An Integrated Model of Facility Location and Transportation Network Design

7/27/2019 An Integrated Model of Facility Location and Transportation Network Design

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An integrated model of facility location and transportation

network design

Sanjay Melkote a,*, Mark S. Daskin b

a Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544, USAb Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston,

IL 60208-3119, USA

Received 25 February 1999; received in revised form 20 December 1999; accepted 14 January 2000

Abstract

Network location models have been used extensively for siting public and private facilities. In this paper,

we investigate a model that simultaneously optimizes facility locations and the design of the underlying

transportation network. Motivated by the simple observation that changing the network topology is often

more cost-eective than adding facilities to improve service levels, the model has a number of applications

in regional planning, distribution, energy management, and other areas. The model generalizes the classical

simple plant location problem. We show how the model can be solved eectively. We then use the model toanalyze two potential transportation planning scenarios. The fundamental question of resource allocation

between facilities and links is investigated, and a detailed sensitivity analysis provides insight into the

model's usefulness for aiding budgeting and planning decisions. We conclude by identifying promising

research directions. 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Facility location; Network design

1. Introduction

Network location models have been used extensively to analyze and determine the locations ofpublic and private facilities. Classical network models include the set covering location problem(Toregas et al., 1971), the maximum covering location problem (Church and ReVelle, 1974),

p-median and p-center problems (Hakimi, 1964), and the uncapacitated facility location problem

Transportation Research Part A 35 (2001) 515538

www.elsevier.com/locate/tra

* Corresponding author. Tel.: +1-609-258-0100; fax: +1-609-258-4363.

E-mail address: [email protected] (S. Melkote).

0965-8564/01/$ - see front matter

2001 Elsevier Science Ltd. All rights reserved.P I I : S0965- 8564( 00) 00005- 7


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(Kuehn and Hamburger, 1963). The last model is also known as the simple plant location problem

and the warehouse location problem and is henceforth referred to as the UFLP. Daskin (1995) andFrancis et al. (1992) provide a comprehensive overview of network location models.

All of these classical models locate facilities on a given network. However, the topology of theunderlying network may have a profound impact on the optimal facility locations. Consider thefollowing problem: a large retailer has outlets in every major city in the US state of Indiana (see,e.g., Rand McNally and Company, 2000 for a highway map of the state). The problem the rm

faces is to locate a new distribution center so as to minimize its total transportation costs to andfrom the outlets. (Assume that the cities shown on the map are the candidate locations andtruckload deliveries to each outlet.) Solving this problem (as a 1-median problem), we would nd

that the optimal decision is to locate at Indianapolis (as we would expect in fact, viewing thesystem of US interstate highways as a network, Indianapolis would have the largest number of

incident links of any node (or the highest degree of any node), hence its nickname ``Crossroadsof America''). Now as simple as this example may appear, it illustrates the impact of the design of

the underlying transportation network (a star network topology in this case) on the optimal fa-cility location. In view of this simple observation, the question posed by Daskin et al. (1993) was:should the underlying network always be treated as given? Specically, they argue (p. 1), ``In some

cases . . . changing the underlying network may be a more cost-eective approach to improvingservice than is adding facilities. Thus, the uncapacitated xed charge facility location problemmay be extended to allow modications of the underlying network. The cost of these modica-

tions would then contribute to the total cost.''Daskin et al. (1993) then introduce such a model. They give some preliminary results showing

the tradeos between locating facilities and constructing links. This model is the topic of our

paper. The problem may be stated as follows. We are given a set of nodes, which represent de-

mand points as well as candidate facility locations, and a set of uncapacitated links. Each link hasa xed construction cost as well as a per unit transport cost, and each node is associated with axed charge for building an uncapacitated facility at that node. All demands must be routed over

the network to the nearest facility. The problem is to nd the network design and the set of facilitylocations that minimize the total system cost: the sum of link and facility construction costs andtransport costs. They contend that such a combined facility location/network design model could

be useful in a number of transportation planning scenarios in which basic tradeos between xedcosts for links, xed facility costs, and operating costs must be made. Specically, they write thatsuch a model could be used in the design of: (1) pipeline distribution systems in which the pumps

or pumping stations are the facilities and the pipelines make up the network, (2) intermodaltransportation systems in which the intermodal rail yards are the facilities and the existinghighway and rail networks may be improved, (3) power transmission networks, where the facilities

are generating stations, switches, and transformers, and the links are transmission and distribu-tion lines (Hingorani and Stahlkopf, 1993), and (4) hub-and-spoke networks, which arise in a

number of transportation contexts (Campbell, 1994). In addition to these applications, we notethat a combined location/network design model could also be used in regional planning eorts orland reuse programs where the government may be simultaneously considering the construction

of a new roadway system as well as the location of public facilities such as post oces, schools,etc. (Rushton, 1984), and in LTL freight distribution system design, in which xed costs are as-sociated with the location of breakbulk facilities and the establishment of a service link between

516 S. Melkote, M.S. Daskin / Transportation Research Part A 35 (2001) 515538


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two points, and variable costs dependent on the amount of freight shipped (Ahuja et al., 1993;

Braklow et al., 1992; Powell and She, 1989).

