General network design: A unified view of combined location and network design problems

European Journal of Operational Research 219 (2012) 680–697

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

General network design: A unified view of combined location and network design problems Ivan Contreras a, Elena Fernández b,⇑ a b

Department of Mechanical and Industrial Engineering, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Canada H3G 1M8 Statistics and Operations Research Department, Technical University of Catalonia, Campus Nord C5-208, Jordi Girona 1-3, 08034 Barcelona, Spain

a r t i c l e

i n f o

Article history: Available online 12 November 2011 Keywords: Network location Network design Network routing Location

a b s t r a c t This paper presents a unified framework for the general network design problem which encompasses several classical problems involving combined location and network design decisions. In some of these problems the service demand relates users and facilities, whereas in other cases the service demand relates pairs of users between them, and facilities are used to consolidate and re-route flows between users. Problems of this type arise in the design of transportation and telecommunication systems and include well-known problems such as location-network design problems, hub location problems, extensive facility location problems, tree-star location problems and cycle-star location problems, among others. Relevant modeling aspects, alternative formulations and possible algorithmic strategies are presented and analyzed. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Location analysis and network design have emerged as two major research areas in network optimization. Location problems typically involve siting facilities at nodes of a network whereas network design consists of activating some of the links. In both cases the aim is to ensure cost-effective flows between pairs of nodes to satisfy user demands. General network design problems (GNDPs) offer a unified view of these two streams of research which we emphasize in this paper. In particular, we will concentrate on classes of problems where both the facility location and the selection of links are predominant and non-trivial. These problems involve design decisions, which are to locate facilities and to activate links in an underlying network, and operational decisions, which are to allocate customers to facilities and to route the users demands. Below we discuss the main modeling aspects of such problems. Design decisions: facility location and link activation  The facility location decisions indicate where to locate the facilities. In principle, facilities may be located at both the nodes and the arcs of the network. It is well known, however, that when Hakimi’s (1964) node optimality property holds, the set of nodes defines a finite dominating set for the optimal facility locations. In other words, there exists ⇑ Corresponding author. E-mail addresses: [email protected] (I. Contreras), edu (E. Fernández).

e.fernandez@upc.

0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.11.009

an optimal solution in which all the facilities are located at nodes. In the problems considered in this paper we assume that the set of potential locations for the facilities coincides with the set of nodes of the underlying network, even if the node optimality property does not necessarily hold. Examples of problems that involve location decisions are the well-known p-median and p-center problems (Hakimi, 1964), the uncapacitated facility location problem (Kuehn and Hamburger, 1963), the location-covering problem (Toregas et al., 1971) and the maximum covering location problem (Church and ReVelle, 1974).  The network design decisions select the links to be activated. These links are used for connecting users and facilities and, possibly, facilities between themselves. Well-known problems involving network design decisions are, for instance, fixed-charge network design problems, which have been studied by Magnanti and Wong (1984), Balakrishnan et al. (1997) and Minoux (1989), among others. Several costs may affect the design decisions. When a node is selected to locate a facility, a facility set-up cost is incurred. Analogously, when a link is activated in a network design decision, a link set-up cost is incurred. Set-up costs are fixed and do not depend on the flows that circulate through the network. In addition, capacity dependent variable costs may be incurred both at the facilities and the links. It is easy to find further examples of network optimization problems that involve design decisions and that can be seen as location or network design problems. For instance, in the

I. Contreras, E. Fernández / European Journal of Operational Research 219 (2012) 680–697

minimum cost spanning tree problem (Boru˚vka, 1926), the minimum cost matching problem (Edmonds, 1965), and the vehicle routing problem (Dantzig and Ramser, 1959; Laporte, 2009), links with fixed set-up costs must be activated. Similarly, in locationvehicle routing problems (Nagy and Salhi, 2007) joint location and network design decisions must be made. The minimum cost Steiner tree problem (Winter, 1987) is a further example where location (the Steiner nodes) and network design decisions (the edges of the tree) have to be made. Operational decisions: allocation and routing  The allocation decisions indicate the facilities that will be used to serve each user. There are two types of allocations. In single allocation, each user is assigned to exactly one facility, whereas in multiple allocation each user may be assigned to more than one facility. Some classical location problems that involve allocation decisions are the p-median and pcenter problems.  The routing decisions dictate the routes that will be used to satisfy the users demands. As is common in network design, we use the term routing to indicate the paths that are used to send flows between pairs of nodes. In this context, the routing and the network design decisions are closely related, as the links that can be used in the paths, are the outcome of the network design decisions. Well-known network optimization problems with routing decisions are the transportation problem (Monge, 1781; Kantorovich, 1942) and, in general, network flow problems (Ahuja et al., 1993). Several operational costs may affect the above decisions. These include costs associated with the allocation of users to facilities, as well as transportation costs incurred when routing the users demands through the selected links. Both allocation and transportation (or routing) costs are variable, as they depend on the amount of demand of each user allocated to each facility and on the total amount of flow routed through each arc, respectively. Note, however, that single allocation can be seen as a network design decision, since the allocation set-up cost is then fixed. There are, indeed, combinations of the above types of decisions that arise in different network optimization problems. Examples of problems that combine design and operational decisions are classical location problems, which involve joint location and allocation decisions, and classical network design problems, which combine network design and routing decisions like the fixed-cost transportation problem (Balinski, 1961), the optimum communication spanning tree problem (Hu, 1974), or the fixed-charge network design problems (Magnanti and Wong, 1984). Observe, however, that these problems do not involve all of the above decisions. For instance, classical location problems involve no network design or routing decisions, as they assume that direct connections between users and their allocated facilities are used to satisfy the users demands. Similarly, fixed-charge network design problems involve no location or allocation decisions, as they focus on deciding the links to be activated and finding the paths to route flows between users. In general, not every network optimization problem with design decisions includes additional operational decisions. For instance, among the above examples, neither the minimum spanning tree nor the minimum cost perfect matching or the minimum cost Steiner tree problem involves allocation or routing decisions. In a different context, vehicle routing problems require, in general, no allocation decisions unless several depots exist and users must be assigned to depots. Table 1 illustrates combinations of design and operational decisions, and their associated costs, involved in some of the problems mentioned above, namely, p-median (p-MP), uncapacitated facility location problem (PLP), the Steiner

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tree (STP), minimum cost spanning tree (MCSTP), transportation (TP), optimum communication tree (OCTP), vehicle routing (VRP) and its multidepot counterpart (MVRP), network design (FCNDP), fixed-cost transportation (FCTP) and location-routing (LRP). A classification of GNDPs based on the type of demand. In all the GNDPs that we study the aim of the location and allocation decisions is to determine the facilities to select and the allocation of users to facilities. The aim of the network design and routing decisions, however, depends on the role of the facilities, which is dictated by the type of service demand required by users. To better discuss the characteristics of the problems we focus on, we propose a classification of GNDPs in two main categories which are based on the type of users demand.  Problems in which service is given at or from the facilities, so that service demand relates users and facilities will be called GNDPs with user-facility (UF) demand. In GNDPs with UF demand the network design and routing decisions determine how to connect users with their allocated facilities. When users travel to the facilities to receive service, the routing decisions consist of finding the minimum cost path from each user to its allocated facility in the network induced by the activated links. When servers travel from facilities to users, the routing decisions may be more involved if the routes contain several users. Classical location problems can be seen as GNDPs with UF demand involving trivial network design and routing decisions. Since the set-up costs of all the links of the underlying network are zero, the network design decisions will activate all the links and, assuming triangle inequality property, the routing decisions will use direct connections between each user and its allocated facilities. GNDPs with UF demand with nontrivial network design decisions include, for instance, the socalled location-network design problems (Melkote and Daskin, 2001a; Contreras et al., 2010c) and location-vehicle routing problems (Nagy and Salhi, 2007).  Problems in which service demand is between pairs of users and the facilities are used as intermediate locations in the routes that connect pairs of users will be called GNDPs with user-user (UU) demand. In GNDPs with UU demand the network design and routing decisions determine how to connect both users and facilities and facilities between themselves. Classical network design problems can be seen as GNDPs with UU demand involving trivial location decisions. Since the set-up costs of all the facilities are zero, the location decisions will activate all the nodes. Classical hub location problems (O’Kelly, 1986; O’Kelly, 1992; Campbell, 1994), can be seen as GNDPs with UU demand, in which the location and allocation decisions determine the routing decisions. Other GNDPs with UU demand with non-trivial location decisions include, for instance, concentrator location problems (Yaman, 2005; Labbé and Yaman, 2006; Gouveia and Saldanha-da-Gama, 2006), tree-star location problems (Contreras et al., 2009b; Contreras et al., 2010a), and cycle-star location problems (Labbé et al., 2004; Labbé et al., 2005a). Fig. 1 presents a classification of various types of GNDPs according to their type of demand. Rows refer to problems with location, network design and combined location-network design decisions. Columns correspond to problems with UF and UU demand, respectively. For each combination a reference of a well-known problem in the class is given along with a schematic illustration that highlights the main decisions and the generic structure of solutions. A square over a node indicates that a facility has been located at the node. Solid lines represent design decisions, whereas dotted lines represent routing decisions.

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Table 1 Decisions in several classical network optimization problems. Locate facilities at nodes Decisions

p-MP PLP STP MCSTP TP OCTP VRP MVRP FCNDP FCTP LRP

Yes Yes Yes No No No No No No No Yes

Select links

Costs

Decisions

Fixed

Variable

No Yes No – – – – – – – Yes

No Yes/No No – – – – – – – Yes/No

No No Yes Yes No No Yes Yes Yes Yes Yes

Costs

Routing

Allocate users to facilities

Costs

Decisions

Costs

Yes Yes No No No No No Yes No No Yes

Yes Yes – – – – – No – – Yes/No

Fixed

Variable

Variable

– – Yes Yes – – Yes Yes Yes Yes Yes

– – No No – – Yes/No Yes/No Yes/No No Yes/No

No No No No Yes Yes No No Yes Yes No

Fig. 1. Classification of problems according to their type of demand.

