40 results on '"QUADRATIC assignment problem"'
Search Results
2. Lower Bounds for the Quadratic Assignment Problem Based upon a Dual Formulation
- Author
-
Peter Hahn and Thomas Grant
- Subjects
Combinatorics ,Linear programming ,Linearization ,Quadratic assignment problem ,Relaxation (approximation) ,Management Science and Operations Research ,Upper and lower bounds ,Assignment problem ,Integer programming ,Interior point method ,Computer Science Applications ,Mathematics - Abstract
A new bounding procedure for the Quadratic Assignment Problem (QAP) is described which extends the Hungarian method for the Linear Assignment Problem (LAP) to QAPs, operating on the four dimensional cost array of the QAP objective function. The QAP is iteratively transformed in a series of equivalent QAPs leading to an increasing sequence of lower bounds for the original problem. To this end, two classes of operations which transform the four dimensional cost array are defined. These have the property that the values of the transformed objective function Z′ are the corresponding values of the “old” objective function Z, shifted by some amount C. In the case that all entries of the transformed cost array are nonnegative, then C is a lower bound for the initial QAP. If, moreover, there exists a feasible solution U to the QAP, such that its value in the transformed problem is zero, then C is the optimal value of Z and U is an optimal solution for the original QAP. The transformations are iteratively applied until no significant increase in constant C as above is found, resulting in the so called Dual Procedure (DP). Several strategies are listed for appropriately determining C, or equivalently, transforming the cost array. The goal is the modification of the elements in the cost array to obtain new equivalent problems that bring the QAP closer to solution. In some cases the QAP is actually solved, though solution is not guaranteed. The close relationship between the DP and the Linear Programming formulation of Adams and Johnson is presented. The DP attempts to solve Adams and Johnson's CLP, a continuous relaxation of a linearization of the QAP. This explains why the DP produces bounds close to the optimum values for CLP calculated by Johnson in her dissertation and by Resende, et al. in their Interior Point Algorithm for Linear Programming. The benefit of using DP within a branch-and-bound algorithm is described. Then, two versions of DP are tested on the Nugent test instances from size 5 to size 30, as well as several other test instances from QAPLIB. These compare favorably with earlier bounding methods.
- Published
- 1998
3. A New Lower Bound for the Quadratic Assignment Problem
- Author
-
P. Carraresi and Federico Malucelli
- Subjects
Discrete mathematics ,Sequence ,Quadratic assignment problem ,Management Science and Operations Research ,Upper and lower bounds ,Algorithm ,Generalized assignment problem ,Computer Science Applications ,Mathematics ,Conjunction (grammar) - Abstract
We introduce a new lower bound for the quadratic assignment problem based on a sequence of equivalent formulations of the problem. We present a procedure for obtaining tight bounds by sequentially applying our approach in conjunction with the A. Assad and W. Xu bound and the N. Christofides and M. Gerrard bound.
- Published
- 1992
4. Computing lower bounds for the quadratic assignment problem with an interior point algorithm for linear programming
- Author
-
Resende, Mauricio G.C., Ramakrishnan, K.G., and Drezner, Zvi
- Subjects
Quadratic programming -- Analysis ,Facility management -- Analysis ,Algorithms -- Usage ,Business ,Mathematics - Abstract
Optimality of quadratic assignment problems (QAP) has been extremely difficult to prove in cases with more than 20 facilities because of the weakness of known lower bounds. To solve this problem, lower bounds for a number of QAPs are estimated using the lower bound examined by Drezner (1995). This lower bound is based on linear programming relaxation of a popular integer programming formulation of the assignment problems. A new best known lower bound is derived on the majority of the tested QAPs. The lower bound is found to be tight in some cases, including those with more than 20 facilities. To solve the linear programs, an interior point code is used which applies a preconditioned conjugate gradient algorithm to estimate interior point directions.
