Embedding economies of scale concepts for hub network design

Journal of Transport Geography 9 (2001) 255±265

www.elsevier.com/locate/jtrangeo

Embedding economies of scale concepts for hub network design Mark W. Horner *, Morton E. O'Kelly Department of Geography, Ohio State University, 1036 Derby Hall, Columbus, OH 43210-1361, USA

Abstract We explore the idea of endogenous hub location on a network. In contrast to much of the literature, we propose that hub networks may emerge naturally out of a set of assumptions and conditions borrowed from equilibrium trac assignment. To this end, we focus on applying a nonlinear cost function that rewards economies of scale on all network links. A model is presented and implemented in a GIS environment using both a 100-node intercity matrix and several synthesized interaction matrices. We compare solutions for di€erent assumptions about network costs, and visualize the results. We ®nd that under discounted conditions, network ¯ow is re-routed to take advantage of the cost savings for amalgamation and that several cities emerge as centers through which large amounts of ¯ow pass. Larger cities such as Los Angeles, New York and Chicago serve gateway functions. We also ®nd that smaller cities such as Oklahoma City, Pittsburgh, Indianapolis, and Knoxville serve major gateway functions because of their locational advantages. Our paper should be of interest to the planner of a surface transportation system, or those interested in nodal concepts such as gateways and transport geography. Results are discussed in light of hub and spoke networks and suggestions are made for future research. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Network analysis; Hub and spoke networks; Nonlinear costs; Freight transportation; Gateways; GIS

1. Introduction Hub and spoke networks are found in air transportation, telecommunications, ground freight transportation and other transportation scenarios. Implementing hub and spoke networks is generally attractive to transport ®rms because of the cost savings derived from concentrating ¯ow density on network links between hub locations. Depending on the degree of discount, there is an incentive to bundle ¯ows, especially on the interhub links given known supply and demand quantities which must be shipped between all origins and destinations (Bryan and O'Kelly, 1999). Many models of hub and spoke networks make three key assumptions that over-simplify the problem. The ®rst assumption, which is related to the bundling of ¯ows on the interhub link, is the exogenous nature of the discount factor. That is, most hub and spoke models assume that the level of interhub discount is not dependent on the amount of ¯ow using the link. The second simplifying assumption is that the number of hubs to be located is determined exogenously. The third assumption is that only ¯ow on interhub links may receive a discount, which *

Corresponding author. Tel.: +1-614-292-1333. E-mail address: [email protected] (M.W. Horner).

is a somewhat arbitrary distinction because discounts could be earned along any portion of a route with sucient volume. More generally, a cost function ought to be responsive to ¯ow, and should reward the higher volume links by giving them a lower rate. With the aim of overcoming some of these limitations, our analysis proposes a partial network design problem given that some quantity (persons or goods) must be transported (minimizing costs) by a single ®rm with being absent other competitors for shipping the quantity. 1 This research is exploratory in nature, yet the results would be useful to the planner of a surface transportation carrier of less than truckload (LTL) shipments or to the planner of a passenger bus carrier (such as Greyhound). The exploratory nature of our research is highlighted by three major di€erences

1 Recent research has considered the implications of competition in the hub location. A recent paper by Marianov et al. (1999) describes a model that locates hubs so as to maximize the ¯ow capture when there are already competitors operating in the market. Their model is intended for air and cargo transport in a competitive environment where a small air carrier might seek to capture the market share from a larger service provider or where a large airline with existing hubs might want to relocate some of their hubs with respect to competitor location and changing demand. Essentially, their model locates hubs by targeting routes where the existing provider is poorly serving the market.

0966-6923/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 6 - 6 9 2 3 ( 0 1 ) 0 0 0 1 9 - 9

