Thomas L. Magnanti's research while affiliated with Massachusetts Institute of Technology and other places
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Publications (126)
Optimization has been one of the most fundamental and extensive contributions of management science/operations research, with an enormous number of contributions and subfields developed by many researchers and practitioners. When the journal Management Science launched in 1954, little was known about optimization, including some results in nonlinea...
Many practical applications of network design, particularly in transportation and logistics, require designing a cost-effective network configuration to meet all demand at total fixed and flow costs, subject to additional constraints on routing decisions to ensure good end-to-end service performance. For instance, in settings such as package delive...
We study a new class of scheduling problems that capture common settings in service environments, in which one has to serve a collection of jobs that have a priori uncertain attributes (e.g., processing times and priorities) and the service provider has to decide how to dynamically allocate resources (e.g., people, equipment, and time) between test...
We discuss an allocation mechanism of capstone projects to senior-year undergraduate students, which the recently established Singapore University of Technology and Design (SUTD) has implemented. A distinguishing feature of these projects is that they are multidisciplinary ; each project must involve students from at least two disciplines. This is...
The maximum flow problem seeks the maximum possible flow in a capacitated network from
a specified source node s to a specified sink node t without exceeding the capacity of any arc.
A closely related problem is the minimum cut problem, which is to find a set of arcs with
the smallest total capacity whose removal separates nodes s and t. The maximu...
The Stochastic User Equilibrium (SUE) model predicts traffic equilibrium flow assuming that users choose their perceived maximum utility paths (or perceived shortest paths) while accounting for the effects of congestion that arise due to users sharing links. Inspired by recent work on distributionally robust optimization, specifically a Cross Momen...
In this article, we focus on relatively new maintenance and operational scheduling challenges that are faced by the United States Air Force concerning low-observable (LO) or stealth aircraft. The LO capabilities of an aircraft degrade stochastically as it flies, making it difficult to make maintenance scheduling decisions. Maintainers can address t...
We study new models of scheduled maintenance management for modular systems, consisting of multiple components with respective cycle limits. The cycle limit of each component specifies the time interval in which this component must be repaired or replaced. The goal is to compute a feasible maintenance schedule that minimizes the cost associated wit...
The vehicles of education have seen significant broadening with the proliferation of new technologies such as social media, microblogs, online references, multimedia, and interactive teaching tools. This paper summarizes research on the effect of using active learning methods to facilitate student learning and describes our experiences implementing...
We introduce an algorithm design technique for a class of combinatorial
optimization problems with concave costs. This technique yields a strongly
polynomial primal-dual algorithm for a concave cost problem whenever such an
algorithm exists for the fixed-charge counterpart of the problem. For many
practical concave cost problems, the fixed-charge c...
We show how to approximate a separable concave minimization problem over a general closed ground set by a single piecewise linear minimization problem. The approximation is to arbitrary 1+ε precision in optimal cost. For polyhedral ground sets in \(\mathbb{R}^n_+\) and nondecreasing cost functions, the number of pieces is polynomial in the input si...
This paper studies a core optimization model--the survivable network design (SND) problem--that incorporates cost and survivability. Given an undirected network G: (N,E) with nodes N, nonnegative costs for each edge in E, and nonnegative integer connectivity requirements specifying the minimum number of edge-disjoint paths needed between pairs of n...
In this paper we study the relationship between the linear programming relaxation of a pure time-dependent formulation (that can be seen as modified version of the well-known Picard and Queyranne formulation for the TSP [11]) and the linear programming relaxation of a well known single-commodity flow model due to Gavish and Graves [6]. In particula...
We consider two types of hop-indexed models for the unit-demand asymmetric Capacitated Vehicle Routing Problem (CVRP): (a) capacitated models guaranteeing that the number of commodities (paths) traversing any given arc does not exceed a specified capacity; and (b) hop-constrained models guaranteeing that any route length (number of nodes) does not...
