Integer programming formulations of discrete hub ...

European Journal of Operational Research 72 (1994) 387-405 North-Holland

387

Theory and Methodology

Integer programming formulations of discrete hub location problems James F. Campbell Department of Management Science and Information Systems. School of Business Administration, Unit~ersity of Missouri-St. Louis, 8001 Natural Bridge Road, St. Louis, MO 63121, USA Received August 1991; revised February 1992

Abstract: Hubs are facilities that serve as transshipment and switching points for transportation and telecommunication systems with many origins and destinations. This paper presents integer programming formulations for four types of discrete hub location problems: the p-hub median problem, the uncapacitated hub location problem, p-hub center problems and hub covering problems. Hub median and uncapacitated hub location problems have received limited attention from researchers. This paper introduces discrete hub center and hub covering problems. Basic formulations and formulations with flow thresholds for spokes are presented.

Keywords: Location; Integer programming; Hub location

I. Introduction Recent surveys of facility location research testify to the breadth of problems considered (Tansel et al., 1983; Hansen et al., 1987; Brandeau and Chiu, 1989; Mirchandani and Francis, 1990). One area that has so far received limited attention is hub location problems. Discrete hub location problems involve locating a set of fully interconnected facilities called hubs, which serve as transshipment and switching points for traffic between specified origins and destinations. A non-negative flow is associated with every origin-destination ( o - d ) pair and an attribute such as distance, time or cost is associated with movement. The term cost is used throughout this paper to represent the attribute of interest. In hub systems, origin-to-destination movements are generally via one or two hubs. As long as the cost of movement is a non-decreasing function of distance, no origin-to-destination movements are via more than two hubs, since hubs are fully interconnected. Hub systems concentrate flows on the spoke links connecting hubs to origins/destinations, and more so on the inter-hub links. Usually, the cost rate for inter-hub movement

to: Dr. James F. Campbell, Department of Management Science and Information Systems. School of Business Administration, University of Missouri-St. Lou,S, 8001 Natural Bridge Road, St. Louis, MO 63121, USA Correspondence

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J.F. Campbell / IP formulations of discrete hub location problems

is less than the cost rate for movement between a hub and an origin/destination due to economies of scale. Hub location problems have important applications in transportation and telecommunication systems. The phenomenal growth of the air express small package delivery business has been linked to the hub-and-spoke networks employed (Chan and Ponder, 1979). Large less-than-truckload motor carriers also employ hub-and-spoke networks, with break-bulk terminals serving as hubs and end-of-line terminals as origins and destinations. Hub location in telecommunication systems is also important (Boorstyn and Frank, 1977; Helme and Magnanti, 1989), for example, in designing backbone networks and locating concentrators. The terminology in this paper is geared toward transportation networks, although the basic formulations may apply to telecommunication networks as well. This paper presents mathematical programming formulations for several discrete hub location problems, which are analogous to four fundamental types of discrete facility location problems: the p-median problem, the uncapacitated facility location problem, the p-center problem and covering problems. Previous discrete hub location research has addressed certain aspects of the p-hub median problem (Campbell, 1990b, 1991; O'Kelly, 1986b, 1987; Klincewicz, 1989, 1991; Aykin, 1990), the uncapacitated hub location problem (O'Kelly, 1992) and hub location with a squared distance objective (O'Kelly, 1990). This paper introduces discrete hub center and hub covering problems. In this paper both the origin/destinations and the potential hub locations are specified as discrete sets of points. Planar hub location problems, where hubs may be located anywhere on a plane have been addressed by O'Kelly (1986a), O'Kelly and Miller (1991), Aykin (1988, 1992), Aykin and Brown (1992) and Campbell (1990a). Hub location problems where the origins and destinations are continuously distributed over a rectangular region are discussed by Campbell (1991). Hub location problems may be classified by the way in which demand points are assigned, or allocated, to hubs. One possibility is single allocation, in which each demand point is allocated to a single hub, i.e. each demand point can send and receive via only a single hub (e.g., O'Kelly, 1986b, 1987, 1992; Klincewicz, 1989, 1991; Aykin, 1990, 1991a). A second possibility is multiple allocation, in which a demand point may send and receive via more than one hub (e.g., Hall, 1989; O'Kelly and Lao, 1991; Campbell, 1990b, 1991). Formulations presented in this paper involve multiple allocation and single allocation. The single and multiple allocation schemes can be augmented by allowing non-stop service between certain origins and destinations, i.e. a demand point may send and receive directly from another demand point without movement via a hub (e.g., Aykin 1991a, b). Non-stop service is not considered in this paper, so all origin-destination movements are via at least one hub. This paper considers discrete hub location problems in which the following five items are given: (1) n demand locations (origin/destinations), (2) r potential hub locations, (3) the flow for the n 2 demand location (o-d) pairs, (4) the per unit cost between all location pairs and (5) the hub-to-hub discount factor a. Hub location problems can be viewed as embedded in an undirected network N = (V, A), where the set of nodes, or vertices, of the network V= {vl, v 2. . . . . %} correspond to origins/destinations and potential hub locations. Thus, hubs are restricted to be located at a subset of the vertices. Associated with link (a, b ) ~ A , which connects vertices va and Vb, is a non-negative weight d(a, b)=-d(b, a) representing its length. This may correspond to travel distance, time, cost or some other attribute. Define Cab to be the length of the shortest path between nodes a and b. The cost for movement on the path from origin i to destination j via hubs at nodes k and m, in that order, is Cik + OtCkm + Cmj , where a is the discount factor for the inter-hub transportation. If k = m, then there is no inter-hub transportation. Associated with each o - d pair (i, j) is a non-negative weight representing the flow from i to j. The following four sections of this paper consider four classes of hub location problems. Each section presents mathematical programming formulations for the associated hub location problems. Section 2 on