1.1. Related literature

The model most closely related to our problem is the uncapacitated xed charge network designproblem, as studied in the classical papers by Magnanti and Wong (1984) and Balakrishnan et al.

(1989). We later show how our problem can be solved eectively by formulating it as a special caseof the xed charge network design problem.

A closely related line of work, which originated in the early 1990s, examines the relationshipbetween facility location and network topology. Berman et al. (1992) show how the utility(measured by transport costs) of existing facilities can be improved by making certain changes to

the conguration of the underlying network. Peeters and Thomas (1993, 1995) investigate theimpact of dierent network topologies on optimal solutions to the p-median problem, nding, notsurprisingly, that the eect is ``signicant''. Bhadury et al. (1998) study the following problem:

Given that a facility is located at a certain node of a network, nd the least expensive spanningtree having that facility as its 1-median. Perhaps the most closely related work in this line is that ofDrezner and Wesolowsky (1998), who investigate heuristics for nding a conguration of one-way

and two-way streets and the location of a single facility in a network to minimize the totaltransportation costs. In contrast, in the problem we consider, multiple facilities may be located,

and we seek to minimize the sum of transport costs, xed network design costs, and xed facilityconstruction costs. Further, we solve our problem to optimality as opposed to heuristically.

Another related line of research deals with location-routing problems. These problems considerthe simultaneous location of facilities and determination of delivery/collection routes or paths of

some sort. Min et al. (1998), ReVelle and Laporte (1996), and Laporte (1988) provide overviewsof these problems. A prototypical location-routing problem is the ``generalized traveling salesmanproblem'' (Laporte et al., 1987), in which we are given a set of nodes that have been partitionedinto m subsets. The problem is to identify one node to be visited in each subset so as to minimize

the length of the Hamiltonian tour through the selected nodes. A recently studied variant of thisproblem is the ``covering tour problem'' (Current and Schilling, 1992; Gendreau et al., 1997), inwhich the objective is to nd a minimum length tour passing through a subset of the nodes such

that each node is within a prespecied distance of the tour. Another type of location-routingmodeling framework is described in Laporte et al. (1988). In their problem, we are given a net-work consisting of nodes and links. The nodes are classied as either candidate depot locations or

customer demands. A xed cost is associated with each link. A number of vehicles may be sta-tioned at each depot, and xed costs are incurred for establishing a depot or using a vehicle. There

is also an upper limit on the cost or length of a vehicle route or on the vehicle capacity. Berger et al.(1996) study a related problem. None of the problems in this line of work, however, is equivalent

to the problem considered in this paper.Another related line of work is the hub location literature. Hub location problems can also be

considered joint location/network design problems in the sense that they simultaneously address:

(1) where to locate the hubs, which serve as consolidation and redirection points for ow, and(2) how to design (a) the hub-level network (how to connect the hubs) and (b) the access-levelnetwork (the allocation of demand points to hubs). See Campbell (1994) or Klincewicz (1998) for

S. Melkote, M.S. Daskin / Transportation Research Part A 35 (2001) 515538 517


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a survey of these models. In these models, however, the denition of a facility and the network

conguration are quite constrained generally the only permissible link connections are from hubto hub or from demand points directly to hubs. In addition, origindestination (O-D) ows are

used and assumed to be known in advance. In contrast, in the model discussed in this paper,facilities are dened in the more general sense of conventional facility location problems, in whichfacilities serve as either customer destinations or origins of shipments to customers, and no re-strictions are imposed on the conguration of the underlying network other than that the selected

links come from a user-specied set of candidate links.The rest of this paper is organized as follows. In the following section, we present the mixed

integer programming (MIP) formulation of Daskin et al. (1993)'s model and discuss some of

its properties. In Section 3, we show how the model can be solved eectively by formulating itas a special case of the xed charge network design problem. Computational results using an

MIP solver are presented. In Section 4, we apply the model to two potential transportationplanning scenarios: a small six-node problem and a budget design problem, with which we

investigate the fundamental question of resource allocation between facilities and links. Adetailed sensitivity analysis is conducted to provide insight into the model's usefulness foraiding budgeting and planning decisions. In Section 5, we draw conclusions and identify re-

search directions.

