hub and spoke network design with single-assignment, capacity decisions and balancing requirements

Applied Mathematical Modelling 35 (2011) 4841–4851

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Applied Mathematical Modelling

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Hub and spoke network design with single-assignment, capacitydecisions and balancing requirements

Isabel Correia a, Stefan Nickel b,c, Francisco Saldanha-da-Gama d,⇑a Departamento de Matemática – CMA, Faculdade de Ciências e Tecnologia, Universidade Nova Lisboa, 2829-516 Caparica, Portugalb Institute for Operations Research, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germanyc Fraunhofer Institute for Industrial Mathematics (ITWM), Kaiserslautern, Germanyd DEIO-CIO, Faculdade de Ciências, Universidade de Lisboa, 1716-049 Lisboa, Portugal

a r t i c l e i n f o

Article history:Received 6 July 2010Received in revised form 14 March 2011Accepted 31 March 2011Available online 12 April 2011

Keywords:Hub and spoke network designBalancing requirementsCapacity choiceSingle-allocation

0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.03.046

⇑ Corresponding author.E-mail address: [email protected] (F. Saldanha-da-

a b s t r a c t

In this paper, an extension of the capacitated single-allocation hub location problem is con-sidered in which the capacity of the hubs is part of the decision making process and bal-ancing requirements are imposed on the network. The decisions to be made comprise (i)the selection of the hubs, (ii) the allocation of the spoke nodes to the hubs, (iii) the flowdistribution through the sub network defined by the hubs and (iv) the capacity level atwhich each hub should operate. In the latter case, for each potential hub, a set of availablecapacities is considered among which one can be chosen. The objective is to minimize thetotal cost, which includes the setup cost for the hubs as well as the flow routing cost. Econ-omies of scale are assumed for the costs. Balancing requirements are imposed to the net-work. In particular, a value is considered for the maximum difference between themaximum and minimum number of spoke nodes that are allocated to the hubs. Twomixed-integer linear programming formulations are proposed and analyzed for this prob-lem. The results of a set of computational experiments using an off-the-shelf commercialsolver are presented. These tests aim at evaluate the possibility of solving the problemto optimality using such a solver with a particular emphasis to the impact of the balancingrequirements. The tests also allow an analysis of the gap of the bounds provided by linearrelaxation.

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1. Introduction

Consider a set of nodes (e.g. population centers, computer terminals) such that some flow (e.g. people, data) has tobe shipped between each pair of them. One possibility could be to send the flow directly between each pair of nodes.Such solution would imply that a complete network was built which, in general is not efficient nor costly attractive.Moreover, the flow to be shipped may require some type of processing. Accordingly, a more rational type of solution isto choose a set of nodes that will consolidate, process and redistribute all the flow. Such nodes are called hubs. Thenodes that are not chosen to be hubs are called spokes. Usually, it is assumed that all the flow should be routed via atleast one hub.

The basic decisions to be made in a hub location problem comprise the selection of the nodes that should become hubsand the way flow should be routed through the network. The goal is to minimize the overall cost which, typically, includes

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the setup cost for the hubs and the total flow routing cost. The latter is often partitioned into three components: the cost forthe flow sent from the spokes to the hubs (collection cost), the cost for the flow shipped between hubs (transfer cost) and thecost for the flow sent from the hubs to the spokes (distribution cost). One important reason for considering hub and spokenetworks is the possibility of taking advantage of economies of scale for the transfer costs due to the large quantities of flowthat typically will flow in the sub network defined by the hubs. For further details, the reader should refer to O’Kelly [1] andCampbell et al. [2].

Applications for hub location problems are found in areas such as public transportation, telecommunications and logis-tics. The reader should refer to Alumur and Kara [3] who present the most recent overview of applications and models in thisarea of research. A more recently application was presented by Yang [4], who consider a stochastic hub location problemwith a case study regarding the air freight market in Taiwan and China.

Hub location problems can be classified according to several features, many of which have been addressed in the litera-ture (see, for instance, Alumur and Kara [5], Contreras et al. [6], Contreras et al. [7], Correia et al. [8], Nickel et al. [9], Tan andKara [10], and Yaman [11]). In particular we can highlight: (i) single-allocation / multiple allocation; (ii) capacitated/ unca-pacitated hubs; (iii) the existence or not of a pre-specified number of hubs to locate; (iv) complete/ incomplete hub network;(v) direct connection allowed or not between spoke nodes.

