A note on capacity models for network design

In network design problems,capacity constraints are modeled in three different ways depending on the application: directed, bidirected and undirected. In the literature, polyhedral investigations for strengthening mixed-integer formulations are done separately for each model. In this note, we examine the relationship between these models to provide a unifying approach and show that one can indeed translate valid inequalities from one to the others.     

Urban rapid transit network capacity expansion

This paper examines a multi-period capacity expansion problem for rapid transit network design. The capacity expansion is realized through the location of train alignments and stations in an urban traffic context by selecting the time periods. The model maximizes the public transportation demand using a limited budget and designing lines for each period. The location problem incorporates the user decisions about mode and route. The network capacity expansion is a long-term planning problem because the network is built over several periods, in which the data (demand, resource price, etc.) are changing like the real problem changes. This complex problem cannot be solved by branch and bound, and for this reason, a heuristic approach has been defined in order to solve it. Both methods have been experimented in test networks.     

A capacity scaling heuristic for the multicommodity capacitated network design problem

In this paper, we propose a capacity scaling heuristic using a column generation and row generation technique to address the multicommodity capacitated network design problem. The capacity scaling heuristic is an approximate iterative solution method for capacitated network problems based on changing arc capacities, which depend on flow volumes on the arcs. By combining a column and row generation technique and a strong formulation including forcing constraints, this heuristic derives high quality results, and computational effort can be reduced considerably. The capacity scaling heuristic offers one of the best current results among approximate solution algorithms designed to address the multicommodity capacitated network design problem.     

A review of urban transportation network design problems

This paper presents a comprehensive review of the definitions, classifications, objectives, constraints, network topology decision variables, and solution methods of the Urban Transportation Network Design Problem (UTNDP), which includes both the Road Network Design Problem (RNDP) and the Public Transit Network Design Problem (PTNDP). The current trends and gaps in each class of the problem are discussed and future directions in terms of both modeling and solution approaches are given. This review intends to provide a bigger picture of transportation network design problems, allow comparisons of formulation approaches and solution methods of different problems in various classes of UTNDP, and encourage cross-fertilization between the RNDP and PTNDP research.     

Multicommodity network design with survivability constraints: Some models and algorithms

We survey the main results presented by the author in his PhD thesis, supervised by F. Malucelli, and defended on the 15th March 2003. The thesis is written in English and is available on the Web page http: //www.elet.polimi.it/upload/belotti/thesis.pdf.gz. We investigate three problems, arising in the field of Telecommunication, of networks design with survivability constraints, and solve them through different approaches on a number of real-world network topologies with up to 40 nodes.Received: April 2004MSC classification: 90B10, 90C57     

Service network design with management and coordination of multiple fleets

We present a new optimization model for the tactical design of scheduled service networks for transportation systems where several entities provide service and internal exchanges and coordination with neighboring systems is critical. Internal exchanges represent border crossings necessitating changes of vehicles, while the coordination with neighboring systems represents intermodal operations. For a given demand, the model determines departure times of the services such that throughput time of the demand in the system is minimized. The model is an extension of the design-balanced capacitated multicommodity network design model that we denote service network design with asset management and multiple fleet coordination to emphasize the explicit modeling of different vehicle fleets. Data from a real-world problem addressing the planning of new rail freight services across borders serves to illustrate the capabilities of the formulation. We analyze how synchronization with collaborating services and removal of border-crossing operations impact the throughput time for the freight. We identify a significant potential for system performance enhancement from synchronization among collaborating services for the problem studied.     

Optimizing capacity,pricing and location decisions on a congested network with balking

In this paper, we consider the problem of making simultaneous decisions on the location, service rate (capacity) and the price of providing service for facilities on a network. We assume that the demand for service from each node of the network follows a Poisson process. The demand is assumed to depend on both price and distance. All facilities are assumed to charge the same price and customers wishing to obtain service choose a facility according to a Multinomial Logit function. Upon arrival to a facility, customers may join the system after observing the number of people in the queue. Service time at each facility is assumed to be exponentially distributed. We first present several structural results. Then, we propose an algorithm to obtain the optimal service rate and an approximate optimal price at each facility. We also develop a heuristic algorithm to find the locations of the facilities based on the tabu search method. We demonstrate the efficiency of the algorithms numerically.     