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between our analysis and most other works on hub and spoke networks found in the literature. First, a hub and spoke network structure is not assumed a priori, rather the location of hubs is endogenously determined by the minimization of transport costs, given a discount function on network links. Second, the level of link discount is modeled as a nonlinear function of link ¯ow volume, which makes the link discount endogenous by rewarding the shipper for greater volumes shipped. Third, ¯owbased discounts are allowed on all network links rather than restricting discounts to only the interhub links. Allowing discounts on all network links is an improvement over the ¯awed assumption that only interhub links may receive volume-based discounts. Incorporating these features into the analysis allows us to take a step back from most hub and spoke research by adopting an evolutionary approach to the network design and location problem. We use the term evolutionary to suggest that a certain network design will emerge based on assumptions that are implicit in the model. Thus, the evolution of the network design should be a function of the degree of ¯ow-based discount, the geography of the node structure, and supply and demand levels associated with the origins and destination locations. Speci®cally, our focus will be on exploring the e€ects of the ¯ow-based discount on network structure, including the detection of hubs. We hypothesize that certain network nodes (cities in our case) will appear as major centers of activity. Visual investigation of trac patterns should provide evidence of a given city's emergence as an activity center. Therefore, it should be pointed that any ``hub locations'' de®ned by this analysis are not directly comparable to the types of hubs selected in a p-hub-modeling construct as one might ®nd in O'Kelly (1987). The methods of ®nding each solution are quite di€erent. Our discussion should add to the general knowledge about activity centers and gateways, as the term gateway has a broad interpretation. Functionally, it captures the basic meaning of the airline hub in the sense that hubs are focal points serving transshipment functions, but the term gateway also may apply to land or sea transport modes. In all modal cases gateways are usually strategic, highly accessible locations that play a key role in transport functions (Taa€e et al., 1996). It has been problematic to identify gateways analytically, and in the past most gateways were awarded their status based on descriptive evidence rather than any systematic measurement (Taa€e et al., 1996). This paper should add to information about gateway identi®cation since our methods are intended to establish a hierarchy among nodes with the gateway or hub at the top of the classi®cation scheme. We should also point out that because a hub and spoke network structure is not assumed a priori, this paper shares certain concepts with other areas of net-

work analysis research. Our work represents a hybrid approach to hub and spoke network design that borrows from other literatures, namely the urban transportation and operations research literature (She, 1985; O'Kelly, 1987, 1998). We can imagine a situation where the central planner of a bus system might structure routes to collect as many passengers as possible, even if that means transporting a majority of the persons slightly o€ their most direct route. The collectivization of passengers would be done to o€set the costs of sending the bus. Similarly, LTL freight transportation ®rms have an incentive to bundle loads together in order to realize truckload savings. The paper is organized as follows. Section 2 touches on some relevant hub and spoke and network analysis literature. In Section 3, the nonlinear cost function is discussed relative to the hub and spoke literature and urban transportation literature. In Section 4, the analysis is conducted. The analysis portion of the paper describes the data on the network structure along with the model formulation, followed by the application of the nonlinear cost function to several numerical examples. In Section 5, we provide a discussion of the results and draw conclusions based on our ®ndings, which include suggestions for future research. 2. Background O'Kelly (1987) uses a quadratic integer program that minimizes network travel costs to site hubs. Interhub discounts are modeled with an exogenous discount factor. The resulting network is a single-allocation (each city connected to only one hub), fully interconnected (all hubs connected to one another) design. Since this initial work, other research has considered multiple assignment formulations, ®xed hub location, endogenous hub location, sensitivity analysis, and other extensions to the basic model (Bryan and O'Kelly, 1999). The following text relates some of these extensions to the aims and assumptions of our analysis. The ®rst de®ning feature of our analysis is that it does not assume an explicit p-hub network, and is motivated by an e€ort to make hub location endogenous. Other research, such as a paper by Jaillet et al. (1996), adopts a similar approach to hub network design by not assuming a hub network a priori. Their approach to achieving agglomerations at nodes and economies of scale is accomplished by routing air passenger ¯ows given known airplane size and capacity on routes linking cities. Restrictions are placed on the number of stops a plane could make on a route since most passengers would be unwilling to make an excessive number of stops. At the conclusion of Jaillet et al.'s analysis, each city is considered a hub candidate, where its suitability for hub designation is assessed based on model outputs. Such