2. The model

The MIP (mixed integer programming) formulation of the uncapacitated facility location/

network design problem (UFLNDP) of Daskin et al. (1993) makes the following assumptions:(1) each node represents a demand point, (2) facilities may be located only on the nodes of the

network, (3) only one facility may be located per node, (4) the network is a customer-to-server

system, in which the demands themselves travel to the facilities to be served, (5) the facilities are

uncapacitated, i.e., they may serve an unlimited amount of demand, and (6) demand is for a singleservice or commodity. Assumption (5) is the standard assumption made in the simple plantlocation problem. This assumption has been successfully applied in many situations in which

facilities typically operate at levels so far below their actual capacities that they are almost nevertaxed.

The input parameters or data are:

N set of nodesL set of undirected candidate links

di demand at node i

M XiPN

di total network demand

tij travel cost per unit flow on link i;j

fi fixed cost of constructing a facility at node i

cij cost of constructing link i;j



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The decision variables are:

Zi 1 if a facility is located at node i;

0 if not;(

Xij 1 if link i;j is constructed; where i < j;

0 if not;

(

Yij; Yji flow of demands on link i;j in the i 3 j and j 3 i directions; respectively;

Wi demand served by a facility at node i:

We assume all parameters are nonnegative. The MIP formulation of Daskin et al. (1993) maythen be stated as follows:

(UFLNDP/W)

minimizeX

i;jPL

tijYij Yji XiPN

fiZi X

i;jPL

cijXij 1

subject toXjPN

Yji di XjPN

Yij Wi; Vi PN; 2

Wi 6MZi; Vi PN; 3

Yij6MXij; Vi;j PL; 4Yji6MXij; Vi;j PL; 5

Yij; YjiP 0; Xij P f0; 1g; Vi;j PL;

WiP 0; Zi P f0; 1g; Vi PN: 6

The objective function minimizes the sum of transportation, facility location, and link con-

struction costs. Eq. (2) is a conservation of ow equation, stating that the inbound ow to a nodemust equal the outbound ow from the node. The inbound ow consists of the total inbounddemand plus the demand at the node, and the outbound ow is the total outbound demand plus

the demand served at the node. (We recall that this is a customer-to-server system.) (3) states thatthe demand served by a node can be positive only if a facility is located at that node. Similarly, (4)

and (5) require that ow in either direction on a link can occur only if the link is constructed. (6)are standard nonnegativity and integrality constraints.

Several aspects of this formulation are worth noting. First, we observe that if we set all link

construction variables Xij to zero, constraints (4) and (5) and the last term of objective (1) dis-appear. The result is simply an alternate formulation of the uncapacitated facility locationproblem (UFLP). Thus the UFLP is just a special case of the UFLNDP in which link additions

are disallowed. Since the UFLP is NP-hard (in the parlance of computational complexity), so isthe more general UFLNDP.



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Property 1. The UFLNDP is NP-hard.

Also, we note that although we do not require the Yij or the Wi to be integral, they are alwaysintegral in an optimal solution as long as the demands are integer-valued. The Yij are integral

because every demand is completely assigned to a single facility (the closest). That is, nothing isgained by ``splitting up'' a demand and sending parts of it to dierent facilities. This is generally

known as the Single-Assignment Property (Krarup and Pruzan, 1990). This property directlyimplies that the Wi must also be integral. We also observe that as long as travel costs are sym-

metric so that tij tji for all links i;j, an optimal solution to this formulation also gives anoptimal solution to the corresponding server-to-customer system, in which the facilities shipcommodities to the demand points. To obtain such a solution, we would simply reverse the di-

rections of the optimal ows.Finally, we note that constraint (3) states that if the demand served by a facility is positive, it

can at most equal the total network demand. This is generally a very weak constraint, as inmost cases a single facility does not serve all demands unless we locate only one facility.

Similarly, constraints (4) and (5) state that a link's ow can at most equal the total networkdemand, another unlikely situation. The result of these ``big M'' constraints is a weak linear

programming (LP) relaxation (hence we refer to this formulation as (UFLNDP/W)). It is wellknown that weak LP relaxations are computationally undesirable (see e.g. Nemhauser and

Wolsey, 1988). However, Daskin et al. (1993) did not investigate solution techniques for theUFLNDP.

3. Model solution

As mentioned above, (UFLNDP/W) suers from a weak LP relaxation. In particular, com-putational results reported in Melkote (1996) show LP/IP ratios of as low as 0.10 for a 21-node,37-link test problem, resulting in solution times of up to 4 h using a standard MIP solver on a SunSPARCstation 10.

3.1. Strong formulation

We now show how the UFLNDP can be solved eectively by formulating it as a specialcase of the classical uncapacitated xed charge network design problem (NDP). The NDP

may be stated as follows. We are given a set of commodities K which must be routedbetween a set of origins and a set of destinations. More specically, the required owbetween a particular O-D pair is denoted a ``commodity.'' It is assumed that one unit of

ow of each commodity kPK must be shipped from its origin Ok to its destinationDk.