An extension to rapid transit network design problem

The rapid transit network design problem consists of the location of train alignments and stations, in a context where the demand makes its own decisions about the mode and route. The originality of this study is to incorporate in the model the line locations constraints with a bounded but variable number of lines, and lines with no predetermined origins and destinations. The computational experiments show the necessity of this extension to solve large networks, principally because of its computational advantage. The project has been supported by Ministerio de Educación y Ciencia (Spain) under project TRA-2005-09068-C03-01/MODAL, and by Ministerio de Fomento (Spain) under project 2005/70029/T05.     

Survivable network design with shared-protection routing

In this paper we study the problem of designing a survivable telecommunication network with shared-protection routing. We develop a heuristic algorithm to solve this problem. Recent results in the area of global re-routing have been used to obtain very tight lower bounds for the problem. Our results indicate that in a majority of problem instances, the average gap between the heuristic solutions and the lower bounds is within 5%. Computational experience is reported on randomly generated problem instances with up to 35 nodes, 80 edges and 595 demand pairs and also on the instances available in SNDlib database.     

A Benders decomposition approach for a distribution network design problem with consolidation and capacity considerations

We develop a model for a strategic freight-forwarding network design problem in which the design decisions involve the locations and capacities of consolidation and deconsolidation centers, and capacities on linehaul linkages as well as the shipment routes from origins to destinations through centers. We devise a solution approach based on Benders decomposition and conduct a computational study that illustrates the efficiency and the effectiveness of the approach.     

Single-commodity network design with random edge capacities

This paper examines the single-commodity network design problem with stochastic edge capacities. We characterize the structures of the optimal designs and compare with the deterministic counterparts. We do this partly to understand what constitutes robust network designs, but also to construct a heuristic for the stochastic problem, leading to optimality gaps of about 10%. In our view, that is a rather good result for problems that otherwise cannot be solved at all.     

A profit-maximizing supply chain network design model with demand choice flexibility

We present a profit-maximizing supply chain design model in which a company has flexibility in determining which customers to serve. The company may lose a customer to competition if the price it charges is too high. We show the problem formulation and solution algorithm, and discuss computational results.     

Enhanced formulations and branch-and-cut for the two level network design problem with transition facilities

We consider a new combinatorial optimization problem that combines network design and facility location aspects. Given a graph with two types of customers and two technologies that can be installed on the edges, the objective is to find a minimum cost subtree connecting all customers while the primary customers are served by a primary subtree that is embedded into the secondary subtree. In addition, besides fixed link installation costs, facility opening costs, associated to each node where primary and secondary subtree connect, have to be paid. The problem is called the Two Level Network Design Problem with Transition Facilities (TLNDF).     

Capacity allocation,ordering, and pricing decisions in a supply chain with demand and supply uncertainties

This paper studies coordinated decisions in a decentralized supply chain that consists of one Original Equipment Manufacturer (OEM), one manufacturer, and one distributor, and possesses uncertainties at both demand and supply sides. These uncertainties emerge, respectively, from random demand the distributor faces and randomness of capacity with which the OEM processes the manufacturer’s outsourced quantity. Sharing supply and demand uncertainty information along the supply chain enables us to develop three models with different coordination efforts—the OEM and manufacturer coordination, the manufacturer and distributor coordination, and the OEM, manufacturer, and distributor coordination—and quantify the coordinated decisions in these three models. Our analysis of these coordination models suggests that coordinating with the OEM improves the manufacturer’s probability of meeting downstream demand and his expected profit, yet coordinating with the manufacturer is not necessarily beneficial to the OEM when downstream coordination is lacking.     