- Published
- 1995
5. A new lower bound for the quadratic assignment problem
- Author
-
Carraresi, Paolo and Malucelli, Federico
- Subjects
Quadratic programming -- Models ,Mathematical optimization -- Research ,Business ,Mathematics - Abstract
A set of new lower bounds for quadratic assignment problems (QAPs) which are patterned after a sequence of equivalent reformulations are presented. The methodology employed on the QAP problem is an improvement of previous findings because it produces tight lower bounds. The QAP reformulations resulted in either bounds that are sharper or bounds that are computationally efficient. The most efficient lower bounds for the QAP problem are obtained through an application of the A. Assad and W. Xu bound to an instance P's pth reformulation. Another favorable solution is that obtained using the N. Christofides and M. Gerrard bound.
- Published
- 1992
6. An Exact Algorithm for the Quadratic Assignment Problem on a Tree
- Author
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Nicos Christofides and Enrique Benavent
- Subjects
Discrete mathematics ,Quadratic assignment problem ,Management Science and Operations Research ,Travelling salesman problem ,Computer Science Applications ,Reduction (complexity) ,Tree (data structure) ,symbols.namesake ,Exact algorithm ,Lagrangian relaxation ,symbols ,Integer programming ,Generalized assignment problem ,Mathematics - Abstract
The Tree QAP is a special case of the Quadratic Assignment Problem (QAP) where the nonzero flows form a tree. No condition is required for the distance matrix. This problem is NP-complete and is also a generalization of the Traveling Salesman Problem. In this paper, we present a branch-and-bound algorithm for the exact solution of the Tree QAP based on an integer programming formulation of the problem. The bounds are computed using a Lagrangian relaxation of this formulation. To solve the relaxed problem, we present a Dynamic Programming algorithm which is polynomially bounded. The obtained lower bound is very sharp and equals the optimum in many cases. This fact allows us to employ a reduction method to decrease the number of variables and leads to search-trees with a small number of nodes compared to those usually encountered in problems of this type. Computational results are given for problems with size up to 25.
- Published
- 1989
7. An exact algorithm for the quadratic assignment problem on a tree
- Author
-
Christofides, Nicos and Benavent, Enrique
- Subjects
Operations research -- Analysis ,Branch and bound algorithms -- Research ,Business ,Mathematics - Abstract
A branch-and-bound algorithm is developed for the exact solution of the Tree Quadratic Assignment Problem built on an integer programming formulation of the problem. The bounds are established using a Lagrangian relaxation of this formulation. A Dynamic Programming algorithm is presented which is polynomially bounded to solve the relaxed problem. Research results indicate that the obtained lower bound is very sharp and equals the optimum in a variety of cases. This finding enables the use of a reduction method to decrease the number of leads and variables to search-trees with a small amount of nodes compared to those normally encountered in a problem of this type. Computational results are also described for problems with size up to 25.
- Published
- 1989
8. Lower bounds for the quadratic assignment problem based upon a dual formulation
- Author
-
Hahn, Peter and Grant, Thomas
- Subjects
Mathematical optimization -- Analysis ,Operations research -- Analysis ,Business ,Mathematics - Abstract
A study was conducted to characterize a novel bounding process for the quadratic assignment problem (QAP). The bounding procedure extends the Hungarian technique for the linear assignment problem to QAP. The QAP was altered to a series of equivalent QAPs resulting to a rise in sequence of lower bounds for the original problem. Two categories of operations that alter the four dimensional cost array were also determined. Several strategies were then utilized to transform the cost array. In addition, the associated benefits of utilizing dual procedure within a branch-and-bound algorithm were discussed. Results suggested the possibility of developing a heuristic that promotes the movement of costs in such a way that the cost elements for the solution can be minimized during increases in other costs.
