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Showing papers on "Linear programming published in 1989"


Book
01 Aug 1989
TL;DR: Mathematical Background Topics from Linear Algebra Single Objective Linear Programming Determining all Alternative Optima Comments about Objective Row Parametric Programming Utility Functions, Nondominated Criterion Vectors and Efficient Points Point Estimate Weighted-sums Approach.
Abstract: Mathematical Background Topics from Linear Algebra Single Objective Linear Programming Determining all Alternative Optima Comments about Objective Row Parametric Programming Utility Functions, Nondominated Criterion Vectors and Efficient Points Point Estimate Weighted-sums Approach Optimal Weighting Vectors, Scaling and Reduced Feasible Region Methods Vector-Maximum Algorithms Goal Programming Filtering and Set Discretization Multiple Objective Linear Fractional Programming Interactive Procedures Interactive Weighted Tchebycheff Procedure Tchebycheff/Weighted-Sums Implementation Applications Future Directions Index.

3,280 citations


Journal ArticleDOI
TL;DR: A model for the transit assignment problem with a fixed set of transit lines is described, formulated as a linear programming problem of a size that increases linearly with the network size that solves the latter problem in polynomial time.
Abstract: We describe a model for the transit assignment problem with a fixed set of transit lines The traveler chooses the strategy that allows him or her to reach his or her destination at minimum expected cost First we consider the case in which no congestion effects occur For the special case in which the waiting time at a stop depends only on the combined frequency, the problem is formulated as a linear programming problem of a size that increases linearly with the network size A label-setting algorithm is developed that solves the latter problem in polynomial time Nonlinear cost extensions of the model are considered as well

753 citations


Posted Content
David Gale1
TL;DR: Gale as mentioned in this paper provides a complete and lucid treatment of important topics in mathematical economics which can be analyzed by linear models, including games, linear programming, and the Neumann model of growth.
Abstract: In the past few decades, methods of linear algebra have become central to economic analysis, replacing older tools such as the calculus. David Gale has provided the first complete and lucid treatment of important topics in mathematical economics which can be analyzed by linear models. This self-contained work requires few mathematical prerequisites and provides all necessary groundwork in the first few chapters. After introducing basic geometric concepts of vectors and vector spaces, Gale proceeds to give the main theorems on linear inequalities—theorems underpinning the theory of games, linear programming, and the Neumann model of growth. He then explores such subjects as linear programming; the theory of two-person games; static and dynamic theories of linear exchange models, including problems of equilibrium prices and dynamic stability; and methods of play, optimal strategies, and solutions of matrix games. This book should prove an invaluable reference source and text for mathematicians, engineers, economists, and those in many related areas.

731 citations


Book ChapterDOI
01 Sep 1989
TL;DR: In this article, the authors present continuous paths leading to the set of optimal solutions of a linear programming problem, which are derived from the weighted logarithmic barrier function and have nice primal-dual symmetry properties.
Abstract: This chapter presents continuous paths leading to the set of optimal solutions of a linear programming problem These paths are derived from the weighted logarithmic barrier function The defining equations are bilinear and have some nice primal-dual symmetry properties Extensions to the general linear complementarity problem are indicated

583 citations


Journal ArticleDOI
01 Oct 1989-Networks
TL;DR: This paper identifies several families of facet defining inequalities for this polytope, the convex hull of solutions to the PCTSP, and uses these inequalities either as cutting planes or as ingredients of a Lagrangean optimand.
Abstract: The following is a valid model for an important class of scheduling and routing problems A salesman who travels between pairs of cities at a cost depending only on the pair, gets a prize in every city that he vitis and pays a penalty to every city that he fails to visit, wishes to minimize his travel costs and net penalties, while visiting enough cities to collect a prescribed amount of prize money We call this problem the Prize Collecting Traveling Salesman Problem (PCTSP) This paper discusses structural properties of the PCTS polytope, the convex hull of solutions to the PCTSP In particular, it identifies several families of facet defining inequalities for this polytope Some of these inequalities are related to facets of the ordinary TS polytope, others to facets of the knapsack polytope They can be used in algorithms for the PCTSP either as cutting planes or as ingredients of a Lagrangean optimand

