On the capacitated concentrator location problem: a reformulation by discretization

Computers & Operations Research 33 (2006) 1242–1258www.elsevier.com/locate/cor

On the capacitated concentrator location problem: a reformulationby discretization

Luís Gouveia∗, Francisco Saldanha-da-Gama

Department of Statistics and Operational Research/Operational Research Centre, Faculty of Science, University of Lisbon,BLOCO C6, Piso 4, 1749-016 Lisboa, Portugal

Available online 25 November 2004

Abstract

In this paper, we present and discuss a discretized model for the two versions of the capacitated concentratorlocation problem: a simple version (SCCLP) and a version with modular capacities (MCCLP). We show that thelinear programming relaxation of the discretized model is at least as good as the linear programming relaxation ofconventional models for the two variations of the problem under study. A technique for deriving valid inequalitiesfrom the equations of the discretized model is also given. We will show that this technique provides inequalitiesthat significantly enhance the linear programming bound of the discretized model. Our computational results showthe advantage of the new models for obtaining the optimal integer solution for the two versions of the problem.� 2004 Elsevier Ltd. All rights reserved.

Keywords:Location; Model reformulation

1. Introduction

Rather informally, we would say that a discretized model of a discrete optimization problem is amodel involving at least one set of multiple-indexed variables where the information associated to oneof the indexes is usually associated to a special variable in standard models of the problem. The so-called time-dependent formulations for the Travelling Salesman Problem (see, for instance,[1–3]) giveone well-known example of discretized models. More recently, Gouveia[4] showed how to transform aso-called single-commodity flow model into an equivalent (in terms of associated linear programmingrelaxations) discretized model. This transformation permits us to rank the discretized model, in terms

∗ Corresponding author.E-mail addresses:[email protected](L. Gouveia),[email protected](F. Saldanha-da-Gama).

0305-0548/$ - see front matter� 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.cor.2004.09.013

http://www.elsevier.com/locate/cor

mailto:[email protected]

mailto:[email protected]

L. Gouveia, F. Saldanha-da-Gama / Computers & Operations Research 33 (2006) 1242–1258 1243

of linear programming bounds, in the context of other formulations for the same problem. One exampleis given in Gouveia and Voß[3] where the same transformation provides a “bridge” between the lin-ear programming bounds of network flow based formulations and time-dependent formulations for theTravelling Salesman Problem.Two advantages of the discretizedmodel have also been briefly suggested by Gouveia[4]: (i) to use the

new variables tomodel non-linear costs and (ii) to suggest valid inequalities which are quite intuitive withthe new set and which tighten the linear programming relaxation of the new model. These two ideas willbe explored in this paper, in the context of a discretizedmodel for a version of the unit-demand capacitatedconcentrator location problem with modular interfaces (MCCLP) where interfaces of different sizes areavailable and more than one module of each type can be located in each potential location. In order tomotivate the transformation that permits us to obtain the discretized model from a standard formulationwe will also study the simple version of the unit-demand capacitated concentrator problem (SCCLP)where there is a fixed cost of installing a concentrator. Our computational results will show the advantageof the new models for obtaining the optimal integer solution for the two versions of the problem.Standardmodels for the SCCLP andMCCLP are discussed in Section 2. The discretized reformulation

for these problems is given in Section 3. A technique for deriving valid inequalities from the equations ofthe discretized model is also given (see Section 4).We will show that this technique provides inequalitiesthat are stronger than similar ones presented in the literature. Section 5 presents computational resultsthat compare the linear programming bounds of the models discussed in the paper.For any modelP we denote byv(P ) its optimal value and for any integer linear programming model

P , we denote byP the corresponding linear programming relaxation.

2. Standard models for the SCCLP and MCCLP

In this subsection we review standard models for the two versions of the capacitated concentratorlocation problem studied in this paper.

2.1. The SCCLP

We let I denote the set of potential concentrator locations andJ denote the set of demand points. Theparameterf (>0) denotes the cost of installing a concentrator (independent from its location),Qdenotesthe capacity of each concentrator andcij denotes the cost of satisfying demand pointj from a concentratorlocated ati (i ∈ I, j ∈ J ). Themodel involves two sets of binary variables: variablesyi indicatingwhethera concentrator is installed in locationi and variablesxij indicating whether the demand pointj (j ∈ J )is served by a concentrator located at nodei (i ∈ I ). A well-known formulation for the SCCLP, whichwill be denoted byS, is as follows:

min f∑i∈I

yi +∑i∈I

∑j∈J

cij xij , (1)

s.t.∑i∈I

xij = 1, j ∈ J, (2)

1244 L. Gouveia, F. Saldanha-da-Gama / Computers & Operations Research 33 (2006) 1242–1258

∑j∈J

xij �Qyi, i ∈ I, (3)

yi ∈ {0,1}, i ∈ I, (4)xij ∈ {0,1}, i ∈ I, j ∈ J. (5)

Constraints (2) indicate that each demand point must be served by one of the concentrators. Constraints(3) guarantee that no concentrator will serve more demand points than its capacity. Constraints (4) and(5) are domain constraints. The objective function measures the total cost, which is the sum of the costassociated with the operating concentrators plus the demand satisfaction cost.The model presented above is similar to the models described in the papers by Pirkul[5], Celani et al.

