Benders Decomposition for Profit Maximizing Hub Location Problems with Capacity Allocation

This paper models capacity allocation decisions within profit maximizing hub location problems to satisfy demand of commodities from different market segments. A strong deterministic formulation of the problem is presented and two exact algorithms based on a Benders reformulation are described to solve large-size instances of the problem. A new methodology is developed to strengthen the Benders optimality cuts by decomposing the subproblem in a two-phase fashion. The algorithms are enhanced by the integration of improved variable fixing techniques. The deterministic model is further extended by considering uncertainty associated with the demand to develop a two-stage stochastic program. To solve the stochastic version, a Monte-Carlo simulation-based algorithm is developed that integrates a sample average approximation scheme with the proposed Benders decomposition algorithms. Novel acceleration techniques are presented to improve the convergence of the algorithms proposed for the stochastic version. The efficiency and robustness of the algorithms are evaluated through extensive computational experiments. Computational results show that large-scale instances with up to 500 nodes and three demand segments can be solved to optimality, and that the proposed algorithms generate cuts that provide significant speedups compared to using Pareto-optimal cuts. The proposed two-phase methodology for solving the Benders subproblem as well as the variable fixing and acceleration techniques can be used to solve other discrete location and network design problems

Hub Location Problems with Profit Considerations

This thesis studies profit maximizing hub location problems. These problems seek to find the optimal number and locations of hubs, allocations of demand nodes to these hubs, and routes of flows through the network to serve a given set of demands between origin-destination pairs while maximizing total profit. Taking revenue into consideration, it is assumed that a portion of the demand can remain unserved when it is not profitable to be served. Potential applications of these problems arise in the design of airline passenger and freight transportation networks, truckload and less-than-truckload transportation, and express shipment and postal delivery. Firstly, mathematical formulations for different versions of profit maximizing hub location problems are developed. Alternative allocation strategies are modeled including multiple allocation, single allocation, and $r$-allocation, as well as allowing for the possibility of direct connections between non-hub nodes. Extensive computational analyses are performed to compare the resulting hub networks under different models, and also to evaluate the solution potential of the proposed models on commercial solvers with emphasis on the effect of the choice of parameters. Secondly, revenue management decisions are incorporated into the profit maximizing hub location problems by considering capacities of hubs. In this setting, the demand of commodities are segmented into different classes and there is available capacity at hubs which is to be allocated to these different demand segments. The decision maker needs to determine the proportion of each class of demand to serve between origin-destination pairs based on the profit to be obtained from satisfying this demand. A strong mixed-integer programming formulation of the problem is presented and Benders-based algorithms are proposed to optimally solve large-scale instances of the problem. A new methodology is developed to strengthen the Benders optimality cuts by decomposing the subproblem in a two-phase fashion. The algorithms are enhanced by the integration of improved variable fixing techniques. Computational results show that large-scale instances with up to 500 nodes and 750,000 commodities of different demand segments can be solved to optimality, and that the proposed algorithms generate cuts that provide significant speedups compared to using Pareto-optimal cuts. As precise information on demand may not be known in advance, demand uncertainty is then incorporated into the profit maximizing hub location problems with capacity allocation, and a two-stage stochastic program is developed. The first stage decision is the locations of hubs, while the assignment of demand nodes to hubs, optimal routes of flows, and capacity allocation decisions are made in the second stage. A Monte-Carlo simulation-based algorithm is developed that integrates a sample average approximation scheme with the proposed Benders decomposition algorithm. Novel acceleration techniques are presented to improve the convergence of the algorithm. The efficiency and robustness of the algorithm are evaluated through extensive computational experiments. Instances with up to 75 nodes and 16,875 commodities are optimally solved, which is the largest set of instances that have been solved exactly to date for any type of stochastic hub location problems. Lastly, robust-stochastic models are developed in which two different types of uncertainty including stochastic demand and uncertain revenue are simultaneously incorporated into the capacitated problem. To embed uncertain revenues into the problem, robust optimization techniques are employed and two particular cases are investigated: interval uncertainty with a max-min criterion and discrete scenarios with a min-max regret objective. Mixed integer programming formulations for each of these cases are presented and Benders-based algorithms coupled with sample average approximation scheme are developed. Inspired by the repetitive nature of sample average approximation scheme, general techniques for accelerating the algorithms are proposed and instances involving up to 75 nodes and 16,875 commodities are solved to optimality. The effects of uncertainty on optimal hub network designs are investigated and the quality of the solutions obtained from different modeling approaches are compared under various parameter settings. Computational results justify the need for embedding both sources of uncertainty in decision making to provide robust solutions