In this work we focus on capacitated single-allocation hub location problems. This means that each spoke is allocatedto one and only one hub and also that there are capacity constraints. In the latter case, we consider a limitation on theamount of unprocessed incoming flow to the hubs. This is the situation occurring when, for instance, hubs perform somekind of sorting operations. In such situation, the flow that is originated at the nodes has to be processed but onceprocessed by some hub, the flow proceeds until the destination without need of further processing. Accordingly, theincoming flow to some hub that comes from another hub does not need to be processed. For a deeper discussion oncapacity issues in hub location problems the reader should refer to Alumur and Kara [3], Campbell et al. [2] and Correiaet al. [12].

It is worth mentioning that in addition to the typical ways of constraining the capacities which refer to limitson quantities of flow, there is another possibility highlighted by Klincewicz [13] which has practical relevance andrefers to the situation in which a limit exists on the number of spoke nodes that can be allocated to a single hub.One situation in which such limits have to be considered regards tracks in a train station. In fact, independentlyfrom the capacity of a train station to process the flow, the number of trains that can stop per period of time isnaturally limited. One possibility for implicitly considering such type of limitation and somehow embed it in theproblem is to require that some balance is reached in terms of the network design. Such balance can be justifiedby efficiency (aiming at a good service level) or reliability arguments. Network design problems arising in telecom-munications are often problems with balancing requirements. Another situation quite common arises in publictransportation in large towns when hubs can be very efficiently connected (e.g. Times Square and the GrandCentral Station in New York City). In such case, it makes sense to balance the number of buses or metro linesheading to each hub.

In order to illustrate the impact of including balancing requirements and capacity choices in the hub and spoke networkdesign problem we are addressing, we present a small example. Consider an instance with 10 nodes scattered in the euclid-ean plane as depicted in Fig. 1. Additionally assume that:

� Each node should send 2 units of flow/traffic to every other node but not to itself. Accordingly, each node originates 18units of flow.

Fig. 1. Example of a hub and spoke network design problem.

I. Correia et al. / Applied Mathematical Modelling 35 (2011) 4841–4851 4843

� A hub can be installed in every node.� Two capacity levels are available for a hub which are equal to 80 and 140 for node 5, 60 and 100 for node 6, and 50 and

100 for all the other nodes. The fixed setup costs for the hubs depend on the capacity level chosen and are equal to 100and 140 monetary units respectively for the first and second capacity level in node 5, 80 and 110 monetary units for thetwo possible capacity levels in node 6 respectively, and finally, 100 and 160 monetary units respectively, for the twocapacity levels available for each of the other nodes.� The cost for sending one unit of flow between two hubs is equal to 0.75 times the distance between the hubs.� The cost for sending one unit of flow between a non-hub node and a hub, as well as, between a hub and a non-hub is equal

to the distance between the nodes involved.� The distance between two nodes in the network is given by the euclidean distance.

In the instance just introduced, the optimal network design is given in Fig. 2. In this figure, the squared nodes are theselected hubs. In the optimal solution, nodes 5 and 6 are selected to become hubs. The former with capacity 140 and thelatter with capacity 60. Nodes 1, 2, 4, 8, 9 and 10 are allocated to hub 5 and the nodes 3 and 7 are allocated to hub 6.The total cost (which is the minimum) is approximately equal to 678,7 monetary units.

The solution depicted in Fig. 2 is clearly unbalanced in terms of the assignment of spokes to hubs. Suppose now that abalancing requirement is imposed such that the difference between the maximum and minimum number of nodes assignedto some hub should be at most equal to 1 node (10% of the number of nodes). In this case, the optimal network designchanges. Fig. 3 depicts the new solution. Now we see that the network is balanced. Moreover, the capacity levels of the oper-ating hubs also change. In particular, hub 6 now operates with its second capacity level (100). The new solution has a totalcost approximately equal to 732,7.

The introduction of the capacity choices in the decision making process of a hub location problem was first proposed byCorreia et al. [12]. However, to the best knowledge of the authors, no work exists in the literature addressing balancingrequirements in network design problems and, in particular, in hub and spoke networks. This is the extension that we ad-dress in this work.

