning
AugmattInsen
AugmmtInsen
Algorithm1
AlgwirhmIl
9.16
7.59
8.05
12.39
13.95
3.91
2.74
1.61
2.52
4.39
4.83
2
1
2
10
3
1
0
2
2
7
16
6
3
0
34.4
n.5
35.1
521
43.9
48.8
51.5
GmStNP Strike
Avasge 96Above ?he Pmblml LOW?Beund AverageRark Among7bc sevalA@nimmr Numberd Pmbkms Aehievina Lower Baud Numberd FmbIansReceiving
Path Sc8lming
Augment
17.91
11.03
5.13
Mege
Bestsdutions worstsolutionin Terms of % Above llte Lower Bound
node is set to 1. Many (over 43O/o)of the arcs have demand exceeding i/3 of the vehicle capacity; some are far away from the depot. Since f(total arc demand)/(vehicle capacity W) I= [212/35 l= 7, the minimum number of vehicles required in this problem is 7. The solution to this problem, generated by the Augment-Insert Algorithm I with C = 10, is summarized in Table 1. This solution consists of seven cycles and the total cost is 474. Comparing this solution value with that generated by the other live procedures-655 560, 573,645 and 566the improvement is signi~cant. 4. COMPUTATIONAL
RESULTS
It should be noted that Augment-Insert Algorithms I and II are developed specifically for solving the CARP with sparse networks and large arc demands. We do not expect them to work well on dense networks with small arc demands. For such networks, the demand arcs are all very close to the depot node. Therefore, the cycle DCYCLE,, found in Phase I includes only a small number of demand arcs with a relatively small total load (compared with the vehicle capacity W). Consequently, the algorithms create many additional cycles in order to cover the whole graph. Therefore, they may not work well on problems with such characteristics. We experimented with Augment-Insert Algorithms I and II on some test problems using six values of C, which were initially set to C = 0, 5, 10, 1520 and 25 (0 < C G maxi,v{d,il). It appears that the solution obtained on these test problems achieved best problem solutions at C = 0, 5 and 10 for both algorithms, and that solution values increased when the value of C exceeded the range of [O, IO]. Therefore, we limited our testing of the two new algorithms to C = 0, 5 and IO. We first ran through the 23 problems found in the literature that have been tested by DeArmon [93, Golden et al. [6] and Pearn [7]. The sizes of the 23 problems range from ‘7to 27 nodes and from I1 to 55 arcs. Many of these problems are dense networks with small arc demands. Table 2 presents a comparison between the two new algorithms and the live existing solution procedures
194
WEN
LEA
PEARN
Table 3. Solution values generated by the two new algorithms using C = 0. 5 and IO
L * Indicates best solution value.
on the 23 problems. The performance of the two new algorithms (Augment-Insert Algorithm I and Augment-Insert Algorithm II) on the 23 problems is considered better than the Construct-Strike algorithm, comparable to the Path-Scanning algorithm and is worse than the other three algorithms. The unsatisfactory performance of the new algorithms on the 23 problems, as we mentioned earlier, is because of the characteristics of the networks-dense with small arc demands. For the purpose of testing the two new algorithms, we generated an additional four sets of test problems. These problems are all sparse networks with some large arc demand. Their sizes range from 13 nodes, 23 arcs to 27 nodes, 51 arcs with density ranging from 15 to 30%. These problems are described in the following: 0 Set A: 0 Set B: l Set C: 0 Set D:
six problems; n = 13, m = 23; density z 30%. eight problems; n = 18, m = 30; density ~20%. six problems; n = 22, m = 45; density 5 20%. ten problems; n = 27, m = 51; density z 15%.