The subject of this paper are problems that involve joint location and network design decisions together with non-trivial routing decisions (and, possibly, additional allocation decisions). These problems are called GNDPs and include, for instance, network location problems with links set-up costs. In this case, users may be connected to their allocated facilities by paths with more than one arc. Thus, the location and allocation decisions are not sufficient to completely determine the solutions, as they do not indicate which links can be used in the paths connecting users and their allocated facilities. Hence, non-trivial network design and routing decisions are needed in addition to the location and allocation decisions. GNDPs also include network design problems with nodes set-up costs, where some nodes have to be selected, in which flows between pairs of users will be consolidated or re-routed. In this case, the network design and routing decisions do not fully determine the solution and non-trivial location and allocation decisions are also needed to determine the nodes to be activated. The term ‘‘general network design problem’’ has already been used by several authors, even if not always consistently. In most cases (see, for instance, Balakrishnan et al., 1997) it has been used

to refer to multi-commodity network design problems, as opposed to other problems in which demand for service is restricted to a given subset of the pairs of users like, for instance, the so-called single-commodity network design problems. As already mentioned we extend the term ‘‘general’’ to network optimization problems with fixed-charge and routing decisions that involve additional location decisions. Reviewing such a general class of problems is, indeed, far beyond the scope and space limitations of this paper, and we thus limit our view to only a subset of them. In particular, we focus on problems where both the location and allocation decisions are non-trivial. This excludes pure location allocation problems as well as pure network design problems. The interested reader is referred to Smith et al. (2009) and references therein for a recent overview of location problems, and to up-to-date references of network design problems like, for instance, Croxton et al. (2009) and Frangioni and Gendron (2009). Furthermore, we focus on problems in which the routing decisions determine simple paths. This excludes location-vehicle routing problems, where active and successful research is taking place. In location-vehicle routing problems the

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routes that serve users are circuits rooted at the selected facilities, and modeling techniques other than those presented here are required. The interested reader is referred to Nagy and Salhi (2007) and references therein for an overview of such problems or to more recent works like, for instance, Belenguer et al. (2011) and Catanzaro et al. (2011). We also address problems with a minsum or minmax objective in which costs are derived from design decisions and from allocation or routing decisions. Finally, we focus only on uncapacitated versions of the problems given that, from the modeling point of view, capacitated extensions can be easily obtained from them. Below we present an example to illustrate how the different modeling hypotheses affect the structure of an optimal solution to GNDPs. Example 1. Consider the nine-node network of Fig. 2 where each node represents one user as well as a potential location for a facility. The length of any vertical link is 1 and the length of any horizontal link is 2. All the other links have Euclidean distances; for example, the length of the link connecting nodes 1 and 6 is pffiffiffi 2 2 ’ 2:83. The objective is to minimize the sum of the facilities and links set-up costs plus the routing costs. We analyze nine scenarios, each corresponding to a different combination of costs and type of demand (see Table 2). All the facilities set-up costs have the same value. These values are 1 (scenarios 2 and 4), 5 (scenarios 1 and 3), 45 (scenario 9), 49 (scenarios 6 and 8) and 51 (scenarios 5 and 7). We have considered two different combinations of link set-up costs and routing costs. In scenarios 1, 3, 5, and 7 the link set-up costs are the lengths of the links in the underlying network and all the routing costs are zero. In scenarios 2, 4, 6, 8 and 9 all links set-up costs are zero and the routing costs are the lengths of the links in the underlying network. Finally, scenarios 1 to 4 correspond to instances with UF demand, whereas in scenarios 5 to 9 we work with UU demand. For the scenarios with UF demand all users have one unit of demand, whereas for the scenarios with UU demand each user must send one unit of demand to every other user. The last two columns of Table 2 give an optimal solution under each scenario and the total optimal cost. This information is complemented in Table 3, which gives the costs of the different solutions of Fig. 2 under each scenario. Optimal values for each scenario are highlighted in boldface. Observe that the routing cost of a solution under different scenarios with identical link routing costs depends on the type of demand. For instance, the routing cost of solution pffiffiffi S2 under scenarios 3 and 4, which have UF demand, is 2ð3 þ 2 5Þ ’ 14:94, whereas its routing cost under scenarios 7–9, which have pffiffiffi the same link routing costs but have UU demand, is 32ð2 þ 3 5Þ ’ 239:11. In scenarios 1 and 2 the routing costs are zero. Thus, once the set of facilities and their allocated users are known, the minimum arc set-up arc costs are attained by finding a minimum spanning

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tree connecting each open facility and its allocated nodes, relative to the links set-up costs. In particular, when the facilities set-up costs are 5, the optimal solution is S1 where only facility 5 is open, whereas when the facilities set-up costs are 1, an optimal solution is S3, with set of open facilities {2, 5, 8} (note that an alternative optimal solution is to open a facility at each node). In scenarios 3 and 4 the arc set-up costs are zero. Once the set of facilities to open is known, minimum routing costs are attained by connecting each user directly with its closest facility (recall we are assuming Euclidean distances). In particular, when the facilities set-up costs are 5, the optimal solution is the star S2 where only facility 5 is open, whereas when the facilities set-up costs are 1, an optimal solution is again S3, with a set of open facilities {2, 5, 8}. When all the routing costs are zero, the only aspect of optimal solutions which is affected by the type of demand is that the subgraph induced by the activated links must be connected in order to allow communication between all pairs of nodes. Hence, for the reasons given above, optimal solutions will be spanning trees relative to the links set-up costs. Thus, the optimal solution to the UU scenarios 5 and 6, is again solution S1. Finally, the optimal solutions to scenarios 7 and 8 are S2 and S4, respectively. Observe that both solutions reflect the tradeoff between the facility set-up costs and the weight of the routing costs in the objective function. If the set-up costs are further decreased to 45, as in scenario 9, the optimal solution is Solution S5. The remainder of the paper is organized as follows. In Section 3, we study GNDPs with UF demand. In particular, we study facility location-network design problems as well as some extensions and related problems. In Section 4, we review GNDPs with UU demand in which flows must be sent between pairs of users and facilities are used as intermediate locations for routing the flow through the underlying network. As illustrated in the above example, in this case the subgraph induced by the open facilities must be connected. We focus on problems in which facilities are connected by means of a complete subgraph, a tree-star, a star-star, and a cycle-star structure. The paper ends in Section 5 with some conclusions and directions for future research. 2. Some preliminaries: modeling flows in network design Before presenting different formulations for several GNDPs, we introduce some notation and recall the usual modeling approaches in network design. Unless otherwise stated, we consider an underlying network N with set of nodes V = {1, 2, . . . , n}. The links of N can be either edges (undirected) or arcs (directed). As usual, the set of edges is denoted by E and the set of arcs is denoted by A. An edge e 2 E connecting pair of nodes k, m 2 V, k < m is denoted by e = (k, m). An arc connecting nodes k, m 2 V in the direction from k to m is represented by an ordered pair a = (k, m). Associated with

Fig. 2. Graph and solutions for Example 1.

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Table 2 Data and solutions for Example 1. Scenario

Input

Output

Set-up costs Facilities

Routing costs

Type of Demand

Solution

Total cost

Links

1 2 3 4

5 1 5 1

dij dij 0 0

0 0 dij dij

UF UF UF UF

S1 S3 S2 S3

15.00 9.00 19.94 9.00

5 6 7 8 9

25 21 25 21 20

dij dij 0 0 0

0 0 0 dij dij

UU UU UU UU UU

S1 S1 S2 S4 S5

35.00 31.00 264.11 259.71 257.21

Table 3 Costs of solutions of Fig. 2 for different scenarios. Scenario

Costs

Solution S1

S2

S3

S4

S5

1

Fac. set-up Link set-up Routing Total

5.00 10.00 0.00 15.00

5.00 14.94 0.00 19.94

15.00 6.00 0.00 21.00

10.00 33.44 0.00 43.44

15.00 27.89 0.00 42.00

2

Fac. set-up Link set-up Routing Total

1.00 10.00 0.00 11.00

1.00 14.94 0.00 15.94

3.00 6.00 0.00 9.00

2.00 33.44 0.00 35.44

3.00 27.89 0.00 30.89

3

Fac. set-up Link set-up Routing Total

5.00 0.00 18.00 23.00

5.00 0.00 14.94 19.94

15.00 0.00 6.00 21.00

10.00 0.00 10.47 20.47

15.00 0.00 6.00 21.00

4

Fac. set-up Link set-up Routing Total

1.00 0.00 18.00 19.00

1.00 0.00 14.94 15.94

3.00 0.00 6.00 9.00

2.00 0.00 10.47 12.47

3.00 0.00 6.00 9.00

5

Fac. set-up Link set-up Routing Total

25.00 10.00 0.00 35.00

25.00 14.94 0.00 39.94

– – – unfeas

50.00 29.66 0.00 79.66

75.00 27.89 0.00 102.89

6

Fac. set-up Link set-up Routing Total

21.00 10.00 0.00 31.00

21.00 14.94 0.00 35.94

– – – unfeas

42.00 29.66 0.00 71.66

63.00 27.89 0.00 90.89

7

Fac. set-up Link set-up Routing Total

25.00 0.00 288.00 313.00

25.00 0.00 239.11 264.11

– – – unfeas

50.00 0.00 217.71 267.71

75.00 0 197.21 272.21

8

Fac. set-up Link set-up Routing Total

21.00 0.00 288.00 309.00

21.00 0.00 239.11 260.11

– – – unfeas

42.00 0.00 217.71 259.71

63.00 0 197.21 260.21

9

Fac. set-up Link set-up Routing Total

20.00 0.00 288.00 308.00

20.00 0.00 239.11 259.11

– – – unfeas

40.00 0.00 217.71 257.71

60.00 0.00 197.21 257.21

each edge e = (k, m) 2 E there are two arcs, (k, m), (m, k) 2 A, one in the direction from k to m and the other in the direction from m to k. In general, network design decisions will be related to edges, whereas routing decisions will be related to arcs. To simplify the notation we will further assume that N is a complete network. In this way, for any given k 2 V, the set of nodes connected with k by some edge of E, is just the set of nodes m – k. In addition, we have the following data:  fi > 0: set-up cost for a facility located at node i 2 V.

 ckm > 0: set-up cost for edge (k, m) 2 E.  tkm > 0: unit routing (service) cost for the arc (k, m) 2 A. Throughout we assume that both edge set-up costs, c, and routing costs, t, satisfy the triangle inequality. In some cases, especially when the objective aims at minimizing the total routing costs, we will assume there is a budget constraint, which limits the total setup costs to at most B > 0. Nodes where demand for service exists will be referred to as users and are denoted by VU # V. The set of potential locations

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for facilities is denoted by VF # V. Observe that this is a general setting which includes as particular cases frequent situations such as VU = VF = V, VU \ VF = ;, and VU [ VF ( V, among others. In some problems, it is assumed that there are origin–destination (O/D) pairs of nodes with communication (demand) between them. In these problems the set of such O/D pairs is denoted by C, and V U ¼ fi 2 V j 9ði; jÞ 2 C for some j 2 Vg. For a given O/D pair ði; jÞ 2 C, we denote by Wij the amount of product (flow) that must be routed from i to j. The triplets (i, j, Wij) with ði; jÞ 2 C are referred to as commodities. In network design there are basically two types of mixed-integer programming (MIP) formulations to model the decisions on the edges to activate for routing the flows between pairs of nodes. Both alternatives use binary variables associated with the edges. In particular, for each e = (k, m) 2 E, let ykm = 1 if edge e is used (activated) and zero otherwise. One additional set of decision variables is needed to model the amount of flow in each possible direction of any edge. In the case of path-based formulations, these are continuous variables (referred to as path-based variables), which are used to trace the paths between O/D pairs. That is, for each ði; jÞ 2 C, let xijkm and xijmk denote the fraction of the flow Wij routed through edge (k, m) 2 E in the direction from k to m and in the direction from m to k, respectively. With the above two sets of decision variables, for any O/D pair ði; jÞ 2 C, a feasible flow from i to j of value Wij can be modeled by means of the constraints

X

xijkm 

m–k

X

xijmk

m–k

xijkm þ xijmk 6 ymk

8 if k ¼ i; > <1 ¼ 1 if k ¼ j; k 2 V; > : 0 otherwise; ðm; kÞ 2 E:

ð1Þ

ð2Þ

Eq. (1) model the paths between given O/D pairs whereas constraints (2) guarantee that the paths use edges that have been activated. Observe that these constraints also model the obvious fact that no optimal solution (relative to service or set-up costs) will use a path in which some edge is traversed in both directions. It is well-known (Balakrishnan et al., 1989) that formulation (1)-(2) can be reinforced by adding constraints

xijkm þ xi‘mk 6 ymk

ðm; kÞ 2 E; ‘ 2 V U with ði; ‘Þ 2 C:

