eulerian location problems

Eulerian Location Problems

Gianpaolo Ghiani, Gilbert Laporte

Centre de recherche sur les transports, Universite de Montreal, C.P. 6128,succursale “Centre-ville”, Montreal, Canada H3C 3J7

Received 20 March 1998; accepted 11 May 1999

Abstract: The problem of locating a set of depots in an arc routing context (with no side constraints) isaddressed. In the case of one depot, it is shown that the problem can be transformed into a Rural PostmanProblem (RPP). In the case of a set of depots, the problem is also reduced to an RPP if there are nobounds on the number of depots to be opened or to a RPP relaxation otherwise. The problem is thensolved to optimality using a branch-and-cut algorithm. Extensive computational results on real-world andon some randomly generated test networks are reported. © 1999 John Wiley & Sons, Inc. Networks 34:291–302, 1999

Keywords: arc routing; rural postman problem; location routing

1. INTRODUCTION

Eulerian location problems consist of simultaneously deter-mining facility sites, or depots, and routes in a graph toserve a specified set of required edges under given opera-tional constraints. Such problems have a wide variety ofapplications in mail and newspaper delivery, garbage col-lection, road gritting, and school bus routing where facilitiesmay be vehicle depots, relay boxes, transfer points, dumpsites, or replenishment points. An extensive literature existson combined node routing and location problems (see, e.g.,Laporte [10]), but there is hardly anything on combined arcrouting and location problems. Our aim was to study someEulerian location problems without side constraints on anundirected graph structure. We call these problemsLocationRural Postman Problems(LRPPs).

The remainder of this paper is organized as follows: TheLRPP is formulated in Section 2. The LRPP with no upperbound on the number of depots is treated in Section 3, whilethe bounded case is analyzed in Section 4. We show that ifthe number of depots is unbounded or equal to 1 then thecorresponding LRPP reduces to a standardRural PostmanProblem (RPP). Otherwise, it is an RPP relaxation. InSection 5, we outline a branch-and-cut algorithm for solvingthese problems. Computational results provided in Section 6confirm the efficiency of the proposed solution methodolo-gies.

2. FORMULATION

The LRPP can be stated as follows: LetG(V, E) be anundirected graph, whereV 5 { v1, . . . , vn} is the vertexset,E is the edge set,ce 5 cij ($ 0) is the cost or lengthof edgee 5 (v i, v j) [ E, R # E is the set of requirededges, that is, the edges that must be serviced, andVR is theset of vertices ofV incident to an edge ofR. Let D # V bea nonempty set of potential depot sites, and letfi($ 0) be

Correspondence to:G. LaporteContract grant sponsor: Ministero dell’ Universita´ e della Ricerca

Scientifica e Tecnologica (MURST)Contract grant sponsor: Canadian Natural Sciences and Engineering

Research Council (NSERC); contract grant number: OGP0039682

© 1999 John Wiley & Sons, Inc. CCC 0028-3045/99/040291-12

291

the operating cost of depotv i [ D. The LRPP is todetermine a set of depots inD and a route for each depot insuch a way the sum of operating and routing costs to serveall edges ofR is minimized. As is commonly done inlocation theory, fixed and operating costs are scaled so thatthey apply to the same planning horizon.

The setR of required edges inducesp subgraphs ofGcalledconnected components.Let Ch(h 5 1, . . . ,p) be theh-th connected component, and letVh be the vertex set ofCh. Denote byDR 5 D ù VR the set of potential depotsthat belong to a connected component, and byDN 5 D\DR,the set of all potential depots that are not incident toR.

A first simplification of the problem can be obtainedthrough a simple reduction rule based on the followingobservation:

Proposition 1. There exists an optimal solution to theLRPP in which all edges belonging to the same componentare on the same route.

Proof. If the edges of a component belong to at least tworoutes, then there is a vertexv in that component at whichtwo routes meet. Using an end-pairing technique [4], thetwo routes (. . . ,vi, v, vj, . . .) and (. . . ,vk, v, v,, . . .) canbe coalesced into a single route (. . . ,v i, v, vk, . . . , v,, v,v j, . . .) having the same cost. ■

It follows that there exists an optimal solution in whichall edges of any given component are served from the samedepot. Therefore, one can eliminate from every componentall depots but the cheapest one. Hence, there exists anoptimal solution containing at most min{uDu, p} depots.

Reduction 1. For eachh [ {1, . . . , p}, drop from D allvertices ofVh ù D except the least expensive one.

After this reduction, letgh be the operating cost of thepotential depot inCh, if any exists. We also transformGfollowing the rules proposed by Christofides et al. [1] for theRPP. In the resulting graph, the only vertices appearing arethose ofVR, those ofDN, and copies of the vertices ofDR.

Transformation 1.

STEP 1. Use Reduction 1 to remove fromDR as manypotential depots as possible.

STEP 2. Add to GR(VR, R) an edge between every pair ofvertices inGR, with a cost equal to the shortest pathlength.