The new problem that we propose is NP-hard as it has the classical single-allocation hub location problem as a partic-ular case which, in turn, is known to be NP-hard (see, for instance, Alumur and Kara [3]). Nevertheless, in this workwe propose two mixed-integer programming formulations for the problem and by making use of a set of computational

Fig. 3. Optimal network design with balancing requirements.

Fig. 2. Optimal network design without balancing requirements.


experiments we show that for small and medium sized instances, it is possible to solve this problem to optimality using acommercial solver.

Note that in this paper we are considering two additional decision dimensions (capacity choices and balancing require-ments) to the classical capacitated single-allocation hub location problem. The literature shows that even this classical prob-lem is already extremely challenging from a computational point of view (e.g. Contreras et al. [6]). Hence, the problem weaddress in this paper should be even more challenging when it comes to solve it to optimality. Accordingly, for large in-stances, the possibilities of obtaining optimal solutions to the problem are extremely scarce and thus, one may have no alter-native but to simply obtain feasible solutions to the problem using some heuristic procedure. However, in this case, it iscrucial to get some information about the quality of such solutions, which can only be achieved if we have (good) lowerbounds for the problem. One of the mixed-integer formulations we proposed was developed with this main purpose: to ob-tain tighter lower bounds to the problem.

The remainder of this paper is organized as follows. In Section 2 we introduce and analyze two mixed-integer linear pro-gramming formulations for the problem. In Section 3 we report and discuss a series of computational tests performed notonly to assert the possibility of solving the problem to optimality using a commercial solver but also to analyze the impactof the balancing requirements. We also report on the quality of the lower bounds produced by the linear relaxation of theproposed models. The paper ends with some conclusions drawn from the work presented and some directions for furtherresearch.

2. Formulations and properties

In this section we propose and discuss two mixed-integer linear programming (MILP) formulations for the single-allocation hub location problem with capacity choices and balancing requirements (BCSAHLP). The following notation isconsidered hereafter.N = {1, . . . ,n} set of nodesQk = {1, . . . ,sk} set of different capacity levels available for a potential hub to be installed at node k (k 2 N)wij flow to be sent from node i to node j (i, j 2 N)dij distance between nodes i and j (i, j 2 N)v cost per unit of flow and per unit of distance between a spoke and a hub. This value is usually known as collection costd cost per unit of flow and per unit of distance between a hub and a spoke. This value is usually known as distribution

costa cost per unit of flow and per unit of distance between hubs. This value is usually known as transfer cost and it is often

assumed that 0 6 a < 1. It is assumed that a is lower than the collection and distribution costscijkl total cost for sending one unit of flow from node i to node j through hubs k and l. This means that the flow follows the

path i ? k ? l ? j and cijkl = vdik + adkl + ddlj (i, j,k, l 2 N)h balancing requirement i.e., maximum value allowed for the difference between the maximum and the minimum

number of spoke nodes that are connected to some hubf qk fixed cost for installing a hub with capacity of level q at node k (k 2 N,q 2 Qk)Cq

k capacity of a hub installed at node k with a level of capacity q (k 2 N,q 2 Qk)Oi ¼

Pj2Nwij total flow originating at node i 2 I

Di ¼P

j2Nwji total flow destined for node i 2 I

It is assumed that the distance matrix [dij]i,j2N is symmetric and also that dii = 0 (i 2 N). Moreover, it is assumed that thedistances satisfy the triangle inequality. Regarding the flow matrix [wij]i,j2N, it should be noted that it does not necessarily hasa null diagonal. Regarding the capacities, for each k 2 N, the following relation is assumed: Cq1

k < Cq2k for q1, q2 2 Qk such that

q1 < q2.It should be noted that the costs f q

k ðk 2 N; q 2 Q kÞ can also include fixed operation costs for the hubs (dependent on thecapacity level) when they exist.

Clearly, a necessary condition for the feasibility of an instance of the problem is thatP

k2NCskk P

Pk2NOk.

2.1. A mixed-integer linear programming formulation

A first formulation can be proposed for the problem by considering two sets of classical decision variables insingle-allocation hub location problems and one additional set regarding the capacity choice that we areconsidering.