ð3Þ

The rationale of constraints (3) extends that of constraints (2), by imposing that if for a given O/D pair ði; jÞ 2 C, one edge e 2 E is traversed in the direction from k to m, none of the commodities having as origin node i will use edge e in the direction from m to k. In the above formulation, the integrality of the y variables and the unimodularity of the coefficients matrix associated with constraints (1) guarantee, for any linear objective, that there exists an optimal solution in which the xijkm variables are binary. Using decision variables y and x, edge set-up costs are modeled as

X

ckm ykm ;

ðk;mÞ2E

whereas routing costs are modeled as

X X

  W ij t km xijkm þ tmk xijmk :

ði;jÞ2C ðk;mÞ2E

One alternative objective is to minimize the routing cost of the commodity associated with the maximum routing cost. This type of objective functions are usually referred to as min–max, bottleneck or center objectives. They have been widely used in location theory, although the literature on center objectives is very scarce in the area of network design. Yet, this objective can be suitable to model network systems where service costs represent displace-

ment times and whose efficiency is measured by the largest service time, or when some final decision cannot be made until all users have been served. There is a very interesting, recent work by Chen et al. (2009) where different types of problems on networks with center objectives are presented and studied. Campbell et al. (2006) have also considered a minmax objective to optimize the maximum travel time in some particular network design problems that involve no location decisions. The center objective can be modeled as the minimization of ði;jÞ2C

  W ij t km xijkm þ tmk xijmk :

X

max

ðk;mÞ2E

The above expression is non-linear, but it can be linearized with the usual technique by defining one additional continuous variable T whose value is to be minimized subject to the additional set of constraints

  W ij t km xijkm þ tmk xijmk ði; jÞ 2 C:

X

TP

ðk;mÞ2E

In the case of flow-based formulations, a different set of continuous variables (referred to as flow-based variables) are used to trace the flows originated at a given node. That is, for each i i i 2 V U ; ðk; mÞ 2 E; hkm and hmk denote the fraction of the flow emanating from node i, which is routed through edge (k, m) 2 E in the direction from k to m and in the direction from m to k, respectively. With these variables, for any node i 2 VU, a feasible flow Oi emanating from i satisfying a demand Wij for each j such that ði; jÞ 2 C, can be modeled by means of the constraints

X

i

hkm 

m–k

i

X

i

hmk

m–k

i

hkm þ hmk 6 ymk

8 > <1 ¼ W ik =Oi > : 0

if k ¼ i; k with ði; kÞ 2 C; k 2 V;

ð4Þ

otherwise;

ðm; kÞ 2 E:

ð5Þ

Now, for a given i 2 VU, Eq. (4) model a flow with of value Oi emanating from i, and destination all nodes k with Wik > 0. As with constraints (2), constraints (5) guarantee that the flows use links that have been activated. Again, the integrality of the y variables and the structure of the coefficients matrix guarantee that, for any linear objective, there is an optimal solution in which the i hkm variables are binary. In this formulation, set-up costs are modeled exactly as before, as only y variables are required, whereas routing costs are modeled as

X X

  i i Oi tkm hkm þ tmk hmk :

i2V U ðk;mÞ2E

It is easy to check that variables x and h are related by means of P P i i the expressions hkm ¼ ði;jÞ2C xijkm , and hmk ¼ ði;jÞ2C xijmk , for all i 2 VU, (k, m) 2 E. In fact, constraints (4) are just an aggregation of constraints (1). Therefore, when the integrality conditions are removed from the y variables the lower bound produced by the path-based formulation will be at least as good as that associated with the flow-based formulation. We next show how path-based and flow-based formulations can be adapted and extended to valid formulations for some GNDPs that incorporate and combine several design and operational decisions. 3. Problems with user-facility demand In this section, we consider GNDPs with UF demand in which facilities are used to provide service to users. In the problems that we present next the location-allocation decisions do not necessarily imply the routing decisions, in contrast to what happens in

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most classical location-allocation problems. Therefore, additional network design and routing decisions are needed to define the paths for routing the users demands to or from their allocated facilities. This class of GNDPs are otherwise known as facility location-network design problems (FLNDP) (see, Daskin et al., 1993; Melkote and Daskin, 2001a; Melkote and Daskin, 2001b; Cocking, 2008; Cocking et al., 2005; Contreras et al., 2010c). In FLNDPs, each user i 2 VU has a demand for service di, and must be allocated to one single open facility, selected from VF, and connected to it by a path. The edges that can be used in these paths must also be decided. If a facility is located at a user-node i 2 VU \ VF, no path is needed for connecting i with its allocated facility, which is also i. Using the notation introduced in Section 2, there are facility set-up costs fi, i 2 VF, and edge set-up costs ckm, (k, m) 2 E. In addition, there are routing costs tkm, for the arcs of the paths connecting users and facilities. FLNDPs can be seen as GNDPs by associating commodities with pairs (i, j), where i is a user and j is the facility i is allocated to, and setting Wij = di, for each i, 2 VU, j 2 VF. Of course, the set of commodities is not known a priori, as the set of facilities has to be established. Several particular cases of FLNDPs correspond to well-known discrete location problems. For instance, if all edge set-up costs are zero, i.e. ckm = 0, (k, m) 2 E, then the network design decisions are trivial, as all edges can be activated at zero cost. In this case, because the triangle inequality property is assumed, minimum cost routing decisions correspond to direct connections between users and their allocated facilities. Thus, if the objective is to minimize the sum of the facilities set-up costs plus the routing costs, we obtain the classical uncapacitated facility location problem (Kuehn and Hamburger, 1963). Other well-known particular cases arise when the facilities set-up costs are all one, i.e. fi = 1, i 2 VF, and the edge set-up costs are all zero, i.e. ckm = 0, (k, m) 2 E. A budget constraint, limiting the total set-up costs to a maximum value p, can then be seen as a constraint to limit the number of open facilities to p. Now we obtain the well-known p-median or p-center problems (Hakimi, 1964), depending on whether we minimize the total routing cost or the maximum routing cost from a user to its allocated facility. FLNDPs, are NP-hard as they have as particular cases the uncapacitated plant location problem, and the p-median and p-center problems, all of which are known to be NP-hard (see, e.g. Kuehn and Hamburger, 1963; Hakimi, 1964; Kariv and Hakimi, 1979; Megiddo and Supowit, 1984). FLNDPs were initially introduced by Daskin et al. (1993) and studied further by Melkote and Daskin (2001a), Melkote and Daskin (2001b), who minimize the sum of the total set-up and routing costs. They mention a number of potential applications related to transportation planning: ‘‘(1) pipeline distribution systems in which the pumps or pumping stations are the facilities and the pipelines make up the network, (2) intermodal transportation systems in which the intermodal rail yards are the facilities and the existing highway and rail networks may be improved, (3) power transmission networks, where the facilities are generating stations, switches, and transformers, and the arcs are transmission and distribution lines’’. Applications in regional planning are mentioned by Melkote and Daskin (2001a), Melkote and Daskin (2001b) as well as by Cocking et al. (2005) and Cocking (2008) in the context of the design of a structure for improving access times to health facilities in the Nouna district of Burkina Faso. We next discuss some properties of the FLNDPs and present alternative formulations. First, observe that any optimal solution to an FLNDP with an objective involving set-up costs or routing costs, induces a rooted forest where the roots of the trees are precisely the open facilities. Indeed, in the network associated with a feasible solution, any arc connecting (i) two nodes allocated to different facilities, (ii) two facilities or, (iii) a node and a facility

different from the one it is allocated to, can be eliminated without deteriorating the value of the associated solution. Moreover, if a solution network contains a cycle, by the triangle inequality assumption, one arc of the cycle can be eliminated without deteriorating the value of the resulting solution. To formulate the location and allocation decisions, for each i 2 VU and j 2 VF, we define binary zij variables as

8 > < 1 if user i 2 V U is allocated to facility j 2 V F ; i – j; zij ¼ 1 if i ¼ j 2 V F and a facility is located at node j; > : 0 otherwise: With the above definition, when a facility is located at a node i 2 VU \ VF, then zii = 1. That is, we assume self-assignment of user i. Variables z must satisfy the constraints

X

zij ¼ 1 i 2 V U ;

ð6Þ

j2V F

zij 6 zjj

i 2 VU;

j 2 V F ; i – j;

ð7Þ

which indicate that each user must be assigned to one single node in which a facility is located. To complete a valid formulation for a FLNDP we can make use of the path-based x variables and the design y variables defined in Section 2. With these variables feasible solutions must satisfy the additional set of constraints

X

xijim ¼ zij

m–i

X

xijkm 

m–k

X

X

xijmk ¼ 0

i 2 V U ; j 2 V F ; i – j;

ð8Þ

i 2 V U ; j 2 V F ; k 2 V; j – i; k;

ð9Þ

m–k

xijmj ¼ zij

m–j

X 

 xijkm þ xijmk 6 ymk

i 2 V U ; j 2 V F ; i – j;

ð10Þ

i 2 V U ; ðm; kÞ 2 E:

ð11Þ

j2V F nfig

Constraints (8)–(10) are an adaptation of Eq. (1), which take into account that now the set of commodities is not known in advance and is given by pairs (i, j) where i 2 VU is a user allocated to facility j 2 VF. The rationale of inequalities (11) is similar to that of constraints (2), although it is now possible to reinforce the left hand side, as for i 2 VU and (k, m) 2 E fixed there is only one index j 2 VF, j – i for which the variables xijkm or xijmk can be non-zero. Inequalities (11) can be reinforced like with constraints (3), yielding the stronger (but larger) set

 X  ij xkm þ x‘jmk 6 ymk

i; ‘ 2 V U ; ðm; kÞ 2 E:

j2V F nfig

As in the path-based formulation for network design, for linear objective functions, the integrality of the z and the y variables guarantees the integrality of the x variables in optimal solutions. Several objective functions have been considered in FLNDPs. Melkote and Daskin (2001a), Melkote and Daskin (2001b) minimize the total sum of set-up and routing costs. That is,

X

fj zjj þ

j2V F

X ðk;mÞ2E

X

ckm ykm þ

X

  di t km xijkm þ tmk xijmk :

i2V U ; j2V F nfig ðk;mÞ2E

One alternative to reach a compromise between routing and set-up costs is to eliminate from the objective function the term corresponding to the latter ones, and to introduce one additional constraint limiting the total set-up costs to a maximum prespecified budget B. Such a constraint can be stated as (Melkote and Daskin, 2001a; Cocking, 2008; Cocking et al., 2005)

X j2V F

fj zjj þ

X

ckm ykm 6 B:

ðk;mÞ2E

Now the objective is to minimize the overall operating costs

I. Contreras, E. Fernández / European Journal of Operational Research 219 (2012) 680–697



 X di tkm xijkm þ t mk xijmk :

X

The minimization of the maximum service costs can be expressed with (12) together with constraints

i2V U ; j2V F nfig ðk;mÞ2E

Budget constraints can also be used in combination with other objectives to reach a tradeoff between set-up and routing costs. For instance, Contreras et al. (2010c) consider an FLNDP with a budget constraint and a center objective, where the maximum routing cost from any user to its allocated facility is minimized. This can be expressed as

min

T

ð12Þ

together with constraints

  tkm xijkm þ xijmk i 2 VU; j 2 VF:

