Single-allocation ordered median hub location problems

The discrete ordered median location model is a powerful tool in modeling classic and alternative location problems that have been applied with success to a large variety of discrete location problems. Nevertheless, although hub location models have been analyzed from the sum, maximum and coverage point of views, as far as we know, they have never been considered under an alternative unifying point of view. In this paper we consider new formulations, based on the ordered median objective function, for hub location problems with new distribution patterns induced by the different users’ roles within the supply chain network. This approach introduces some penalty factors associated with the position of an allocation cost with respect to the sorted sequence of these costs. First we present basic formulations for this problem, and then develop stronger formulations by exploiting properties of the model. The performance of all these formulations is compared by means of a computational analysis.     

Formulating and solving splittable capacitated multiple allocation hub location problems

It is only recently that good formulations and properties for the basic versions of the hub location problem have become available. Now, versions closer to reality can be tackled with greater guarantees of success. This article deals with the case in which the capacity of the hubs is limited. The focus is on the following interpretation of this capacity: there is, for each hub, an upper bound on the total flow coming directly from the origins. Our problem has the so-called multiple allocation possibility, i.e., there is no hub associated to each node; on the contrary, flows with, say, the same origin but different destinations, can be sent through different routes. Moreover, it is assumed that the flow between a given origin–destination pair can be split into several routes; if this is not the case, the problem becomes quite different and cannot be approached by means of the techniques used in this paper.Tight integer linear programming formulations for the problem are presented, along with some useful properties of the optimal solutions which can be used to speed up the resolution.The computational experience shows that instances of medium size can be solved very efficiently using the new method, which outperforms other methods given in the literature.     

The uncapacitated hub location problem in networks under decentralized management

We propose a new hub location model defined by the minimization of costs. The main contribution of this work is to permit the analysis of a hub-and-spoke network operated under “decentralized management”. In this type of network, various transport companies act independently, and each makes its route choices according to its own criteria, which can include cost, time, frequency, security and other factors, including subjective ones. Therefore, due to the diversity of the various companies’ criteria, one can expect that between each origin–destination pair, a fraction of the flow will be carried through hubs and a fraction will be carried by the direct route. to resolve this problem, it becomes necessary to determine the probability that any network user will choose the hub route for each trip to be made (or for each load to be carried). We present an integer programming formulation, subject the new model to experiments with an intermodal general cargo network in Brazil and address questions regarding the solution of the problem in practice.     

Capacitated single allocation hub location problem—A bi-criteria approach

A different approach to the capacitated single allocation hub location problem is presented. Instead of using capacity constraints to limit the amount of flow that can be received by the hubs, we introduce a second objective function to the model (besides the traditional cost minimizing function), that tries to minimize the time to process the flow entering the hubs. Two bi-criteria single allocation hub location problems are presented: in a first model, total time is considered as the second criteria and, in a second model, the maximum service time for the hubs is minimized. To generate non-dominated solutions an interactive decision-aid approach developed for bi-criteria integer linear programming problems is used. Both bi-criteria models are tested on a set of instances, analyzing the corresponding non-dominated solutions set and studying the reasonableness of the hubs flow charge for these non-dominated solutions. The increased information provided by the non-dominated solutions of the bi-criteria model when compared to the unique solution given by the capacitated hub location model is highlighted.     

A branch-and-cut algorithm for the hub location and routing problem

We study the hub location and routing problem where we decide on the location of hubs, the allocation of nodes to hubs, and the routing among the nodes allocated to the same hubs, with the aim of minimizing the total transportation cost. Each hub has one vehicle that visits all the nodes assigned to it on a cycle. We propose a mixed integer programming formulation for this problem and strengthen it with valid inequalities. We devise separation routines for these inequalities and develop a branch-and-cut algorithm which is tested on CAB and AP instances from the literature. The results show that the formulation is strong and the branch-and-cut algorithm is able to solve instances with up to 50 nodes.     

