Showing papers on "Linear programming published in 1981"
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01 Jan 1981
TL;DR: Formulation skills for problems as a deterministic linear mathematical model, application of software to solve the problems and extensive sensitivity analysis to answer “what if” questions.
Abstract: Formulation skills for problems as a deterministic linear mathematical model (linear programming, goal (multi-objective) programming, integer programming, transportation, transshipment) Travel Sales Person (TSP) problems Simplex method for solving linear programming (LP) Dual of an LP and its application Extensive sensitivity analysis to answer “what if” questions (for all models) Application of software to solve the problems
1,514 citations
TL;DR: Issues have, in general, not been well understood, including the exact character of the ellipsoid method and of Khachiyan's result on polynomiality, its practical significance inlinear programming, its implementation, its potential applicability to problems outside of the domain of linear programming, and its relationship to earlier work.
Abstract: In February 1979 a note by L. G. Khachiyan indicated how an ellipsoid method for linear programming can be implemented in polynomial time. This result has caused great excitement and stimulated a flood of technical papers. Ordinarily there would be no need for a survey of work so recent, but the current circumstances are obviously exceptional. Word of Khachiyan's result has spread extraordinarily fast, much faster than comprehension of its significance. A variety of issues have, in general, not been well understood, including the exact character of the ellipsoid method and of Khachiyans result on polynomiality, its practical significance in linear programming, its implementation, its potential applicability to problems outside of the domain of linear programming, and its relationship to earlier work. Our aim is to help clarify these important issues in the context of a survey of the ellipsoid method, its historical antecedents, recent developments, and current research.
297 citations
TL;DR: In this paper, an optimization technique is presented for use in planning and designing looped water distribution systems, which is iterative, employing linear programming and a gradient procedure; and it produces a locally optimal solution.
Abstract: An optimization technique is presented for use in planning and designing looped water distribution systems. The technique is iterative, employing linear programming and a gradient procedure; and it produces a locally optimal solution. A demonstration using data for the New York City water supply system, shows a potential cost saving of 13 percent for the solution obtained using the new method, in comparison to the best design developed in an earlier study.
272 citations
TL;DR: The present paper unifies two studies of linear programming problems for which the greedy algorithm works, and establishes the converse of each theorem.
268 citations
TL;DR: In this article, a surrogate constraints algorithm for nonlinear programming, nonlinear integer programming, and nonlinear mixed integer programming problems is presented, which contains a new technique for generating a succession of vector values of surrogate multiplier (i.e., surrogate problems).
Abstract: This paper presents a surrogate constraints algorithm for solving nonlinear programming, nonlinear integer programming, and nonlinear mixed integer programming problems. The algorithm contains a new technique for generating a succession of vector values of surrogate multiplier (ie, surrogate problems). By using this technique, a computer can keep a polyhedron, which is a vector space of surrogate multipliers to be considered at a certain time, in its memory. Furthermore it can cut the polyhedron by a given hyperplane, and produce the remaining space as the next polyhedron. Simple examples are included.
205 citations
TL;DR: A new approach to the one-dimensional cutting stock problem is described and compared to the classical model for which Gilmore and Gomory have developed a special column-generation technique, characterized by a dynamic use of simply structured cutting patterns.
Abstract: A new approach to the one-dimensional cutting stock problem is described and compared to the classical model for which Gilmore and Gomory have developed a special column-generation technique. The new model is characterized by a dynamic use of simply structured cutting patterns. Nevertheless, it enables the representation of complex combinations of cuts. It can be advantageous in practical applications where many different stock lengths or a relatively large number of order lengths have to be dealt with. The new approach is applied to a real problem where the “trim loss” is not valueless, since it can be used for further demands arising in later planning periods.
174 citations
TL;DR: This paper presents a method of obtaining a time schedule of velocities and accelerations along the path that the manipulator may adopt to obtain a minimum traveling time, under the constraints of composite Cartesian limit on linear and angular velocity and acceleration.