X

TP

ð13Þ

ðk;mÞ2E

Apart from Contreras et al. (2010c), we are not aware of any other GNDP with UF demand in which both location and network design decisions must be made while considering a center objective function. Flow-based variables can also be used to model FLNDPs. We can proceed quite similarly to Section 2, taking now into account the additional z variables. We can, however, obtain a better formulation as follows. Let us extend the original network N and consider a new pseudo-node, denoted by 0, which is connected with an edge with all potential facilities j 2 VF. Furthermore, with each new edge (0,j), j 2 VF, we associate a set-up cost c0j = fj and routing costs t0j = tj0 = 0. The set of commodities is redefined as C ¼ fði; 0Þ j i 2 V U g with demands Wi0 = di, i 2 VU. By definition of the extended network each feasible set of flows for C, corresponds to a set of paths connecting each user with one open facility to which it is allocated (the node of the path connected with 0). As the routing costs of the new arcs are all zero, the routing costs of the paths in the enlarged network coincide with the routing costs of the sub-path connecting each non-facility node to the facility to which it is allocated. Hence, feasible solutions to FLNDPs can also be modeled by means of the following constraints, which are an adaptation of constraints (4), and (5):

X

i

him ¼ 1

m2V[f0gnfig

X

i hkm

X

X



m2Vnfig

i 2 VU; i hmk

¼ 0 i 2 V U ; k 2 V; i – k;

ð14Þ ð15Þ

m2Vnfkg

i hm0

¼1

i 2 VU;

ð16Þ

m2V F i

i

hkm þ hmk 6 ymk

i 2 V U ; ðm; kÞ 2 E [ fð0; jÞ : j 2 V F g: ð17Þ

In (14)–(17) open facilities are associated with nodes j 2 VF coni nected with the pseudo-node, i.e. such that hj0 ¼ 1 for some i 2 VU. Constraints (14)–(16) guarantee that each user is allocated to one open facility and define the paths connecting users and their allocated facilities. The role of constraints (17) is to ensure that the arcs of the paths have been activated. Again, they can be reinforced similarly to constraints (3). As before, the integrality of the y variables guarantees the integrality of the h variables. Using the above set of variables, the minimization of the total routing cost can be expressed as

min

X X

  i i di t km hkm þ tmk hmk ;

i2V U ðk;mÞ2E

whereas the minimization of the total set-up costs can be stated as

min

X j2V F

c0y y0j þ

X

687

ckm ykm :

ðk;mÞ2E

When a budget constraint is added to limit the total set-up costs to a maximum value B, the integrality of the y variables is no longer enough to guarantee that the h variables are binary, and this condition must be stated explicitly.

  i i di tkm hkm þ tmk hmk i 2 VU:

X

TP

ðk;mÞ2E

Next, we exploit the rooted forest structure of the solution graph associated with optimal solutions to FLNDPs to obtain a formulation in which the paths connecting users with their allocated facilities are not traced explicitly and, thus, it uses fewer variables. The set of nodes is partitioned in three groups: (i) the nodes which are roots of some tree (where the facilities are located), (ii) the leaves of the trees, and (iii) intermediate nodes (which are neither facilities nor leaves). A root of a tree is a node with at least one arc entering it but no arc leaving it. A leaf of a tree is a node with exactly one arc leaving it but no arc entering it. An intermediate node has one arc leaving it and at least one arc entering it. We consider design variables y to represent the arcs used in the rooted forests. As opposed to the design variables used in all previous formulations, now the y variables have a direction. In particular, ykm = 1 if and only if the arc (k, m) 2 A is in the path from some user to its allocated facility. We also use allocation variables z to identify non-intermediate nodes (leaves or roots). For j 2 VF, zjj = 1 if and only if node j is the root of some tree (a facility), whereas for i 2 VU, j 2 VF, zij = 1 if and only if i is a leaf of some tree with root at node j. Note that the meaning of the z variables is not like in previous formulations, as all the allocation variables associated with intermediate nodes take the value zero. Finally, intermediate nodes are associated with binary variables q. For i 2 V, qi = 1 if and only if i is an intermediate node of some rooted tree. The formulation of a rooted forest in the underlying network N is as follows:

X

zij þ qi þ zii ¼ 1

j2V F nfig

zij þ qj 6 X

X

ykj

ðk; jÞ2A

X

zij þ qi ¼

j2V F nfig

yij

i 2 V;

ð18Þ

i; j 2 V;

ð19Þ

i 2 V;

ð20Þ

i; j 2 V; j – i:

ð21Þ

j2Vnfig

yij 6 qj þ zjj

Constraints (18) partition nodes into roots (facilities), leaves, and intermediate nodes. Inequalities (19) ensure that, when j 2 V is an intermediate node or a facility, then some arc entering node j is activated. Eq. (20) model the condition that if i is a leaf or an intermediate node, then some arc must leave i. Constraints (21) state that if arc (i, j) is used, then j must be either a root or an intermediate node. Note that in constraints (18) and (20) the term P j2V F nfig zij only exists if i 2 VU. Also, when i R VF, neither constraints (18) nor (21) contain the term zii. In order to model the routing costs we define one additional set of decision variables that we denote by g. For i 2 VU, gi is an upper bound on the unit routing cost from i to its allocated facility, so that digi is an upper bound on the routing cost associated with user i. These new variables are related to the design variables y by the constraints

g i P ðg j þ t ij Þyij ;

ð22Þ

which indicate that if arc (i, j) belongs to the rooted forest, then the unit routing cost from node i to its allocated facility must be at least gj + tij. In principle, digi is an upper bound on the routing cost from i to its allocated facility. However, when the objective aims to minimize the routing costs, digi will coincide with the exact value of the routing cost associated with i. The difficulty due to the non-linearity of inequalities (22) can be overcome by expressing them as

g i P g j þ tij yij  Mð1  yij Þ;

ð23Þ

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I. Contreras, E. Fernández / European Journal of Operational Research 219 (2012) 680–697

where, as usual, M is a large enough constant (in this case, M P maxi,j2V,i–j tij). Using the above decision variables, the minimization of the total routing cost can be expressed as

min

X

3.2. Extensions and related problems

di g i ;

i2V U

whereas the minimization of the maximum routing cost is formulated with (12), together with the additional set of constraints

T P di g i

many variables and cannot be solved using a general purpose solver, due to memory limitations.

i 2 VU:

3.1. Comparison of formulations In general, the number of variables in MIP formulations is a valuable indicator of the difficulty to solve them. In the case of the FLNDP, the path-based formulation has OðjVj4 Þ variables whereas the flowbased formulation has OðjVj3 Þ. Thus, from this point of view, the latter should be preferred. However, it is also known that formulations with fewer variables may lead to very weak linear programming (LP) relaxations, which, in practice, may translate into an excessive computational burden for solving the problems to optimality. Hence, the quality of the LP relaxation bound is also a valuable indicator of the effectiveness of MIP formulations. As expected, any feasible solution to the LP relaxation of the pathbased formulation defines a feasible solution to the LP relaxation of the flow-based formulation. In particular, if (x, y, z) is a solution to the LP relaxation of (6)–(11), then the solution (h, y) with the same vecP i i tor y and hkm ¼ j2V F nfig xijkm for all i 2 VU, k, m 2 V, m – i,k and hj0 ¼ zij for all i 2 VU, j 2 VF, is feasible to the LP relaxation of (14)–(17). Observe that the reverse is also true. That is, a feasible solution for the LP relaxation of (14)–(17) defines a feasible solution to the LP relaxation of (6)–(11). In particular, if (h, y) is a feasible solution to the LP relaxation of (14)–(17), then the solution (x, y, z) with the i i i same vector y and xijkm ¼ hkm hj0 ; zij ¼ hj0 defines a feasible solution to the LP relaxation of the flow-based formulation. Hence, when the same objective function is considered, the LP relaxations of both formulations yield the same lower bound. Therefore, for the FLNDP, the flow-based formulation should be preferred to the path-based formulation. Yet, in terms of the number of variables, the flow-based formulation for the FLNDP is dominated by the formulation with 2index variables. Unfortunately, the LP bound associated with the formulation with the 2-index variables is extremely weak, as the example of Fig. 3 illustrates. In this example with V = VU = VF = {1,2,3}, the non-zero design variables are y12 ¼ y23 ¼ y31 ¼ 12. The remaining non-zero variables are q1 ¼ q2 ¼ q3 ¼ 12 and z11 ¼ z22 ¼ z33 ¼ 12. The reader can check that the above fractional solution satisfies constraints (18)–(21). Furthermore, it is easy to check that this vector y, satisfies constraints (23) together with gi = 0 for all i 2 V. Hence, when the objective is to minimize the routing costs, the LP bound will be zero. Therefore, we cannot conclude that the formulation with 2-index variables should be preferred to the flow-based formulation. However, it can be considered as an alternative for large size instances for which the flow-based formulation already requires too

Several other GNDPs with UF demand have been proposed in the literature. Melkote and Daskin (2001b) present the capacitated facility location - network design problem (CFLNDP), which extends the FLNDP to consider a limited amount of capacity at the facilities. Drezner and Wesolowsky (2003) deal with another class of FLNDPs in which the arcs of the network can either be one-way or two-way and the transportation costs consider round trips from each open facility to its allocated nodes. Murawski and Church (2009) study a FLNDP in which the objective is to maximize the demand that is captured subject to a budget constraint on the total set-up costs of the selected arcs. Ravi and Sinha (2006) present polynomial time approximation algorithms for integrated logistic problems that combine elements of facility location and network design. In particular, they introduce the capacitated cable facility location with unit demands, which generalizes both the Steiner tree and the uncapacitated facility location problems. They also consider a capacitated p-median problem, which restricts the maximum number of open facilities to p. Campbell et al. (2006) present a model to minimize the maximum travel time in specific network design problems that involve no location decisions. They consider a series of problems that involve the selection of q arcs in already established networks so as to minimize the total diameter of the upgraded network. Melo et al. (2009) present a survey with more than 120 references of facility models involving location decisions arising in the design of supply chain networks. The reviewed models cover a wide range of issues, including multi-layer facilities, multiple commodities, single/multiple period(s), and deterministic/stochastic parameters. Some of these models include additional decisions such as capacity design, inventory control, procurement, production planning and routing. Several of these problems can be seen as FLNDPs with additional constraints. Cordeau et al. (2006) and Cordeau et al. (2008) propose an integrated model for a logistic network design problem involving strategic and tactical decisions. According to our classification, that problem is a GNDP with UF demand, which can be seen as an FLNDP with additional requirements. 4. Problems with user-user demand In this section, we consider GNDPs with UU demand, in which the facilities are used as intermediate locations where flows between users are consolidated and re-routed to their final destinations. Like in previous GNDPs, these problems involve location, allocation, network design and routing decisions. The location decisions select the nodes to locate facilities. The allocation decisions indicate the facilities used to send or receive the flows with origin or destination at the users. The network design decisions activate two types of edges: those connecting facilities between

Fig. 3. Very week LP solution to FLNDP with 2-index variables.

