Showing papers on "Linear programming published in 1968"

Programming with linear fractional functionals

Abstract: Charnes and Cooper [1] showed that a linear programming problem with a linear fractional objective function could be solved by solving at most two ordinary linear programming problems. In addition, they showed that where it is known a priori that the denominator of the objective function has a unique sign in the feasible region, only one problem need be solved. In the present note it is shown that if a finite solution to the problem exists, only one linear programming problem must be solved. This is because the denominator cannot have two different signs in the feasible region, except in ways which are not of practical importance.

The fixed charge problem

Abstract: : A fundamental unsolved problem in the programming area is one in which various activities have fixed charges (e.g., set-up time charges) if operating at a positive level. Properties of a general solution to this type problem are discussed in this paper. Under special circumstances it is shown that a fixed charge problem can be reduced to an ordinary linear programming problem.

Optimum open pit mine production scheduling

Abstract: : The multi-period open pit mine production scheduling problem is formulated as a large scale linear programming problem using the block concept. A solution procedure is developed through decomposition and partitioning of the subproblem into elementary profit routing problems for which an algorithm is presented. Many of the traditional mine planning concepts are discussed and suggestions for improvement through use of the techniques developed in this thesis are given. In the development of the solution procedure, those constraints which govern the mining system are considered as the master problem. The constraints which dictate the sequence of extraction are used as the subproblem. The properties of the single period subproblem and its dual are discussed, and the dual problem is shown to be equivalent to a bipartite maximum flow problem for which an algorithm is given. The Multi-period subproblem algorithm is developed by partitioning by stages and using the properties of the single period subproblem. This treatment allows optimization of the complete mining-concentrating-refining system over the entire planning horizon and permits the system to dictate how and when to process a block of material.

Discrete Stochastic Programming

Abstract: A method is presented for solving linear programming problems where (any number of) the functional, restraint, and input-output coefficients are subject to discrete; probability distributions. The objective function is formulated in terms of variance and/or expectation. The procedure involves the simultaneous generation of all (mutually exclusive) possible outcomes and hence the transference of all variability into the objective function of a very much enlarged linear program.

Pattern Classifier Design by Linear Programming

Abstract: —A common nonparametric method for designing linear discriminant functions for pattern classification is the iterative, or "adaptive," weight adjustment procedure, which designs the discriminant function to do well on a set of typical patterns. This paper presents a linear programming formulation of discriminant function design which minimizes the same objective function as the "fixed-increment" adaptive method. With this formulation, as with the adaptive methods, weights which tend to minimize the number of classification errors are computed for both separable and nonseparable pattern sets, and not just for separable pattern sets as has been the emphasis in previous linear programming formulations.

Multichain Markov Renewal Programs

Abstract: : Two methods are presented for computing optimal decision sequences and their cost functions. The first method, called 'policy iteration,' is an adaption of an iterative scheme that is widely used for sequential decision problems. The second method is to specify a linear programming problem whose solution determines an optimal policy and its cost function. A third solution technique is shown to apply to certain, but not all, of these Markov renewal programs. As a byproduct of the development, new techniques are provided for solving a broad class of shortest-route problems. (Author)

Linear programming applied to water quality management

Abstract: Linear programming is applied to the management of water quality in a river basin. The charge is to select the efficiencies of the treatment plants on the river that will achieve the dissolved oxygen standards at a minimum cost. The objective function is structured in terms of the costs of the treatment plants. The principal constraints prevent violation of the dissolved oxygen standards. A simplified version of the Willamette River in Oregon is studied, using the linear programming formulation, and the results are compared with those obtained by dynamic programming. The effects of changes in the dissolved oxygen standards are explored by use of the dual variables.

Algorithm for discovering the set of all the solutions of a linear programming problem

Abstract: IN this paper two versions of a canonical algorithm for discovering all the optimal solutions of a linear programming problem with the condition of non-negativeness of the variables are presented: the first for the case of canonical notation, the second for the standard notation. The algorithm is obtained by applying the results of [1] (respectively [2]) to the following problem which is obviously equivalent to the linear programming problem indicated: for a finite system of linear equations (inequalities), one of the free terms of which is a parameter, to find the greatest value of this parameter for which the system is consistent, and all its non-negative solutions for this greatest value of the parameter.