- Published
- 1998
9. A constructive method for improving lower bounds for a class of quadratic assignment problems.
- Author
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Chakrapani, Jaishankar and Skorin-Kapov, Jadranka
- Subjects
QUADRATIC assignment problem ,QUADRATIC equations ,HEURISTIC ,MATRICES (Mathematics) ,EIGENVALUES ,MATHEMATICS ,COMBINATORIAL optimization ,MATHEMATICAL analysis ,ALGEBRA - Abstract
A new approach to evaluate lower bounds for a class of quadratic assignment problems (QAP) is presented. An instance of a QAP of size n is specified by two n × n matrices D and F, and is denoted by QAP(D, F). This approach is presented for QAPs where D is the matrix of rectilinear distances between points on a regular grid. However, it can easily be generalized to a wider class of problems. Two matrices F
opt and Fres are constructed such that F = Fopt + Fres , and the optimal solution to QAP(D, Fopt ) is known. Any existing lower bound can then be applied to QAP(D, Fres ), which in sum with the optimal value for QAP(D, Fopt ), provides a lower bound to QAP(D, F). Thus, this method could serve as a reduction process to possibly improve the results from a variety of bounding techniques. This approach is tested on two bounds from the literature, the Gilmore-Lawler bound (GLB) and an eigenvalue bound (PB), for various problems of size ranging from 6-49. Computational results show a good improvement in bounds for all the test problems. An extension of our method to a more general class of QAPs is also presented. [ABSTRACT FROM AUTHOR]- Published
- 1994
- Full Text
- View/download PDF
10. A General Framework for Designing Approximation Schemes for Combinatorial Optimization Problems with Many Objectives Combined into One
- Author
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Andreas S. Schulz, Shashi Mittal, Massachusetts Institute of Technology. Operations Research Center, Sloan School of Management, and Schulz, Andreas S.
- Subjects
Vector optimization ,Mathematical optimization ,Optimization problem ,L-reduction ,Job shop scheduling ,Quadratic assignment problem ,Test functions for optimization ,Combinatorial optimization ,Management Science and Operations Research ,Multi-objective optimization ,Computer Science Applications ,Mathematics - Abstract
In this paper, we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function. Examples of such problems include those in which the objective function is a product or ratio of two linear functions, parallel machine scheduling problems with the makespan objective, robust versions of weighted multiobjective optimization problems, and assortment optimization problems with logit choice models. The main idea behind our approximation schemes is the construction of an approximate Pareto-optimal frontier of the functions that constitute the given objective. Using this idea, we give the first fully polynomial-time approximation schemes for the max-min resource allocation problem with a fixed number of agents, combinatorial optimization problems in which the objective function is the sum of a fixed number of ratios of linear functions, or the product of a fixed number of linear functions, and assortment optimization problems with logit choice model.
- Published
- 2013
11. Generating Experimental Data for the Generalized Assignment Problem
- Author
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Dolores Romero Morales and H. Edwin Romeijn
- Subjects
Linear bottleneck assignment problem ,Mathematical optimization ,Augmented assignment ,Stochastic modelling ,Quadratic assignment problem ,media_common.quotation_subject ,Management Science and Operations Research ,Infinity ,Computer Science Applications ,Greedy algorithm ,Generalized assignment problem ,Weapon target assignment problem ,Mathematics ,media_common - Abstract
The Generalized Assignment Problem (GAP) is the problem of finding the minimal cost assignment of jobs to machines such that each job is assigned to exactly one machine, subject to capacity restrictions on the machines. We propose a new stochastic model for the GAP. A tight condition on this stochastic model under which the GAP is feasible with probability one when the number of jobs goes to infinity is derived. This new stochastic model enables us to analyze the adequacy of most of the random generators given for the GAP in the literature. We demonstrate that the random generators commonly used to test solution procedures for the GAP tend to create easier problem instances when the number of machines m increases. We describe a greedy heuristic for the GAP, and use it to illustrate the results from the paper.
- Published
- 2001
12. A Branch-and-Price Algorithm for the Generalized Assignment Problem
- Author
-
Martin W. P. Savelsbergh
- Subjects
Linear bottleneck assignment problem ,Mathematical optimization ,Augmented assignment ,Quadratic assignment problem ,Management Science and Operations Research ,Computer Science Applications ,Computer Science::Multiagent Systems ,Column generation ,Algorithm ,Assignment problem ,Integer programming ,Weapon target assignment problem ,Generalized assignment problem ,Mathematics - Abstract
The generalized assignment problem examines the maximum profit assignment of jobs to agents such that each job is assigned to precisely one agent subject to capacity restrictions on the agents. A new algorithm for the generalized assignment problem is presented that employs both column generation and branch-and-bound to obtain optimal integer solutions to a set partitioning formulation of the problem.
