Computers & Operations Research

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1 Computers & Operations Research 36 (2009) Contents lists available at ScienceDirect Computers & Operations Research journal homepage: Hub location for time definite transportation James F. Campbell College of Business Administration, University of Missouri St. Louis, One University Blvd., St. Louis, MO , USA A R T I C L E I N F O A B S T R A C T Available online 30 January 2009 Keywords: Hub location Hub median Hub arc location Time definite transportation Motor carriers Time definite motor carriers provide very reliable scheduled truck transportation service between specified terminals. They provide service competitive with airfreight carriers over continental-scale distances at a much lower cost. This paper provides time definite models for multiple allocation p-hub median problems and hub arc location problems. Service levels are imposed by limiting the maximum travel distance via the hub network for each origin destination pair. Computational results are presented to demonstrate the effects of the time definite service levels on practical network design for truck transportation in North America Elsevier Ltd. All rights reserved. 1. Introduction Over the past two decades, hub location researchers have addressed a wide range of problems motivated by air, ground and water transportation systems. Hub facilities are used in transportation networks to provide a transshipment, consolidation and break-bulk function that allows discounted transport costs via the economies of scale in transportation. Hub location problems involve locating hub facilities and discounted transportation links, allocating origin and destinations nodes (e.g., cities) to hubs, and routing flows through the network. Much research has focused on discrete hub median and related models that seek to minimize total transportation costs, along with fixed costs for hubs in some cases. Hub center problems seek to locate hubs to minimize a maximum measure related to transportation distance or cost. While hub median-type models focus on economic objectives, and hub center and covering models focus more on service level objectives, models that combine both dimensions could provide valuable insights for designing transportation hub networks. In this paper we present cost minimizing hub location models for time definite transportation, where each origin destination pair has a specified level of service that must be met. These service levels are expressed in terms of the travel distance via the hub network from the origin to the destination. This research was motivated by our work with a major North American time definite motor carrier. Time definite trucking firms offer very reliable, scheduled service between major cities, generally for freight shipments of several hundred to several thousand pounds. Tel.: ; fax: address: campbell@umsl.edu. Given business operating cycles, the delivery schedules generally conform to either overnight, second day, third day, etc. delivery, with pickup from a terminal in the early evening (e.g., 6:00 pm) and delivery to the terminal in the morning (e.g., by 8:00 or 9:00 am). The combination of higher service levels than general motor carriers, and lower cost than airfreight carriers provides a strong competitive advantage in an environment of decreasing cycle times and increasing uncertainty. The results in this paper provide insights into how time definite transportation firms can configure their networks to be both efficient (low cost) and effective (provide high levels of service). Recent reviews of hub location research by Alumur and Kara [1] and Campbell et al. [7] show the majority of research has addressed cost minimizing (e.g., hub median) models, although more attention recently has been devoted to hub center and hub covering models [14 16,27]. The classic models in hub location research have assumed that the hub nodes are fully connected by hub arcs with discounted transportation cost. These hub arcs reflect the economies of scale in transportation that provide a major incentive for hub networks. These classic hub location models have a number of attractive theoretical features, but the assumption of a fully connected set of hub arcs can lead to some unrealistic results where hub arcs may carry a much lower flow than non-hub arcs, yet the transportation cost is discounted on the hub arcs. Several authors have noted this anomaly (see [18] for a discussion) and alternative hub location models have been proposed to more properly discount transportation costs, including models with flow-dependent costs [22], flow thresholds [25] and hub arc location models [8 10]. Hub arc location models relax the assumption that all hub nodes are connected by hub arcs and seek to locate the discounted hub arcs, whose endpoints are hubs, to minimize the total transportation cost. This more /$ - see front matter 2009 Elsevier Ltd. All rights reserved. doi: /j.cor

