optimal location of intermodal freight hubs

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Transportation Research Part B 39 (2005) 453–477

Optimal location of intermodal freight hubs

Illia Racunica a, Laura Wynter b,*

a Institut Eurecom-Edite, 2229 Route des Cretes, B.P. 193, 06904 Sophia Antipolis, Franceb IBM Watson Research Center, P.O. Box 704, Yorktown Heights, NY 10598, United States

Abstract

We present an optimization model that has been developed to address the problem of increasing the

share of rail in intermodal transport through the use of hub-and-spoke type networks for freight rail.

The model defined is a generalization of the hub location problem in that it allows for non-linear and con-

cave cost functions on different segments. A linearization procedure along with two efficient variable-reduc-

tion heuristics was developed for its resolution, making use of recent results on polyhedral properties of thisclass of problems. Computational experience and a qualitative analysis from a case study on the Alpine

freight network is provided.

� 2004 Elsevier Ltd. All rights reserved.

1. Introduction

The most recent scenarios under study for integrating freight transportation in Europe effi-ciently and with minimum social and environmental cost involve extensive use of intermodaltransport. The intent of these new transport scenarios is to make maximum use of rail transport,not only for long haul and low value freight distribution, as has been the case, but over medium-length distances as well.

To permit rail to be competitive with road haulers in Europe, one of the few policies that re-mains viable is the use of a subnetwork of dedicated or semi-dedicated (that is, in which freight

0191-2615/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.trb.2004.07.001

* Corresponding author. Tel.: +1 315 268 3861; fax: +1 315 268 4410.

E-mail address: [email protected] (L. Wynter).

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454 I. Racunica, L. Wynter / Transportation Research Part B 39 (2005) 453–477

is given priority) freight rail lines. Indeed, it has been observed that complete integration with pas-senger rail services has rendered it difficult to increase the market share of rail in the freight trans-port sector, since freight slots are generally given to trains only at night, or between passengertrains during the day. Because of that, travel times associated with medium and long-haul freighttransport by rail are high, due to the restrictions on daytime freight travel.

Since the freight trains should furthermore be absent from the network during the busy morn-ing passenger travel period, they are being driven at somewhat higher speeds than they otherwisewould during the night hours, and so as to ensure safety, are therefore loaded less fully than theycould be. The overall result is then that most freight rail movements, and particularly the longer,international movements, witness long delays and high costs, due to inefficient use of traincapacities.

A dedicated or semi-dedicated freight rail network would permit operators to add new servicesand configurations to the palette of rail freight transport types available. In particular, the incor-poration of high-speed shuttle-type services, in conjunction with a hub-and-spoke network struc-ture, similar in concept to those that exist in airline networks, present an innovative way to reducelead times on the train routes and increase flexibility for the customer. In the case of rail, shuttletrains are defined by a fixed and pre-determined number of wagons. As in the airline industry,shuttles would operate on fixed schedules, offering frequent service. While this implies less flexi-bility for the train operator, it benefits the client, who has a wider range of departure times avail-able, and would not be required to make advance reservations.

Fixed-length shuttle services also mean reduced terminal times, since trains need not be recon-figured. Shunting costs and terminal costs are reduced (to an estimated 15–20%), and since itin-eraries do not vary, complexities associated with assigning conductors to trains (who must befamiliar with the itineraries and their signals) and of routing and wagon repositioning are avoided.

Of further interest is the fact that shuttle services would permit the use of a viable hub-and-spoke network configuration for rail freight transport, in that rapid and reliable hub-to-hub trans-fers could be included in freight itineraries. These itineraries could then include high-speed freighttrain services on a few corridors. The combination of hub-and-spoke networks and shuttle serv-ices would allow for the consolidation of flows, higher equipment rotation, reduced staffing costs,and higher train frequencies. An essential feature of shuttle services, and hub-and-spoke networksin general, is the economies of scale that can be gained by consolidation as well as by reducingcosts.

This paper is concerned with devising a model for the conception of such an innovative hub-and-spoke network for intermodal freight transport on dedicated or semi-dedicated freight raillines which could make use of shuttle trains between hubs. Of particular interest is the incorpo-ration of the scale economies resulting from freight consolidation at hub terminals.

In the next section, we present some particularities of the cost function, followed by the form ofthe model, a comparison with related models from the literature, how we model the scale econo-mies, a linearization of that model, and finally, the polyhedral properties of the model. Then, inSection 3, we present the heuristic solution method developed and compare it with other methodsfor solving related problems. In Section 4, we first validate the model and method on a small sub-set of the data, comparing results with exact solutions, then provide an analysis of the model on afull-scale data set of the Alpine region in Europe. The paper is concluded with some directions inwhich this work could be extended in the future.

I. Racunica, L. Wynter / Transportation Research Part B 39 (2005) 453–477 455

2. Modeling intermodal freight hubs

Desired results of the model include information on the optimal hub locations: How muchfreight can each (potential) hub capture? What percentage of the market can the shuttle servicestake from both the current direct complete block train service, and from the share of the roadhaulers? Which corridors can become competitive enough to warrant the construction costs asso-ciated with high-speed freight train services? How would an evolution of freight flows effect theseconclusions?

Since it is necessary to evaluate both the market share of the hubs themselves and of the lines(shuttle services versus direct train service and road transport), coupled with the need to explicitlyinclude economies of scale, the choice of a model representing each train path explicitly appearsjudicious, as opposed to a more compact arc-based formulation of the network flow problem.Furthermore, the importance of the construction and development costs in converting marshalingyards and terminals into ‘‘mega-hub’’ nodes, capable of handling shuttle services, leads us toadopt a network design approach. An illustration of part of such a network is provided in Fig. 1.

Concerning the maximum number of hubs that are viable on a single path it was concluded,based on cost estimates combined with survey data, that single hub itineraries (that is, origin-hub–destination) could be efficient from a threshold of roughly 740km onwards, and bi-hub itin-eraries (origin-hub 1–hub 2-destination), based on present-day costs, from around 950km. Bi-hubpaths are especially interesting as they could justify the development of high-speed shuttle linksbetween the hubs. Handling paths with more than two hubs would increase substantially the num-ber of paths in the model, and it is believed that such paths would attract too little demand toappear in any optimal solution; therefore we limit our model to include 0-, 1-, and 2-hub paths.