J.F. Campbell / IP formulations of discrete hub location problems

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the p - h u b median problem includes the definitions of the decision variables and parameters that are used throughout this paper. The final section of the p a p e r is a brief summary.

2. The p-hub median problem The p-median problem has been a fundamental problem in locational research since its inception (Hakimi, 1964, 1965). Campbell (1991) defined a p-hub median analogous to a p-median (Hakimi, 1965). This definition is included in the Appendix. The p-hub median problem has straightforward applications to transportation and telecommunication networks in which the objective is to minimize the total cost of movement. The recent development of hub-based air passenger and freight networks is just one example of the importance of this problem. Define the following variables and p a r a m e t e r s to be used throughout this paper: Xijkm Fraction of flow from location (origin) i to location (destination) j that is routed via hubs at locations k and rn in that order. Yk = 1 if location k is a hub and 0 otherwise. Zik = 1 if location i is allocated to the hub at location k and 0 otherwise. Wq = Flow from location i to location j. cij = Standard cost per unit from location i to j. =

Cijkm ~ Cik "~- Crnj "~ OlCkm.

Decision variables Xijkm and Zik determine the allocation. Decision variable Yk indicates hub locations. Usually cii is proportional to the distance between i and j. C~/~,n is the cost per unit from origin i to destination j via hubs k and m, in that order. In the remainder of this p a p e r i and j are used to index origins and destinations respectively, and k and m are used to index potential hub locations. Unless otherwise specified, the summations for i and j are from one to n and the summations for k and m are from 1 to r. Figure 1 illustrates 1-, 2- and 3-hub medians for a simple example in which hubs may be located at any of the nodes of the network and the per unit costs for movement between nodes are indicated on the links. The 1-hub median is node 2. T h e 2-hub median is node pair (1, 3) for 0 < a < 0.8 and node pairs (1, 2) or (2, 3) for 0.8 < a < 1.0. The 3-hub median also depends on the value of a. For a = 0, node triplets (1, 2, 3) and (1, 3, 4) are 3-hub medians. For a = 0.5, node triplet (1, 3, 4) is the 3-hub median. For a = 1.0, any three of the four hubs constitutes a 3-hub median.

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Campbell/ IPformulationsof discretehub locationproblems

2.1. Basic formulations Campbell (199l) formulated the p-hub median problem as follows: p-HM Minimize

E E E E WijXijkmCijkm i j k m

subject to

EY~ =p,

(1)

k

0 < Yk < 1 and integer for all k,

(2)

0 ~_~Xijkm_~ 1 for all i,j,k,m,

(3)

E ESijkm

for all i,j,

(4)

Xijkm ~-~Yk for all i,j,k,m, Sijkm ~-~Ym for all i,j,k,m.

(5)

k

=

1

rn

(6)

The objective function in p - H M sums the transportation cost over all o - d pairs. Constraint (1) establishes exactly p hubs. Constraint (2) restricts Yk to be zero or one. Constraint (3) limits the range of X,vkm. Constraint (4) assures that the flow for every o - d pair is routed via some hub pair. Constraints (5) and (6) assure that flows are routed via locations that are hubs. Constraints (5) and (6) may alternately be expressed

Y'~ Y', EXijkm < n(n - p + 1)Yk for all k i

j

2 2 ~Yijkm