minimizing the maximum travel time in a combined model of facility location and network design
TRANSCRIPT
Omega 40 (2012) 847–860
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Omega
0305-04
doi:10.1
n Corr
E-m
ivan.con
gerhard
journal homepage: www.elsevier.com/locate/omega
Minimizing the maximum travel time in a combined model of facilitylocation and network design
Ivan Contreras a,n, Elena Fernandez b, Gerhard Reinelt c
a Concordia University and Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT), Montreal, Canadab Statistics and Operations Research Department, Technical University of Catalonia, Barcelona, Spainc Institute of Computer Science, University of Heidelberg, Heidelberg, Germany
a r t i c l e i n f o
Article history:
Received 27 January 2011
Accepted 23 January 2012
Processed by Associate Editor Pesch.maximum customer-facility travel time. The model generalizes the classical p-center problem and has
various applications in regional planning, distribution, telecommunications, emergency systems, and
Available online 24 February 2012Keywords:
Facility location
Network design
Integer programming
Valid inequalities
83/$ - see front matter & 2012 Elsevier Ltd. A
016/j.omega.2012.01.006
esponding author. Tel.: þ1 514 8484 2424x3
ail addresses: [email protected],
[email protected] (I. Contreras), e.fernandez@u
[email protected] (G. Rei
a b s t r a c t
This paper presents a combined Facility Location/Network Design Problem which simultaneously
considers the location of facilities and the design of its underlying network so as to minimize the
other areas. Two mixed integer programming formulations are presented and compared. Several valid
inequalities are derived for the most promising of these formulations to strengthen its LP relaxation
bound and to reduce the enumeration tree. Numerical results of a series of computational experiments
for instances with up to 100 nodes and 500 candidate links are reported.
& 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Network location models cover a wide range of applicationswithin the design of transportation and telecommunication sys-tems. These models are frequently used to analyze and determinethe location of public and private facilities within a prespecifiednetwork. There exist a variety of network location models whichcan be categorized according to the type of objective that isconsidered. The min-sum models, such as the classical p-medianproblem [17] and the uncapacitated facility location problem([19]), focus on minimizing the overall set-up or operational cost.The min–max models, such as the p-center problem [17], focus onminimizing the largest customer-facility distance. The coveringmodels, such as the set covering location problem [32] and themaximum covering location problem [4], focus on finding theleast number of facilities to cover all customers or the maximumnumber of customers to cover within a prespecified distance,respectively.
All these location models assume the underlying network to begiven as an input. However, in some contexts the topology of thenetwork plays a crucial role for the optimal location of facilities(see [11]). Economically it may be more attractive to modify theunderlying network than to add facilities. For this reason, com-bined facility location network design models play an important
ll rights reserved.
130.
pc.edu (E. Fernandez),
nelt).
role in situations in which there exists a non-trivial tradeoffbetween set-up costs for the location of facilities and links of thenetwork as well as operating costs for providing service tocustomers. These models have a broad range of applicationsrelated to the design of distribution systems, air and groundtransportation networks, telecommunication networks, hub-and-spoke networks, and regional planning efforts (see [23] fordetails).
Facility Location/Network Design Problems (FLNDPs) are a chal-lenging class of combinatorial optimization problems that com-bine two types of decisions. The location decisions consist ofselecting a set of nodes to locate facilities and designing theallocation pattern of nodes to facilities. The network designdecisions consist of selecting a set of links to provide theconnection of nodes to their allocated facilities. Several costsmay affect the structure of optimal solutions such as set-up costsof facilities, set-up costs of links to connect the nodes to facilities,and operating (or service) costs to transport the demand throughthe network. FLNDPs were initially introduced by Daskin et al.[11] as generalizations of classical facility location problems. TheUncapacitated Facility Location/Network Design Problem (UFLNDP)was proposed by Daskin et al. [11] as a generalization of theuncapacitated facility location problem and further studied byMelkote and Daskin [23]. The UFLNDP considers the minimizationof the overall operational costs as well as the set-up costs offacilities and links. As the magnitudes of set-up and service costsin UFLNDP may be non-homogeneous, adding the above threeterms into one single objective may be meaningless. One possi-bility for considering all these costs simultaneously is to minimize
I. Contreras et al. / Omega 40 (2012) 847–860848
the operating costs while requiring that the overall set-up costssatisfy a given budget constraint. This approach is often applied innetwork design problems (see, e.g. [15,33]). Adding a budgetconstraint to FLNDP gives rise to the so-called Facility Location/
Network Design Problem with Budget constraint (FLNDB), whichwas introduced by Melkote and Daskin [23] and further studiedby Cocking et al. [5] and Cocking [6]. Both UFLNDP and FLNDPassume that the amount of demand that can be served byfacilities is unbounded. This is valid in situations in which it isknown a priori that the facilities will operate significantly belowtheir capacity levels. Nevertheless, there exist situations in whichthe capacities of the facilities are very constraining. Melkote andDaskin [24] presented the Capacitated Facility Location/Network
Design Problem (CFLNDP), which extends the UFLNDP to considera limited amount of capacity at the facilities. Drezner andWesolowsky [13] deal with another class of min-sum FLNDPs inwhich the links can either be one-way or two-way and the servicecosts consider round trips from the facility to each node and back.
Like in many network location models, the min-sum objectiveis typically useful in cases in which the objective is to minimizethe average service cost of the network without taking intoconsideration if some customers have not been properly allocatedin terms of the distance to the facilities. However, there existcontexts in which a cost-based objective is not an appropriatemeasure of the quality of the solution. This is the case whendesigning networks for emergency services such as the location ofhealth facilities or emergency units. Service costs representcustomer-facility travel times, which are critical for each indivi-dual customer and should not be overlooked using a cost-basedobjective.
In this paper we introduce the Center Facility Location/Network
Design Problem with Budget constraint (CFLNDB), which consist ofselecting a set of nodes to locate facilities and a set of links toallow the connection between customers and facilities. TheCFLNDB considers both service costs and design (or set-up) cost.Service costs are taken into account in the objective functionwhich aims at minimizing the maximum travel time between anynode to their allocated facilities, rather than minimizing theoverall service costs as in the previously proposed FLNDPs. Theoverall design costs are taken into account by means of a budgetconstraint which limits the overall set-up costs to a maximumbudget. It is assumed that the amount of demand that can beserved from the facilities is unbounded. In order to build a moreflexible network, we let the model decide the best number offacilities and links to install within a prespecified budget, ratherthan specifying a priori the number of them. The CFLNDB is ageneralization of the classical p-center problem, which has beenextensively studied (see [27,28]). However, to the best of theauthors’ knowledge the CFLPNDB has not been studied so far.
The CFLNDB involves one additional difficulty with respect toclassical center location problems, because for tracing the pathsthat determine the value of the objective function, the location ofthe facilities and the allocation of nodes to facilities must beknown, in addition to the selected arcs.
Potential transportation applications of FLNDPs where a min–max approach is required arise in regional planning and landreuse programs. In this case, the government may be simulta-neously considering the construction of new roads as well as thelocation of public facilities such as hospitals, police and firestations, and schools. Given that these facilities provide essentialand emergency services, it is strongly desirable to minimize themaximum travel time from any potential customer to its closestfacility. A concrete example of an application of the CFLNDB to aregional planning program appears in the design of a roadstructure for improving the travel times to health facilities inthe Nouna District of Burkina Faso, studied in Cocking et al. [5]
and Cocking [6]. In Nouna, travel times of population to healthservices become extremely high during the rainy season, sincemany roads are unusable. Cocking [6] approaches this problem bydeciding the number and location of the health facilities over apotential set of villages and the roads to be built or improve toconnect the health facilities with a set of nodes representing thepopulation. A min-sum objective is considered and the problem isformulated as a particular case of the FLNDB. However, in thisparticular application a min–max objective function is moreappropriate for the reasons already described. Other papers, suchas Syam and Cote [31] and Murawski and Church [26], have alsostudied the location of health services, however, they havefocused on other types of objectives such as min-sum or max-covering.
Additional applications of FLNDPs where a min–max approachis required arise in the design of less-than-truckload (LTL)distribution systems. In this case, transportation times throughthe network are critical in achieving delivery deadlines. Therefore,LTL carrier companies are interested in designing a transportationnetwork such that it minimizes the maximum travel time fromthe breakbulk facilities to the distribution points, subject tobudgetary limitations on the design costs for the location ofbreakbulks and the establishment of a service link between twopoints.
1.1. Related literature
Combined problems of facility location and network design havebeen studied under different names, according to their particularapplications. Contreras and Fernandez [7] presents a unified frame-work for the general network design problem which encompassesseveral classical problems involving combined location and networkdesign decisions. Recent surveys on discrete facility location such asReVelle and Eiselt [27], Klose and Drexl [20], ReVelle et al. [28], andSmith et al. [29], point out some location models in which thereexist network design decisions. Melo et al. [25] presents a surveycontaining more than 120 references of facility location modelsarising in the design of supply chain networks. The reviewedmodels include a combination of strategic features such as multi-layer facilities, multiple commodities, single/multiple period(s), anddeterministic/stochastic parameters. Some of these models alsoinclude tactical and operational supply chain decisions such ascapacity design, inventory control, procurement, production plan-ning and routing. Srivastava [30] introduces an integrated frame-work for the location-allocation of facilities to design cost-effectivereverse logistic networks. Cordeau et al. [9] propose an integratedmodel for logistic network design involving strategic decisionsregarding the number, location, capacity and technology of manu-facturing plants and warehouses. It also considers tactical decisionssuch as the selection of suppliers, product range assignment,distribution channel, and transportation modes. Some operationaldecisions, like the routing of raw materials, semi-finished andfinished products through the network, are also consider inthe model.