I. Contreras, E. Fernández / European Journal of Operational Research 219 (2012) 680–697

themselves, and those connecting users and their allocated facilities. The routing decisions define paths going through facilities to send the demand of users. In some cases, the network design and routing decisions are implied by the location-allocation decisions. In other cases, the network design and routing decisions affect the structure of optimal solutions and, therefore, are needed. As before, V denotes the set of nodes, VU # V the set of demand users, and VF # V the set of potential locations for the facilities. Each pair of users i, j 2 VU, i – j defines an O/D pair and Wij denotes the amount of flow that must be sent from i to j. It is assumed that flows must be routed through paths in which all intermediate nodes are facilities. The set-up cost for a facility located at node i 2 VF is denoted by fi, and the set-up cost for edge (k, m) 2 E is denoted by ckm. In addition, unit routing costs tkm are associated with the arcs of the paths connecting users between themselves. The type of allocation of users influences the solutions to the different problems of this section. In single allocation models, it is assumed that all the flow leaving or entering a user is routed through the same facility. In multiple allocation models, the flow leaving or entering a user may be sent through a different facility, depending on the O/D nodes of the commodities. Fig. 4 illustrates the difference between the two types of allocation patterns. From a modeling point of view the possibility of having single allocation or multiple allocation causes no difficulty and both cases can be handled similarly. In particular, for single allocation we can consider binary allocation variables z, such that for k 2 VF, zkk = 1 if and only a facility is located at node k; and, for i 2 VU, k 2 VF, i – k, zik = 1 if and only if user i is allocated to facility k. This definition coincides with that of the allocation variables for the flow-based and path-based formulations for the FLNDP, and again assumes self-assignment when a facility is located at a node i 2 VU \ VF. For multiple allocation we can consider continuous allocation variables, also denoted by z, such that zik represents the fraction of the total flow leaving node i 2 VU, routed via facility k 2 VK, k – i, and for k 2 VF, zkk = 1 has the same meaning as above. In both cases the sets of constraints

X

zik ¼ 1 i 2 V U ;

ð24Þ

k2V F

zik 6 zkk

i 2 VU;

k 2 VF

ð25Þ

is needed, so the only difference is the domain of the variables. GNDPs with UU demand can be further classified according to the topological structure induced by the facilities and the edges of the underlying network used to connect them. We focus on structures that have received the most attention in the literature, which are of the following types: (i) cliques, i.e. facilities are fully interconnected, (ii) trees, i.e. facilities are connected by means of a tree, and (iii) cycles, i.e. the edges that connect the facilities define a cycle. For trees and cycles, we will further assume single allocation, so that the feasible solutions induce tree-star and cycle-star topologies, respectively. Basically, the only differences among

689

problems of the same type refer to the objective function. That is, some problems consider either service or design costs, whereas other problems consider both. 4.1. Fully interconnected facilities We next review several problems in which facilities are fully interconnected. This assumption simplifies the network design decisions given that now it is only required to select the edges connecting users and their allocated facilities. Under this assumption, and because of the triangle inequality property, the paths that route the flows between users contain at most two intermediate facilities. That is, the path from i to j has the form (i, k, m, j) where k and m respectively denote the facilities i and j are allocated to. When a facility is located at i, then k = i. Similarly, when a facility is located at j, k = j. Also, when i and j are allocated to the same facility, m = k. Note that with single allocation, for a given user i 2 VU, all feasible paths leaving i go through the same facility k, independently of the destination node j, while in multiple allocation the facility k may differ depending on the destination node j. A similar situation occurs with the intermediate facility m relative to node j. Potential applications of models where facilities are fully interconnected arise in a broad range of applications related to the design of networks in telecommunications, air and ground transportation, postal delivery, and computer systems. Applications in telecommunications occur in a variety of contexts where facilities correspond to electronic devices (such as switches, multiplexors, and concentrators), and demand corresponds to data transmissions routed over a variety of physical media (i.e. fiber optic cables or telephone lines) or through the air (i.e. satellite channels or microwave links). The economies of scale in transmission and utilization costs encourage the use of structures with full interconnection between facilities. In the case of transportation networks, applications of these models appear in package delivery systems, air freight travel, air passenger travel, rapid transit systems, large trucking systems, and postal operations. In this class of applications, facilities correspond to transportation terminals or sorting centers, and commodities (i.e. passengers, express packages, mail, goods) are routed in vehicles (i.e. trucks, trains, airplanes). High transportation costs promote the use of networks where facilities are fully interconnected to benefit from the use of economies of scale. Moreover, due to the huge amounts of flows to be routed in most transportation networks, facilities are usually fully interconnected. In the case of single assignments, the relation between the path-based and allocation variables is,

xijkm ¼ zik zjm

i; j 2 V U ; i – j; k; m 2 V F :

This is a non-linear relation that can be easily linearized as

Fig. 4. Types of allocation: (a) Single allocation; (b) Multiple allocation.

690

X

I. Contreras, E. Fernández / European Journal of Operational Research 219 (2012) 680–697

xijkm ¼ zik

i; j 2 V U ; j – i;

k 2 VF;

ð26Þ

xijkm ¼ zjm

i; j 2 V U ; j – i;

m 2 VF;

ð27Þ

m2V F

X

k2V F

which, together with constraints (24) and (25) guarantee that each commodity is routed through the arcs connecting the users and the facilities. Note that, because of the single assignment assumption, the allocation and routing subproblems remain difficult problems, even if the location of facilities is known. However, when the allocation of users to facilities becomes also known, the paths connecting O/D pairs associated with each commodity can be easily determined, since for each pair i, j 2 VU, i – j, xijkm ¼ 1 if and only if, i is allocated to k 2 VF and j is allocated to m 2 VF. In the case of multiple allocation, for a given pair of users i, j 2 VU, i – j, the relation xijkm ¼ zik zjm no longer holds. Now the location/allocation and routing variables can be related using the two set of constraints

X

xijkm 6 zkk

i; j 2 V U ;

k 2 VF;

ð28Þ

m2V F

X

xijkm 6 zmm

i; j 2 V U ;

m 2 VF;

ð29Þ

which guarantee that all commodities are routed through the open facilities. It is known (Hamacher et al., 2004; Marı´n, 2005a) that such paths can also be modeled with the following set of tighter and more compact constraints,

X

xijkm þ

m2V F

xijmk 6 zkk

i; j 2 V U ; k 2 V F :

ð30Þ

m2V F nfkg

Oi r ikj ¼ W ij

k2V F nfjg

X

rikj 6

i2V U

X

W ij zkk

i; j 2 V U ; j 2 VU; k 2 VF:

i2V U

Feasible flows must now satisfy the set of constraints

X

i

hkm þ

X j2V U nfig

r ikj ¼ zik þ

X

i

h‘k

i 2 VU; k 2 VF:

ð32Þ

‘2V F nfkg

The explanation of (32) is quite similar to that of constraints (31), except for the fact that now both the left and right hand sides represent the fraction of flow emanating from user i that leaves and enters facility k, respectively. With multiple allocation, the relation between the x and the h variables is the same as with single allocation. In addition, the r variables are also related to some x variables according to the expression r ikj ¼ Oi xijkk . The advantages and limitations of each type of formulation are the same as with single allocation.

m2V F nfkg

In contrast to the single allocation case, when multiple allocation is allowed and the location of facilities is already known, the allocation-routing subproblem is reduced to finding the best paths between the O/D pairs associated with each commodity, which can easily be obtained. Flow-based formulations can also be used to formulate GNDPs with UU demand. Let us redefine the flow variables h defined in i Section 2, so that now hkm denotes the fraction of the flow emanating from i 2 VU that is routed through consecutive facilities k, m 2 VF, k – m. In the case of single allocation, feasible flows routing the commodities Wij, i, j 2 VU through paths (i, k, m, j) can be modeled with constraints

X

X

m2V F nfkg

k2V F

X

for medium size instances, due to the number of x variables. In this case, flow-based formulations should be preferred. In the case of multiple allocation, constraints (31) do not model correctly the balance of flow through the underlying network, as the term Wijzjk does not give the value Wij for 0 < zij < 1, when all the flow Wij arrives at j directly from facility k. Now, additional flow-based variables are needed to model the flows from users to facilities. In particular, for i, j 2 VU, j – i, and k 2 VF, let rikj now denote the fraction of the flow emanating from user i which is routed through the path (i, k, j). The rationale of these variables and their relation with the z variables is modeled by the two set of constraints

i

Oi hkm þ

X j2V U nfig

W ij zjk ¼ Oi zik þ

X

i

Oi h‘k

i 2 VU; k 2 VF;

‘2V F nfkg

ð31Þ which are a restatement of constraints (5) to this particular case. In constraints (31), for i 2 VU, k 2 VF given, the right hand side represents the flow originated at user i that enters facility k, whereas the left hand side represents the flow originated at user i leaving or staying at facility k. When i is allocated to k then zik = 1. In this P i case, ‘2Vnfkg Oi h‘k ¼ 0 and Oizik is the flow arriving at k from i. The left hand side is split in the sum of the demands Wij routed to their destination directly from facility k (when j is also allocated to k so zjk = 1), plus the flow emanating from i routed through conP i secutive facilities k, m 2 VF, m – k, which is precisely m2V F nfkg Oi hkm . When i is not allocated to k, then all the flow emanating from i arriving at facility k must be routed through some other facility ‘ – k. The explanation of the left hand side is the same as before. It is clear that flow variables h can be expressed as P i hkm ¼ j2V U nfig xijkm . Thus, flow-based formulations are aggregations of path-based formulations, and therefore, the former usually lead to tighter LP bounds. However, when using general purpose solvers, path-based formulations cannot be handled efficiently even

4.1.1. Hub location Hub Location Problems (HLPs) are by far the most studied problems among GNDPs with UU demand. In these problems, hub facilities are used as consolidation, switching or transshipment points so as to reduce set-up costs, centralize commodity sorting and handling operations and, most importantly, to achieve economies of scale on service costs through the consolidation of flows. There is a large literature on these problems and many different models have been proposed considering different assumptions and characteristics. We do not intend to review all such problems, but rather to highlight how most usual hub location models fit within the framework of GNDPs with UU demand. Readers interested in an overview of HLPs are referred to the survey by Campbell et al. (2002) or the more recent one by Alumur and Kara (2008). In general, HLPs constitute a challenging class of NP-hard problems. Their main difficulty lies in the inherent interrelation between the location and routing decisions. In the case of HLPs with multiple assignments and no capacity constraints, once the location of hubs is known, the allocation-routing subproblems can be efficiently solved by an all-pairs shortest path problem (Ernst and Krishnamoorthy, 1998b). However, in the case of HLPs with single assignments, even with no capacity constraints, when the location of hubs is fixed, the resulting allocation-routing subproblem is equivalent to a quadratic semi-assignment problem, known to be NP-hard (Sahni and Gonzalez, 1976). Next we discuss HLPs under the usual requirement that the paths that route the flows between users contain two intermediate hubs at the most. The implications of this requirement and alternatives for formulations have already been discussed. Furthermore, in HLPs the use of arcs connecting two hubs is usually enhanced by applying economies of scale to their routing costs. This is modeled by scaling the routing costs of such arcs with a discount factor 0 6 a 6 1. In a more general setting it is also possible that correction factors v and d apply to the collection and distribution costs, respectively. Now, the unit routing cost through a path (i, k, m, j)

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can be computed in advance as vtik + atkm + dtmj. Hence, the cost for routing the flow Wij through the path (i, k, m, j) is F ijkm ¼ W ij ðvtik þ atkm þ dtmj Þ. Therefore, using the path-based variables x, the overall routing costs, for both single and multiple allocation, can be expressed as

X

X

F ijkm xijkm :

i; j2V U k;m2V F

With single allocation, when flow-based variables are used, the routing costs can be expressed as