- Published
- 1997
13. Using Confidence Limits for the Global Optimum in Combinatorial Optimization
- Author
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Ulrich Derigs
- Subjects
Mathematical optimization ,Optimization problem ,Branch and bound ,Quadratic assignment problem ,Cross-entropy method ,Combinatorial optimization ,Management Science and Operations Research ,Generalized assignment problem ,Combinatorial explosion ,Weapon target assignment problem ,Computer Science Applications ,Mathematics - Abstract
Let z* be the (unknown) optimal value of a combinatorial optimization problem (in minimization form) and let z be a constant satisfying Prob(z* ≥ z) ≥ 1 − α for a fixed α ∈ (0, 1). Then z is called a statistical or probabilistic lower bound at significance level 1 − α. We discuss how statistical lower bounds can be used to measure the effectiveness of an approximate solution and to fathom subproblems in a branch and bound approach. After reviewing some methods for obtaining statistical lower bounds, we report on extensive computational experience for two notoriously difficult combinatorial optimization problems—the traveling salesman problem and the quadratic assignment problem. From our experience we conclude that statistical bounds are highly useful for quadratic assignment problems. In this problem context, the quality of simple analytic formulas is comparable to the quality of the maximum likelihood formula. In general, using statistics to aid in the solution of combinatorial optimization problems should be most useful when such problems are “large.”
- Published
- 1985
14. Branch-and-Bound Methods: A Survey
- Author
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D. E. Wood and Eugene L. Lawler
- Subjects
Mathematical optimization ,Branch and bound ,Quadratic assignment problem ,Branch and price ,Constrained optimization ,Quadratic programming ,Management Science and Operations Research ,Algorithm ,Integer programming ,Computer Science Applications ,Domain (software engineering) ,Mathematics ,Nonlinear programming - Abstract
The essential features of the branch-and-bound approach to constrained optimization are described, and several specific applications are reviewed. These include integer linear programming (Land-Doig and Balas methods), nonlinear programming (minimization of nonconvex objective functions), the traveling-salesman problem (Eastman and Little, et al. methods), and the quadratic assignment problem (Gilmore and Lawler methods). Computational considerations, including trade-offs between length of computation and storage requirements, are discussed and a comparison with dynamic programming is made. Various applications outside the domain of mathematical programming are also mentioned.
- Published
- 1966
15. Solving the Assignment Problem by Relaxation
- Author
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Walter O. Rom and Ming S. Hung
- Subjects
Linear bottleneck assignment problem ,Mathematical optimization ,Augmented assignment ,Quadratic assignment problem ,Management Science and Operations Research ,Flow network ,Assignment problem ,Multi-commodity flow problem ,Generalized assignment problem ,Weapon target assignment problem ,Computer Science Applications ,Mathematics - Abstract
This paper presents a new algorithm for solving the assignment problem. The algorithm is based on a scheme of relaxing the given problem into a series of simple network flow (transportation) problems for each of which an optimal solution can be easily obtained. The algorithm is thus seen to be able to take advantage of the nice properties in both the primal and the dual approaches for the assignment problem. The computational bound for the algorithm is shown to be 0(n3) and the average computation time is better than most of the specialized assignment algorithms.
- Published
- 1980
16. Technical Note—A Polynomial Simplex Method for the Assignment Problem
- Author
-
Ming S. Hung
- Subjects
Linear bottleneck assignment problem ,Discrete mathematics ,Combinatorics ,Polynomial ,Simplex algorithm ,Quadratic assignment problem ,Management Science and Operations Research ,Extreme point ,Assignment problem ,Generalized assignment problem ,Weapon target assignment problem ,Computer Science Applications ,Mathematics - Abstract
We present a polynomial primal simplex algorithm for the assignment problem. For n × n assignment problem with integer cost coefficients, the algorithm generates at most n3 ln ▵ bases prior to reach the optimal basis, where ▵ is the difference in objective value between an initial extreme point and the optimal extreme point.