STEP 3. Reintroduce the vertices ofDN. Create one dummycopy of each vertex ofDR and callD9R the set of suchcopies. Connect every dummy vertexv [ D9R to thecorresponding vertex inVR with a zero-cost edge. Also,connect every vertexv of DN to all verticesv i of com-

ponentCh(h 5 1, . . . , p) if there is no potential depotwithin Ch or if Ch has its own potential depot andgh

. fv (this is a direct consequence of Proposition 1). Thecost of each such edge is that of a least-cost path betweenv andv i. Let D9 5 D9R ø DN.

STEP 4. Eliminate from the augmented network the nonre-quired edges for which a parallel required edge havingthe same cost exists and the nonrequired edgese 5 (vi,v j) for which the edge costcij is equal tocik 1 ckj forsome vertexvk [ VR.

It is worth observing at this stage that only the potentialdepots ofDR are duplicated. Indeed, early in Step 2, onlyGR (and, therefore, onlyDR) is considered. The depotsubsetDN is reintroduced only later in Step 3, together withcopies of the potential depots ofDR.

To illustrate Transformation 1, consider the graph de-picted in Figure 1. There are two connected components ofrequired edges,C1 and C2, and three potential depots be-longing to two sets:DR 5 {2}, DN 5 {8, 9}, where thefixed costs aref2 5 4, f8 5 3 and f9 5 5. In Step 1, nopotential depot can be removed. In Step 2, a full graph isconstructed with the vertices ofGR. The two original po-tential depots 8 and 9 are reintroduced in Step 3, togetherwith the dummy depot 29. These depots are linked toGR inStep 3: The edge (2, 29) has a cost of 0 (note that there isonly one edge incident to 29); vertex 8 is connected to thevertices ofC1 andC2; and vertex 9 is connected to everyvertex ofC2 but not to those ofC1 sinceC1 contains a lessexpensive depot.

Proposition 2. There is at least one optimal solution to theLRPP in which the degree of a depotv [ D9 is 2, if thedepot is selected, and 0, otherwise.

Proof. The thesis follows from the observation that if thesolution contains the vertex subsequence (. . . ,v i, vk,v j, . . .), where v i, vj [ VR and vk [ D9, then thissubsequence can be replaced with (. . . ,v i, vj, . . .) withoutdeteriorating the objective function, as long as the requirednumber of depots appear elsewhere in the solution, sinceedge costs represent shortest path lengths and thereforesatisfy the triangle inequality. ■

In what follows, a vertexvi [ VR is said to beR-odd(R-even) if and only if an odd (even) number of requirededges are incident to it. Define integer variablesxe repre-senting the number of times edgee [ E is traversed in asolution and binary variablesyv equal to 1 if and only ifdepot v [ D9 is selected. (Because of the cost structureintroduced in the transformation, if a dummy depotv [ D9Ris selected, this amounts to using the corresponding depot ofDR.) For S # V, let d(S) 5 {( vi, v j) [ E : vi [ S, vj ¸S or v j [ S, vi ¸ S}. Denote d({ v}) by d(v). The basic

292 GHIANI AND LAPORTE

LRPP can then be formulated as an integer problem asfollows:

~LRPP! minimize Oe[E

cexe 1 Ov[D9

fvyv ,

subject to

Oe[d~v!

xe 5 0 mod~2! ~v [ VR is R-even! (1)

Oe[d~v!

xe 5 1 mod~2! ~v [ VR is R-odd! (2)

Fig. 1. Graph transformation.

EULERIAN LOCATION PROBLEMS 293

Oe[d~v!

xe 5 2yv ~v [ D9! (3)

Ov[D9

yv $ 1 (4)

Oe[d~S!

xe $ 2~1 2 Ov[W

yv!~S5 ~ ø i[PVi!

ø W, A Þ P , $1, . . . , p%, W # D9!(5)

xe $ 0 and integer~e [ E! (6)

yv 5 0 or 1 ~v [ D9!. (7)

In (LRPP), constraints (1) and (2) are the usual degreeconstraints. Constraints (3) state that any potential depot ofD9 has a degree of 0 or 2 in the optimal solution. In the latter

case, the vertex of the original graph to which this dummydepot corresponds is effectively chosen as a depot. Con-straint (4) forces at least one depot to be selected. Con-straints (5) can be explained as follows: Consider a setSconsisting of all vertices of some connected componentsand possibly some depots ofD9. If none of these depotsbelongs to the solution, then the right-hand side of (5) isequal to 2, forcingS to be connected to some other part ofthe graph (in such a case, (5) reduces to the connectivityconstraints in the formulation proposed by Corbera´n andSanchis, [2]); otherwise, constraints (5) are redundant. Forexample, using the example of Figure 1 withS 5 {1, 2, 3,4} ø {2 9}, constraint (5) reads

x3,8 1 x4,8 1 x1,5 1 x4,6 $ 2~1 2 y29!. (59)

If S 5 {1, 2, 3, 4}, then constraint (5) reduces to

Fig. 1. (Continued from the previous page)


x2,29 1 x3,8 1 x4,8 1 x1,5 1 x4,6 $ 2, (50)

sinceS does not contain any potential depot.If an upper boundr on the number of depots to be opened

exists, the following constraint must be added to the model:

Ov[D9

yv # r . (8)

In particular, if only one depot is to be located, constraints(4) and (8) can be merged into the equality

Ov[D9

yv 5 1. (9)

3. THE LRPP WITH NO BOUND ON THENUMBER OF DEPOTS

If the number of depots to be located is not subject to anupper-bound constraint, the LRPP can be transformed intoan RPP as follows:

Transformation 2. Arbitrarily create a loop of requirededges linking the vertices ofD9. Assign to each of theseedges a very large costM. This loop constitutes a newconnected component ofR. Denote byGD the resultinggraph.