Keywords
Applications
Distribution Problems
Airplane Hopping Problem
Directed Chinese Postman Problem
Preliminaries
Assumptions
Graph Notation
Residual Network
Order Notation
Cycle-Canceling Algorithm
Successive Shortest Path Algorithm
Network Simplex Algorithm
See also
References
Keywords
Applications
Capacity of Physical Networks
Feasible Flow Problem
Matrix Rounding Problem
Preliminaries
Residual Network
Flow across an s − t-Cut
Generic Augmenting Path Algorithm
Generic Preflow-Push Algorithm
Combinatorial Implications of the Max–Flow Min–Cut Theorem
Network Connectivity
Matchings and Covers
See also
References
The function need not be continuous; it can have positive or negative jumps, though we do assume that the function is lower semi-continuous, that is, g a (x a ) lim inf x # a #xa g a (x # a ) for 1 any sequence x # a that approaches x a . Without loss of generality, we also assume, through a simple translation of the costs if necessary, that g a (0...
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 pres...
"7-102-77." Includes author index. Cover title. Supported in part by the U.S. Deaprtment of Transportation, Transportation Advanced Research Program (TARP) contract no. DOT-TSC-1058 by Bruce L. Golden and Thomas L. Magnanti.
In a previous paper, Gouveia and Magnanti (2003) found diameter-constrained minimal spanning and Steiner tree problems to
be more difficult to solve when the tree diameter D is odd. In this paper, we provide an alternate modeling approach that views problems with odd diameters as the superposition
of two problems with even diameters. We show how to...
By adding a set of redundant constraints, and by iteratively refining the approximation, we show that a commercial solver is able to routinely solve moderate-size strategic safety stock placement problems to optimality. The speed-up arises because the solver automatically generates strong flow cover cuts using the redundant constraints.
This paper is an edited transcription of the author’s lecture given on February 25th, 2006, in honor of Saul Gass.
The network design problem with connectivity requirements (NDC) models a wide variety of celebrated combinatorial optimization problems including the minimum span- ning tree, Steiner tree, and survivable network design problems. We develop strong for- mulations for two versions of the edge-connectivity NDC problem: unitary problems re- quiring conn...
We introduce a general adaptive line search framework for solving fixed point and variational inequality problems. Our goals are to develop iterative schemes that (i) compute solutions when the underlying map satisfies properties weaker than contractiveness, for example, weaker forms of nonexpansiveness, (ii) are more efficient than the classical m...
In a previous article, using underlying graph theoretical properties, Gouveia and Magnanti (2003) described several network flow-based formulations for diameter-constrained tree problems. Their computational results showed that, even with several enhancements, models for situations when the tree diameter D is odd proved to be more difficu...
This paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry. In this problem, given point-topoint demand between various pairs of nodes of a network must be met by installing (loading) capacitated facilities on the arcs. The facilities are chosen from a small set of alt...
Within the extensive variational inequality literature, researchers have developed many algorithms. Depending upon the problem setting, these algorithms ensure the convergence of (i) the entire sequence of iterates, (ii) a subsequence of the iterates, or (iii) averages of the iterates. To establish these convergence results, the literature repeated...
We develop and study averaging schemes for solving fixed point and variational inequality problems. Typically, researchers have established convergence results for solution methods for these problems by establishing contractive estimates for their algorithmic maps. In this paper, we establish global convergence results using nonexpansive estimates....
Survivability is becoming an increasingly important criterion in network design. This paper studies formulations, heuristic worst-case performance, and linear programming relaxations for two classes of survivable network design problems: the low connectivity Steiner (LCS) problem for graphs containing nodes with connectivity requirement of 0, 1, or...
Linear approximation and linear programming duality theory are used as unifying tools to develop saddlepoint, Fenchel and local duality theory. Among results presented is a new and elementary proof of the necessity and sufficiency of the stability condition for saddlepoint duality, an equivalence between the saddlepoint and Fenchel theories, and na...
In this paper, we establish global convergence results for projection and relaxation algorithms for solving variational inequality problems, and for the Frank-Wolfe algorithm for solving convex optimization problems defined over general convex sets. The analysis rests upon the condition of f-monotonicity,which we introduced in a previous paper, and...
