Showing papers on "Linear programming published in 1970"

How good is the simplex algorithm

Abstract: : By constructing long 'increasing' paths on appropriate convex polytopes, It is shown that the simplex algorithm for linear programs (at least with its most commonly used pivot rule) is not a 'good algorithm' in the sense of J. Edmonds. That is, the number of pivots or iterations that may be required is not majorized by any polynomial function of the two parameters that specify the size of the program. (Author)

Transmission Network Estimation Using Linear Programming

Abstract: One aspect of long-range planning of electric power systems involves the exploration of various designs for the bulk power transmission network. The use of linear programming for network analysis to determine where capacity shortages exist and, most importantly, where to add new circuits to relieve the shortages is presented. The new method of network estimation produces a feasible transmission network with near-minimum circuit miles using as input any existing network plus a load and generation schedule. An example is used to present the two steps of the method: 1) linear flow estimation and 2) new circuit selection. The method has become a fundamental part of computer programs for transmission network synthesis.

Cash Flows in Networks

Abstract: The concept of maximising present value is applied to the timing of activities in a network. The mathematical form of the problem is that of maximising a nonlinear function subject to linear constraints and can be solved as a succession of linear programmes. By application of duality principles the problem can be treated as a form of maximum value flow problem in which discounted cash flows are distributed along arcs from pay events to receipt events. The solution is aided by the “equilibrium theorem” of dual linear programming in that in the optimum condition flows occur only along arcs whose corresponding activity has zero float. The flows which occur in the optimally scheduled solution are directly proportional to the marginal cost which would be incurred by lengthening the activity corresponding to the arc along which the flow occurs. Some implications derived from the model are discussed and a number of possible applications are proposed.

A Network Flow Solution to a Rectilinear Distance Facility Location Problem

Abstract: The problem of locating new facilities is considered with respect to existing facilities so as to minimize a sum of costs which consists of costs proportional to the rectilinear distances between new and existing facilities, and costs proportional to the rectilinear distances among new facilities. The location problem decomposes into two independent sub-problems, each of which is equivalent to a linear programming problem which is essentially the dual of a minimal cost network flow problem. Fulkerson's out-of-kilter algorithm provides an efficient means of solving each of the network flow problems as well as the location problem. The dual variables in each of the optimum tableaus to the two flow problems give the x and y coordinates respectively of the optimum locations of the new facilities. Several alternative approaches to solving the equivalent linear programming problems are also discussed, and some research questions are identified.

Solving Certain Nonconvex Quadratic Minimization Problems by Ranking the Extreme Points

Abstract: Certain types of quadratic programs with linear constraints have the property that an extreme point of the convex set of feasible solutions is an optimal solution. This paper presents a procedure for solving these problems, it involves determining a related linear program having the same constraints, the extreme-point-ranking approach of Murty is then applied to this linear program to obtain an optimum solution to the quadratic program.

Power-system load scheduling with security constraints using dual linear programming

Abstract: The optimisation of power-system operating conditions is formulated as a dual linear programming problem. This suboptimal model allows fast solutions to be obtained dependably by applying the revised simplex l.p. method to the problem of minimising total generation costs subject to the constraints imposed. These constraints include the network equations, the inequalities restricting generator loading, runningspare capacity and transmission-line loading under normal and outage conditions. The fast speed of solution and low computer-storage requirements result from the reduced mathematical model developed by means of the variable eleimination and the computing strategy used. The computational procedure automatically adjusts the size of the problem to be solved according to indications obtained of the likely critical lineoutage security constraints, a small number in relation to the prohibitively large number of possible outage constraints. A sample application of the method is given for a 2700MW, 275/132kV system of 23 busbars, 30 lines and transformers, supplied by 24 generators. Using ALGOL 60 on the Atlas computer, solutions were obtained in 5s neglecting line-outage security, and 11.5s including security under all possible single-line-outage conditions. For accuracy, comparisons are also made with the network-flow technique and the full nonlinear programming solutions.

An Enumeration Algorithm for Knapsack Problems

Abstract: Enumeration techniques have been shown to be successful for solving integer linear programming problems. The purpose of this paper is to apply the enumeration philosophy to the classical knapsack problem; it shows that this approach applies quite naturally to this type of integer linear program when combined with the Fourier-Motzkin elimination method for solving linear inequalities. Some computational results are reported.

