Standard Simplex Method Research Articles

Solving Fuzzy Time–Cost Trade-Off in Construction Projects Using Linear Programming

The time–cost trade-off has been recognized as a very significant aspect of construction management. Generally, time–cost trade-off can be modeled as a fuzzy linear programming problem with symmetric or non-symmetric fuzzy numbers. However, it was successfully solved when the fuzzy membership functions are only symmetric. In the present work, a novel approach is introduced to solve fuzzy linear programming problem with non-symmetric fuzzy membership functions by transforming it to its corresponding nearest symmetric one. The transformed problem is then converted to its crisp linear programming problem and then solved by the standard primal simplex method. Two examples are presented to show the effectiveness of the proposed approach, and the results are discussed.

Self-Regulating Artificial-Free Linear Programming Solver Using a Jump and Simplex Method

An enthusiastic artificial-free linear programming method based on a sequence of jumps and the simplex method is proposed in this paper. It performs in three phases. Starting with phase 1, it guarantees the existence of a feasible point by relaxing all non-acute constraints. With this initial starting feasible point, in phase 2, it sequentially jumps to the improved objective feasible points. The last phase reinstates the rest of the non-acute constraints and uses the dual simplex method to find the optimal point. The computation results show that this method is more efficient than the standard simplex method and the artificial-free simplex algorithm based on the non-acute constraint relaxation for 41 netlib problems and 280 simulated linear programs.

Social Network Monitoring for Bursty Cascade Detection

Social network services have become important and efficient platforms for users to share all kinds of information. The capability to monitor user-generated information and detect bursts from information diffusions in these social networks brings value to a wide range of real-life applications, such as viral marketing. However, in reality, as a third party, there is always a cost for gathering information from each user or so-called social network sensor. The question then arises how to select a budgeted set of social network sensors to form the data stream for burst detection without compromising the detection performance. In this article, we present a general sensor selection solution for different burst detection approaches. We formulate this problem as a constraint satisfaction problem that has high computational complexity. To reduce the computational cost, we first reduce most of the constraints by making use of the fact that bursty cascades are rare among the whole population. We then transform the problem into an Linear Programming (LP) problem. Furthermore, we use the sub-gradient method instead of the standard simplex method or interior-point method to solve the LP problem, which makes it possible for our solution to scale up to large social networks. Evaluating our solution on millions of real information cascades, we demonstrate both the effectiveness and efficiency of our approach.

LS-SVM Regression for Structural Damage Diagnosis Using the Iterated Improved Reduction System

A damage detection and estimation method is proposed for structural health monitoring using incomplete modal data and least squares support vector machine (LS-SVM). To accommodate the use of incomplete modal data, the iterated improved reduction system (IIRS) method has been used to condense the mass and stiffness matrices of the structure. The first two incomplete mode shapes and natural frequencies of a damaged structure are used as input data to the LS-SVM. The coupled simulated annealing (CSA) and standard simplex method using 10-fold cross-validation techniques are adopted to determine the optimal tuning parameters in the LS-SVM model. Three illustrative examples with and without noise in modal data are prepared to evaluate the performance of the proposed method. The results indicated that this method can be reliably used to identify the damages of structures with good accuracy.

Rapid optimization of tension distribution for cable-driven parallel manipulators with redundant cables

The solution of tension distributions is infinite for cable-driven parallel manipulators(CDPMs) with redundant cables. A rapid optimization method for determining the optimal tension distribution is presented. The new optimization method is primarily based on the geometry properties of a polyhedron and convex analysis. The computational efficiency of the optimization method is improved by the designed projection algorithm, and a fast algorithm is proposed to determine which two of the lines are intersected at the optimal point. Moreover, a method for avoiding the operating point on the lower tension limit is developed. Simulation experiments are implemented on a six degree-of-freedom(6-DOF) CDPM with eight cables, and the results indicate that the new method is one order of magnitude faster than the standard simplex method. The optimal distribution of tension distribution is thus rapidly established on real-time by the proposed method.

WE‐G‐207‐09: A Practical Bowtie Ring Artifact Correction Algorithm for Cone‐Beam CT

Purpose:Shaded ring artifact in cone‐beam CT (CBCT) is caused by scatter contamination from bowtie‐modulated primary beam and presents a unique ring pattern in CBCT image. Previous shading correction methods, which suppress the artifact as conventional scatter correction in projection domain, are complicated due to the non‐trivial scatter estimation. In this work, we propose a practical and readily implementable algorithm to correct for the severe shaded ring in CBCT without relying on prior information and directly in image domain.Methods:Due to the correlation between the ring pattern and bowtie modulator, an initial bowtie mask was reconstructed from bowtie‐modulated air‐scan projections as if they were acquired from a flat‐field exposure penetrating the bowtie modulator. The shape of the bowtie mask matches well with that of the shaded ring in CBCT, while its intensity needs to be scaled to fully compensate for the shading ring. To find a correct scaling factor, we start from the anatomical knowledge that the same tissue has comparable CT number. This knowledge indicates a sharp peak in histogram of that specific tissue. One way to achieve this goal is to maximize the peak value of the histogram of that tissue. The above concept is formulated as a mathematical optimization problem which is solved using a standard Simplex method. The shaded ring artifact in CBCT is finally corrected for by adding a scaled bowtie mask.Results:The proposed method is evaluated on one pelvis patient. Severe shaded ring artifact is greatly suppressed. Our method reduces the CT number error from >200 HU to be ∼50 HU, and increases the spatial uniformity by 1.3 times.Conclusion:We propose a practical image‐domain algorithm for shaded ring artifact correction in CBCT. It is computationally efficient and does not rely on prior knowledge. It is thus attractive for clinical use.This work is supported by the National Science Foundation of China (NSFC Grant No. 81201091), National High Technology Research and Development Program of China (863 program, Grant No. 2015AA020917), and Fund Project for Excellent Abroad Scholar Personnel in Science and Technology.