Proposition 3. An optimal solution to the RPP onGD isalso optimal for the LRPP onG.

Proof. Due to the connectivity requirements and theattachment of theM coefficients to the loop edges, an RPPoptimal solution onGD is such that (a) there is exactly onecopy of each loop edge and (b) the nonrequired edgesincident to each potential depot are 2, if the depot is se-

lected, or 0 otherwise. Every RPP solution satisfying con-ditions (a) and (b) corresponds to a feasible LRPP solutionand vice versa. Furthermore, an LRPP solution has the costof the corresponding RPP solution, minus the constantMuDu. ■

The idea of creating a loop of potential depots andlinking it to customer vertices by large cost edges was alsoused by Goemans and Williamson [8] in a vertex–location–routing context.

Fig. 2. Transformation of the multiple-depot case with anupper bound on the number of depots. Nos. within thecircles correspond to potential depot indices.

Fig. 3. Three possible solutions to the RPP relaxation aris-ing when r 5 3. The selected depots are depicted by blackvertices.


4. THE LRPP WITH A BOUND ON THENUMBER OF DEPOTS

If an upper boundr on the number of depots to be locatedis given, the LRPP can be transformed into a suitablerelaxation of the RPP by creatingr loops of required edgeswith large positive costs. These loops are then linked to oneanother and toVR with appropriate edges.

Transformation 3.

STEP 1. Creater 2 1 additional copiesuis (s 5 2, . . . , r )

of each potential depotui1 [ D9R. For s 5 2, . . . , r ,

creater 2 1 nonrequired zero-cost edges (uis, u# i), where

u# i is the vertex ofVR corresponding to the dummy depotui1.

STEP 2. Creater 2 1 additional copieswis (s 5 2, . . . , r )

of each potential depotwi1 [ DN. For each edge (wi

1,v j)(vj [ VR), creater 2 1 edges (wi

s, v j)(s 5 2, . . . ,r ) having the same cost as that of (wi

1, v j).

STEP 3. Creater loops of uD9u required edges each, con-necting the verticesui

s andwis with the same upper index

s, by means of edges having cost equal toM. Theseloops make upr additional components called the “depotcomponents”C,(, 5 p 1 1, . . . , p 1 r ).

STEP 4. Increase byM the cost of every edge (v i, vk)incident to a “loop vertex”vk(vi [ VR).

STEP 5. Link together ther copies of the same depot (ver-ticesui

s andwis with the same lower indexi ) by means of

a loop of nonrequired edges having a cost equal to 2M.

Let GD be the graph resulting from this transformation(Fig. 2). In the RPP relaxation, we allow a setS of compo-nents to be disconnected from the remaining part of thegraph if S contains at least both an “original” componentCh(1 # h # p) and a depot componentCh( p 1 1 # h # p1 r ). This means that, in the RPP formulation by Christ-ofides et al. [1], instead of

Oe[d~S!

xe $ 2 ~S5 ø i[PVi, A Þ P , $1, . . . ,p 1 r%!,

(10)

the following inequalities are imposed:

Oe[d~S!

xe $ 2 ~S5 ø i[PVi, A Þ P , $1, . . . , p 1 r %,

Sù ~ ø i[$1,· · ·,p%Vi! 5 A or Sù ~ ø i[$ p11,· · ·,p1r%Vi! 5 A!.

(11)

Proposition 4. An optimal solution to the RPP relaxationon GD is an optimal solution to the LRPP on G.

Proof. In every feasible solution of the RPP relaxation,at least two edges are incident to each depot loopCh(h[ { p 1 1, . . . , p 1 r }). Since these edges have largecosts, an optimal solution will contain exactly two suchedges for each depot loop. Let {D1, D2, . . . , Dq} be amaximal partition of {1, 2, . . . ,r } such that the depotloops inDj( j 5 1, . . . , q) are linked to one another in anoptimal solution. Loops inDj are linked as follows: (a) toone another by means ofuDj u 2 1 edges (having cost equalto 2M) joining verticesv i

s with the same lower indexi ; (b)to two verticesvs, vt [ VR by means of two edges havingcost equal toM 1 ( fi/ 2) 1 cis and M 1 ( fi/ 2) 1 cit,respectively.

Hence, the optimal solution of the RPP relaxation can betranslated into a feasible LRPP solution by substituting eachsetDj with the corresponding depotvi. In such a way, theappropriate cost is yielded because the cost difference be-tween the two solutions is equal to the cost of the requirededges ofCh(h [ { p 1 1, . . . , p 1 r }) plus ¥i51

q

[2M(uDi u 2 1) 1 2M] 5 2M ¥i51q uDi u 5 2Mr . Vice

versa, every feasible LRPP solution can be translated into afeasible solution of the RPP relaxation satisfying conditions(a) and (b). In Figure 3, we illustrate the case that can occurfor r 5 3. ■

Observe that a further simplification can be obtained inthe case wherer 5 1. Before applying Transformation 3,the following reductions can be applied wheneverDR Þ A:

Reduction 2. Eliminate all potential depots inDR, exceptthe least expensive one, denoted byw.