Development of a new approach for deterministic supply chain network design

This paper proposes a mixed integer linear programming model and solution algorithm for solving supply chain network design problems in deterministic, multi-commodity, single-period contexts. The strategic level of supply chain planning and tactical level planning of supply chain are aggregated to propose an integrated model. The model integrates location and capacity choices for suppliers, plants and warehouses selection, product range assignment and production flows. The open-or-close decisions for the facilities are binary decision variables and the production and transportation flow decisions are continuous decision variables. Consequently, this problem is a binary mixed integer linear programming problem. In this paper, a modified version of Benders’ decomposition is proposed to solve the model. The most difficulty associated with the Benders’ decomposition is the solution of master problem, as in many real-life problems the model will be NP-hard and very time consuming. In the proposed procedure, the master problem will be developed using the surrogate constraints. We show that the main constraints of the master problem can be replaced by the strongest surrogate constraint. The generated problem with the strongest surrogate constraint is a valid relaxation of the main problem. Furthermore, a near-optimal initial solution is generated for a reduction in the number of iterations.     

The implications of costs, capacity, and competition on product line selection

We consider a situation in which a manufacturer has to select the product(s) to sell as well as the selling price and production quantity of each selected product. There are two substitutable products in the consideration set, where product 2 has a higher quality and reservation price than that of product 1. By considering the cannibalization effect that depends on the selling price of each product, the manufacturer needs to evaluate the profit function associated with three different product line options: sell both products or only one of the 2 products. In order to examine the impact of costs, capacity, and competition on the optimal product line selection, optimal price, and optimal production quantity analytically, we present a stylized model in this paper so that we can determine the conditions under which a particular option is optimal.     

Formulations and Benders decomposition algorithms for multidepot salesmen problems with load balancing

This paper describes new models and exact solution algorithms for the fixed destination multidepot salesmen problem defined on a graph with n nodes where the number of nodes each salesman is to visit is restricted to be in a predefined range. Such problems arise when the time to visit a node takes considerably longer as compared to the time of travel between nodes, in which case the number of nodes visited in a salesman’s tour is the determinant of their ‘load’. The new models are novel multicommodity flow formulations with O(n²) binary variables, which is contrary to the existing formulations for the same (and similar) problems that typically include O(n³) binary variables. The paper also describes Benders decomposition algorithms based on the new formulations for solving the problem exactly. Results of the computational experiments on instances derived from TSPLIB show that some of the proposed algorithms perform remarkably well in cases where formulations solved by a state-of-the-art optimization code fail to yield optimal solutions within reasonable computation time.     

Location coverage models with demand originating from nodes and paths: Application to cellular network design

Location covering problems, though well studied in the literature, typically consider only nodal (i.e. point) demand coverage. In contrast, we assume that demand occurs from both nodes and paths. We develop two separate models – one that handles the situation explicitly and one which handles it implicitly. The explicit model is formulated as a Quadratic Maximal Covering Location Problem – a greedy heuristic supported by simulated annealing (SA) that locates facilities in a paired fashion at each stage is developed for its solution. The implicit model focuses on systems with network structure – a heuristic algorithm based on geometrical concepts is developed. A set of computational experiments analyzes the performance of the algorithms, for both models. We show, through a case study for locating cellular base stations in Erie County, New York State, USA, how the model can be used for capturing demand from both stationary cell phone users as well as cell phone users who are in moving vehicles.     

Simpler analysis of LP extreme points for traveling salesman and survivable network design problems

We consider the Survivable Network Design Problem (SNDP) and the Symmetric Traveling Salesman Problem (STSP). We give simpler proofs of the existence of a -edge and 1-edge in any extreme point of the natural LP relaxations for the SNDP and STSP, respectively. We formulate a common generalization of both problems and show our results by a new counting argument. We also obtain a simpler proof of the existence of a -edge in any extreme point of the set-pair LP relaxation for the element connectivitySurvivable Network Design Problem ().     

An MPEC formulation and its cutting constraint algorithm for continuous network design problem with multi-user classes

Continuous network design problem (CNDP) is to determine the set of link capacity expansions and the corresponding equilibrium flows for which the measures of performance index for the network is optimal. Conventionally, CNDP assumed users to be homogeneous, that is, all travelers on the same link of the network are identical insofar as congestion effect and they have the same value of time (VOT). In fact, it does not accord with the real situation that all have the same VOT. So, multiple user classes with different VOT should be considered. This paper examines the CNDP with different VOT for multiple user classes, which is generally expressed as a mathematical programming with equilibrium constraint (MPEC). Then, the cut constraint algorithm (CCA) is presented to solve the problem. The numerical experiments on the examples from the literature are illustrated to demonstrate that our model and algorithm are feasible.