Allocation strategies in hub networks

In this paper, we study allocation strategies and their effects on total routing costs in hub networks. Given a set of nodes with pairwise traffic demands, the p-hub median problem is the problem of choosing p nodes as hub locations and routing traffic through these hubs at minimum cost. This problem has two versions; in single allocation problems, each node can send and receive traffic through a single hub, whereas in multiple allocation problems, there is no such restriction and a node may send and receive its traffic through all p hubs. This results in high fixed costs and complicated networks. In this study, we introduce the r-allocation p-hub median problem, where each node can be connected to at most r hubs. This new problem generalizes the two versions of the p-hub median problem. We derive mixed-integer programming formulations for this problem and perform a computational study using well-known datasets. For these datasets, we conclude that single allocation solutions are considerably more expensive than multiple allocation solutions, but significant savings can be achieved by allowing nodes to be allocated to two or three hubs rather than one. We also present models for variations of this problem with service quality considerations, flow thresholds, and non-stop service.     

A basic variable neighborhood search heuristic for the uncapacitated multiple allocation <Emphasis Type="Italic">p</Emphasis>-hub center problem

The uncapacitated multiple allocation p-hub center problem (UMApHCP) consists of choosing p hub locations from a set of nodes with pairwise traffic demands in order to route the traffic between the origin-destination pairs such that the maximum cost between origin-destination pairs is minimum. It is assumed that transportation between non-hub nodes is possible only via chosen hub nodes. In this paper we propose a basic variable neighborhood search (VNS) heuristic for solving this NP hard problem. In addition we apply two mathematical formulations of the UMApHCP in order to detect limitations of the current state-of-the-art solver used for this problem. The heuristics are tested on benchmark instances for p-hub problems. The obtained results reveal the superiority of the proposed basic VNS over the state-of-the-art as well as over a multi-start local search heuristic developed by us in this paper.     

Modeling fuzzy capacitated p-hub center problem and a genetic algorithm solution

Hub and spoke networks are used to switch and transfer commodities between terminal nodes in distribution systems at minimum cost and/or time. The p-hub center allocation problem is to minimize maximum travel time in networks by locating p hubs from a set of candidate hub locations and allocating demand and supply nodes to hubs. The capacities of the hubs are given. In previous studies, authors usually considered only quantitative parameters such as cost and time to find the optimum location. But it seems not to be sufficient and often the critical role of qualitative parameters like quality of service, zone traffic, environmental issues, capability for development in the future and etc. that are critical for decision makers (DMs), have not been incorporated into models. In many real world situations qualitative parameters are as much important as quantitative ones. We present a hybrid approach to the p-hub center problem in which the location of hub facilities is determined by both parameters simultaneously. Dealing with qualitative and uncertain data, Fuzzy systems are used to cope with these conditions and they are used as the basis of this work. We use fuzzy VIKOR to model a hybrid solution to the hub location problem. Results are used by a genetic algorithm solution to successfully solve a number of problem instances. Furthermore, this method can be used to take into account more desired quantitative variables other than cost and time, like future market and potential customers easily.     

Optimizing fuzzy p-hub center problem with generalized value-at-risk criterion

In this study, we reduce the uncertainty embedded in secondary possibility distribution of a type-2 fuzzy variable by fuzzy integral, and apply the proposed reduction method to p-hub center problem, which is a nonlinear optimization problem due to the existence of integer decision variables. In order to optimize p-hub center problem, this paper develops a robust optimization method to describe travel times by employing parametric possibility distributions. We first derive the parametric possibility distributions of reduced fuzzy variables. After that, we apply the reduction methods to p-hub center problem and develop a new generalized value-at-risk (VaR) p-hub center problem, in which the travel times are characterized by parametric possibility distributions. Under mild assumptions, we turn the original fuzzy p-hub center problem into its equivalent parametric mixed-integer programming problems. So, we can solve the equivalent parametric mixed-integer programming problems by general-purpose optimization software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the efficiency of the proposed solution methods.     