M.W. Horner, M.E. O'Kelly / Journal of Transport Geography 9 (2001) 255±265

outputs are the city's number of departing aircraft, the city's number of originating passengers, the number of persons using the city as a connecting stop, and other trac-related measures. Bryan (1998) also demonstrates a way to endogenously determine hub locations. Bryan's model selects hub locations from a predetermined candidate list. Any number of hubs could be opened as long as the links between the opened hubs receive some predetermined minimum level of ¯ow (a minimum threshold constraint must be satis®ed). In short, the principal conceptual di€erence between Bryan's (1998) method of endogenously determining hub location and Jaillet et al.'s (1996) approach is that hub location is a direct output from Bryan's (1998) model. The second feature of our analysis is its explicit consideration of the volume discount factor. As mentioned brie¯y in the preceding section, models seeking to locate a predetermined number of hubs usually rely on some constant factor in their formulations to discount interhub ¯ows (Aykin and Brown, 1992). Other recent research has attempted to implement more realistic discounts on interhub links. Rather than use an exogenously determined interhub discount factor within a hub location model, O'Kelly and Bryan (1998) use a piecewise linear function to determine the discount on the interhub link based on the amount of ¯ow carried. The function appearing in O'Kelly and Bryan (1998) is designed to approximate a nonlinear function that rewards larger interhub ¯ows with larger discounts. The presence of discounts on all network links is the third feature of our analysis. The work of Bryan (1998) appears to be the ®rst research to allow discounts on all link ¯ows. She points out that potential exists for all network links to amalgamate ¯ow, meaning all links should be rewarded for doing so. Using the piecewise linear function found in O'Kelly and Bryan (1998), Bryan (1998) extends the method of discounting interhub link ¯ows to spoke link ¯ows. 2 In this fashion, all network links receive some level of ¯ow-based discount. Other research is relevant to the general scope and purpose of this paper, particularly research that has looked at the hub location problem in an e€ort to impart more realism to the resulting network structure. For example, Kuby and Gray (1993) construct a model of the western half of the Federal Express network to optimize aircraft routes by minimizing total air transportation costs for the system, subject to the constraint that all packages get to the hub within the allotted time. Kuby and Gray's (1993) formulation allows for multiple stops along routes and uses di€erent types of aircraft to construct routes, which is in contrast to more traditional, stylized hub representations seen in the literature.

Their model allows for smaller aircraft ¯ying ``feeder'' routes, in addition to standard sized aircraft carrying larger loads. Each aircraft type has its own cost factor. As a second example, Jourquin and Beuthe (1996) design a shipping model to predict the choice of modes and routes that would result from cost minimization. Their analysis is conducted in a geographic information system (GIS) on a hypothetical network where links are representative of various modes with di€erent costs. Total freight costs are linear functions of distance and a commodity is shipped among one of the three modes (rail, truck, or water). Aside from exploring the minimum cost solution given modal transport costs, Jourquin and Beuthe also test the sensitivity of the modal split to changes in transport costs. This section has related some of the features of our analysis to previously reported literature. The following section introduces a nonlinear cost function that will be used later in the analysis. 3. The nonlinear cost function In urban transportation planning, planners perform trip generation and distribution to determine the travel, or ¯ow between places. Following these two steps, and perhaps after considering the mode of travel, ¯ow is assigned to a road network using user equilibrium or some other assignment algorithm. 3 Closely tied to the assignment solution of urban transportation network analysis are link performance functions (LPFs) that penalize links for high volume-to-capacity ratios (She, 1985). Hence, in urban transportation, congestion is an adverse e€ect while in hub and spoke networks, congestion is considered to be advantageous up until some reasonable notion of link capacity, particularly on the interhub links. In analytical terms, O'Kelly (1998) points out that the cost function in urban transportation networks is an increasing function of link ¯ow, as opposed to hub and spoke networks, where the cost function should be a decreasing function of link ¯ow. Interestingly, in terms of cost functions, our exploratory hub and spoke problem (concave costs) and the urban transportation problem (convex costs) are mirror opposites of one another. One of the most common LPFs used in urban transportation is the Bureau of Public Roads (BPR) function (She, 1985). The BPR function is as follows:  b ! xl Pl ˆ 1 ‡ h : …1† Kl

3

2

Discounts for spoke ¯ows are less than interhub link ¯ows because the functions are de®ned di€erently.

257

User equilibrium assignments adhere to Wardrop's ®rst principle, that is, the travel times on used routes connecting an origin/destination pair of interest must be equal.