Several problems in the theory of combinatorial geometries (or matroids) are solved by means of algorithms which involve the notion of "abstract pivots". The main example is the Edmonds-Fulkerson partition theorem, which is applied to prove a number of generalized exchange properties for bases. Supported in part by the U.S. Army Research Office (Du...
This paper studies a new multi-facility network synthesis problem, called the Multi-level Network Design (MLND) problem, that arises in the topological design of hierarchical communication, transportation, and electric power distribution networks whose nodes have varying levels of importance:the more critical or higher level nodes require higher gr...
We model Pup Matching, the logistics problem of matching or pairing semitrailers known as pups to cabs able to tow one or two pups simultaneously, as an NP-complete version of the Network Loading Problem (NLP). We examine a branch and bound solution approach tailored to the NLP formulation through the use of three families of cutting planes and fou...
We study a class of models, known as overlay optimization problems, with a "base" subproblem and an "overlay" subproblem, linked by the requirement that the overlay solution be contained in the base solution. In some telecommunication settings, a feasible base solution is a spanning tree and the overlay solution is an embedded Steiner tree (or an e...
We develop an optimization-based approach for a point-to-point route planning problem that arises in many large scale delivery systems(for example, less-than-truckload freight, rail, mail and package delivery, communications). In these settings, a firm which must ship goods between many origin and destination pairs on a network needs to specify a r...
We consider generalizations of the steepest descent algorithm for solving asymmetric systems of equations. We first show that if the system is linear and is defined by a matrix M, then the method converges if M2 is positive definite. We also establish easy to verify conditions on the matrix M that ensure that M is positive definite, and develop a s...
Given an undirected network with L possible facility types for each edge, and a partition of the nodes into L levels, the Multi-level Network Design (MLND) problem seeks a fixed cost minimizing design that spans all the nodes and connects the nodes at each level by facilities of the corresponding or higher type. This problem generalizes the well-kn...
In this paper, we consider algorithms, duality and sensitivity analysis for optimization problems, called fractional, whose objective function is the ratio of two real valued functions. We discuss a procedure suggested by Dinkelbach for solving the problem, its relationship to certain approaches via variable transformations, and a variant of the pr...
We study a specialized version of network design problems that arise in telecommunication, transportation and other industries. The problem, a generalization of the shortest path problem, is defined on an undirected network consisting of a set of arcs on which we can install (load), at a cost, a choice of up to three types of capacitated facilities...
The survivable network design (SND) problem seeks a minimum-cost robust network configuration that provides a specified number of alternate (edge-disjoint) paths between nodes of the network. For this problem, we present a family of new mixed-integer programming formulations whose associated linear programming relaxations can be tighter than that o...
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 theor...
The Diameter-Constrained Minimum Spanning Tree Problem seeks a least cost spanning tree subject to a (diameter) bound imposed on the number of edges in the tree between any node pair. A traditional multicommodity flow model with a commodity for every pair of nodes was unable to solve a 20-node and 100-edge problem after one week of computation. We...
We develop integer programming formulations and solution methods for addressing operational issues in merge-in-transit distribution systems. The models account for various complex problem features, including the integration of inventory and transportation decisions, the dynamic and multimodal components of the application, and the nonconvex piecewi...
The network restoration problem is a specialized capacitated network design problem requiring the installation of spare capacity to fully restore disrupted network flows if any edge in a telecommunications network fails. We present a new mixed-integer programming formulation for a line restoration version of the problem using a single type of capac...
We study a scheduling problem with changeover costs and capacity constraints. The problem is NP-complete, and combinatorial algorithms for solving it have not performed well. We identify a general class of facets that subsumed as special cases some known facets from the literature. We also develop a cutting-plane-based procedure and reformulation f...
gically ordered". page 80, line-15. "set of path and cycle flow" should be "set of path and cycle flows". page 80, line-15. "cycle fow" should be "set of path and cycle flows". page 80, line-7. "path and cycle flow" should be "a path and cycle flow". page 83, line 22. "cx = cx 0 + cx " should be "x = x 0 + x,,. page 85, line 1. "used" should be "us...
