Revised Simplex Method Research Articles

Network flows with age dependent decay rates

This paper considers the problem of maximizing the output flow in a multicommodity network in which flow entering an arc experiences a decay rate which is a function of three factors: the arc, the commodity, and the age of the commodity as it enters the arc. An arc-chain linear programming formulation of the problem is given. The algorithm for solving the problem involves a novel column generation scheme for basis entry embedded in the revised simplex algorithm. An efficient algorithm for generating, at each iteration, such a column is provided and illustrated with a numerical example.

Leakage reduction by optimised control of valves in water networks

The problem of minimisation of leakages due to overpressure in a water distribution network is considered, and an algorithm for computation of the optimal valve controls based on the sparse revised Simplex method is presented. Computational experience indicates that the method has low memory and processor time requirements and is suitable for online implementation on minicomputer systems. The economy of the optimal valve control policy is shown by comparison of the volume of leakages for uncontrolled, manually controlled and optimally controlled networks.

Comparison of Four Goal Programming Algorithms

A major limitation in the use of goal programming has been the lack of an efficient algorithm for model solution. Schniederjans and Kwak recently published a proposal for more efficient solution of goal programming models utilizing dual simplex procedures. A goal programming algorithm based upon that method has been coded, as well as a revised, simplex-based algorithm. These algorithms are compared in terms of accuracy and time requirements with algorithms previously presented by Lee and by Arthur and Ravindran. Solution times for a series of 12 goal programming models are presented. The dual simplex method appears to have superior computational times for models with a large proportion of positive deviational variables in the solution. The revised simplex algorithm appears more consistent in time and accuracy for general goal programming models.

Large Scale Reactive Power Planning

A new methodology for planning future reactive compensation for large scale systems is presented. The planning technique is based on a restructuring of the network model and a special formulation of the revised simplex method. A preprocessor is incorporated for correcting systems which are not able to converge using conventional power flow or optimal power flow programs. A 600 bus example is provided which considers 285 transmission contingencies.

Large Scale Reactive Power Planning

A new methodology for planning future reactive compensation for large scale systems is presented. The planning technique is based on a restructuring of the network model and a special formulation of the revised simplex method. A preprocessor is incorporated for correcting systems which are not able to converge using conventional power flow or optimal power flow programs. A 600 bus example is provided which considers 285 transmission contingencies.

Ein effektives simplexverfahren mit teiltableauwahl

We consider the revised simplex algorithm with multiple pricing. The optimal size of the subproblem is given. Numerical examples show the efficiency of the algorithm.

Economic dispatch of active power with constraint relaxation

The economic dispatch of active power in large-scale power systems is considered and an algorithm for the solution of the problem by means of the sparse dual revised Simplex method is presented. Computational experience with large-scale problems, including dynamic dispatch over several time steps, is given, which indicates that the method has very low memory and processor time requirements and is suitable for implementation on minicomputer systems. The algorithm is extended to allow for hierarchical constraint relaxation and removal in cases where an infeasible problem has been specified inadvertently. The availability of such a constraint relaxation strategy permits the application of approximate methods for the inclusion of transmission losses and spare capacity constraints.

Linear programming and limit analysis of shell roofs

Limit analysis for cylindrical shell roofs has been formulated as a linear programming problem based on lower bound theorem. The differential equations of equilibrium for a circular cylindrical shell element are transformed into algebraic equations by finite differences. The equilibrium equations and the linearized non-linear yield conditions at various points of the shell are linear functions of the stress resultants. These form the linear constraints of the problem. The load parameter is taken as an objective function and it is maximized using revised simplex method. For a shell of given geometry, stress resultants at various points are obtained to give the optimum collapse load. Thus the versatile technique avoids various trial solutions to achieve best lower bound for complicated shell problems.

Constrained Economic Dispatch of Multi-Area Systems Using the Dantzig-Wolfe Decomposition Principle

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.

Zur lösung linearer ungleichungssysteme bei störimg der rechten seite und der koeffizientenmatrix

A condition is given for the compatibility of a solution of a system of linear inequalities with given error bounds for the coefficients and right hand sides. This condition is a generalization of a lemma of PRAGER/OETTLI. Application of the condition are given for optimization problems with linear inequality constraints and the revised simplex algorithm of linear optimization.