547 citations


Book ChapterDOI
01 Jan 1989
TL;DR: In this paper, a nonparametric approach to the measurement of productive efficiency can be specified either through a flexible form of the production function which satisfies the efficiency hypothesis, or a set-theoretic characterization of an efficient isoquant.
Abstract: The nonparametric approach to the measurement of productive efficiency can be specified either through a flexible form of the production function which satisfies the efficiency hypothesis, or a set-theoretic characterization of an efficient isoquant. In the first case the production frontier can be of any general shape satisfying some very weak conditions like quasi-concavity or monotonicity, although in most empirical and applied work piecewise linear or, log-linear functions have been frequently used. Thus both Farrell and Johansen applied linear programming models in the specification of the production frontier. Farrell’s efficiency measure is based on estimating by a sequence of linear programs (LPs) a convex hull of the observed input coefficients in the input space. Two features of Farrell efficiency make it very useful in applied research. One is that it is completely data-based i.e., it uses only the observed inputs and outputs of the sample units while assuming production functions to be homogeneous of degree one. Hence it has many potential applications for the public sector units, where for most of the inputs and outputs the price data are not available. For example consider educational production functions for public schools, where outputs such as test scores in achievement tests are only proxy variables for learning; inputs such as average class size, experience of teachers or ethnic background of students do not have observed market prices. Secondly, Farrell’s method uses a set of LP models to estimate the efficiency parameters, so that the production frontier appears as piecewise linear functions. Nonnegativity conditions on the parameter estimates can therefore be easily incorporated.

477 citations


Journal ArticleDOI
TL;DR: A primal-dual interior point algorithm for linear programming problems which requires a total of O(n L) number of iterations to find the Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the logarithmic barrier function problem.
Abstract: We describe a primal-dual interior point algorithm for linear programming problems which requires a total of $$O\left( {\sqrt n L} \right)$$ number of iterations, whereL is the input size. Each iteration updates a penalty parameter and finds the Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the logarithmic barrier function problem. The algorithm is based on the path following idea.

458 citations


Journal ArticleDOI
TL;DR: A cone ratio data envelopment analysis (DEA) model that substantially generalizes the Charnes-Cooper-Rhodes (CCR) model and characterizes its efficiency classes is developed and studied as discussed by the authors.
Abstract: A new ‘cone ratio’ data envelopment analysis (DEA) model that substantially generalizes the Charnes-Cooper-Rhodes (CCR) model and the Charnes-Cooper-Thrall approach characterizing its efficiency classes is developed and studied. It allows for infinitely many decision-making units (DM Us) and arbitrary closed convex cones for the virtual multipliers as well as the cone of positivily of the vectors involved. Generalizations of linear programming and polar cone equalizations arc the analytical vehicles employed.

445 citations


Journal ArticleDOI
TL;DR: A probabilistic version of the maximal covering location problem is introduced here, structured here as a zero--one linear programming problem and solved on a medium-sized transportation network representing Baltimore City.
Abstract: A probabilistic version of the maximal covering location problem is introduced here. The maximum available location problem (MALP) positions p servers in such a way as to maximize the population which will find a server available within a time standard with α reliability. The maximum availability problem builds on the probabilistic location set covering problem in concept and on backup covering and expected covering models in technical detail. MALP bears the same relation to the probabilistic location set covering problem as the deterministic maximal covering problem bears to the deterministic location set covering problem. The maximum availability problem is structured here as a zero–one linear programming problem and solved on a medium-sized transportation network representing Baltimore City.