Reduction 3. Eliminate fromDN all potential depots hav-ing a cost at least equal to that ofw.

5. BRANCH-AND-CUT ALGORITHM

The transformations presented in the previous sections al-low us to reformulate the LRPP to optimality as an RPP or

TABLE I. Albaida1 graph, single-depot case

Succ Root LB/z* Nodes Connect Cocirc Depots Seconds

10 10 1 1 11.1 154.6 1 8.7


as an RPP relaxation. We use a formulation containing edgevariablesxe partitioned into two sets. The first set,E012,contains variables taking the values 0, 1 or 2 in an optimalsolution. The second set,E01, contains binary edge vari-ables. Ghiani and Laporte [7] showed thatE012 can be takenas the edge set of a minimum spanning tree on an auxiliarygraphGC containing a vertex for each connected compo-nent and edges (i , j ) of cost equal to the length of aleast-cost path between componentsCi andCj. Each edgeof E012 is then replaced by two edges to which are associ-ated binary variables. Denote byE# the edge set resultingfrom this transformation. The RPP formulation is then

~RPP! minimize Oe[E#

cexe,

subject to

Oe[d~v!\F

xe $ Oe[F

xe 2 uFu 1 1 (12)

~v [ V, F # d~v!, uFu is odd if v is R-even,

uFu is even ifv is R-odd!

Oe[d~S!

xe $ 2SS5 øi[P

Vi, P , $1, . . . ,p%, P Þ AD ~13!

xe 5 0 or 1 ~e [ E# !. (14)

In this formulation, constraints (12) are called cocircuitinequalities. They state that an even (odd) number of edgesare incident to eachR-even (R-odd) vertexv, while con-straints (13) are the usual connectivity constraints. Thisformulation can be used directly to solve the LRPP with anunbounded number of depots or with only one depot. If thenumber of depots is bounded above by a number differentfrom one, then constraints (13) are replaced by the weakerconstraint set (11). Both the RPP or is relaxation obtainedby the use of constraints (11) can be solved by the followingbranch-and-cut procedure.

5.1. Branch-and-Cut Procedure

STEP 1. (Upper bound). Compute an upper boundz# on theoptimal solution valuez*.

TABLE II. Albaida1 graph, multidepot case with no bound on the number of depots

fv Succ Root LB/z* Nodes Connect Cocirc DepotsFixedcost

RoutingCost

Totalcost Seconds

f 5 0 10 9 0.999 1.4 11.7 165.4 1.8 0 10,328 10,328 9.7

f 5 2cmin 10 5 0.999 3.6 11.9 225.3 1.8 115 10,328 10,332 10.2

f 5 4cmin 10 8 0.999 1.8 11.5 172.3 1.6 205 10,352 10,557 9.3

f 5 8cmin 10 8 0.999 1.8 11.6 173.1 1.5 384 10,377 10,761 9.1

f 5 16cmin 10 6 0.996 10.1 11.3 246.6 1 512 10,599 11,111 12.4

TABLE III. Randomly generated graphs, single-depot case

uVu p Comp Succ Root LB/z* Connect Cocirc Seconds

50 0.3 8.4 5 5 1 11.8 33.6 0.6

0.5 8 5 3 0.997 12.2 72.4 1

0.7 6.6 5 3 0.992 11.4 128.4 1.4

100 0.3 19 5 3 0.995 26.6 164.4 4.6

0.5 14.8 5 3 0.997 19.8 151.2 6.2

0.7 6 5 4 0.998 11 234.2 12.2

150 0.3 29 5 1 0.997 42.2 257.8 14

0.5 19.6 5 2 0.998 25.4 296.8 31.4

0.7 8.6 5 1 0.998 16.6 472.6 60.8

200 0.3 38 5 2 0.995 91.8 2021.4 204.2

0.5 22 5 3 0.998 29.8 882.6 171.6

0.7 9.4 5 2 0.999 12.4 628.6 185.6


STEP 2. (Graph transformation). Construct the transformedgraphGD as explained in Sections 3 and 4.

STEP 3. (First node of the search tree). Define a first sub-problem by the linear program containing a connectivityinequality for each single component and a cocircuitinequality withF 5 A for eachR-odd vertex. Insert thisproblem into a list.

STEP 4. (Termination check). If the list is empty, stop.Otherwise, select a subproblem from the list according tothe best lower-bound strategy.

STEP 5. (Subproblem solution). Solve the subproblem usingCPLEX [3]; let z be the solution value. Ifz $ z#, go toStep 4.

STEP 6. (Constraint elimination). Among all constraints,eliminate those that have been ineffective for 20 execu-tions of Step 5.