258

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Eq. (1) is an increasing function stating that the penalty, P, on a given link, l, is dependent on link ¯ow, xl , divided by link capacity, Kl , raised to some power, b, and multiplied by a scalar, h. In practice, BPR parameter choices are h ˆ 0:15 and b ˆ 4 (She, 1985). Using these parameters, one can see that the function in Eq. (1) increases very rapidly, adding large link penalties when xl > Kl . Simply changing the sign associated with the scalar h, and restricting it to enter the LPF as a positive number yields the following decreasing function of link ¯ow:  b ! xl Dl ˆ 1 h : …2† Kl Link discount, Dl , is given by Eq. (2). Just as in Eq. (1), both parameters …h; b†, enter Eq. (2) as positive numbers. A similar link discount is demonstrated by O'Kelly and Bryan (1998), but they actually use a piecewise linear approximation of the function in (2) to discount interhub ¯ows. In order to understand how discounts are applied to our analysis, let Wij be a matrix of reported interactions, or ¯ows between the ith origin and jth destination. The elements of Wij are scaled by the sum of the total interactions in the matrix. Wij is the scaled interaction matrix de®ned as Wij Wij ˆ P P i

j

Wij

8…i; j†:

Eq. (5) is simply the discount function of Eq. (4) multiplied by link ¯ow, xl . Although we allow the model to apply increasingly discounted travel costs as a function of ¯ow, care must be taken to ensure that the cost per mile function (5) is monotonically increasing on the interval …0; 1Š. To ensure this, we di€erentiate Eq. (5) with respect to xl and evaluate the derivative for xl ! 1 (or a reasonable estimate of the maximum possible link ¯ow). This yields the valid combinations of b and h. If xl ! 1, dF =dx P 0 implies …b ‡ 1† 6 1=h:

…6†

For empirically observed maximum link ¯ow volumes, xmax , we reevaluate the derivative of Eq. (5) for l xl ! xmax , then l dF P0 dx Xl ˆX max l

implies h…b ‡ 1†xmax b P 0; l

1

…b ‡

b † 1†…xmax l

6 1=h:

…7† …8†

The combinations of parameters used in our numerical calculations satisfy the condition (8). Given that xmax < 1, the allowable magnitude of b in (8) will generally be larger than the restriction imposed in (6).

…3†

PP Obviously then, i j Wij ˆ 1. The diagonals of Wij are set to zero for intercity analyses. The scaled ¯ows, or ¯ow proportions contained in Wij , are assigned to network links using an assignment algorithm, and are subsequently quanti®ed by xl . Since xl depends on Wij the largest possible value of xl is 1. Only in the case where one link in the network is carrying all of the ¯ow would xl ˆ 1. While we cannot predetermine the maximum ¯ow, it is highly improbable that any link would approach the limit of xl ! 1. Eq. (3) is now rewritten to re¯ect the key di€erence between our analysis and conventional urban transportation analysis. The new discount function is given by
Dl ˆ 1 hxbl : …4† The capacity term, Kl , in Eq. (2) is set to 1 for all links in the network, (Kl ˆ 1 8l), which assumes an uncapacitated network. Removing the capacity term ensures that only xl , h, and b determine link discounts. Based on Eq. (4), the discount at xl ˆ 1 is given by Dl ˆ 1 h, while decay of the function for all other values of xl is governed by b. The cost per mile function corresponding to Eq. (4) is
…5† F …x† ˆ 1 hxbl xl :

4. Analysis The data used in the analysis consist of air passenger ¯ights for the largest 100 US cities as reported by the 1970 Civil Aeronautics Board (CAB). 4 Even though there is an obvious mismatch between using the CAB aviation data in the context of surface transportation, the CAB data are representative of realistic interurban demand for one period in time. Due to the historical nature of the data, it should be emphasized that the analytical results should only be interpreted as illustrative of the modeling approach. To facilitate the modeling and visualization using the CAB data, our analysis is implemented in a GIS environment (TransCAD 3.2 is used). The CAB database is converted to a GIS database consisting of the point locations of the 100 largest US cities. A scaled matrix, Wij , containing passenger ¯ows between the cities is also imported into the GIS. A preexisting GIS layer containing a representation of the United States' Interstate system is used as the network over which surface ¯ows are assigned. Interstate network link lengths (noted as fl ) are used as the free-¯ow link impedances. Under low discounts, we expect the ¯ow to traverse its direct route. 4