426 citations


Journal ArticleDOI
TL;DR: Based on a continuous version of Karmarkar's algorithm, two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested and compares favorably with the simplex codeMinos 4.0.
Abstract: This paper describes the implementation of power series dual affine scaling variants of Karmarkar's algorithm for linear programming. Based on a continuous version of Karmarkar's algorithm, two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested. Linear programs are expressed in an inequality form, which allows for the inexact computation of the algorithm's direction of improvement, resulting in a significant computational advantage. Implementation issues particular to this family of algorithms, such as treatment of dense columns, are discussed. The code is tested on several standard linear programming problems and compares favorably with the simplex codeMinos 4.0.

386 citations


Journal ArticleDOI
TL;DR: In this paper, a necessary and sufficient condition for linear quadratic stabilization of linear uncertain systems when both the dynamic as well as the control matrix are subject to uncertainty is given. And a constructive numerical procedure is defined to check the condition and furthermore provides a stabilizing linear feedback gain.

Journal ArticleDOI
TL;DR: It is shown in particular that, given any desired product mix, it is possible to start the system with enough jobs in process so that some machines will be fully utilized in steady-state and the productivity is optimal.
Abstract: Timed event-graphs, a special class of timed Petri nets, are used for modelling and analyzing job-shop systems. The modelling allows the steady-state performance of the system to be evaluated under a deterministic and cyclic production process. Given any fixed processing times, the productivity (i.e., production rate) of the system can be determined from the initial state. It is shown in particular that, given any desired product mix, it is possible to start the system with enough jobs in process so that some machines will be fully utilized in steady-state. These machines are called bottleneck machines, since they limit the throughput of the system. In that case, the system works at the maximal rate and the productivity is optimal. The minimal number of jobs in process allowing optimal functioning of the system is further specified as an integer linear programming problem. An efficient heuristic algorithm is developed to obtain a near-optimal solution. >

Proceedings ArticleDOI
P.M. Vaidya1
30 Oct 1989
TL;DR: An algorithm for solving linear programming problems that requires O((m+n)/sup 1.5/nL) arithmetic operations in the worst case is presented, which improves on the best known time complexity for linear programming by about square root n.
Abstract: The author presents an algorithm for solving linear programming problems that requires O((m+n)/sup 15/nL) arithmetic operations in the worst case, where m is the number of constraints, n the number of variables, and L a parameter defined in the paper This result improves on the best known time complexity for linear programming by about square root n A key ingredient in obtaining the speedup is a proper combination and balancing of precomputation of certain matrices by fast matrix multiplication and low-rank incremental updating of inverses of other matrices Specializing the algorithm to problems such as minimum-cost flow, flow with losses and gains, and multicommodity flow leads to algorithms whose time complexity closely matches or is better than the time complexity of the best known algorithms for these problems >

Journal ArticleDOI
TL;DR: In this paper, a technique for computing rigorous upper bounds on limit loads under conditions of plane strain is described, which assumes a perfectly plastic soil model and employs finite elements in conjunction with the upper bound theorem of classical plasticity theory.
Abstract: This paper describes a technique for computing rigorous upper bounds on limit loads under conditions of plane strain. The method assumes a perfectly plastic soil model, which is either purely cohesive or cohesive-frictional, and employs finite elements in conjunction with the upper bound theorem of classical plasticity theory. The computational procedure uses three-noded triangular elements with the unknown velocities as the nodal variables. An additional set of unknowns, the plastic multiplier rates, is associated with each element. Kinematically admissible velocity discontinuities are permitted along specified planes within the grid. The finite element formulation of the upper bound theorem leads to a classical linear programming problem where the objective function, which is to be minimized, corresponds to the dissipated power and is expressed in terms of the velocities and plastic multiplier rates. The unknowns are subject to a set of linear constraints arising from the imposition of the flow rule and velocity boundary conditions. It is shown that the upper bound optimization problem may be solved efficiently by applying an active set algorithm to the dual linear programming problem. Since the computed velocity field satisfies all the conditions of the upper bound theorem, the corresponding limit load is a strict upper bound on the true limit load. Other advantages include the ability to deal with complicated loading, complex geometry and a variety of boundary conditions. Several examples are given to illustrate the effectiveness of the procedure.