2 3108 J.F. Campbell / Computers & Operations Research 36 (2009) general model allows the hub arcs to be used where appropriate to achieve the specified objective. Travel distances and times have been used to measure the level of service in transportation network design and location science since the introduction of covering models and the distance constrained p-median problem [29]. More recently, research has addressed service network design issues, including routing, scheduling and fleet sizing for express delivery services and time definite transportation (e.g., see [3,4,17,19]). In hub location research, O'Kelly [21] and O'Kelly and Lao [23] considered maximum distance constraints for the travel between hubs and origins/destinations in networks with a mini-hub. Other hub location research that incorporates service level constraints on path lengths includes hub covering models with a single maximum origin destination travel time (or distance) constraint [2,28], hop constrained models [24], penalty-minimizing models with tree structures [11] and p-hub center models with stochastic travel times [27]. While each of these models includes aspects of cost and service, none of them include locating hubs with the objective of minimizing transportation costs under distance-dependent service level constraints. In general, hub center and hub covering models have focused on worst-case service (based on the maximum origin destination distance or travel time), while ignoring the total cost for transportation. These models often use a discounted inter-hub travel time (analogous to the discounted travel cost in hub median models) to reflect the use of faster vehicles between hubs. However, the underlying practical motivation for many-to-many hub center and hub covering models seems less compelling than that for the one-to-many (nonhub) center and covering models, which have strong motivations from locating public sector emergency service facilities [29]. Manyto-many transportation carriers (airlines, trucking companies, etc.) are generally private sector firms for whom the cost of transport cannot be easily ignored. Thus, models that integrate both cost and service may provide better insights into practical transportation networks. In this paper we present time definite hub location models for multiple allocation p-hub median problems (HMP) and multiple allocation hub arc location problems. These models blend the costoriented hub median models that minimize the total transportation cost with level of service constraints to ensure origin destination travel times are competitive. The remainder of this paper is organized as follows. Section 2 provides background on time definite motor carriers. Section 3 presents mixed integer programming models for the time definite p-hmp and a time definite hub arc location model. Section 4 presents computational results and Section 5 is the conclusion. 2. Background Trucking is the most important mode of freight transportation in the US, and LTL (less-than-truckload) firms are the largest part of the motor carrier industry. LTL carriers consolidate many small shipments (each generally between 100 and 10,000 pounds) from many different shippers to make efficient vehicle loads. LTL carriers route shipments via a network of consolidation and break-bulk terminals. Each terminal collects shipments from its local service region. Shipments are sorted at the terminal and loaded into line-haul trucks, which carry the shipments to terminals near their destinations. The freight is then transshipped from the line-haul truck to a local delivery truck for transport to the destination. See [12,26] for details on trucking operations and [6] for a review of motor carrier network design. Time definite motor carriers are a small, but growing segment of the LTL carriers that provide very reliable scheduled service between specified terminals. These carriers compete with airfreight carriers for higher value, lower weight products that need fast, reliable transportation, but do not require the immediacy of emergency or next-flight-out service. Typical freight includes electronic and electrical equipment (especially telecommunications and high technology equipment), trade show materials, machinery and parts, and clothing and fashion accessories. Time definite trucking firms use a network of terminals to consolidate small shipments into economic truckloads, as do other more general LTL carriers. However, the lack of strict service schedules for regular LTL carriers allows them more flexibility in constructing routes. For example, a regular LTL carrier may achieve lower costs via greater consolidation by routing a shipment over a more circuitous path, or by holding shipments at a terminal until a large load is accumulated. These measures may reduce costs, but will increase the time until delivery and degrade service. In contrast, the networks for time definite trucking firms must respond to the competing pressures of better service (faster deliveries via smaller shipments, less circuitous routes and smaller delays at terminals) and lower costs (from larger shipments providing economies of scale). Thus, time definite trucking firms should locate terminals (hubs) and construct their networks to minimize cost while meeting specified service levels. 