Exact computational approaches to a stochastic uncapacitated single allocation p-hub center problem

The stochastic uncapacitated single allocation p-hub center problem is an extension of the deterministic version which aims to minimize the longest origin-destination path in a hub and spoke network. Considering the stochastic nature of travel times on links is important when designing a network to guarantee the quality of service measured by a maximum delivery time for a proportion of all deliveries. We propose an efficient reformulation for a stochastic p-hub center problem and develop exact solution approaches based on variable reduction and a separation algorithm. We report numerical results to show effectiveness of our new reformulations and approaches by finding global solutions of small-medium sized problems. The combination of model reformulation and a separation algorithm is particularly noteworthy in terms of computational speed.     

Solving large single allocation p-hub problems with two or three hubs

In this paper we present an efficient approach for solving single allocation p-hub problems with two or three hubs. Two different variants of the problem are considered: the uncapacitated single allocation p-hub median problem and the p-hub allocation problem. We solve these problems using new mixed integer linear programming formulations that require fewer variables than those formerly used in the literature. The problems that we solve here are the largest single allocation problems ever solved. The numerical results presented here will demonstrate the superior performance of our mixed integer linear programs over traditional approaches for large problems. Finally we present the first mixed integer linear program for solving single allocation hub location problems that requires only O(n²) variables and O(n²) constraints that is valid for any number of hubs.     

Avoiding local optima in thep-hub location problem using tabu search and GRASP

In the discretep-hub location problem, various nodes interact with each other by sending and receiving given levels of traffic (such as telecommunications traffic, data transmissions, airline passengers, packages, etc.). It is necessary to choosep of the given nodes to act as hubs, which are fully interconnected; it is also necessary to connect each other node to one of these hubs so that traffic can be sent between any pair of nodes by using the hubs as switching points. The objective is to minimize the sum of the costs for sending traffic along the links connecting the various nodes. Like many combinatorial problems, thep-hub location problem has many local optima. Heuristics, such as exchange methods, can terminate once such a local optimum is encountered. In this paper, we describe new heuristics for thep-hub location problem, based on tabu search and on a greedy randomized adaptive search procedure (GRASP). These recently developed approaches to combinatorial optimization are capable of examining several local optima, so that, overall, superior solutions are found. Computational experience is reported in which both tabu search and GRASP found optimal hub locations (subject to the assumption that nodes must be assigned to the nearest hub) in over 90% of test problems. For problems for which such optima are not known, tabu search and GRASP generated new best-known solutions.     

A geneal vaiable neighbohood seach fo solving the uncapacitated $$$$-allocation $$p$$p-hub median poblem

The \(p\)-hub median problem consists of choosing \(p\) hub locations from a set of nodes with pairwise traffic demands in order to route the traffic between the origin-destination pairs at minimum cost. We accept general assumption that transportation between non-hub nodes is possible only via \(r\)-hub nodes, to which non-hub nodes are assigned. In this paper we propose a general variable neighborhood search heuristic to solve the problem in an efficient and effective way. Moreover, for the first time full nested variable neighborhood descent is applied as a local search within Variable neighborhood search. Computational results outperform the current state-of-the-art results obtained by GRASP based heuristic.     

A general variable neighborhood search for solving the uncapacitated single allocation p-hub median problem

We present a new general variable neighborhood search approach for the uncapacitated single allocation p-hub median problem in networks. This NP hard problem is concerned with locating hub facilities in order to minimize the traffic between all origin-destination pairs. We use three neighborhoods and efficiently update data structures for calculating new total flow in the network. In addition to the usual sequential strategy, a new nested strategy is proposed in designing a deterministic variable neighborhood descent local search. Our experimentation shows that general variable neighborhood search based heuristics outperform the best-known heuristics in terms of solution quality and computational effort. Moreover, we improve the best-known objective values for some large Australia Post and PlanetLab instances. Results with the new nested variable neighborhood descent show the best performance in solving very large test instances.     