A genetic algorithm for the uncapacitated single allocation planar hub location problem

Given a set of n interacting points in a network, the hub location problem determines location of the hubs (transfer points) and assigns spokes (origin and destination points) to hubs so as to minimize the total transportation cost. In this study, we deal with the uncapacitated single allocation planar hub location problem (PHLP). In this problem, all flow between pairs of spokes goes through hubs, capacities of hubs are infinite, they can be located anywhere on the plane and are fully connected, and each spoke must be assigned to only one hub. We propose a mathematical formulation and a genetic algorithm (PHLGA) to solve PHLP in reasonable time. We test PHLGA on simulated and real life data sets. We compare our results with optimal solution and analyze results for special cases of PHLP for which the solution behavior can be predicted. Moreover, PHLGA results for the AP and CAB data set are compared with other heuristics.     

Solving the hub location problem with modular link capacities

This paper deals with a capacitated hub location problem arising in the design of telecommunications networks. The problem is different from the classical hub location problem in two ways: the cost of using an edge is not linear but stepwise and the capacity of a hub restricts the amount of traffic transiting through the hub rather than the incoming traffic. In this paper both an exact and a heuristic method are presented. They are compared and combined in a heuristic concentration approach to investigate whether it is possible to improve the results within limited computational times.     

Tight bounds from a path based formulation for the tree of hub location problem

This paper considers the tree of hub location problem. We propose a four index formulation which yields much tighter LP bounds than previously proposed formulations, although at a considerable increase of the computational burden when obtained with a commercial solver. For this reason we propose a Lagrangean relaxation, based on the four index formulation, that exploits the structure of the problem by decomposing it into independent subproblems which can be solved quite efficiently. We also obtain upper bounds by means of a simple heuristic that is applied at the inner iterations of the method that solves the Lagrangean dual. As a consequence, the proposed Lagrangean relaxation produces tight upper and lower bounds and enable us to address instances up to 100 nodes, which are notably larger than the ones previously considered in the literature, with sizes up to 20 nodes. Computational experiments have been performed with benchmark instances from the literature. The obtained results are remarkable. For most of the tested instances we obtain or improve the best known solution and for all tested instances the deviation between our upper and lower bounds, never exceeds 10%.     

A tabu-search based heuristic for the hub covering problem over incomplete hub networks

Hub location problems deal with finding the location of hub facilities and with the allocation of demand nodes to these located hub facilities. In this paper, we study the single allocation hub covering problem over incomplete hub networks and propose an integer programming formulation to this end. The aim of our model is to find the location of hubs, the hub links to be established between the located hubs, and the allocation of non-hub nodes to the located hub nodes such that the travel time between any origin–destination pair is within a given time bound. We present an efficient heuristic based on tabu search and test the performance of our heuristic on the CAB data set and on the Turkish network.     

A hub location problem with fully interconnected backbone and access networks

This paper considers the design of two-layered fully interconnected networks. A two-layered network consists of clusters of nodes, each defining an access network and a backbone network. We consider the integrated problem of determining the access networks and the backbone network simultaneously. A mathematical formulation is presented, but as the linear programming relaxation of the mathematical formulation is weak, a formulation based on the set partitioning model and column generation approach is also developed. The column generation subproblems are solved by solving a series of quadratic knapsack problems. We obtain superior bounds using the column generation approach than with the linear programming relaxation. The column generation method is therefore developed into an exact approach using the branch-and-price framework. With this approach we are able to solve problems consisting of up to 25 nodes in reasonable time. Given the difficulty of the problem, the results are encouraging.     