3. Models Consider a complete graph G=(V, E) with node set V ={1, 2,..., N}, where nodes correspond to origins and destinations (i.e., city terminals) and potential hub locations. Each arc (i, j) isoflengthd ij and W ij is the flow (e.g., volume of freight) to be transported from i to j. Distances are assumed to satisfy the triangle inequality. This paper provides two hub location models for time definite transportation that have the same objective of minimizing the total transportation cost. The first is based on the p-hub median model and the second is based on the first type of hub arc location model presented in [8], denoted HAL1. Both models assume each origin destination flow must be routed via at least one hub. Thus, each origin destination flow includes collection from the origin to a hub via arc (i, k) and distribution from a hub to the destination via arc (m, j). If the origin or destination is also a hub, then the collection or distribution component may be at the origin or destination node (via a degenerate arc from i to i). Paths may also include transfer between two hubs on hub arc (k, m). The cost for transportation along a hub arc is discounted by the parameter 0 α 1, presumably to capture the economies of scale from consolidated transportation. The p-hub median model locates p hubs that are fully connected by p(p 1)/2 hub arcs. As a consequence of the full connectivity of the hub arcs in the p-hub median model, each path will have at most one hub arc and at most three arcs. The hub arc location problem HAL1 locates q hub arcs with no restrictions on their locations or connectivity. In both models the nodes at the ends of each hub arc are hubs but in HAL1 every pair of hubs need not be connected by a hub arc. To maintain a similar level of service to the p-hub median model, HAL1 limits each origin destination path to at most three arcs and one central hub arc. (Other hub arc location models in [8,9] allow longer and more complex paths.) Thus, HAL1 requires that the collection and distribution components of an origin destination path occur on arcs other than hub arcs. This can lead to the use of non-discounted bridge arcs that join two hubs and may coincide with hub arcs [8,9]. Fig. 1 provides examples of some of the differences between hub median networks and HAL1 networks. Fig. 1(a) shows a 3-hub median solution with hubs at nodes 1, 3 and 4 (indicated by circles). Paths in Fig. 1(a) may involve three arcs, for example , if this provides a lower cost via the discounted hub arc than the shorter

3 J.F. Campbell / Computers & Operations Research 36 (2009) Fig. 1. (a) 3-Hub median solution; (b) solution for HAL1 with three hub arcs. path Fig. 1(b) shows a solution for HAL1 with three hub arcs and five hubs. Note that the path from nodes 1 to 6 would be , and path is not allowed since it includes two adjacent non-hub arcs. Also, note that the path is possible as long as it includes one hub arc and one bridge arc coinciding with a hub arc, since two hub arcs in a path are not allowed in HAL1. By relaxing the restriction that the hub arcs form a complete subgraph on the hub nodes, HAL1 has greater flexibility to utilize the hub arcs to produce a network with lower transportation costs (for a given number of hub arcs). Since both hub location models use three-part paths for collection, transfer and distribution (even though one or two parts may be the degenerate arc from a node to itself), we write the cost for an origin destination path as C ijkm = d ik + αd km + d mj, where (k, m) is a hub arc for the transfer component and (i, k) and (m, j) are non-hub (possibly degenerate) arcs for collection and distribution. Note that each arc in the model may correspond to travel on multiple arcs (e.g., roads) in the underlying physical transport network. Let p be the given number of hubs to locate and q be the given number of hub arcs to locate. Hub median models have been formulated with a variety of approaches [7,20] and the HAL1 model was originally formulated in a more general form by Campbell et al. [9]. The formulations below are based on the UM8 model in [20], which uses the four-subscript notation originally presented in [5]. The decision variables in the models are: X ijkm is the fraction of flow from origin i to destination j on path i k m j (in that order), Z km = 1 if there is a hub arc between nodes k and m, andy k = 1 if node k is a hub and 0 otherwise. While a very large number of paths may be feasible in the basic p-hub median and HAL1 models, a key feature of time definite models is the elimination of potential paths (and flow variables) based on the service level time restrictions. We use the set xfeas to represent the set of indices (i.e., paths) that are feasible for a particular problem. Let S be the number of service levels defined in terms of pairs of distances where dd(1,2,...,s) is the set of direct origin destination upper limit