[6], Darby-Dowman and Lewis[7], Barcelo et al.[8,9], Holmberg[10] and Cortinhal and Captivo[11].Capacitated versions of the problem similar to the problem described here, were studied by Raja andHan[12] which consider two types of capacity constrains namely, total connection ports and maximumdata processing rate, and Filho and Galvão[13] who propose a tabu search procedure to obtain feasiblesolutions to the problem. Uncapacitated versions of the problem are described in Migdalas and Narula[14], and Soltys et al.[15]. The last paper considers a cost function, which depends on the volume oftraffic homed to each concentrator.All the papers mentioned in this paragraph, except the last one, handlenon-unit demands.

2.2. The MCCLP

Let f l >0 (l ∈ {1, . . . , L}) denote the unitary cost for each interface sizel (l ∈ {1, . . . , L}) and let{C1, C2, . . . , CL} denote the set of available sizes. Typically, the costsf l (l ∈ {1, . . . , L}) are subject

to economies of scale, i.e.f1

C1 >f 2

C2 > · · ·> fL

CL. Again, we useQ to denote the maximum capacity of

each concentrator andcij to denote the cost of satisfying demand pointj from a concentrator located ati (i ∈ I, j ∈ J ). Besides the binary variablesxij (i ∈ I, j ∈ J ) that are used in the previous model,we shall also use the integer nonnegative variablesuli (i ∈ I, l ∈ {1, . . . , L}) to indicate the number ofinterfaces of sizel located ati. The decision is to know the number of interfaces of each type (chosenamongst a set of available sizes) to be located in each potential concentrator location. The MCCLP canbe formulated as follows:

min∑i∈I

L∑l=1

f luli +∑i∈I

∑j∈J

cij xij , (6)

s.t. (2), (5) and∑j∈J

xij �L∑l=1

Cluli, i ∈ I, (7)

∑j∈J

xij �Q, i ∈ I, (8)

uli ∈ Z+0 , i ∈ I, l ∈ {1, . . . , L}. (9)

We denote byM the previous model. Constraints (7) state that each location cannot serve more demandpoints than its operating capacity; Constraints (8) are capacity constraints and (9) are domain constraints.


Note that if we setL= 1 andC1=Q, the modelM becomes the modelS of the SCCLP discussed in theprevious subsection.The models discussed inYoo and Tcha[16], Holmberg[17], Holmberg and Ling[18], Sridharan[25]

and Correia and Captivo[19] consider a situation similar to the one described here, however handlenon-unit demands and require the additional condition that at most one interface of each type can belocated in each available location. Poliscastro and Ukovich[20] consider the possibility of locating morethan one interface but they consider only one type of interface. Shulman[26] and Antunes and Peeters[21] consider multi-periodic versions of the problem involving facilities whose capacity can be expandedover time by the addition of interfaces to the facilities already operating.

3. The discretized model

In this section we present a discretized model for the SCCLP and the MCCLP. The main idea of thediscretizedmodel is that it uses discretized variablesz

qi (i ∈ I, q ∈ {1, . . . ,Q}) indicating the number of

demand points served by the concentrator at locationi (i ∈ I ). A general discretized model is as follows:

min∑i∈I

Q∑q=1

�qi zqi +

∑i∈I

∑j∈J

cij xij (10)

s.t.∑i∈I

xij = 1, j ∈ J, (2)

∑j∈J

xij =Q∑q=1

qzqi , i ∈ I, (11)

Q∑q=1

zqi �1, i ∈ I, (12)

xij ∈ {0,1}, i ∈ I, j ∈ J, (5)

zqi ∈ {0,1}, i ∈ I, q ∈ {1, . . . ,Q}. (13)

Inequality (12) is a consistency constraint and states that ifzqi = 1 for a givenq, thenzpi = 0 for all

p �= q. Equality (11) states that the indexq of the positive discretized variable associated to a nodei

must be equal to the number of demand points assigned toi. The interest of the formulation arises fromthe generality of the objective function because the coefficients�

qi can be as general as we want them

to be.We discuss next, how to model the SCCLP and the MCCLP with the discretized model.

3.1. Modelling the SCCLP

We obtain a valid formulation for the SCCLP by considering�qi = f for all i andq. We denote byDS,

the formulation obtained in this way. Following Gouveia[4], we can show that the two formulations,S


andDS, produce the same linear programming bound. First, note that the “concentrator” variables in thetwo formulations are related as follows:

yi =Q∑q=1

zqi i ∈ I. (14)

Result 1. v(S)= v(DS).Proof. (i) If {xij , zqi } is feasible forDS then the solution{xij , yi} with yi given by (14) is feasible forS

(note that∑j∈J xij =∑Q

q=1 qzqi �Q

∑Qq=1 z

qi =Qyi (the last equality follows from (14))). Clearly, both

solutions have the same cost. This shows thatv(DS)�v(S). The sequence of inequalities given abovesuggests thatDS might be stronger thanS. The second part of our proof shows that this is not the case.(ii) Let {xij , yi} be a feasible solution forS. One way of building a solution that is feasible forDS and

with the same cost is as follows. For eachi, if∑j∈J xij /yi is integer then we simply setzli = yi with

l=∑j∈J xij /yi andzqi =0, q=1, . . . ,Q andq �= l (such a solution satisfies (14) and is clearly feasible

for DS). If∑j∈J xij /yi is non-integer, then leta =

⌊∑j∈J xij /yi

⌋andb =

⌈∑j∈J xij /yi

⌉and we set

zqi = 0, q = 1, . . . ,Q, q �= a, b. The values for the variableszai andz

bi can be obtained by solving the

systemzai + zbi = yi (satisfying (12));azai + bzbi =∑j∈J xij (satisfying (11)). Solving the system gives

zai =(byi −

∑j∈J xij

)(b − a) and zbi =

(∑j∈J xij − ayi

)(b − a) .