Abstract: To assure a successful completion of an assigned task without interruption, such as the collision with fixtures, the hand of a mechanical manipulator often travels along a preplanned path. An advantage of requiring the path to be composed of straight-line segments in Cartesian coordinates is to provide a capability for controlled interaction with objects on a moving conveyor. This paper presents a method of obtaining a time schedule of velocities and accelerations along the path that the manipulator may adopt to obtain a minimum traveling time, under the constraints of composite Cartesian limit on linear and angular velocities and accelerations. Because of the involvement of a linear performance index and a large number of nonlinear inequality constraints, which are generated from physical limitations, the “method of approximate programming (MAP)” is applied. Depending on the initial choice of a feasible solution, the iterated feasible solution, however, does not converge to the optimum feasible point, but is often entrapped at some other point of the boundary of the constraint set. To overcome the obstacle, MAP is modified so that the feasible solution of each of the iterated linear programming problems is shifted to the boundaries corresponding to the original, linear inequality constraints. To reduce the computing time, a “direct approximate programming algorithm (DAPA)” is developed, implemented and shown to converge to optimum feasible solution for the path planning problem. Programs in FORTRAN language have been written for both the modified MAP and DAPA, and are illustrated by a numerical example for the purpose of comparison.
173 citations
TL;DR: This work proposes an alternative solution to the discriminant problem that requires little more than a minimum familiarity with linear programming and shows promise for eliminating the complexities of conventional statistical approaches without sacrificing the essential power of existing methods.
Abstract: We propose an alternative solution to the discriminant problem, one that requires little more than a minimum familiarity with linear programming. The approach shows promise for eliminating the complexities of conventional statistical approaches without sacrificing the essential power of existing methods.
171 citations
TL;DR: In this paper, a distribution planning model is formulated which considers existing and potential substation locations, their capacities and costs, together with the primary feeder network represented by small area demand locations to represent non-uniform loads, and feeder segments having variable distribution costs and limited capacities.
Abstract: A distribution planning model is formulated which considers existing and potential substation locations, their capacities and costs, together with the primary feeder network represented by small area demand locations to represent non-uniform loads, and feeder segments having variable distribution costs and limited capacities. A branch and bound search method is described which utilizes a shortest path table to obtain lower bounds and solutions from a transshipment linear programming model for upper bounds. The solution of a small example is presented in detail, and computational results for several larger problems are summarized.
156 citations
TL;DR: In this paper, the allocation of different categories of resources in a network project under multiple conflicting criteria including project duration or maximum lateness, and several cost criteria is discussed, where the resource requirements of the splittable activities are assumed to be discrete, i.e., assuming arbitrary values from given finite sets for particular resource types and categories.
143 citations
TL;DR: A constraint generation algorithm and a branch-and-bound algorithm that uses linear programming relaxations that uses greedy heuristics to produce feasible solutions, which, in turn, are used to generate upper bounds.
Abstract: We consider integer programming formulations of problems that involve the maximization of submodular functions. A location problem and a 0–1 quadratic program are well-known special cases. We give a constraint generation algorithm and a branch-and-bound algorithm that uses linear programming relaxations. These algorithms are familiar ones except for their particular selections of starting constraints, subproblems and partitioning rules. The algorithms use greedy heuristics to produce feasible solutions, which, in turn, are used to generate upper bounds. The novel features of the algorithms are the performance guarantees they provide on the ratio of lower to upper bounds on the optimal value.
TL;DR: The problem of cyclic staff scheduling is solved by a linear programming round-off heuristic for which a bound on the absolute error is established and the quality of the bound improves as the matrix of resource availability approximates the property of consecutive ones.
Abstract: The problem of cyclic staff scheduling is solved by a linear programming round-off heuristic for which a bound on the absolute error is established. The quality of the bound is independent of the resource requirements. Moreover, the quality of the bound improves as the matrix of resource availability approximates the property of consecutive ones. The appropriateness of the heuristic is further established by showing that cyclic staff scheduling is NP-complete.
TL;DR: Three algorithms for solving linear programming problems in which some or all of the objective function coefficients are specified in terms of intervals, which are most suitable to linear programs in which the objectivefunction coefficients are deterministic but are likely to vary from time period to time period.