As before, Xij is a binary variable that indicates whether or not link i;j is chosen. Also, let Ykij

denote the fraction of commodity kthat ows on the directed link i;j. Then the network designproblem can be formulated as follows:



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(NDP)

minimize

Xi;jPLXkPKtkijY

kij t

kjiY

kji Xi;jPL

cijXij 7

subject to

XjPN

Ykji XjPN

Ykij

1; Vi PN; VkPK : i Ok;

1; Vi PN; VkPK : i Dk;

0 otherwise;

8>: 8

Ykij 6Xij; Vi;j PL; VkPK; 9

Ykji 6Xij; Vi;j PL; VkPK; 10

Ykij ; YkjiP 0; Xij P f0; 1g; Vi;j PL; VkPK: 11

The objective function minimizes the sum of transport costs and xed network design costs.Eqs. (8) are conservation of ow constraints. (9) and (10) permit ow only on links selected to bein the design, and (11) are integrality and nonnegativity constraints. It is known that this for-

mulation has a strong LP relaxation (Balakrishnan et al., 1989).To formulate the UFLNDP as a pure network design problem, we rst introduce a ``supern-

ode'' to the network of interest (see Fig. 1). We also add a candidate ``superlink'' connecting eachcandidate facility node i to the supernode. Each such superlink has construction cost fi and zerotravel costs, so that building it is equivalent to constructing a facility at i. Next, set the ``re-

quirement'' at the supernode equal to M, the total demand in the network, and the ``supply'' ateach original node i equal to its demand di. Now the problem may be stated as follows: Find theset of links that minimizes the sum of link construction costs and travel costs between each de-

mand node and the supernode so that all demands are served. This problem is a special case of theNDP with a single destination for all demands.

Now since each O-D pair is a demand nodesupernode pair, we can dene a commodity k to

simply be the demand originating at node k. (For now we assume that all demands are strictlypositive. We later show how this assumption can easily be relaxed.) Then we can scale the size ofall commodities to unity, as is required in the preceding formulation of the NDP, and dene

t0ij travel cost per unit flow on link i;j;

Fig. 1. Conversion to pure network design problem.



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so that link travel costs are now commodity-specic:

tkij travel cost of commodity k on link i;j t0ijdk:

As before, Ykij represents the fraction of the ow of commodity kon a ``normal'' link i;j. Denotingthe supernode as node S, we now let Wki denote the fraction of commodity k that ows on the

superlink i; S, or equivalently, the fraction of commodity k that is served by a facility at node i.Also, we now assume the Xijs represent directed link variables. Thus for each link i;j PL, we

add the link j; i. Both of these links have the same construction cost as the original link i;j.The ``direction'' of a link only species the direction of the ow it carries, and does not imply that

directed links are actually built, as we shall see later.As a result of the UFLNDP's special structure, the conservation of ow constraints (8) may be

simplied. For the rst case in (8), we note that each ``normal'' node is an origin of ow. Thus the

outbound ow has a magnitude of 1, and outbound ow is served using either a normal link or asuperlink (i.e., it is served by a facility at that node). This gives

Wii XjPN

Yiij 1; Vi PN: 12

For the second case, observing that there is a single destination (the supernode), we haveXiPN

Wki 1; VkPN: 13

This equation states that the inbound ow of each commodity into the supernode must sum to 1;i.e., each demand must be completely served.

For the third case of trans-shipped demand, we have

XjPN

Ykji Wki

XjPN

Ykij

! 0; Vi; kPN : i T k: 14

This equation is identical to (12) with the exception of the inbound ow term.After its transformation to a pure network design problem, the UFLNDP is a single destination

(the supernode) network ow problem, as we have already noted. Further, it is a concave cost ow

problem. To see this, observe that the cost of using each normal link i;j is zero whenPkPN Y

kij 0 and cij tij

PkPN Y

kij when

PkPN Y

kij > 0. Similarly, the cost of using a superlink

i; S is zero whenP

kPNWki 0 and fi when

PkPNW

ki > 0. Thus the cost of using any link is a

concave function of the total ow on the link. A well-known result on concave cost ow problems(Ahuja et al., 1993) then implies that the UFLNDP has an optimal solution which is a directed,rooted spanning tree with the supernode as the root node. We can visualize this structure if we

refer to Fig. 2 and imagine a superlink emanating from each facility and directed into a singlesupernode. By denition, all nodes of a directed, rooted spanning tree except the root node have

exactly one outbound link (or outdegree of one). The following identities are a consequence of thisproperty:

Yiij Xij; Vi;j PL; 15

Wii Zi; Vi PN: 16



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These identities state that at optimality, the demand at each node is served via a single outbound

(super)link. If the link is a superlink, then this is of course equivalent to constructing a facility atthat node. We may substitute these identities into the preceding constraints and delete the Yiij andWii variables from the formulation altogether.