A Man-Machine Interactive Sensitivity Analysis Feature for the Kinki Regional Model

There is a great need for the heuristic manipulation capability of the model and heuristic building of scenario pictures in an integrated regional development planning area. Therefore, a computer-assisted interactive manmachine capability model is constructed to seek a desirable socio-economic development pattern with the aim of giving decision/policy makers a dynamic sensitivity analysis feature for comparing simulation results under various constraints. The model structure is to maximize the gross national product by allocating both capital and labor force with the help of a revised simplex method assuming the future composition ratio of expenditure. The results are displayed interactively in graphical or tabular format for a variety of possible Policy factors in which the decision/policy maker might be interested. The sensitivity analysis is performed in like manner, but by displaying any combination of two simulated policy scenarios.

A mathematical programming updating method using modified Givens transformations and applied to LP problems

An efficient and numerically stable method is presented for the problem of updating an orthogonal decomposition of a matrix of column (or row) vectors. The fundamental idea is to add a column (or row) analogous to adding an additional row of data in a linear least squares problem. A column (or row) is dropped by a formal scaling with the imaginary unit, √-1, followed by least squares addition of the column (or row). The elimination process for the procedure is successive application of the Givens transformation in modified (more efficient) form. These ideas are illustrated with an implementation of the revised simplex method. The algorithm is a general purpose one that does not account for any particular structure or sparsity in the equations. Some suggested computational tests for determining signs of various controlling parameters in the revised simplex algorithm are mentioned. A simple means of constructing test cases and some sample computing times are presented.

Computational methods for solving two-stage stochastic linear programming problems

Approximating a given continuous probability distribution of the data of a linear program by a discrete one yields solution methods for the stochastic linear programming problem with complete fixed recourse. For a procedure along the lines of [8], the reduction of the computational amount of work compared to the usual revised simplex method is figured out. Furthermore, an alternative method is proposed, where by refining particular discrete distributions the optimal value is approximated.

A revised simplex algorithm for the absolute deviation curve fitting problem

(1979). A revised simplex algorithm for the absolute deviation curve fitting problem. Communications in Statistics - Simulation and Computation: Vol. 8, No. 2, pp. 175-190.

Large-scale linearly constrained optimization

An algorithm for solving large-scale nonlinear programs with linear constraints is presented. The method combines efficient sparse-matrix techniques as in the revised simplex method with stable quasi-Newton methods for handling the nonlinearities. A general-purpose production code (MINOS) is described, along with computational experience on a wide variety of problems.

Testing of a large-scale network optimization program

This paper describes the experimental results of testing a “large-scale” program for solving minimum-cost network flow problems. With this program, general structure transshipment problems with over ten thousand nodes and thirty thousand arcs have been easily solved without resorting to auxiliary storage. The algorithm is a variant of the primal revised simplex method; the computer code is called LPNET illustrating the close connection between linear programming and network graphs. This approach substantially improves computer processing timeand core storage, especially for relatively large network problems. The results of these experiments are provided. It is emphasized that an organized experimental design and a detailed series of empirical tests are crucial for an efficient implementation.

Pricing for sparsity in the revised simplex method

Pricing for sparsity in the revised simplex method

An algorithm for determining all extreme points of a convex polytope

An algorithm for determining all the extreme points of a convex polytope associated with a set of linear constraints, via the computation of basic feasible solutions to the constraints, is presented. The algorithm is based on the product-form revised simplex method and as such can be readily linked onto standard linear programming codes. Applications of such an algorithm are reviewed and limited computational experience given.

Multistage linear programming for discrete optimal control with distributed-lags

A multistage decomposition scheme is developed for optimizing discrete-time dynamic systems, which include distributed and/or multiple pure delays. The discrete optimal control problem in this paper consists of a system dynamics described by a multidimensional linear difference equation of high-order which is called the distributed-lag model, a linear objective function, and linear state and control constraints. This problem may be solved as a linear program by, for example, a revised simplex method. However, this leads to excessive storage requirement for large problems. Instead, by taking advantage of the staircase-structure of equality constraints (system equation), Dantzig-Wolfe decomposition principle is applied repeatedly in each stage, and an effective multistage decomposition algorithm for distributed-lag models is obtained. Significant advantage of the optimization technique in this paper is that it can handle any number of delay terms in the system without reducing the multidimensional high-order system equation to a conventional larger dimensional first-order system equation (state equation of normal form). Therefore, a substantial reduction of computational burden, the so called curse of dimensionality, in the existing discrete optimal control algorithms, is obtained. A numerical example of a congested urban road traffic control problem with many delays is included.

A primal algorithm for solving a capacitated network flow problem with additional linear constraints

AbstractA variant of the revised simplex method is given for solving capacitated network flow problems with additional linear constraints. The mathematical development, computer implementation and computational experience are presented. The algorithm described uses a working basis of size equal to the number of additional linear constraints for pivoting, pricing, and inversion, which for large networks, can result in a substantial reduction of computation over a standard linear programming approach.