Abstract: This paper presents three algorithms for solving linear programming problems in which some or all of the objective function coefficients are specified in terms of intervals. Which algorithm is applicable depends upon a the number of interval objective function coefficients, b the number of nonzero objective function coefficients, and c whether or not the feasible region is bounded. The algorithms output all extreme points and unbounded edge directions that are “multiparametrically optimal” with respect to the ranges placed on the objective function coefficients. The algorithms are most suitable to linear programs in which the objective function coefficients are deterministic but are likely to vary from time period to time period as for example in blending problems.
TL;DR: A branch-and-bound algorithm for solving fixed charge transportation problems where not all cells exist that exploits the absence of full problem density in several ways, yielding a procedure which is especially applicable to solving real world problems which are normally quite sparse.
Abstract: This paper presents a branch-and-bound algorithm for solving fixed charge transportation problems where not all cells exist. The algorithm exploits the absence of full problem density in several ways, thus yielding a procedure which is especially applicable to solving real world problems which are normally quite sparse. Additionally, streamlined new procedures for pruning the decision tree and calculating penalties are presented. We present computational experience with both a set of large test problems and a set of dense test problems from the literature. Comparisons with other codes are uniformly favorable to the new method, which runs more than twice as fast as the best alternative.
01 Jan 1981
TL;DR: In this article, the running time of the algorithm is polynomial in the number of digits of the coefficients, which can be applied to solve linear programs in poly(n) time.
Abstract: L.G. Khachiyan’s algorithm to check the solvability of a system of linear inequalities with integral coefficients is described. The running time of the algorithm is polynomial in the number of digits of the coefficients. It can be applied to solve linear programs in polynomial time.
01 Jan 1981
TL;DR: The special partitioning method, called the simplex special ordered network (SON) procedure, applies to LP problems that contain both non-network rows and non- network columns, with no restriction on the form of the rows and columns that do not exhibit a network structure.
Abstract: This paper develops a special partitioning method for solving LP problems with embedded network structure. These problems include many of the large-scale LP problems of practical importance, particularly in the fields of energy, scheduling, and distribution. The special partitioning method, called the simplex special ordered network (SON) procedure, applies to LP problems that contain both non-network rows and non-network columns, with no restriction on the form of the rows and columns that do not exhibit a network structure.
TL;DR: In this paper, the authors considered the problem of irreducibly inconsistent linear inequalities and provided necessary and sufficient conditions for a system to be irreduceibly inconsistent, which can be found by applying a simplex algorithm.
TL;DR: This paper describes an effort based on state-of-the-art, modular linear programming software (IBM's MPSX/370), which indicates that much improvement is possible through advanced implementations and careful selection of computational strategies.
Abstract: Since the original work of Dantzig and Wolfe in 1960, the idea of decomposition has persisted as an attractive approach to large-scale linear programming However, empirical experience reported in the literature over the years has not been encouraging enough to stimulate practical application Recent experiments indicate that much improvement is possible through advanced implementations and careful selection of computational strategies This paper describes such an effort based on state-of-the-art, modular linear programming software (IBM's MPSX/370)
TL;DR: In this paper, the economic dispatch problem with constraints on line flows and spinning reserve is formulated as a linear optimization program, which is then solved using the Dantzig-Wolfe decomposition principle.
Abstract: The economic dispatch problem with constraints on line flows and spinning reserve is formulated as a linear optimization program. Such a linear program is then solved using the Dantzig-Wolfe decomposition principle. The sub-programs of the decomposition may correspond to physical areas of the power network, and are individually solved employing the revised simplex method with upper bounds. A fast decoupled load flow algorithm is used to calculate penalty and sensitivity factors. Several illustrative examples are included.
TL;DR: It appears that all the standard algorithms terminate by constructing primal and dual feasible solutions of equal value, i.e., by satisfying generalised optimality conditions.