STEP 7. (Constraint generation). Identify up to 60 violatedinequalities, starting with connectivity constraints andthen cocircuit inequalities. If no inequality can be gen-erated, go to Step 8. Otherwise, generate the most vio-lated inequalities, up to 40, and go to Step 5.

STEP 8. (Branching). Branch on the variablexe nearest to0.5; if the variable corresponds to an edgee [ E01, twoson subproblems are generated settingxe 5 0 and xe

5 1. If the variable corresponds to an edgee [ E012,three son subproblems are generated settingxe9 5 0 and

TABLE IV. Randomly generated graphs, multidepot case with no bound on the number of depots (fv 5 0)

uVu p Comp Connect Succ Root LB/z* Nodes Connect Cocirc Depots Seconds

50 0.3 8.4 12 5 4 0.999 1.8 12 48.4 4.2 0.4

0.5 8 11.4 5 5 1 1 11.4 62 4.8

0.7 6.6 9.6 5 4 0.999 1.8 9.6 88.2 4.4

100 0.3 19 26.2 5 5 1 1 26.2 130.4 4.8 3.6

0.5 14.8 21.2 5 5 1 1 21.2 136.6 4.8 5.8

0.7 8.4 8.4 5 5 1 1 8.4 126 2.4 6.4

150 0.3 29 37.8 5 4 0.999 1.8 37.8 234.2 9.6 16.4

0.5 19.6 24.8 5 4 0.999 1.4 24.8 200.2 6.4 31.8

0.7 8.6 11.6 5 3 1 2.2 11.6 316.6 4 48.8

200 0.3 38 47.6 5 2 0.999 4.2 47.6 335.6 9.4 56.4

0.5 22 28 5 2 0.999 4.2 28 428.8 7 126.2

0.7 9.4 13.8 5 2 0.999 3.8 13.8 435.6 4 149.8

TABLE V. Randomly generated graphs, multidepot case with no bound on the number of depots (fv 5 2cmin)

uVu p Comp Succ Root LB/z* Nodes Connect Cocirc DepotsFixedCost

RoutingCost

TotalCost Seconds

50 0.3 8.4 5 2 0.997 3.4 12.4 63.2 2.8 217 5186 5403 0.6

0.5 8 5 1 0.997 6.2 13 120 3.4 261 7431 7692 1.2

0.7 6.6 5 1 0.986 33 12.6 226.2 2.2 249 9574 9823 3.6

100 0.3 19 5 0 0.994 19.8 30.4 240.6 3.2 149 8660 8809 7.2

0.5 14.8 5 2 0.999 3.4 21.6 237 3.6 102 10,908 11,010 7.2

0.7 5.5 4 2 0.998 4 8.2 253.5 1.3 112 14,461 14,573 8.3

150 0.3 29 5 1 0.997 24.2 45.8 498.4 8.4 182 10,107 10,289 34.6

0.5 19.6 5 2 0.998 9.8 25.8 317.6 5.8 130 13,613 13,743 39.4

0.7 8.6 5 1 0.998 31 16 538.6 2.6 109 18,831 18,940 83.4

200 0.3 38 5 0 0.997 24.2 51.6 457.8 8.2 179 11,473 11,652 71.2

0.5 22 5 0 0.997 27.8 32.2 896.4 6.2 123 16,603 16,726 245.4

0.7 9.4 5 0 0.997 52.2 16.8 868.8 2.8 111 21,170 21,281 236.8


xe0 5 0, xe9 5 1 andxe0 5 0, xe9 5 1 andxe0 5 1(the subproblemxe9 5 0 andxe0 5 1 can be obviouslyomitted). Insert the subproblems into a list and go toStep 4.

5.2. Additional Computational Aspects

In the LRPP with a single depot and in the LRPP withoutbound, the feasible solution required in Step 1 can becomputed by applying the procedure outlined by Frederick-son [6] toGD. Furthermore, in these two cases, the separa-tion problems associated with connectivity and cocircuitinequalities can be solved as in Ghiani and Laporte [7].

In the LRPP with a bound on the number of depots, threeaspects need to be fully described:

Determination of the 0/1/2 Edges

Construct an auxiliary graphGc as above. Introduce inGc asingle additional vertexv9d representing a depot loop. Forevery original componentCi, insert edgee 5 (v9d, v9i)corresponding to the least cost edge inGD between a depotloop andCi. Then, as shown in [7], the 0/1/2 edges belongto the minimum spanning tree onGC [if edge (v9d, v9i)belongs to the minimum spanning tree, ther copies of thisedge that connect ther depot loops toCi in GD are 0/1/2].