Since the UFLNDP has an optimal solution in which ow on any constructed link is in onedirection only, we may write

Xij Xji6 1; Vi;j PL: 17

This inequality was also derived in Magnanti and Wong (1984) and Balakrishnan et al. (1989).

If we wish to provide the user with control over the number of facilities to locate, we may addthe following constraint:X

iPN

Zi p: 18

With this constraint, the model becomes a generalization of the p-facility location problem, alsoknown as the p-UFLP, in which the network topology is endogenously determined. If we omit the

xed charge term from the objective, the model becomes an extension of the p-median problem.Summarizing our discussion, the strong formulation of the UFLNDP is

(UFLNDP/S)

minimizeX

i;jPL

XkPN:kTi

tkijYkij

Xi;jPL

tiijXij X

i;jPL

cijXij XiPN

fiZi 19

subject to

Zi XjPN

Xij 1; Vi PN; 20

Xki X

jPN:jTk

Ykji XjPN

Ykij Wki ; Vi; kPN : i T k; Vk; i PL; 21

Fig. 2. Structure of an optimal solution.



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XjPN:jTk

Ykji XjPN

Ykij Wki ; Vi; kPN : i T k; Vk; i TPL; 22

Zk X

iPN:iTkW

k

i 1; VkPN; 23

Ykij 6Xij; Vi;j PL; VkPN : i T k; 24

Wki 6Zi; Vi; kPN : i T k; 25

Xij Xji6 1; Vi;j PL; 26

YkijP 0; Xij P f0; 1g Vi;j PL; VkPN : kT i;

Wki P 0; Zi P f0; 1g Vi; kPN : kT i:

Note that we have eliminated the Yi

ij

and Wi

i

variables by performing the substitution discussed

above. Also observe that zero-demand or Steiner nodes can easily be accommodated in the for-mulation by specifying that constraints (20) and (23) do not apply to these nodes.

Earlier we noted that an optimal solution to the UFLNDP is a directed, rooted spanning treewith the supernode as its root node. Since the augmented network (Fig. 1) has jNj 1 nodes, anyspanning tree has jNj links. If p of these links are superlinks (the links i; S), then jNj p linksof the tree are ``normal'' links. This gives the following property.

Property 2. An optimal solution to the UFLNDP consists of p facilities and jNj p links.

This property quanties our intuition about the tradeo between constructing facilities and links;

i.e., as we build more facilities, fewer links are needed. The property also has implications in theidentication of polynomially solvable cases, as is discussed in Melkote and Daskin (1997).

3.2. Computational results

We now show that after reformulating the UFLNDP as a pure network design problem, it canbe solved eectively to optimality using a standard MIP solver.

3.2.1. Test problems

To generate test networks, we use an approach similar to that used in Balakrishnan et al.

(1989)'s study of large-scale pure network design problems. (They use a set of randomly generatedtest problems ranging in size from 20 to 45 nodes with varying link densities.) Specically, we use

the following procedure. First the desired number of nodes are randomly generated on a100 100 grid from a uniform distribution. Then the desired number of candidate links arerandomly selected and added to the network, with a bias towards shorter links to emulate

transportation networks. Euclidean distances are computed for each link and rounded to thenearest integer. The demand and xed facility charges at each node are sampled fromUniform 10; 40 and 1200; 1500 distributions respectively and are also rounded to the nearestinteger. These parameters were chosen because of their similarity to those used in the six-nodenetwork of Section 4.1, for which we will observe an interesting relationship between facility, link,



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and transportation costs. We introduce a parameter u, which is dened as the cost of constructingone unit length of a link, so that each link construction cost cij utij. In this way, the constructioncost of each link is directly proportional to its size, which is often the case in real-life scenarios

(Melkote, 1996). To test a range of problem diculties, we use three dierent values of u. Thesevalues are identical to the xed cost/variable cost ratios used by Balakrishnan et al. We also vary

the number of facilities to locate. (Hence constraint (18) is added to the formulation.) Thecomplete experimental design is shown in Table 1. For each of the 3 3 3 3 81 combi-nations of values of the factors, one test problem is generated.