General variable neighborhood search for the uncapacitated single allocation <Emphasis Type="Italic">p</Emphasis>-hub center problem

In this paper we propose a general variable neighborhood search heuristic for solving the uncapacitated single allocation p-hub center problem (USApHCP). For the local search step we develop a nested variable neighborhood descent strategy. The proposed approach is tested on benchmark instances from the literature and found to outperform the state-of-the-art heuristic based on ant colony optimization. We also test our heuristic on large scale instances that were not previously considered as test instances for the USApHCP. Moreover, exact solutions were reached by our GVNS for all instances where optimal solutions are known.     

A new distributionally robust p-hub median problem with uncertain carbon emissions and its tractable approximation method

The p-hub median problem is to determine the optimal location for p hubs and assign the remaining nodes to hubs so as to minimize the total transportation costs. Under the carbon cap-and-trade policy, we study this problem by addressing the uncertain carbon emissions from the transportation, where the probability distributions of the uncertain carbon emissions are only partially available. A novel distributionally robust optimization model with the ambiguous chance constraint is developed for the uncapacitated single allocation p-hub median problem. The proposed distributionally robust optimization problem is a semi-infinite chance-constrained optimization model, which is computationally intractable for general ambiguity sets. To solve this hard optimization model, we discuss the safe approximation to the ambiguous chance constraint in the following two types of ambiguity sets. The first ambiguity set includes the probability distributions with the bounded perturbations with zero means. In this case, we can turn the ambiguous chance constraint into its computable form based on tractable approximation method. The second ambiguity set is the family of Gaussian perturbations with partial knowledge of expectations and variances. Under this situation, we obtain the deterministic equivalent form of the ambiguous chance constraint. Finally, we validate the proposed optimization model via a case study from Southeast Asia and CAB data set. The numerical experiments indicate that the optimal solutions depend heavily on the distribution information of carbon emissions. In addition, the comparison with the classical robust optimization method shows that the proposed distributionally robust optimization method can avoid over-conservative solutions by incorporating partial probability distribution information. Compared with the stochastic optimization method, the proposed method pays a small price to depict the uncertainty of probability distribution. Compared with the deterministic model, the proposed method generates the new robust optimal solution under uncertain carbon emissions.     

New formulation and a branch-and-cut algorithm for the multiple allocation p-hub median problem

This article deals with the uncapacitated multiple allocation p-hub median problem, where p facilities (hubs) must be located among n available sites in order to minimize the transportation cost of sending a product between all pairs of sites. Each path between an origin and a destination can traverse any pair of hubs.     

Hub location–allocation in intermodal logistic networks

Within the context of intermodal logistics, the design of transportation networks becomes more complex than it is for single mode logistics. In an intermodal network, the respective modes are characterized by the transportation cost structure, modal connectivity, availability of transfer points and service time performance. These characteristics suggest the level of complexity involved in designing intermodal logistics networks. This research develops a mathematical model using the multiple-allocation p-hub median approach. The model encompasses the dynamics of individual modes of transportation through transportation costs, modal connectivity costs, and fixed location costs under service time requirements. A tabu search meta-heuristic is used to solve large size (100 node) problems. The solutions obtained using this meta-heuristic are compared with tight lower bounds developed using a Lagrangian relaxation approach. An experimental study evaluates the performance of the intermodal logistics networks and explores the effects and interactions of several factors on the design of intermodal hub networks subject to service time requirements.     

Exact and heuristic algorithms for the uncapacitated multiple allocation p-hub median problem

In this paper new MILP formulations for the multiple allocation p-hub median problem are presented. These require fewer variables and constraints than those traditionally used in the literature. An efficient heuristic algorithm, based on shortest paths, is described. LP based solution methods as well as an explicit enumeration algorithm are developed to obtain exact solutions. Computational results are presented for well known problems from the literature which show that exact solutions can be found in a reasonable amount of computational time. Our algorithms are also benchmarked on a different data set. This data set, which includes problems that are larger than those used in the literature, is based on a postal delivery network and has been treated by the authors in an earlier paper.     