Averaging methods for solving fixed point problems combine the underlying fixed point map T with some “well-behaved” map g. The map g might, for example, be contractive or might be a nonexpansive map whose fixed points include those of the original map T. One class of averaging methods (inside averaging) averages any current iterate with its image...
An intuitive solution-doubling argument establishes well known results concerning the worst-case performance of spanning tree-based heuristics for the Steiner network problem and the traveling salesman problem. This note shows that the solution-doubling argument and its implications apply to certain more general Low Connectivity Steiner (LCS) probl...
![Table 4 . Computational results for random [0,1] SCP problems](profile/Anantaram_Balakrishnan/publication/220461415/figure/tbl2/AS:669155707596811@1536550561299/Computational-results-for-random-0-1-SCP-problems_Q320.jpg)
To ensure uninterrupted service, telecommunication networks contain excess (spare) capacity for rerouting (restoring) traffic in the event of a link failure. We study the NP-hard capacity planning problem of economically installing spare capacity on a network to permit link restoration of steady-state traffic. We present a planning model that incor...
Edited by Saul I. Gass and Carl M. Harris
The Encyclopedia provides decision makers and problem solvers in business, government and academia with a comprehensive overview of the concepts and methodologies that combine to form the fields of operations research and management science (OR/MS). More than 228 separate entries show how OR/MS applies scie...
The survivable network design (SND) problem seeks a minimum cost set of edges that meet prescribed node connnectivity requirements. We present a new family of strong mixed integer programming formulations for this problem, examine the tightness of the associated linear programming relaxations, and then use the relaxations to analyze heuristics for...
As the computer, communication, and entertainment industries begin to integrate phone, cable, and video services and to invest in new technologies such as fiber optic cables, interruptions in service can cause considerable customer dissatisfaction and even be catastrophic. In this environment, network providers want to offer high levels of servicei...
We study averaging methods for solving variational inequalities whose underlying maps are nonexpansive and for solving systems of (asymmetric) equations. Our goal is to establish global convergence results using weaker assumptions than are traditional in the literature. We examine averaging schemes for relaxation algorithms and for their specializa...
We introduce an approach, called the orthogonality theorem, for establishing the convergence of several algorithms for solving variational inequalities. This theorem, as well as several basic convergence theorems from the literature, impose the condition of strong-f-monotonicity on the problem function. We analyze and introduce some new results con...
We study a class of models, known as overlay optimization problems, with a "base" subproblem and an "overlay" subproblem, linked by the requirement that the overlay solution be contained in the base solution. In some telecommunication settings, a feasible base solution is a spanning tree and the overlay solution is an embedded Steiner tree (or an e...
Given a treeG = (V, E) and a weight function defined on subsets of its nodes, we consider two associated problems. The first, called the “rooted subtree problem”, is to find a maximum weight subtree, with a specified root, from a given set of subtrees.
The second problem, called “the subtree packing problem”, is to find a maximum weight packing of...
In this paper, we propose a concept of polynomiality for variational inequality problems and show how to find a near optimal solution of variational inequality problems in a polynomial number of iterations. To establish this result we build upon insights from several algorithms for linear and nonlinear programs (the ellipsoid algorithm, the method...
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...
Given an undirected network with L possible facility types for each edge, and a partition of the nodes into L levels or grades, the Multi-level Network Design (MLND) problem seeks a fixed cost minimizing design that spans all the nodes and connects the nodes at each level by facilities of the corresponding or higher grade. This problem generalizes...
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...
We study a specialized version of network design problems that arise in telecommunications, transportation, and other industries. The problem, a generalization of the shortest path problem, is defined on an undirected network consisting of a set of arcs on which we can install (load), at a cost, a choice of up to three types of capacitated faciliti...
"August 1988. Revised: December, 1988."