service level distances and dmax(1,2,...,s) is the set of maximum travel distances for each service level. For example, (1) if all origins and destinations within 400 miles of each other need to be served by a path of length at most 600 miles, then dd(1) = 400 and dmax(1) = 600; and (2) if all origins and destinations between 400 and 1000 miles of each other need to be served by a path of length at most 1200 miles, then dd(2)=1000 and dmax(2)=1200; etc. The set xfeas is then defined based on the direct origin destination distances d ij and the path distances d ik +d km +d mj associated with each flow variable X ijkm. In the formulations below we use the set xfeas to limit the size of the formulation by restricting sets of constraints and summations to only the feasible flow variables. The formulation for the p-hmp to minimize total transportation costs is HMP: Minimize (W ij + W ji )C ijkm X ijkm i<j,k,m xfeas Subject to X ijkm = 1 (i, j, k, m) xfeas, i < j, (1) k m Y k = p, (2) k V X ijkk + ijkm + X ijmk ) Y k (i, j, k, m) xfeas, i < j, (3) m k(x Y k {0, 1} k V, (4) X ijkm 0 (i, j, k, m) xfeas, i < j, (5) The objective is the total cost for collection, distribution and transfer. Constraint (1) ensures that each origin destination flow is sent via some hub pair (possibly a single hub as in X ijkk ). Constraint (2) requires that exactly p hubs are selected. Constraint (3) ensures that hubs are opened for all routings of flows. Constraints (4) and (5) restrict the variables appropriately. The hub arc location problem HAL1 can be formulated similarly to the HMP using the four subscript flow variables X ijkm. However, it requires additional constraints to track the use of hub arcs. As in [8,9] we assume the hub arcs are undirected to allow discounted flow in either direction. The formulation for the hub arc location problem (HAL1) to minimize total transportation costs is HAL1: Minimize (W ij + W ji )C ijkm X ijkm i<j,k,m xfeas Subject to X ijkm = 1 (i, j, k, m) xfeas, i < j, (1) k m Y k p, (2b) k V X ijkk + m k(x ijkm + X ijmk ) Y k (i, j, k, m) xfeas, i < j, (3) X ijkm Z km (i, j, k, m) xfeas, i < j, k m, (6) Z km = q, (7) k V m V m>k Y k Z mk + Z km k V, (8) m<k m>k Y k {0, 1} k V, (4) X ijkm 0 (i, j, k, m) xfeas, i < j, (5) Z km {0, 1} k, m V. (9)

4 3110 J.F. Campbell / Computers & Operations Research 36 (2009) The objective and constraints (1) and (3) are the same as for HMP. Constraint (2b) is a relaxed version of constraint (2) to allow connected hub arcs (with fewer hubs) if that reduces the cost. Constraint (6) ensures there is a hub arc established for each transfer flow. Constraint (7) requires exactly q hub arcs to be selected. Constraint (8) ensures that each hub node has an adjacent hub arc. This prevents the use of isolated hubs (hubs with no adjacent hub arc) as discussed in [8]. Without constraint (8) isolated hubs can arise when an optimal solution with q hub arcs does not use all p hubs. In this case, there is an incentive to locate additional hubs without hub arcs to reduce costs. Constraint (9) restricts the hub arc variables appropriately. 4. Results This section provides results for time definite hub network design with the HMP and HAL1 models using the CAB dataset commonly used in hub location research, along with two augmented versions of this dataset. The original CAB dataset, denoted CAB25, includes 25 US cities numbered 1 25 as shown in Fig. 2. Because this dataset reflects the concentration of population in the eastern US, an augmented dataset denoted CAB was created by adding 15 additional origins/destinations that provide a more uniform distribution of nodes and potential hub locations across an approximately rectangular service region. (Note that these new nodes do not necessarily correspond to US cities.) The locations of the 40 cities in CAB are shown in Fig. 3 and the flow W ij in and out of the 15 new cities was set at a low value of 0.2 (the flows between the first 25 cities range from to 24.0). With small flows, these 15 new cites are not attractive hubs from the standpoint of minimizing costs for originating or terminating flows, but they may be attractive from a geographical standpoint due to their intermediate locations. A third dataset, denoted CAB E, was created by using the same locations as in CAB , but setting all flows equivalent (W ij = 0.2). This dataset represents a more idealized even distribution of demand and removes the attractiveness of the larger cities in the other datasets. Because the service levels are based on those for a major time definite carrier in North America, results based on the CAB25 dataset should represent a realistic application. We provide results with two levels of inter-hub transportation discount α = 0.2 and 0.6. The small value of α reflects a strong degree of consolidation and economies of scale that corresponds to the use of much larger trucks on hub arcs than on non-hub arcs to generate the large relative cost savings. The value α = 0.6 reflects a lesser degree of consolidation and may be Fig. 2. CAB25 cities Fig. 3. CAB cities.