The Consistent Assessment and Fairing of Preference Functions

Abstract: When a decision maker is assessing a preference (utility) function for assets (wealth), it is natural for him to start by making some quantitative assessments of the certainty equivalents of a few simple gambles and some qualitative statements specifying any regions in which he feels risk-averse or risk-seeking and any regions in which he feels decreasingly or increasingly risk-averse or risk-seeking. Several questions then arise. Does any preference function exist which satisfies all the quantitative and qualitative restrictions simultaneously, that is, are the restrictions consistent? If so, how far do they determine the preference function? How might one fair a "smooth" function satisfying the restrictions? This paper is addressed to these questions. First the problem is introduced in some detail, and the concepts involved reviewed. Then the case is considered where the qualitative restrictions only specify regions of risk-aversion or risk-seeking. It turns out in this case that all the restrictions are linear in certain quantities, so that the existence problem is essentially one of satisfying linear constraints. Furthermore, finding the maximum or minimum solution at a specified point is exactly a linear programming problem. Also discussed briefly are the possibility that some smoothing problems might simply introduce a nonlinear objective function (though the general smoothing problem is more complicated) and the problem of making the derivative of the preference function continuous (which is not always possible). If regions of increasing or decreasing risk-aversion are also given, the problem becomes much more difficult.

Allocation of Transportation Units to Alternative Trips—A Column Generation Scheme with Out-of-Kilter Subproblems

Abstract: The problem of allocating transportation units to alternative trips can be solved by linear programming. However, the linear programming formulation has three shortcomings: (1) the optimal solution obtained does not have an integer number of trips; (2) the optimal solution may have sets of disconnected cycles; and (3) problems of considerable size will be intractable because of the vast number of variables and constraints. In some applications, it has been found that shortcomings 1 and 2 are not serious, but the third one—the size of the problem—presents considerable difficulty. This paper describes an approach—a column generation scheme with out-of-kilter subproblems—that reduces the size of the problem considerably. An algorithm for the method is given, and it is proved that the algorithm solves the problem in a finite number of steps. Computational experience thus far with this method has been limited, but encouraging.

Linear Programming as an Aid in Planning Kilovar Requirements

Abstract: In the planning and design of a high-voltage transmission network it is sometimes desirable to install controllable kilovar ar capacity at several locations to support bus voltages during emergencies. This problem arises when transmission line or generation outages cause bus voltage magnitudes to decrease below desirable limits. The problem of selecting where and how much kilovar capacity is required has many feasible solutions which satisfy the conditions imposed. A method for locating that solution with the minimum total installed capacity is presented. This nonlinear programming problem will be solved by using a linear approximation, solving the linear programming problem, and then correcting the linear approximation through the use of differences between the linear and nonlinear results. The method is illustrated by an application to a portion of a large high-voltage network.

An Iterative Chebyshev Approximation Method for Network Design

Abstract: One of the most important problems of computeraided network design is the optimization of network characteristics by iterative calculation. In this paper, the problem of realizing a network whose transmission characteristics approximate a given function in Chebyshev sense is treated as a nonlinear programming problem, and a method of solving this problem by successively solving linear programming problems, which are derived by locally linearizing the original nonlinear programming problem, is proposed. An improvement of the method for reducing the computation time is also considered and is proved to be practical and very effective by many design examples.

On minimization of trajectory sensitivity

Abstract: The paper explores the engineering potential of incorporating quadratic sensitivity terms in linear optimization problems as a means of sensitivity reduction to plant-parameter deviations in linear feedback control systems.

