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1.
The Sample Average Approximation Method Applied to Stochastic Routing Problems: A Computational Study 总被引:1,自引:0,他引:1
Bram Verweij Shabbir Ahmed Anton J. Kleywegt George Nemhauser Alexander Shapiro 《Computational Optimization and Applications》2003,24(2-3):289-333
The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps.We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 21694 scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time. 相似文献
2.
Christofides [1] proposes a heuristic for the traveling salesman problem that runs in polynomial time. He shows that when the graphG = (V, E) is complete and the distance matrix defines a function onV × V that is metric, then the length of the Hamiltonian cycle produced by the heuristic is always smaller than 3/2 times the length of an optimal Hamiltonian cycle. The purpose of this note is to refine Christofides' worst-case analysis by providing a tight bound for everyn 3, wheren is the number of vertices of the graph. We also show that these bounds are still tight when the metric is restricted to rectilinear distances, or to Euclidean distances for alln 6.This work was supported, in part, by NSF Grant ENG 75-00568 to Cornell University. This work was done when the authors were affiliated with the Center for Operations Research and Econometrics, University of Louvain, Belgium. 相似文献
3.
We study a special case of a structured mixed integer programming model that arises in production planning. For the most
general case of the model, called PI, we have earlier identified families of facet–defining valid inequalities: (l, S) inequalities (introduced for the uncapacitated lot–sizing problem by Barany, Van Roy, and Wolsey), cover inequalities, and
reverse cover inequalities. PI is 𝒩𝒫–hard; in this paper we focus on a special case, called PIC. We describe a polynomial
algorithm for PIC, and we use this algorithm to derive an extended formulation of polynomial size for PIC. Projecting from
this extended formulation onto the original space of variables, we show that (l, S) inequalities, cover inequalities, and reverse cover inequalities suffice to solve the special case PIC by linear programming.
We also describe fast combinatorial separation algorithms for cover and reverse cover inequalities for PIC. Finally, we discuss
the relationship between our results for PIC and a model studied previously by Goemans.
Received: December 13, 2000 / Accepted: December 13, 2001 Published online: October 9, 2002
RID="★"
ID="★" Some of the results in this paper have appeared in condensed form in Miller et al. (2001).
Key words. mixed integer programming – polyhedral combinatorics – production planning – capacitated lot–sizing – fixed charge network
flow – setup times
This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian
State, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the authors.
This research was also supported by NSF Grant No. DMI-9700285 and by Philips Electronics North America. 相似文献
4.
An optimization approach for planning daily drayage operations 总被引:1,自引:0,他引:1
Yetkin Ileri Mokhtar Bazaraa Ted Gifford George Nemhauser Joel Sokol Erick Wikum 《Central European Journal of Operations Research》2006,14(2):141-156
Daily drayage operations involve moving loaded or empty equipment between customer locations and rail ramps. Our goal is to
minimize the cost of daily drayage operations in a region on a given day. Drayage orders are generally pickup and delivery
requests with time windows. The repositioning of empty equipment may also be required in order to facilitate loaded movements.
The drayage orders are satisfied by a heterogeneous fleet of drivers. Driver routes must satisfy various operational constraints.
We present an optimization methodology for finding cost-effective schedules for regional daily drayage operations. The core
of the formulation is a set partitioning model whose columns represent routes. Routes are added to the formulation by column
generation. We present numerical results for real-world data which demonstrate that our methodology produces low cost solutions
in a reasonably short time. 相似文献
5.
Min-cut clustering 总被引:1,自引:0,他引:1
We describe a decomposition framework and a column generation scheme for solving a min-cut clustering problem. The subproblem to generate additional columns is itself an NP-hard mixed integer programming problem. We discuss strong valid inequalities for the subproblem and describe some efficient solution strategies. Computational results on compiler construction problems are reported.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.This research was supported by NSF grants DMS-8719128 and DDM-9115768, and by an IBM grant to the Computational Optimization Center, Georgia Institute of Technology. 相似文献
6.
We present a strong cutting plane/branch-and-bound algorithm for node packing. The cutting planes are obtained from cliques and lifting odd hole inequalities. Computational results are reported. 相似文献
7.
Given a graphG=[V, E] with positive edge weights, the max-cut problem is to find a cut inG such that the sum of the weights of the edges of this cut is as large as possible. Letg(K) be the class of graphs whose longest odd cycle is not longer than2K+1, whereK is a nonnegative integer independent of the numbern of nodes ofG. We present an O(n
4K) algorithm for the max-cut problem for graphs ing(K). The algorithm is recursive and is based on some properties of longest and longest odd cycles of graphs.
This research was supported by National Science Foundation Grant ECS-8005350 to Cornell University. 相似文献
8.
The analysis of data concerning the deterioration of pavement over time yielded a problem of aggregating the data in a manner that preserved independence of the aggregated data points and maximized the number of points. We show that this problem can be modeled as a maximum cardinality vertex packing problem on a proper internal graph and thus can be solved very efficiently by a greedy algorithm. 相似文献
9.
We consider a bottleneck location problem on a graph and present an efficient (polynomial time) algorithm for solving it. The problem involve the location of K noxious facilities that are to be placed as far as possilbe from the other facilities, and the objective is to maximize the minimum distance from the noxious facilities to the others. We then show that two other bottleneck (min-max) location problems, finding K-centers and absolute K-centers of a graph appear to be very difficult to solve even for reasonably good approximate solutions. 相似文献
10.
We consider a binary integer programming formulation (VP) for the weighted vertex packing problem in a simple graph. A sufficient “local” optimality condition for (VP) is given and this result is used to derive relations between (VP) and the linear program (VLP) obtained by deleting the integrality restrictions in (VP). Our most striking result is that those variables which assume binary values in an optimum (VLP) solution retain the same values in an optimum (VP) solution. This result is of interest because variables are (0, 1/2, 1). valued in basic feasible solutions to (VLP) and (VLP) can be solved by a “good” algorithm. This relationship and other optimality conditions are incorporated into an implicit enumeration algorithm for solving (VP). Some computational experience is reported. 相似文献