New simple and efficient heuristics for the uncapacitated single allocation hub location problem

Hub-and-spoke networks are widely studied in the area of location theory. They arise in several contexts, including passenger airlines, postal and parcel delivery, and computer and telecommunication networks. Hub location problems usually involve three simultaneous decisions to be made: the optimal number of hub nodes, their locations and the allocation of the non-hub nodes to the hubs. In the uncapacitated single allocation hub location problem (USAHLP) hub nodes have no capacity constraints and non-hub nodes must be assigned to only one hub. In this paper, we propose three variants of a simple and efficient multi-start tabu search heuristic as well as a two-stage integrated tabu search heuristic to solve this problem. With multi-start heuristics, several different initial solutions are constructed and then improved by tabu search, while in the two-stage integrated heuristic tabu search is applied to improve both the locational and allocational part of the problem. Computational experiments using typical benchmark problems (Civil Aeronautics Board (CAB) and Australian Post (AP) data sets) as well as new and modified instances show that our approaches consistently return the optimal or best-known results in very short CPU times, thus allowing the possibility of efficiently solving larger instances of the USAHLP than those found in the literature. We also report the integer optimal solutions for all 80 CAB data set instances and the 12 AP instances up to 100 nodes, as well as for the corresponding new generated AP instances with reduced fixed costs.     

A traffic grooming problem considering hub location for synchronous optical network-wavelength division multiplexing networks

In this paper, we deal with a traffic demand clustering problem for designing SONET-WDM rings. The objective is to minimize the total cost of optical add-drop multiplexers (OADMs) and inter-ring hub equipments, while satisfying intra-ring and inter-ring capacities. Also, the minimum number of nodes, for example three, for each ring should be satisfied. We develop an integer programming (IP) formulation for the problem and develop some valid inequalities that provide a tight lower bound for the problem. Dealing with the inherent computational complexity of the problem, we also devise an effective tabu search procedure for finding a feasible solution of good quality within reasonable computing time. Computational results are provided to demonstrate the efficacy of the lower and upper bound procedures for solving the problem.     

The heuristic concentration-integer and its application to a class of location problems

We propose a new metaheuristic called heuristic concentration-integer (HCI). This metaheuristic is a modified version of the heuristic concentration (HC), oriented to find good solutions for a class of integer programming problems, composed by problems in which p   elements must be selected from a larger set, and each element can be selected more than once. These problems are common in location analysis. The heuristic is explained and general instructions for rewriting integer programming formulations are provided, that make the application of HCI to these problems easier. As an example, the heuristic is applied to the maximal availability location problem (MALP), and the solutions are compared to those obtained using linear programming with branch and bound (LP+B&B)$(\mathrm{LP}+\mathrm{B}\&\mathrm{B})$. For one-third of the instances of MALP, LP+B&B$\mathrm{LP}+\mathrm{B}\&\mathrm{B}$ can be allowed to run until the computer is out of memory without termination, while HCI can find good solutions to the same instances in a reasonable time. In one such case, LP-IP was allowed to run for nearly 100 times longer than HCI and HCI still found a better solution. Furthermore, HCI found the optimal solution in 33.3% of cases and had an objective value gap of less than 1% in 76% of cases. In 18% of the cases, HCI found a solution that is better than LP+B&B. Therefore, in cases where LP+B&B$\mathrm{LP}+\mathrm{B}\&\mathrm{B}$ is unreasonable due to time or memory constraints, HCI is a valuable tool.     

Hub location for time definite transportation

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.     

An interactive possibilistic programming approach for a multi-objective hub location problem: Economic and environmental design

This paper presents a new multi-objective mathematical model for a multi-modal hub location problem under a possibilistic-stochastic uncertainty. The presented model aims to minimize the total transportation and traffic noise pollution costs. Furthermore, it aims to minimize the maximum transportation time between origin-destination nodes to ensure a high probability of meeting the service guarantee. In order to cope with the uncertainties and the multi-objective model, we propose a two-phase approach, including fuzzy interactive multi-objective programming approach and an efficient method based on the Me measure. Due to the NP-hardness of the presented model, two meta-heuristic algorithms, namely hybrid differential evolution and hybrid imperialist competitive algorithm, are developed. Furthermore, a number of sensitivity analyses are provided to demonstrate the effectiveness of the presented model. Finally, the foregoing meta-heuristics are compared together through different comparison metrics.     