Journal ArticleDOI
TL;DR: An investigation was conducted of the qualitative properties of a class of neural networks described by a system of first-order linear ordinary differential equations defined on a closed hypercube of the state space with solutions extended to the boundary of the hypercube.
Abstract: An investigation was conducted of the qualitative properties of a class of neural networks described by a system of first-order linear ordinary differential equations which are defined on a closed hypercube of the state space with solutions extended to the boundary of the hypercube. When solutions are located on the boundary of the hypercube, the system is said to be in a saturated mode. The class of systems considered retains the basic structure of the Hopfield model but is easier to analyze, synthesize, and implement. An efficient analysis method is developed which can be used to determine completely the set of asymptotically stable equilibrium points and the set of unstable equilibrium points. The latter set can be used to estimate the domains of attraction for the elements of the former set. The class of systems considered can easily be implemented in analog integrated circuits. The applicability of the results is demonstrated by means of several examples. >

Proceedings ArticleDOI
05 Nov 1989
TL;DR: It is demonstrated that the ratio cut algorithm can locate the clustering structures in the circuit and as much as 70% improvement over the Kernighan-Lin algorithm in terms of the proposed ratio metric.
Abstract: A partitioning approach called ratio cut is proposed. The authors demonstrate that the ratio cut algorithm can locate the clustering structures in the circuit. Finding the optimal ratio cut is NP-complete. However, in certain cases the ratio cut can be solved by linear programming techniques via the multicommodity flow problem. They also propose a fast heuristic algorithm running in linear time with respect to the number of pins in the circuit. Experiments show good results in all tested cases, and as much as 70% improvement over the Kernighan-Lin algorithm in terms of the proposed ratio metric. >

Journal ArticleDOI
TL;DR: A new method for solving linear programming problems with fuzzy parameters in the objective function where the information contained in the membership functions can be used to any extent by a method called ‘α-level related pair formation’.

Journal ArticleDOI
TL;DR: In this paper, a new methodology is developed for determining the optimal (minium-cost) design of water distribution systems based on a generalized reduced gradient model to solve a problem that is reduced in size and complexity by implicitly solving the conservation of mass and energy equations.
Abstract: A new methodology is developed for determining the optimal (minium‐cost) design of water distribution systems. The components that can be sized are the pipe network, pumps or pump station, and tanks. In addition, the optimal settings for control and pressure‐reducing valves can be determined. This methodology couples nonlinear programming techniques with existing water distribution simulation models. Previous methodologies have typically simplified the system hydraulics to be able to solve the optimization problem. This new methodology retains the generality of the hydraulic simulation model so that the problem is only limited by the ability of the simulation model rather than the optimization model. The methodology uses a generalized reduced gradient model to solve a problem that is reduced in size and complexity by implicitly solving the conservation of mass and energy equations using the hydraulic simulator and an augmented Lagrangian approach to incorporate pressure head bounds in the objective functi...

Journal ArticleDOI
TL;DR: An extension of Karmarkar's linear programming algorithm for solving a more general group of optimization problems: convex quadratic programs, based on the iterated application of the objective augmentation and the projective transformation, followed by optimization over an inscribing ellipsoid centered at the current solution.
Abstract: We present an extension of Karmarkar's linear programming algorithm for solving a more general group of optimization problems: convex quadratic programs. This extension is based on the iterated application of the objective augmentation and the projective transformation, followed by optimization over an inscribing ellipsoid centered at the current solution. It creates a sequence of interior feasible points that converge to the optimal feasible solution in O(Ln) iterations; each iteration can be computed in O(Ln3) arithmetic operations, wheren is the number of variables andL is the number of bits in the input. In this paper, we emphasize its convergence property, practical efficiency, and relation to the ellipsoid method.

