[Show abstract][Hide abstract] ABSTRACT: 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 Moment (CMM) choice model, we develop a new SUE model that uses the mean and covariance information on path utilities but does not assume the particular form of the distribution. Robustness to distributional assumptions is obtained in this model by minimizing the worst-case expected cost over all distributions with fixed two moments. We show that under mild conditions, the CMM-SUE (Cross Moment-Stochastic User Equilibrium) exists and is unique. By combining a simple projected gradient ascent method to evaluate path choice probabilities with a gradient descent method to find flows, we show that the CMM-SUE is efficiently computable. CMM-SUE provides both modeling flexibility and computational advantages over approaches such as the well-known MNP-SUE (Multinomial Probit-Stochastic User Equilibrium) model that require distributional (normality) assumptions to model correlation effects from overlapping paths. In particular, it avoids the use of simulation methods employed in computations for the distribution-based MNP-SUE model. Preliminary computational results indicate that CMM-SUE provides a practical distributionally robust alternative to MNP-SUE.
Transportation Research Part B Methodological 03/2015; DOI:10.1016/j.trb.2015.01.005 · 2.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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 group activities for a calculus course for first year university students at Singapore University of Technology and Design, a new design-centric university established in collaboration with Massachusetts Institute of Technology (MIT). We describe the educational impact of different pedagogical techniques, such as real-time response tools, hands on activities, mathematical modeling, visualization activities and motivational competitions on students with differing learning preferences in a unique cohort classroom setting. Based on faculty reflection and survey data, we provide guidelines on how to adopt the right set of active and group learning techniques to handle the changing learning preferences in the current and future generation of students.
2013 IEEE Global Engineering Education Conference (EDUCON); 03/2013
[Show abstract][Hide abstract] ABSTRACT: 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 counterpart is a well-studied
combinatorial optimization problem. Our technique preserves constant factor
approximation ratios, as well as ratios that depend only on certain problem
parameters, and exact algorithms yield exact algorithms.
Using our technique, we obtain a new 1.61-approximation algorithm for the
concave cost facility location problem. For inventory problems, we obtain a new
exact algorithm for the economic lot-sizing problem with general concave
ordering costs, and a 4-approximation algorithm for the joint replenishment
problem with general concave individual ordering costs.
[Show abstract][Hide abstract] ABSTRACT: We study the problem of minimizing a nonnegative separable concave function
over a compact feasible set. We approximate this problem to within a factor of
1+epsilon by a piecewise-linear minimization problem over the same feasible
set. Our main result is that when the feasible set is a polyhedron, the number
of resulting pieces is polynomial in the input size of the polyhedron and
linear in 1/epsilon. For many practical concave cost problems, the resulting
piecewise-linear cost problem can be formulated as a well-studied discrete
optimization problem. As a result, a variety of polynomial-time exact
algorithms, approximation algorithms, and polynomial-time heuristics for
discrete optimization problems immediately yield fully polynomial-time
approximation schemes, approximation algorithms, and polynomial-time heuristics
for the corresponding concave cost problems.
We illustrate our approach on two problems. For the concave cost
multicommodity flow problem, we devise a new heuristic and study its
performance using computational experiments. We are able to approximately solve
significantly larger test instances than previously possible, and obtain
solutions on average within 4.27% of optimality. For the concave cost facility
location problem, we obtain a new 1.4991+epsilon approximation algorithm.
[Show abstract][Hide abstract] ABSTRACT: 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 nodes, the SND problem seeks the minimum cost network that satisfies all the connectivity requirements. Although simple to describe, this model captures, as special cases, many classic, but difficult, optimization problems like the Steiner tree problem and the traveling salesman problem. In this paper, we develop a family of new formulations for the SND problem, and show that these formulations are stronger than the traditional cutset formulation of the problem. Our proposed connectivity-splitting formulations have intuitive special cases that motivate the design and analysis of specialized heuristics for the survivable network design problem
[Show abstract][Hide abstract] ABSTRACT: 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 exceed a given value. The latter might, in turn, be divided into two classes: (b1) those restricting the length of the path from the depot to any node k, and (b2) those restricting the length of the circuit passing through any node k. Our results indicate that formulations based upon circuit lengths (b2) lead to models with a linear programming relaxation that is tighter than the linear programming relaxation of models based upon path lengths (b1), and that combining features from capacitated models with those of circuit lengths can lead to formulations for the CVRP with a tight linear programming bound. Computational results on a small number of problem instances with up to 41 nodes and 440 edges show that the combined model with capacities and circuit lengths produce average gaps of less than one percent. We also briefly examine the asymmetric travelling salesman problem (ATSP), showing the potential use of the ideas developed for the vehicle routing problem to derive models for the ATSP with a linear programming relaxation bound that is tighter than the linear programming relaxation bound of the standard Dantzig, Fulkerson and Johnson [G. Dantzig, D. Fulkerson, D. Johnson, Solution of large-scale travelling salesman problem, Operations Research 2 (1954) 393–410] formulation.