In a two-capacitated spanning tree of a complete graph with a distinguished root vertex v, every component of the induced subgraph on V\{v} has at most two vertices. We give a complete,non-redundant characterization of the polytope defined by the convex hull of the incidence vectors of two-capacitated spanning trees. This polytope is the intersecti...
Growing demand, increasing diversity of services, and advances in transmission and switching technologies are prompting telecommunication companies to rapidly expand and modernize their networks. This paper develops and tests a decomposition methodology to generate cost-effective expansion plans, with performance guarantees, for one major component...
In this paper we study the behavior of the local solutions of perturbed variational inequalities, governed by perturbations to both the variational inequality function and the feasible region. Assuming appropriate second-order and regularity conditions, we show that the perturbed local solution set is Lipschitz continuous and directionally differen...
This paper studies a multi-facility network synthesis problem, called the Two-level Network Design (TLND) problem, that arises in the topological design of hierarchical communication, transportation, and electric power distribution networks. We consider an undirected network containing two types of nodes – primary and secondary – and fixed costs fo...
The literature on network flow problems is extensive, and over the past 40 years researchers have made continuous improvements to algorithms for solving several classes of problems. However, the surge of activity on the algorithmic aspects of network flow problems over the past few years has been particularly striking. Several techniques have prove...
This paper was also printed as a Working Paper at the Yale School of Organization and Management, Series B, No. 103. We develop an optimization-based approach for a point-to-point route planning problem that arises in many large scale delivery systems(for example, less-than-truckload freight, rail, mail and package delivery, communications). In the...
Changeover costs (and times) are central to numerous manufacturing operations. These costs arise whenever work centers capable of processing only one product at a time switch from the manufacture of one product to another. Although many researchers have contributed to the solution of scheduling problems that include changeover costs, due to the pro...
We study the polyhedral structure of two related core combinatorial problems: the subtree cardinalityconstrained minimal spanning tree problem and the identical customer vehicle routing problem. For each of these problems, and for a forest relaxation of the minimal spanning tree problem, we introduce a number of new valid inequalities and specify c...
In this paper we examine computational complexity issues and develop algorithms for a class of "shoreline" single-vehicle routing and scheduling problems with release time constraints. Problems in this class are interesting for both practical and theoretical reasons. From a practical perspective, these problems arise in several transportation envir...
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 prob...
The dynamic economic lot sizing model, which lies at the core of numerous production planning applications, is one of the most highly studied models in all of operations research. And yet, capacitated multi-item versions of this problem remain computationally elusive. We study the polyhedral structure of an integer programming formulation of a sing...
Variational inequalities have often been used as a mathematical programming tool in modeling various equilibria in economics and transportation science. The behavior of such equilibrium solutions as a result of the changes in problem data is always of concern. In this paper, we present an approach for conducting sensitivity analysis of variational...
In satellite communications networks, distinctive facilities called homing stations perform special transmission functions. Local demand nodes clustered around each homing station communicate with each other via a local switch at the homing station; demand nodes in different clusters communicate with each other via satellite earth stations at the h...
Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitated plant location problem. Our purpose is to identi...
Publisher Summary This chapter discusses the fundamental ideas of network optimization. Networks provide a concrete setting for testing and devising new theories and serves as the major prototype for several theoretical domains and as the core model for a variety of min/max duality results in discrete mathematics. The chapter discusses (1) differen...
The traditional perturbation (or lexicographic) methods for resolving degeneracy in linear programming impose decision rules that eliminate ties in the simplex ratio rule and, therefore, restrict the choice of exiting basic variables. Bland's combinatorial pivoting rule also restricts the choice of exiting variables. Using ideas from parametric lin...
The model achieves a practical compromise between behavioral and computational aspects of modeling the problem. It is formulated as an equivalent convex optimization problem, yet it is behaviorally richer than other models that can be cast as equivalent convex programs. Although the model is not as behaviorally rich as the most general equilibrium...