- Published
- 1983
17. Technical Note—Rounding Symmetric Traveling Salesman Problems with an Asymmetric Assignment Problem
- Author
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A. Volgenant, J. A. Van Der Velde, Roy Jonker, and G. De Leve
- Subjects
Discrete mathematics ,Traveling purchaser problem ,Quadratic assignment problem ,Cross-entropy method ,Management Science and Operations Research ,2-opt ,Travelling salesman problem ,Computer Science Applications ,Combinatorics ,Computer Science::Data Structures and Algorithms ,Bottleneck traveling salesman problem ,Assignment problem ,Generalized assignment problem ,Mathematics - Abstract
For the distance matrix of symmetric traveling salesman problems a simple transformation into an equivalent asymmetric one is given. Assignment algorithms yield sharper lowerbounds and less subtours from the transformed distance matrix. This implies a better performance for traveling salesman algorithms based on the assignment relaxation.
- Published
- 1980
18. An Additive Bounding Procedure for Combinatorial Optimization Problems
- Author
-
Matteo Fischetti and Paolo Toth
- Subjects
Mathematical optimization ,Optimization problem ,Linear programming ,Quadratic assignment problem ,Cross-entropy method ,Combinatorial optimization ,Relaxation (approximation) ,Management Science and Operations Research ,Upper and lower bounds ,Algorithm ,Travelling salesman problem ,Computer Science Applications ,Mathematics - Abstract
We know that the effectiveness of the branch-and-bound algorithms proposed for the solution of combinatorial optimization problems greatly depends on the tightness of the available bounds. In this paper, we consider optimization problems with a linear objective function. We propose an additive approach for computing lower bounds that yields an increasing sequence of values. An application to the traveling salesman problem with precedence constraints is presented to exemplify the technique.
- Published
- 1989
19. Signature Methods for the Assignment Problem
- Author
-
Michel Balinski
- Subjects
Linear bottleneck assignment problem ,Mathematical optimization ,Augmented assignment ,Quadratic assignment problem ,Component (UML) ,Management Science and Operations Research ,Algorithm ,Assignment problem ,Weapon target assignment problem ,Signature (logic) ,Generalized assignment problem ,Computer Science Applications ,Mathematics - Abstract
The “signature” of a dual feasible basis of the assignment problem is an n-vector whose ith component is the number of nonbasic activities of type (i, j). This paper uses signatures to describe a method for finding optimal assignments that terminates in at most (n − 1)(n − 2)/2 pivot steps and takes at most O(n3) work.
- Published
- 1985
20. Optimality Properties of a Special Assignment Problem
- Author
-
Roger Wets and S. C. Parikh
- Subjects
Linear bottleneck assignment problem ,Discrete mathematics ,Matrix (mathematics) ,Mathematical optimization ,Linear programming ,Quadratic assignment problem ,Management Science and Operations Research ,Assignment problem ,Matrix similarity ,Generalized assignment problem ,Weapon target assignment problem ,Computer Science Applications ,Mathematics - Abstract
In this paper, it is shown that if the cost matrix of an assignment problem has the following property cij = |j − i| then any basic feasible solution is optimal if and only if its unit components belong to two well-defined symmetric regions. The matrix with above mentioned property is called the “reordering matrix,” because it arose for the first time in the reordering of nodes of a critical path and other acyclic network problems. One deals with similar matrices in some problems related to order statistics.
- Published
- 1964
21. Letter to the Editor—The Multidimensional Assignment Problem
- Author
-
William P. Pierskalla
- Subjects
Linear bottleneck assignment problem ,Mathematical optimization ,Tree (data structure) ,Quadratic assignment problem ,Management Science and Operations Research ,Variety (universal algebra) ,Assignment problem ,Generalized assignment problem ,Weapon target assignment problem ,Computer Science Applications ,Mathematics ,Dual (category theory) - Abstract
The multidimensional assignment problem is a higher dimensional version of the standard (two-dimensional) assignment problem in the literature. The higher dimensions can be thought of as time or space dimensions or both. An algorithm is proposed for the solution of the multi-index assignment problem. The algorithm is based on a tree search technique of the branch-and-bound variety. It uses dual subproblems to provide easily computed bounds for the primal assignment problem.