Consolidation, and the cost efficiencies that follow consolidation, are the means of justifyinginvestments associated with introducing new services like shuttle trains and direct block trains(which are not reconfigured at intermediate terminals). Traditional hub location models do notaccurately represent these cost efficiencies, as they use purely linear cost terms on each link,and apply a so-called discount factor on the interhub links of the network, so that the per-unitprice on interhub links is lower than that on extremal links of the network. In our application,the threshold at which a shuttle link becomes cost efficient is a very important feature of the model;linear functions do not capture this threshold effect.

origin, imega-hub, k

destination, l

mega-hub, j

Direct route

Origin-to-mega-hub

Shuttle service (between mega-hubs)

Mega-hub-to-destination

Fig. 1. Direct versus hub-based itineraries for a typical origin–destination pair.


Consequently, we require non-linear and concave increasing costs on certain links. In particu-lar, scale economies are assumed to be potentially significant on interhub links, due to reducedshunting costs at both ends of the link, and also on hub-to-destination links, although to a lesserdegree; all other links are modeled with linear increasing costs.

We were thus lead to generalize the costs of the standard hub location model to allow for non-linear and, in particular, concave increasing functions that model scale economies.

Explicit capacity constraints on the hubs and shuttle and direct block services were not includedin this model, as one desired result of the model is to evaluate the maximum average frequencyattainable by rail on any freight corridor. That is, we solve for the maximal possible frequenciesthat these services could attain, permitting the planners to then evaluate the cost effectiveness ofthe proposal tested.

Fixed costs of converting existing terminals and marshaling yards into mega-hubs are difficultto obtain with any precision, which led us to use high, medium, and low estimates for these costs,rather than assigning them any single value.

In order to correctly model the influence of train frequency on choice decisions and on travelcosts, we have chosen to implicitly include frequency effects through a calibration of the cost func-tion, rather than through the use of a dynamic model with time as a parameter. This model istherefore similar to that of frequency network design with frequencies as derived output, withinthe context of the classification proposed by Crainic (2000).

To summarize, then, the objective of the model is to minimize a linear combination of hubdevelopment and traction costs, where the latter take into account the effect of freight consolida-tion and their scale economies, on interhub and hub-destination itineraries. Insufficient potentialfor consolidation between origin nodes and hub nodes has led us to use linear cost functions onthose links. Similarly, the cost of direct (origin-to-destination) block train service is taken to be alinear function of distance, since the number of wagons is directly proportional to the tonnagecarried. The resulting model is a non-linear, mixed-integer program.

2.1. Estimating the effects of weekly frequency

In order to take into account the effect of the weekly frequency on the travel costs, onecould formulate a fully dynamic model in which time is explicitly included as a parameter,and, for example, over a weekly horizon. However, the resulting complexity of the modelwould be significant. Instead, we have chosen to implicitly take into account the effect of timewithin a static model through an appropriate calibration of the non-linear term in the costfunction.

Since both direct (block train) itineraries and itineraries passing through hub nodes are main-tained in the model, we define a generic concave cost function, and calibrate its parameters so thatthe interhub shuttle trip becomes more economical than the direct service at the threshold mini-mum weekly frequency needed to make the shuttle service viable.

Consider Fig. 1 which represents the itinerary choices for a given origin–destination pair, (i, l).Fig. 2 illustrates the calibration of the concave cost curves with respect to the linear, direct (blocktrain) cost curves on a typical example. The two curves should intersect precisely where the shuttleservices becomes advantageous, referred to as point a in the figure. Quantitative data on this pointof intersection permits calibration of the cost function parameters.

a

direct link cost function

inter-hub or hub-to-destination cost function

freight volume, x

variable costs, c(x)

Fig. 2. Cost function calibration between a direct and a hub-based trip leg.


Let us define the graph G = (N,A) where N is the set of all terminal nodes, and the set of poten-tial hub nodes is H � N. The set of origin–destination (o–d) pairs is W � N2. We define the flowvariables as x = xijkl, where i and l the origin and destination terminals, respectively, and j and k

represent intermediate hub nodes.Then, let the concave interhub discount function be given by

aX

ði;lÞ2Wxijkl

!b

ð1Þ

for some value of the parameter b such that 0 6 b 6 1. To determine the value of a which willprovide the shape of the concave cost curve for a fixed value of b, we set

aX

ði;lÞ2W�xijkl

!b

¼ �cX

ði;lÞ2W�xijkl ð2Þ

where �c is the known cost of direct block train service between the pair of nodes (j,k). The term�xjk ¼

Pði;lÞ2W �xijkl is the amount of flow in tons corresponding to the minimum number of complete

block trains needed to make the (shuttle) service viable, where an average conversion factor (e.g.,480T per complete block train) is used. The resulting calibration gives the following function c1jkð Þas the non-linear concave cost of the interhub link (j,k) with constant terms a1jk ¼ a where a is thevalue obtained above, and b is set to 0.5

c1jk ¼ a1jkX

ði;lÞ2Wxijkl

" #0:5ð3Þ

for each hub–hub pair (j,k) 2 H2. An explicit form is obtained for the hub-to-destination discountcost function, c2 in an analogous manner where b was set to 0.6, given as

c2kl ¼ a2klXi2N

Xj2H

xijkl

" #0:6ð4Þ


for each hub–destination node pair (k, l) 2 HxN. Note that the exponent of the discount functionc2 is higher than that of c1, corresponding to a lower potential for scale economies on these paths.

2.2. The hub location model with general costs

The maximum number of hubs open is given by the cardinality of the set H. The weight of eacharc in the graph, a 2 A, is its travel time. Paths in the graph are identified as a sequence of thenodes traversed, where this number is limited to at most two hub nodes per path by the definitionof the path variables, that is, a path (i, j,k, l) originates at terminal i, stops at hub nodes j and k,and then terminates at node l.

The uncapacitated hub location model with general flow costs is then given as follows:

minx;z

F ðx; zÞ ¼ WðxÞ þ UðzÞ

¼Xi2N

Xj2H

cijðxÞ þXj2H

Xk2H

c1jkðxÞ

þXk2H

Xl2N

c2klðxÞ þXj2H

fjzj ð5Þ

subject to Xj2H

Xk2H

xijkl ¼ dil 8ði; lÞ 2 W ; ð6Þ

Xj2H

xijkl 6 dilzk 8k 2 H ; ði; lÞ 2 w; ð7Þ

Xk2H

xijkl 6 dilzj 8j 2 H ; ði; lÞ 2 w; ð8Þ

xijkl P 0 8j; k 2 H ; ði; lÞ 2 W ; ð9Þ

06 zj P 1 8j 2 H ; ð10Þ

zj 2 f0; 1g 8j 2 H : ð11Þ

where the decision variables are the following: the flow variables xijkl 2 RjN j2xjH j2þ with i and l the

origin and destination terminals, respectively, and j and k the intermediate hub nodes. The vari-able z = {zj} 2 {0,1}jHj is the vector of binary decision variables indicating whether a hub is to beopened or not.