Markov Models for Flow Regulation

Abstract: The problem of defining alternative operating policies for a single multipurpose reservoir was examined through the use of several pairs of discrete stochastic linear- and dynamic-programming models. The net flows into the reservoir were assumed to be serially correlated, their probabilistic sequence defined by first-order Markov chains. Each linear programming model was shown to correspond to a dynamic programming model. The solutions and computational efficiencies of each of the models were compared using a simplified numerical example based on an actual reservoir operating problem. Although the policies obtained from each pair of corresponding models were identical, the time required to solve the dynamic programming models was less than that required for the linear programming models.

a Comparison of Some Dynamic, Linear and Policy Iteration Methods for Reservoir Operation

Abstract: Within the past few years, a number of papers have been published in which stochastic mathematical programming models, incorporating first order Markov chains, have been used to derive alternative sequential operating policies for a multiple purpose reservoir. This paper attempts to review and compare three such mathematical modeling and solution techniques, namely dynamic programming, policy iteration, and linear programming. It is assumed that the flows into the reservoir are serially correlated stochastic quantities. The design parameters are assumed fixed, i.e., the reservoir capacity and the storage and release targets, if any, are predetermined. The models are discrete since the continuous variables of time, volume, and flow are approximated by discrete units. The problem is to derive an optimal operating policy. Such a policy defines the reservoir release as a function of the current storage volume and inflow. The form of the solution and some of the advantages, limitations and computational efficiencies of each of the models and their algorithms are compared using a simplified numerical example.

Applications of Mathematical Programming to ℓp Approximation

Abstract: The problem of determining a best lp approximation to discrete data is recast as a nonlinear program. For a linear approximating function the resulting program involves linear constraints and a nonlinear objective function. This objective function is concave for 0 < p < 1 and convex for 1 < p < ∞. For p = 1 or p = ∞ the determination of best approximations can be accomplished by linear programming. Computational aspects of these formulations are discussed, and some numerical and theoretical results are presented.

The transportation problem and the vogel approximation method

Abstract: This paper shows that the so-called “transportation problem” possesses the four characteristics which define linear programming (LP) type problems. Thus, the transportation problem can always be solved by using the Simplex Method, a well-known but tedious technique for dealing with any linear programming problem. A special procedure for solving the transportation problem is the so-called “Transportation Method,” which involves three steps. However, due to intricacies in steps 2 and 3, this Method can, like Simplex, become quite tedious and time-consuming. A short-cut approach to solving the transportation problem is the Vogel Approximation Method (VAM), which is a very simple means of performing step 1 of the Transportation Method. Application of VAM to a given problem does not guarantee that an optimal solution will result. However, a very good solution is invariably obtained, and is obtained with comparatively little effort. For many purposes, using VAM to carry out step 1 of the Transportation Method eliminates (or all but eliminates) the need for performing steps 2 and 3. The mechanics of the Vogel Approximation Method are illustrated with reference to a particular transportation problem.

Game theory-an extension of its application to farm planning under uncertainty

Abstract: Current game theoretic models for farnt planning are limited in scope and restrictive in their assumptions. This paper reviews the current status of such models, compares them to the quadratic programming model, and then combines the desirable features of the game theoretic and quadratic Programming approaches by means of parametric linear programming

Letter to the Editor—An Experimental Investigation and Comparative Evaluation of Flow-Shop Scheduling Techniques

Abstract: This note is concerned with the solution of the flow-shop scheduling problem where all jobs have the same machine ordering. Because of the combinatorial nature of this problem, most practical situations remain unsolved. Various techniques such as switch and check, branch and bound with and without backtracking, modified decomposition, and rounded linear programming have been proposed by several investigators. However, no comparative evaluation of these procedures has been previously made. This note investigates the solutions obtained by these procedures considering both the quality of the solutions and the computational efficiency. Extensive experimentation has been conducted and significant results are reported. The effects of changes in the size of problems on the above criteria are also included.

Adaptive Linear Classifier by Linear Programming

Abstract: A linear classifier based on linear programming which is adaptive to a change in the set of input vectors is discussed. Different from other linear classifiers, this one maintains the maximum reliability of its operation, provided that the set of pattern vectors is linearly separable. A procedure of deriving an optimum structure of the linear classifier for a change in the set of input vectors is a modification of the ordinary simplex method and yields an optimum structure in much fewer iterations than the straightforward application of the ordinary simplex method does. The adaptive procedure is then extended to the case in which a linear classifier maintains the minimum number of erroneously classified input vectors even if the set of input pattern vectors is not linearly separable. This is based on Gomory's algorithm for integer linear programming. The feasibility and efficiency of these linear classifiers are computationally proved by some examples.

