A branch-and-cut algorithm for two-level survivable network design problems

This paper approaches the problem of designing a two-level network protected against single-edge failures. The problem simultaneously decides on the partition of the set of nodes into terminals and hubs, the connection of the hubs through a backbone network (first network level), and the assignment of terminals to hubs and their connection through access networks (second network level). We consider two survivable structures in both network levels. One structure is a two-edge connected network, and the other structure is a ring. There is a limit on the number of nodes in each access network, and there are fixed costs associated with the hubs and the access and backbone links. The aim of the problem is to minimize the total cost. We give integer programming formulations and valid inequalities for the different versions of the problem, solve them using a branch-and-cut algorithm, and discuss computational results. Some of the new inequalities can be used also to solve other problems in the literature, like the plant cycle location problem and the hub location routing problem.     

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.     

An optimization algorithm for the inventory routing problem with continuous moves

The typical inventory routing problem deals with the repeated distribution of a single product from a single facility with an unlimited supply to a set of customers that can all be reached with out-and-back trips. Unfortunately, this is not always the reality. We focus on the inventory routing problem with continuous moves, which incorporates two important real-life complexities: limited product availabilities at facilities and customers that cannot be served using out-and-back tours. We need to design delivery tours spanning several days, covering huge geographic areas, and involving product pickups at different facilities. We develop an integer programming based optimization algorithm capable of solving small to medium size instances. This optimization algorithm is embedded in local search procedure to improve solutions produced by a randomized greedy heuristic. We demonstrate the effectiveness of this approach in an extensive computational study.     

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

We introduce the Partitioning-Hub-Location-Routing Problem (PHLRP), a hub location problem involving graph partitioning and routing features. The PHLRP consists of partitioning a given network into sub-networks, locating at least one hub in each sub-network and routing the traffic within the network at minimum cost. This problem finds applications in deployment of an Internet Routing Protocol called Intermediate System–Intermediate System (ISIS), and strategic planning of LTL ground freight distribution systems. We present an Integer Programming (IP) model for solving exactly the PHLRP and explore possible valid inequalities to strengthen it. Computational experiments prove the effectiveness of our model which is able to tackle instances of PHLRP containing up to 20 vertices.     

A column generation and branch-and-cut algorithm for the channel assignment problem

We present an exact algorithm for solving the channel assignment problem in cellular telephony networks. This problem consists of assigning sets of channels to the network cells in such a way that the channel demand is satisfied, while avoiding co-channel interference and adjacent channel interference and respecting the channel spacing constraint for each antenna. In a previous article [Jaumard B, Marcotte O, Meyer C, Vovor T. Comparison of column generation models for channel assignment in cellular networks. Discrete Applied Mathematics 2002; 118:299–322], we formulated this problem as a covering problem in two different ways and compared these two formulations and another formulation both from a theoretical and computational point of view (by solving their linear relaxations). In the present article we focus on the best set covering formulation, where the subsets are sets of cells that can be assigned the same channel, and we actually solve two versions of the integer program. In the first version, we seek to minimize the unmet demand, while in the second, we seek to minimize the overall interference while assigning the required number of channels to each cell. In either version, the integer program is solved by an algorithm combining the column generation technique and a branch-and-cut scheme. Finally, we present the solutions produced by these algorithms for some instances of European networks and real-life instances supplied by the Bell Mobilité company.     

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.     

Time constrained maximal covering salesman problem with weighted demands and partial coverage

In a routing framework, it may not be viable to visit every single customer separately due to resource limitations or efficiency concerns. In such cases, utilizing the notion of coverage; i.e., satisfying the demand of multiple customers by visiting a single customer location, may be advantageous. With this motivation, we study the time constrained maximal covering salesman problem (TCMCSP) in which the aim is to find a tour visiting a subset of customers so that the amount of demand covered within a limited time is maximized. We provide flow and cut formulations and derive valid inequalities. Since the connectivity constraints and the proposed valid inequalities are exponential in the size of the problem, we devise different branch-and-cut schemes. Computational experiments performed on a set of problem instances demonstrate the effectiveness of the proposed valid inequalities in terms of strengthening the linear relaxation bounds as well as speeding up the solution procedure. Moreover, the results indicate the superiority of using a branch-and-cut methodology over a flow-based formulation. Finally, we discuss the relation between the problem parameters and the structure of optimal solutions based on the results of our experiments.     

The vendor location problem

The vendor location problem is the problem of locating a given number of vendors and determining the number of vehicles and the service zones necessary for each vendor to achieve at least a given profit. We consider two versions of the problem with different objectives: maximizing the total profit and maximizing the demand covered. The demand and profit generated by a demand point are functions of the distance to the vendor. We propose integer programming models for both versions of the vendor location problem. We then prove that both are strongly NP-hard and we derive several families of valid inequalities to strengthen our formulations. We report the outcomes of a computational study where we investigate the effect of valid inequalities in reducing the duality gaps and the solution times for the vendor location problem.     

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%.     

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.     

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 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.     

Aggregation in hub location problems

Because of their widespread use in real-world transportation situations, hub location models have been extensively studied in the last two decades. Many types of hub location problems are NP-hard and remain unmanageable when the number of nodes exceeds 200. We present a way to tackle large-sized problems using aggregation, explore the resulting error, and show how to reduce it. Furthermore, we develop a heuristic based on aggregation for k-hub center problems and present computational results.     

A branch-and-cut algorithm for the minimum labeling Hamiltonian cycle problem and two variants

This paper proposes a mathematical model, valid inequalities and polyhedral results for the minimum labeling Hamiltonian cycle problem. This problem is defined on an unweighted graph in which each edge has a label. The aim is to determine a Hamiltonian cycle with the least number of labels. We also define two variants of this problem by assigning weights to the edges and by considering the tour length either as an objective or as a constraint. A branch-and-cut algorithm for the three problems is developed, and computational results are reported on randomly generated instances and on modified instances from TSPLIB.     

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.     

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.     

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.     

The pickup and delivery problem: Faces and branch-and-cut algorithm

This paper formulates the pickup and delivery problem, also known as the dial-a-ride problem, as an integer program. Its polyhedral structure is explored and four classes of valid inequalities developed. The results of a branch-and-cut algorithm based on these constraints are presented.