Two genetic algorithms for solving the uncapacitated single allocation p-hub median problem

This paper deals with the Uncapacitated Single Allocation p-Hub Median Problem (USApHMP). Two genetic algorithm (GA) approaches are proposed for solving this NP-hard problem. New encoding schemes are implemented with appropriate objective functions. Both approaches keep the feasibility of individuals by using specific representation and modified genetic operators. The numerical experiments were carried out on the standard ORLIB hub data set. Both methods proved to be robust and efficient in solving USApHMP with up to 200 nodes and 20 hubs. The second GA approach achieves all previously known optimal solutions and achieves the best-known solutions on large-scale instances.     

The capacitated p-hub median problem with integral constraints: An application to a Chinese air cargo network

Consolidation at hubs in a pure hub-and-spoke network eliminates partial center-to-center direct loads, resulting in savings in transportation costs. In this research, we propose a general capacitated p-hub median model, with economies of scale and integral constraints on the paths. This model requires the selection of a specific p among a set of candidate hubs so that the total cost on the resulting pure capacitated hub-and-spoke network is minimized while simultaneously meeting origin–destination demands, operational capacity and singular path constraints. We explored the problem structure and developed a genetic algorithm using the path for encoding. This algorithm is capable of determining local optimality within less than 0.1% of the Lagrangian relaxation lower bounds on our Chinese air cargo network testing case and has reasonable computational times. The study showed that designating airports with high pickups or deliveries as hubs resulted in a high percentage of origin–destination pairs (ODs) in direct deliveries. Furthermore, the more hubs there are, the higher the direct share and the less likely for double rehandles. Sensitivity analysis on the discount rate showed that the economies of scale on trunk lines of hub-and-spoke networks may have a substantial impact on both the operating costs and the route patterns.     

The obnoxious p facility network location problem with facility interaction

Three heuristics are proposed to solve the maximin formulation for siting p facilities on a network considering a pollution dispersion equation and facility interaction. Initially, the single facility problem is approached by building up polygons which model pollution spread about the nodes of the network. This is extended in the first heuristic to the p facility problem. The second combines both the p-maximin and p-maxisum objectives in a lexicographic manner. The third is based on recent developments of Simulated Annealing. The proposed heuristics are evaluated for up to six facilities on a set of randomly generated networks having 20 to 40 nodes and low or medium arc density.     

A capacitated hub location problem in freight logistics multimodal networks

In this paper we deal with a capacitated hub location problem arising in a freight logistics context; in particular, we have the need of locating logistics platforms for containers travelling via road and rail. The problem is modelled on a weighed multimodal network. We give a mixed integer linear programming model for the problem, having the goal of minimizing the location and shipping costs. The proposed formulation presents some novel features for modelling capacity bounds that are given both for the candidate hub nodes and the arcs incident to them; further, the containerised origin-destination (\(o-d)\) demand can be split among several platforms and different travelling modes. Note that here the network is not fully connected and only one hub for each \(o-d\) pair is used, serving both to consolidate consignments on less transport connections and as reloading point for a modal change. Results of an extensive computational experimentation performed with randomly generated instances of different size and capacity values are reported. In the test bed designed to validate the proposed model all the instances up to 135 nodes and 20 candidate hubs are optimally solved in few seconds by the commercial solver CPLEX 12.5.     

Formulations and decomposition methods for the incomplete hub location network design problem with and without hop-constraints

The incomplete hub location problem with and without hop-constraints is modeled using a Leontief substitution system approach. The Leontief formalism provides a set of important theoretical properties and delivers formulations with tight linear bounds that can explicitly incorporate hop constraints for each origin-destination pair of demands. Furthermore, the proposed formulations are amenable to a Benders decomposition technique which can solve large scale test instances. The performance of the devised algorithm is primarily due to a new general scheme for separating Benders feasibility cuts. The novel cuts render a stabilizing effect that is directly responsible for the solution of instances up to 80 nodes.