The literature on problems involving network decisions withmin–max objective is scarce. Campbell et al. [3] present a modelto minimize the maximum travel time in specific network designproblems that involve no location decisions. They consider aseries of problems that involve the selection of q arcs in alreadyestablished networks so as to minimize the total diameter of theupgraded network. In [1], several hub location problems withmin–max objectives are described. However, these problemsconsider the requirement that hub facilities induce a completegraph, making the evaluation of the objective function mucheasier. We are not aware of any work that considers a min–maxobjective with joint location and network design decisions
I. Contreras et al. / Omega 40 (2012) 847–860 849
without making additional requirements on the topology inducedby the selected arcs.
Recently, some papers have appeared dealing with combinedmodels of facility location and network design. Murawski andChurch [26] study a FLNDP in which the objective is to maximizethe demand that is captured subject to a budget constraint on thetotal installation cost of the selected arcs. Costa et al. [10] presenta network design model with location ingredients arising inelectrical distribution systems. The problem consists of selectinglinks to locate two types of edge facilities as well as locating nodefacilities to allow the connection of edge facilities. The objective isto minimize the sum of design costs of edge and node facilitiesand the operational costs. Gollowitzer and Ljubic [16] studiesconnected facility location problems, which combine facilitylocation and Steiner trees. These problems consider the locationof facilities, the assignment of customers and the connection ofopen facilities via a Steiner tree. The objective is to minimize thedesign cost for the facilities and the links of the Steiner tree aswell as the service costs.
The rest of the paper is organized as follows. In the followingsection, the CFLNDB is formally defined and its complexityestablished. In Section 3, we present two different mixed integerprogramming formulations. One of them is based on multicom-modity type variables to trace the paths in the network and theother is based on the fact that optimal solutions to CFLNDB areassociated with rooted forests. Given that the former formulationturns out to be the most promising when used in a generalpurpose solver, in Section 4 we show how it can be successivelyimproved by means of different types of valid inequalities. Thedescription of the computational results and the analysis of theobtained results are given in Section 5. Before presenting theobtained numerical results, we analyze the effect of employingtight upper bounds to reinforce some constraints of the formula-tion, and for simple but effective reduction tests. Section 6 drawssome conclusions and some comments on possible lines for futureresearch.
2. The Center Facility Location/Network Design Problem
Let D¼ ðV ,AÞ be a directed graph with node set V ¼ f1;2, . . . ,ngand arc set A. Let f iZ0 denote the set-up (or design) cost ofselecting node i as a facility. For each arc (i,j), let cijZ0 and tijZ0denote its construction cost and travel time, respectively. Further-more, BZ0 gives the budget on the overall installation costs(facilities plus arcs). Feasible solutions to the Center Facility
Location/Network Design Problem with Budget constraint (CFLNDB)consist of:
–
a set of nodes of V to locate facilities, – an allocation of non-facility nodes to facilities, – a subset of arcs that connect each node to its allocated facility,such that the total set-up cost of the facilities plus the construc-tion cost of the selected arcs does not exceed the budget B. Thevalue of a feasible solution to CFLNDB is given by the maximumtravel time from any node to its allocated facility, in the graphinduced by the selected arcs. Then the problem is to find a feasiblesolution of minimum cost.
In the case of FLND and FLNDB, when all set-up and transpor-tation costs are non-negative, it is known that there exists anoptimal solution which is a rooted forest (see [23]). As we will seenext, this property also holds for CFLNDB. Observe first that theonly arcs that are required in a feasible solution are the ones thatdefine the paths Pi from each node iAV to its allocated facility. ForiAV the length of Pi determines the travel time from i to its root
node. As a consequence, any arc not belonging to any such pathcan be removed from any feasible solution without affecting itsfeasibility or its objective function value. Moreover, given that it isassumed that nodes where facilities are located are self-assignedand there are no capacity constraints, no path Pi will contain morethan one facility. Therefore, arcs between two facilities will neverbe part of an optimal solution. This implies that there exists anoptimal solution to CFLNDB that does not contain any arcconnecting two facilities or a non-facility node and a facilitydifferent from the one it is allocated to. Thus, each component ofsuch a solution contains one single facility which is its root node.In addition, if one such component contains a cycle, say C, thenthere exists one arc aAC not belonging to any of the paths Pi, iAV .Therefore, arc a can also be removed without affecting thefeasibility or deteriorating the objective function value of theresulting solution. As a consequence there exists an optimalsolution which is a rooted forest.
Let R denote the set of rooted forests of D, and RB the subsetof rooted forests that satisfy the budget constraint, i.e.,
RB ¼ RAR :X
Tk AR
f rkþX
Tk AR
Xði,jÞAAk
cijrB
8<:
9=;:
For a rooted forest R we write R¼ T1,T2, . . . ,Trf g, where
–
Tk ¼ ðVk,Ak,rkÞ is a rooted tree with VkDV , AkDA, and rootnode rkAVk, for k¼ 1, . . . ,r,–
fV1,V2, . . . ,Vkg defines a partition of V.For RAR and iAV given, we call the root of i the root node of theunique Vk such that iAVk, and we denote by tði,RÞ the travel timefrom node iAVk to its root node rk, which is the total travel timeof the unique path Pi from i to rk in R. Then, the objective functionvalue of RAR is given by the maximum travel time from any nodeiAV to its root node
maxiAV
tði,RÞ:
Therefore, using the above notation, CFLNDB can be stated as,
ðCFLNDBÞ minRARB
maxiAV
tði,RÞ:
Observe that the well known p-center problem defined on acomplete graph, with assignment costs dij, i,jAV satisfying thetriangle inequality, is a particular case of CFLNBD with designcosts fi¼1, iAV , construction costs cij ¼ 0, i,jAV , travel timestij ¼ dij, i,jAV , and budget B¼p. Since the p-center problem isNP-hard (see, e.g. [18,22]), we can also establish the NP-hardnessof CFLNDB.
Note that without the budget constraint the node optimalityproperty can be proved for CFLNBD using similar arguments as forthe center location problem [17]. That is, even if facilities can belocated at the edges of the network, an optimal solution exists inwhich all the facilities are located at the nodes of the network.However, like in other center location problems with additionalknapsack type constraints, after introducing the budget con-straint, the node optimality property no longer holds. In addition,in order to allow facilities to be located at the edges of thenetwork, the facility set-up cost function should be defined for allthe points of the edges. In any case, like it is usual in most locationproblems on networks, we assume in the following that facilitiescan only be located at the nodes of the network.
3. Mathematical programming formulations of the CFLNBD
In classical location problems service costs, representing theassignment costs of customers to facilities, are known in advance.
I. Contreras et al. / Omega 40 (2012) 847–860850
This makes it possible to model the center objective as atelescopic sum [14] or by means of covering type constraints,which limit the maximum assignment cost of a customer to itsallocated facility [12]. In contrast, in bottleneck network designproblems, the above techniques become extremely involved,since service costs are not known in advance. These now repre-sent communication costs between pairs of customers, whichdepend on the selected arcs, and the objective is to minimize themaximum communication cost given by the diameter of thenetwork induced by the selected arcs (see, e.g. [3]). Moreover,in order to evaluate the objective it is not enough to know whichare the selected arcs, as the paths connecting the different pairs ofnodes must be traced. In CFLNDP service costs represent accesstimes of nodes to their allocated facilities in the network inducedby the selected arcs, and the objective is to minimize themaximum service cost. Therefore, these problems involve oneadditional difficulty, because for tracing the paths that determinethe value of the objective function, the location of the facilitiesand the allocation of nodes to facilities must be known, inaddition to the selected arcs.
In this section we present two different formulations forCFLNBD. The first one makes use of multicommodity typedecision variables, like it is common in other network designproblems (see for instance [7,21]). In the second formulation,which exploits the structure induced by optimal solutions, deci-sion variables are defined to represent feasible rooted treesRARB.
3.1. Formulation with multicommodity type decision variables
We define several sets of decision variables to representfeasible solutions to the CFLNBD. The location of the facilities isrepresented by the binary variables
zi ¼1 if a facility is located at node i,
0 otherwise,
(
for each iAV .The assignment of nodes to facilities is modeled with binary
variables x, where for each i,jAV with ja i,
xij ¼1 if node i is assigned to facility j,
0 otherwise:
(
The arcs selected to connect the nodes with their allocatedfacilities are modeled with binary variables y, where for eachðk,mÞAA,
ykm ¼1 if arc ðk,mÞ is constructed,
0 otherwise:
(
The paths from nodes to their allocated facilities, which needto be traced in order to evaluate the objective function, aremodeled by means of multicommodity type binary variables X,where for each i,jAV with ja i, and ðk,mÞAA, we set
Xijkm ¼
1 if arc ðk,mÞ is on the path from i to j,
0 otherwise:
(
Finally, we define a continuous variable T that gives themaximal accumulated travel time for all paths. Then, CFLNDBcan now be written as the (mixed) integer program
min T , ð1Þ
Xðk,mÞAA
tkmXijkmrT 8i,jAV , ð2Þ
XjAV\fig
xijþzi ¼ 1 8iAV , ð3Þ
Xði,kÞAA
Xijik ¼ xij 8i,jAV ,ja i, ð4Þ
Xðk,mÞAA
Xijkm�
Xðm,kÞAA,ka i
Xijmk ¼ 0 8i,j,mAV ,ja i,m, ð5Þ
Xðk,jÞAA
Xijkj ¼ xij 8i,jAV ,ja i, ð6Þ
xijrzj 8i,jAV ,ja i, ð7Þ
Xijkmrykm 8i,j,AV ,ja i,ðk,mÞAA, ð8Þ
XiAV
f iziþXðk,mÞAA
ckmykmrB, ð9Þ
XijkmAf0;1g 8i,j,AV ,ja i 8ðk,mÞAA, ð10Þ
xij,ziAf0;1g 8i,j,AV , ð11Þ
ykmAf0;1g 8ðk,mÞAA: ð12Þ
Constraints (2) keep track of the longest accumulated traveltime among all possible paths. The set of equalities (3) states that,if node i is not a facility, then it has to be assigned to one.Constraints (4)–(6) are a particular case of flow conservationconstraints, by which a path connecting nodes i and j is con-structed, only if node i is assigned to a facility located at node j. Asusual, in this type of constraints, for each pair i,jAV one equalityis redundant and can be eliminated. Inequalities (7) ensure thatnodes are assigned to open facilities, whereas constraints (8)guarantee that all the arcs of the paths are constructed. Thebudget limitation on the total set-up costs for the location offacilities and the construction of links is imposed in (9). Finally,((10)–(12) are non-negativity and integrality conditions on thedecision variables. In the following formulation (1)–(12) above isreferred to as formulation F.