These values are nonnegative because we always have∑j∈J xij �byi and

∑j∈J xij �ayi with a andb

defined as above.�

3.2. Modelling the MCCLP

To formulate the MCCLP, we can use the same discretized model as we have used at the beginning ofSection 3,providedthat we useadequatecosts�qi for the discretized variablesz

qi (i ∈ I, q ∈ {1, . . . ,Q}).

If a concentrator in a given location,i, is servingq demand points then the best choice for the number ofinterfaces of each type to be located at that point can be obtained by solving the minimization version ofthe well-known knapsack problem:Kp(i, q)

min f 1u1i + f 2u2i + · · · + f LuLi ,s.t. C1u1i + C2u2i + · · · + CLuLi �q,

u1i , u2i , . . . , u

Li ∈ Z+

0 ,

i.e. �qi = v(Kp(i, q)). The problemKp(i, q) can be easily solved by Dynamic Programming (see, for

instance,[22]). We denote by DM the discretized model obtained with the costs derived in this way.We also denote by�li (q) the value of the variableuli in the optimal solution ofKp(i, q). The following


equality relates the “concentrator” variables from the two types of formulations:

uli =Q∑q=1

�li (q)zqi , i ∈ I, l ∈ L. (15)

Result 2. v(M̄)�v(DM).

Proof. Let {xij , zqi } be a feasible solution forDM and consider the solution{xij , uli} with uli given by(15). We show next that this solution is feasible forM̄. Note that (8) follows from (11) and (12). To seethat (7) are satisfied, note thatC1�1i (q)+ C2�2i (q)+ · · · + CL�Li (q)�q because{�li (q)} is feasible forKp(i, q). By (11) and the previous inequality we obtain

∑j∈J

xij �Q∑q=1

(L∑l=1

Cl�li (q)

)zqi =

Q∑q=1

L∑l=1

(Cl�li (q)zqi )=

L∑l=1

Cl Q∑

q=1

�li (q)zqi

=

L∑l=1

Cluli

and (7) are satisfied by{xij , uli}.It remains to show that the two solutions have the same cost. The value of the solution{xij , zqi } inDM

is given by

∑i∈I

∑j∈J

cij xij +∑i∈I

∑l∈L

f l Q∑

q=1

�li (q)zqi

.

Thisobservation follows from the fact thatv(Kp(i, q)=∑l∈L f l�li (q). Using (15), thepreviousexpressioncan be rewritten as follows:∑

i∈I

∑j∈J

cij xij +∑i∈I

∑l∈L

f luli,

i.e. the value of the solution{xij , uli} in M̄. �

For all of the instances tested in our computational experiments, the linear programming bounds givenby the twomodels are equal. This suggests that the previous result may also hold as an equality (similarlyto what happens for the simple case). However, the following example shows that we may have strictinequality in the previous result.Consider an instance with 2 possible site locations and three demand points, i.e.|I | = 2 and|J | = 3.

LetQ= 3 (the problem is uncapacitated), only one interface andC1 = 2. The following solution (all thedemand points are satisfied by one concentrator):

u11 = 1.5, x11 = x12 = x13 = 1

and other variables equal to zero, is an extreme point of the set of feasible solutions of the linear pro-gramming relaxation of the modelM (this can be easily established by solving the linear programmingrelaxation ofM with adequate costs: small costs for assigning the three demand points to concentrator 1,


volume

cost

Fig. 1. An example of a cost structure[15].

high costs for the remaining three assignment costs andmedium costs for interface installation). Note alsothat for such assignment of costs, the optimal integer solution is obtained from the linear programmingoptimal solution, by raising the value of variableu11 from 1.5 to 2.The interesting point is that we cannot find an “equivalent” solution in thez

qi variables (i.e., a solution

that is feasible for the linear programming relaxation ofDM and is related with the given solution byusing (15)). To see this, consider (15) and (11) for the first concentrator

1.5= z11 + z21 + 2z31,

3= z11 + 2z21 + 3z31.

Combining the two equations givesz21 + z31 = 1.5 which contradicts (12).Although, results 1 and 2 state that we can use the discretized models to replicate the bounds (or

improve, in a few cases) given by the standard models, the discretized models use many more variables,which suggests that solving their linear programming relaxations should consumemore time than solvingthe linear programming relaxation of the original models (and this is confirmed by our computationalresults). However, we will show in Section 4 that the extra index permits us to generate new sets of validinequalities that are quite effective when added to the discretized model.We conclude this section by remarking again that the objective function of the discretizedmodel is quite

general and permit us to model situations that do not appear to be easily modelled with standard models.One example that comes to mind is the problem described in Soltys et al.[15] where the costs associatedwith the concentrators depend on the traffic assigned to them. Volume discounts are considered, whichmeans that costs may have a structure as the one inFig. 1.

4. Valid inequalities

In this section we discuss inequalities that can tighten the linear programming relaxation of the modelsdiscussed in the previous two sections. In Section 4.1, we review an inequality known from the literatureand their direct analog using the discretized variables. In Section 4.2 we present new inequalities that arederived directly from the new variables.


