The hierarchical network design problem

In this paper the authors introduce the hierarchical network design problem (HNDP). The object of the HNDP is to identify the least cost, two-level hierarchical network. The network must include a primary path from a predetermined starting node to a predetermined terminus node. In addition, each node not on the primary path must be connected to some node on that path by means of a secondary path. The problem is initially formulated as an integer linear program. An heuristic is then presented which employs a K shortest path algorithm, and a minimum spanning tree algorithm. Heuristic results of two sample problems are presented and compared to the results obtained by solving the integer LP formulation. Potential applications of the formulation are also discussed.     

The hierarchical network design problem with multiple primary paths

In this paper we develop a suboptimal solution for the hierarchical network design problem (HNDP) with multiple primary paths. In practice some networks may require solutions to the HNDP with more than one origin/destination pair to be connected by primary paths. This paper develops a dynamic programming solution to find a suboptimal solution to the problem.     

A branch and cut algorithm for the hierarchical network design problem

The Hierarchical Network Design Problem consists of locating a minimum cost bi-level network on a graph. The higher level sub-network is a path visiting two or more nodes. The lower level sub-network is a forest connecting the remaining nodes to the path. We optimally solve the problem using an ad hoc branch and cut procedure. Relaxed versions of a base model are solved using an optimization package and, if binary variables have fractional values or if some of the relaxed constraints are violated in the solution, cutting planes are added. Once no more cuts can be added, branch and bound is used. The method for finding valid cutting planes is presented. Finally, we use different available test instances to compare the procedure with the best known published optimal procedure, with good results. In none of the instances we needed to apply branch and bound, but only the cutting planes.     

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.     

An improved Benders decomposition applied to a multi-layer network design problem

Benders decomposition has been widely used for solving network design problems. In this paper, we use a branch-and-cut algorithm to improve the separation procedure of Gabrel et al. and Knippel et al. for capacitated network design. We detail experiments on bi-layer networks, comparing with Knippel’s previous results.     

An approximation algorithm for the k-median warehouse-retailer network design problem

We study the generalizedk-median version of the warehouse-retailer network design problem(kWRND).We formulate the k-WRND as a binary integer program and propose a 6-approximation randomized algorithm based on Lagrangian relaxation.     

An efficient approximation algorithm for the survivable network design problem

The survivable network design problem (SNDP) is to construct a minimum-cost subgraph satisfying certain given edge-connectivity requirements. The first polynomial-time approximation algorithm was given by Williamson et al. (Combinatorica 15 (1995) 435–454). This paper gives an improved version that is more efficient. Consider a graph ofn vertices and connectivity requirements that are at mostk. Both algorithms find a solution that is within a factor 2k – 1 of optimal fork 2 and a factor 2 of optimal fork = 1. Our algorithm improves the time from O(k ³n⁴) to O ). Our algorithm shares features with those of Williamson et al. (Combinatorica 15 (1995) 435–454) but also differs from it at a high level, necessitating a different analysis of correctness and accuracy; our analysis is based on a combinatorial characterization of the redundant edges. Several other ideas are introduced to gain efficiency. These include a generalization of Padberg and Rao's characterization of minimum odd cuts, use of a representation of all minimum (s, t) cuts in a network, and a new priority queue system. The latter also improves the efficiency of the approximation algorithm of Goemans and Williamson (SIAM Journal on Computing 24 (1995) 296–317) for constrained forest problems such as minimum-weight matching, generalized Steiner trees and others. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.A preliminary version of this paper has appeared in the Proceedings of the Third Mathematical Programming Society Conference on Integer Programming and Combinatorial Optimization, 1993, pp. 57–74.Research supported in part by NSF Grant No. CCR-9215199 and AT & T Bell Laboratories.Research supported in part by Air Force contracts AFOSR-89-0271 and F49620-92-J-0125 and DARPA contracts N00014-89-J-1988 and N00014-92-1799.This research was performed while the author was a graduate student at MIT. Research supported by an NSF Graduate Fellowship, Air Force contract F49620-92-J-0125, DARPA contracts N00014-89-J-1988 and N00014-92-J-1799, and AT & T Bell Laboratories.     

A branch-and-price approach to the feeder network design problem

In this paper we consider the problem of designing a container liner shipping feeder network. The designer has to choose which port to serve during many rotations that start and end at a central hub. Many operational characteristics are considered, such as variable leg-by-leg speeds and cargo transit times. Realistic instances are generated from the LinerLib benchmark suite. The problem is solved with a branch-and-price algorithm, which can solve most instances to optimality within one hour. The results also provide insights on the cost structure and desirable features of optimal routes. These insights were obtained by means of an analysis where scenarios are generated varying internal and external conditions, such as fuel costs and port demands.     

