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.     

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.     

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.     

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.     

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 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.     

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.     

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.     

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.     

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.     

A parallel local search approach to solving the uncapacitated warehouse location problem

The uncapacitated warehouse location problem (UWLP) has been studied by many researchers. It has been solved using various approaches, including branch and bound linear programming, tabu search, simulated annealing, and genetic algorithms. This study presents a new local search (LS) approach to the UWLP that is quite simple and robust and is efficient in some cases. The algorithm was tested against standard OR Library benchmarks and M^* instances, which have already been used to test other approaches. The results show that the only disadvantage of the algorithm is the exponential growth of its computation time with the problem size. However, the multi-search design suggested here enables the algorithm to run under multi-processor or multi-core systems, which are currently provided as part of standard PC configurations.     

Solving the uncapacitated facility location problem using tabu search

A tabu search heuristic procedure is developed to solve the uncapacitated facility location problem. Tabu search is used to guide the solution process when evolving from one solution to another. A move is defined to be the opening or closing of a facility. The net cost change resulting from a candidate move is used to measure the attractiveness of the move. After a move is made, the net cost change of a candidate move is updated from its old value. Updating, rather than re-computing, the net cost changes substantially reduces computation time needed to solve a problem when the problem is not excessively large. Searching only a small subset of the feasible solutions that contains the optimal solution, the procedure is computationally very efficient. A computational experiment is conducted to test the performance of the procedure and computational results are reported. The procedure can easily find optimal or near optimal solutions for benchmark test problems from the literature. For randomly generated test problems, this tabu search procedure not only obtained solutions completely dominating those obtained with other heuristic methods recently published in the literature but also used substantially less computation time. Therefore, this tabu search procedure has advantage over other heuristic methods in both solution quality and computation speed.     

Facility location and scale decision problem with customer preference

This paper addresses the facility location problem that aims to optimize the location and scale of a new facility in consideration of customer restrictions, including customer preference and the minimum number of customers required to open the facility. In a classic covering problem, the customer is assumed to be covered if he/she is located within the critical distance zone around the facility and is otherwise not covered. This problem is caused by customer facility selection, which differs from the classic covering problem in which services are determined only by proximity. This paper proposes a mixed integer programming formulation based on customer restrictions and also develops a heuristic solution procedure using Lagrangian relaxation. The suggested solution procedure is shown to yield acceptable results in a reasonable computation time.     

Planar geometric location problems

We present anO(n ² log³ n) algorithm for the two-center problem, in which we are given a setS ofn points in the plane and wish to find two closed disks whose union containsS so that the larger of the two radii is as small as possible. We also give anO(n ² log⁵ n) algorithm for solving the two-line-center problem, where we want to find two strips that coverS whose maximum width is as small as possible. The best previous solutions of both problems requireO(n ³) time.Pankaj Agarwal has been supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), an NSF Science and Technology Center, under Grant STC-88-09648. Micha Sharir has been supported by the Office of Naval Research under Grants N00014-89-J-3042 and N00014-90-J-1284, by the National Science Foundation under Grant CCR-89-01484, by DIMACS, and by grants from the US-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. A preliminary version of this paper has appeared inProceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, 1991, pp. 449–458.