A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers

Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency. Although precise data are fundamentally indispensable in conventional LP problems, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We convert the FLP problem into an equivalent crisp LP problem and solve the crisp problem with the standard primal simplex method. We show that the method proposed in this study is simpler and computationally more efficient than two competing FLP methods commonly used in the literature.

A Note on an Integer Programming Problem That Has a Linear Programming Solution

It is well-known that solutions to integer programming problems usually cannot be obtained by simply solving the corresponding linear programming relaxation. There are, however, examples of integer programming problems whose solutions can be obtained by simply solving the linear program and ignoring the integer constraints. Proving that these particular models have this trait is generally beyond the scope of a beginning course in operations research. In this paper an integer programming model, with only two constraints, is presented whose solution can be directly obtained using the standard simplex method. A proof is provided that makes a connection between analysis and operations research.

Highly Scalable Parallelization of Standard Simplex Method on a Myrinet-Connected Cluster Platform

The simplex method has been successfully used in solving linear programming problems for many years. Parallel approaches have also extensively been studied due to the intensive computations required, especially for the solution of large linear problems (LPs). In this paper we present a highly scalable parallel implementation framework of the standard full tableau simplex method on a highly parallel (distributed memory) environment. Specifically, we have designed and implemented a suitable column distribution scheme as well as a row distribution scheme and we have entirely tested our implementations over a considerably powerful distributed platform (linux cluster with myrinet interface). We then compare our approaches (a) among each other for variable number of problem size (number of rows and columns) and (b) to other recent and valuable corresponding efforts in the literature. In most cases, the column distribution scheme performs quite/much better than the row distribution scheme. Moreover, both schemes (even the row distribution scheme over large-scale problems) lead to particularly high speedup and efficiency values, which are considerably better in all cases than the ones achieved in other similar research efforts and implementations. Moreover, we further evaluate our basic parallelization scheme over very large LPs in order to validate more reliably the high efficiency and scalability achieved.

Construction of aggregation functions from data using linear programming

This article examines the construction of aggregation functions from data by minimizing the least absolute deviation criterion. We formulate various instances of such problems as linear programming problems. We consider the cases in which the data are provided as intervals, and the outputs ordering needs to be preserved, and show that linear programming formulation is valid for such cases. This feature is very valuable in practice, since the standard simplex method can be used.

On pivot rules for simplex method of interior points, and their investigation on Klee-Minty cube

In [25] it was proposed a parametric linear transformation, which is a "convex" combination of the Gauss transformation of elimination method and the Gram-Schmidt transformation of modified orthogonalization process. Using this transformation, in particular, elimination methods were generalized, Dantzig's optimality criterion and simplex method were developed [26]. The essence of the simplex method development is the following. At each sth step the pivot (positive) vector of length Ks is selected, that allows us to move to improved feasible solution after the step of the generalized Gauss-Jordan complete elimination method. In this method the movement to the optimal point takes place over pseudobases, i.e., over interior points. This method is parametric and finite. Since the method is parametric there are various variants to choose the pivot vectors (rules), in the sense of their lengths and indices. In this article we propose three rules, which are the development of Dantzig's first rule. These rules are investigated on the Klee-Minty cube (problem) [14, 31]. It is shown that for two rules the number of steps necessary equals to 2n, and for third rule we obtain the standard simplex method with the largest coefficient rule, i.e., the number of steps for solving this problem is 2n - 1.

Solving the undirected multicommodity flow problem using a shortest path‐based pricing algorithm

AbstractThe undirected multicommodity flow problem is encountered in the solution to traffic‐scheduling problems related to computer, communication, railroad, and other networks. We show how to formulate this problem as a piecewise linear optimization problem (with piecewise linear convex functions in both the objective and the constraints). We discuss how EMNET, a primal basis partitioning simplex algorithm, was modified so as to efficiently solve these problems using a pricing procedure that we call “shortest path pricing.” Extremely good solution times for some very large problems are presented. The solutions are not only obtained quickly, but also with a fraction of the number of pivots that are needed for the standard simplex method. © 2001 John Wiley & Sons, Inc.