5 J.F. Campbell / Computers & Operations Research 36 (2009) Table 1 Service levels. Level of service dd dmax High Medium Low a more reasonable value for truck transportation where the trucks used on hub arcs may be one-and-a-half to two times larger than the trucks used on the non-hub arcs. (Note that for the Australia Post hub location dataset [13], the value of α for inter-hub transportation is between 0.25 and 0.375, relative to the value for transportation on the non-hub arcs.) We derived service levels for time definite transportation from the service schedules of a major North American time definite motor carrier. The direct (straight line) distance between all city pairs being served was calculated and compared to the published delivery service schedules of next day, second day, third day, etc. From this comparison we determined that with rare exceptions, city pairs separated by 400 miles or less received next day service; those separated by miles received second day service; and those separated by miles received third day service. City pairs with greater separation received fourth day service. The options for achieving a specified level of service with ground transportation are limited by the available time (e.g., from availability of shipments at the origin terminal at 6:00 pm to delivery at the destination terminal at 9:00 am the following day), truck speeds, and federal hours-of-service regulations that limit the number of consecutive hours that truck drivers may drive. Many of the arcs that make up each path will be short enough for a single driver to handle within the allowed time (e.g., 11 hours). Travel on longer arcs may use trucks with two drivers to increase the distance that can be covered in a given time. We leave the details of assigning drivers to a more tactical decision maker. Also, note that while air transportation can increase the level of service by extending the distances reached in a given time, this comes at a very substantial cost and the focus in this paper is on the use of truck transportation to provide a high level of service comparable to that from air transport. Hub location models that combine air and truck transport to achieve higher levels of service are an important area for future research. We consider three levels of service in this paper as shown in Table 1. These are designed for truck transportation in a region the size of the continental US where the maximum distance between cities is less than 3000 miles. The high level of service corresponds to the distance limits derived from the operations of a large North American time definite motor carrier. The medium level of service relaxes the maximum distance limits by 200 miles (about 4 hours of driving). The low level of service corresponds to no constraints on the travel distance and allows comparison with basic hub median and hub arc location models. All problems were solved to optimality using the GAMS integrated development environment with CPLEX on a 3.2 GHz Pentium processor with 1 GB RAM. Preprocessing was used to eliminate some variables by setting the flow variable X ijkm equal to zero whenever C ijmk < C ijkm, C ijmm < C ijkm or C ijkk < C ijkm. Results with the CAB25 dataset are presented in Tables 2 4 for p-hmp with up to Table 2 p-hub median problem results with CAB25 and the low service level. α p q Cost Hubs #A cpu , , 17, , 12, 17, , 7, 12, 14, , 6, 7, 12, 14, , 6, 7, 12, 14, 17, , 4, 6, 7, 12, 14, 17, , 4, 6, 7, 8, 12, 14, 17, , 4, 6, 7, 8, 12, 14, 17, 22, , , 12, , 4, 12, , 7, 12, 14, , 7, 12, 14, 17, , 7, 12, 14, 17, 20, , 7, 8, 12, 14, 17, 20, , 4, 7, 8, 12, 14, 17, 20, , 4, 6, 7, 8, 12, 14, 17, 22, hubs and in Tables 5 7 for hub arc location problems with up to five hub arcs and 10 hubs. The column #A provides the number of non-hub arcs in the solutions, and cpu time is in seconds. Each table shows how the cost, the hub network and the number of non-hub arcs change as the number of hubs and hub arcs vary. A comparison across tables demonstrates the effects of changing the level of service. Note that the higher levels of service cannot be met with small numbers of hubs and hub arcs: the medium service level requires at least five hubs and the high service level requires at least nine hubs. Table 7 for the HAL1 model provides solutions with five hub arcs and both nine and 10 hubs, where the nine hub solution necessarily involves a pair of connected hub arcs (4 18 and 14 18). Figs. 4 and 5 are plots of the direct origin destination distances versus the corresponding travel distances via the hub network for the HAL1 solution with five hub arcs, 10 hubs and α = 0.2. Fig. 4 displays results for the low service level, along with lines showing the medium and high service levels. The points above the service