3.2.2. Results

Preliminary computational results (Melkote, 1996) indicated that the LP relaxation of

(UFLNDP/S) is very strong, but the largest problems initially took over 4 h to solve (on a SunSPARCstation 10). Close inspection of the execution of the simplex method indicated that ourproblems are highly degenerate (i.e., the algorithm tends to get ``stuck'' at an extreme point). Thereason for this degeneracy is likely to be the large number of variable upper bound (VUB) con-

straints in our formulation of the general form Ykij 6Xij. These constraints typically make upapproximately 80% of our problem's rows. According to Todd (1982), ``massive degeneracy'' istypically inherent in VUB constraints. He shows how the simplex method may be modied to

circumvent this problem, but unfortunately no large-scale implementations of this algorithm arecurrently available (Fourer, 1996). To resolve the degeneracy, we tried a number of dierent

approaches. It is well known that: (1) interior-point, or barrier methods do not suer from de-generacy, (2) duals of highly degenerate problems are not necessarily degenerate, and (3) the

pricing scheme, which determines the way variables are brought into the basis, may aect de-generacy. To this end, we rst tried using the barrier algorithms of the CPLEX and OSL solvers.However, this was not successful as our machine did not have enough memory for the problem's

Cholesky factorization. We then tried running the primal and dual simplex with their dierentpricing options using both CPLEX and OSL, as well as running primal simplex on the duals of theproblems.

CPLEX 3.0's MIP solver, using the dual simplex algorithm with the default hybrid reduced-cost/normalized reduced-cost pricing scheme to solve the LP relaxations, gave the best results.Table 2 gives the CPU times for our set of test problems on a Sun SPARCstation 10. We observe

that all problems are now solved in less than 2 min, as compared with several hours previously.Branch-and-bound nodes (other than the root node of the branch-and-bound tree) were generatedfor only six out of the 81 test problems, the maximum number of such nodes being only eight.

We also observed that all LP relaxations are within 1% of the corresponding optimal integer

Table 1

Experimental design (81 problems total)

Factor No. of levels Values of levels

jNj 3 20, 30, 40jLj 3 2jNj; 3jNj; 4jNju 3 2, 10, 15

p 3 1, 5, 10



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solutions, indicating that (UFLNDP/S) is very strong, as anticipated. The solution diculty in-creases with the size and density of the network, as we may expect. We also notice that the valueof u does not appear to aect the CPU time.

4. Example applications

Now that we are able to solve the UFLNDP, we use the model to analyze two potentialtransportation planning scenarios: a small six-node problem and a budget design problem.

4.1. Six-node problem

Consider the six-node network shown in Fig. 3, taken from Daskin et al. (1993). In thisproblem, we are given a minimum road infrastructure connecting all nodes (i.e., the minimum

spanning tree). This type of situation often occurs in sparsely populated areas or in developingcountries (Francis et al., 1992). Candidate link additions are shown in dashed lines. The questionswe ask are: (1) where should we locate two facilities, and (2) is considering network improvements

simultaneously benecial? For our analysis, we again use a unit link construction cost u, so thateach link construction cost cij utij. This construction cost applies only to the ``dashed'' links,i.e., the ones that do not currently exist.

Fig. 4 shows the optimal solution to the pure facility location problem; i.e., when we do notconsider network improvements. (In fact, the integrated model gives this solution foruP 21:69.) The facilities are located at nodes C and F, and the objective function value as wellas its breakdown into its component investment and transport costs are shown. The cases forwhich it is feasible to consider link improvements are shown in Figs. 5 and 6. For the range

17:366 u6 21:69, we see the facilities have shifted to nodes A and D and that link (B,D)is added. To see why, consider the specic instance u 18. For this value, total costs are

Table 2

CPU times (in seconds)

Nodes

jN

j

u Links

2jNj 3jNj 4jNj

p: 1 5 10 1 5 10 1 5 10

20 2 3.13 1.32 0.82 4.05 2.27 1.27 4.65 2.05 1.45

10 2.52 1.30 0.88 4.35 2.23 1.23 4.20 2.00 1.57

15 2.67 1.35 0.87 5.45 2.20 1.37 5.72 2.02 1.55

30 2 9.13 4.47 3.57 20.57 6.77 4.43 30.23 8.32 6.28

10 9.78 4.68 3.73 20.23 6.77 4.85 28.22 8.78 6.68

15 10.58 5.17 3.87 20.7 6.77 5.08 30.37 8.72 6.90

40 2 43.52 14.42 10.53 42.08 26.28 10.67 82.98 35.13 16.77

10 40.48 15.23 10.18 52.45 27.93 11.30 108.62 30.93 16.43

15 44.68 13.97 10.82 51.90 27.12 11.87 113.57 33.58 17.02



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4563 26 18 $5031, approximately 2% below the total cost of the pure location problem.This comes from the increased capital investments in links and facilities (2670 26 18 $3138) being more than oset by the large (% 26%) drop in transport costs. Now for the range06 u6 17:36, we see that links (A,B) and (E,F) are added instead, and that the facility at D hasshifted to F. Consider the instance of u 5. The objective function value in this case is4320 40 5 $4520, a 12% decrease compared to the pure location problem. Facility andlink investment costs are 2475 40 5 $2675, or about 3% more than the xed costs in thepure location problem, but the transport costs of $1845 are 27% lower. Why does the facility at

D shift to F? Observe that: (1) the facility at F is less costly; and (2) F has more than double thedemand of D. The lower link costs enable us to build (E,F) and take advantage of the attractivecharacteristics of node F.