A. Stewart Fotheringham provided the 100-node city data.

M.W. Horner, M.E. O'Kelly / Journal of Transport Geography 9 (2001) 255±265

At high discounts, we hypothesize that the direct connections will be abandoned in the interest of collectivization along indirect routes to achieve ¯ow-based discounts. To operationalize the model, a system optimal (SO) network assignment model is used. The solution to the SO assignment is such that no network ¯ow may be rerouted without increasing the system travel cost (SO solves for minimum system costs). The SO assignment is borrowed from urban transport analysis, but is consistent with the notion of hub and spoke cost minimization. Furthermore, the model may be formulated and solved given the specialized nonlinear ¯ow discount. The formulation of the SO assignment is similar to She (1985) except that the link costs, Cl , are based on the nonlinear discount function in Eq. (4) instead of the congestion penalty in Eq. (1). In our model, note that the scaled interaction matrix, Wij , is assigned, as opposed to the unscaled matrix assigned in She's (1985) formulation. X Min…Z† Z ˆ xl Cl …xl † …9† l

subject to X ij fk ˆ Wij

k fkij

P0

8i; j;

8i; j; k;

…10† …11†

where xl is the ¯ow on link l; Cl is the cost of the traveling link l. It is a product of the discount function, Dl (Eq. (4)) and the link length, fl . fkij is the ¯ow on path k connecting origin i and destination j; Wij is the scaled ¯ow from origin i to destination j. Eq. (9) gives the objective function of the SO assignment. The urban transportation SO assignment is similar to the formulation above in that the objective function (9) is speci®ed in terms of link ¯ows, while the constraints (10) and (11) are speci®ed in terms of path ¯ows. The variable, fkij , is used to denote the ¯ow on path k connecting origin i and destination j. Incidence relations presented by She (1985) allow conversion from the previously de®ned link ¯ows, xl , to path ¯ows, fkij . X X X ij ij fk dl;k 8l 2 L: …12† xl ˆ i

j

k

The variable, d, is an indicator variable equal to 1 if the link l is on path k connecting origin i to destination j, and is equal to zero otherwise. Similarly, path costs may be expressed in link notation. X Cl …xl †dijl;k 8k 2 K; 8i 2 I; 8j 2 J : …13† Ckij ˆ l

Returning to the SO assignment, Eq. (10) constrains the assignment such that the total ¯ow on all paths linking an origin±destination, (OD), pair is equal to the corre-

259

sponding OD ¯ow speci®ed in Wij . Non-negative path ¯ows are ensured by inequality (11). When the SO model is formulated and solved using our concave cost function as opposed to the convex cost function found in the urban transportation literature, the solution space is characterized by many local minima which make ®nding an optimal solution very dicult (Zangwill, 1968). Thus it is important to note that the solutions we present to the problem are feasible, but are not necessarily global optima. However, the nonoptimality of these solutions is not especially problematic given our emphasis on the exploratory aspects of the technique. Furthermore, the general trends we found in the solutions relative to the parameter choices and the resulting visualized ¯ow patterns con®rmed that the model performed as would be expected. 4.1. Analysis 1: CAB-based models Our ®rst analysis uses the 1970 CAB data in a series of numerical examples. In the experiment, values of theta, h, are chosen to be 0.25, 0.50, 0.75 and 0.90. These range from very low discounts (a 25% discount at x ˆ 1) to very high discounts (a 90% discount at x ˆ 1). Choices of beta, b, for each h are taken between 0.1 and 1 in increments of 0.1. Selecting such a wide range of parameter values is done to explore the nature of the solutions. The model (Eq. (9)) is solved for each of these choices and the results are reported in Table 1. Some important trends in the solutions are worth noting. First, for any given h, we ®nd that smaller values of b result in greater average miles traveled (scaled ¯ow multiplied by its assigned route length) and lower average costs of travel (discounted average miles). Second, holding b constant, greater values of h produce lower average travel costs and consequently produce more average miles of travel. Thus, the most signi®cant amalgamation of ¯ow occurs when h is greatest and b is lowest. The results show that higher discounts encourage alternate ¯ow routings other than the direct paths between locations, which also result in lower average travel costs. The exceptions to these trends are when b ˆ 0:1 and h ˆ …0:25; 0:50; 0:75†. For these three cases, the model does not ®nd a solution as amalgamated as the solution found for b ˆ 0:2 and h ˆ …0:25; 0:50; 0:75†. Average system miles traveled actually decreases for these three cases even though they represent greater discounts than the prior cases. However, average system miles traveled behaves as expected for h ˆ 0:9 when b is decreased. This occurrence is explained by the aforementioned diculty of solving the concave cost problem to optimality, and reporting what can only be guaranteed to be locally optimal solutions (Zangwill, 1968; Bazlamacci and Hindi, 1996). For higher values of b, ¯ow is not suciently rewarded for bundling, so it does not group itself as