TABLE VI. Randomly generated graphs, multidepot case with no bound on the number of depots (fv 5 4cmin)


RoutingCost

TotalCost Seconds

50 0.3 8.4 5 2 0.991 5.8 13 62.6 2.8 434 5154 5588 0.8

0.5 8 5 0 0.988 13.4 13.8 148.8 2.6 330 7574 7904 1.8

0.7 6.6 5 0 0.987 18.6 12.6 257.6 1.2 314 9674 9988 5

100 0.3 19 5 0 0.987 202.6 62 897.4 2.6 240 8709 8949 85.6

0.5 14.8 5 0 0.996 13 22.6 381.6 3.6 204 10,908 11,113 12

0.7 6 5 0 0.998 5.8 10 228.4 1 199 14,531 14,730 9.6

150 0.3 29 5 0 0.994 69.4 50 626 6 230 10,197 10,427 102.2

0.5 19.6 5 0 0.996 21.8 28.8 362.4 5.8 260 13,613 13,873 43.8

0.7 8.6 5 0 0.996 105.4 29.8 808.4 2 170 18,863 19,034 136.8

200 0.3 38 5 0 0.994 163 74 983 6.8 281 11,531 11,813 337.4

0.5 22 4 0 0.997 21.6 31.3 756.3 5.3 149 17,241 17,390 185.3

0.7 9.4 5 0 0.997 27 17.2 982.8 1.8 125 21,235 21,361 249.4

TABLE VII. Randomly generated graphs, multidepot case with no bound on the number of depots (fv 5 8cmin)


RoutingCost

TotalCost Seconds

50 0.3 8.4 5 3 0.998 3 13 51.8 2.2 555 5336 5891 0.2

0.5 8 5 1 0.988 57.4 17.4 167 2 512 7709 8221 6.8

0.7 6.6 5 1 0.995 6.2 12.4 176 1.2 629 9593 10,223 1.8

100 0.3 18.75 4 1 0.991 16 27.75 283.2 2 330 8957 9287 7.5

0.5 14.8 5 0 0.993 13.8 22.2 363.6 2.8 242 11,025 11,267 10.4

0.7 6 5 4 0.998 5.8 9.8 216 1 399 14,531 14,930 10

150 0.3 29 5 0 0.992 353.4 107.4 2676.8 4.6 291 10,319 10,610 478.4

0.5 18.75 4 1 0.994 183.5 57.2 862.7 2.75 224 13,722 13,946 155.2

0.7 8.6 5 0 0.997 27 16 741.8 1.4 246 18,921 19,167 107.8

200 0.3 37.75 5 0 0.993 578.5 207 3776.5 4 276 12,123 12,399 1380

0.5 21.25 5 0 0.994 76.5 34.2 1283 4 242 17,090 17,332 454

0.7 9.4 5 0 0.998 32.6 18.8 1281.8 1.2 181 21,275 21,456 417.4


Heuristic

An upper boundz# on the optimal solution valuez* is deter-mined using a two-step procedure: First, a solution is com-puted as in the multidepot case without bounds. Second, if thenumber of depots in this solution is larger thanr, some depotsare dropped and the corresponding routes are merged. For thispurpose, a saving-based technique is employed.

Separation Problem Associated to ModifiedConnectivity Inequalities (11)

Although this separation problem is well known to be solvablein O(uVu3) time, we use the heuristic procedure proposed byFischetti et al. [5]. Using Prim’s [11] algorithm, a maximumspanning tree is determined on an auxiliary graphG# , whose

vertices v9h correspond to connected componentsCh, andwhose edgese9 5 (v9h, v9k) have a cost equal to the sum ofvariablesxe associated with edgese5 (vi, vj) such thatvi [ Ch

andvj [ Ck. At any stage of the construction of this tree, letSbe the set of connected components ofR just introduced. IfSyields a violated constraint (11), this constraint is generated.Once the spanning tree is complete, another check for violatedconnectivity constraints is made by removing, in turn, eachedge of the tree.

6. COMPUTATIONAL RESULTS

The algorithm described in Section 5 was coded inC, usingthe CPLEX 3.0 optimization routines [3] for the solution of

TABLE VIII. Randomly generated graphs, multidepot case with no bound on the number of depots (fv 5 16cmin)


RoutingCost

TotalCost Seconds

50 0.3 8.4 5 3 0.985 9.4 13.6 68.6 1.4 918 5495 6413 1

0.5 8 5 2 0.988 19 13.4 150.8 1.2 810 7849 8659 2.6

0.7 6.6 5 2 0.995 5.4 11.8 154.2 1.2 1258 9432 10,689 1.6

100 0.3 19 5 0 0.984 307.8 54.8 1498.8 1.4 506 8927 9433 164

0.5 14.8 5 0 0.992 29 23 482.2 2.2 378 11,085 11,463 25.8

0.7 6 5 4 0.998 5.8 9.8 216 1 797 14,531 15,328 10.4

150 0.3 29.5 4 0 0.992 73 51.5 817.5 3.5 328 10,329 10,657 113.5

0.5 19.6 5 1 0.993 68.2 35.4 1434 2 266 13,970 14,236 518.2

0.7 8.6 5 1 0.998 10.6 14.8 485.6 1 390 18,997 19,387 58.4

200 0.3 39 4 0 0.990 701 183.3 2963.6 3 448 11,847 12,295 2054.5

0.5 22.5 5 0 0.998 13 33.2 1000.4 1 448 16,832 17,280 290

0.7 9.4 5 0 0.997 16.6 15.2 819 1 333 21,300 21,633 238.8

TABLE IX. Randomly generated graphs, multidepot case with no bound on the number of depots (fv 5 32cmin)