We consider generalizations of the steepest descent algorithm for solving asymmetric systems of equations. We first show that if the system is linear and is defined by the matrix M, then the method converges if M ² is positive definite. We also establish easy to verify conditions on the matrix M that ensure that M ² is positive definite, and develo...
Because of its imbedded network flow structure, the generic network design problem is an attractive candidate for integer
programming decomposition. This paper studies the application and acceleration of Benders decomposition for uncapacitated
models from this problem class and illustrates the potential flexibility of the Benders solution strategy....
We provide worst case error bounds for several approximation methods (heuristics, product aggregation, and partitioning of the planning horizon) for the uncapacitated dynamic lot size problem. We propose two managerially oriented heuristics and show that they have a relative wont case error bound equal to two, and develop similar analyses for metho...
Citations
... SND belongs to a general network design problem in transportation planning, logistics, telecommunication, and production systems [12][13][14][15]. The objective of the general network design problem consists of finding a minimum cost network decision, i.e., a network configuration to enable the flow of commodities while minimizing the total cost [16]. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/276903782043653-1443030414204_Q64/Prakash-Mirchandani-2.jpg)
... Proof The optimal value function of a minimization linear program is a piecewise linear convex function of the right-hand sides (see e.g., Martin 2012;Vaidyanathan et al. 2016). Therefore, ϕ(d) which is the optimal value function for PS( y, d) is piecewise linear and convex. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/272394367533066-1441955286159_Q64/James-Orlin.jpg)
... In the Appendix, we show that it can be reformulated into a minimum-cost flow problem (MCFP). Such types of linear programs have the property of having an integral flow if all arc capacities and supply/demands of the nodes are integers (Integrality Theorem) [27]. The optimal routing problem can therefore be solved as if it were a linear program (LP relaxation), i.e., f (u, v) ∈ R, as opposed to an ILP, ensuring that the solution can be efficiently found also for large datasets. ...
... Only some recent work considered the performance of these rules when some of these parameters are unknown. In the line of work on scheduling with testing [7,16], marginal job holding costs and mean processing times of jobs are a-priori unknown, but their values for a job become known by testing this job. The optimal policy combines testing the jobs up to certain time and serving the jobs according to the cµ rule policy. ...
Reference: Learning to Schedule
![[object Object]](https://i1.rgstatic.net/ii/profile.image/278637371772930-1443443734078_Q64/Retsef-Levi.jpg)
... Even in situations where students need to be assigned to groups for project work, problems with matching relevant attributes become complex and time consuming when the number of students increases. (Anwar & Bahaj, 2003;Calvo-Serrano, Guillen-Gosablez, & Kohn, 2017;Magnanti & Natarajan, 2018). Optimization problems can be computationally intensive and there is no universal solution method. ...
... The exact computation of Wasserstein distances can be done by network simplex methods that take O(D 3 ) (D is the feature dimension of X 1 and X 2 ) (Ahuja et al., 1995). Muzellec et al. (2020) uses Sinkhorn iterations (Cuturi, 2013) with the entropic regularisation to compute the 2-Wasserstein distances in O(D 2 log D) (Altschuler et al., 2017;Dvurechensky et al., 2018): ...
... Network design models arise in large applications in telecommunications, transportation, logistics and production planning Balakrishnan et al. (1997Balakrishnan et al. ( , 1991; Minoux (1989); Gavish (1991). The multicommodity capacitated fixed-charge network design problem (MCND), is an NPhard discrete optimization problem Magnanti and Wong (1984). ...
... 548 Cf. Magnanti and Wong (1984), Magnanti (1996), Diruf (1999, pp. 378-380) or Crainic (2000). ...
... However, this representation of the flow constrains assumes that none of the detections are false positives. For a more general formulation of the flow conservation constraints we refer the reader to [AMO93] and the supplemental material from [BL20]. ...
... This algorithm has O(|V | 3 ) complexity 1 when it is implemented with proper data structures [21]. There are some newer algorithms with different complexities, but they utilize the maximum weights on the edges [2,12,24]. These newer algorithms are not suitable for an incremental version of the problem. ...
Reference: Incremental assignment problem