An approximation algorithm for the hierarchical median problem

The hierarchical median problem asks for a hierarchical sequence of solutions to the k-median problems of growing cardinality. The best algorithm known for this problem in the general metric case has competitive ratio 20.71. In the paper, the case is under study that the clients and facilities lie on the real line, as well as the case of a Euclidean space. An algorithm is proposed with competitive ratio 8 in the case of the real line, and 8 + 4√2 (approximately 13.66), in the Euclidean case.     

The multi-layered network design problem

We address the problem of designing a network built on several layers. This problem occurs in practical applications but has not been studied extensively from the point of view of global optimisation, since the problem of designing a single-layered network is complex. An example of an application is the design of a virtual network (Internet Protocol) built on a sparse optical transport network.     

Algorithms for the non-bifurcated network design problem

In this paper we consider the non-bifurcated network design problem where a given set of cities must be connected by installing on a given set of links integer multiples of some base capacity so that a set of commodity demands can be routed. Each commodity flow is constrained to run through a single path along the network. The objective is to minimize the sum of capacity installation and routing costs. We present an exact algorithm and four new heuristics. The first heuristic generates an initial feasible solution, then it improves it until a necessary condition for optimality is satisfied. Two heuristics are partial enumeration methods and the last one iteratively applies a tabu search method to different initial feasible solutions. Computational results over a set of test problems from the literature show the effectiveness of the proposed algorithms.     

An exact algorithm for a cross-docking supply chain network design problem

This paper proposes a branch-and-price algorithm as an exact algorithm for the cross-docking supply chain network design problem introduced by one of the authors of this paper. The objective is to optimally locate cross-docking (CD) centres and allocate vehicles for direct transportation services from the associated origin node to the associated CD centre or from the associated CD centre to the associated destination node so as to satisfy a given set of freight demands at minimum cost subject to the associated service (delivery) time restriction. A set-partitioning-based formulation is derived for the problem for which some solution properties are characterized. Based on the properties, a branch-and-price algorithm is derived. The properties can also be used in deriving any efficient local search heuristics with the move operation (neighbourhood search operation) of modifying assignment of some freight demands from current CD centres to other CD centres. Computational experiments show that the branch-and-price algorithm is effective and efficient and also that the solution properties contribute to improve the efficiency of the local search heuristics.     

An algorithm to solve the proportional network flow problem

The proportional network flow problem is a generalization of the equal flow problem on a generalized network in which the flow on arcs in given sets must all be proportional. This problem appears in several natural contexts, including processing networks and manufacturing networks. This paper describes a transformation on the underlying network that reduces the problem to the equal flow problem; this transformation is used to show that algorithms that solve the equal flow problem can be directly applied to the proportional network flow problem as well, with no increase in asymptotic running time. Additionally, computational results are presented for the proportional network flow problem demonstrating equivalent performance to the same algorithm for the equal flow problem.     

An alternate approach to solve two-level hierarchical time minimization transportation problem

This paper discusses a two-level hierarchical time minimization transportation problem, which is an important class of transportation problems arising in industries. This problem has been studied by various researchers (Sharma et al. in Eur J Oper Res 246:700–707, 2015; Sonia and Puri in TOP 12(2):301–330, 2004; Xie et al. in Comput Oper Res 86:124–139, 2017) and therefore, a number of polynomial time iterative algorithms are available to find its solution. All the existing algorithms, though efficient, have some shortcomings. The current study proposes an alternate solution algorithm for the problem that is more efficient in terms of computational time than the existing algorithms. The results justifying the underlying theory of the proposed algorithm are given. Further, a detailed comparison of the computational behaviour of all the algorithms for randomly generated instances of this problem, of different sizes validates the efficiency of the proposed algorithm.

     

A dual ascent approach to the fixed-charge capacitated network design problem

In this paper we consider the problem of constructing a network over which a number of commodities are to be transported. Fixed costs are associated to the construction of network arcs and variable costs are associated to routing of commodities. In addition, one capacity constraint is related to each arc. The problem is to determine a network design that minimizes the total cost; i.e., it balances the construction and operating costs. A dual ascent procedure for finding improved lower bounds and near-optimal solutions for the fixed-charge capacitated network design problem is proposed. The method is shown to generate tighter lower bounds than the linear programming relaxation of the problem.     

A note on the subadditive network design problem

We study approximation algorithms for generalized network design where the cost of an edge depends on the identities of the demands using it (as a monotone subadditive function). Our main result is that even a very special case of this problem cannot be approximated to within a factor 2^log1−ε|D| if D is the set of demands.