Journal ArticleDOI
TL;DR: It is proven that the mathematical expression of the GOF is independent of the choice of the sets of loops and paths along which the head constraints are formulated, contrary to the claim made by I. C. Goulter et al. (1986).
Abstract: A theoretical analysis of the linear programming (LP) gradient method for optimal design of water distribution networks is presented. The method was first proposed by A. Alperovits and U. Shamir (1977) and has received much attention in the last 10 years. It consists of two stages that are solved in alteration: (1) a LP problem is solved for a given feasible flow distribution and (2) a search is conducted in the space of flow variables, based on the gradient of the objective function (GOF). In this paper a matrix formulation is given for both stages using well-known graph theory matrices. It is proven that the mathematical expression of the GOF is independent of the choice of the sets of loops and paths along which the head constraints are formulated. This is contrary to the claim made by I. C. Goulter et al. (1986). The original GOF expression is shown to have been an approximation of the steepest direction, but still gives good results. Finally, the search procedure is improved by using the projected gradient method.

Proceedings ArticleDOI
P.M. Vaidya1
30 Oct 1989
TL;DR: A by-product of the algorithm is an algorithm for solving linear programming problems that performs a total of O(mn/sup 2/L+M(n)nL) arithmetic operations in the worst case, which gives an improvement in the time complexity of linear programming for m>n/Sup 2/.
Abstract: An algorithm for minimizing a convex function over a convex set is given. The notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope is central to the algorithm. The algorithm has a much better rate of global convergence than the ellipsoid algorithm. A by-product of the algorithm is an algorithm for solving linear programming problems that performs a total of O(mn/sup 2/L+M(n)nL) arithmetic operations in the worst case, where m is the number of constraints, n the number of variables, and L a certain parameter. This gives an improvement in the time complexity of linear programming for m>n/sup 2/. >

Book ChapterDOI
01 Jan 1989
TL;DR: This chapter describes a short-step penalty function algorithm that solves linear programming problems in no more than O(n 0.5 L) iterations and follows the path of optimal solutions for the penalized functions as in a predictor-corrector homotopy algorithm.
Abstract: This chapter describes a short-step penalty function algorithm that solves linear programming problems in no more than O(n 0.5 L) iterations. The total number of arithmetic operations is bounded by O(n 3 L), carried on with the same precision as that in Karmarkar’s algorithm. Each iteration updates a penalty multiplier and solves a Newton-Raphson iteration on the traditional logarithmic barrier function using approximated Hessian matrices. The resulting sequence follows the path of optimal solutions for the penalized functions as in a predictor-corrector homotopy algorithm.

Journal ArticleDOI
TL;DR: Preliminary computational results indicate that this implementation compares favorably with a comparable implementation of a dual affine interior point method, and with MINOS 5.0, a state-of-the-art implementation of the simplex method.
Abstract: The purpose of this paper is to describe in detail an implementation of a primal-dual interior point method for solving linear programming problems. Preliminary computational results indicate that this implementation compares favorably with a comparable implementation of a dual affine interior point method, and with MINOS 5.0, a state-of-the-art implementation of the simplex method. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Journal ArticleDOI
TL;DR: In this article, the authors describe the convex hull of the incidence vectors of stable matchings and propose a linear program to solve the optimal stable marriage problem as a linear programming problem.

Journal ArticleDOI
TL;DR: This work presents computational experience with a cutting plane algorithm for 0–1 quadratic programming without constraints based on a reduction of this problem to a max-cut problem in a graph and on a partial linear description of the cut polytope.
Abstract: We present computational experience with a cutting plane algorithm for 0–1 quadratic programming without constraints. Our approach is based on a reduction of this problem to a max-cut problem in a graph and on a partial linear description of the cut polytope.