level lines in Fig. 4 indicate that 34 city pairs cannot meet the medium service levels and 69 city pairs cannot meet the high service levels for this instance. For example, cities 5 and 16 are 700 miles apart, but use the 1729 mile long path due to the cost savings on the hub arc Note that some of these city pairs might be able to meet the service levels via a shorter, but more expensive path using the same set of hubs. However, the large discount (α = 0.2) encourages circuitous paths to exploit the reduced cost hub arcs. Fig. 5 shows that the corresponding high service level solution results in all points clustered beneath the service level line. This is achieved by relocating six hubs and adjusting four of the five hub arcs accordingly (see Tables 5 and 7). Results with the CAB dataset are presented in Tables 8 and 9 for hub median and hub arc location problems, respectively. The SL column indicates the service level. With this larger dataset, 16 hubs are needed to meet the high service level requirements. Table 9 shows solutions with eight hub arcs (the minimum number with 16 hubs) and with 60 hub arcs (half the maximum number used in the hub median solutions in Table 8). Rather than show all 60 hub arcs in Table 9, the notation A(B) indicates that hub city A is adjacent to B hub arcs. Hub arc location problems with the low and medium service levels for α = 0.6 could not be solved due to exceeding the memory limits. Hub median results with the CAB E dataset are presented in Table 10. The hub arc location versions of these problems could not be solved due to memory limitations. In general, the results show that moving from low to high service leads to a shortening of the hub arcs and more centralization of the

6 3112 J.F. Campbell / Computers & Operations Research 36 (2009) Table 3 p-hub median problem results with CAB25 and the medium service level. α p q Cost Hubs #A cpu , 10, 12, 14, , 7, 12, 13, 14, , 6, 7, 12, 13, 14, , 6, 7, 12, 13, 14, 17, , 6, 8, 10, 12, 14, 17, 21, , 4, 6, 7, 8, 12, 14, 17, 21, , 10, 12, 14, , 7, 12, 13, 14, , 7, 12, 13, 14, 17, , 7, 12, 13, 14, 17, 20, , 4, 7, 11, 12, 14, 17, 20, , 4, 7, 8, 12, 14, 17, 20, 21, Table 4 p-hub median problem results with CAB25 and the high service level. α p q Cost Hubs #A cpu , 5, 10, 11, 13, 14, 18, 19, , 4, 6, 10, 11, 13, 14, 17 19, , 5, 10, 11, 13, 14, 18, 19, , 4, 10, 11, 13, 14, 17, 19, 20, Table 5 HAL1 results with CAB25 and the low service level. α p q Cost Hubs Hub arcs #A cpu , 4, 12, , , 12, 14, 17, 21, , 14 25, , 7, 12, 14, 17, 18, 20, , 7 22, 12 20, , 7, 12, 14, 17, 18, 20, 21, , 7 20, 12 21, 14 18, , 7, 12, 14, 17, 18, 20, 21, , 7 20, 12 21, 14 18, , 4, 12, , , 7, 12, 14, 17, , 7 20, , 6, 7, 12, 14, 17, 22, , 6 12, 7 22, , 6, 7, 12, 14, 17, 21, 22, , 6 22, 7 12, 12 21, , 6, 7, 12, 14, 17, 19, 21, 22, , 6 22, 7 19, 14 25, Table 6 HAL1 results with CAB25 and the medium service level. α p q Cost Hubs Hub arcs #A cpu , 10, 12, 14, 17, , 12 21, , 10, 12, 14, 17, 18, 20, , 10 20, 12 21, , 4, 5, 10, 11, 12, 14, 17, , 5 10, 17 22, 11 12, , 4, 5, 10, 11, 12, 14, 17, , 5 10, 17 22, 11 12, , 7, 12, 13, 14, , 7 13, , 10, 12, 14, 17, 21, 22, , 4 17, 12 21, , 4, 7, 12, 14, 17, 20, 21, , 4 17, 12 21, 14 17, , 4, 7, 12, 14, 17, 18, 20, 21, , 4 17, 12 21, 14 18, Table 7 HAL1 results with CAB25 and the high service level. α p q Cost Hubs Hub arcs #A cpu , 5, 10, 11, 13, 14, 18, 19, , 4 18, 5 22, 10 13, , 4, 5, 10, 11, 13, 14, 17, 19, , 4 17, 5 22, 10 13, , 5, 10, 11, 13, 14, 18, 19, , 5 22, 10 13, 11 19, , 5, 10, 11, 13, 14, 17, 19, 22, , 5 22, 10 13, 11 19,

7 J.F. Campbell / Computers & Operations Research 36 (2009) Fig. 4. Direct and network travel distances for the low service level HAL1 solution with CAB25 when p = 10, q = 5andα = 0.2. Fig. 5. Direct and network travel distances for the high service level HAL1 solution with CAB25 when p = 10, q = 5andα = 0.2. Table 8 p-hub median problem results with CAB for p = 16. α SL p q Cost Hubs #A cpu 0.2 L , 3, 4, 6, 7, 8, 12, 14, 15, 16, 17, 19, 21, 22, 23, M , 3, 4, 6, 7, 8, 12, 14, 15, 17, 21, 22, 25, 29, 34, H , 4, 6, 7, 8, 12, 14, 16, 18, 19, 21, 23, 30, 32, 33, L , 3, 4, 6, 7, 8, 12, 14, 15, 16, 17, 19, 21, 22, 23, M , 4, 6, 7, 8, 12, 14, 15, 17, 19, 21, 22, 23, 25, 32, H , 4, 7, 8, 12, 14, 16, 18, 19, 20, 21, 23, 30, 32, 33, hub nodes. As an example, Fig. 6 displays the HAL1 solutions for the CAB25 dataset with five hub arcs, 10 hubs and α = 0.2 for the three different levels of service. The transition from the low to the medium level of service changes three hub arcs, as does the transition from the medium to the high level of service. In each transition, the net result is a set of shorter hub arcs. This pattern of centralization of hubs is also evident for other hub arc and hub median problems for all three datasets. For example, the lower two solutions in Fig. 7 show the low and high service level solutions for the p-hub median solution with p = 16 and α = 0.2 for the CAB E dataset. Comparison of the transportation cost with different service levels shows that the increase in cost needed to improve the level of service decreases as the number of hubs and the number of hub arcs increases, and it decreases as α increases. With the CAB25 dataset, the cost increase to move from the low to the high service level ranges from 27.2% for the HMP with CAB25 when p = 5andα = 0.2