The demands for transport are given by the vector d ¼ fdilg 2 RjW jþ over the set W � N2 of ori-

gin–destination (o–d) pairs. Eqs. (7) and (8) ensure that hub k (respectively, j) is open for the flowthrough that hub to be non-zero. Costs on the path (i, j,k, l) are given by the (possibly non-linear)functions c•• (x).

The fixed costs associated with converting the terminal j into a hub node as well as the cost ofsetting up the high-speed shuttle service at hub j are referred to as fj for each hub node j. Note thatfixed costs of non-hub services are not included explicitly, since they will exist regardless of thechoice of hubs and shuttle links.


2.3. Comparison with other hub location models

The problem of optimally located of hubs in a network has received attention over the past dec-ade due to its importance in air transportation, and in telecommunications. See, for example,Bryan and O�Kelly (1999), Campbell (1994), or Campbell (1996). The objective in all cases is todetermine the number of hubs to be opened and the paths used in the network, where a hub isopened only if it is profitable to do so. As in the model we presented, the definition of ‘‘profitable’’is given in terms of hub opening costs and travel time savings, where the latter are, in principle,due both to sufficient consolidation and as well as trip time reduction. As such, the hub locationproblem can be seen as a variant of the network design and facility location problems, which seekto determine flows on a network while taking into account the fixed costs of opening facilities,necessary for routing the flows. Those models and this one then determine the optimal numberof facilities, as well an estimate of their usage levels.

The principal difference then, between this class of hub location models and the above twomodels, in addition to the applications that they address, are the types of path flows that are de-fined by the decision variables. In the case of hub location with 0, 1, or 2 possible hubs per path,decision variables will contain four indices. While this presents no conceptual difficulty, itmeans that in practice, even for a relatively small number of nodes, the dimension of the problemin path flow variables will be very large. Due to the nature of hub location applications, such asours, it is generally necessary to explicitly maintain the path flows in the formulation, as opposedto making use of the much more compact arc-based formulation, often used in network designproblems.

The first hub location model was defined as a quadratic integer program, and was presented byO�Kelly (1987) and later by Klincewicz (1991). Since then, the model has evolved to its presentform, which resembles the facility location problem, and has been studied extensively and somealgorithms proposed in the references O�Kelly et al. (1995), Klincewicz (1996), Skorin-Kapovet al. (1996), Ernst and Krishnamoorthy (1998), Nickel et al. (2000), Sohn and Park (2000) andHamacher et al. (2000). In all of these references, the objective function is linear in the flowand binary hub-opening variables.

O�Kelly and Bryan (1998) and Guldmann and Shen (1997) proposed a non-linear version of thehub location problem, in which the interhub cost term was a concave increasing function of flow.Our model is similar to theirs, with the modification of including non-linear terms on the finalhub-to-destination links as well as on the interhub links, and the use of a different non-linear func-tion, further described below, to capture the efficiency threshold of the interhub and hub-to-des-tination links.

2.4. Taking into account economies of scale

In order to take into account the cost reductions that are obtained by consolidation at hubnodes, the technique used in the standard, linear, hub location model has been to apply a so-calleddiscount factor on the interhub links of the network, so that the per-unit price on interhub links islower than that on extremal links of the network. See, for example, Ernst and Krishnamoorthy(1998), Hamacher et al. (2000), O�Kelly et al. (1995), or Skorin-Kapov et al. (1996) where the


authors use the following flow-related term in their objective function, making use of the discountfactor, a 2 (0,1):

WðxÞ ¼ cijxij þ acjkxjk þ cklxkl; ð12Þ
where xij ¼
Pk2HP

l2Nxijkl and similarly for variables xjk and xkl. That is, the unit cost per distanceis lower between pairs of hubs than otherwise, but the marginal costs on all links are constant withflow.

It is clear, however, that the use of a linear cost function as given byW (x) above does not modelscale economies, which require that the marginal price decreases with increasing flow, in whichcase the cost function must be strictly concave increasing, rather than linear. Clearly, this simpli-fication is costly in terms of accuracy of the solution since large and small flow values all receivethe same discount.

To deal with this deficiency, O�Kelly and Bryan (1998), generalize the definition of the interhubcost term in W (x) above, that is, c1jkðxÞ, to a concave increasing function. We follow this approachand apply it as well to the hub-to-destination cost, c2jkðxÞ.

Indeed, shuttle services between two (mega-) hub nodes are designed to operate with a fixedcomposition, so as to reduce shunting costs; consequently, the marginal costs decrease consider-ably with increasing flow on hub-to-hub shuttle lines. Similarly, for non-shuttle services betweenmega-hub terminals and destination terminals, sufficient consolidation, along with somewhat re-duced shunting costs at the mega-hub node, are believed to allow for some economies of scale.

Our definition of the non-linear term varies somewhat from that of O�Kelly and Bryan (1998).In O�Kelly and Bryan (1998), the non-linear term is intended as a pure discount: that is, it has theform cijklxijkl (1 � d (x)), where the second term d (x) = h (x•jk•/x)

b so that 0 6 d (•) 6 1, where x

represents the entire network flow. Therefore, in their interhub function, the cost on the interhublink is always less than the linear cost; the Fig. 3 illustrates their function for different values of thetwo parameters h and b where the bold line is the linear function cijklxijkl. In our setting, we areinterested in modelling the efficiency threshold of a hub-based itinerary, which is non-zero; that is,the interhub cost should be higher than the linear (or block train itinerary) cost up to the thresh-old, and less costly thereafter. Our non-linear cost function, which is of this latter type, and there-fore permits calibration of this efficiency threshold, is represented in Fig. 4. (In the figure, theefficiency threshold is set to 1; the bold line is the linear cost cijklxijkl. Other differences betweenO�Kelly and Bryan�s non-linear term and ours include the itinerary-dependent parameters thatwe use in our model, and the fact that our non-linear term acts directly on flow on the link, whiletheirs acts on the ratio of interhub link flow to total network flow; this latter feature essentiallyranks interhub links and gives a discount directly proportional to the rank of that hub–hub pair.In our case, different hub pairs have different cost characteristics and different efficiency thresh-olds; consequently each function is calibrated individually and does not depend upon the calibra-tion of the other interhub cost terms.

In addition, since a primary objective of the model is to evaluate the market share of the hubswith respect to currently existing direct, block train itineraries, it is necessary to include paths notpassing through the hub nodes. Flow on a direct path from i to l is therefore represented by thenew flow variable xiill, where nodes i and l are not hub nodes. Note that itineraries passingthrough exactly one hub are defined in the original model, where the flow is given as xijkl withhubs j = k.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

14

16

18

20c(

x)

x

Fig. 3. Inter-hub discount function in O�Kelly and Bryan (1998).