[Show abstract][Hide abstract] ABSTRACT: 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) = 0. Such a piecewise linear function can be fully characterized by its segments. On each arc a, each segment s of the function has a non-negative variable cost, c a (the slope), a non-negative fixed cost, f a (the intercept), and upper and lower bounds, b a and b a , on the flow of that segment. Since the total flow on each arc can always be bounded from above by either the arc capacity or the total demand flowing through the network, we assume that there is a finite number of segments on each arc a, which we represent by the set S a . We further introduce the following notation: K denotes the set of commodities, N is the |V
Operations Research 02/2007; 55(1):146-157. DOI:10.1287/opre.1060.0314 · 1.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: "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.
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: 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 tighten the resulting formulation to develop a model with a stronger linear
programming relaxation. The linear programming gaps for the tightened model are very small, typically less than 0.5–, and
are usually one third to one tenth of the gaps of the best previous model described in Gouveia and Magnanti (2003). Moreover,
the new model permits us to solve large Euclidean problem instances that are not solvable by prior approaches.
Annals of Operations Research 09/2006; 146(1):19-39. DOI:10.1007/s10479-006-0049-0 · 1.22 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
Operations Research Letters 03/2006; 34(2-34):228-238. DOI:10.1016/j.orl.2005.04.004 · 0.62 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper is an edited transcription of the author’s lecture given on February 25th, 2006, in honor of Saul Gass.
Perspectives in Operations Research, 12/2005: pages 77-98;
[Show abstract][Hide abstract] ABSTRACT: 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 connected network designs, and nonunitary problems permitting non-connected networks as solutions. We (i) present a new directed formulation for the unitary NDC problem that is stronger than a natural undirected formulation, (ii) project out several classes of valid inequalities—partition inequalities, odd-hole inequalities, and combi- natorial design inequalities—that generalize known classes of valid inequalities for the Steiner tree problem to the unitary NDC problem, and (iii) show how to strengthen and direct nonunitary problems. Our results provide a unifying framework for strengthening formulations for NDC problems, and demonstrate the strength and power of ßow-based formulations for net- work design problems with connectivity requirements.
[Show abstract][Hide abstract] ABSTRACT: 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 methods even when the underlying map is contractive, and (iii) unify and extend several convergence results from the fixed point and variational inequality literatures. To achieve these goals, we introduce and study joint compatibility conditions imposed upon the underlying map and the iterative step sizes at each iteration and consider line searches that optimize certain potential functions. As a special case, we introduce a modified steepest descent method for solving systems of equations that does not require a previous condition from the literature (the square of the Jacobian matrix is positive definite). Since the line searches we propose might be difficult to perform exactly, we also consider inexact line searches.
[Show abstract][Hide abstract] ABSTRACT: 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 route for each origin-destination pair so as to minimize transportation costs and/or transit times. Typically, the cost structure is very complicated. The approach discussed in this paper exploits the structure of the problem to decompose it into two smaller subproblems, each amenable to solution by a combination of optimization and heuristic techniques. One subproblem is an 'assignment' problem with capacity constraints. The other subproblem is a mixed-integer multicommodity flow problem. We propose solution methods based on Lagrangian relaxation for each subproblem. Computational results with these methods and with a heuristic procedure for the multicommodity flow problem on a problem met in practice are encouraging and suggest that mathematical programming methods can be successfully applied to large-scale problems in delivery systems planning and other problems in logistical system design.
[Show abstract][Hide abstract] ABSTRACT: 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. After first establishing convergence for a general iterative scheme for computing fixed points, we consider applications to projection and relaxation algorithms for solving variational inequality problems and to a generalized steepest descent method for solving systems of equations. As part of our development, we also establish a new interpretation of a norm condition typically used for establishing convergence of linearization schemes, by associating it with a strong-f-monotonicity condition. We conclude by applying our results to transportation networks.