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M.W. Horner, M.E. O'Kelly / Journal of Transport Geography 9 (2001) 255±265

Table 1 Assignment model results

h ˆ 0:25

h ˆ 0:50

h ˆ 0:75

h ˆ 0:90

a

Iter.a

b

Average miles traveled

Average cost of travel

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

1146.710 1146.755 1146.758 1146.811 1146.813 1146.882 1147.037 1147.313 1147.344 1147.052

1137.898 1134.391 1129.509 1122.567 1112.781 1098.788 1078.724 1049.904 1008.482 948.405

3 3 3 3 3 3 3 4 4 4

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

1146.811 1146.813 1146.857 1147.322 1147.449 1147.763 1148.093 1148.843 1150.048 1148.518

1128.887 1121.919 1112.081 1097.665 1077.769 1049.416 1009.054 951.154 867.826 748.736

3 3 3 4 4 4 4 4 6 4

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

1146.913 1147.144 1147.396 1147.593 1148.055 1149.674 1151.579 1155.663 1157.759 1156.343

1119.913 1109.001 1093.001 1072.514 1042.294 998.415 936.308 845.738 721.079 544.289

3 4 4 4 4 5 5 5 6 5

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

1147.070 1147.372 1147.535 1147.923 1149.617 1151.342 1155.754 1170.744 1190.700 1197.407

1114.160 1101.034 1082.885 1057.185 1019.598 966.813 888.485 773.744 615.335 407.678

3 4 4 4 6 5 5 11 11 9

Models allowed 500 iterations, convergence of 0.00001.

tightly. Irrespective of h, Table 1 shows that the average miles traveled for the solutions based on the higher values of b are very close to the simple non-discounted shortest path solution (1146.673 miles). Clearly, these low-discount solutions represent situations where ¯ow is being routed in a fashion similar to the non-discounted direct path. Three sets of ¯ow patterns are visualized in Fig. 1. These maps represent three distinct variations in spatial ¯ow patterns. Map 1 depicts the model solution using

no discount in the link performance function. In terms of trac assignments, the SO assignment `collapses' into an all-or-nothing assignment (AON) when there are no incentives to bundle ¯ow. Analogously, ¯ow between origins and destinations also takes its most direct path in urban transportation planning when congestion penalties are ignored. Map 2 (h ˆ 0:5, b ˆ 0:5) shows more bundling of ¯ows than the simple non-discounted case depicted in Map 1. Even with the modest discounts provided by these parameter settings (modest relative to the possible range of discounts allowed in our experiment), we observe that ¯ow along the east/west routes traversing the Rocky Mountain States begins to group. Several of the smaller links carrying trac in the nondiscounted case depicted in Map 1 do not carry trac in Map 2. Similarly, there are links in the eastern portion of the US that are not utilized when a discount is given, albeit a modest one. Looking at more general spatial trends, we can already discern the large ¯ows on certain network links between major cities such as Los Angeles and New York. In fact, New York's relative strength as a center is apparent in both Maps 1 and 2. Other areas show grouping of ¯ow, particularly in the Southeast Piedmont region and the Midwest, but these ¯ow streams may be more a function of actual demand as opposed to any alternative routing encouraged by the modest discounts. Map 3 (h ˆ 0:9, b ˆ 0:1) represents a solution with very large discounts and shows very strong evidence of ¯ow collectivization around some urban centers. Oklahoma City, Los Angeles, Pittsburgh, Indianapolis and Knoxville all appear as major interchanges in Map 3. In comparing these results to the idea of hub networks, the networks displayed in the maps point to a tendency of this network design method to concentrate ¯ows along a few select network links as discounts increase. As the patterns emerge, prominent ``pipelines'' form through the mid-section of the country, and in the Southeast. Given that strings of cities form backbonelike networks that are not necessarily consistent with hub networks, discerning hub locations from the results is not easy since many centers appear as likely hub locations. 5 What these ¯ow patterns more closely resemble are the major trunklines traversing the US. Trunklines are certain heavily traveled routes that historically connected major urban centers and coincide with the complete interstate highway network we use in our analysis (Taa€e et al., 1996). It is interesting that the two or three distinct east±west trunklines depicted in Map 2 merge into one major route traversing the southern portion of the country, passing through Santa Fe, Oklahoma City