Denote by xik, (i,k 2 N) a binary variable equal to 1 if node i is assigned to hub k and 0 otherwise. For k 2 N, xkk = 1 indi-cates that node k is a hub. Regarding the flows, we consider continuous variables yi

kl ði; k; l 2 NÞ representing the amount offlow with origin at i that goes through hubs k and l. Finally, due to the capacity choice that is included in the decisionmaking process we consider zq

k ðk 2 N; q 2 Q kÞ as a binary variable equal to 1 if node k receives a hub with capacity levelq and 0 otherwise. In order to consider the balancing requirements we also consider auxiliary variables A and B that rep-resent respectively, a lower and an upper limit on the minimum and maximum number of spokes that are allocated tosome hub.

The problem can be formulated as follows:


MinXi2N

Xk2N

dik vOi þ dDið Þxik þXi2N

Xk2N

Xl2N

adklyikl þ

Xk2N

Xq2Qk

f qk zq

k; ð1Þ

s:t: :Xk2N

xik ¼ 1 i 2 N; ð2Þ

xik 6 xkk i; k 2 N; ð3ÞXl2N

yikl �

Xl2N

yilk ¼ Oixik �

Xj2N

wijxjk i; k 2 N; ð4Þ

Xl2N

yikl 6 Oixik i; k 2 N; ð5Þ

Xi2N

Oixik 6Xq2Qk

Cqkzq

k k 2 N; ð6Þ

Xq2Qk

zqk 6 1 k 2 N; ð7Þ

If xkk ¼ 1 then

A 6Pi2N

xik; k 2 N;

B PPi2N

xik; k 2 N;

8><>: ð8Þ

B� A 6 h; ð9Þxik 2 f0;1g i; k 2 N; ð10Þ

yikl P 0 i; k; l 2 N; ð11Þ

zqk 2 f0;1g k 2 N; q 2 Q k; ð12Þ

A;B P 0 and integer: ð13Þ

The objective function (1) minimizes the costs for collecting, transferring and distributing the flow and the costs of estab-lishing the hubs. Constraints (2) assure that all nodes are hubs or are assigned to a single hub. Constraints (3) ensure thatspoke nodes can only be allocated to the opened hubs. Flow conservation is ensured by constraints (4), which are, in fact,divergence equations for commodity i at node k. Constraints (5) ensure that a yi

kl can only be different from 0 if xik is equalto one and in this case, all the flow originated in node i is sent to hub k. (6) are the capacity constraints. Inequalities (7) guar-antee that for each hub at most one size is chosen. Constraints (8) and (9) ensure the balancing requirements. Finally, (10)–(13) are domain constraints.

A mixed-integer linear programming formulation for the problem can be obtained by considering a big M and replacing(8) by

A 6Xi2N

xik þMð1� xkkÞ; k 2 N; ð14Þ

B PXi2N

xik �Mð1� xkkÞ; k 2 N; ð15Þ

In fact, A and B are defined considering the nodes that are hubs and the number of spoke nodes allocated to them.We denote by P1 model

Min ð1Þ;s:t: ð2Þ—ð7Þ;

ð14Þ; ð15Þ;ð9Þ—ð13Þ:

2.2. An alternative formulation

An alternative formulation can be obtained to the BCSAHLP by replacing the former network design decision variables xik

and zqk by a new set of variables namely, tq

ik ði; k 2 N; q 2 QkÞ denoting a binary variable equal to 1 if node i is assigned to hub kwhich has capacity level q and 0 otherwise. tq

kk ¼ 1 ðk 2 NÞ indicates that node k is a hub with capacity at level q.The relation between the former variables xik and zq

k and the new variables is straightforward:

xik ¼Xq2Qk

tqik i; k 2 N ð16Þ

and

zqk ¼ tq

kk k 2 N; q 2 Q k: ð17Þ


By replacing the x- and z-variables in formulation P1 according to relations (16) and (17) and disaggregating the projectedconstraints (6) i.e., disaggregating the capacity constraints (6) transformed using the relations above, we obtain the followingalternative formulation for the problem.