Hub Network Design Problem with Capacity, Congestion and Stochastic Demand Considerations

Our study introduces the hub network design problem with congestion, capacity, and stochastic demand considerations (HNDC), which generalizes the classical hub location problem in several directions. In particular, we extend state-of-the-art by integrating capacity acquisition decisions and congestion cost effect into the problem and allowing dynamic routing for origin-destination pairs. Connecting strategic and operational level decisions, HNDC jointly decides hub locations and capacity acquisitions by considering the expected routing and congestion costs. A path-based mixed-integer second-order cone programming (SOCP) formulation of the HNDC is proposed. We exploit SOCP duality results and propose an exact algorithm based on Benders decomposition and column generation to solve this challenging problem. We use a specific characterization of the capacity-feasible solutions to speed up the solution procedure and develop an efficient branch-and-cut algorithm to solve the master problem. We conduct extensive computational experiments to test the proposed approach’s performance and derive managerial insights based on realistic problem instances adapted from the literature. In particular, we found that including hub congestion costs, accounting for the uncertainty in demand, and whether the underlying network is complete or incomplete have a significant impact on hub network design and the resulting performance of the system

Lagrangian Relaxation for q-Hub Arc Location Problems

The topic of this Master thesis is an in-depth research study on a specific type of network systems known as hub-and-spoke networks. In particular, we study q-Hub Arc Location Problems that consist, at a strategical level, of selecting q hub arcs and at most p hub nodes, and of the routing of commodities through the so called hub level network. We propose strong formulations to two variants of the problem, namely the q-hub arc location problem and the $q$-hub arc location problem with isolated hub nodes. We present a Lagrangian relaxation that exploits the structure of these problems by decomposing them into |K|+2 independent easy-to-solve subproblems and develop Lagrangian heuristics that yield high quality feasible solutions to both models. We, further, provide some insights on the structure of the optimal solutions to both models and investigate the cost benefit of incomplete hub networks with and without isolated hub nodes. Finally, computational results on a set of benchmark instances with up to 100 nodes are reported to assess the performance of the proposed MIP formulations and of our algorithmic approach

A new bi-objective model of the urban public transportation hub network design under uncertainty

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Relay Network Design in Logistics and Telecommunications: Models and Solution Approaches

Strategic network design has significant impacts on the operational performance of transportation and telecommunications industries. The corresponding networks are typically characterized by a multicommodity ow structure where a commodity is defined by a unique origin-destination pair and an associated amount of ow. In turn, multicommodity network design and hub location models are commonly employed when designing strategic networks in transportation and telecommunications applications. In this dissertation, these two modeling approaches are integrated and generalized to address important requirements in network design for truckload transportation and long-distance telecommunications networks. To this end, we first introduce a cost effective relay network design model and then extend this base model to address the specific characteristics of these applications. The base model determines relay point (RP) locations where the commodities are relayed from their origins to destinations. In doing this, we explicitly consider distance constraints for the RP-RP and nonRPRP linkages. In truckload transportation, a relay network (RP-network) can be utilized to decrease drivers' driving distances and keep them within their domiciles. This can potentially help alleviate the high driver turnover problem. In this case, the percentage circuitry, load-imbalance, and link-imbalance constraints are incorporated into the base model to control related performance metrics that are affected by the distance constraints. When compared to the networks from other modeling approaches, the RP-network is more effective in controlling drivers' tour lengths and capable of controlling the empty mileage to low levels without adding a large amount of additional travel distance. In telecommunications, an RP-network can be beneficial in long-distance data transfers where the signals' delity must be improved/regenerated at RPs along their travel paths. For this setting, we extend the base model to include fixed link setup costs and capacities. From our computational results, our models provide better network configuration that is cost effective and facilitates a better service quality (shorter delays and better connectivity). Concerning methodology, we develop effcient exact solution algorithms based on Benders decomposition, Lagrangean decomposition, and Lagrangean relaxation. The performance of the typical solution frameworks are enhanced via numerous accelerating techniques to allow the solution of large-sized instances in reduced solution times. The accelerating techniques and solution approaches are transferable to other network design problem settings with similar characteristics