Adding these characteristics to the objective function (5), we obtain

WðxÞ ¼Xi2N

Xj2H

cijXk2H

Xl2N

xijkl þXj2H

Xk2H

c1jkðxÞ þXk2H

Xl2N

c2klðxÞ; ð13Þ

where c1 (x) and c2 (x) are the non-linear, concave discount functions.Then, the objective function can be expressed as

minx;x;z

F ðx; x; zÞ ¼ WðxÞ þX

ði;lÞ2W nðHxHÞcilxiill þ

Xj2H

fjzj ð14Þ

Note that the constraint (6) must be replaced by
Xj2H
Xk2H

xijkl þ xiill ¼ dil 8ði; lÞ 2 W ; ð15Þ

so as to include flows on direct routes.We first make the following assumption.

Assumption 1. The functions c1jk : RjW jþ 7!Rþ (respectively, c2kl : R

jNxH jþ 7!Rþ) are continuous,

concave increasing functions for all non-negative x, for all pairs of nodes (j,k) 2 H2, (respectively,(k, l) 2 HxN). Then, we have the following property of the objective function.

Proposition 1. The objective function F ðx; x; zÞ is a continuous, non-negative, and concave func-tion of (•,•,z) for all feasible z, and affine in ðx; x; �Þ, for all non-negative x; x.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

14

16

18

20

x

c(x)

Fig. 4. Inter-hub discount function in our model.


The resulting concave-cost hub location model is given then by Eqs. (13) and (14) and constraintset (7)–(11) and (15).

2.5. A linearized model

The concave increasing cost terms on the interhub and hub-to-destination portions of each itin-erary are approximated by a piecewise-linear function so as to permit the use of linear program-ming solvers for its resolution. To this end, the concave functions c1jkð�Þ and c2klð�Þ are dividedeach into a number M of equal unit pieces and the flows are normalized to this scale. Additionalbinary variables are introduced to represent the flow between each kink in the piecewise-linearapproximation, defined as

hhjkm ¼ 1; if the mth portion of flow is captured by the hub shuttle service between

hubs j and k; for m ¼ 1; . . . ;M ; and

hdklm ¼ 1; if the mth portion of flow is captured by the hub shuttle service between

hubs j and k; for m ¼ 1; . . . ;M ; and

and 0 otherwise, for all (j,k) 2 H2 and for all (k, l) 2 HxN. Then, the slopes of each linear unit-length piece of the functions c1jkð�Þ and c2klð�Þ can be expressed as /hh

m ¼ m0:5 � ðm� 1Þ0:5 and


/hdm ¼ m0:6 � ðm� 1Þ0:6, respectively. Recall that the constants 0.5 and 0.6 were determined in Sec-

tion 2.1.The costs in the linearized objective function are then expressed as

Wðx;hh;hdÞ ¼Xi2N

Xj2H

cijXk2H

Xl2N

xijkl þXj2H

Xk2H

a1jkX

m¼1;...;M

uhhm hhjkm þ

Xk2H

Xl2N

a2klX

m¼1;...;M

/hdm hdklm

ð16Þ
and the linearized objective function is then given by
minx;x;hh;hd;z

Wðx; hh; hdÞ þX

ði;lÞ2W nðHxHÞcilxiill þ

Xj2H

fjzj: ð17Þ

The pieces of the piecewise linear approximations must be filled in order from the most costlymarginal cost in the first piece, to the least costly in the last piece; this restriction is given by thefollowing constraints for the interhub links:

hhjkm 2 f0; 1g; j 2 H ; k 2 H ; m ¼ 1; . . . ;M ; ð18Þ

hhjkm P hhjk;mþ1; j 2 H ; k 2 H ; ð19Þ

so that hhjk2 = 1 only if hhjk1 = 1 and so on, and

hdklm 2 f0; 1g; k 2 H ; l 2 N ; m ¼ 1; . . . ;M ; ð20Þ

hdklm P hdkl;mþ1; k 2 H ; l 2 N ; ð21Þ
for the hub-to-destination links, hdklm. The following definitional constraints link the flow varia-bles with the binary approximation variables: X
ði;lÞ2Wxijkl � chh

Xm¼1;...;M

hhjkm ¼ 0; j 2 H ; k 2 H ; ð22Þ

Xi2N

Xj2H

xijkl � chdX

m¼1;...;M

hdklm ¼ 0; k 2 H ; l 2 N ; ð23Þ

where the constants chh and chd are used to normalize flow values to the unit scale used in thepiecewise-linear approximation.

The resulting linearized concave-cost hub location model is then given by objective function(16) and (17), subject to constraints Eqs. (15), (7)–(11), (18)–(21) and (22) and (23).

2.6. Polyhedral properties of the model

In this section, we provide some polyhedral properties of the above model. In Hamacher et al.(2000), the hub location polytope was studied by lifting facets from the polytope associated withthe facility location problem. It was shown that a set of inequalities define facets for the hub loca-tion problem when the definition of the flow variable represents percentages of the OD demand.(That is, the formulation is such that the sum of the flows between each OD pair over all inter-mediate hub nodes is equal to one

Xðj;kÞ2H2
xijkl ¼ 1 for each ði; lÞ 2 W :

This requires a normalization of the constraints (6)–(10) with respect to the OD demand dil. Tothis end, let the constraints (6) be given instead by
X
j2H

Xk2H

xijkl ¼ 1 8ði; lÞ 2 W ð24Þ

and remove the constant term dil from (7) and (8). Then the following result holds.

Proposition 2. Let constraint (24) replace constraint (6) in the hub location model (4)–(11), wherethe constant Q is then no longer needed in the constraints (7) and (8). Let the number of potential hubnodes, jHj P 3. Then the following inequalities are valid and define facets of the convex hull ofinteger solutions of (7)–(11), 24 and with the constant term dil removed from (7) and (8)

xijkl P 0; j 2 H ; k 2 H ; ði; lÞ 2 W ; ð25Þ

zj 6 1; j 2 H ; ð26ÞXj2H

xijkl þX

j2Hnfkgxijkl 6 zk; k 2 H ; ði; lÞ 2 W : ð27Þ

Proof. See Corollary 3.5 of Hamacher et al. (2000). h

Note that this set of inequalities differs from the basic model in that rather than dividing thehub opening constraints into two inequalities, that is, (7) and (8), a sum of the two terms is pref-erable, since it is tighter.

In the case of our model, in which demands are explicitly included, we modify the constraints(27) as follows:
X
j2Hxijkl þ

Xj2Hnfkg

xijkl 6 dilzk; k 2 H ; ði; lÞ 2 W : ð28Þ

Next, we show that (28) is both valid and facet-defining for the model (4), (15), (7)–(11).