X

X

Oi

i2V U

atkm hikm þ

X X

ðvOi þ dDi Þzik ;

i2V U k2V F

k;m2V F ;k – m

whereas with multiple allocation the routing costs can be evaluated with

X i2V U

Oi

X

atkm hikm þ

k;m2V F

X i2V U

Oi

X X

ðvtik þ dtkj Þr ikj :

j2V U nfig k2V F

Similarly to previous models, the set-up costs for the selected hubs can be expressed using the allocation variables. However, to model the set-up costs of the activated edges, it is again necessary to introduce additional design variables, y, with ykm = 1 if and only if an arc associated with edge (k, m) 2 E is used in some path. Constraints (2) or (3) can be used to link the y variables with the x variables. As in the case of FLNDPs, the expression of the overall set-up costs can, thus, be modeled as

X i2V F

fi zii þ

X

ckm ykm :

ðk; mÞ2E

Alternatively, hub set-up costs can be taken into account, by means of a cardinality constraint limiting the number of hubs to a maximum value p. Indeed, this is equivalent to adding a budget constraint with B = p, fi = 1 for all i 2 VF, and ckm = 0 for all (k, m) 2 E. Different authors have proposed path-based formulations for various HLPs with fully interconnected hubs. Among them we can mention Campbell (1994), Skorin-Kapov et al. (1996), Hamacher et al. (2004), Marı´n (2005a), and Marı´n et al. (2006) for multiple allocation variants, and Campbell (1994), and Skorin-Kapov et al. (1996) for single allocation variants. Among the works that have used flow variables to model HLPs we can mention Ernst and Krishnamoorthy (1998a), Boland et al. (2004), and Marı´n (2005b), for multiple allocation versions and Ernst and Krishnamoorthy (1996), Ernst and Krishnamoorthy (1999), and Correia et al. (2010) for single allocation versions. When using general purpose solvers for solving HLPs, pathbased formulations are usually able to optimally solve instances involving no more than 50 nodes, due to memory limitations. Flow-based formulations are able to approach slightly larger instances. However, because the usually weak lower bounds associated with these formulations instances with more than 75 nodes can not be solved. In order to deal with more realistic, large-scale instances, several decomposition methods have recently been developed, which exploit the strengths of path-based formulations. In the case of HLPs with multiple assignments, Contreras et al. (2012) present an exact algorithm, based on Benders decomposition, able to solve uncapacitated instances with up to 500 nodes. In the case of of HLPs with single assignments, Contreras et al. (2011a) present an exact algorithm, based on column generation and Lagrangean relaxation, able to optimally solve capacitated instances with up to 200 nodes. To the best of our knowledge, these are the largest HLPs instances that have been solved so far.

691

their incoming or outgoing flow, or that there is a limit for the amount of flow that can be routed through the arcs of the underlying network. Some related works for multiple allocation case are Campbell (1994), Ebery et al. (2000), Boland et al. (2004), Marı´n (2005b). For the single allocation case we can mention Campbell (1994), Labbé et al. (2005b), Yaman (2008),Contreras et al. (2009a), and Contreras et al. (2011a). Some extensions, in which the size (or capacity) of the hub facilities is part of the decision process, have also been studied Correia et al. (2010). Alternative objectives have been proposed for hub location models. Campbell (1994) presents several variants of HLPs dealing with center objectives so as to minimize: (i) the maximum routing cost for any origin–destination pair, (ii) the maximum routing cost associated with any arc on the hub network, (iii) the maximum routing cost between a hub and an O/D point. In Alumur and Kara (2008), several HLPs with minmax objectives are also described. For instance, Yaman et al. (2007) present the latest arrival hub location problem with stopovers, in which the objective is to minimize the maximum travel time on the network, assuming that paths between O/D points may include more than one non-hub node. Campbell (1994) also presents several variants of HLPs dealing with cover type of objective functions. In hub covering problems, demand is covered if both O/D points are within a specified distance from a hub. Campbell (1994) proposed three different types of coverage criteria. An O/D pair (i, j) is covered by hubs k and m if: (i) the routing cost of the path from i to j via hubs k and m is within a specified value, (ii) the routing cost of each arc in the path from i to j via hubs k and m does not exceed a predetermined value, or (iii) the routing costs of the origin-hub and hub-destination arcs do not exceed some given values. Other extensions and more general models have been considered in the literature. Campbell et al. (2005a,b) present several hub arc location problems, in which it is desired to locate hub arcs rather than hub nodes. In these problems it is not necessary to have a fully interconnected network between the hub nodes. In fact, hub arcs may not even define a single connected component. Several authors have also studied other hub location problems requiring no particular topological structure for the connection of the facilities like, for instance, the design of incomplete hub networks (O’Kelly and Miller, 1994; Alumur et al., 2009). HLPs in which a dynamic or multiperiodic nature is considered, have been studied by Gelareh (2008) and Contreras et al. (2011b). These problems consider a finite time horizon in which it is necessary to decide when and where to locate hub facilities and when they should be closed. Other related problems are the so-called concentrator location problems, arising mainly in telecommunications. These problems can be seen as a particular case of HLPs with no routing costs for the arcs connecting two facilities (or concentrators). That is, the objective function is the sum of the facilities set-up costs plus the routing costs between user nodes and the facilities. In practice, since full interconnection of concentrators implies no additional cost, it is possible to formulate the problem as a variation of the well-known single source plant location problem with or without capacity constraints (see, for instance, Cornuejols et al., 1990; Dı´az and Fernández, 2002). Concentrator location problems have been studied in (Yaman, 2005 and Labbé and Yaman, 2006), where polyhedral properties of these problems have been studied. Gouveia and Saldanha-da-Gama (2006) have also given a reformulation by discretization for a capacitated version of the problem. 4.2. Facilities connected with a tree

4.1.2. Extensions and Related problems Capacitated hub location models have also been studied. In these problems, it is assumed that hubs have a finite capacity on

In this section, we review GNDPs with UU demand in which facilities are connected by means of a tree. There are several

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problems in the literature, under different names, in which facilities have to be located on a network and connected with a tree structure, and additional allocation or routing decisions are involved. In the case of single assignment, which is the case that we consider in this section, the allocation of users to facilities induces a forest consisting of a series of stars, each of them rooted at an open facility. As the topology induced by open facilities is a tree, feasible solutions induce tree-star topologies. The assumption that the facilities are connected by means of a tree notably increases the difficulty of the problems, since the routing decisions are no longer trivial. Now the paths connecting pairs of users may traverse several consecutive facilities, and cannot be predicted in advance. Potential applications of models where facilities are connected by means of a tree occur when the edges set-up costs are very high, so full interconnection is prohibitive. Since all the commodities must be routed, a path must exist between each O/D pair, so a connected graph must be built. Due to the high connection costs, connectivity must be achieved using the minimum number of edges. Specific applications of such problems arise in telecommunications (see, e.g. Hu, 1974 or Nguyen and Knippel, 2007) and in transportation (see, for instance, Chen et al., 2008, for an excellent description of the practical relevance of tree-backbone problems in small package delivery systems). Other examples of applications of treestar location problems appear in the design of high-speed train networks or rapid transit systems for urban areas where flows of users (citizens) must travel between O/D pairs. More often than not, such systems are designed with a tree structure, where the users are allocated to the closest station (facility). 4.2.1. The tree of hubs location problem In the Tree of Hubs Location Problem (THLP), p hubs have to be located at the nodes of a network and connected by means of a tree. Each user must be allocated to a hub, and only two types of arcs can be used to route the commodities: (i) arcs connecting users and their allocated hubs, and (ii) arcs connecting two hubs. Similarly to previous models we assume self-assignment when a facility is located at a user node i 2 VU \ VF, so in this case no arc is used to connect the user and its allocated facility. Like in other hub location problems, a discount factor a is applied to the routing costs of the arcs connecting two hubs, and factors v and d are applied to the collection and distribution costs between users and facilities, respectively. The TLPH limits the number of open hubs to exactly p. The objective is to minimize the total routing costs. The TLPH has been studied by Contreras et al. (2009b, 2010a). As before, the cardinality constraint on the number of hubs can be seen as a budget constraint, which can be easily modeled using the same z allocation binary variables as before. Now, in addition to constraints (24), and (25), the following condition must be imposed

X

zkk ¼ p:

ð33Þ

k2V F

Like in classical HLPs, the routing aspect of the THLP is affected by both the connection of users to hubs (which is determined by the allocation decisions) and the connections between hubs. Since the paths connecting the end-nodes of commodities may now traverse more than two hubs, formulations for this problem must trace such paths. In particular, we can use binary path-based variables x where, for each pair i; j 2 V U ; i – j; xijkm ¼ 1 if and only if the flow Wij traverses arc (k, m) connecting hubs k, m 2 VF. As before, when k = m such variable takes the value 1 when both i and j are allocated to the same hub k, so the path connecting i and j is (i, k, j). Feasible paths connecting pairs of users must now satisfy the flow balance equalities

X

X

xijkm þ zjk ¼ zik þ

m2V F nfkg

xijmk

i; j 2 V U ; i – j; k 2 V F ;

ð34Þ

m2V F nfkg

which are an adaptation of constraints (1). Indeed, for i, j 2 VU, i – j, and k 2 V given, the left hand side of (34) takes the value one if Wij leaves hub k. This may occur either because some arc (k, m) is used in the path from i to j or because user j is allocated to hub k. The right hand side of the constraint takes value one if Wij enters hub k. This may occur either because some arc (m, k) is used in the path from i to j or because user i is allocated to hub k. Hence, constraints (34) define paths connecting pairs of users. To model the additional condition that hubs are connected by a tree we resort to design variables y, which are linked to the x variables by the usual constraints (2) or (3). In Contreras et al. (2009b) it is proven that, together with the previous constraints, the new equality

X

ykm ¼ p  1;

ð35Þ

ðk;mÞ2E

guarantees that the edges connecting pairs of hubs define a tree. This implies that classical subtour elimination constraints are not required to prevent that paths between O/D pairs contain cycles. With the above variables, the objective function that minimizes the total routing costs can be modeled as

min

X X X

X

aW ij tkm xijkm þ

i2V U j2V U nfig k2V F m2V F nfkg

X X

ðOi vtik þ Di dt ki Þzik :

i2V U k2V K

ð36Þ The THLP can also be modeled using the flow-based variables h i defined in Section 4.1 such that hkm denotes the fraction of the flow originated at user i which is routed via consecutive hubs k, m 2 VF. Like in HLPs with full interconnection between hubs, flows between pairs of users are modeled by constraints (31). Again, constraint (35) is needed to ensure that a correct number of edges are used in the tree connecting the hubs. Now the following constraints relate the design y variables to the allocation and the h variables:

zij þ yij 6 zjj

ði; jÞ 2 E with i 2 V U ; j 2 V F ;

ð37Þ

zij þ yji 6 zjj

ðj; iÞ 2 E with i 2 V U ; j 2 V F ;

ð38Þ

ðk; mÞ 2 E:

ð39Þ

i hkm

þ

i hmk

6 ymk

The structure of constraints (31) together with constraints (37)– (39) guarantee that in optimal solutions, the paths that route the flows between users, and hence the paths connecting pairs of hubs, contain no cycles (see Contreras et al., 2009b for details). However, the additional assumption that the graph induced by the flows between pairs of users is connected is needed to guarantee the correctness of the above formulation. Otherwise, the subgraph induced by the open hubs could have several connected components. Using the flow-based variables, the objective function can now be expressed as

min

X X

X

i2V U k2V F m2V F nfkg

aOi tkm hikm þ

X X

ðOi vt ik þ Di dtki Þzik :