RoutingCost

TotalCost Seconds

50 0.3 8.4 5 3 0.986 63.4 14.8 170.6 1.2 1594 5638 7232 6.6

0.5 8 5 2 0.989 19.0 13.6 151.2 1.2 1619 7849 9468 3.2

0.7 6.6 5 2 0.995 4.6 11.6 154.8 1 1869 9763 11,632 1.8

100 0.3 19 5 1 0.989 175.4 45.6 1209.8 1.2 896 8993 9889 127.6

0.5 14.8 5 1 0.991 49.8 25.8 299 1 544 11,254 11,798 18.2

0.7 6 5 4 0.998 5.8 9.8 216 1 1594 14,531 16,125 10.2

150 0.3 29 5 0 0.995 21.0 43.4 366.4 1.4 403 10,655 11,058 31.4

0.5 19.6 5 1 0.995 25.0 29.4 500.4 1.8 442 14,026 14,468 61.6

0.7 8.6 5 1 0.998 10.6 14.8 485.6 1 781 18,997 19,778 58.4

200 0.3 37.7 5 0 0.994 85.5 64.25 917.7 1.25 296 12,469 12,765 311.2

0.5 22.5 5 0 0.998 13.0 30.7 659 1 448 16,832 17,280 290

0.7 9.4 5 0 0.997 16.6 15.2 819 1 666 21,301 21,967 238.8


the linear programs. We ran all tests on a PC with 133 MHzPentium Processor and 32 Mbytes RAM.

Tests were performed on the Albaida1 network of Corb-eran and Sanchis [2] and on randomly generated instances.The Albaida1 network contains 113 vertices and 171 edges.As there appear to be some inconsistencies in this network,it is modified, as suggested by Hertz et al. [9], by includingthe two nonrequired edges (26, 29) and (29, 33), instead ofthe nonrequired edge (26, 33); also both edges (17, 21) and(25, 26) should be required. For the randomly generatedinstances, we generated type 3 graphs as in [9]. For this,uVuvertices are first generated in the unit square according to acontinuous uniform distribution, and alluVu(uVu 2 1)/ 2edges are defined. The costcij betweenv i and vj is theEuclidean distance multiplied by 10,000 and rounded up ordown to the closest integer. Then, starting from an arbitraryvertex, a first Hamiltonian cycle is randomly generated. Alledge costs of this tour are set to infinity and a second touris computed in the same way. LetE be the set of all edgeson the two tours. The unit square is then divided into fourequal smaller squares andR is defined as the set of all edgesof E entirely contained in a small square, with probabilityp.

This procedure ensures that the graph induced byR isdisconnected. In practice, most vertices of the graphG5 (V, E) have a degree of 4. In our tests, we generated inthis manner 12 sets of five graphs each, for all combinationsof uVu 5 50, 100, 150, 200, andp 5 0.3, 0.5, 0.7.

We then generated potential depots setsT for each ofthese graphs. For the Albaida1 graph, 10 different depot setswere generated. For each of the randomly generated graphs,one depot set was generated. For both types of graphs, wegenerateduTu 5 uVu/4 potential depots. Depot fixed costswere generated as follows:

Unbounded case.We successively set all potential depotfixed costs equal tofv 5 0, 2cmin, 4cmin, 8cmin, 16cmin,32cmin, and 64cmin, wherecmin 5 mini,j { cij }. This cre-ates a variety of situations ranging between the two extremecases where each component has its own depot and whereonly one depot is located in the entire graph.

Single-depot case.This is a particular case of the un-bounded case withfv 5 64cmin.

Bounded case.An upper boundr on the number ofdepots was determined as follows: We first solved the

TABLE X. Randomly generated graphs, multidepot case with no bound on the number of depots (fv 5 64cmin)


RoutingCost

TotalCost Seconds

50 0.3 8.4 5 3 0.987 65.4 14.6 156.2 1.2 3187 5545 8732 6.4

0.5 8 5 2 0.981 19.8 13.6 149.4 1 3085 7999 11,084 3

0.7 6.6 5 2 0.996 4.6 11.6 154.8 1 3738 9763 13,501 1.6

100 0.3 19 5 2 0.992 169.4 50.4 1341.8 1 1638 9098 10,736 149.4

0.5 14.8 5 1 0.994 13 21.2 226.8 1 1088 11,254 12,342 9.2

0.7 6 5 4 0.998 5.8 9.8 216 1 3187 14,531 17,718 10.2

150 0.3 29 5 0 0.993 35.4 45 509.6 1.4 806 10,655 11,461 80.6

0.5 19.6 5 1 0.993 33 30.6 753.4 1.6 858 14,046 14,904 112.8

0.7 8.6 5 1 0.998 10.6 14.8 485.6 1 1562 18,997 20,559 58.4

200 0.3 37.7 5 1 0.997 6.5 50.5 539.7 1 479 12,499 13,010 128

0.5 22 5 1 0.996 30.6 33.4 915.8 1.2 819 16,966 17,786 381

0.7 9.4 5 0 0.997 16.6 15.2 819 1 1331 21,300 22,631 238.8

TABLE XI. Randomly generated graphs, multidepot case with an upper bound on the number of depots (r 5 p*/2)