Proposition 3. The inequality (28) is valid and defines a facet for the convex hull of constraints (7)–(11), and (15).

Proof. The first part of the proposition is straightforward, since the sum of the OD flow can neverexceed the sum of the OD flow through the hub k and the OD flow sent on a direct route. Thesecond follows as a direct extension of Proposition 2 above. h

3. Solution method

For linear hub and facility location models, a classic resolution method consists in performing aBenders decomposition and solving independently the 0–1 programming problem over the binary


variables z, which indicate whether or not a hub is to be opened, and the linear network flowproblem for a fixed set z of open hubs. In Benders decomposition, however the first phase integerprogramming master problem is generally very difficult to solve if there is no particular structureto exploit. For convex non-linear facility location, adaptations of the Benders scheme exist, as doa number of Lagrangian heuristics; the latter seek to obtain good lower bounds and then performheuristics or branch and bound from the dual solutions to obtain very good or exact primal solu-tions. Other methods have been presented in the literature for solving the linear hub locationproblem, based on the use of specialized heuristics (Klincewicz, 1991) and meta-heuristics, suchas tabu search (Skorin-Kapov and Skorin-Kapov, 1994), dual procedures and linear program-ming relaxations (Klincewicz, 1996; Skorin-Kapov et al., 1996, respectively).

However, the related problem of network flow with concave costs has been considered in theliterature significantly less than its linear and convex counterparts. This relative dearth of work,as compared to linear and convex network flow problems in general, is due most likely to the fun-damental intractability of concave-cost network problems. Indeed, concave minimization prob-lems posses a large number of local optima––for some concave functions, all of the extremepoints could qualify––and therefore require global optimization techniques to obtain exact solu-tions. Concave network minimizations are particularly intractable, as a consequence of the highdimension of the feasible region in networks.

For this reason, solution methods used in practice for concave-cost network flow problems tendto be heuristic. An important contribution in this respect is that of Minoux (1989), who provideda path flow exchange-type algorithm based on repeated shortest path calculations, and shifts offlow across the shortest paths. This algorithm, which has the added benefit of being very easyto implement, is still used today in many implementations of concave-cost network flow modelsin practice. In Balakrishnan and Graves (1989), upper and lower-bounding procedures were pro-posed to provide a good estimate of the solution to a concave-cost network minimization prob-lem. Holmqvist et al. (1998) suggested solving the concave-cost network flow problem with therandomized meta-heuristic GRASP. Kim and Pardalos (2000) propose a specialized heuristic tosolve the concave cost network flow problem by successive linear underestimation of the piece-wise-linearized cost function. One might consider further combining their method with anupper-bounding procedure, as in Balakrishnan and Graves (1989).

A classic reference of algorithms for the exact solution of concave minimization problems de-fined over polytopes is that of Horst and Tuy (1996). In this book, numerous versions are pre-sented for partitioning or covering the polytope of feasible solutions by polyhedral cones, andthen solving one-dimensional problems over each cone. While some variants of this idea havebeen proven to converge in finite time, the method assumes only continuous variables, and fur-thermore works much like a branch-and-bound scheme, enumerating wisely the extreme pointslikely to give a global solution to the problem; consequently, it is inappropriate for solvinglarge-scale concave-cost network problems.

Verter and Dincer (1995) propose a method capable of solving the concave-cost facility locationproblem exactly that also makes use of the extreme point nature of the solution. Their methodrelies on a transformation of the (concave cost) facility location problem into a standardmixed-integer facility location problem, where the number of facilities is multiplied by the numberof segments approximating the concave cost function. The authors claim that the method workswell on small problems, and can be used as a heuristic on larger facility location problems. In the


hub location problem, this would require multiplying the number of hubs by the number of seg-ments with which we would like to approximate the non-linear functions; as such, the methoddoes not appear to translate well to the non-linear hub location problem, since the number ofpaths is already much larger than in standard facility location.

In terms of concave-cost hub location models, the only reference of which we are aware is thatof O�Kelly and Bryan (1998), who test their model on a 20-node, 2–4 hub network using standardmixed-integer linear programming software. In their example, however, each piecewise-linearapproximation to their concave cost function has only two pieces. For larger-scale problems,MIP software cannot be used directly.

Rather than decompose the problem as in Bender�s decomposition, and use the techniques de-scribed above for the network problem, such as the algorithm of Minoux (1989), or apply trans-formations multiplying the size of the network by an order of magnitude, we have chosen to makeuse of recent polyhedral properties of the linear hub location problem and solve the problem as asingle, large mixed-integer program. Indeed, the preliminary results in Hamacher et al. (2000)have shown the cuts to be quite efficient at removing non-integer solutions of the hub-locationvariables, zi.

Note that for this type of approach to be efficient, the flow problem should be relatively easy tosolve; however, our problem in the flow variables is still a very difficult problem to solve. This isdue, on the one hand, to the piecewise linear relaxation of the concave cost curves, and on theother hand to the very large number of paths that must be considered in a hub-location frame-work on a real-size network. Indeed, the number of binary variables associated with the segmentsof the piecewise linear curves is on the order of several thousands. In order to approximate exactlyeach concave cost curve, it is necessary to impose the constraint that the second piece of eachpiecewise-linear function is used (that is, hhjk2 = 1 only if the first piece is used (that is, hhjk1 = 1)and similarly for the third piece (hhjk3 = 1 only if hhjk2 = 1) and so on for all remaining pieces (cf.(19) and (21)). and that each of these variables (hh and hd) is binary (cf. (18) and (20)).

To solve the model in practice, then, we have devised a variable-reduction heuristic whichsolves a sequence of relaxed subproblems, in which constraints (18) are replaced by

hhjkm 2 ½0; 1�; j 2 H ; k 2 H ; m ¼ 1; . . . ;M ð29Þ

and similarly for constraints (20)

hdklm 2 ½0; 1�; k 2 H ; l 2 N ; m ¼ 1; . . . ;M : ð30Þ

The following variable-reduction heuristic technique successively reduces the number of free var-iables, thereby forcing the initial pieces to 1, allowing only the final used piece to be fractional, asshould be the case.

3.1. A variable-reduction heuristic: heuristic 1

1. Set bMjk ¼ M and bMkl ¼ M for each (j,k) 2 H2 and each (k, l) 2 HxN.2. Solve the (relaxed) subproblem with constraints (27) and (28) replacing (18) and (20).3. For each arc (j,k) 2 H2 with fractional values of hhjkm.


3.1. Calculate the total amount of flow assigned to the arc. That is, obtain yjk ¼Pm¼1;...;bM jk

fhhjkm j hhjkm > eg for some user-specified tolerance 0 < e � 1.