Fig. 3. Six-node network (Daskin et al., 1993).

Fig. 4. Optimal solution to pure facility location problem.



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Returning to our question of whether considering link additions is benecial, the potentialbenet of a link, as we have seen, is to: (1) reduce transport costs; and (2) lower facility invest-ments. So the answer to the question is armative, if the cost of a link is less than its benet.

When this is true, the integrated model is clearly useful in identifying better solutions.

4.2. Budget design problem

In this section, we consider a dierent scenario, a budget design problem. In particular, we wish

to address the fundamental question of resource allocation: when is constructing links more cost-eective than building facilities (and vice versa)? We motivate our analysis with the followingscenario. To design a new infrastructure, we have a xed budget Bwhich can be used to construct

facilities or links on the network in Fig. 7. Then B is called an investment or capital expenditure

budget. Given this budget, we wish to design the infrastructure so as to keep operating, or

Fig. 5. Optimal solution, 17:366 u6 21:69.

Fig. 6. Optimal solution, 06 u6 17:36.



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transportation costs, to a minimum. So our new objective function is simply (using the notation of

Section 2)

minimizeX

i;jPL

tijYij:

And we add the budget constraintXiPN

fiZi X

i;jPL

cijXij6B:

Fig. 7. 21-node network.



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The 21-node network of Fig. 7 is a widely used test problem (Daskin, 1987; Hodgson and

Rosing, 1992; Simchi-Levi and Berman, 1988), with the travel costs scaled up by a factor of 10.The demands, which are shown in parentheses beside each node, are normalized so that they sum

to 1000. The xed facility location costs are taken from a Uniform 500; 1500 distribution andnormalized so that their mean is $1000. These costs are assigned to the demand nodes in ascendingorder; that is, they increase with the amount of demand. This assignment is intended to ap-proximate the phenomenon that land values tend to increase with the population density of towns

or cities. The travel cost tij of each candidate link may be interpreted as its length. We assume thatall travel costs are symmetric so that tij tji for all i;j. Also, all distances on this network satisfythe triangle inequality. We again assume that we have a unit link construction cost u and thatcij utij.

We wish to investigate how we can best allocate our budget to the construction of facilities and

links. This decision critically depends on two factors: (1) the size of the budget, and (2) the relativecosts of building links and facilities. By varying the values ofBand the unit link construction costu, we can examine the eect of these factors.

4.2.1. Sensitivity to B

Fig. 8 clearly shows the tradeo between investment and operating costs. As expected, the

transportation costs decrease as the investment budget increases. How is the reduction in oper-ating costs achieved? Figs. 9 and 10 reveal that we start with few facilities but build more facilitiesas the investment budget increases (and consequently, fewer links are built).

In fact, facility expenditures appear to grow linearly with the budget. But what occurs at a moremicroscopic level, between facility additions? Figs. 1114 illustrate the changes in the optimalnetwork conguration for u 2 when we start with the minimum feasible budget of $2000 and

Fig. 8. Transportation costs vs. investment budget.



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increase it in increments of $200. The arrows on the links indicate the direction of ow. We canclearly observe what is happening. When B $2000, we build a tree network consisting of onefacility and 20 links (Fig. 11). This design is very cheap but inecient in terms of travel costs. Notethat this set of links constitutes a minimum spanning tree (MST) for the network. Most demands

have to travel through many nodes to reach the facility very few short paths and direct con-nections exist. Now as we slowly increase the budget, longer (and hence more expensive) links

Fig. 10. Link construction expenditures vs. investment budget.

Fig. 9. Facility location expenditure vs. investment budget.



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providing direct connections to the facility and trans-shipment nodes become feasible, makingthe conguration more ecient by reducing the overall transportation costs. This is observed in

Figs. 12 and 13. Finally, we come to a point at which no more marginal improvements can bemade by changing the set of links the conguration is the best we can achieve with only onefacility. The next improvement can only be attained by building an additional facility (see Fig. 14).

Again, the two resulting trees are inexpensive but inecient. We may expect that when we furtherincrease the budget, more direct and expensive links will replace existing ones until we reach the

best attainable conguration with two facilities, and the next improvement can occur only bybuilding three facilities, and so on.

Fig. 11. Optimal solution for B $2000. (Transport costs $155; 770).

Fig. 12. Optima solution for B $2200. (Transport costs $98; 600).