Uncapacitated single allocation p-hub maximal covering problem

The hub location problem is to find a set of hub nodes on the network, where logistics transportation via the hubs is encouraged because of the cost or distance savings. Each node that has a specified amount of demands can be connected to one of p hubs. The uncapacitated single allocation p-hub maximal covering problem is to maximize the logistics covered, where the logistics of demand is said to be covered if the distance between two nodes is less than or equal to the specified range in consideration of the distance savings between hubs. The aim of our model is to locate the hub, and to allocate non-hub nodes to the located hub nodes; the hub can maximize the demand covered by deadline traveling time. It is presented an integer programming formulation for the new hub covering model, and a computational study based on several instances derived from the CAB (Civil Aeronautics Board) data set. Two heuristics, distance based allocation and volume based allocation methods, are suggested with a computational experiment on the CAB data set. Performances of heuristics are evaluated, and it is shown that good solutions are found in a relatively reasonable computation time for most of instances.     

A conditional p-hub location problem with attraction functions

We formulate the competitive hub location problem in which customers have gravity-like utility functions. In the resulting probabilistic model, customers choose an airline depending on a combination of functions of flying time and fare. The (conditional) follower's hub location problem is solved by means of a heuristic concentration method. Computational experience is obtained using the Australian data frequently used in the literature. The results demonstrate that the proposed method is viable even for problems of realistic size, and the results appear quite robust with respect to the leader's hub locations.     

Applying two efficient hybrid heuristics for hub location problem with fully interconnected backbone and access networks

This paper considers the design of two-layered networks with fully interconnected backbone and access networks. The problem, a specific application of hub location to network design, is known as fully interconnected network design problem (FINDP). A novel mathematical programming formulation advantageous over an earlier formulation is presented to model the problem. Two hybrid heuristics are proposed to solve the problem, namely SA_VNS and TS_VNS which incorporate a variable neighborhood search (VNS) algorithm into the framework of simulated annealing (SA) and tabu search (TS). The proposed algorithms are able to easily obtain the optimal solutions for 24 small instances existing in the literature in addition to efficiently solve new generated medium and large instances. Results indicate that the proposed algorithms generate high quality solutions in a quite short CPU time.     

The gravity multiple server location problem

Models for locating facilities and service providers to serve a set of demand points are proposed. The number of facilities is unknown, however, there is a given number of servers to be distributed among the facilities. Each facility acts as an M/M/k queuing system. The objective function is the minimization of the combined travel time and the waiting time at the facility for all customers. The distribution of demand among the facilities is governed by the gravity rule. Two models are proposed: a stationary one and an interactive one. In the stationary model it is assumed that customers do not consider the waiting time at the facility in their facility selection decision. In the interactive model we assume that customers know the expected waiting time at the facility and consider it in their facility selection decision. The interactive model is more complicated because the allocation of the demand among the facilities depends on the demand itself. The models are analyzed and three heuristic solution algorithms are proposed. The algorithms were tested on a set of problems with up to 1000 demand points and 20 servers.     

A hybridization of an evolutionary algorithm and a parallel branch and bound for solving the capacitated single allocation hub location problem

In this study, we propose a hybrid optimization method, consisting of an evolutionary algorithm (EA) and a branch-and-bound method (BnB) for solving the capacitated single allocation hub location problem (CSAHLP). The EA is designed to explore the solution space and to select promising configurations of hubs (the location part of the problem). Hub configurations produced by the EA are further passed to the BnB search, which works with fixed hubs and allocates the non-hub nodes to located hubs (the allocation part of the problem). The BnB method is implemented using parallelization techniques, which results in short running times. The proposed hybrid algorithm, named EA-BnB, has been tested on the standard Australia Post (AP) hub data sets with up to 300 nodes. The results demonstrate the superiority of our hybrid approach over existing heuristic approaches from the existing literature. The EA-BnB method has reached all the known optimal solutions for AP hub data set and found new, significantly better, solutions on three AP instances with 100 and 200 nodes. Furthermore, the extreme efficiency of the implementation of this hybrid algorithm resulted in short running times, even for the largest AP test instances.