3.2. Set hhjkð�yjkþiÞ ¼ 0 for i ¼ 2; . . . ; bMjk � �yjk, where �yjk ¼ yjk � ðyjkmod1Þ is the integer part ofthe assigned flow on arc (j,k). Set bMjk ¼ �yjk þ 1.

4. Go to step 2 and perform the same procedure for hdklm, for each (k, l) 2 HxN.5. Check whether either all hhjk1 2 {0,1} and hdkl1 2 {0,1} or hhjk1 2 (0,1) and hhjk2 = 0 [respec-

tively, hdjk1 2 (0,1) and hdjk2 = 0]. Otherwise, return to Step 2.

A few remarks are in order.

Remark 1. In Step 3(a), the portions of the piecewise-linear curve that should have been unusedin the case of binary hh (respectively, hd) are forced to zero. The process is then repeated, thereby

eliminating at each iteration the number of free variables in the relaxed subproblem, that is, bMdecreases at each iteration. That is, at each iteration of the heuristic, there is a reduction in thenumber of variables in the problem.

One iteration of the heuristic is illustrated in Fig. 6, on the example of Fig. 5. Since the totalflow on the pair is equal to 2.5, and should therefore use the first three (unit-length) pieces only,the remaining pieces, 4 and 5 in this case, are fixed and removed from the problem in the nextiteration.

Remark 2. Note that, in the first iteration, if any hhjkm > e and fractional, then all pieces satisfyprecisely hhjk1 ¼ hhjk2 ¼ � � � ¼ hh

jkbM due to the constraints (19) and (21). (See Fig. 5 for an illustra-

tion.) Therefore, setting the last piece to zero has the effect of ‘‘forcing’’ the higher (marginal)cost pieces to be used, or shifting the flow to a different path. This characteristic holds for theremaining iterations, with the index bM reduced at each iteration for which there are fractional flows.

Remark 3. As a stopping criterion, it is thus sufficient to check the value of pieces m = 1 andm = 2 only, since either each piece i 2 bMt

(where bMtis the number of free variables remaining

at iteration t) has an equal fractional value, or else only the last piece is fractional.

c(x)

piecewise-linear cost curveconcave cost curve

1 2 3 4 M=5

hh-m is 0.5 for each piece m, in 1,...M

flow on this pair, x = 2.5

Fig. 5. Before variables are fixed, the (fractional) flow on each piece is equal.

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Services passing through hub 11 only (1mm: <1000 KT/year; 2mm: 1000-3000 KT/year; 3mm: >3000 KT/year)

Shuttle service between 2 hubs, including hub 11 (3mm: <1000 KT/year; 4mm: 1000-3000 KT/year; 5mm: >3000 KT/year)

Fig. 6. Results of the trans-Alpine case study, centered on the Mannheim Terminal.


For example, if one considers Fig. 5 to represent the output of the relaxed subproblem reso-lution at some iteration, then Step 3 of the variable-reduction heuristic would add the followingconstraints (letting the decision variable for each piece be expressed as hhm, hhm = 0 for m = 4and m = 5, since the total flow on the path is 2.5, and the piecewise curve is defined by fivepieces. In other words, variables hh4 and hh5 are removed from the problem in subsequentiterations.

3.2. A faster variable-reduction heuristic: heuristic 2

The following heuristic speeds up the variable reduction process of Step 3 in heuristic 1, at theexpense of solution accuracy. In particular, in Step 3(c), the additional fractional pieces are fixedto at most the value of the fractional part of the total flow present on the route. (In Fig. 5, then,the piece m = 3 would have the following constraint added to those defined by heuristic 1:hh3 6 0.5.)


1. Set bMjk ¼ M and bMkl ¼ M for each (j,k) 2 H2 and (k, l) 2 HxN.2. Solve the (relaxed) subproblem with constraints (27) and (28) replacing (18) and (20).3. For each arc (j,k) 2 H2 with fractional values of hhjkm

(a) Calculate the total amount of flow assigned to the arc. That is, obtain

Table

Variab

Opera

Shunt

Tracti

Tracti

Marsh

Splitti

yjk ¼X

m¼1;...;M

fhhjkm j hhjkm > eg

for some user-specified tolerance 0 < e � 1.(b) Set hhjkð�yjkþiÞ ¼ 0 for i ¼ 2; . . . ; bMjk � �yjk, where �yjk ¼ yjk � ðyjkmod1Þ is the integer part of

the assigned flow on arc (j,k). Set bMjk ¼ �yjk þ 1.(c) Set hhjkð�yjkþiÞ 6 yjkmod1.

4. Go to step 2 and perform the same procedure for hdklm for each (k, l) 2 HxN.5. Check whether either all hhjk1 2 {0,1} and hdkl1 2 {0,1} or hhjk1 2 (0,1) and hhjk2 = 0 [respec-

tively, hdkl1 2 (0,1) and hdkl2 = 0]. Otherwise, return to Step 2.

Heuristic 2 therefore converges faster, since the problem is more constrained at each iteration,the price to pay being naturally that the additional constraints cut better solutions from the fea-sible set. The choice across the two heuristics therefore depends upon the user�s preference ofspeed in obtaining a good solution versus quality of the solution. The next section examines thistradeoff on some concrete examples.

4. Computational experience and analysis

The case study was performed on the 32 terminals in the Alpine region with the largest inter-national intermodal freight flows. Other data needed by the model concerns the different fixed andvariable costs on the network. The principal variable costs are summarized in Table 1. Fixed costsof hub development were estimated individually and three scenarios were developed: high, med-ium, and low. Pre-processing steps were taken to eliminate some clearly non-optimal hub andbi-hub itineraries from the set of possible paths.

Next, the two variable-reduction heuristics are validated on a subset of the data.

4.1. Validation of the variable-reduction heuristics

In this section, the two variable-reduction heuristics are tested on a subset of the data smallenough to permit an exact resolution of the linearized subproblem by branch and bound. The

1

le costs hub- and bi-hub itineraries

tion Cost (Euro/container)

ing 57

on to/from hubs 163

on on direct block trains 0.5/km

alling at hub nodes 8

ng 0.5

Table 2

Validation of the variable-reduction heuristics

#hubs, #nodes, #pieces Heuristic 1 solution Heuristic 2 solution Exact solution

3, 4, 25 8277 (0.3%) 9744 (18.1%) 8252

3, 5, 25 25,273 (3.9%) 28,223 (16.1%) 24,315

3, 6, 25 37,899 (4.9%) 39,610 (9.6%) 36,138

4, 5, 15 23,352 (4.8%) 26,024 (16.8%) 22,274

4, 6, 15 38,130 42,337 –

5, 6, 15 36,249 39,649 –


Table 2 provides the results of these tests. The first column of the table provides the characteristicsof the data set, in terms of number of possible hub nodes, total number of nodes (where each pairof nodes has a non-zero demand to every other pair), and the number of segments used in theapproximation of the concave cost curve. The integer programming solver of CPLEX (version6) was used to obtain the exact solution. Similarly, the CPLEX linear programming solver wasused to solve the linear subproblems in the two heuristics.