- Published
- 1968
22. Minimizing the Number of Operations in Certain Discrete-Variable Optimization Problems
- Author
-
Francesco Brioschi and Shimon Even
- Subjects
Continuous optimization ,Mathematical optimization ,Vector optimization ,Optimization problem ,Quadratic assignment problem ,Discrete optimization ,Combinatorial optimization ,Management Science and Operations Research ,Metaheuristic ,Multi-objective optimization ,Computer Science Applications ,Mathematics - Abstract
This paper deals with the solution to a discrete optimization problem by decomposition. It develops a criterion for ranking the decomposition procedures, discusses the properties of the optimal decompositions, and gives an algorithm for finding the best decomposition in the case of no storage limitation.
- Published
- 1970
23. On a Linear-Programming, Combinatorial Approach to the Traveling-Salesman Problem
- Author
-
George B. Dantzig, Selmer Martin Johnson, and D. R. Fulkerson
- Subjects
Mathematical optimization ,Quadratic assignment problem ,Branch and price ,Cross-entropy method ,Combinatorial optimization ,Criss-cross algorithm ,Management Science and Operations Research ,Bottleneck traveling salesman problem ,Randomized rounding ,Computer Science Applications ,Mathematics ,Linear-fractional programming - Abstract
This paper elaborates a method of attack on traveling-salesman problems, proposed by the authors in an earlier paper, in which linear programming is used to reduce the combinatorial magnitude of such problems. To illustrate the method, a step-by-step solution of Barachet's ten-city example is presented.
- Published
- 1959
24. A Probabilistic Approach to Solving Assignment Problems
- Author
-
Mehmet I. Celebiler
- Subjects
Linear bottleneck assignment problem ,Mathematical optimization ,Quadratic assignment problem ,Probabilistic logic ,Set theory ,Function (mathematics) ,Management Science and Operations Research ,Finite set ,Generalized assignment problem ,Weapon target assignment problem ,Computer Science Applications ,Mathematics - Abstract
Assignment problems arise whenever elements of a finite set have to be paired with elements of another finite set in such a way as to maximize a certain function of the pairing. In this paper three different formulations of such a problem are given. They are analyzed and a method for obtaining a low-cost solution is developed.
- Published
- 1969
25. Letter to the Editor—An Algorithm for Ranking all the Assignments in Order of Increasing Cost
- Author
-
Katta G. Murty
- Subjects
Linear bottleneck assignment problem ,Mathematical optimization ,Sequence ,Augmented assignment ,Ranking ,Quadratic assignment problem ,Hungarian algorithm ,Management Science and Operations Research ,Algorithm ,Generalized assignment problem ,Weapon target assignment problem ,Computer Science Applications ,Mathematics - Abstract
The Hungarian method gives an efficient algorithm for finding the minimal cost assignment. However, in some cases it may be useful to determine the second minimal assignment (i.e., the best assignment after excluding the minimal cost assignment) and in general the kth minimal assignment for k = 1, 2, …. These things can easily be determined if all the assignments can be arranged as a sequence in increasing order of cost. This paper describes an efficient algorithm for such a ranking of all the assignments. The maximum computational effort required to generate an additional assignment in the sequence is that of solving at most (n − 1) different assignment problems, one each of sizes 2, 3, …, n.
- Published
- 1968
26. Technical Note—A 'Hard' Assignment Problem
- Author
-
Robert E. Machol and Michael Wien
- Subjects
Linear bottleneck assignment problem ,Mathematical optimization ,Quadratic assignment problem ,Technical note ,Construct (python library) ,Management Science and Operations Research ,Assignment problem ,Generalized assignment problem ,Weapon target assignment problem ,Computer Science Applications ,Mathematics - Abstract
We construct numerical examples that force Hungarian-method algorithms for the assignment problem to their worst-case time bounds. These bounds are thereby clarified.
- Published
- 1976
27. Problem Decomposition and Data Reorganization by a Clustering Technique.
- Author
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McCormick Jr., William T., Sehweitzer, Paul J., and White, Thomas W.