4.1. “Strong-model” inequalities

It is well known that theS andM formulations have, in general, a weak linear programming bound.One way of improving the linear programming bound ofS is by adding the following constraints:

xij �yi, i ∈ I, j ∈ J. (16)

These inequalities improve considerably the linear programming bound whenQ is close ton but becomeless effective whenQ becomes smaller. To see this, observe that in the optimal solution of the corre-sponding linear programming relaxation, inequalities (3) must be satisfied as equalities. Then, (16) areeffective only when for certain values ofi, we have max[j ] {xij }>(1/Q)∑j xij . WhenQ is small, theprevious strict inequality does not usually hold.By using (14) we can use a similar set of constraints

xij �Q∑q=1

zqi , i ∈ I, j ∈ J (17)

in the context of the discretized modelDS. We denote byS∗ the formulationS augmented with (16) andwe denote byDS∗ the formulationDS augmented with (17). It can be easily seen that the proof of Result1 holds when these constraints are included in the two types of models, i.e.v(S∗)= v(DS∗).In a similar way, the linear programming relaxation ofM can be improved by adding the following

constraints:

xij �L∑l=1

uli, i ∈ I, j ∈ J (18)

toM. We denote byM∗ this strengthened model. It is important to point out that the inequalities (18)may not be as effective for the MCCLP as inequalities (16) are for the SCCLP. To see this, consider thesolution described in Section 3. As noted, we haveu11 = 1.5 in the optimal linear programming solutionandu11=2 in the optimal integer solution. Including (18) does not violate the given solution.An inequalitylike (18) may be effective only when the value of

∑l=1,...,L u

li is less than 1 in the linear programming

relaxation of the modelM. Then, the addition of some of these inequalities may help to raise the value of∑l=1,...,L u

li and as a consequence, of some of the variablesuli . Inequalities (18) are not effective when

the value of∑l=1,...,L u

li is already greater than 1 because the value of the variablesxij is bounded above

by 1 and thus, (18) has no effect. With respect to our small example, we would like to find an inequalitythat would permit us to raise the value of the variableu11 from 1.5 to 2. In fact, it appears that such a typeof inequality might be crucial to help solving more general network loading problems as described inMagnanti et al.[23] and Croxton et al.[24].Inequalities (17) are also valid for theDM model and we denote byDM∗ the modelDM augmented

with such inequalities. Result 2 implies thatv(M∗)�v(DM∗). However, as our computational resultswill show, there are several cases for which we havev(M) = v(DM) andv(M∗)< v(DM∗), i.e., thelinear programming bound of the discretized modelDM does not improve on the linear programmingbound of the original modelM but the linear programming bound of the improved discretized modelDM∗ improves on the linear programming bound of the improved modelM∗. As a sort of explanation


for this result, note that the inequality (17) is stronger than the inequality that is obtained by combining(15) with (18) because

xij �Q∑q=1

zqi �

Q∑q=1

(L∑l=1

�li (q)

)zqi =

L∑l=1

uli .

4.2. New valid inequalities

As pointed out before, valid inequalities are suggested immediately by the discretized variables. Fol-lowing Gouveia[4], the following easily proved inequality gives an upper bound on the number ofconcentrators handlingq or more demand points exactly

∑i∈I

∑p�q

zpi �

⌊ |J |q

⌋q = 1, . . . ,Q. (19)

The point is that we can still tighten the previous inequalities by increasing, inmany cases, the coefficientsof some the variables arising on the left-hand side. However, an intuitive rule to derive such inequalitiesis far from obvious (even less, proving its validity).In this section, we show how to use the equalities of the discretized model in order to generate (by

integer programming arguments) two classes of inequalities (one class contains inequalities that arestronger than the previously presented inequalities). The validity of the proposed inequalities followsimmediately from the way we derive the inequalities.If we add constraints (11) for alli ∈ I and use (2) we obtain the following equality:

∑i∈I

Q∑p=1

pzpi = |J |.

FromthepreviousequalitywecanobtainQequivalentequalitiesbydividingeachmemberbyq=1, . . . ,Qleading to

∑i∈I

Q∑p=1

p

qzpi = |J |

q, q = 1, . . . ,Q.

Although the previous equalities are equivalent, we show next that from each one, we can derive up totwo different inequalities.

4.2.1. The� inequalitiesBy roundingdown thecoefficients on the left-handsideof the inequalities andsubsequently by rounding

down the corresponding right-hand sides, we obtain (and prove the validity of) the following set ofinequalities:

∑i∈I

∑p�q

⌊p

q

⌋zpi �

⌊ |J |q

⌋, q = 1, . . . ,Q. (20)


Note that the indexation on the second summation term simply follows from the fact thatp/q is lessthan 1 whenp<q. Inequalities (20) give an upper bound on a “weighted” number of concentrators thatwill be necessary to handleq demand points or more for eachq ∈ {1, . . . ,Q} and imply constraints (19)given at the beginning of this section.

4.2.2. The� inequalitiesBy rounding up the coefficients on the left-hand side of the equalities and subsequently by rounding

up the corresponding right-hand sides, we obtain (and prove the validity of) the following similar set ofinequalities:

∑i∈I

Q∑p=1

⌈p

q

⌉zpi �

⌈ |J |q

⌉, q = 1, . . . ,Q. (21)

whenq = Q, the previous inequality corresponds (by using (14)) to the well known inequality whichgives a lower bound on the number of concentrators of capacityQ needed to handle|J | demand points.It is important to point out that not all of these inequalities are relevant (when rounding on the right-

hand sides has no effect) and that in some cases, some of the� inequalities are equivalent to some ofthe� inequalities. As an example consider an instance withQ = 3. The two inequalities forq = 2 canbe shown to be equivalent by using (11). Thus, we only need to consider three new inequalities: the�inequality forq = 3, the� inequality forq = 3 and one of them forq = 2.As we show in the computational results section, the new sets of inequalities tighten considerably the

linear programming relaxation of the discretized formulations described in Section 3. We will also showthat the new inequalities play an important role in solving many instances to optimality.