ðP2Þ MinXi2N

Xk2N

dikðvOi þ dDiÞXq2Qk

tqik

!þXi2N

Xk2N

Xl2N

adklyikl þ

Xk2N

Xq2Qk

f qk tq

kk; ð18Þ

s:to :Xk2N

Xq2Qk

tqik ¼ 1 i 2 N; ð19Þ

Xq2Qk

tqik 6

Xq2Qk

tqkk i; k 2 N; ð20Þ

Xl2N

yikl �

Xl2N

yilk ¼ Oi

Xq2Qk

tqik �

Xj2N

wij

Xq2Qk

tqjk

!i; k 2 N; ð21Þ

Xl2N

yikl 6 Oi

Xq2Qk

tqik i; k 2 N; ð22Þ

Xi2N

Oitqik 6 Cq

ktqkk k 2 N; q 2 Q k; ð23Þ

A 6Xi2N

Xq2Qk

tqik þM 1�

Xq2Qk

tqkk

!k 2 N; ð24Þ

B PXi2N

Xq2Qk

tqik �M 1�

Xq2Qk

tqkk

!k 2 N; ð25Þ

B� A 6 h; ð9Þyi

kl P 0 i; k; l 2 N; ð11Þtq

ik 2 f0;1g i; k 2 N; q 2 Q k; ð26ÞA;B P 0 and integer: ð13Þ

Note that the transformed constraints (7), (10), and (12) are redundant.

2.3. Comparison between the models

As far as the dimension is concerned, model P1 has Oðn2 þ n� sÞ binary variables, n3 continuous variables and 3n2 + 5n + 1constraints while model P2 has Oðn2 � sÞ binary variables, n3 continuous variables and Oð3n2 þ 3nþ n� sþ 1Þ constraints.This means that model P2 has clearly more binary variables than model P1. In terms of continuous variables both modelshave the same. Regarding the number of constraints, model P1 will have in general more than model P2.

One important aspect to analyze when more than one MILP formulation is proposed for a minimization problem refers tothe lower bounds provided by the linear relaxations namely, how they do compare with each other. In fact, a sharp linearrelaxation bound not only gives a mean for evaluating the quality of a feasible solution obtained for the problem (e.g. usingan heuristic approach) but also can be very helpful speeding up a branching procedure devised for solving the problem tooptimality.

The following result holds:

Result 1. Model P2 dominates model P1 in terms of the linear relaxation bound i.e., the lower bound provided by the linearrelaxation of model P2 is at least as good as the bound provided by the linear relaxation of P1.

Proof. Taking into account that the balancing requirement constraints are exactly the same in both models, the result fol-lows directly from results 3 and 4 presented in Correia et al. [12]. h

The previous result assures that the polyhedral description of the feasibility set of the BCSAHLP is never worst using mod-el P2 than using model P1.

The practical usefulness of the previous result can only be asserted empirically, which is done in the following section.

3. Computational experience

In this section we report the computational tests performed in order not only to evaluate the possibility of solvingBCSAHLP to optimality using an off-the-shelf commercial solver but also to analyze the impact of the balancing requirementsin the optimal solutions. We also analyze the quality of the lower bounds provided by the linear relaxation of the modelsproposed in the previous section.

Table 1Summary of the results.

h n Time opt (s) Time LR (s) Gap LR (%)

P1 P2 P1 P2 P1 P2

10 0.57 1.53 0.01 0.01 36.72 15.3720 3.73 10.74 0.04 0.07 27.53 5.65

b0.05nc 25 35.00 69.15 0.08 0.27 19.46 4.6140 145.20 704.33 0.67 2.22 17.05 4.8850 4618.34 12074.41 2.16 9.36 17.75 4.45Avg. 960.57 2572.03 0.59 2.38 23.70 6.99

10 0.17 0.22 0.01 0.01 30.72 7.7020 1.81 2.15 0.03 0.07 26.44 4.27

b0.10nc 25 8.09 16.88 0.08 0.25 18.73 3.7540 144.85 463.92 0.69 2.21 16.55 4.2950 5495.64 6000.00 2.16 8.91 17.56 4.24Avg. 645.05 699.36 0.59 2.29 22.00 4.85

10 0.15 0.20 0.00 0.02 30.72 7.7020 1.40 1.74 0.03 0.08 25.86 3.49

b0.15nc 25 6.95 13.06 0.07 0.25 18.60 3.6040 93.88 303.55 0.65 2.25 16.08 3.7650 1602.76 4450.43 1.97 10.60 17.24 3.86Avg. 274.62 769.76 0.54 2.64 21.70 4.48