Problem generation
These problems were generated from the problems found in the open literature 16, 7, 93 with some modifications, i.e. by arbitrarily: (1) removing demand arcs from the original network (so
Augment-Insert Table 4. Solution
algorithms
values generated
195
for the CARP
by the five existing
algorithms
~
A3
719
611
793
674
589
422
A4
613
560
714
541
530
413
1
634
1
648
1
635
)
646
1
337 1
B8
1
-
Cl
1
889
1
686
1
683
1
715
1
680
1
453
C2
1
807
1
630
1
658
1
643
1
644
1
478 1
C3
1
-
1
686
1
689
1
685
1
659
1
454 1
I
816
I
684
I
686
I
701
I
668
I
450 I
1
719
620
1
449
c4 C5 C6
I
I
I
1
I
1
I
I
640
1 I
I
Dl
I
638 592
1 I
I
I
617
1 I
I
I
1
634
1 I
I
622
641
569
440
444
419
402
432
279
D2
-
447
401
436
428
280
D3
-
519
501
489
479
283
D4
1
-
1
415
1
370
1
367
1
380
1
280 1
D5
1
-
1
5%
1
551
1
598
1
562
1
371 1
D6
1
-
I
426
1
436
1
-
1
412
1
287 I
D7
1
-
1
452
1
413
1
-
1
425
1
280 (
I
I
D8
I
D9I
I
I
DlO
438
-
I
1 1
I
484 I
I
I
484 I
286
482
342
I
485 I
422 I
465
463 1
530
I
443
402 I
I
-
I
486 I
285 I
I
that the density is no greater than 30%); (2) changing depot nodes; (3) changing arc cost; (4) changing arc demand; and (5) reducing the vehicle capacity W, so that the problems generated are significantly different from the original networks. Many of the demand arcs in the new networks are “far away from the depot”, and, therefore, the vehicles must travel along many other arcs before returning to the depot node. Table 3 displays the solution values of the four sets of test problems generated by the two new algorithms using C = 0,s and 10. For example, if we run Augment-Insert Algorithm I on problem 1 with C = 0, 5 and 10, then three solution values-484, 474 and 474- will be generated. Table 4 displays solution values of the four sets of test problems generated by the other five solution procedures. The performance comparisons of the two new algorithms using C = 0, 5 and IO, and the other five algorithms in terms of the average percentage above the problem’s lower bound, the
196
WEN LEA PEARN Table 5. C = 0: performance comparisons of the seven algorithms
56.78
s1.32
130.9
94.8
1
101.5
91.7
1
77.9
1
863
Table 6. C = 5: performance comparisons of the seven algorithms
~
71.95
1 57.91 1 56.78 1 51.32
bwaBotmd Numbaof FmblanaBeDBivin8 Be!uttiam
WmtSOlU!iOOin T-of%Abow TbsLowcrBound
0
1
1
1
13
2
14
I 12lX2 j
101.5 /
130.9 1
94.8
91.1
1
74.4
1
74.s
Table 7. C = 10: performance comparisons of the seven algorithms
LowaBowid Number of
Pmblan#Rec&iq Bat Sdutiav
0
wart solutial in TmnrofZAbove TlwLowa Bound
120.2
I
2
I
11
I 101.5
130.9
94.8
91.7
14 I
74.4
79.5
number of problems receiving the best solution values, and the worst solution in terms of percentage above the lower bound, are displayed in Table 5 (C = 0), Table 6 (C = 5) and Table 7 (C = IO). The performance of the two new algorithms is reasonably good in that: (I) in most (over 80%) of the problems tested, the new algorithms received best problem solutions among all algorithms we examined regardless of which parameter value we chose (in particular, C = 5); (2) the new algorithms outperformed the other five procedures by roughly 26, 12, 11, 12 and 5% of the problem lower bound; (3) the worst solutions in terms of percentage above the problem lower bound obtained by the new algorithms were under 80% (comparing this with that of 91-120% by the other five procedures, the improvement is significant). Further improvement was attempted by taking the best of the solutions generated by the two algorithms using C = 0, 5 and IO. The results of the comparisons are displayed in Table 8. Unfortunately, the improvement is insignificant considering the additional amount of computer time it takes. Problem lower bound
The lower bound used in this paper, for assessing the deviation of the heuristic solutions from optimality, is obtained by using both the Matching Lower Bound [l] and the Node Scanning
Augment-Insert
algorithms for the CARP
Table 8. C = 0, 5 and 10: performance comparisons of the seven algorithms
Table 9. Run time comparisons of the seven algorithms
DIO
27
51
-
10.5
54.9
996.4
40.0
42.3
Lower Bound [2]. These two lower bounding procedures have been shown to perform poorly for many instances of problems, in particular, for problems with sparse network and large arc demand. For the four sets of problems we tested, the networks are much more perverse than those tested by DeArmon [9], Golden et al. [6] and Pearn [7]. It might be that which caused such a large deviation of the heuristic solutions from the problem’s lower bounds. Computer
run time