Journal ArticleDOI
TL;DR: In this article, the authors introduce a general framework that guides the management scientist's formulation of deterministic models of production processes and reformulate the constraints of familiar linear programming-based planning models to specifically treat components of production lead time, thereby realizing a more accurate representation of the production process.
Abstract: We introduce a general framework that guides the management scientist's formulation of deterministic models of production processes. Using the framework, we reformulate the constraints of familiar linear programming-based planning models to specifically treat components of production lead time, thereby realizing a more accurate representation of the production process. In addition, the reformulation accommodates noninteger values for lead times as well as unequal-length planning periods. Manufacturing Resources Planning (MRP) and the Critical Path Method (CPM) are recast in terms of the framework, revealing opportunities for model generalization and extension, and their relationship to linear programming models.

Journal ArticleDOI
TL;DR: In this article, the authors present linear programming in infinite-dimensional spaces and show that linear programming can be used to solve problems in the real world as well as in the virtual world.
Abstract: (1989). Linear Programming in Infinite-Dimensional Spaces. Journal of the Operational Research Society: Vol. 40, No. 1, pp. 109-110.

Journal ArticleDOI
TL;DR: A maximin model for IRT-based test design is proposed that serves as a constraint subject to which a linear programming algorithm maximizes the information in the test.
Abstract: A maximin model for IRT-based test design is proposed. In the model only the relative shape of the target test information function is specified. It serves as a constraint subject to which a linear programming algorithm maximizes the information in the test. In the practice of test construction, several demands as linear constraints in the model. A worked example of a text construction problem with practical constraints is presented. The paper concludes with a discussion of some alternative models of test construction.

Journal ArticleDOI
TL;DR: In this article, the affine scaling vector fields are defined for linear programs of a special form, called standard form and canonical form, respectively, and the trajectories obtained by integrating these vector fields, called P-trajectories, are studied using a nonlinear change of variables called Legendre transform coordinates, which is a projection of the gradient of a logarithmic barrier function.
Abstract: Karmarkar's projective scaling algorithm for solving linear programming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem. This paper studies the set of trajectories obtained by integrating this vector field, called P-trajectories, as well as a related set of trajectories, called A-trajectories. The A-trajectories arise from another linear programming algorithm, the affine scaling algorithm. The affine and projective scaling vector fields are each defined for linear programs of a special form, called standard form and canonical form, respectively. These trajectories are studied using a nonlinear change of variables called Legendre transform coordinates, which is a projection of the gradient of a logarithmic barrier function. The Legendre transform coordinate mapping is given by rational functions, and its inverse mapping is algebraic. It depends only on the constraints of the linear program, and is a one-to-one mapping for canonical form linear programs. When the polytope of feasible solutions is bounded, there is a unique point mapping to zero, called the center. The A-trajectories of standard form linear programs are linearized by the Legendre transform coordinate mapping. When the polytope of feasible solutions is bounded, they are the complete set of geodesics of a Riemannian geometry isometric to Euclidean geometry. Each A-trajectory is part of a real algebraic curve. Each P-trajectory for a canonical form linear program lies in a plane in Legendre transform coordinates. The P-trajectory through 0 in Legendre transform coordinates, called the central P-trajectory, is part of a straight line, and is contained in the A-trajectory through 0, called the central A-trajectory. Each P-trajectory is part of a real algebraic curve. The central A-trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form (c, x) = ,u . It is also the set of minima of a parametrized family of logarithmic barrier functions. Power-series expansions are derived for the central A-trajectory, which is also the central P-trajectory. These power-series have a simple recursive form and are useful in developing "higher-order" analogues of Karmarkar's algorithm. A-trajectories are defined for a general linear program. Using this definition, it is shown that the limit point x,0 of a central A-trajectory on the boundary of the feasible solution polytope P is the center of the unique face of P containing x,0 in its relative interior. Received by the editors October 8, 1986 and, in revised form, June 9, 1987 and March 25, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 90C05; Secondary 52A40, 34A34. The first author was partially supported by ONR contract N00014-87-K0214. ( 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page