A zero-one programming approach to scheduling with limited resources,

Abstract: : A zero-one linear programming formulation of scheduling problems that is more versatile and efficient than any other known LP formulation. It is the first formulation designed to accommodate multiple resource constraints. Bowman's formulation can be extended to do so, but in a sample comparison, it required 72 variables and 125 constraints for a 3-project, 8-job, 3-resource problem compared with 33 variables and 48 constraints for the new formulation. The new method also showed substantial improvement when compared with schedules based on two common dispatching rules: first-come-first-served and earliest-completion-date. Execution time on an IBM 7044, using the Geoffrion computer code was 3 seconds. Other objective functions are formulated in the study and additional constraints are modeled. Examples of larger problems rewritten for standard LP computer codes showed that the number of variables and constraints involved are probably within no more than one order-of-magnitude of the maximum number that could be handled with available zero-one computer codes. (Author)

The numerical solution of network problems using the out-of-kilter algorithm

Abstract: : Network flow problems arise in the solution of transportation and scheduling problems. Divided into four substantially independent sections, this Memorandum: (1) reviews the types of problems that are representable as capacitated network problems; (2) explains (with diagrams) the out-of-kilter algorithm and techniques for implementing it on a computer; (3) describes modification of the algorithm to a two-phase algorithm; (4) presents a method for labeling the nodes by means of a scan list. Tentative conclusions are that the two-phase algorithm is undesirable, and that the labeling procedure shortens computer time at the cost of using more memory.

A Note on the Application of Linear Programming to Capital Budgeting

Abstract: The application of linear programming techniques to the problem of capital budgeting has repeatedly been proposed in the literature. However, while the potential of linear programming models for this important area of business decisions is generally recognized, practical applications still face some severe limitations. This note focuses on one particular problem peculiar to the application of programming techniques to capital budgeting, namely the mutual dependence between the optimal solution of the linear programming model and the discount rate used to calculate the coefficients of its objective function.

Letter to the Editor—A Note on the Generalized Lagrange Multiplier Solution to an Integer Programming Problem

Abstract: The problem of applying Generalized Lagrange Multipliers (GLM) to 0-1 integer programming problems is investigated. It is shown that GLM can produce optimal solutions if and only if these solutions are extreme points of the convex set of linear equations. The presentation is based on a capital allocation problem recently discussed by Kaplan. Kaplan's procedure for finding optimal solutions that are not extreme points is discussed in terms of linear programming.

A numerical investigation of the simplex method

Abstract: : The report analyzes the behavior of the round-off errors associated with three different computer implementations of the simplex method of linear programming. One of the three is representative of computer implementations in common use, and it is shown that the standard method of updating the basic- matrix inverse is numerically unstable. The remaining two implementations are suggested by the author and use triangular decompositions of the basic matrix as substitutes for its inverse. The implementations are shown to be stable, and one of them is shown to be competitive in speed with the standard simplex-method computer implementations. Error bounds which may be calculated from intermediate results are developed for each of the three implementations, and their use during computation for error monitoring and control is discussed.

Automatic load scheduling in a multiarea power system

Abstract: In the paper an algorithm is described for solving the problem of secure economic load scheduling on a large power system. The method is an extension of that of Wells, which solves the problem as a linear program by means of the dual simplex algorithm. A dual form of the decomposition principle of Dantzig and Wolfe is used in order to reduce the size of the problem, so that it can be solved efficiently on a computer system. This is correlated with a corresponding decomposition of the power system into autonomous areas, co-ordinated by the specification of recommended boundary transfers and spinning spare capacity. Further levels of decomposition are incorporated in a similar fashion, allowing three or more levels, say, for the CEGB supply system. Although the entire problem can be solved on a single computer, for application to online control a more complex computing system is anticipated.