10 0.06 0.09 0.01 0.00 28.64 4.9420 1.10 1.36 0.03 0.06 25.78 3.39

n 25 6.45 10.21 0.07 0.23 18.58 3.5740 49.28 101.14 0.65 1.90 15.37 2.9550 2643.49 3710.45 1.85 9.36 16.96 3.54Avg. 540.08 609.61 0.52 2.31 21.07 3.68


As far as the data for the experiments is concerned, we consider, as a starting point, a subset of the instances generated inCorreia et al. [12]. In particular, we consider the basic set of 20 instances with the tighter setup costs in the upper level capac-ity and the value 1.1 for the factor defining the economy of scale for the setup costs (the reader should refer to the abovementioned reference for further details). Accordingly, we are considering instances with 10, 20, 25, 40 and 50 nodes as wellas with 2, 3, 4 and 5 capacity levels and thus we have sk 2 {2,3,4,5}, k 2 N. For each instance 4 possibilities were consideredfor h: b0.05nc, b0.10nc, b 0.15nc and n. The last value reduces our problem to the situation in which there are no balancingrequirements. This allow us to evaluate the impact of imposing those requirements. In total we are considering 80 instances.

For solving models P1 and P2 and the corresponding linear relaxations the general solver CPLEX 11.2 (ILOG CPLEX User’sManual (2009)) was used. No change was made in the default values of the solver parameters apart from the time limitwhich was set to 21600 s (6 h). The tests were run on a machine with an INTEL processor with 2.9 GHz and 2 GB of RAM.

Table 1 summarizes the results obtained. The detailed results can be seen in Tables 2–5 in the Appendix. In Table 1, thefirst two columns indicate the combination of h and n being considered. For both models P1 and P2 we present results regard-ing the CPU time required to solve the problem to optimality and also to solve the linear relaxation. We also present the per-centage gap of the linear relaxation, which is defined by 100 � (vOpt � vOptLR)/vOpt, where vOpt is the optimal value of theproblem or the value of the best feasible solution found (in case the problem could not be solved to optimality) and vOptLR isthe optimal value of the linear relaxation.

Each line of results headed by a number of nodes in Table 1 contains average results for the instances defined by thatnumber of nodes and by the value of h associated with the corresponding block of results. For each value of h a line headedby ‘‘Avg.’’, is also presented which gives the total average for the instances sharing that h.

The time limit of 6 h was never achieved. However, several out-of-memory errors were obtained namely in some of thelargest instances. Using model P1 such error occurred 3 times namely, in the 50-node instances with h = 0.10n and 2 and 5capacity levels, and in the 50-node instance with h = 0.15n and 2 capacity levels. The gap between the best lower and upperbounds upon termination was 0.01%, 1.18% and 1.28% respectively, for each of these three instances. Using model P2 five out-of-memory errors occurred again in 50-node instances namely in the instances defined by h = 0.10n and 2, 3 and 5 capacitylevels, h = 0.15n and 2 capacity levels and h = n and 2 capacity levels. In this case, the final gaps upon termination were 1.37%,0.76%, 1.36%, 1.25% and 1.37%. It should be noted that in all these cases, the final gap achieved was already quite acceptable.

3.1. Model P1 versus model P2

As far the resolution of the problem to optimality is concerned, the results presented in Table 1 show that model P1 is lesstime consuming than model P2. This can be explained by the size of the models. In fact, model P2 contains more binary vari-ables than model P1. Nevertheless, as we will observe, model P2 has some competitive advantage when compared with mod-el P1. We can also observe in Table 1 that the complexity of the problem increases when the number of nodes increases,


which, again is not surprising due to the increase in the dimensionality of the models. Nevertheless, we were able to solve allthe instances with up to 40 nodes. A few 50-node instances could not be solved to optimality. In particular, using model P1, 3out of 16 instances could not be solved while using model P2 there were 5 out of 16 unsolved instances. These figures areextremely encouraging since, as we emphasized in the introduction, we are adding two decision dimensions to the classicalcapacitated single-allocation hub location problem, which, to the best of the authors knowledge, can only be solved to opti-mality using a commercial solver for instances with up to 50 nodes (e.g. Contreras et al. [6]). Accordingly, the results ob-tained for the problem we are addressing, can be considered quite positive.