5. Computational experience

In this sectionwe report on some computational experience to evaluate the quality (linear programmingbound values and time to reach the optimum) of the new models.With respect to the SCCLP, we have tested with several randomly generated instances that are obtained

by combining the parameters given next:

(i) The number of locations was chosen in the set{5,10,20,40}.(ii) The number of demand points was chosen in the set{50,100}.(iii) The fixed operation cost for the concentrators was generated randomly according to a Uniform

distribution U∼ [100,1000].(iv) The costs for satisfying the demandwere generated according to aUniformdistributionU∼ [1,100].With respect toMCCLP, we have used a similar pattern. Besides using (i), (ii) and (iv) as described above,we have also considered the following rules:

(v) One, two, and three different types of interfaces.(vi) The sizes of each interface type are chosen in the set{1,2,3,4,6,8,12}.(vii) For the instances with one interface type, the operation costs for the modules were generated ac-

cording to a Uniform distribution U∼ [100, 1000]. In the case of two interfaces, the fixed operation


cost of smallest size interface was randomly generated according to the same distribution. For thelargest size interface, we have used the formulaf 2 = 0.75∗f 1∗C2/C1. For the case with threeinterface sizes, the operating cost for the smallest module was generated according to a uniformU ∼ [100, 1000]), and the formulasf 2 = 0.75∗f 1∗C2/C1, f 3 = 0.75∗f 1∗C3/C1 were used togenerate the operating cost for the second and third type of interfaces (thus, the per-unit cost ofmodule 2 is 75% of the same cost for the module 1 and the similar relationship between module 1and 3 is equal to 60%).

All the formulations described in the previous sections were implemented using the modelling languageILOG Concert 1.3. All the instances were solved with standard mathematical programming softwarenamely ILOG CPLEX 8.0, on a Pentium II PC with 300MHz processor and 512MB RAM. The integerprogramming problems were solved using the branch-and-bound algorithm of the mentioned package.The standard options of CPLEX were taken when solving the instances.Table 1presents our computational results obtained for SCCLP. The first column gives the values for

|I |, |J |, andQ. For each combination of|I | and|J |, we have considered the three smallest feasible valuesforQ apart from the cases when|J | is a multiple of|I |. When this happens, the second, third and fourthsmallest feasible values forQ were considered. The remaining columns give results for each one of themodels discussed in Sections 2 and 3, and eventually a model with the valid inequalities discussed inSection 4. Each entry depicts three values namely, the gap of the linear relaxation and, in brackets andbelow the gap value, theCPU time (in seconds) required to obtain the optimumof the linear relaxation andthe integer problem, respectively (for a modelP we denote byP + (.) the modelP with the constraintsbetween the brackets).The results show the effectiveness of the valid inequalities (21) in enhancing the linear relaxation bound

of the weak model as well as the linear programming bound of the strong discretized formulation. Theinequalities (20) were also able to strengthen these bounds although not as much as inequalities (21). Thecombination of the inequalities (20) and (21) either separately or together with the strong formulationinequalities leads to the best lower bounds. We emphasize that this is accomplished without a significantincrease in the execution times.The time required to obtain the linear programming relaxation bound was not significant no matter the

formulation that is considered. In terms of the time required to obtain the integer optimal, the inclusionof inequalities (21) were crucial for speeding up the computations.For instances withn=20/m=100 andn=40/m=100 the models without the new inequalities were

difficult to solve. In fact, for two of these instances, the optimal solution was still unknown after 21,600 sof computations.In 10 out of the 15 instances of CCLP solved, the strong discretized model enhanced with inequalities

(21) produced a linear relaxation bound equal to the optimal integer value. When these inequalities wereincluded in the weak discretized formulation only 2 (out of 15) linear relaxation bounds were equal tothe optimal integer value of the corresponding instance.Tables 2, 3, and4 present some computational tests for instances of the modular problem with one,

two, and three different sizes, respectively. In the first two columns, the parameters defining the in-stance are presented. All the other columns refer to one model or to one model with valid inequal-ities. As for the SCCLP, three values are presented for each instance in each of these columns: thelinear programming gaps and the CPU times (in seconds) to obtain the linear and integer optimum,respectively.

L.Gouveia,F.S

aldanha-da-Gama/Computers&OperationsResearch

33(2006)1242–1258

1253

Table 1Computational results for CCLP|I |, |J |,Q S S∗ DS DS∗ DS + (20) DS∗ + (20) DS + (21) DS∗ + (21) DS + (20,21) DS∗ + (20,21)5,50,11 6.21% 5.35% 6.21% 5.35% 5.46% 4.74% 0.00% 0.00% 0.00% 0.00%

(<1, <1) (<1, <1) (<1,2) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1)5,50,12 11.38% 10.13% 11.38% 10.13% 9.76% 8.73% 0.00% 0.00% 0.00% 0.00%

(<1, <1) (<1, <1) (<1,19) (<1, <1) (<1,56) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1)5,50,13 3.88% 1.12% 3.88% 1.12% 2.78% 0.63% 1.48% 0.00% 1.48% 0.00%

(<1, <1) (<1,1) (<1, <1) (<1,2) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1)10,50,6 7.03% 6.38% 7.03% 6.38% 5.35% 4.80% 0.32% 0.00% 0.32% 0.00%