ð40Þ

i2V U k2V K

In Contreras et al. (2009b) the THLP is introduced and a flow-based formulation is presented. Several valid inequalities are also presented and a cut-and-branch algorithm is developed to solve instances with up to 25 nodes to optimality. Contreras et al. (2010a) propose a path-based formulation, yielding tighter LP bounds than the flow-based formulation, and present a Lagrangean relaxation, yielding tight upper and lower bounds for instances with up to 100 nodes. 4.2.2. The star-star hub location problem In the star-star Hub Location Problem (SHLP), hubs are directly connected to a central node and each user (terminal node) is

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directly connected to a hub node. The set of potential locations for the hubs is the set of terminal nodes, i.e. VF = VU and V = VU [ {0}, where 0 denotes the central node. The SHLP considers hub set-up costs, assignment costs of terminals to hubs, and costs for routing flows from the selected hubs to the central node. (No cost is incurred for routing flows through arcs not incident with the central node.) The objective of the SHLP is to minimize the sum of location costs, terminal assignment costs, and routing costs. Labbé and Yaman (2008) propose two small-size formulations that exploit the special structure of the problem. Both formulations use binary allocation z variables that satisfy constraints (24) and (25), and differ in the characterization of feasible flows between pairs of terminals. The first formulation uses additional binary path-based variables uikj that take value 1 if and only if the flow between terminals i, j 2 VU, i – j is routed via hub k 2 VF and the central node. The authors note that the (non-linear) relation between the z and the u variables is

ukij ¼ zik ð1  zjk Þ þ zjk ð1  zik Þ ¼ jzik  zjk j; which can be easily linearized by imposing constraints

ukij P zik  zjk

i; j; k 2 V U ; i – j;

ð41Þ

ukij

i; j; k 2 V U ; i – j:

ð42Þ

P zjk  zik

In particular, since the objective is to minimize

X

fk zkk þ

k2V U

X

cik zik þ

i;k2V U ;i–k

X

tk0

k2V U

X

ðW ij þ W ji Þukij ;

i; j2V U ;i–j

any optimal solution in the domain (24), (25), (41), (42) with binary z variables, will satisfy

ukij ¼ maxfzik  zjk ; zjk  zik g ¼ jzik  zjk j: Note that it is not necessary to impose integrality on the u variables as it is implied by the integrality of the assignment vector z. The second formulation uses a set of variables denoted by w, instead of the u variables. For each node k 2 VU, wk represents the total flow routed between hub k and the central node. Labbé and Yaman (2008) prove that, since the objective is the minimization of

X

fk zkk þ

k2V U

X i;k2V U ;i – k

cik zik þ

X

t k0 wk ;

k2V U

an optimal solution can be obtained by imposing constraints (24) and (25) plus the set of constraints

wj P

X

ðW ij þ W ji Þðzik  zjk Þ S # V U  V U n fði; iÞ j i 2 V U g: ð43Þ

ði; jÞ2S

The above formulation has a reduced number of variables, OðjV U j2 Þ, at the expense of having an exponential number of constraints. Problems related to the SHLP have also been considered by Yaman (2008, 2009). The first work studies an extension with modular capacities at the arcs of the network. Yaman (2009) studies the problem of designing a three level hub network, where the top level consists of a complete network connecting the central hubs, and the second and third layers are unions of star networks connecting the remaining hubs to central hubs and the users to hubs or central hubs, respectively. 4.2.3. Connected facility location The Connected Facility Location Problem (CFLP) is a tree-star GNDP with UU demand, in which the tree is a Steiner tree spanning all the selected facilities and, possibly, a subset of Steiner nodes. As usual, the stars correspond to the single allocation of users to open facilities. It is assumed that the set of Steiner nodes, denoted by VS, contains the set of potential facilities and that no user is a Steiner

node. That is, V = VU [ VS with VU \ VS = ; and VF # VS. The cost of a solution is the sum of the facilities and edges set-up costs of the tree-star induced by the solution. The CFLP is to find the set of facilities to open, the Steiner tree spanning all the selected facilities, and the allocation of users to facilities of minimum total cost. The CFLP has two main differences with respect to other treestar location models. One affects the topology of the solutions, since now the tree connecting the facilities may have additional nodes. The second difference is that the objective function, only takes into account set-up costs and does not consider any routing costs. This implies that, as opposed to previous models, if the set of selected facilities is given, the allocation of users of open facilities is trivial. In particular, each user must be allocated to the closest open facility, relative to the set-up costs c. Gollowitzer and Ljubic´ (2011) present and theoretically compare several existing and new formulations. All the formulations build an arborescence directed away from a root node in which users are the leaf nodes and the open facilities are connected by means of a Steiner tree. Let us denote by 0 2 VF the root node. It is easy to see that an artificial root, with set-up cost f0 = 0, may be added to VS if one does not exist, and connected with any node j 2 VS by a directed arc (0, j) with set-up cost c0j = 0. In this case, the number of arcs emanating from the root must be limited to one. The formulations compared by Gollowitzer and Ljubic´ (2011) are of the following types: cut-set formulations based on that of Gupta et al. (2001), reinforcements of flow-based formulations using common-flow variables (see, Polzin and Daneshmand, 2001) and formulations based on the Miller-Tucker-Zemlin constraints for subtour elimination. The analysis of Gollowitzer and Ljubic´ (2011) showed that the strongest formulation is the one called common flow between root and customers which we describe next. In addition to the usual binary allocation variables satisfying constraints (24), and (25), and the binary design variables y, two sets of additional variables are used. The first set consists of i flow-based variables h where hkm denotes the total flow emanating from root node 0 with destination user i 2 VU routed through consecutive nodes k and m, with k 2 VS, m 2 VS [ VU, m – k. These variables must satisfy the following sets of constraints, which are an adaptation of constraints (4) and (5)

X

i

h‘k 

X

i

hkm

m2V S [V U

‘2V S

i

i

hkm þ hmk 6 ykm

8 if k ¼ i; > <1 ¼ 1 if k ¼ 0; k 2 V; i 2 V U ; > : 0 otherwise;

ðk; mÞ 2 E;

i 2 VU:

ð44Þ

ð45Þ ij h km

 where The second set contains path-based variables h, denotes the common flow towards users i, j 2 VU, i – j which is routed through consecutive nodes k and m, with k, m 2 VS, m – k. These variables must satisfy the following sets of constraints ( 1 if k ¼ 0; X ij X ij   hkm  h‘k 6 k 2 V S ; i; j 2 V U ; i – j; ð46Þ 0 otherwise; m2V S ‘2V S n o ij 6 min hi ; hj h k; m 2 V S ; k – m; i; j 2 V C ; i – j; km km km

ð47Þ i hkm

þ

j hkm

þ

i hmk

þ

j hmk

6 ykm þ

ij h km

k; m 2 V S ; k – m; i; j 2 V C ; i – j: ð48Þ

The objective can be expressed in terms of the design variables y and the zii variables used to represent nodes where facilities are located as

min

X j2V F

fj zjj þ

X ðk;mÞ2E

ckm ykm :

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A considerable amount of literature on the CFLP presents approximation algorithms and studies the quality of the approximations (see, for instance, Jung et al., 2008; Williamson and van Zuylen, 2007; Eisenbrand et al., 2008). Swamy and Kumar (2004) present a primal–dual approximation based on the integer programming formulation of Gupta et al. (2001). Chimani et al. (2009) study the case in which the facilities are 2-interconnected, and present complexity results for this type of problems as well as cut-based formulations to solve several variants of these problems. Bardossy and Raghavan (2010) propose a dual-based local search heuristic for several problems that combine facility location and connectivity decisions, including the CFLP. Also Costa et al. (2011) study a network design model with location ingredients arising in electrical distribution systems. The problem consists of selecting edges to locate two types of edge facilities as well as locating node facilities to allow the connection of edge facilities. The objective is to minimize the sum of the facilities and edges set-up costs plus the operational costs. 4.2.4. Related problems Additional GDNPs with UU demand in which facilities are connected with a tree have been studied in the literature. The reviews by Labbé et al. (1998), Mesa and Boffey (1998), Klincewicz (1998), and Campbell et al. (2002), describe several models and applications related to the location of tree structures. Kim and Tcha (1992) presented a problem with a tree-star topological configuration for hierarchical networks. Lee et al. (1994, 1998b) consider the problem of locating a set of hubs and connecting them by means of a tree, in the context of digital data service networks. However, none of these problems considers the routing costs. Recently, Marı´n (2007) studied rapid transit network design problems that locate lines on rail transit systems. Particular cases of GNDPs with UU demand in which facilities are connected by means of a tree, are problems in which facilities are connected by means of a path. Such problems have been studied by some authors. The reader is referred to Slater (1982), Hakimi et al. (1993), Labbé et al. (1998), for a description of models and applications of this class of problems. Other related problems study the location on a network of the so-called extensive facilities, which are subgraphs with a treeshaped topology. In these problems, the central facility takes the form of a subtree of the underlying network and provides service to users located at the nodes of the network. The facility set-up cost is proportional to the total costs of the facility (that is, the sum of the edges set-up costs in the tree). Allocation costs of users to the facility are also taken into account. Hakimi et al. (1993) study the complexity of solving various versions of these problems on both tree and general networks. They show that several of the problems considered can be solved in polynomial time if the underlying network is a tree, but are NP-hard if the network is arbitrary. Kim et al. (1996) focus on the case in which the underlying network is a tree and develop a dynamic programming algorithm for a general model, containing several problems as particular cases. Tamir and Lowe (1996) study the complexity of the generalized p-forest problem when dealing with tree networks and provide a polynomial algorithm to solve this problem. Puerto and Tamir (2005) study the location of tree-shaped facilities with an ordered median objective function which generalizes both min-sum and min–max objectives. They give complexity results for various cases of discrete and continuous versions of the problem. Finally, other related problems include some difficult allocation or routing subproblems that might appear when the set of facilities is fixed. For instance, in the THLP when the location of the hubs and the allocation of users are given, the remaining routing subproblem can be easily transformed into an Optimum Communication