- Subjects
CLUSTER analysis (Statistics) ,STATISTICAL correlation ,MULTIVARIATE analysis ,ALGORITHMS ,MATHEMATICS ,OPERATIONS research - Abstract
A new cluster-analysis method, the bond energy algorithm, has been developed recently; it operates upon a raw input object-object or object-attribute data array by permuting its rows and columns in order to find informative variable groups and their interrelations. This paper describes the algorithm and illustrates by several examples its use for both problem decomposition and data reorganization. [ABSTRACT FROM AUTHOR]
- Published
- 1972
- Full Text
- View/download PDF
28. Using confidence limits for the global optimum in combinatorial optimization
- Author
-
Derigs, Ulrich
- Subjects
Mathematical optimization -- Analysis ,Branch and bound algorithms -- Analysis ,Combinatorial analysis -- Research ,Business ,Mathematics - Abstract
Statistical lower bounds can be used to evaluate the effectiveness of estimated solutions for combinatorial optimization problems. This technique is discussed, along with the use of statistical lower bounds as indicators of subproblems in branch and bound processing. The statistical lower bound techniques developed are then applied to two extremely difficult optimization problems: the traveling salesman problem and the quadratic assignment problem. These applications demonstrate the usefulness of statistical methods when solving significantly large combinatorial optimization problems.
- Published
- 1985
29. 123 ... QAP!
- Subjects
Quadratic functions -- Analysis ,Linear programming -- Analysis ,Business ,Mathematics - Abstract
The quadratic assignment problem (QAP) is one of the most famous (and difficult in practice) NP-hard optimization problems. A challenging feature of the problem is that a collection of small [...]
- Published
- 2012
30. On the Minimum Chordal Completion Polytope
- Author
-
Bergman, David, Cardonha, Carlos H., Cire, Andre A., and Raghunathan, Arvind U.
- Subjects
Heuristic programming -- Usage ,Graph theory -- Analysis ,Polytopes -- Research ,Business ,Mathematics - Abstract
A graph is chordal if every cycle with at least four edges contains a chord--that is, an edge connecting two nonconsecutive vertices of the cycle. Several classical applications in sparse linear systems, database management, computer vision, and semidefinite programming can be reduced to finding the minimum number of edges to add to a graph so that it becomes chordal, known as the minimum chordal completion problem (MCCP). We propose a new formulation for the MCCP that does not rely on finding perfect elimination orderings of the graph, as has been considered in previous work. We introduce several families of facet-defining inequalities for cycle subgraphs and investigate the underlying separation problems, showing that some key inequalities are NP-hard to separate. We also identify conditions through which facets and inequalities associated with the polytope of a certain graph can be adapted in order to become facet defining for some of its subgraphs or supergraphs. Numerical studies combining heuristic separation methods and lazy-constraint generation indicate that our approach substantially outperforms existing methods for the MCCP. Funding: The research was partially funded by a Natural Sciences and Engineering Research Council Discovery grant. Supplemental Material: The online supplement is available at https://doi.org/10.1287/opre.2018.1783. Keywords: networks/graphs * applications: networks/graphs * programming: integer * algoritms: cutting plane/facet, 1. Introduction Given a simple graph G = (V, E), the minimum chordal completion problem (MCCP) asks for the minimum number of edges to add to E so that the [...]
- Published
- 2019
- Full Text
- View/download PDF
31. New formulations for the conflict resolution problem in the scheduling of television commercials
- Author
-
Giallombardo, Giovanni, Jiang, Houyuan, and Miglionico, Giovanna
- Subjects
Television advertising -- Analysis ,Conflict management -- Analysis ,Problem solving -- Analysis ,Integer programming -- Analysis ,Business ,Mathematics - Abstract
We consider the conflict-resolution problem arising in the allocation of commercial advertisements to television program breaks. Because of the competition-avoidance requirements issued by advertisers, broadcasters aim to allocate any pairs [...]
- Published
- 2016
- Full Text
- View/download PDF
32. Improving community cohesion in school choice via correlated-lottery implementation
- Author
-
Ashlagi, Itai and Shi, Peng
- Subjects
Business ,Mathematics - Abstract
In school choice, children submit a preference ranking over schools to a centralized assignment algorithm, which takes into account schools' priorities over children and uses randomization to break ties. One [...]
- Published
- 2014
- Full Text
- View/download PDF
33. Exploiting erraticism in search
- Author
-
Fischetti, Matteo and Monaci, Michele
- Subjects
Mathematical optimization -- Analysis ,Self-denial -- Analysis ,Integer programming -- Analysis ,Business ,Mathematics - Abstract
High sensitivity to initial conditions is generally viewed as a drawback of tree search methods because it leads to erratic behavior to be mitigated somehow. In this paper we investigate [...]
