Thomas L. Magnanti

Network Design and Transportation Planning: Models and Algorithms

Numerous transportation applications as diverse as capital investment decision-making, vehicle fleet planning, and traffic light signal setting all involve some form of (discrete choice) network design. In this paper, we review some of the uses and limitations of integer programming-based approaches to network design, and describe several discrete and continuous choice models and algorithms. Our objectives are threefold---to provide a unifying view for synthesizing many network design models, to propose a unifying framework for deriving many network design algorithms, and to summarize computational experience in solving design problems. We also show that many of the most celebrated combinatorial problems that arise in transportation planning are specializations and variations of a generic design model. Consequently, the network design concepts described in this paper have great potential application in a wide range of problem settings.

Accelerating Benders Decomposition: Algorithmic Enhancement and Model Selection Criteria

This paper proposes methodology for improving the performance of Benders decomposition when applied to mixed integer programs. It introduces a new technique for accelerating the convergence of the algorithm and theory for distinguishing “good” model formulations of a problem that has distinct but equivalent mixed integer programming representations. The acceleration technique is based upon selecting judiciously from the alternate optima of the Benders subproblem to generate strong or pareto-optimal cuts. This methodology also applies to a much broader class of optimization algorithms that includes Dantzig-Wolfe decomposition for linear and nonlinear programs and related “cutting plane” type algorithms that arise in resource directive and price decomposition. When specialized to network location problems, this cut generation technique leads to very efficient algorithms that exploit the underlying structure of these models. In discussing the “proper” formulation of mixed integer programs, we suggest criteria for comparing various mixed integer formulations of a problem and for choosing formulations that can provide stronger cuts for Benders decomposition. From this discussion intimate connections between the previously disparate viewpoints of strong Benders cuts and tight linear programming relaxations of integer programs emerge.

Implementing Vehicle Routing Algorithms

Heuristic programming algorithms frequently address large problems and require manipulation and operation on massive data sets. The algorithms can be improved by using efficient data structures. With this in mind, we consider heuristic algorithms for vehicle routing, comparing techniques of Clarke and Wright, Gillett and Miller, and Tyagi, and presenting modifications and extensions which permit problems involving hundreds of demand points to be solved in a matter of seconds. In addition, a multi-depot routing algorithm is developed. The results are illustrated with a routing study for an urban newspaper with an evening circulation exceeding 100,000.

A Dual-Ascent Procedure for Large-Scale Uncapacitated Network Design

The fixed-charge network design problem arises in a variety of problem contexts including transportation, communication, and production scheduling. We develop a family of dual-ascent algorithms for this problem. This approach generalizes known ascent procedures for solving shortest path, plant location, Steiner network and directed spanning tree problems. Our computational results for several classes of test problems with up to 500 integer and 1.98 million continuous variables and constraints show that the dual-ascent procedure and an associated drop-add heuristic generate solutions that, in almost all cases, are guaranteed to be within 1 to 4% of optimality. Moreover, the procedure requires no more than 150 seconds on an IBM 3083 computer. The test problems correspond to dense and sparse networks, including some models that arise in freight transport.

Equilibria on a Congested Transportation Network

Network equilibrium models arise in applied contexts as varied as urban transportation, energy distribution, spatially separated economic markets, electrical networks and water resource planning. In this paper, we propose and study an equilibrium model for one of these applications, namely for predicting traffic flow on a congested transportation network. The model is quite similar to those that arise in most contexts of network equilibria, however, and the methods that we use are applicable in these other settings as well.Our transportation model includes such features as (i) multiple modes of transit, (ii) link interactions and their effect on congestion, (iii) limited choices (or perceptions) of paths for flow between any origin-destination pair, (iv) generalized cost or disutility for travel, and (v) demand relationships for travel between origin-destination pairs that depend upon the travel time (cost) between all other origin-destination pairs. Using Brouwer’s fixed-point theorem, we establish exist...

Modeling and Solving the Two-Facility Capacitated Network Loading Problem

This paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry. In this problem, given point-to-point communication demand in a network must be met by installing loading capacitated facilities on the arcs: Loading a facility incurs an arc specific and facility dependent cost. This paper develops modeling and solution approaches for loading facilities to satisfy the given demand at minimum cost. We consider two approaches for solving the underlying mixed integer program: a Lagrangian relaxation strategy, and a cutting plane approach that uses three classes of valid inequalities that we identify for the problem. We show that a linear programming formulation that includes these inequalities always approximates the value of the mixed integer program at least as well as the Lagrangian relaxation bound. Our computational results on a set of prototypical telecommunication data show that including these inequalities considerably improves the gap between the integer programming formulation and its linear programming relaxation: from an average of 25% to an average of 8%. These results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry.

The convex hull of two core capacitated network design problems

The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed point-to-point demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs at a specified arc dependent cost. The problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost.This paper studies two core subproblems of the NLP. The first problem, motivated by a Lagrangian relaxation approach for solving the problem, considers a multiple commodity, single arc capacitated network design problem. The second problem is a three node network; this specialized network arises in larger networks if we aggregate nodes. In both cases, we develop families of facets and completely characterize the convex hull of feasible solutions to the integer programming formulation of the problems. These results in turn strengthen the formulation of the NLP.

Combinatorial optimization and vehicle fleet planning : perspectives and prospects

As a well-structured and costly activity that pervades industries in both the public and private sector, vehicle fleet management would appear to be a splendid candidate for model-based planning and optimization. And yet, until recently the combinatorial intricacies of vehicle routing and of vehicle scheduling have precluded the widespread use of optimization (exact) methods for this problem class. Our discussion in this paper identifies the extent and nature of these problem complexities and draws contrasts with other applications of combinatorial optimization. It also summarizes a number of successful uses of optimization for vehicle fleet planning and highlights potentially fruitful avenues for algorithmic development. In particular, we describe several alternative models and novel algorithms for the vehicle routing problem, show how various modeling approaches for this problem are intimately related, and illustrate the interplay between model formulations and the algorithms that they suggest. This discussion shows that prospects for applying exact methods, possibly in conjunction with heuristics, are far from fully realized and points to vehicle fleet planning as a tempting target of opportunity for further investigation.

A Comparison of Mixed-Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems

We study a generic minimization problem with separable nonconvex piecewise linear costs, showing that the linear programming (LP) relaxation of three textbook mixed-integer programming formulations each approximates the cost function by its lower convex envelope. We also show a relationship between this result and classical Lagrangian duality theory.

Models for planning capacity expansion in local access telecommunication networks

The rapid progress of communications technology has created new opportunities for modeling and optimizing the design of local telecommunication systems. The complexity, diversity, and continuous evolution of these networks pose several modeling challenges. In this paper, we present an overview of the local telephone network environment, and discuss possible modeling approaches. In particular, we (i) discuss the engineering characteristics of the network, and introduce terminology that is commonly used in the communications industry and literature; (ii) describe a general local access network planning model and framework, and motivate different possible modeling assumptions; (iii) summarize various existing planning models in the context of this framework; and (iv) describe some new modeling approaches. The discussion in this paper is directed both to researchers interested in modeling local telecommunications systems and to planners interested in using such models. Our goal is to present relevant aspects of the engineering environment for local access telecommunication networks, and to discuss the relationship between engineering issues and the formulation of economic decision models. We indicate how changes in the underlying switching and transmission technology affect the modeling of the local telephone network. We also review various planning issues and discuss possible optimization approaches for treating them.