(<1,7) (<1,1) (<1,93) (<1, <1) (<1,184) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1)10,50,7 10.21% 9.00% 10.21% 9.00% 9.94% 8.76% 0.65% 0.00% 0.65% 0.00%

(<1,7) (<1, <1) (<1,1104) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1)10,50,8 10.65% 8.63% 10.65% 8.63% 9.61% 7.74% 1.29% 0.06% 1.29% 0.06%

(<1,9) (<1,1) (<1,2171) (<1,2) (<1,5981) (<1,3) (<1,1) (<1,1) (<1,1) (<1, <1)20,50,3 2.02% 1.88% 2.02% 1.88% 0.11% 0.00% 0.11% 0.00% 0.11% 0.00%

(<1,153) (<1, >21600) (<1,97) (<1, >21600) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1)20,50,4 4.09% 3.57% 4.09% 3.58% 2.86% 2.38% 0.38% 0.00% 0.38% 0.00%

(<1,2730) (<1, >21600) (<1, >21600) (<1, >21600) (<1,7) (<1, >21600) (<1, <1) (<1, <1) (<1, <1) (<1, <1))20,50,5 1.49% 0.10% 1.49% 0.10% 1.49% 0.10% 1.49% 0.10% 1.49% 0.10%

(<1,2) (<1,7) (<1,3) (<1,11) (<1,4) (<1,9) (<1,3) (<1,10) (<1,3) (<1,8)20,100,6 1.44% 1.03% 0.33% 0.00% 0.33% 0.00%

(<1,83) (<1,2) (<1,2) (<1, <1) (<1,2) (<1,1)20,100,7 4.86% 3.81% 0.75% 0.00% 0.75% 0.00%

(<1,o.m.) (1,3) (<1,1) (<1, <1) (<1,1) (<1,1)20,100,8 4.66% 2.82% 1.67% 0.06% 1.67% 0.06%

(<1,o.m.) (1,31) (<1,7) (<1,24) (<1,7) (<1,25)40,100,3 1.53% 1.45% 0.08% 0.00% 0.08% 0.00%

(<1,1318) (1, >21600) (<1,3) (1,2) (<1,2) (1,2)40,100,4 0.35% 0.01% 0.35% 0.01% 0.35% 0.01%

(<1,15634) (2,25) (<1,7) (1,8) (<1,5) (1,16)40,100,5 0.67% 0.01% 0.67% 0.01% 0.67% 0.01%

(<1,9) (2,8) (<1,3) (2,8) (<1,3) (2,6)

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Table 2Computational results for instances with one type of module|I |, |J |,Q C M M∗ DM DM∗ DM + (20) DM∗ + (20) DM + (21) DM∗ + (21) DM + (20,21) DM∗ + (20,21)10,50,6 2 0.01% 0.01% 0.01% 0.01% 0.01% 0.01% 0.01% 0.01% 0.01% 0.01%

(<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1)10,50,6 3 1.88% 1.88% 1.88% 1.88% 0.02% 0.02% 0.02% 0.02% 0.02% 0.02%

(<1,8) (<1,45) (<1,58) (<1, >21600) (6,12) (4,12) (<1, <1) (<1, <1) (<1, <1) (<1, <1)10,50,9 3 1.88% 1.88% 1.88% 1.88% 0.02% 0.02% 0.02% 0.02% 0.02% 0.02%

(<1,689) (<1,3474) (<1,9226) (<1, >21600) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1)10,50,12 4 3.64% 3.64% 3.64% 3.64% 2.44% 2.44% 0.05% 0.05% 0.05% 0.05%

(<1,760) (<1,3230) (<1, >21600) (<1, >21600) (<1,18130) (<1, <1) (<1, <1) (<1, <1) (<1,5) (<1, <1)10,50,12 6 7.05% 6.38% 7.05% 6.38% 5.37% 4.80% 0.34% 0.00% 0.34% 0.00%

(<1,205) (<1, <1) (<1,15842) (<1, <1) (<1,11829) (<1, <1) (<1, <1) (<1, <1) (<1,4) (<1, <1)20,50,4 2 0.09% 0.05% 0.09% 0.05% 0.09% 0.05% 0.09% 0.05% 0.09% 0.05%

(<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1)20,50,6 2 0.09% 0.05% 0.09% 0.05% 0.09% 0.05% 0.09% 0.05% 0.09% 0.05%

(<1, <1) (<1,1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1,1) (<1, <1) (<1,1)20,50,6 3 2.13% 1.96% 2.13% 1.96% 0.22% 0.08% 0.22% 0.08% 0.22% 0.08%

(<1,o.m.) (<1, >21600) (<1,o.m.) (<1, >21600) (<1, <1) (<1, <1) (<1, <1) (<1,1) (<1, <1) (<1, <1)20,50,9 3 2.13% 1.96% 2.13% 1.96% 0.22% 0.08% 0.22% 0.08% 0.22% 0.08%

(<1,o.m.) (<1, >21600) (<1,o.m.) (<1, >21600) (<1, <1) (<1, <1) (<1,1) (<1,3) (<1, <1) (<1,1)20,100,6 3 0.04% 0.04% 0.04% 0.04%

(<1,2) (<1,5) (<1,4) (<1,5)20,100,12 3 0.06% 0.06% 0.06% 0.06%

(<1,7) (<1,16) (<1,5) (<1,11)20,100,12 4 0.14% 0.08% 0.14% 0.08%

(<1,10) (<1,7) (<1,4) (<1,8)40,100,6 3 0.12% 0.03% 0.12% 0.03%

(<1,9) (1,20) (<1,5) (1,5)40,100,9 3 0.12% 0.03% 0.12% 0.03%

(<1,16) (1,21) (<1,10) (1,27)40,100,10 5 0.67% 0.01% 0.67% 0.01%

(<1,2) (2,11) (<1,29) (2,53)