uVu p r Comp Succ Root LB/z* Nodes Connect Cocirc Depots Total cost Seconds

50 0.3 2.4 8.4 5 1 0.958 113.4 17.2 449.2 2.4 5411 57.8

0.5 2.8 7.3 5 0 0.976 266.5 17 743 2.5 7761 188.7

0.7 2.4 6.6 5 0 0.991 55.4 12.4 438.4 2 9561 12

100 0.3 2.3 18.8 5 0 0.989 41 27.5 521 2.3 8904 89.2

0.5 2 15 5 0 0.989 106 24.5 853.5 2 11,085 146.2

0.7 1.4 6 5 2 0.994 48.6 10.6 446 1.4 14,491 32.6


unbounded case withfv 5 0 and noted the numberp* ofdepots in the optimal solution. We then setr 5 p*/ 2 andr 5 p*/4. The fixed depots costs were set equal to 0.

Our computational results are summarized in TablesI–XII. Statistics are computed over the number of successfulinstances, that is, those that could be solved to optimalitywithin 10,000 seconds. The column headings are as follows:

Succ number of successful instances

Root number of instances solved to optimality atthe root of the search tree

LB/z* lower bound computed at the root of thesearch tree divided by the optimal solu-tion value

Nodes number of nodes in the search tree

Depots number of depots in the optimal solution

Fixed costs sum of thedepot costs in the optimal solution

Routing cost routing cost of the optimal solution

Total cost fixed cost plus routing cost

Seconds computation time in seconds

fv potential depot fixed costs

uVu number of vertices in the graph

p edge-selection probability in type 3 graphs

Comp number of connected components in the graph

Connect number of connectivity constraints gener-ated in the branch-and-cut process

Cocirc number of cocircuit inequalities generated inthe branch-and-cut process

Computational results over the various families of testproblems indicate that the algorithm generally succeeds insolving to optimality instances involving up to 200 verticeswithin relatively short computing times. Several instancesare solved at the root of the search tree. Most instancesderived from the Albaida1 graph are solved with little or nobranching within 10 seconds. Single-depot instances andmultidepot instances with no bounds on the number ofdepots are relatively easy to solve.

Problems with low fixed costs are generally easier. This

is reflected by a better LB/z* ratio, by a higher success rate,and by shorter computation times. Multidepot instanceswith an upper bound on the number of depots represent themost difficult class of problem. Here, the transformationintroduces a fair amount of degeneracy and the LB/z* ratiotends to be lower than in the unbounded case.

Thanks are due to an anonymous referee for several valuablecomments.

REFERENCES

[1] N. Christofides, V. Campos, A. Corbera´n, and E. Mota, Analgorithm for the rural postman problem, Imperial CollegeReport, IC.O.R.81,5, London, 1981.

[2] A. Corberan and J.M. Sanchis, A polyhedral approach to therural postman problem, Eur J Oper Res 79 (1994), 95–114.

[3] CPLEX Optimization Inc., using the CPLEX callable li-brary and CPLEX mixed integer library, Incline Village,NV, 1993.

[4] J. Edmonds and E.L. Johnson, Matching, Euler tours, andthe Chinese postman problem, Math Program 5 (1973),88–124.

[5] M. Fischetti, J.J. Salazar, and P. Toth, A branch-and-cutalgorithm for the symmetric generalized traveling salesmanproblem, Oper Res 45 (1997), 378–394.

[6] G.N. Frederickson, Approximation algorithms for somepostman problems, J ACM 26 (1979), 538–554.

[7] G. Ghiani and G. Laporte, A branch-and-cut algorithmfor the undirected rural postman problem, Math Program,forthcoming.

[8] M.X. Goemans and D.P. Williamson, Approximating min-imum-cost graph problems with spanning tree edges, OperRes Lett 16 (1994), 183–189.

[9] A. Hertz, G. Laporte, and P. Nanchen-Hugo, Improvementprocedures for the undirected rural postman problem,INFORMS J Comput 11 (1999), 53– 62.

[10] G. Laporte, “Location-routing problems,” Vehicle routing:Methods and studies, B.L. Golden and A.A. Assad, (Edi-tors), North-Holland, Amsterdam, 1988, pp. 163–198.

[11] R.C. Prim, Shortest connection networks and some gener-alizations, Bell Syst Tech J 36 (1957), 1389–1401.

TABLE XII. Randomly generated graphs, multidepot case with an upper bound on the number of depots (r 5 p*/4)

uVu p r Comp Succ Root LB/z* Nodes Connect Cocirc Depots Total cost Seconds

50 0.3 1.4 8.4 5 0 0.952 55.0 14.4 187.4 1.4 5622 7.8

0.5 1.8 8 5 0 0.979 43.8 15.2 397.4 1.8 7734 11.8

0.7 1.5 6.5 5 0 0.985 43.0 11.75 303 1.5 9673 6.5

100 0.3 1 17.3 5 0 0.978 437.6 27 1020.333 1.0 9053 918

0.5 1.6 14.8 5 0 0.987 87.4 23.4 885.2 1.6 11,104 151.4

0.7 1 6 5 2 0.995 10.2 9 347.8 1.0 14,498 12


eulerian location problems

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