3.2. Alternative formulation
Formulation F uses multicommodity type decision variables,which are commonly used in network design models (see, e.g. [2]).This type of variables is indeed very intuitive but, on the otherhand, requires four indices, which most often limits the size of theinstances that can be solved efficiently. For this reason we havedeveloped an alternative formulation for CFLNDB which usesfewer variables and exploits the rooted tree structure of optimalsolutions. We use the same variables z and y as in F and definesome new decision variables:
�
Non-facility nodes are partitioned in two categories:leaves and intermediate nodes. Leaves are modeled usingtwo-index binary variables h which also indicate the root node(facility) to which the leaf is allocated. For each i,jAV withja i, we sethij ¼1 if node i is a leaf assigned to facility j,
0 otherwise:
(
Intermediate nodes are modeled with binary variables q. Forthese nodes we do not explicitly indicate the facility they are
I. Contreras et al. / Omega 40 (2012) 847–860 851
assigned to. For each iAV we define
qi ¼1 if node i is an intermediate node of some path;
0 otherwise:
(
�
Finally, we define continuous variables g for the travel times.For every jAV , gj gives the maximal accumulated travel timeof any path of R from a leaf to node j.In any rooted forest, for each jAV , at least one path starting ata leaf and containing j exists. If a facility is located at j, then j isa root of the forest, and the path terminates at j. Otherwise, if jis not a facility, the path continues towards a root node. In anycase, gj represents the maximal accumulated travel time of anypath that starts at a leaf and enters node j. As travel times areassumed to be non-negative, in any feasible solution, themaximum of all the gj values corresponds to the total traveltime from some leaf node to the facility it is allocated to, andindicates the objective function value of the solution.
CFLNDB can now be written as the (mixed) integer program
min maxjAV
gj,
ðgiþtijÞyijrgj 8ði,jÞAA, ð13Þ
XjAV\fig
hijþqiþzi ¼ 1 8iAV , ð14Þ
XjAV\fig
hijþqi ¼Xði,jÞAA
yij 8iAV , ð15Þ
hijþqjrXðk,jÞAA
ykj 8i,jAV , ð16Þ
yijrzjþqj 8ði,jÞAA, ð17Þ
hijrzj 8i,jAV , ja i, ð18Þ
XiAV
f iziþXðk,mÞAA
ckmykmrB, ð19Þ
giZ0 8iAV , ð20Þ
hij,zi,qiAf0;1g 8i,jAV , ð21Þ
ykmAf0;1g 8ðk,mÞAA: ð22Þ
Constraints (13) keep track of the longest accumulated traveltime among all paths that terminate at node jAV . The set ofconstraints (14) establishes a partition of the set of nodes of thenetwork: every node iAV is either the beginning of some path, anintermediate node, or it is used to locate a facility. The paths fromnodes to facilities are defined by constraints (15)–(17). When iAV
is a leaf or an intermediate node, constraints (15) ensure thatexactly one arc leaving node i exists. However, when iAV is afacility, constraints (15) ensure, together with constraints (14),that there is no arc leaving node i. Inequalities (16) ensure that,when jAV is an intermediate node or a facility, then some arcentering node j exists. The set of complementary constraints (17)guarantees that the end node of an arc is either an intermediatenode or a facility. Alternatively, taking constraints (14) intoaccount, inequalities (17) can be expressed as
Pðj,kÞAAhjkþ
yijr1, indicating that when jAV is a leaf, no arc may enter node j.Constraints (18) ensure that no leaf is assigned to a node
which is not a facility. Together with constraints (15), they alsoguarantee that no path exists which ends at a node which is not afacility. Constraint (19) is the budget constraint that guaranteesthat the total set-up costs for the location of facilities and the
construction of links does not exceed a pre-specified value.Finally, constraints (20)–(22) are the non-negativity and integral-ity constraints, respectively.
Formulation (14)–(22) is a min–max non-linear mixed integerprogram that cannot be used directly by any commercial solver toobtain optimal solutions to CFLNDB. Thus, we linearize con-straints (13) with the classical big-M technique, where M is anupper bound on the value of any feasible solution (e.g. the sum ofall the travel costs). Then, we obtain the following mixed integerlinear programming formulation:
ðCP0Þ min T ,
TZgi 8iAV , ð23Þ
gjZgiþtijyij�Mð1�yijÞ 8ði,jÞAA,
ð14Þ2ð22Þ: ð24Þ
Note that, by constraints (23), variable T is the maximumaccumulated travel time among all nodes jAV and, thus, itrepresents the optimal value to the problem.
Let CP denote the formulation that results from CP0 when theintegrality conditions hijAf0;1g are substituted by hijZ0, i,jAV .An optimal solution to CP0 can be obtained by solving CP. If in CPthe variables z, q and y remain binary, then constraints (14) and(15) imply that in any feasible solution to CP,
PjAV\fighij takes the
value 0 or 1, for all iAV . Moreover, in such a solution the values ofthe variables h do not determine the objective function value.Thus, if in a feasible solution hij1
¼D140 and hij2¼D240 for
some iAV , then the solution with hij1¼D1þD240 and hij2
¼ 0and all other variables as before, is also feasible and has the sameobjective function value. Therefore, in the following we will focuson the solution to CP.
3.3. Comparison of formulations
We next perform a comparison, between the multicommodityflow-based formulation F and the alternative formulation CP, toobserve their effectiveness and limitations for solving CFLNDBinstances. Table 1 summarizes the numerical results obtainedwith both formulations on two sets of randomly generatedinstances (see Section 5 for a detailed description of the instancesgeneration process), each of them with 16 instances of 10 and 20nodes, respectively. All instances were solved with the MIP Xpressoptimizer, under a Windows environment on a VAIO VGN-SR19XN, with a processor Intel(R) Core(TM)2 P8400 at 2.26 GHzand 3 GB of RAM. We enabled the generation of cuts in order toobserve the potential improvement of the LP relaxation bound byusing them. Default values were used for all other parameters.A CPU time limit of 1 h was set.
Table 1 has two blocks of columns, one for each dimensionnAf10;20g. In turn, each block has two columns, one for each ofthe two formulations F and CP. The first three rows give theaverage gaps in percent between the optimal value (or best-known value, when the optimal value was not known at termina-tion) and the value of (1) the LP bound of the formulation, (2) theLP bound of the formulation enhanced with the cuts generated atthe root node of the search tree, and (3) the best lower bound attermination. The row labeled ‘‘max % g’’ gives the maximum gapsat termination among all the instances of the correspond-ing group. Row labeled ‘‘Avg. time’’ depicts the average comput-ing times (in s), while rows labeled ‘‘mint’’ and ‘‘maxt’’ givethe minimum and maximum computing times among all theinstances of each subset, respectively. Entries in row labeled ‘‘Avg.nodes’’ are the average number of nodes explored in the searchtree. Finally, the row labeled ‘‘Opt. solutions’’ gives the number of
Table 1Comparison of formulations CP and F.
n¼10 n¼20
CP F CP F
Avg. % gap LP 100.00 91.96 100.00 94.48
Avg. % gap LPþcuts 41.45 45.72 43.51 84.79
Avg. % gap 1 h 0.00 21.51 4.65 78.43
max % g 0.00 85.19 37.35 100.00
Avg. time 6.82 1783.33 1065.93 3530.17
mint 1.06 46.31 70.48 2834.66
maxt 52.63 3600.00 3600.00 3600.00
Avg. nodes 3274.23 3379.42 157,118.87 67.02
Opt. solutions 16/16 10/16 13/16 2/16
I. Contreras et al. / Omega 40 (2012) 847–860852
instances of each group that could be solved to optimality withinthe time limit.
The results of Table 1 provide clear evidence of the inherentdifficulty of CFLNDP, even when considering small size instances.First, the LP bounds obtained with both formulations are extre-mely weak. The value of the initial LP bound of formulation CP is0 in all of the 32 considered instances. The initial gaps offormulation F are over 90% in all of the 32 instances. Second,the addition of cuts generated by Xpress reduces the LP gap up to40%, in both formulations. Third, none of the formulations allowsus to solve optimally all the considered instances within 1 h ofcomputing time.
The results of Table 1 also provide clear evidence of thesuperiority of formulation CP over formulation F, in terms of theireffectiveness for solving CFLNDP instances to optimality. With CPwe obtain the optimal solution of all the 10 nodes instances,whereas with F we only obtain the optimal solution of 10 out ofthe 16 instances. The average gap at termination of the 6 unsolvedinstances is close to 22%, with a maximum gap at terminationslightly over 85%. Also, observe that the average time to solve the10 nodes instances is less than 7 s for CP while this average isabove 30 min for F. Furthermore, these CPU times range between1 and 52 s for CP, and between 46 and 3600 s for F. With thelarger instances, the superiority of CP is even more evident. CPwas able to solve to optimality 13 out of the 16 instances of 20nodes, whereas F was only able to solve to optimality 2 of them.Moreover, at termination the average gap is less than 5% for CPand a little more than 78% for F.
In view of these results we concentrate on improving CP inorder to solve larger CFLNDP instances. In the next sections, wepresent some complementary techniques to obtain a stronger CPformulation. First, we derive several types of valid inequalities.Second, we indicate how tight upper bounds can be used both tofurther tighten some of the valid inequalities, and to fix theoptimal values of some variables.