5 In visualizing the prototypical hub and spoke network, we tend to think of groups of star-like formations.

M.W. Horner, M.E. O'Kelly / Journal of Transport Geography 9 (2001) 255±265

261

Fig. 1. Interstate ¯ow visualization.

and St. Louis in Map 3. There is the emergence of a strong eastern north±south trunkline (Map 2) down the seaboard, which shifts inland through the Piedmont to pick up additional ¯ow in the more highly discounted case (Map 3). The ¯ows depicted in Map 3 give some indication as to the urban centers that are serving gateway functions.

One would expect that the larger cities such as Los Angeles, New York and Chicago would serve gateway functions. Perhaps it is more surprising that smaller cities such as Oklahoma City, Pittsburgh, Indianapolis, and Knoxville serve major gateway functions as well (due to their relative locations with respect to other cities). In Map 3, we can see that each of these cities

262

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Fig. 2. E€ects of discounting on route structure.

plays a transshipment role for ¯ow originating from more than one stream. Of course, the descriptions of the model results are based on a spatial pattern from 1970. If more recent data were used, it is indeed likely that other cities would serve gateway functions. To further demonstrate the e€ects and signi®cance of the discount function, we show the shortest path route connecting San Francisco, to Chicago, to New York and then to Atlanta. Fig. 2 shows the shortest path tree between the cities using the actual link lengths (labeled natural route) as the travel costs. Using actual link lengths, the total miles traveled are 3833.29. The second route depicts the solution where the link travel costs are discounted by the ¯ow traveling over them in the highdiscount case of (h ˆ 0:9, b ˆ 0:1). Not only does the route change visually, but also the route length connecting the same four cities increases almost 500 miles to 4305.11 in order to take advantage of the lower costs available on the more traveled route. 4.2. Analysis 2: Synthetic data models In our second analytical experiment, we adopt the CAB data to be used in a more generic modeling situation. Taking the marginals of the scaled interaction matrix (Wij ) we construct ®ve new interaction matrices with varying degrees of long distance travel using the basic gravity model. The general model is of the form: Tij ˆ k

Pi Pj : dijc

…14†

The arbitrary scalar k is ignored since each Tij is scaled by its own total interaction. Pi and Pj are masses corresponding to the marginals of matrix Wij . Network

distance corresponds to dij and its e€ect on interaction is governed by the parameter c. In our experiment, the parameter c is systematically varied (1, 1.25, 1.5, 1.75, 2) and used to calculate ®ve interaction matrices. Increases in the parameter, c, result in a diminished long distance interaction among urban centers. Each of the matrices is input twice into the SO assignment; once using a low discount parameter choice (h ˆ 0:25, b ˆ 1), and once using a high-discount parameter choice (h ˆ 0:9, b ˆ 0:1) for a total of 10 model runs. Table 2 reports the results of the model runs. At higher values of c, long distance interaction is diminished and more localized ¯ow takes place, which is evident from the generally lower average miles traveled. Interestingly, for larger c, there is not as much opportunity for the bundling of ¯ows over long distance routes. Greater bundling of ¯ows occurs when distance is less of a deterrent to interaction, as illustrated by the column labeled di€erence in average miles traveled. We note that the absolute di€erence in the two solutions' objectives is greater when the e€ects of distance are lessened in terms of the magnitude of c. Two sets of solutions are visualized in Figs. 3 and 4. Fig. 3 depicts the low (h ˆ 0:25, b ˆ 1) and high (h ˆ 0:9, b ˆ 0:1) discount solutions using c ˆ 2. Maps 1 and 2 illustrate how the large interaction parameter limits long distance travel. Not much cross-country ¯ow takes place, rather a majority of the ¯ow is localized, especially in the Northeastern US where many large cities are close together. Still, the maps in Fig. 3 point to the tendency for ¯ow to collect itself under discounted conditions. Fewer of the smaller links are used when comparing the high-discount case to the low discount case. Fig. 4 reveals many of the same patterns found in