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The interesting part of these observations is that the model appears to combine the behavior of

budget-constrained facility location and network design problems. On a macroscopic level, wehave seen that we build more facilities as the budget increases, as is true for the budget-con-

strained version of the UFLP. However, on a microscopic level (between facility additions), ourmodel behaves as the budget-constrained xed charge network design problem; i.e., changing

from a (minimum) spanning tree type of conguration to a shortest path tree design as the budgetincreases.

To address our question of resource allocation, our analysis suggests that if a planner wishes

to keep ``up front'' expenditures to a minimum, it is optimal to invest in few facilities and

Fig. 13. Optima soltuion for B $2400. (Transport costs $96; 590.)

Fig. 14. Optimal solution for B $2600. (Transport costs $81; 140.)



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consequently, a large number of links, as specied by Property 2. The tradeo, of course, is in theform of high operating costs. However, by increasing the expenditure level slightly (by increments

of $1000), we achieve large reductions in transportation costs. Fig. 15 illustrates this tradeo forthe case u 2. But the marginal decrease in operating costs diminishes as B increases, so there is

an expenditure level after which further outlays do not result in substantial savings. For our data,with u 2, this level appears to be around $6000. This solution, with ve facilities, 16 links, andtransportation costs of $26,610, may represent a good compromise between infrastructure in-

vestments and operating costs. From the plot we see that the minimum total costs are achieved ata budget of $18,000, with total costs of $19,329. Finally, we note that this plot bears a strongsimilarity to the cost tradeo curves for the UFLP (e.g., see Daskin, 1995).

4.2.2. Sensitivity to u

Figs. 810 show that for a given budget, larger values of u generally correspond to greater

transportation costs, higher link expenditures, and lower facility expenditures. We may interpretthis as follows. For a given budget, suppose a particular facility/network conguration is

optimal. If we are now faced with an increase in the unit link construction cost, we are forced toselect the design more judiciously. This means: (1) reducing the number of facilities or movingthem to cheaper locations (which also contain less demand) and (2) choosing shorter, less

expensive links that provide fewer direct connections to facility and trans-shipment nodes.Clearly, these changes are likely to result in greater operating costs. However, we observe thatas B 3

Pi fi, the model's sensitivity to u diminishes. The reason is that as the budget increases,

we build fewer links (as we have already noted), so a change in u no longer aects our designdecision as much.

Fig. 15. Example transport, investment, and total costs.



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4.2.3. Outliers

The results of our analysis are aected by the fact that the problem data come from uniform (oruniform-like) distributions, which are good approximations in many real-life scenarios. However,

if there is much variation in the data or if clear outliers are present, as may sometimes be the case,the trends discussed above may not hold.

For example, consider the three-node network shown in Fig. 16. When the investment budget is$22, it is optimal to build two facilities (at the bottom nodes), giving transport costs of $1000.

When the budget increases to $102, however, it is optimal to construct only one facility located atthe top node, giving transport costs of $3. So in this case increasing the budget results in de-creasing the number of facilities and increasing the number of links. The culprit, of course, is the

large demand at the top node.As another example, consider the four-node network of Fig. 17. In this instance, clearly, we will

never add the center crossing links between facility additions as the investment budget increases.In other words, we will never attain a shortest path tree conguration at the end of the ``marginal

improvement'' process between facility additions, a trend we observed above.

Fig. 16. Three-node network.

Fig. 17. Four-node network.



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5. Conclusions and research directions

In this paper we have investigated a model that simultaneously optimizes facility locations and

the design of the underlying network. This model has a number of important applications inregional planning, distribution, energy management, and other areas. By formulating the modelas a special case of the NDP, we are able to solve realistically sized problem instances in a verysmall amount of computer time. We discussed some of the fundamental properties of the model.

The model's benet over the classical simple plant location problem was demonstrated using asmall six-node network. We gained insight into the question of resource allocation between fa-cilities and links by using the model to analyze a budget design problem. Our analysis suggested

that, if the data are fairly uniform with few outliers, at small investment levels it is optimal toinvest in few facilities and many links. As we increase the budget, however, more facilities and

fewer links are constructed, and transport costs appear to decrease in a piecewise-linear convexmanner.

Several extensions of the model are possible to enhance its applicability to a variety of real-lifetransportation planning scenarios. In Melkote and Daskin (1998), a generalization of the maxi-mum covering location problem in which the network topology is determined endogenously is

investigated. This extension is particularly relevant in regional planning contexts. Another variantof the model with capacitated facilities is examined in Melkote and Daskin (2000). Other possibleextensions include incorporation of capacitated or congested links, multi-period/dynamic prob-

lems, and models with facility interactions.

Acknowledgements

We thank Dr Arthur C. Hsu of United Technologies, Inc. and Professor Robert Fourer of

Northwestern University for their valuable advice on computational strategies. We also thank twoanonymous referees for their helpful comments.

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An Integrated Model of Facility Location and Transportation Network Design

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