Conclusiones

Hub location problems are concerned with locating hubs and assigning the nodes of a physical network to these hubs in such a way that the total cost of the fixed hub loca-tions and transportation are minimized over the network. An example hub location

Figure 5 Hub Location Problem

problem is shown in Figure 5. In Figure 5, small circles represent clients that need to be assigned to hubs indicated by large circles. Notice that the subgraph induced by the hubs is complete. Generally, the transportation cost for inter-hub transfer is discounted (due to the economies-of-scale for inter-hub transfers) which provides the motivation for locating hubs. “Hub” is a general term used to refer to location or a point where a commodity or information from several sources gets consolidated to go either to another hub or to its final destination. Common examples are hub and spoke network in air transportation, LTL transportation, and telecommunications.

We refer the reader to Campbell (1994), Campbell et al. (2002) and O’Kelly and Miller (1994) for exhaustive surveys of the hub location literature.

There are several variants of hub location problems, such as the single/multiple allocation hub location problem, the capacitated or uncapacitated hub location prob-lem, and the p-hub location problem. Also, there may, or may not, be a fixed charge for locating the hubs. We refer the reader to O’Kelly and Miller (1994) who develop a hub network classification system based on characteristics of service nodes, hubs and arcs.

Different models are motivated by different applications, and below we present a generic model for a capacitated single allocation hub location problem adapted from Ernst and Krishnamoorthy (1999) with some notational changes.

II.1.1. The Model

We will use the notation given below to describe the mathematical formulation. Let I = {1, . . . , n} be the set of nodes and H = {1, . . . , m} be the set of candidate hub locations in the network.

Decision Variables:

z_ij 1 if a node i is allocated to the hub located at node j,0 o.w.

zkk 1 if a node k is selected as a hub, 0 o.w.

y_klⁱ total flow emanating from node i routed between hubs k and l.

Parameters:

wij flow between nodes i and j.

dij the distance between nodes i and j.

Oi P

j∈Iwij

Di P

j∈Iwji

fk fixed cost of locating hub at k.

bk capacity of hub k.

In the objective function given by expression (2.1), the first term represents the total transportation cost for the collection and distribution operations and the second and third terms represent the total transportation cost for the inter-hub transfers and the fixed cost of locating the hubs, respectively. Constraint set (2.2) ensures that each commodity is assigned to exactly one hub. Constraints set (2.3) ensures that a hub is located if the node is assigned to itself, and constraint set (2.4) ensures flow conserva-tion at the hubs and constraint set (2.5) implies capacity restricconserva-tion. Constraint sets (2.6) and (2.7) impose standard binary restrictions and non negativity restrictions on decision variables z and y respectively. The parameters χ, δ, α represent the per unit per mile cost of transportation for collection, distribution and inter-hub transfer.

Generally, the inter-hub costs are discounted, i.e. α < χ and α < δ. Depending upon the application, there may be restriction on the maximum number of hubs a commodity can visit before it reaches its destination. There are several alternate formulations for various hub location problems and we refer the reader to Campbell (1994) for details. We do not cover the p-Hub median problem in which the number of hubs to be located is pre-specified. However, it can be included in the formulation presented above.

Among the several hub location problems, uncapacitated and capacitated hub location problems have received more attention from the research community and are also relevant to our problem. Hub location problems are in general very difficult to solve, significantly more difficult than the classical facility location problem. As observed by Campbell (1994), even a 50 node and 5 hub problem can pose a significant solution challenge. Further the capacitated models are significantly more challenging than the uncapacitated ones. Although it is possible to obtain exact solutions to small problems, researchers have turned to heuristics for the solution of large size problems.

Traditional hub location models assume a complete subgraph formed by arcs between the hubs. According to Campbell et al. (2005a), this assumption imposes a topological and cost structure that may not be desired or realistic in many settings such as LTL network design. They propose a new model called the hub arc location model to overcome the restrictions due to the assumptions. In the first part, they introduce this new model, and examine four special cases in detail. In a companion paper, Campbell et al. (2005b) provide an integer programming formulation for hub arc problems and solution algorithms.

Our problems TNDP and ONDP, however, do not involve hub (or center) lo-cation decisions. Further, in all three levels of problems ONDP, TNDP and SNDP-we consider explicit commodity-based routing decisions, which is not the case in hub location problems. More specifically, we are interested in assigning commodities to consolidation and deconsolidation centers as opposed to hub location problems where the assignment of a node to a hub implicitly determines the assignment of the com-modities originating from, or destined to, a particular node to that hub. Third, we explicitly consider capacity issues on transfer links by incorporating the assign-ment of capacitated trucks to linehaul links and the assignassign-ment of commodities to these trucks. Finally, in hub location models, the transportation costs between hubs typically involve the use of discounted per unit per mile costs. In our case, we con-sider consolidation into TL shipments and the associated costs explicitly. Therefore, despite operational similarities, the problems considered in this dissertation have fun-damental differences from the hub location problems, and these differences hinder the efficient use of the modelling and solution approaches devised for these problems.