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263

Table 2 Gravity model (synthetic) assignment results c

1.00 1.25 1.50 1.75 2.00

Low discount (h ˆ 0:25, b ˆ 1)

High discount (h ˆ 0:9, b ˆ 0:10)

Average miles traveled

Average cost of travel

Average miles traveled

Average cost of travel

691.74 562.64 451.89 361.65 291.40

688.27 560.06 449.90 360.02 289.97

706.37 572.95 458.14 364.21 293.69

274.02 228.99 188.18 153.63 124.18

Di€erence in miles traveled

14.63 10.31 6.24 2.56 2.29

Fig. 3. Synthetic data visualization (Gamma ˆ 2).

the initial maps of the assigned CAB data. Because c is now suciently low to allow more long distance interaction than in the previous case, many of the same urban centers discussed relative to Fig. 1 (Oklahoma City, Knoxville, and Indianapolis) appear again as major interchanges.

5. Conclusions Our analysis has taken a systematic approach to exploring the e€ects of various concave cost structures on a network design problem. The analysis was conducted with the intent of making link discounts and hub

264

M.W. Horner, M.E. O'Kelly / Journal of Transport Geography 9 (2001) 255±265

Fig. 4. Synthetic data visualization.

location endogenous. A general conclusion from the results of the analysis is that large discounts are necessary to encourage a signi®cant bundling of ¯ows. Only at those large discount levels where there were agglomerations, densities and other interesting structures were present in the network. Visual evidence and numerical results support the conclusion that under high discounts, certain cities do appear to be major places of interchange. These interchanges seem consistent with the literature's de®nition of a gateway (Taa€e et al., 1996). Models using the synthetic input matrices con®rmed the expected relationships between the discount function parameters and long distance interaction. That is, bundling of ¯ows seems to make the maximum sense in long distance transportation operations. Owing to the propensity for pipeline/trunkline network formulation, conceptually this approach to network design seems well suited for delivery systems such as the US Postal Service. Pipeline networks imply mul-

tiple stops that would not be feasible for an air passenger carrier, but would well work for packages (Kuby and Gray, 1993). It is also possible to imagine this type of network used by long distance interurban bus carriers such as Greyhound, where stops are being made at cities to board and alight passengers along a heavily traveled route. Similarly, a freight-trucking carrier would ®nd interest in arranging ¯ows in such a way as to take advantage of the cost savings of bundled ¯ows. In short, the cities pointed out as `gateways' in our analysis could very well be stops along the route structure a transport carrier's central planner would construct. In summary, this paper has documented some of the basic ideas of taking an ``evolutionary'' approach to network design. Visualization demonstrates how altering the discount parameters in the model leads to differing spatial network structures. The approach described here is focussed on one aspect of network design, albeit for exploratory reasons. In practice,

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network modeling and planning scenarios are more complex and inclusive. 6 Many more areas exist for additional work. For example, in this paper, cities as centers of activity were identi®ed based on visual evidence. A more thorough approach would be to attempt to discern routes and count trac at individual cities by analytical means. We might also consider the e€ects of link capacity since the analysis here considered all the network links to be uncapacitated. Perhaps by introducing some capacity restrictions, the network design might follow a hub network more closely or a di€erent network type altogether. Other empirical works could be undertaken to ®nd more realistic model parameters (h, b). Our approach was to simply select values re¯ecting a wide range of possible discounts. Finally, the literature on hub and spoke networks is very much interested in single versus multiple allocation (O'Kelly and Bryan, 1998). This analysis largely ignored allocation because the model assumes multiple allocations. Some rules could be devised in the future for what constitutes actual allocation, which in turn could be used to identify hub locations or further clarify gateway locations. Acknowledgements The authors would like to acknowledge the National Science Foundation's (NSF) support of this research (BCS-9876455), and the support of The Ohio State University's Urban and Regional Analysis Initiative (URAI). A portion of this research has also been funded as part of the NCRST-F project to the OSU Center for Mapping from the US DOT. The authors appreciate the helpful comments of the anonymous reviewers. Responsibility for all errors and omissions lies with the authors.

6

Readers are referred to Crainic (2000), who provides a detailed review of the relevant decision making issues for service network design in freight transportation. He also includes formulations for a multicommodity, capacitated, ®xed cost network design problem and other such problems.

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