Finding minimal cost-time ratio circuits,

Abstract: : An efficient method is given for finding minimal cost-time ratio circuits in routing problems through the use of the out-of-kilter algorithm. The problem of finding a cycle in a network having a minimal cost-to-time ratio has been considered previously, and column generators have been used to introduce, into the basis of the master problem, the solution that corresponds to this cycle. The subproblem is of independent interest and corresponds to deterministic single chain Markov renewal programming. The main difference between earlier approaches and the present one is that where others have used a shortest path method corresponding to the simplex method (except that steepest descent is not used), here the flow circulation problem for optimal cost-time tradeoff is solved parametrically by the out-of-kilter algorithm. (Author)

Derivation of optimal livestock rations using quadratic programming

Abstract: Linear programming has been used extensively as a procedure for formulating least-cost livestock rations. More recently attention has heen focused on the incorporation of information on animal performance into the linear programming derivation of optimum livestock rations. Where the production is a quadratic function of nutrient inputs and where there are a large number of possible ration ingredients, parametric quadratic programming is shown to be an efficient computational technique for the derivation of “expansion path” rations. The region of economic interest along the expansion path is identified for the liveweight gain interval described by the production function. The rise of information contained in programming solutions to evaluate the economic significance of nutrient requirements and to aid in the search for possible ration ingredients is briefly discussed.

Algorithms: Algorithm 341: solution of linear programs in 0-1 variables by implicit enumeration

Abstract: The following procedures are based on the Cooley-Tukey algor i thm [1] for comput ing the finite Fourier t r ans fo rm of a complex da ta vector; the dimension of the da t a vector is assumed here to be a power of two. Procedure COMPLEXTRANSFORM computes ei ther the complex Fourier t ransform or its inverse. Procedure REALTRANSFORM computes ei ther the Fourier coefficients of a sequence of real da ta points or eva lua tes a Fourier series wi th given cosine and sine coefficients. The number of ar i thmet ic operat ions for ei ther procedure is proport ional to n logs n, where n is the number of da ta points. Procedures FFT2, REVFFT2, REORDER, and REAL TRAN are building blocks, and are used in the two complete procedures ment ioned above. The fas t t r ans fo rm can be computed in a number of different ways, and these bui lding block procedures were wri t ten so as to make practical the comput ing of large t rans forms on a system wi th vir tual memory. Using a method proposed by Singleton [2], d a t a is accessed in sub-sequences of consecutive a r ray elements , and as m u ch comput ing as possible is done in one section of the d a t a before moving on to another. Procedure FFT2 computes the Fourier t r ans fo rm of da ta in normal order, giving a resu l t in reverse b ina ry order. Procedure REVFFT2 computes the Fourier t r ans fo rm of da ta in reverse b ina ry order and leaves the resul t in normal b inary order. Procedure REORDER permutes a complex vector f rom b inary to reverse b ina ry order or f rom reverse b inary to b inary order; this procedure also permutes real da ta in prepara t ion for efficient use of the complex Fourier t ransform. Procedures FFT2, REVFFT2, and REORDER m a y also be used to compute mul t iva r i a te Fourier t ransforms. The procedure R E A L T R A N is used to unscramble and combine the complex t rans forms of the even and odd numbered e lements of a sequence of real d a t a points . This procedure is not restr ic ted to powers of two and can be used whenever the number of da t a points is even.

An Approach to Zero-One Integer Programming

Abstract: By starting with an all-integer zero-one linear programming problem, it is possible to develop a modified, possibly linear, programming problem that provides a characterization of the basis corresponding to a feasible zero-one solution to the integer problem. This characterization is based on the number of variables equal to one in the feasible solution. This paper develops an approach to zero-one programming based on this characterization. The method uses the criterion function of the original problem as a constraint, and then generates a sequence of feasible zero-one solutions, each with a greater value of the objective function. The solution technique is terminated when no more feasible solutions can be found, indicating that the last feasible solution determined is the optimum.

A Simple Class of Parametric Linear Programming Problems

Abstract: To find changes in coefficients that can occur without affecting the choice of optimal feasible basic variables is a well known problem in linear programming sensitivity theory. This paper shows that there is a very simple solution to this problem when the m × m matrix A of optimal basis vectors is varied parametrically, provided the matrix B of variations is chosen so that the inverse of A + λ B is linear in λ. It gives a general expression for such matrices B, which allows a considerable degree of arbitrariness; in particular, there exist B's, that can have rank as high as m/2. Extensions to include variations in the nonbasic part of the coefficient matrix and in the objective function coefficients are briefly described.