# Uncapacitated r-Allocation p-Hub Median Problem

Peiró J., Corberán A., Martí R. (2013)

Optsicom project, University of Valencia (Spain)

## Problem Description

The Uncapactitated *r*-Allocation *p*-Hub Median Problem (UrApHMP) is a variant of the classical *p*-hub problem. Hub location problems arise when given a set of nodes with pairwise traffic demands, we have to choose *p* of them as hub locations and route all the traffic through these hubs at a minimum cost. For each pair of nodes *i* and *j*, there is a traffic *t _{ij}* that needs to be transported. It is generally assumed that direct transportation between non-hub nodes is not possible, and the

*t*traffic travels on a path

_{ij}*i→h*, where

_{i}→h_{j}→j*h*and

_{i}*h*are hubs assigned to

_{j}*i*and

*j*, respectively.

Recently, Yaman (2011) proposed the UrApHMP, a very interesting variant of this classical location problem in which each node can be connected to at most *r* of the *p* hubs. The motivation of this variant comes from the fact that the single allocation version, in which a node is connected (assigned) to a single hub is too restricted for real-world situations, while the multiple allocation variant, where each node can use any of the *p* hubs to route its traffic, results in high fixed costs and complicated networks. The *r*-allocation *p*-hub problem, being *r≤p*, generalizes both versions of the *p*-hub median problem. When *r=1*we are at the single allocation version, whereas if *r=p*, we have the multiple allocation version.

In mathematical terms, given a network with a set of nodes *N* and a set of arcs *A*, let *t _{ij}* be the amount of traffic to be routed from node

*i*to node

*j*, i.e., through the arc

*(i,j)*, and let

*d*be its associated unit routing cost. The UrApHMP is then formulated (Yaman, 2011) in terms of the following variables: Given a node

_{ij}*k*,

*z*if the node is a hub (i.e., if a hub is set or located at this node), and

_{kk}=1*z*otherwise. Given a non-hub node

_{kk}=0*i*and a hub

*k*,

*z*if node

_{ik}=1*i*is assigned or allocated to node

*k*, and 0 otherwise. Finally,

*f*is the proportion of the traffic

_{ijkl}*t*from node

_{ij}*i*to node

*j*that travels along the path

*i→k→l→j*, where

*k*and

*l*are hubs. With these variables, the problem can be formulated as follows:

The *p*-hub median problem belongs to the class of *NP*-hard problems. Even when the set of hubs is given, the sub-problem of optimal allocation of non-hub nodes to hubs is also *NP*-hard (Love, Morris and Wesolowski, 1988).

## State of the Art Methods

A GRASP in which we consider three local search procedures. We also implement a filtering mechanism to discard low-quality constructions to help saving running time for large instances.

A scatter search algorithm in which we consider seven diversification generation methods, and path-relinking as an extension of the classical combination method.

## Instances

We have tested our algorithms on three sets of instances:

- The
**CAB**(Civil Aviation Board) data set. It is based on airline passenger flows between some important cities in the United States. It consists of a data file, presented by O’Kelly in 1987, with the distances and flows of a 25 nodes graph. From this original file, 75 instances with 25 nodes and*p= 1,…, 5*, and*r= 1,…, p*have been generated by several authors. The following parameter values have been widely used:*χ=1,δ=1*, and*α= 0.2, 0.4, 0.6, 0.8*, and*1*. - The
**AP**(Australian Post) data set. It is based on real data from the Australian postal service and was presented by Ernst and Krishnamoorthy in 1996. The size of the original data file is 200 nodes. Smaller instances can be obtained using a code from ORLIB. As with CAB, many authors have generated different instances from the original file. We have extended this set of instances by generating 360 instances with*n*= 40, 50, 70, 75, 80, 85, 90, 95, 100, 150 and 200 nodes. For those instances with 40≤*n*≤50,*p*ranges from 1 to 5. For those with 70≤*n*≤95,*p*ranges from 1 to 8, and for those with 100≤*n*≤200,*p*takes values between 1 and 20. In all these cases,*r<∈{1,…,p}*. According with previous articles, cost parameter values are χ=3, α=0.75 and δ=2. Regarding the flows between nodes, these instances do not have symmetric flows (i.e., for a given pair of nodes*i*and*j,t*is not necessarily equal to_{ij}*t*). Moreover, flows from one node to itself can be positive (i.e.,_{ji}*t*can be strictly positive for a given_{ii}*i*). - The
**USA423**data set. This is a new family of instances that we introduce here based on real airline data. It consists of a data file concerning 423 cities in the United States, where real distances and passenger flows for an accumulated 3 months period are considered. From the original data, 30 instances have been generated with*p*∈{3,4,5,6,7} and*2≤r≤p-1*. For each combination of parameters*p*and*r*, two different values for χ,α,δ have been used: 0.1, 0.07, 0.09, and 0.09, 0.075, 0.08, respectively.

You can download the instances here.

## Computational Experience

We performed extensive computational experiments with 465 instances. The best values for the hardest instances can be downloaded here.

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