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Table 3Computational results for problems with two different sizes for the modules

|I |, |J |,Q C1, C2 M M∗ DM DM∗ DM + (20) DM∗ + (20) DM + (21) DM∗ + (21) DM + (20,21) DM∗ + (20,21)10,50,6 1,3 1.29% 1.29% 1.29% 1.29% 0.00% 0.00% 0.97% 0.97% 0.00% 0.00%

(<1,5) (<1, >21600) (<1,161) (<1,377) (<1, <1) (<1, <1) (<1,141) (<1, >21600) (<1, <1) (<1, <1)10,50,6 1,6 1.37% 1.20% 1.37% 1.20% 0.08% 0.08% 0.85% 0.77% 0.08% 0.08%

(<1,6) (<1,1) (<1,9) (<1,1) (<1, <1) (<1,1) (<1,6) (<1, >21600) (<1, <1) (<1, <1)10,50,6 3,6 3.83% 3.37% 3.83% 3.37% 2.04% 1.64% 0.24% 0.01% 0.24% 0.01%

(<1,10) (<1,100) (<1,53) (<1,1) (<1,6) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1)10,50,6 2,6 1.44% 1.12% 1.44% 1.12% 0.19% 0.02% 0.19% 0.02% 0.19% 0.02%

(<1,4) (<1, <1) (<1,7) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1)10,50,12 2,6 1.45% 1.12% 1.45% 1.12% 0.19% 0.02% 0.19% 0.02% 0.19% 0.02%

(<1,51) (<1, <1) (<1,98) (<1,1) (<1,2) (<1, <1) (<1,3) (<1, <1) (<1,2) (<1, <1)10,50,12 2,12 3.12% 0.99% 3.12% 0.99% 1.88% 0.50% 1.88% 0.50% 1.88% 0.50%

(<1,8) (<1,69) (<1,13) (<1,9) (<1,4) (<1,12) (<1,5) (<1,12) (<1,5) (<1,11)10,50,12 3,12 5.99% 2.97% 5.99% 2.97% 3.49% 0.67% 2.49% 0.52% 2.49% 0.52%

(<1,46) (<1,331) (<1,359) (<1, >21600) (<1,6) (<1,27) (<1,7) (<1,12) (<1,6) (<1,13)10,50,12 6,12 12.76% 7.51% 12.76% 7.51% 10.58% 5.67% 3.62% 0.20% 3.62% 0.20%

(<1,97) (<1,1507) (<1,5181) (<1, >21600) (<1,2092) (<1, >21600) (<1,3) (<1,7) (<1,5) (<1,11)20,50,6 1,3 1.38% 1.30% 1.38% 1.30% 0.08% 0.05% 1.06% 1.01% 0.08% 0.05%

(<1,o.m.) (<1, >21600) (<1,o.m.) (<1, >21600) (<1, <1) (<1,2) (<1,o.m.) (<1, >21600) (<1, <1) (<1,2)20,50,6 3,6 5.05% 3.50% 5.05% 3.50% 3.21% 1.75% 1.37% 0.12% 1.37% 0.12%

(<1, >21600) (<1, >21600) (<1,o.m.) (<1, >21600) (<1,725) (<1, >21600) (<1,3) (<1,10) (<1,4) (<1,19)20,100,6 1,6 0.57% 0.48% 1.03% 0.98% 0.49% 0.43

(<1,284) (<1,10) (<1,3414) (<1,14) (<1,455) (<1,13)20,100,6 3,6 1.35% 1.08% 0.23% 0.00% 0.23% 0.00%

(<1,683) (<1,2824) (<1,2) (<1, <1) (<1,2) (<1,1)20,100,12 1,6 0.62% 0.48% 1.04% 0.98% 0.50% 0.43%

(<1, >21600) (<1,23) (<1,o.m.) (<1,20) (<1, >21600) (1,10981)20,100,12 3,6 1.41% 1.08% 0.27% 0.00% 0.27% 0.00%

(<1, >21600) (<1, >21600) (<1,7) (<1,1) (<1,10) (<1,1)20,100,12 2,12 1.99% 0.49% 1.49% 0.42% 1.49% 0.42%

(<1,409) (2,3716) (<1,60) (1,197) (<1,64) (1,140)40,100,6 1,3 0.05% 0.01% 0.05% 0.01% 0.05% 0.01%

(<1,7) (<1,16) (<1,7) (1,20) (<1,5) (1,16)40,100,6 2,6 0.97% 0.54% 0.45% 0.03% 0.45% 0.03%

(<1,o.m.) (2, >21600) (<1,42) (3,156) (<1,40) (2,193)40,100,6 3,6 1.88% 1.15% 0.74% 0.03% 0.74% 0.03%

(<1,o.m.) (2, >21600) (<1,28) (2,68) (<1,27) (3,107)

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Table 4Computational results for instances with three different sizes

|I |, |J |,QC1, C2, C3 M M∗ DM DM∗ DM + (20) DM∗ + (20) DM + (21) DM∗ + (21) DM + (20,21) DM∗ + (20,21)10,50,6 1,2,3 0.97% 0.97% 0.97% 0.97% 0.01% 0.01% 0.33% 0.33% 0.01% 0.01%