4. Valid Inequalities for CP
In this section we derive some valid inequalities for CP. Theaddition of these inequalities does not change the feasible domainof CP, since each of them must be satisfied by all feasible integersolutions. However, they yield a reinforced formulation of the LPrelaxation of CP, as fractional solutions to the linear programmingrelaxation of CP might fail to satisfy them, thus, producingimproved LP bounds. As we will see, the addition of theseinequalities allows to solve the CFLNDP more efficiently. Through-out the following, Q denotes the convex hull of feasible solutionsto CP associated with rooted forests.
The travel times of the arcs used in feasible solutions yieldtrivial lower bounds on the accumulated travel times to reach theend-nodes of such arcs.
Proposition 1. For every ðj,kÞAA, a valid inequality for Q is
gkZtjkyjk: ð25Þ
These inequalities can be strengthened to take the travel timeassociated with chains defined by two consecutive arcs intoaccount. In particular, in any feasible solution to CP, for given(i,j) and (j,k), the value tijðyijþyjk�1Þ is positive only when the twoconsecutive arcs ði,jÞ and ðj,kÞ are used. Clearly, in this case, theaccumulated travel time to reach node k will be at least tijþtjk.Thus, we have
Proposition 2. For every (i,j) and ðj,kÞAA, a valid inequality for Q is
gkZtijðyijþyjk�1Þþtjkyjk: ð26Þ
It is clear that inequalities (26) extend inequalities (25) topaths with two arcs. These inequalities can be extended further tolonger paths. However, the number of inequalities increases withthe number of arcs of the path and their contribution to theimprovement of the lower bound does not compensate for theadditional computational burden due to the increase of the size ofthe formulation.
Proposition 3. For every iAV , a valid inequality for Q is
TZgiþXði,jÞAA
tijyij: ð27Þ
Proof. Consider iAV . By constraints (14) and (15), in any feasiblesolution to CP, the out-degree of any node iAV is at most 1, sothat
Pði,jÞAAyijr1. If
Pði,jÞAAyij ¼ 0, then inequality (27) reduces to
constraint (23). Otherwise, if yij ¼ 1 for some ði,jÞAA, theninequality (27) reduces to TZgiþtij, which is implied by con-straints (24). &
Since constraints (23) are dominated by inequalities (27), theycan be directly replaced by them.
In any feasible solution to CP we haveP
jAV hijr1 for any iAV .Therefore, the following family of inequalities is also valid.
Proposition 4. Let aij denote the length of the shortest path, relative to
travel times t, from i to j in D. For every iAV , a valid inequality for Q is
TZ
XjAV
aijhij: ð28Þ
For jAV , let bj ¼minftrjþtjk9ðr,jÞ,ðj,kÞAAg. If j is an intermediatenode in a feasible solution to CP, there is (at least) one arc enteringnode j, and (exactly) one arc leaving node j. Therefore, the maximumtravel time T must be at least bj. Therefore, we have:
Proposition 5. For jAV , let bj ¼minftrjþtjk9ðr,jÞ,ðj,kÞAAg. Then, a
valid inequality for Q is
TZbjqj: ð29Þ
Taking into account that no leaf can be at the same time anintermediate node, inequalities (28) and (29) can be combinedinto one single family.
Corollary 1. Let bj ¼minftrjþtjk9ðr,jÞ,ðj,kÞAAg and let aij denote the
length of the shortest path from i to j in D. For every iAV , a valid
inequality for Q is
TZ
XjAV
aijhijþbiqi: ð30Þ
Table 2Results of formulation CP reinforced with inequalities (25), (30) and (31).
n Opt % gap maxg Nodes maxnod Time maxt
10 16 45.11 66.14 490.06 2975 1.64 7.45
20 15 45.41 72.48 35,215.38 319,671 376.42 3600.00
30 12 45.60 76.93 86,045.88 332,584 1418.95 3600.00
I. Contreras et al. / Omega 40 (2012) 847–860 853
Inequalities (29) can be somewhat strengthened to take thosearcs into account which actually define a given rooted forest asstated in the following result. Let r1 and k1, respectively, denotethe indices of the predecessor and the successor of a given node j,in the incoming and outgoing arcs which determine the value ofbj (with ties broken arbitrarily).
Proposition 6. For every jAV , a valid inequality for Q is
TZbjqjþXðj,iÞAA
ðtji�tjk1Þyji: ð31Þ
Proof. Consider again a rooted forest associated with a feasiblesolution to CP and a node jAV . If j is an intermediate node of therooted forest,
Pði,jÞAAyijZ1 and
Pðj,iÞAAyji ¼ 1. If i0AV is any of the
predecessors of j in the rooted forest, then the actual travel timefrom i0 to the (unique) successor of j can be expressed as
ti0 jþXðj,iÞAA
tjiyji ¼ ti0jþtjk1þXðj,iÞAA
ðtji�tjk1Þyji
Ztr1jþtjk1þXðj,iÞAA
ðtji�tjk1Þyji ¼ bjþ
Xðj,iÞAA
ðtji�tjk1Þyji:
Therefore, when jAV is an intermediate node, inequality (31)holds since in this case qj¼1. Note also that, when jAV is not anintermediate node, the inequality also holds. If j is a facility, thereexists no arc leaving j in the rooted forest, so that the right handside of the inequality takes the value zero. Otherwise, j is a leaf,and the right hand side of the inequality is a trivial lower boundon the value of the arc that leaves node j in the solution. &
Another reinforcement of constraints (29) gives rise to thefamily of inequalities
TZbjqjþðtij�tr1 jÞyij, 8i,jAV : ð32Þ
The rationale for inequalities (32) is quite similar to that ofinequalities (31). Now we focus on the arcs entering node j in thecurrent rooted forest. Given that several such arcs may existsimultaneously, it is no longer correct to accumulate the traveltime of all of them to the travel time from node j to its successor.Therefore, each potential predecessor of node j gives rise to onevalid inequality.
Proposition 7. Define gij ¼ tijþminftriþtjk9ðr,iÞ,ðj,kÞAAg, for every
ði,jÞAA. Then, a valid inequality for Q is
TZgijðqiþqjþyij�2Þ: ð33Þ
The validity of inequalities (33) follows as the right hand sideof the inequality is positive only if both i and j are intermediatenodes and, in addition, arc ði,jÞ is used. In this case, it is clear thatthe accumulated travel time must be at least gij.
Proposition 8. For every (i,j), (j,k) and ðk,mÞAA a valid inequality
for Q is
TZtijyijþtjkyjkþtkmðykmþyjk�1Þ: ð34Þ
Inequality (34) is valid since its right hand side accumulatesthe travel times of all possible subpaths of the path ði,j,k,mÞ whensome of its arcs, ði,jÞ, ðj,kÞ, or (k,m) are used consecutively in afeasible solution to CP.
Inequalities (33) and (34) could also be extended to considerlonger paths. However, similarly to inequalities (25), the compu-tational burden caused by the increase in the size of theformulation does not compensate for the improvement on thelower bound associated with the linear programming relaxation.
Proposition 9. Let 0¼H1i oH2
i o � � �oHWi denote the different
possible values of the travel times from i to other nodes in the original
graph D, ordered by increasing values and let Sri ¼ fjAV ,ja i9aijoHr
i g.For iAV and r¼ 2, . . . ,Wi a valid inequality for Q is
TZgiþHri qiþ
XjAV
hij�XjASr
i
zj
0@
1A: ð35Þ
Proof. By constraints (14), for a given iA I we have
qiþXjAV
hij�XjASr
i
zj ¼ 1�zi�XjA Sr
i
zj:
As Sri is the set of nodes that, in the original digraph, can be
reached from i in a total travel time strictly smaller than HWi
i , theabove expression takes the value 1 in a feasible solution to CPonly if, in the associated rooted tree, the travel time from node i toits allocated root node is at least Hr
i . Therefore, inequality (35) isvalid, because gi represents the maximal accumulated time tonode i, from any leave that is connected with its root node with apath going through to node i. &
4.1. Effectiveness of the valid inequalities
For evaluating the effect of the valid inequalities we haveincorporated them into formulation CP and run new series ofcomputational experiments with the same test instances asbefore, plus 16 additional instances with 30 nodes.
Since the rationale of the cover type inequalities (35) is quitedifferent from all the other ones, we have run the new experi-ments in two separated phases, in order to evaluate the effect ofinequalities (35) independently. Some preliminary testing indi-cated that the best combination of inequalities for the first phasewas (25), (30) and (31), since for all other combinations the addedimprovement did not compensate for the increase in the CPUtime. Therefore, for the first phase of the new experiments,formulation CP has been reinforced with inequalities (25), (30)and (31), while the second phase uses inequalities (35) inaddition. Table 2 gives the results of the first phase. The meaningof the columns is as follows. Column ‘‘Opt’’ gives the number ofinstances (out of the 16 instances in the corresponding group)that could be solved optimally within 1 h. Column ‘‘% gap’’ depictsthe average gaps between the LP and optimal values of CP inpercent, i.e., 100ðOpt�LPÞ=Opt. (The value of the best-knownsolution was taken instead of the optimal value, for instancesthat could not be solved to optimality within the maximum timelimit.) Columns ‘‘nodes’’ and ‘‘maxnod’’ indicate, respectively, theaverage and maximum number of nodes explored in the branchand bound tree, while columns ‘‘time’’ and ‘‘maxt ’’ give theaverage and maximum running time in seconds over the 16instances, respectively. Finally, column ‘‘maxg ’’ depicts the max-imum gap between the optimal and the LP value, among all theinstances of the group.
As can be seen, a considerable improvement can be obtainedby just including these families of inequalities. Even if the gapbetween the LP bound and the optimal value is, on the average,
I. Contreras et al. / Omega 40 (2012) 847–860854
still very big (over 45%), the computational cost for exactlysolving the instances has decreased considerably. In particular,for the instances with n¼10, the average CPU time required tosolve the instances has been reduced from 7 s to less than 2 s. Theimprovement can be further appreciated for n¼20. Now all butone instance could be solved within the maximal time limit. Thisimprovement is also reflected in the average number of explorednodes, which has been reduced by one order of magnitude, aswell as in the average required CPU time, which has decreasedfrom above 15 min to a little more than 6 min.