Table 1 shows that the CPU time required by the linear relaxation is not significant. However, in terms of the gap providedby such relaxation we can observe that model P2 is far better than model P1. A deeper insight on this fact is achieved whenwe look at Figs. 4–7. Each of these figures depicts the linear relaxation gaps for all instances defined by a value of h.

Observing Table 1 and Figs. 4–7 we realize that model P2 gives much stronger linear relaxation bounds than model P1,which means that it is clearly superior to model P1 in terms of the polyhedral description of the feasibility set of the problem.Moreover, the lower bounds provided by model P2 are rather insensible to the dimension of the instances namely, the num-ber of nodes, which is also an important feature because it shows some ‘robustness’ in terms of the polyhedral description ofthe feasibility set.

The results show that model P2 is clearly worth using for obtaining tighter lower bounds to the problem in a computa-tional time that is not significant. As mentioned in the introduction, this aspect is crucial for large instances when hardly, theoptimal solution can be obtained and thus some heuristic procedure must be considered.

3.2. The impact of the balancing requirements

The inclusion of balancing requirements clearly leads to a more difficult problem. In fact, when we compare the resultsobtained with h = n (i.e., without balancing requirements) with those obtained for more binding values of h we see a cleardeterioration of the results as the value of h decreases. In particular, if we compare the most extreme situations that we

Fig. 4. Linear relaxation gap (%) – h = b0.05nc.



Fig. 7. Linear relaxation gap (%) – h = n.


considered (h = b0.05nc and h = n) we see in Table 1 that the CPU time required to solve the problem to optimality increasedsignificantly. The difference is also clear in terms of the gaps of the linear relaxation. This shows that the introduction ofbalancing requirements seems to increase the combinatorial nature of the problem and thus, adds difficulty to a problemwhich base version is already a hard problem to solve.

4. Conclusion

In this paper we studied an extension of the classical single-allocation hub location problem in which dimensioning deci-sions and balancing requirements were introduced in the decision making process. We proposed two mathematical pro-gramming formulations for the problem. These models were evaluated using an off-the-self commercial package. Theresults show that for small and medium size instances, the problem can be solved to optimality using the chosen solver.The results also confirm the superiority of model P2 in terms of the polyhedral description of the feasibility set. Empirically,it is possible to conclude that the inclusion of balancing requirements adds difficulty to the problem. Two lines of researchcan be drawn from here. One regards the development of heuristic procedures for giving good quality feasibly solutions forlarge instances of the problem. Another line of research regards the improvement of the polyhedral description of the fea-sibility set in an attempt to improve the lower bounds obtained by linear relaxation. With this respect, model P2 seems to bea better starting point.

Acknowledgement

This research has been partially supported by the Portuguese Science Foundation, POCTI - ISFL - 1-152, POCTI - ISFL - 1-297and SFRH/BSAB/799/2008.


Appendix

In this appendix, detailed results are presented regarding the resolution of models P1 and P2 proposed in Section 2. In par-ticular, Tables 2–5 detail the results that were summarized in Table 1.

Table 2Results for h = b0.05nc.

Instance Time opt (s) Time LR (s) Gap LR (%)

n sk P1 P2 P1 P2 P1 P2

10 2 0.44 0.77 <0.01 <0.01 38.24 7.613 0.58 2.55 <0.01 <0.01 37.81 14.424 0.67 1.58 <0.01 <0.01 36.11 17.915 0.58 1.22 0.02 0.02 34.73 21.54

20 2 3.56 12.47 0.03 0.06 30.99 4.783 2.23 1.91 0.03 0.06 27.50 3.954 3.16 6.14 0.05 0.06 25.82 5.545 5.98 22.45 0.05 0.09 25.82 8.33

25 2 8.16 10.28 0.08 0.19 19.47 2.973 30.70 46.67 0.08 0.27 19.47 4.024 41.25 90.09 0.08 0.28 19.45 5.285 59.88 129.56 0.08 0.34 19.45 6.16

40 2 137.53 526.28 0.64 1.36 18.71 5.933 86.08 212.56 0.67 2.00 16.50 4.234 167.06 690.97 0.64 2.84 16.50 4.535 190.11 1387.52 0.73 2.66 16.50 4.82