A Production Scheduling Model by Bivalent Linear Programming

Abstract: The paper focuses on multi-stage production systems. If such a system is of the more general type than a production line and many products have to be manufactured according to different technological sequence restrictions then two problems arise: (1) scheduling and (2) sequencing. The formulation of that problem requires the simultaneous consideration of the scheduling and sequencing aspect. The paper integrates these two functions into one single model using zero-one programming.

Optimal multicommodity network flows with resource allocation

Abstract: The problem of determining multicommodity flows over a capacitated network subject to resource constraints may be solved by linear programming; however, the number of potential vectors in most applications is such that the standard arc-chain formulation becomes impractical This paper describes an approach—an extension of the column generation technique used in the multicommodity network flow problem—that simultaneously considers network chain selection and resource allocation, thus making the problem both manageable and optimal The flow attained is constrained by resource availability and network capacity A minimum-cost formulation is described and an extension to permit the substitution of resources is developed Computational experience with the model is discussed

The Product-Mix Problem under Stochastic Seasonal Demand

Abstract: The product-mix problem of production planning is to determine the best quantity of each product to manufacture, over a complete range of products competing for a number of limited resources. This paper examines the problem when uncertainty of demand is a major factor, making any differences between the penalties of over-and under-production important. In these circumstances the basic linear programming technique is insufficient, and a model has been developed as a multi-item newsboy problem in which a number of linear resource constraints affect the decision variables. The formulation is one of linear programming under uncertainty. A marginal analysis approach is used to obtain a detailed insight into the structure of the solution and to develop an effective computational procedure. Very little additional data is required, beyond that needed for a conventional linear program.

Linear Programming Solutions to Ratio Games

Abstract: This paper develops a method for finding the minimax solutions to a game whose payoff function is of the form XtAY/XtBY, where X, Y are mixed-strategy vectors and A, B > 0 are matrices. Such a payoff function arises in stochastic games, economic models of an expanding economy, and some nonzero-sum game formulations. The game is transformed into a linear program with a parameter in the constraint set. By successive solutions of this program with appropriate values of the parameter, the value and optimal strategies of the game can be approximated to any desired degree of accuracy. A new approach of perturbing linear programs is used to prove convergence of the computational technique and to bound the rate of convergence.

Some Algorithms Based on the Principle of Feasible Directions

Abstract: A number of algorithms will be described which are based on the principle of feasible directions. Special problems like linear programming, unconstrained optimization, optimization subject to linear equality constraints, quadratic programming and linearly constrained nonlinear programming will be briefly dealt with.

The optimum two-dimensional allocation of irregular, multiply-connected shapes with linear, logical and geometric constraints

Abstract: : The optimum two-dimensional allocation problem consists in taking some two-dimensional resource, such as a piece of cloth, a sheet of steel, or a parcel of land, and cutting it up into a number of two-dimensional forms, such as clothing patterns, sheet-metal parts, or parking spaces, in such a way that some objective, such as minimum waste of material or maximum total number of pieces, is achieved This thesis describes the results of an investigation into methods of handling this type of problem when linear, logical, and geometric constraints, in addition to the usual area and nonoverlapping constraints, are imposed on the allocations The investigation is concerned with two-dimensional shapes that can be irregular and either simply- or multiply-connected (Author)

A finiteness proof for modified dantzig cuts in integer programming

Abstract: Let be a basic solution to the linear programming problem subject to: where R is the index set associated with the nonbasic variables. If all of the variables are constrained to be nonnegative integers and xu is not an integer in the basic solution, the linear constraint is implied. We prove that including these “cuts” in a specified way yields a finite dual simplex algorithm for the pure integer programming problem. The relation of these modified Dantzig cuts to Gomory cuts is discussed.

On the unlimited number of faces in integer hulls of linear programs with a single constraint

Abstract: The convex hull of the feasible integer points to a given integer program is a convex polytope I. The feasible set obtained by relaxing the integrality requirements is another convex polytope L. Cutting-plane algorithms essentially try to remove part of L − I. Hence the more complicated the relationship between L and I, the more difficult (in some sense) the integer program. This paper shows one such complexity: specifically, we construct a series of programs such that I has arbitrarily many faces even though L is a triangle. We also indicate the existence of a large class of problems that exhibit the same behavior.