Despite the remarkable effect of the inequalities added in thefirst phase of the new experiments, in general, the improvementthat was obtained was not good enough to solve all instanceswith n¼30 and nearly no instance with n¼40 in the maximumtime limit of 1 h. On the contrary, as illustrated in Table 3, whenthe cover type inequalities (35) are also used, the additionalimprovement allows not only to solve the smaller instances muchmore efficiently, but also to solve instances on up to n¼50(see again Section 5 for the description of the instances) withinthe time limit.
It is interesting to observe that inequalities (35) have a verysmall influence on the quality of the LP gap which, on average, isreduced by less than 5%, but an important impact on the overalleffort needed to exactly solve the instances. Now the averagenumber of explored nodes has decreased to less than 10% of thenumber of nodes for the instances of Table 2, resulting in averagereductions on the required CPU times over 85%. Observe also thatthe computational effort required to solve exactly the new sets of40 and 50 nodes instances is quite small, since the maximumnumber of explored nodes is less than 16,000 resulting in amaximum CPU time slightly over 15 min.
Unfortunately, the reinforcement of formulation CP obtainedwith the different types of inequalities was still not good enoughto allow us to handle instances of larger sizes. In the next sectionwe describe alternative tools that we have used to furtherreinforce the formulation.
5. Computational study
For our experiments all programs have been coded in C andrun under Windows on a VAIO VGN-SR19XN, with a processorIntel(R) Core(TM)2 P8400 at 2.26 GHz and 3 GB of RAM. Thesolver Xpress v2.2.0 has been used to solve formulation CP.Default parameters were used, except for the branching ruleswhere we first branched on the z variables, then on the q variables
Table 3Results of the additional reinforcement with inequalities (35).
n Opt % gap maxg Nodes maxnod Time maxt
10 16 43.33 65.01 3.19 14 1.35 2.95
20 16 41.32 65.74 148.88 1409 13.55 33.84
30 16 40.99 53.62 210.00 2143 74.02 276.98
40 16 42.52 50.40 502.50 4947 264.08 1230.34
50 16 42.65 53.63 2693.31 13,541 1018.13 2955.41
Table 4
Instance generation for the symmetric case. kAf2;10g; r�U½�0:2,0:2�.
n m B
10r,r¼ 1, . . . ,10 2n, 3n, 4n, 5n 300n, 400n
and, finally, on the y variables. The time limit was set to 10 h ofCPU time (36,000 s).
5.1. Test instances
Our generation procedure is largely based on that of Melkoteand Daskin [23,24] which, in turn, is based on that of Balakrish-nan et al. [2]. We have generated networks with a number ofnodes nAf10;20,30;40,50;60,70;80,90;100g and, for each fixednumber of nodes, all possible combinations with a number oflinks mAf2n,3n,4n,5ng and a budget BAf300n,400ng. For fixedvalues of n and m, first we randomly generate the (x,y)-coordi-nates of the nodes from a uniform distribution in ð½0;100�,½0;100�Þ,and the facility set-up costs from an integer uniform distributionin ½1200;1700�. Then, the desired number of links is randomlyselected, and for each such link we calculate the Euclideandistance between its end nodes and round it to the nearestinteger dij. The construction costs of the links are symmetricand are taken to be cij ¼ cji ¼ kdijð1þrÞ, where the proportionalityfactor varies in f2;10g, and r is randomly generated from auniform distribution in ½�0:2,0:2�. This procedure generates 16instances for each fixed number of nodes, giving a total of 160data sets. From each data set we have generated two differentinstances, one with symmetric travel times and another one withasymmetric travel times. In the first case, the travel time of eacharc is given by tij ¼ tji ¼ dijð1þrÞ where again r is randomlygenerated from a uniform distribution in ½�0:2,0:2�. In the secondcase, the travel times in each of the two possible directions aretij ¼ dijð1þr1Þ and tji ¼ dijð1þr2Þ where, as before, r1,r2 arerandomly generated from a uniform distribution in ½�0:2,0:2�. Intotal, we have generated a set of 320 instances. Each instance hasbeen labeled as ‘‘InMkB’’, where ‘‘I’’¼ ‘‘S’’ for symmetric instancesand ‘‘I’’¼ ‘‘A’’ for asymmetric ones; n is the number of nodes, ‘‘M’’takes the values 2, 3, 4 or 5 depending on the relation m=n
between the number of arcs and the number of nodes; ‘‘k’’ takesthe value of the parameter k; and ‘‘B’’¼ ‘‘T’’ when the budget is300n, whereas ‘‘B’’¼ ‘‘L’’ when the budget is 400n. Thus, forexample, the instance labeled as S050410L has been generatedwith symmetric travel times, n¼50 nodes, m¼ 4� 50 arcs, ak¼10, and a budget B¼ 400� 50. A summary of the design forthe symmetric case is shown in Table 4.
5.2. Preliminary tests: effect of the upper bound
As it is usual in min–max formulations, the quality of a knownupper bound U is crucial to improve formulation CP in twocomplementary aspects. On the one hand, such a bound allowsto obtain a tighter expression of constraints (24), as we cansubstitute the big M value on the right hand side of constraints(24) with the value U. On the other hand, we can apply simpleelimination tests, derived from the validity of inequalities (25),(28) and (29), to reduce the size of the instances. Let U denote thevalue of the solution obtained with a heuristic.
(i)
If Uotij, then yij ¼ 0. (ii) If Uoaij, then hij ¼ 0.(iii)
If Uobi, then qi¼0. (iv) If there is no jAV , ja i such that tijrM, then zi¼1.f c t
[1200, 1700] kdij tij ¼ tji ¼ dijþr
I. Contreras et al. / Omega 40 (2012) 847–860 855
Table 5 summarizes the effect of using a good upper bound U,both to reinforce constraints (24) and in the elimination tests.Columns denoted by ‘‘fix z’’, ‘‘fix q’’, ‘‘fix h’’, and ‘‘fix y’’, indicatethe average number of z, q, h and y variables that have been fixed,respectively. In our experiments with instances on up to 50 nodes,the upper bound U is obtained with the GRASP heuristic ofContreras et al. [8].
Table 5Effectiveness of the upper bound.
n Opt % gap maxg Nodes maxnod T
10 16 36.87 65.00 1.25 5
20 16 27.84 56.36 44.71 589
30 16 33.48 52.51 31.36 280
40 16 30.81 44.75 413.25 3237 1
50 16 37.99 42.63 332.38 2561 3
Table 6Computational results for medium-size symmetric instances.
Opt U LP % dev heur % gap LP
S050202L 16 19 9.21 15.79 42.44
S050210L 19 19 12.25 0.00 35.53
S050302L 17 17 10.00 0.00 41.18
S050310L 17 21 10.09 19.05 40.65
S050402L 14 16 10.16 12.50 27.43
S050410L 16 17 9.18 5.88 42.63
S050502L 17 18 11.22 5.56 34.00
S050510L 18 20 11.67 10.00 35.17
S050202T 20 21 11.71 4.76 41.45
S050210T 21 21 14.37 0.00 31.57
S050302T 20 21 11.73 4.76 41.35
S050310T 21 21 12.95 0.00 38.33
S050402T 19 20 11.45 5.00 39.74
S050410T 19 19 11.02 0.00 42.00
S050502T 22 23 13.25 4.35 39.77
S050510T 21 23 13.73 8.70 34.62
S060202L 18 19 10.38 5.26 42.33
S060210L 17 17 11.11 0.00 34.65
S060302L 18 19 10.74 5.26 40.33
S060310L 19 19 10.46 0.00 44.95
S060402L 17 26 11.75 34.62 30.88
S060410L 18 23 10.39 21.74 42.28
S060502L 22 25 11.59 12.00 47.32
S060510L 23 26 12.92 11.54 43.83
S060202T 20 22 11.97 9.09 40.15
S060210T 20 20 12.07 0.00 39.65
S060302T 24 27 12.32 11.11 48.67
S060310T 25 26 12.67 3.85 49.32
S060402T 17 26 11.75 34.62 30.88
S060410T 23 23 12.57 0.00 45.35
S060502T 28 29 14.04 3.45 49.86
S060510T 29 30 15.76 3.33 45.66
S070202L 18 19 10.36 5.26 42.44
S070210L 16 18 10.13 11.11 36.69
S070302L 15 18 8.74 16.67 41.73
S070310L 20 21 10.96 4.76 45.20
S070402L 17 18 9.93 5.56 41.59
S070410L 16 19 9.78 15.79 38.88
S070502L 19 21 10.50 9.52 44.37
S070510L 19 26 11.32 26.92 40.42
S070202T 21 23 11.82 8.70 43.71
S070210T 19 21 11.61 9.52 38.89
S070302T 20 21 10.76 4.76 46.20
S070310T 23 27 13.23 14.81 42.48
S070402T 20 24 11.46 16.67 42.70
S070410T 22 22 11.66 0.00 47.00
S070502T 26 29 13.14 10.34 49.46
S070510T 26 31 13.93 16.13 46.42
The obtained results indicate that taking the value of the upperbound into account considerably reduces the overall computa-tional burden required to solve the instances to optimality. Ingeneral, the improvement in the LP bound is moderate and the LPgaps remain above 30%. In contrast, the required CPU times havedecreased by at least one order of magnitude and the maximumCPU time for any of the considered instances has decreased from
ime maxt Fix z Fix q Fix h Fix y
0.15 0.52 0.13 3.50 45.75 9.88
0.80 4.80 0.29 7.75 286.00 22.29
2.58 9.12 0.29 10.62 709.65 30.86
0.68 66.43 0.50 14.93 1360.50 47.43
1.66 152.06 0.63 20.38 2204.63 53.25
Time Nodes Fix z Fix q Fix h Fix y
23.32 235 1 21 2218 50
5.29 1 1 27 2258 54
20.47 56 1 24 2256 54
70.83 4453 0 10 2200 16
15.21 1 1 29 2260 98
21.37 98 1 19 2242 102
12.37 1 0 27 2256 94
25.68 334 2 24 2228 80
29.09 552 1 16 2176 32
7.16 1 0 22 2212 26
9.38 1 0 14 2200 24
46.04 2023 0 5 2074 6
41.23 641 0 20 2186 46
9.92 1 0 15 2190 70
25.72 138 0 17 2162 36
106.87 1490 0 15 2166 38
979.57 18,976 0 21 3214 68
34.75 229 0 31 3264 94
47.49 286 1 16 3200 72
42.06 418 1 18 3160 80
12.40 1 0 9 3138 6
33.77 65 0 16 3206 12
7074.11 249,202 0 5 3192 0
72.56 1403 0 11 3156 2
71.90 1160 0 13 3116 38
32.11 26 0 17 3180 52
949.28 48,930 0 5 2982 14
2003.80 53,104 0 5 2946 28
13.28 1 0 9 3138 6
24.27 1 0 16 3206 12
569.72 16,810 0 2 3096 0
90.22 2372 0 7 3068 0
85.08 540 1 16 4370 52
29.36 1 1 20 4424 64
157.40 4502 0 14 4392 76
2606.60 54,717 0 18 4334 54
46.89 116 0 21 4496 20
39.95 1 0 17 4488 8
63.15 314 0 10 4432 2
50.56 74 0 4 4292 2
175.70 2387 0 10 4222 18
238.32 3512 0 10 4298 32
138.47 1414 0 9 4302 38
6532.77 1,152,500 0 11 4088 18
54.18 134 0 9 4358 8
212.00 3777 0 10 4402 4
133.57 2279 0 2 4218 0
227.42 4764 0 1 4132 0
I. Contreras et al. / Omega 40 (2012) 847–860856
nearly 50 min to 152 s. These results suggest that the value of theLP gap is not a meaningful indicator of the actual difficulty of theproblem, since instances with large LP gaps are solved efficientlyin small computing times. Additional computational tests indicatethat, when an upper bound is known, tightening inequalities (24)is equally important as applying the elimination tests since theimprovement is considerably smaller when only inequalities (24)are reinforced or when only the elimination tests are applied.