In our testing, the run times for problems with the same size and vehicle capacity are very much the same. However, run times are quite different for problems with the same size but different vehicle capacity. In fact, our testing shows that the run time increases when the vehicle capacity decreases. This is because the vehicle capacity W determines the minimal number of cycles required for the solution and hence determines the number of program iterations. Therefore, instead of showing the average run time, we only display eight test problems (two from each set) in CPU set on the CDC CYBER 720 system (Table 9) as an indication of the relative efficiency of the seven algorithms. Since the solutions generated by the Path-Scanning algorithm, Random Path-Scanning algorithm and Augment-Merge algorithm were obtained by selecting the best among a set of solutions, the run times for these algorithms were calculated by totaling each single execution time. In the Modified Construct-Strike algorithm, however, the first single solution was taken as the solution from this approach. Therefore, only one execution was counted. The four sets of problems we tested are all sparse networks with some large arc demand (or with relatively small vehicle capacity). Each vehicle can only service a small number of demand arcs while it has to travel along many other arcs before returning to the depot. This generates a long cycle. Therefore, the procedures must be repeated as many times as are required to cover all demand arcs. Consequently, it results in taking a great amount of computer time to obtain a complete
198
WEN LEA PEARN
solution. In particular, the Construct-Strike algorithm and the Modified Construct-Strike algorithm involve the Minimal Cost l-Matching applications, which encountered more than 16 odd-degree nodes when the size of the network went up to 27 nodes, 51 arcs; and the matching procedure has to be iteratively repeated so many times that each single execution of the program took longer than 500 CPU sec. It should be noted that both new algorithms are faster than all the existing algorithms (except the Path-Scanning algorithm); in particular, the two new algorithms are IO-20 times faster than the Construct-Strike algorithm, the Modified Construct-Strike algorithm and the Augment-Merge algorithm while the new algorithms outperform all the other live procedures. 5. CONCLUSIONS
In this paper, we have introduced two fast algorithms using tour-augment and insertion ideas. They are very straightforward and are easy to implement. We have tested them on many problems with sparse network and large arc demands, and compared them with live other procedures. The computational results have shown that: (1) both of the two new algorithms significantly outperform the other five algorithms by roughly 26, 12, 11, 12 and 5% of the problem lower bound for all parameter values (in particular, C = 5) we tested; (2) the worst solutions in terms of percentage above the problem’s lower bound obtained by the two new algorithms were all under 80%, as opposed to 91-120% by the other five existing methods; (3) the number of best solutions obtained by the two new algorithms was roughly equal to 14 (46%) and 12 (41%), respectively, as opposed to 1 or 2 (c 7%) by the other five algorithms; (4) the two new algorithms are lo-20 times faster than most existing procedures. While the characteristics of the network may well influence the performance of certain algorithms, we have limited our testing to problems where the underlying networks-are sparse (density is no greater than 30%) with some large arc demands. This is certainly the case in many real-world applications when the operating vehicles have relatively small capacity. We have made no claims that the new algorithms presented in this paper always do better than the other five algorithms on problems with such characteristics. However, we do believe that the relative performance of these seven algorithms on the sample problems in this paper, which were arbitrarily generated from existing test problems (with some modifications) found in the literature, is indicative of their relative performance on the CARP, with such characteristics (sparse network with some large demands), and the new algorithms should be applied along with other existing solution procedures to obtain near-optimal solutions. Acknowledgement-The author would like to thank the anonymous referees for their careful reading of the paper and several valuable suggestions for improvements. REFERENCES 1. B. Golden and R. Wong, Capacitated arc routing problems. Networks 11(3), 305-318 (1981). 2. A. Assad, W. Pearn and B. Golden, Capacitated Chinese postman problem: lower bounds and solvable cases. Am. J. math. Mgmt Sci. 7( I-2), 63-88 (1987). 3. W. Pearn, A. Assad and B. Golden, Transforming arc routing into node routing problem. Computers Ops Res. 14(4). 285-288 (1987). 4. N. Christofides, The optimal traversal of a graph. Omega 1, 719-732 (1973). 5. H. Stem and M. Dror, Routing electric meter readers. Computers Ops Res. 6, 209-223 (1979). 6. B. Golden, J. DeArmon and E. Baker, Computational experiments with algorithms for a class of routing problems. Compurers Ops Res. 10, 47-69 (1983). 7. W. Pearn, Approximate solutions for the capacitated arc routing problem. Compurers Ops Res. 16, 589-600 (1989). 8. L. Chapleau, J. Ferland, G. Lapalme and J. Rousseau, A parallel insert method for the capacitated arc routing problem. Ops Res. Lerr. 3, 95-99 (1984). 9. J. DeArmon, A comparison of heuristics for the capacitated Chinese postman problem. Master’s Thesis, Univ. of Maryland at College Park (1981). IO. C. Clarke and J. White, Scheduling of vehicles from a central depot to a number of delivery points. Ops Res. 12(4), 568-58 I (1964). Il. S. Roy and J. Rousseau, The capacitated Canadian postman problem. INFOR 27(l), 58-73 (1989).