- Published
- 2014
- Full Text
- View/download PDF
34. Generating applicable synthetic instances for branch problems
- Author
-
Lopes, Leo and Smith-Miles, Kate
- Subjects
Approximation theory -- Usage ,Mathematical optimization -- Usage ,Data mining -- Analysis -- Research ,Data warehousing/data mining ,Business ,Mathematics - Abstract
Generating valid synthetic instances for branch problems--those that contain a core problem like knapsack or graph coloring, but add several complications--is hard. It is even harder to generate instances that [...]
- Published
- 2013
35. A 0-1 LP model for the integration and consolidation of air cargo shipments
- Author
-
Leung, Lawrence C., Hui, Yer Van, Wang, Yong, and Chen, Gang
- Subjects
Air freight -- Management -- Analysis ,Linear programming -- Usage ,Business ,Mathematics ,Company business management ,Management ,Analysis ,Usage - Abstract
This paper addresses the problem of determining the optimal integrations and consolidations of air cargo shipments. A freight forwarder arranges for the execution of many jobs (shipments) on behalf of [...]
- Published
- 2009
36. The black and white traveling salesman problem
- Author
-
Ghiani, Gianpaolo, Laporte, Gilbert, and Semet, Frederic
- Subjects
Choice of transportation -- Analysis ,Business ,Mathematics ,Analysis - Abstract
The black and white traveling salesman problem (BWTSP) is defined on a graph G whose vertex set is partitioned into black and white vertices. The aim is to design a [...]
- Published
- 2006
37. Enhanced model formulations for optimal facility layout
- Author
-
Sherali, Hanif D., Fraticelli, Barbara M.P., and Meller, Russell D.
- Subjects
Management science -- Analysis ,Business ,Mathematics ,Analysis - Abstract
This paper presents an improved mixed-integer programming (MIP) model and effective solution strategies for the facility layout problem and is motivated by the work of Meller et al. (1999). This [...]
- Published
- 2003
38. On tightening the relaxations of miller-tucker-zemlin formulations for asymmetric traveling salesman problems
- Author
-
Sherali, Hanif D. and Driscoll, Patrick J.
- Subjects
Management science -- Analysis ,Business ,Mathematics ,Analysis - Abstract
This paper is concerned with applying the Reformulation-Linearization Technique (RLT) to derive tighter relaxations for the Asymmetric Traveling Salesman Problem (ATSP) formulation that is based on the Miller-Tucker-Zemlin (MTZ) subtour [...]
- Published
- 2002
39. Hybrid Fiber Coaxial network design
- Author
-
Patterson, Raymond A. and Rolland, Erik
- Subjects
Electrical wire and cable industry -- Product information -- Equipment and supplies ,Coaxial cables -- Usage -- Equipment and supplies -- Product information ,Telecommunications equipment industry -- Product information -- Equipment and supplies ,Network architecture -- Design and construction -- Product information -- Equipment and supplies ,Broadband transmission -- Equipment and supplies -- Product information -- Design and construction ,Telecommunications services industry -- Equipment and supplies -- Product information ,Communications industry -- Equipment and supplies -- Product information ,Business ,Mathematics ,Telecommunications services industry ,Network architecture ,Broadband Internet ,Telecommunications equipment industry ,Design and construction ,Usage ,Product information ,Equipment and supplies - Abstract
Much interest exists in broadband network services to deliver a variety of products to consumers, such as Internet access, telephony, interactive TV, and video on demand. Due to its cost [...]
- Published
- 2002
40. Generating experimental data for the generalized assignment problem
- Author
-
Romeijn, H. Edwin and Morales, Dolores Romero
- Subjects
Machine-shop practice -- Management ,Machinery industry -- Management ,Machine-tools ,Tool industry -- Management ,Machinists' tools ,Business ,Mathematics ,Company business management ,Management - Abstract
The Generalized Assignment Problem (GAP) is the problem of finding the minimal cost assignment of jobs to machines such that each job is assigned to exactly one machine, subject to [...]
- Published
- 2001
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