8 3114 J.F. Campbell / Computers & Operations Research 36 (2009) Table 9 HAL1 results with CAB for p = 16. α SL p q Cost Hubs Hubs arcs #A cpu 0.2 L , 4, 5, 6, 7, 8, 12, 14, 17, 21, 23, , 4 17, 4 24, 5 23, 6 8, 12 17, 12 21, M , 2, 4, 6, 7, 12, 14, 17, 21, 29, 33, 34, , 4 17, 6 36, 7 34, 12 17, 12 21, 14 17, H , 10, 11, 12, 13, 14, 15, 17, 19, 25, 29, 30, 31, 34, 35, , 10 34, 11 19, 12 25, 13 36, 14 35, 15 29, H , 4, 6, 7, 8, 12, 14, 16, 18, 19, 21, 23, 30, 32, 33, 36 1(9), 4(11),6(10), 7(11), 8(9), 12(12), 14(9), 16(6), 18(12), 19(5), 21(7), 23(7), 30(1), 32(5), 33(5), 36(1) M 16 8 Mem 0.6 H , 8, 10, 12, 13, 14, 17, 19, 21, 23, 25, 30, 32, 33, 35, , 8 13, 10 35, 12 21, 14 25, 19 36, 23 33, H , 4, 7, 8, 12, 14, 16, 18, 19, 20, 21, 23, 30, 32, 33, 36 1(9), 4(13), 7(9), 8(10), 12(11), 14(7), 16(6), 18(13), 19(6), 20(10), 21(7), 23(5), 30(1), 32(6), 33(6), 36(1) Table 10 p-hub median problem results with CAB Eforp = 16. α SL p q Cost Hubs #A cpu 0.2 L , 10, 18, 19, 21, 22, 24, 26, 27, 28, 29, 33, 34, 35, 36, M , 7, 16, 18, 19, 21, 22, 24, 26, 28, 29, 33, 34, 35, 36, H , 2, 6, 7, 12, 14, 15, 16, 19, 21, 29, 30, 31, 33, 34, L , 7, 9, 13, 19, 21, 22, 24, 26, 28, 29, 33, 34, 35, 36, M , 7, 9, 13, 19, 21, 22, 24, 26, 28, 29, 33, 34, 35, 36, H , 5, 10, 11, 12, 13, 15, 19, 24, 29, 30, 31, 34, 35, 36, to only 2.3% for HAL1 with CAB25 when p = 10, q = 5andα = 0.6. Comparison of Tables 8 and 10 shows that with equal demands, the increase in cost required to move from the low to the high service level is much less with the more uniform demand in CAB E. Note that although the costs of hub arc and hub median solutions can differ considerably (with the hub median solutions providing a lower transportation cost due to the greater number of hub arcs), some of the hubs used by the HMP and HAL1 solutions are identical or very similar. This commonality of hubs seems stronger for the CAB25 dataset than in the larger datasets. For example, each of the CAB25 HAL1 solutions with α = 0.2, p = 10 and q = 5 shown in Fig. 6 use between 6 and 8 of the same 10 hubs as in the corresponding HMP solutions (in Tables 2 4). It is also interesting that in some HAL1 problems with small α, the optimal solution does not use all the hub nodes allowed. For example, the CAB25 solutions in Tables 5 and 6 with α = 0.2, p = 10 and q=5 useonlyninehubs (notthe10possible) andthecab25+15 solutions in the first two rows of Table 9 (α = 0.2, p = 16 and q = 8) use only 12 and 13 hubs, respectively. These examples achieve the minimum cost with fewer than the maximum number of hub nodes by deploying adjacent hub arcs at nodes 4, 12 and 17. The number of non-hub allocation arcs (or spokes ), as shown by #A in Tables 2 10 generally increases with an increasing level of service as flows may be shifted from lower cost circuitous routes to more direct routes to reduce the travel distance. The length of these allocation arcs can be used to provide the cut-off times for providing freight at the origins and the delivery times at the destinations. The results also show that hubs consistently appear at certain cities or in certain geographic regions. For the CAB25 dataset, all HAL1 solutions use hubs at nearby eastern cities 17 or 18, west-coast city 12 or 22, and central city 4 (except nearby city 6 in one solution). For q 3, nearby cities 7 or 10 are always used. Similarly, all hub median solutions use cities 12, 4 (except nearby city 21 in one case), and either city 17, 18 or 2 (all along the east coast). However, there are differences in the networks, even with high service levels when the path length limits restrict the set of feasible hubs. In the larger 40 node datasets, fewer hubs are commonly used across solutions, with only cities 4, 12, and 14 appearing in all solutions for CAB25+15 (Tables 8 and 9). Fig. 7 shows the differential use of new cities as hubs from three hub median solutions in Tables 8 and 10. The circles with crosses represent hubs at new cities not in CAB25. The top solution uses no new cities as hubs, while the central solution for equal demands with CAB E has nine of the 16 hubs at new cities. Comparison of the lower two solutions in Fig. 7 show how the even distribution of hubs with