Spanning Tree Problem (OCSTP) (Hu, 1974; Wu, 2004; Contreras et al., 2010b). Also, the Minimum Sum Violation Tree Problem (MSVTP) (Chen et al., 2008) is an interesting problem which involves only routing decisions in which the objective is to minimize the sum of the violations of the delivery due dates in transportation systems. 4.3. Facilities connected with a cycle In this section, we review GNDPs with UU demand in which facilities are connected by a cycle. Like in the previous subsection we assume single allocation. Therefore, the subgraph induced by the assignment of users to facilities consists of a series of stars and the topology induced by feasible solutions is a cycle-star. As in the case of tree-star topologies, even if the location of the facilities is known, the routing decisions are non-trivial, and the resulting subproblems are, in general, difficult. Several problems have been studied in the literature that combine decisions on the location of facilities, which must be connected with a cycle, and the allocation of users to selected facilities (see, for instance, Laporte and Rodrı´guez-Martı´n, 2007). As in the case of tree-star location problems, potential applications of models where facilities are connected with a cycle arise when the edges set-up costs are very high. In the design or reliable networks, however, cycle topologies may be preferred to tree topologies. If an edge connecting two facilities fails for some reason, a cycle topology guarantees connectivity of the remaining subnetwork, and allows the commodities to be routed through alternative paths. Specific applications of these problems arise mainly in the design of telecommunication networks and in rapid transit systems planning. In the former case, models usually connect terminal nodes to concentrators (or facilities) by point-topoint arcs, resulting in a star structure, and locate a set of facilities which are interconnected by a ring (or cycle) (see Xu et al., 1999 for an example in digital data service design). In the latter case, models usually consist of selecting a set of facilities, which are served by a single-vehicle route and assigning users to their closest facility. The reader is referred to Labbé et al. (1998) for a review of applications considering the location of extensive facilities connected by a cycle. 4.3.1. The ring-star problem and the median cycle problem The Ring-Star Problem (RSP) and the Median Cycle Problem (MCP) are two closely related cycle-star GNDPs with UU demand in which a set of facilities must be located on a network and connected by a cycle. The set of potential locations for the facilities is the set of terminal nodes, i.e. VF = VU [ {0}, where 0 represents a root node that must be contained in the cycle. A solution to both RSP and MCP is a simple cycle passing through a subset of VF including the root node and at least two other nodes. Each user must be assigned to the closest facility and therefore, the assignment decisions induce a series of stars. Similarly to hub location problems, it is assumed that if a facility is located at a user node i 2 VU \ CF, the user is self-assigned. There is a set-up cost ckm for the edge (k, m) 2 E connecting two facilities and an assignment cost tij associated with the allocation of user i 2 VU to facility j 2 VF. The RSP is to determine a solution for which the sum of ring set-up costs and assignment costs is minimized. The MCP consists of locating a cycle of minimum set-up cost such that the assignment costs do not exceed a given budget constraint. These problems were introduced by Labbé et al. (2004, 2005a). The RSP and the MCP do not consider any routing costs. Therefore, they can be modeled using only the network design variables ykm and the location-assignment variables zik. The allocation of users to facilities can be modeled with constraints (24). We use

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constraints to ensure that the degree of a node contained in the cycle is 2,

X

ykm ¼ 2zii

i 2 VF:

ð49Þ

ðk;mÞ2dðiÞ

Finally, constraints

X

ykm P 2

X

zij

S  V F ; i 2 S; 0 R S;

ð51Þ

j2S

ðk;mÞ2dðSÞ

are connectivity constraints that state that S must be connected to its complement by at least two edges of the cycle whenever S contains at least one facility. The objective of the RSP is expressed in terms of the design variables ykm and the assignment variables zik as

min

X

ckm ykm þ

XX

t ij zij :

i2V U j2V F

ðk;mÞ2E

In the case of the MCP, the objective function is expressed only in terms of the design variables as

min

X

ckm ykm ;

ðk;mÞ2E

and incorporates the assignment costs of users to facilities as a budget constraint

XX

t ij zij 6 B:

i2V U j2V F

In Labbé et al. (2004), Labbé et al. (2005a) polyhedral studies are presented for the the RSP and the MCP, respectively. The proposed formulations are strengthened with several families of valid inequalities and branch-and-cut algorithms are developed. RSP instances with up to 300 nodes and MCP instances with up to 150 nodes are solved to optimality. Some heuristic algorithms have been developed for these problems by Renaud et al. (2004) and Moreno-Pérez et al. (2003). Several problems related to the RSP and the CMP have also been studied in the literature. Current and Schilling (1994) and Gendreau et al. (1997) study covering variants of the RSP in which all users must be within a prespecified distance from the cycle. Baldacci et al. (2007) introduce the capacitated m-ring star problem, which is to locate a set of m cycles that pass through a central node, and to assign each user to a facility. In this problem there is an upper bound on the number of users visited and assigned to a ring. The authors propose two formulations and several families of valid inequalities, and develop a branch-and-cut algorithm able to solve instances with up to 137 nodes to optimality. Lee et al. (1998a) propose the Steiner ring star problem, where the ring only contains Steiner nodes selected from a given set. They propose a branchand-cut method that is able to solve instances with up to 90 Steiner nodes and 50 user nodes. Xu et al. (1999) develop a tabu search procedure for the same problem and test it on instances with up to 300 Steiner nodes and 300 user nodes. Current and Schilling (1994) introduce the median tour problem, in which a cycle with p nodes has to be located. It is a bicriterion problem, which focuses on the minimization of the cycle length, and the minimization of the total travel distances of users to their closest facilities. Liefooghe et al. (2010) present a bi-objective ring star problem, in which the setup cost of the cycle and the cost for assigning users to facilities are in different objectives. For additional models and applications related to the location of cycle structures on a network, the interested reader is referred to the surveys of Labbé et al. (1998) and Laporte and Rodrı´guez-Martı´n (2007). 4.3.2. The Cycle Hub Location Problem The Cycle Hub Location Problem (CHLP) is a GNDP with UU demand, in which p hubs must be located on a network and

695

connected by a cycle. Each user has to be allocated to exactly one hub and if a hub is located at a user node i 2 VU \ CF, the user is self-assigned. Two types of arcs exist that can be used to route commodities through the network: (i) arcs connecting users to facilities, and (ii) arcs connecting the hub nodes. As in other hub location problems, a discount factor a is applied to the routing costs of the arcs connecting two hubs, and factors v and d are applied to the collection and distribution routing costs, respectively. The aim of the CHLP is to minimize the total routing cost for sending the commodities through the network. The main difference between the CHLP and the RSP and MCP is that the CHLP considers routing costs, making it more difficult to formulate and to solve. In terms of modeling aspects, the CHLP also shares several similarities with the tree of hubs location problem. In particular, there is a cardinality constraint on the number of hubs to locate that, with the usual location-allocation variables z, is modeled with constraint (33). The allocation of users to open hubs can also be modeled with constraints (24), and (25). The routing decisions of the CHLP are affected by both the allocation and the network design decisions. Therefore, formulations for this problem have to keep track of the paths connecting the end-nodes of the commodities. To do this, we may use path-based variables xijkm to model feasible paths between end-nodes of commodities passing through consecutive hub nodes. Variables x are related to the design and allocation variables with constraints (2) and (34). i Alternatively, we can use flow-based variables hkm to model the flow between commodities end-nodes by using constraints (31), together with constraints (37)–(39). The condition that the hubs are connected by means of a cycle must be guaranteed. Therefore, in every feasible solution to the CHLP: (i) there exist p edges connecting pairs of hub nodes; (ii) every hub node is connected with exactly two other hub nodes; and (iii) the graph induced by the hubs does not contain cycles smaller than p. These conditions can be formulated with constraints (33) and (49) together with

X

ykm P 2ðzii þ zjj  1Þ S  V F ;

i 2 S; j R S; 2 6 jSj 6 p:

ðk;mÞ2dðSÞ

ð52Þ Observe that the set of constraints (33), (49) and (52), depend only on the ykm and zii variables and, therefore, can be used in both path-based and flow-based formulations. If the graph induced by the flows between pairs of users is connected, constraints (33) and (52) are enough to guarantee that hubs are connected by means of a cycle. In the case of the path-based formulation the total routing costs for sending the flow though the network is modeled by expression (36), whereas in the flow-based formulation it is modeled by (40). 5. Conclusions We have presented and discussed General Network design Problems (GNDPs), which combine design decisions to locate facilities and to select links on an underlying network, with operational allocation and routing decisions to satisfy the users demands. GNDPs define a large class of difficult problems which have been presented under a unifying framework. They can be classified in two main categories, according to the type of demand required by the users. In user-facility problems users demand must be satisfied from or at the facilities, whereas in user-user problems the service demand relates pairs of users between them. GNDPs include as particular cases well-known network optimization problems such as facility location problems, location-routing problems, extensive facility location problems, and hub location problems, among others. All these problems share most

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of their essential aspects. However, ‘‘small’’ differences among them may have have important effects on the type of formulations that can be used and, thus, on the actual difficulty to solve them. We have focused mostly on modeling aspects and on alternative formulations of the problems, trying to highlight their common aspects and to point out the main differences. There are, indeed, families of problems that have not been reviewed, which can be seen under the same unifying framework. Some of them have been mentioned as related to the ones we have reviewed. Among them we should recall all location-routing problems in which routes define circuits (Nagy and Salhi, 2007), as well as other problems where no assumption is made on the topological structure for the connection of the facilities like, for instance, the design of incomplete hub networks (O’Kelly and Miller, 1994; Alumur et al., 2009) and hub arc location problems (Campbell et al., 2005a; Campbell et al., 2005b). We believe there are also interesting extensions of problems with user-facility demand in which connections among facilities could also exist. We have reviewed deterministic, single period, static models. Active research is also taking place on extensions that include some of the above elements. These include multiperiod or stochastic hub location problems like, for instance, Contreras et al. (2011b), Contreras et al. (2011c) to mention just a few. GNDPs are challenging, not only for their inherent theoretical interest but also for their wide range of applications and for the difficulty in solving actual instances of realistic sizes. Therefore, in our opinion, this is a very promising area for future research, not only from a modeling point of view but also in terms of devising efficient solution methods for these classes of problems. Acknowledgements The authors are grateful to Gilbert Laporte for his comments and suggestions, which have greatly contributed to improve the presentation of the paper. Thanks are also due to an anonymous referee. This research has been partially supported by Grant MTM2009-14039-C06-05 of the Spanish Ministry of Education and Science and by ERDF funds. The research of the first author has partly funded by the Canadian Natural Sciences and Engineering Research Council under Grants 227837-09 and 39682-10. The research of the second author has been partially supported through Grant PR2008-0116 of the Spanish Ministry of Science and Education. This support is gratefully acknowledged. References Ahuja, R.K., Magnanti, T.L., Orlin, J.B., 1993. Network Flows: Theory, Algorithms and Applications. Prentice-Hall, Inc., Englewood Cliffs, New Jersey. Alumur, S., Kara, B.Y., 2008. Network hub location problems: State of the art. European Journal of Operational Research 190, 1–21. Alumur, S., Kara, B.Y., Karasan, O.E., 2009. The design of single allocation incomplete hub networks. Transportation Research Part B 43, 936–951. Balakrishnan, A., Magnanti, T.L., Mirchandani, P., 1997. Network design. In: Dell’Amico, M., Maffioli, F., Martello, S. (Eds.), Annotated Bibliographies in Combinatorial Optimization. John Wiley and Sons, New York, pp. 311–334. Balakrishnan, A., Magnanti, T.L., Wong, R.T., 1989. A dual ascent procedure for largescale uncapacitated network design. Operations Research 37, 716–740. Baldacci, R., Dell’Amico, M., Salazar-González, J.J., 2007. The capacitated m-ring-star problem. Operations Research 55, 1147–1162. Balinski, M.L., 1961. Fixed-cost transportation problems. Naval Research Logistics Quarterly 8, 41–54. Bardossy, M.G., Raghavan, S., 2010. Dual-based local search for the connected facility location and related problems. INFORMS Journal on Computing 22, 584– 602. Belenguer, J.M., Benavent, E., Prins, C., Prodhon, C., Wolfler-Calvo, R., 2011. A Branch-and-cut method for the capacitated location-routing problem. Computers and Operations Research 38, 931–941. Boland, N., Krishnamoorthy, M., Ernst, A.T., Ebery, J., 2004. Preprocessing and cutting for multiple allocation hub location problems. European Journal of Operational Research 155, 638–653. Boru˚vka, O., 1926. O jistém problému minimálním (About a certain minimal problem) (in Czech). Práce mor. prˇr´odoveˇd. spol. v Brneˇ III 3, 37–58.

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