50 2 5260.03 12137.28 2.00 6.27 19.06 4.903 2023.84 14860.41 2.33 8.31 17.67 3.884 3295.20 10437.55 2.17 10.91 17.14 3.935 7894.30 10862.41 2.14 11.95 17.14 5.10




10 2 0.19 0.13 0.02 <0.01 36.46 4.963 0.23 0.38 <0.01 <0.01 33.69 8.754 0.17 0.23 <0.01 0.02 28.98 8.765 0.08 0.13 <0.01 0.02 23.74 8.33

20 2 2.63 2.86 0.02 0.05 30.34 3.913 1.53 1.86 0.03 0.06 27.50 3.954 1.42 1.50 0.03 0.06 24.18 3.445 1.64 2.36 0.03 0.09 23.75 5.77

25 2 6.50 6.67 0.08 0.17 19.47 2.973 8.09 11.55 0.08 0.25 19.04 3.514 6.45 11.05 0.08 0.25 18.21 3.825 11.31 38.25 0.06 0.33 18.21 4.71

40 2 137.97 163.5 0.66 1.34 18.13 5.263 49.45 116.64 0.69 1.98 16.02 3.674 207.80 395.02 0.66 2.44 16.02 3.985 184.17 1180.50 0.73 3.08 16.02 4.27

50 2 Out mem. Out mem. 2.00 6.28 a18.77 a4.553 8509.38 Out mem. 2.27 8.75 17.51 3.714 2481.89 9956.44 2.19 10.14 17.01 3.785 Out mem. Out mem. 2.19 10.47 a16.96 a4.90

a Computed using the best known integer solution.




10 2 0.16 0.19 <0.01 0.02 36.46 4.963 0.23 0.23 <0.01 0.02 33.69 8.75

Table 4 (continued)



4 0.14 0.17 <0.01 0.02 28.98 8.765 0.08 0.19 <0.01 0.02 23.74 8.33

20 2 1.97 1.55 0.03 0.05 29.26 2.423 1.39 1.95 0.03 0.06 27.13 3.474 1.13 1.36 0.03 0.08 23.83 3.005 1.09 2.09 0.02 0.11 23.20 5.08

25 2 3.52 4.75 0.06 0.17 18.99 2.403 7.83 24.81 0.08 0.25 18.99 3.454 6.48 9.05 0.08 0.28 18.22 3.825 9.98 13.61 0.06 0.31 18.22 4.71

40 2 98.16 347.69 0.63 1.33 17.67 4.723 92.73 207.47 0.64 1.86 16.02 3.674 55.64 239.00 0.63 2.70 15.31 3.175 128.97 420.02 0.70 3.11 15.31 3.46

50 2 Out mem. Out mem. 1.91 6.42 a18.68 a4.463 1856.94 6670.52 2.08 10.19 17.25 3.414 1983.45 4818.14 1.95 12.61 16.97 3.745 967.89 1862.63 1.92 13.19 16.05 3.85

a Computed using the best known integer solution.

Table 5Results for h = n.



10 2 0.08 0.11 <0.01 <0.01 35.50 3.523 0.06 0.13 0.02 <0.01 30.32 4.114 0.06 0.08 <0.01 <0.01 26.97 6.175 0.03 0.02 0.02 <0.01 21.76 5.94

20 2 1.33 1.06 0.03 0.06 28.96 2.013 1.23 1.63 0.02 0.05 27.13 3.474 0.94 1.30 0.03 0.06 23.83 3.005 0.91 1.45 0.03 0.08 23.20 5.08

25 2 3.84 3.80 0.06 0.14 18.99 2.403 6.55 10.48 0.06 0.23 18.90 3.344 6.84 9.33 0.06 0.25 18.22 3.825 8.58 17.22 0.06 0.31 18.22 4.71

40 2 32.63 41.94 0.58 1.05 16.48 3.343 38.92 100.78 0.81 1.45 15.31 2.864 22.72 124.30 0.63 2.06 14.86 2.655 102.86 137.52 0.59 3.02 14.86 2.94

50 2 7107.14 Out mem. 1.81 6.11 18.30 4.013 1625.70 4005.08 1.84 8.00 17.07 3.194 1215.16 4935.00 1.81 9.92 16.63 3.355 625.95 2191.27 1.95 13.41 15.86 3.63


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hub and spoke network design with single-assignment, capacity decisions and balancing requirements

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