Next we will analyze in detail the numerical results that wehave obtained in the computational experiments with instanceson up to 100 nodes.
5.3. Computational experiments
The algorithm that we apply in our computational experi-ments is the following. First, we apply the heuristic of Contrerasand Fernandez [8] to obtain an upper bound U, and we applythe variable elimination tests for fixing as many variables aspossible. Then, we consider the formulation CP associated with
Table 7Computational results of large symmetric instances.
Opt U LP % dev heur % gap LP
S080202L 16 18 9.06 11.11 43.38
S080210L 15 16 8.99 6.25 40.07
S080302L 17 21 10.45 19.05 38.53
S080310L 17 19 9.96 10.53 41.41
S080402L 15 17 9.83 11.76 34.47
S080410L 16 17 9.43 5.88 40.06
S080502L 20 22 11.00 9.09 45.00
S080510L 21 25 11.92 16.00 43.24
S080202T 19 20 10.52 5.00 44.63
S080210T 19 21 10.17 9.52 46.47
S080302T 22 25 12.28 12.00 41.52
S080310T 21 24 11.91 12.50 43.29
S080402T 19 23 10.90 17.39 42.63
S080410T 20 23 10.74 13.04 46.30
S080502T 23 27 13.23 14.81 42.48
S080510T 28 32 14.35 12.50 48.75
S090202L 16 17 8.87 5.88 44.56
S090210L 14 18 8.96 22.22 36.00
S090302L 18 18 9.45 0.00 47.50
S090310L 17 19 9.23 10.53 42.13
S090402L 15 19 8.73 21.05 41.80
S090410L 15 18 9.18 16.67 38.80
S090502L 19 21 10.09 9.52 46.89
S090510L 17 21 10.38 19.05 38.94
S090202T 19 21 10.04 9.52 47.16
S090210T 19 21 10.23 9.52 39.82
S090302T 18 23 10.80 21.74 43.16
S090310T 23 23 10.72 0.00 53.39
S090402T 18 21 10.03 14.29 44.28
S090410T 18 19 10.54 5.26 41.44
S090502T 23 30 11.96 23.33 48.00
S090510T 24 25 12.14 4.00 47.22
S100202L 16 17 8.63 5.88 46.06
S100210L 16 20 8.86 20.00 44.63
S100302L 18 18 8.81 0.00 51.06
S100310L 17 18 9.14 5.56 46.24
S100402L 14 17 8.28 17.65 40.86
S100410L 16 18 8.28 11.11 48.25
S100502L 20 22 10.05 9.09 49.75
S100510L 17 22 9.44 22.73 44.47
S100202T 17 19 9.58 10.53 43.65
S100210T 19 22 10.22 13.64 46.21
S100302T 19 21 10.20 9.52 46.32
S100310T 20 22 10.54 9.09 47.30
S100402T 17 18 9.34 5.56 45.06
S100410T 18 20 9.39 10.00 47.83
S100502T 24 25 10.06 4.00 50.71
S100510T 22 24 11.06 8.33 49.73
the resulting instance, reinforced with inequalities (25), (30), (31)and (35), where the value M in right hand side of constraints (24)is substituted by the upper bound U. Finally the solver ofXpress v2.2.0 is called for solving to optimality the reinforcedformulation.
It is clear that a valid upper bound can also be obtained byXpress. However, we have observed that usually, it is not untilmany nodes have been explored in the search tree that the solverof Xpress produces relatively good bounds. Since for the overalloptimization process it is important to obtain as soon as possiblethe benefits of having a reasonable upper bound, we use theheuristic of Contreras and Fernandez [8] and reinforce theformulation CP before resorting to Xpress.
The results of our experiments with the symmetric instancesare summarized in Tables 6 and 7, whereas the results of theasymmetric instances are summarized in Tables 8 and 9. Themeaning of the columns is as follows: columns ‘‘Opt’’ show thevalues of the optimal/best-known solutions. The following twocolumns, ‘‘U’’ and ‘‘LP’’, give the values of the feasible solution
Time Nodes Fix z Fix q Fix h Fix y
240.02 2087 0 23 5766 80
86.77 235 0 30 5872 72
204.41 1338 0 21 5700 44
437.64 4990 0 12 5652 30
27.85 1 0 25 5918 12
84.33 165 0 18 5882 8
861.06 18,706 0 11 5728 8
147.87 1596 0 10 5724 4
3574.00 18,328 0 15 5668 46
381.62 5785 0 13 5738 24
568.89 8446 0 8 5460 18
7438.72 149,912 0 4 5502 12
91.59 368 0 10 5734 4
113.04 446 0 9 5730 2
6777.90 152,500 0 8 5444 2
36,000.00 719,447 0 4 5496 0
1428.05 11,212 0 21 7404 62
5752.97 72,881 1 21 7370 50
36,000.00 789,410 0 14 7198 10
3756.47 44,109 0 13 7420 8
155.16 747 0 20 7510 6
58.23 1 0 11 7442 4
416.24 4468 0 12 7396 2
219.02 1655 0 8 7328 4
3797.82 54,227 0 14 7198 10
13,439.35 124,458 1 5 7224 28
36,000.00 689,189 0 6 7002 14
36,000.00 269,909 0 3 7076 14
110.01 262 0 14 7484 4
87.53 185 0 2 7090 0
1658.25 16,911 0 5 7214 2
18,581.75 314,288 0 22 9188 56
2374.91 18,424 2 15 8918 44
36,000.00 124,563 0 19 9018 38
36,000.00 110,178 0 15 9112 26
36,000.00 325,003 1 13 9264 8
208.25 1111 0 16 9300 8
898.67 4968 1 10 9212 8
36,000.00 418,168 0 6 9172 2
8238.82 100,946 0 6 9138 2
4520.20 15,343 0 11 9040 22
13,944.22 121,565 1 11 8774 34
36,000.00 384,265 0 12 8766 14
36,000.00 630,934 0 8 8834 8
958.89 9268 1 10 9212 8
36,000.00 425,134 0 11 9220 2
29,324.69 404,316 0 2 9044 2
535.74 3264 0 3 9028 0
Table 8Computational results of medium-size asymmetric instances.