Abstract: We survey some recent developments in duality theory with the idea of explaining and unifying certain basic duality results in both nonlinear and integer programming. The idea of replacing dual variables (prices) by price functions, suggested by Everett and developed by Gould, is coupled with an appropriate dual problem with the consequence that many of the results resemble those used in linear programming. The dual problem adopted has a (traditional) economic interpretation and dual feasibility then provides a simple alternative to concepts such as conjugate functions or subdifferentials used in the study of optimality. In addition we attempt to make precise the relationship between primal, dual and saddlepoint results in both the traditional Lagrangean and the more general duality theories and to see the implications of passing from prices to price functions. Finally, and perhaps surprisingly, it appears that all the standard algorithms terminate by constructing primal and dual feasible solutions of equal value, i.e., by satisfying generalised optimality conditions.
01 Dec 1981
TL;DR: In this article, a linear programming (LP) approach is developed for control problems which contain inequality constraints on state and control variables, and a suboptimal feedback control policy results since the LP calculations are repeated at each sampling instant.
Abstract: A linear programming (LP) approach is developed for control problems which contain inequality constraints on state and control variables. A suboptimal feedback control policy results since the LP calculations are repeated at each sampling instant. Simulation results illustrate that the on-line computational requirements are modest and that the new LP approach compares favorably with alternate approaches.
TL;DR: The XVERT, WYNN, EXCHANGE, CONSIM, and CADEX algorithms are shown to be useful aids in constructing linear and quadratic model designs when the region of feasible blends is restricted by single-component and multiple-component constraints.
Abstract: The construction of gasoline blending models is discussed to illustrate some of the practical problems encountered in mixture experimentation. Attention is focused on the use and modification of the simplex and extreme vertices designs in the development of blending models. The XVERT, WYNN, EXCHANGE, CONSIM, and CADEX algorithms are shown to be useful aids in constructing linear and quadratic model designs when the region of feasible blends is restricted by single-component and multiple-component constraints. The evaluation of competing models and the use of the quadratic blending model in conjunction with linear programming calculations are also discussed. The methodology is general and can be used in all types of mixture experiments and product formulation studies, Examples are included to illustrate the use of the design algorithms and models.
TL;DR: A set of staircase linear programming test problems is made available for computational experiments to solve the problem of how to program a staircase in a linear fashion.
Abstract: A set of staircase linear programming test problems are made available for computational experiments.
TL;DR: This work gives explicit sparsity-preserving SOR (successive overrelaxation) algorithms for the solution of separable quadratic and linear programming problems that preserve the sparsity structure of the problem and do not require the computation of the product of the constraint matrix by its transpose.
TL;DR: A new and simple algorithm for the least absolute value regression problem based on the notion of “edge” descent along the surface of the objective function that is comparable or better in computational efficiency to current linear programming approaches for roughly 4 or fewer independent variables.
Abstract: This paper presents a new and simple algorithm for the least absolute value regression problem. It is based on the notion of “edge” descent along the surface of the objective function. It is comparable or better in computational efficiency to current linear programming approaches for roughly 4 or fewer independent variables.
TL;DR: Theoretical aspects of the programming problem of maximizing the minimum value of a set of linear functionals subject to linear constraints are explored and an optimality condition is developed.
Abstract: Theoretical aspects of the programming problem of maximizing the minimum value of a set of linear functionals subject to linear constraints are explored. Solution strategies are discussed and an optimality condition is developed. An algorithm is also presented.
TL;DR: It is shown that two prototype models of complementary pivot and fixed point theory and the corresponding path following solution methods are conceptually equivalent.
Abstract: It is our purpose here to show that two prototype models of complementary pivot and fixed point theory and the corresponding path following solution methods are conceptually equivalent.
TL;DR: This paper first reformulates the model as a linear complementarity problem and then applies the parametric principal pivoting algorithm for its solution, leading to the study of an “arc—arc weighted adjacency matrix” associated with a simple digraph having weights on the nodes.
Abstract: This paper presents a parametric linear complementarity technique for the computation of equilibrium prices in a single commodity spatial model We first reformulate the model as a linear complementarity problem and then apply the parametric principal pivoting algorithm for its solution This reformulation leads to the study of an “arc—arc weighted adjacency matrix” associated with a simple digraph having weights on the nodes Several basic properties of such a matrix are derived Using these properties, we show how the parametric principal pivoting algorithm can be greatly simplified in this application Finally, we report some computational experience with the proposed technique for solving some large problems