the low service level is replaced by a more clustered pattern with fewer hubs at new cities under the high service level. In general, hub arc location solutions (Table 9) consistently use more new cities as hubs than hub median solutions (Table 8) and the even demand in CAB E encourages the use of new cities as hubs (46 out of 96 hubs in Table 10 are at new cities). With the CAB25+15 dataset, the large cities attract hubs and create a more clustered pattern of hubs that reflects the underlying population distribution with concentrations in the eastern US. In contrast, the solutions with CAB E provide a more uniform distribution of hubs by utilizing many of the new cities not in the original CAB25 dataset. The cpu times show that all HMP and the hub arc location problems with CAB25 were solved in very reasonable times (with the HAL1 problems solved much more quickly than in [9] due to the improved formulation). Solution times for HAL1 with the CAB dataset were over 2800 seconds in one instance, and several problems could not be solved due to memory limitations. This is not surprising with the four subscript notation (as noted in [20]) and a more compact formulation (as in [9]) may reduce the memory requirements, at the cost of increased solution times. Note though that the results presented here are the largest hub arc problems solved to date. 5. Conclusions This paper presents models for time definite hub location and network design motivated by time definite trucking in the US. The results document the impact on network design of different levels of service, and highlight the relative importance of different terminals or geographic regions in achieving high levels of service. The results allow a comparison of optimal networks for time definite hub arc location solutions and hub median solutions. The models also can help evaluate the ability of firms to improve their level of service without significant changes in their network. The results show that an increase in the service level may require modifying a given hub network by relocating hubs (and hub arcs) or by adding hubs and hub arcs. Adding hubs and hub arcs

9 J.F. Campbell / Computers & Operations Research 36 (2009) Low service level HAL1 solution CAB25+15 with Low service level Medium service level HAL1 solution CAB25+15-E with Low service level High service level HAL1 solution Fig. 6. Time definite HAL1 solutions for CAB25 with α = 0.2, p = 10, and q = 5. CAB25+15-E with High service level would increase fixed costs for the new facilities and assets, while decreasing transportation costs. The hub arc location model used in this paper (HAL1) requires that each hub be adjacent to a hub arc, but another option to meet higher levels of service would be to add isolated hubs that are not adjacent to a hub arc. Isolated hubs may improve the level of service at a more moderate cost compared to adding hub arcs (and associated hubs), and this is an area of ongoing research. Other areas for future research include consideration of other hub arc location models from [8,9] that allow more complex paths, incorporation of service level constraints in other hub location models especially models with fixed costs for hubs, and models with multiple transportation modes (e.g., air and truck). Alternate forms of the service level restrictions could also be explored to allow service levels based on a percentage or fixed amount of travel in excess of the direct distance (as might be more relevant for passenger transportation). Service levels might also be relaxed or tightened for certain origin destination pairs to reflect competitive pressures. Note though that service level constraints have the advantage of reducing problem size and thus may allow solution of more realistic problems of larger size and with more practical levels of service. Fig. 7. Time definite HMP solutions with α = 0.2 and p = 16. Acknowledgment The author thanks the reviewers and the editors for their many helpful comments and suggestions that substantially improved this paper. References [1] Alumur S, Kara BY. Network hub location problems: the state of the art. European Journal of Operational Research 2008;190:1 21. [2] Alumur S, Kara BY. A hub covering network design problem for cargo applications in Turkey. Journal of the Operational Research Society 2008; in press, doi: /jors [3] Barnhart C, Krishnan N, Kim D, Ware K. Network design for express shipment delivery. Computational Optimization and Applications 2002;21: [4] Barnhart C, Schneur R. Air network design for express shipment service. Operations Research 1996;44: [5] Campbell JF. Integer programming formulations of discrete hub location problems. European Journal of Operational Research 1994;72:

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