Opt U LP % dev heur % gap LP Time Nodes Fix z Fix q Fix h Fix y
A050202L 16 18 9.95 11.11 37.81 16.75 147 0 17 2213 22
A050210L 21 24 12.25 12.50 41.67 36.44 520 0 9 2160 4
A050302L 18 21 11.65 14.29 35.28 19.89 68 0 19 2243 18
A050310L 23 27 12.81 14.81 44.30 23.32 51 0 8 2124 0
A050402L 18 21 11.58 14.29 35.67 343.31 7053 0 18 2209 27
A050410L 19 22 12.01 13.64 36.79 22.14 1 0 12 2193 37
A050502L 15 17 10.33 11.76 31.13 14.71 1 0 24 2243 106
A050510L 23 24 13.12 4.17 42.96 74.07 2430 0 11 2116 67
A050202T 17 18 12.13 5.56 28.65 7.21 1 0 17 2213 22
A050210T 26 30 14.23 13.33 45.27 71.23 2299 0 4 2061 1
A050302T 22 27 12.49 18.52 43.23 34.55 545 0 5 2155 4
A050310T 28 29 15.64 3.45 44.14 47.92 2003 0 6 2092 0
A050402T 21 23 12.80 8.70 39.05 36.15 310 0 14 2176 12
A050410T 23 27 13.79 14.81 40.04 35.60 141 0 5 2099 13
A050502T 19 19 10.91 0.00 42.58 26.60 174 0 17 2208 79
A050510T 27 31 15.08 12.90 44.15 1077.96 36,957 0 5 1928 25
A060202L 17 20 10.74 15.00 36.82 44.16 409 0 19 3247 11
A060210L 23 24 12.63 4.17 45.09 125.36 4156 0 11 3094 9
A060302L 18 21 10.80 14.29 40.00 35.40 134 0 16 3244 9
A060310L 22 22 12.90 0.00 41.36 29.63 31 0 12 3214 8
A060402L 15 19 9.57 21.05 36.20 288.95 3186 0 14 3190 49
A060410L 18 20 10.92 10.00 39.33 68.87 894 0 15 3152 24
A060502L 17 19 9.95 10.53 41.47 33.14 90 0 20 3184 90
A060510L 17 20 9.33 15.00 45.12 40.37 205 0 10 3144 86
A060202T 21 22 12.83 4.55 38.90 26.33 1 0 18 3212 9
A060210T 27 30 15.34 10.00 43.19 256.07 12,189 0 6 2915 2
A060302T 20 23 12.40 13.04 38.00 16.72 1 0 11 3198 2
A060310T 27 29 15.93 6.90 41.00 83.41 2487 0 7 3066 2
A060402T 19 22 10.94 13.64 42.42 37.14 36 0 8 3099 33
A060410T 22 26 12.74 15.38 42.09 310.64 19,301 0 5 2957 3
A060502T 18 21 11.65 14.29 35.28 38.03 101 0 15 3131 70
A060510T 21 25 11.05 16.00 47.38 300.67 12,726 0 0 2971 30
A070202L 15 19 9.26 21.05 38.27 36.27 1 0 13 4507 6
A070210L 19 23 10.89 17.39 42.68 51.98 152 0 10 4377 1
A070302L 17 19 9.93 10.53 41.59 68.05 414 0 21 4465 18
A070310L 17 22 9.97 22.73 41.35 61.82 150 0 8 4375 3
A070402L 15 17 8.84 11.76 41.07 46.91 106 0 16 4463 70
A070410L 18 19 9.79 5.26 45.61 386.32 9635 0 11 4393 35
A070502L 15 17 10.11 11.76 32.60 57.67 161 0 28 4468 125
A070510L 15 18 8.63 16.67 42.47 576.29 10,043 0 12 4382 86
A070202T 17 21 11.05 19.05 35.00 30.07 1 0 11 4451 0
A070210T 24 30 13.55 20.00 43.54 179.32 4856 0 0 4184 0
A070302T 20 22 11.43 9.09 42.85 358.32 7811 0 12 4382 8
A070310T 24 26 12.59 7.69 47.54 502.74 15,982 0 6 4271 2
A070402T 17 22 10.35 22.73 39.12 131.27 1318 0 9 4302 29
A070410T 22 25 11.82 12.00 46.27 2660.00 81,932 0 3 4189 10
A070502T 18 21 11.30 14.29 37.22 117.70 925 0 13 4332 60
A070510T 21 23 10.57 8.70 47.15 36,000.00 1,690,759 0 4 4195 34
I. Contreras et al. / Omega 40 (2012) 847–860 857
obtained with the heuristic and the values of the LP relaxation,respectively. Columns ‘‘% dev heur’’ give the values of the percentdeviation between the best-known solution and the optimalvalue, i.e., 100ðU�OptÞ=U, whereas ‘‘% gap LP’’ provide the valuesof the percent deviation of the optimal/best-known solution andthe value of the LP relaxation, i.e., 100ðOpt�LPÞ=Opt. The meaningof the remaining columns is the same as in the tables described inthe previous sections. For the instances that did not reach thetime limit of 36,000 s, the entries in columns ‘‘Opt’’, are theiroptimal values, whereas for the other instances, these entries arethe values of the best solution found, although optimality of suchsolutions could not be proven. As can be seen, we solved allmedium-size instances, both symmetric and asymmetric (exceptfor instance A070510T), although as the sizes of the instancesincreased the time limit was not enough, especially for the larger100 nodes instances. For all the instances that were not solvedoptimally after 1 h of CPU time, we recorded the value of the best-known solution and the lower bound at that point. Thus, in thecases that the instances were still not solved to optimality at
termination after 10 h, we could compare the above bounds withthe final ones. We observed that, in all cases, the improvement ofthe lower bound was nearly negligible (only in one case theincrease was more than one unit), whereas in most of the casesthere was some improvement in the value of the upper bound.Therefore, we assume that, probably, the best-known solutionsare not optimal ones. However, in all cases, the gap between thebest-known solution and the lower bound never exceeded twounits, so best-known solutions are very close to the optimal ones.
Next we discuss briefly how the characteristics of theinstances may affect their difficulty for being solved. In ourcomputational experiments we have observed that the increasein the difficulty for solving the instances as their dimensions growdepends strongly on the type of instance. In particular, sometypes of instances can still be solved, in general, within small CPUtimes even when their dimensions increase, whereas other typesof instances become very time consuming already for not so largedimensions. Roughly speaking, it seems that asymmetricinstances are somewhat easier to solve than symmetric ones.
Table 9Computational results of large asymmetric instances.
Opt U LP % dev heur % gap LP Time Nodes Fix z Fix q Fix h Fix y
A080202L 15 19 8.32 21.05 44.53 72.62 2242 1 7 5787 8
A080210L 17 19 10.24 10.53 39.76 102.98 6658 0 14 5833 9
A080302L 15 19 9.22 21.05 38.53 167.56 13,766 0 14 5849 2
A080310L 16 22 9.52 27.27 40.50 116.67 7597 0 7 5741 3
A080402L 15 20 9.14 25.00 39.07 58.31 11 0 7 5668 17
A080410L 17 22 9.73 22.73 42.76 1443.24 237,274 1 7 5511 12
A080502L 14 17 8.77 17.65 37.36 147.18 10,248 0 23 5787 102
A080510L 20 23 11.44 13.04 42.80 1251.97 175,137 0 6 5537 25
A080202T 18 21 9.75 14.29 45.83 104.83 659 1 6 5704 5
A080210T 22 27 12.24 18.52 44.36 86.67 50 0 3 5570 0
A080302T 19 22 10.51 13.64 44.68 102.16 439 0 8 5738 1
A080310T 21 25 11.36 16.00 45.90 109.16 584 0 3 5620 1
A080402T 18 21 10.64 14.29 40.89 90.54 274 0 4 5625 14
A080410T 21 25 11.91 16.00 43.29 2313.24 64,834 1 4 5371 7
A080502T 15 18 9.97 16.67 33.53 894.19 4405 0 18 5724 78
A080510T 20 23 11.44 13.04 42.80 1236.80 17,513 0 6 5537 25
A090202L 14 19 8.53 26.32 39.07 103.93 151 0 10 7415 7
A090210L 18 20 9.47 10.00 44.29 8590.71 139,324 0 6 7383 6
A090302L 16 18 8.78 11.11 45.13 103.41 273 0 11 7490 12
A090310L 17 22 10.23 22.73 39.82 194.25 1412 0 4 7329 4
A090402L 13 17 7.80 23.53 40.00 125.67 435 1 12 7362 30
A090410L 16 19 9.38 15.79 41.38 4856.88 36,610 0 11 7388 30
A090502L 13 15 8.47 13.33 34.85 183.21 648 1 25 7403 122
A090510L 17 18 9.09 5.56 46.53 36,000.00 358,280 0 13 7295 72
A090202T 17 20 9.81 15.00 42.29 125.21 478 0 9 7371 7
A090210T 21 25 11.22 16.00 46.57 529.04 5353 0 0 7134 3
A090302T 18 22 10.02 18.18 44.33 2653.62 24,709 0 6 7322 3
A090310T 22 24 12.08 8.33 45.09 1014.78 12,390 0 2 7235 3
A090402T 16 18 8.98 11.11 43.88 266.60 2353 1 10 7294 22
A090410T 20 24 10.76 16.67 46.20 36,000.00 997,708 0 4 7102 3
A090502T 15 18 9.33 16.67 37.80 747.63 9558 0 14 7212 75
A090510T 20 21 10.48 4.76 47.60 36,000.00 394,389 0 8 7081 30
A100202L 15 19 8.18 21.05 45.47 241.32 958 0 7 9147 9
A100210L 15 18 8.66 16.67 42.27 128.31 130 0 10 9218 14
A100302L 13 17 8.23 23.53 36.69 973.27 5622 0 17 9218 5
A100310L 16 20 8.96 20.00 44.00 1121.65 11,124 0 9 9182 0
A100402L 13 17 7.79 23.53 40.08 189.66 732 0 15 9159 34
A100410L 17 19 8.61 10.53 49.35 36,000.00 422,411 0 7 8961 19
A100502L 12 15 7.75 20.00 35.42 150.51 211 0 20 9260 117
A100510L 15 19 8.34 21.05 44.40 2612.67 23,059 0 10 8909 45
A100202T 17 21 9.29 19.05 45.35 299.64 1407 0 5 9037 7
A100210T 19 25 9.96 24.00 47.58 28,087.39 329,204 0 3 8850 1
A100302T 15 20 9.27 25.00 38.20 217.79 583 0 8 9056 0
A100310T 21 26 10.55 19.23 49.76 527.26 5219 0 1 8855 0
A100402T 15 19 8.73 21.05 41.80 985.44 7466 0 8 9019 16
A100410T 18 22 9.77 18.18 45.72 33,978.30 308,327 0 3 8726 11
A100502T 14 17 8.60 17.65 38.57 412.39 1698 0 13 9140 75
A100510T 19 20 9.64 5.00 49.26 36,000.00 391,615 0 10 8814 35
Symmetric Instances
0.00
500.00
1000.00
1500.00
2000.00
2500.00
3000.00
m=2n,k=2
m=2n,k=10
m=3n,k=2
m=3n,k=10
m=4n,k=2
m=4n,k=10
m=5n,k=2
m=5n,k=10
LT
Asymmetric instances
0.00
500.00
1000.00
1500.00
2000.00
2500.00
3000.00
m=2n,k=2
m=2n,k=10
m=3n,k=2
m=3n,k=10
m=4n,k=2
m=4n,k=10
m=5n,k=2
m=5n,k=10
LT
Fig. 1. Comparison of average CPU times between instances ‘‘L’’ and ‘‘T’’.