(<1,8) (<1,23) (<1,57) (<1, >21600) (10,10) (<1, <1) (<1,5) (<1,2) (<1, <1) (<1, <1)10,50,6 1,2,6 1.08% 0.87% 1.08% 0.87% 0.12% 0.02% 0.12% 0.02% 0.12% 0.02%

(<1,2392) (<1, <1) (<1,5) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1,1)10,50,6 2,3,6 2.67% 2.28% 2.67% 2.28% 0.84% 0.50% 0.23% 0.03% 0.23% 0.03%

(<1,3) (<1,2) (<1,3) (<1,2) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1) (<1, <1)10,50,12 2,3,4 2.50% 2.50% 2.50% 2.50% 1.58% 1.58% 0.05% 0.05% 0.05% 0.05%

(<1,1330) (<1,2042) (<1,16969) (<1, >21600) (7,12501) (<1,1) (<1, <1) (<1,2) (<1, <1) (<1,1)10,50,12 2,3,6 2.68% 2.28% 2.68% 2.28% 0.85% 0.50% 0.24% 0.03% 0.24% 0.03%

(<1,198) (<1,55) (<1,1180) (<1,1) (<1,15) (<1,26) (<1,2) (<1, <1) (<1,3) (<1, <1)10,50,12 3,4,6 3.84% 3.26% 3.84% 3.26% 2.08% 1.58% 0.31% 0.01% 0.31% 0.01%

(<1,211) (<1, <1) (<1,1468) (<1,1) (<1,99) (<1, <1) (<1,2) (<1, <1) (<1,3) (<1, <1)10,50,12 4,6,12 10.90% 6.34% 10.90% 6.34% 8.63% 4.36% 3.49% 0.51% 3.49% 0.51%

(<1,127) (<1,4898) (<1,4491) (<1, >21600) (13,987) (<1, >21600) (<1,6) (<1,8) (<1,10) (<1,11)20,50,6 1,2,3 1.10% 1.00% 1.10% 1.00% 0.12% 0.04% 0.45% 0.37% 0.12% 0.04%

(<1,o.m.) (23, >21600) (8,o.m.) (98, >21600) (10,10) (<1,1) (<1,28) (<1,20) (<1, <1) (<1,1)20,50,6 2,3,4 2.90% 2.45% 2.90% 2.45% 1.95% 1.55% 0.37% 0.05% 0.37% 0.05%

(<1, >21600) (1229, >21600) (11,o.m.) (89, >21600) (11,82) (<1,2) (<1,1) (<1, <1) (<1, <1) (<1,2)20,50,6 1,3,6 3.13% 2.47% 3.13% 2.47% 1.23% 0.59% 2.12% 1.58% 1.01% 0.47%

(<1,o.m.) (25, >21600) (10,o.m.) (129, >21600) (9, >21600) (135, >21600) (<1,o.m.) (78, >21600) (9,7959) (1,38)20,50,6 2,4,6 3.71% 2.38% 3.71% 2.38% 1.83% 0.59% 1.21% 0.14% 1.21% 0.14%

(<1,7900) (<1, >21600) (12,o.m.) (103, >21600) (11,137) (<1,702) (<1,3) (<1,13) (<1,3) (<1,14)40,100,6 1,2,3 0.72% 0.66% 0.72% 0.66% 0.48% 0.41% 0.07% 0.02% 0.07% 0.02%

(<1, >21600) (152, >21600) (47,o.m.) (144, >21600) (63,o.m.) (109,135) (<1,7) (2,24) (<1,4) (2,17)


In order to evaluate the importance of the value ofQ for solving the instances, we considered severalvalues forQ for each combination (|I |, |J |).We considered the smallest or second smallest feasible value(depending onwhether or not|I | divides|J | aswhen|I | divides|J | the problem becomes a transportationproblem and is easy to solve). Other values ofQ that are bigger than this lower limit were also consideredin some cases. For instance, for|I | = 10 and|J | = 50, the smallest feasibleQ would be 5. However,because 10 divides 50 we consideredQ=6 as the smallest and interesting value forQ. For this instance,we have also consideredQ=12, which is sufficiently greater than 6 to permit us to understand what mayhappen whenQ increases.The efficiency of the new cuts (20) and (21) in enhancing the linear relaxation is, again, clear.Moreover,

the execution times required to obtain the linear relaxation bounds are not significant for the instancestested. For the situationwith one type ofmodule, inequalities (21) never performedworse than inequalities(20). For the instances with two and three different sizes, inequalities (20) performed better than (21) forthe cases where the smallest module has capacity 1 while (21) is in general better for all the others.We note again, that the use of the new inequalities (20) and (21) were crucial for obtaining a lower

bound close to the optimum, which in turn led to drastic reductions in CPU times.We could also say thatour results are not clear in terms of using the strong location inequalities (17). In fact, and apart from a fewcases, the time required to obtain the integer optimum increases when these inequalities were included.In some cases this increase was significant (as, for instance, the case withn= 40,m= 100, andQ= 6in Table 3).Concerning the effect of the value ofQ in the linear programming relaxation bounds, we note that for

non-discretized models there appears to be an increase in the gap whenQ also increases. Interestinglyenough, this is not the case for the discretized models.To conclude, our experiments show that the new models help to solve unit demand instances of the

capacitated concentrator problem with and without modular interfaces that are difficult to handle withstandard models.

Acknowledgements

The authors thank the referees for their helpful comments.

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On the capacitated concentrator location problem: a reformulation by discretization

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