Stochastic hub and spoke networks

Transportation systems such as mail, freight, passenger and even telecommunication systems most often employ a hub and spoke network structure since correctly designed they give a strong balance between high service quality and low costs resulting in an economically competitive operation. In addition, consumers are increasingly demanding fast and reliable transportation services, with services such as next day deliveries and fast business and pleasure trips becoming highly sought after. This makes finding an efficient design of a hub and spoke network of the utmost importance for any competing transportation company. However real life situations are complicated, dynamic and often require responses to many different fixed and random events. Therefore modeling the question of what is an optimal hub and spoke network structure and finding an optimal solution is very difficult. Due to this, many researchers and practitioners alike make several assumptions and simplifications on the behavior of such systems to allow mathematical models to be formulated and solved optimally or near optimally within a practical timeframe. Some assumptions and simplifications can however result in practically poor network design solutions being found. This thesis contributes to the research of hub and spoke networks by introducing new stochastic models and fast solution algorithms to help bridge the gap between theoretical solutions and designs that are useful in practice. Three main contributions are made in the thesis. First, in Chapter 2, a new formulation and solution algorithms are proposed to find exact solutions to a stochastic p-hub center problem. The stochastic p-hub center problem is about finding a network structure, where travel times on links are stochastic, which minimizes the longest path in the network to give fast delivery guarantees which will hold for some given probability. Second, in Chapter 3, the stochastic p-hub center problem is looked at using a new methodological approach which gives more realistic solutions to the network structures when applied to real life situations. In addition a new service model is proposed where volume of flow is also accounted for when considering the stochastic nature of travel times on links. Third, in Chapter 4, stochastic volume is considered to account for capacity constraints at hubs and, de facto, reduce the costs embedded in excessive hub volumes. Numerical experiments and results are conducted and reported for all models in all chapters which demonstrate the efficiency of the new proposed approaches

Hub Network Design and Discrete Location: Economies of Scale, Reliability and Service Level Considerations

In this thesis, we study three related decision problems in location theory. The first part of the dissertation presents solution algorithms for the cycle hub location problem (CHLP), which seeks to locate p-hub facilities that are connected by means of a cycle, and to assign non-hub nodes to hubs so as to minimize the total cost of routing flows through the network. This problem is useful in modeling applications in transportation and telecommunications systems, where large setup costs on the links and reliability requirements make cycle topologies a prominent network architecture. We present a branch and-cut algorithm that uses a flow-based formulation and two families of mixed-dicut inequalities as a lower bounding procedure at nodes of the enumeration tree. We also introduce a greedy randomized adaptive search algorithm that is used to obtain initial upper bounds for the exact algorithm and to obtain feasible solutions for large-scale instances of the CHLP. Numerical results on a set of benchmark instances with up to 100 nodes confirm the efficiency of the proposed solution algorithms. In the second part of this dissertation, we study the modular hub location problem, which explicitly models the flow-dependent transportation costs using modular arc costs. It neither assumes a full interconnection between hub nodes nor a particular topological structure, instead it considers link activation decisions as part of the design. We propose a branch-and-bound algorithm that uses a Lagrangean relaxation to obtain lower and upper bounds at the nodes of the enumeration tree. Numerical results are reported for benchmark instances with up to 75 nodes. In the last part of this dissertation we study the dynamic facility location problem with service level constraints (DFLPSL). The DFLPSL seeks to locate a set of facilities with sufficient capacities over a planning horizon to serve customers at minimum cost while a service level requirement is met. This problem captures two important sources of stochasticity in facility location by considering known probability distribution functions associated with processing and routing times. We present a nonlinear mixed integer programming formulation and provide feasible solutions using two heuristic approaches. We present the results of computational experiments to analyze the impact and potential benefits of explicitly considering service level constraints when designing distribution systems