I. Contreras et al. / Omega 40 (2012) 847–860858
Figs. 1–3 allow for a better comparison among the differentclasses of instances, symmetric (‘‘S’’) and asymmetric (‘‘A’’), andfor a deeper analysis of the influence of the choice of parameterson the difficulty of the instances. In all cases, the left chart
corresponds to ‘‘S’’ instances and the right one to instances ofclass ‘‘A’’, so it is possible to observe whether or not the influenceof the parameter’s choice is similar on both classes. The figuresrepresent average CPU times of instances generated with fixed
Symmetric Instances
0.00
500.00
1000.00
1500.00
2000.00
2500.00
3000.00
m=2n,L
m=2n,T
m=3n,L
m=3n,T
m=4n,L
m=4n,T
m=5n,L
m=5n,T
k=2k=10
Asymmetric instances
0.00
500.00
1000.00
1500.00
2000.00
2500.00
3000.00
m=2n,L
m=2n,T
m=3n,L
m=3n,T
m=4n,L
m=4n,T
m=5n,L
m=5n,T
k=2k=10
Fig. 2. Comparison of average CPU times between instances with k¼2 and k¼10.
Symmetric Instances
0.00
500.00
1000.00
1500.00
2000.00
2500.00
3000.00
k=2,L
k=10,L
k=2,T
k=10,T
m=2nm=3nm=4nm=5n
Asymmetric instances
0.00
500.00
1000.00
1500.00
2000.00
2500.00
3000.00
k=2,L
k=10,L
k=2,T
k=10,T
m=2nm=3nm=4nm=5n
Fig. 3. Comparison of average CPU times between instances with k¼2 and k¼10.
I. Contreras et al. / Omega 40 (2012) 847–860 859
sets of parameters over all dimensions from 50 to 100. Further-more, all instances requiring more than 3600 s of CPU have beenconsidered as ‘‘difficult’’, so that for calculating these averages,CPU times over 1 h have been ‘‘truncated’’ to 3600 s. In this way,all values lie in a rather small range and relatively smalldifferences can be appreciated at the same scale. Fig. 1 illustratesthe influence of the value of the budget B. For ‘‘S’’ instances theaverage CPU times of the ‘‘T’’ instances are always bigger thanthose of the ‘‘L’’ instances, for each considered set of parameters,whereas for the ‘‘A’’ instances the higher difficulty of the ‘‘T’’instances is not so evident, although, in general also ‘‘T’’ instancesappear to require more computational time.
Fig. 2 illustrates the influence of the parameter k, that relatesthe length of an arc with its travel time. In the class ‘‘A’’, instanceswith k¼10 are, in general, more difficult, although this is not soevident in the class ‘‘S’’.
Finally, Fig. 3 illustrates the influence of the number m of arcsdensity on the difficulty of an instance. Now, instances withm¼ 4n seem to be easier, and those with m¼ 3n seem to beharder for the ‘‘S’’ class, whereas instances with m¼ 3n seem to beeasier and instances with m¼ 4n or m¼ 5n seem to be harder forthe ‘‘A’’ class.
6. Conclusions and remarks
In this paper we have studied the center Facility Location/Network Design Problem. We have presented two integerprogramming formulations for the problem, which have beencompared empirically. The multicommodity-based formulationwas clearly outperformed by the formulation that exploits thestructure of the problem, which has been successively reinforcedwith different types of valid inequalities. We have generated a set
of benchmark instances with both symmetric and asymmetrictravel costs with different characteristics and varying number ofnodes up to 100. The numerical results of the computationalexperiments indicate that, in general, symmetric instances aresomewhat more difficult to solve than the asymmetric ones.However, in both cases, the vast majority of instances on up to80 nodes were solved optimally within 1 h of CPU time. Forinstances with 90 and 100 nodes in many cases more CPU timewas required, although most of them could be solved to optim-ality within 10 h of CPU time. Future research will focus ondifferent formulations for the problem and especially thosesuitable for being addressed with column generation methodol-ogy. Another interesting topic for further research is to considercapacities both in the arcs or the facilities.
Acknowledgments
This research has been partially supported by grant MTM2009-14039-C06-05 of the Spanish Ministry of Education and Scienceand by ERDF funds. The research of the first author has beenpartially supported through grant 197243/217997 from theNational Council of Science and Technology, Mexico and throughcontract DFG Re776/10-1 of the University of Heidelberg. Theresearch of the second author has been partially supportedthrough grant PR2008-0116 of the Spanish Ministry of Scienceand Education. This support is gratefully acknowledged. Thanksare due to two referees for their valuable comments.
References
[1] Alumur S, Kara BY. Network hub location problems: the state of the art.European Journal of Operational Research 2008;190:1–21.
I. Contreras et al. / Omega 40 (2012) 847–860860
[2] Balakrishnan A, Magnanti TL, Wong RT. A dual ascent procedure for large-scale uncapacitated network design. Operations Research 1989;37:716–40.
[3] Campbell AM, Lowe T, Zhang L. Upgrading arcs to minimize the maximumtravel time in a network. Networks 2006;47:72–80.
[4] Church R, ReVelle C. The maximal covering location problem. RegionalScience Association Papers 1974;32:101–18.
[5] Cocking C, Fleßa S, Reinelt G. Locating health facilities in Nouna District,Burkina Faso. In: Haasis H-D, Kopfer H, Schnberger J, editors. Operationsresearch proceedings; 2005.
[6] Cocking C. Solutions to facility location–network design problems. PhDthesis. Germany: University of Heidelberg; 2008.
[7] Contreras I, Fernandez E. General network design: a unified view of combinedlocation and network design problems. European Journal of OperationalResearch, doi:10.1016/j.ejor.2011.11.009, in press.
[8] Contreras I, Fernandez E, Reinelt G. Heuristic solutions for the center facilitylocation network design problem with budget constraint, in preparation.
[9] Cordeau J-F, Pasin F, Solomon MM. An integrated model for logistics networkdesign. Annals of Operations Research 2006;144:59–82.
[10] Costa A, Franc-a PM, Lyra Filho C. Two-level network design with intermedi-ate facilities: an application to electrical distribution systems. Omega2011;39:3–13.
[11] Daskin MS, Hurter AP, Van Buer MG. Toward an integrated model of facilitylocation and transportation network design. Working Paper. Evanston, IL,USA: The Transportation Center, Northwestern University; 1993.
[12] Daskin MS. Network and discrete location: models, algorithms and applica-tions. New York: John Wiley and Sons, Inc.; 1995.
[13] Drezner Z, Wesolowsky G. Network design: selection and design of links andfacility location. Transportation Research Part A 2003;37:241–56.
[14] Elloumi S, Labbe M, Pochet Y. New formulation and resolution method for thep-center problem. INFORMS Journal on Computing 2004;16:84–94.
[15] Ghiani G, Laporte G, Musmanno R. Fixed-charge, network design models. In:Introduction to logistics systems planning and control. Wiley; 2004.
[16] Gollowitzer S, Ljubic I. MIP models for connected facility location: atheoretical and computational study. Computers & Operations Research2011;38:435–49.
[17] Hakimi SL. Optimum locations of switching centers and the absolute centersand medians of a graph. Operations Research 1964;12:450–9.
[18] Kariv O, Hakimi SL. An algorithmic approach to network location problems, I:the p-centers. SIAM Journal of Applied Mathematics 1979;37:513–37.
[19] Kuehn AA, Hamburger MJ. A heuristic program for locating warehouses.Management Science 1963;9:643–66.
[20] Klose A, Drexl A. Facility location models for distribution system design.European Journal of Operational Research 2005;162:4–29.
[21] Li X, Aneja YP, Huo J. Using branch-and-price approach to solve the directednetwork design problem with relays. Omega 2012;40:672–9.
[22] Megiddo N, Supowit KJ. On the complexity of some common geometriclocation problems. SIAM Journal on Computing 1984;13:182–96.
[23] Melkote S, Daskin MS. An integrated model of facility location and tran-sportation network design. Transportation Research Part A 2001;35:515–38.
[24] Melkote S, Daskin MS. Capacitated facility location/network design problems.European Journal of Operational Research 2001;129:481–95.
[25] Melo MT, Nickel S, Saldanha-da-Gama F. Facility location and supply chainmanagement—a review. European Journal of Operational Research 2009;196:401–12.
[26] Murawski L, Church RL. Improving accessibility to rural health services: themaximal covering network improvement problem. Socio-Economic PlanningSciences 2009;43:102–10.
[27] ReVelle CS, Eiselt HA. Location analysis: a synthesis and survey. EuropeanJournal of Operational Research 2005;165:1–19.
[28] ReVelle CS, Eiselt HA, Daskin MS. A bibliography for some fundamentalproblem categories in discrete location science. European Journal of Opera-tional Research 2008;184:817–48.
[29] Smith HK, Laporte G, Harper PR. Location analysis: highlights of growth tomaturity. Journal of the Operational Research Society 2009;60:140–8.
[30] Srivastava SK. Network design for reverse logistics. Omega 2008;36:535–48.[31] Syam S, Cote MJ. A location-allocation model for service providers with
application to not-for-profit health care organizations. Omega 2010;38:157–66.
[32] Toregas C, Swain R, ReVelle C, Bergmann L. The location of emergency servicefacilities. Operations Research 1971;19:1363–73.
[33] Wong RT. Probabilistic analysis of a network design problem heuristic.Networks 1985;15:347–63.