Note that, for test sets of more than three hub nodes, the integer programming solver was una-ble to provide a solution, even with the number of segments reduced to 15 for each concave costcurve. As concerns the quality of the two heuristics, the first heuristic is clearly superior to thesecond, more constrained heuristic. The first heuristic provides solutions within five percent ofthe optimal value, for those test sets for which an optimal value was obtained. On the other hand,the second, more constrained heuristic provided, on average, solutions on the order of 15% morecostly than the optimal solution. Since the computation time of the both heuristics is very reason-able (less than 0.5s for each of the test sets above), the first heuristic is clearly preferable. It is thisheuristic that was therefore used in the code for solving the overall hub location problem. Numer-ical results on the overall problem are presented below.

4.2. Results and analysis on the full data set

The purpose of the model developed is to permit studying the potential of a hub-and-spoke typenetwork of intermodal freight terminals in Europe. One key corridor for freight transportimprovement in Western Europe is the Alpine crossing. The numerical results presented in thissection are taken from the application of the model to this important corridor.

The number of nodes in the graph is 32, representing the major sources of emission and recep-tion of intermodal freight traffic in Europe. Of these, 14 terminals were selected as potential mega-

hub nodes; that is nodes which are candidates for construction of efficient container swappingequipment and fixed-composition freight shuttle service. All tests in this section were thereforerun on this full set of 32 nodes, 14 potential hub nodes, and a maximum of 100 linear piecesfor each of the concave cost curves.

The CPLEX linear programming solver was used to solve the overall problem along with var-iable-reduction heuristic 1. The addition of the constraint (28) was sufficient in eliminating non-integer solutions on very small data sets, on the order of those presented in Table 2.

To test the model and solution method on the full-size data set, initially, two sets of randomhub opening costs were generated. The results are provided in Table 3. The first column provides

Table 3

Results on the full-size data set

Costs fj Hubs open {i:zi = 1} {i:zi = 0} I ¼ fi : zi 2 ð0; 1Þg; lðzIÞRandom 1 0,1,3,4,5,6,7,8,9,10,11,13 – j I j¼ 2; l ¼ 0:04Random 2 0,1,3,11,13 5,6 7, l = 0.05

High est. 0 12 12, l = 0.16

Med. est. 1,6,11 13 10, l = 0.08

Low est. 0, 5, 6, 11 10, 13 8, l = 0.07


the type of hub-opening costs used in the test set. The second column gives the number of hubswith values zi identically 1, and then those with values equal to 0. The last column provides thenumber of hubs with fractional values of zi and the average fractional value.

The main thing to observe from Table 3 is that, when fractional values were present in the re-sults, the numerical values were very close to zero, particularly in the tests with randomly gener-ated hub opening costs. In the last three tests, realistic hub-opening costs were used (with high,medium, and low estimates of the costs). The quality of the polyhedral information is good,but could be stronger. This is seen from the fact that the average value of the non-integer resultsis significantly higher than in the first two results, particularly in the test set ‘‘High est’’. In thisrange, rounding heuristics tend to perform very poorly, and branch-and-bound can require sub-stantial computing time from such points.

Nevertheless, if one takes two averages in that set, the first over the non-integer values of zj suchthat zj 2 [0.5,1) (call it l>) and the second over the zj for which zj 2 (0,0.5) (call it l<), we obtainl> = 0.85l< = 0.07; that is, the non-integer values are in fact quite close to 0 and 1. This is impor-tant since for values of zj < 0.08, a single branch and bound node often provides integer solutions.

Qualitative results with the model run on the case study provided here show that when hubs 0,6, and 11 (Munich, Verona, and Mannheim) are of great importance in reducing transportationcosts through consolidation across the trans-Alpine region, since they are opened for both lowand medium-level hub development costs, and hub 5 (Milan) of importance when the openingcosts are sufficiently low.

Fig. 7 illustrates the shuttle services proposed by the model to and from the Mannheim terminalwhen hub development costs are low. It is of interest to note that the Mannheim hub is used heav-ily towards Italy in the north-to-south direction. However, with the exception of flows from Turinto Mannheim, which most likely regroup flows originating in France and the Iberian peninsula,the trans-Alpine flows towards central Germany are not well balanced. This means that the effi-ciency of shuttle services crossing the Alps would be compromised somewhat by relatively heavyuse from north to south, and significantly lower usage from south to north.

Fig. 8 shows qualitative results from the trans-Alpine study on the same data set (low hubdevelopment costs), this time focused on flows in and out of the Munich terminal. Indeed, themega-hub at Munich is shown to consolidate freight flows on the heavily used North–SouthAlpine corridor, that is to Verona and Naples, in particular, as well as from North-east Europeto West Europe. Furthermore, the hub at Munich is shown to consolidate flow to other Germancities such as Hamburg and Nurenburg. The high-speed shuttle service proposal on the corri-dor Mannheim–Milan (hub 11 – hub 5), according to the model solution, would consolidate

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Fig. 7. Results of trans-Alpine case study around the Munich terminal.


significant flow from north to south, as well flow with destinations in Spain and France. The pairMilan–Verona (hubs 5, 6) would be important in consolidating flow across Italy, such as to Genoaand Bologna.

Freight flows that continue to be sent on direct block trains as opposed to taking shuttles orsingle-hub itineraries illustrate the corridors where consolidation is insufficient to render thehub-based itineraries economical, at least with the present cost savings estimates. An examinationof these flows shows that some of the Italy-towards-Germany flows are in this category, includingRome to Mannheim, Rome to Munich, Naples to Mannheim, and Naples to Munich.

4.2.1. Empirical sensitivity analysis: reduced scale economiesAn empirical sensitivity analysis was performed by adjusting the coefficients of the discount

functions; the coefficient was maintained on hub-to-hub links, but the scale economies that couldbe obtained from hub to non-hub destination terminals were decreased by increasing the param-eter from 0.6 to 0.8. Based on high, medium, and low hub development costs, as before, we illus-trate the results on the hub variables in Table 4.

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Fig. 8. Results of the trans-Alpine case study, centered on the Mannheim Terminal.

Table 4

Perturbing the hub–destination parameter on the full-size data set

Costs, fj Hubs open {i:zi = 1} {i:zi = 0} j I j; lðzIÞ; j I<0:1 jHigh est. – 2,4,10,12,13 9, l<0.1 = 0.03, 7

Med. est. – 10,12,13 11, l<0.1 = 0.05, 8

Low est. 0,5,11 10,12,13 8, l<0.1 = 0.06, 5


In this table, the first, second, and third columns are as before. The last column provides thenumber of hubs with fractional values of zi and the average fractional value of those less than0.1, followed by the number of zi such that zi 2 (0,0.1).

We observe that the model appears to be quite sensitive to the parameter that adjusts the degreeof scale economies. With medium-level hub development costs, the reduction of scale economiesled to a solution in which no hubs were to be opened. Three potential hubs had z values identicallyzero, i.e. closed, and 11 hub opening variables were between 0 and 1, with eight having values of


0.1 or less, and a mean among those values of 0.05. High costs led to two additional hubs having zvalues identically zero, and the indeterminate values were, on average, lower. Low hub develop-ment costs meant that three hubs should be opened, three closed, eight indeterminate, but five ofthose had small values (0.06 on average); this last simulation then suggested a lower use of hubsand shuttle services than did the original tests.

This set of simulations, therefore, indicate that the model is indeed sensitive to those parame-ters, which means furthermore that they must be set with great care. In terms of making any qual-itative evaluation from the sensitivity analysis, one would conclude only that, given low hubdevelopment costs, the terminals at Munich, Milan, and Mannheim (0, 5, and 11) stand out asefficient, since they remain so even when scale economies on some paths are reduced.

A map indicating the flows and shuttle services in and out of Mannheim, when the scale econ-omies are reduced on the hub-to-destination links, and for low hub development costs, is providedin Fig. 8. The illustration of flows and services in and out of this important hub confirms theobservation made from examining the initial values for the z variables. High-speed shuttle servicesare still present, but given the reduced scale economies on hub-to-destination links, even low hubdevelopment costs outweigh the cost advantages on many corridors.

4.2.2. Comparison with a cruder linear approximation

A second empirical sensitivity analysis tested the behavior of the model when the piecewise-lin-ear approximation was made very crudely. That is, only three pieces were used to approximateeach concave-cost curve, and the original simulation was re-run. This test is important since algo-rithms proposed in the literature for concave minimizations (such as Kim and Pardalos (2000),O�Kelly and Bryan (1998)) make use of approximating functions with only two or three pieces.A crude approximation of this type reduces significantly the computational burden (since a binaryvariable is needed per piece for every (hub-originating) link). In our case, on the full-size data set,computation time was decreased from roughly 1.5h to 20min when the three-piece approximationwas used.

However, we predict that such a crude approximation will also reduce the quality of the solu-tion. Indeed, within each piece, costs increase linearly; if, therefore, successive iterations of ouralgorithm test shifting flow to a hub-based itinerary, but costs remain within a single piece ofthe curve, then the hub-based itinerary is likely not to be chosen, whereas with a more preciseapproximation it may be.

Table 5 summarizes the results of running the model with a 3-piece approximation of the non-linear cost curves and original parameter values (from Table 3).

In this table, the last column provides the number of hubs with fractional values of zi and theaverage fractional value of all fractional zi, since all fractional zi have values less than 0.07.

Table 5

Using a crude three-piece linear approximation on the original data set

Costs fj Hubs open {i:zi = 1} {i:zi = 0} j I j; lðzIÞHigh est. – 1,2,3,4,7,8,9,10,12,13 4, l = 0.04

Med. est. – 2,4,10, 12,13 9, l = 0.03

Low est. 6,11 10,12,13 9, l = 0.02


Indeed, we observe that, compared to results from Table 3, the number of hubs open is reduced,the number of hubs to be closed is increased, and of the intermediate values, we find that all aresignificantly under the 0.1 threshold.

If we compare the results of approximating concave costs with only three pieces to results fromTable 4 above, we see that, although scale economies in the present simulation are significant, theeffect of the crude approximation is stronger than reducing scale economies to the level of Table 4.Indeed, more hubs are to be closed in the present simulation than in that of Table 4, for each sce-nario of hub development costs.

We can conclude from this experiment that, in this type of network model, which is very sen-sitive to the degree of scale economies on a path, a more precise approximation of those scaleeconomies can be worth the added computational expense. This is all the more true if, in bothcases, heuristic methods will need to be used.

5. Conclusions

We have presented an application of locating the optimal configuration of intermodal freighttransport hubs and obtaining their usage levels. The model we propose for this application isbased on the uncapacitated hub location problem. We further add to this model an accurate rep-resentation of the economies of scale due to consolidation; this is accomplished through explicituse of concave cost functions for the interhub (and hub-to-destination) portions of each trip.

In terms of modelling and solving real-life problems of this type, taking into account time-of-day through a dynamic model, over a one or several-day period, would improve substantially therealism of the model, since it would permit comparison of dedicated freight lines with the currentpassenger-priority rule that relegates freight flows to the night hours. Another practical contribu-tion could come from differentiation of freight flows by product type.

In order to solve in practice this complex model on a static network, we proposed two heuristicsfor solving a piecewise approximation of the non-linear, concave cost curves that permit handlingeven very large problems quickly. The efficiency of the heuristics is such that the piecewise linearapproximation need not lose much; indeed the number of pieces considered for each curve can belarge, and problems having 30 nodes are still easily solvable. We compared empirically resultsfrom a much cruder (three-piece) piecewise-linear approximation, which showed that the qualityof the solution is indeed effected significantly by this simplification.

Comparisons with exact solutions on smaller test sets show that the percentage deviation of theheuristic on our tests was within five percent of optimal. The use of recent results on polyhedralproperties of the model enable us to obtain quasi-integer solutions with a mixed-integer formula-tion of the problem, with better results on small-scale problems.

Interesting extensions of the algorithmic side of this work could involve the use of non-linearprogramming techniques, such as non-linear column generation, to obtain good lower bounds,in conjunction with some upper-bounding scheme. It would be interesting as well to perform acomparative study of the proposed heuristic with other techniques for handling concave costs,such as the linear programming-based methods. Our results have shown as well that furtherwork on the polyhedral properties of the problem would be of benefit on problems of mediumand large size.


Acknowledgments

This work was supported in part by the European Union DG-7 projects IQ, and SCENES, andcarried out while the second author was at INRIA and the Universite de Versailles, France. Datawas supplied by the Department of Transport Economics (DEST), INRETS, Arcueil, France. Thesecond author thanks Dr. Tim Sonneborn of ITWM, Kaiserslautern, for valuable comments. Theremarks of Professor Teodor Crainic on an earlier version of this work, as well as those ofthe Associate Editor and the anonymous referees, are gratefully acknowledged.

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