Interior-point methods for reduced Hessian successive quadratic programming

Typical chemical process optimization problems have a large number of equations, but relatively few degrees of freedom. A reduced Hessian successive quadratic programming (rSQP) algorithm has been shown to be successful in solving these types of models efficiently. While the rSQP algorithm reduces the dimension of the QP subproblem solved at each iteration, the number of inequality constraints could still become quite large. As a result, there is still a potentially large combinatorial expense to solving large NLPs with rSQP, as the solution of the QP subproblem is the bottle-neck of the SQP algorithm. To overcome this combinatorial expense, interior-point (IP) methods have been shown to solve large linear programs much faster than the standard simplex method. These interior-point techniques have also been extended to solve quadratic programming problems. For these reasons, we describe an IP method to solve the QP subproblem for rSQP at each iteration. The primal-dual interior-point method is described as well as some higher-order strategies designed to improve convergence. These higher order strategies are evaluated, and also a comparison of several linear equation solvers is made, since the solution of a linear system is required for each interior-point iteration. Finally, we evaluate and compare the new interior-point QP formulation with two standard active-set solvers, QPKWIK and QPOPT, on some industrial-strength NLP problems.

An exterior-point method for linear programming problems

This paper proves the convergence of an algorithm for solving linear programming problems inO(mn 2) arithmetic operations. The method is called an exterior-point procedure, because it obtains a sequence of approximations falling outside the setU of feasible solutions. Each iteration consists of a single step within some constraining hyperplane, followed by one or more projections which force the new approximation to fall within some envelope aboutU. The paper also discusses several numerical applications. In some types of problems, the method is considerably faster than a standard simplex method program when the size of the problem is sufficiently large.

Stochastic Dedication: Designing Fixed Income Portfolios Using Massively Parallel Benders Decomposition

Drawing on recent developments in discrete time fixed income options theory, we propose a stochastic programming procedure, which we call stochastic dedication, for managing asset/liability portfolios with interest rate contingent claims. The model uses scenario generation to combine deterministic dedication techniques with stochastic duration matching methods, and provides the portfolio manager with a risk/return Pareto optimal frontier from which a portfolio may be selected based on individual risk attitudes. We employ a fixed income risk metric that can be interpreted as the fair market value of a collection of interest rate options that eliminates bankruptcy risk from the asset/liability portfolio. We incorporate this metric into a risk/return stochastic optimization model, using a binomial lattice sampling procedure to construct interest rate paths and cash flow streams from an arbitrage-free term structure model. The resulting parametric linear program has a particularly simple subproblem structure, and we have been able to solve it using resource-directed decomposition on a massively parallel computer system, the Connection Machine CM-2. We take a novel approach that uses a standard serial simplex method to solve the master problem, but generates scenarios and Benders cuts in a massively parallel manner. We discuss the performance of this implementation and present the results for a simple pension fund immunization problem.

Description of the optimal solution set of the linear programming problem and the dimension formula

We give a definition of the normal form of an optimal solution of a linear programming problem and propose an algorithm to reduce the optimal solution to its normal form. The number of steps required to execute the proposed algorithm is slightly higher than in the standard simplex method. However, the algorithm enables us to describe the entire optimal solution set and find its dimension. One important particular case of this problem is the description and determination of the dimension of the set of points belonging to a convex polyhedron specified by linear constraints.

Making progress during a stall in the simplex algorithm

All the standard methods for avoiding cycling in the simplex algorithm use row selection rules for resolving degeneracy. The method proposed in this paper does not. Instead, the Gass-Saaty parametric method applied to the objective (cost) form is used to choose the incoming column. The pivot-row choice among blocking rows, i.e., those tied for pivot, is arbitrary. A simple anticycling device is used which avoids dual degeneracy of the parametrized objective with “probability one”. Tests were run on nine highly degenerate practical test problems ranging in size from small to large. Using MINOS software, the standard simplex method required 31,195 iterations, of which 20,504 (or 66%) were blocked pivots. Using the parametric scheme required 13,812 iterations, of which 67% were blocked. The reduction in the number of iterations for this set of highly degenerate test problems was 56%. The CPU-time reduction was 48%.

Implementation of a Double-Basis Simplex Method for the General Linear Programming Problem

The basis handling procedures of the simplex method are formulated in terms of a “double basis.” The basis is expressed as a matrix product, one of the factors being the basis matrix of the last refactorization. Forward and backward transformations and update are presented for each of two implementations of the double-basis method. The double-basis update is restricted to a matrix of dimension limited by the refactorization frequency and two permutation matrices. This can lead to a saving in storage space and updating time. The cost is that the time for the forward and backward transformations is about double. Computational comparisons of storage and speed are made with the standard simplex method on problems of up to 1,480 constraints. Generally, the double-basis method performs best on larger, denser problems. Density seems to be the more important factor, and the problems with large nonzero growth between refactorizations are the better ones for the double-basis method. Storage saving in the basis inverse representation versus the standard method is as high as 36%, whereas the double-basis run times are 1.2 or more times greater.

Compact simplex methods

The following documented algorithm solves the standard linear programming problem of optimizing a linear form subject to linear inequality or equality constraints and nonnegativity conditions. The solution procedure incorporates the standard simplex method with no embellishments. There is only one loop corresponding to the basic simplex iteration.

New Numerical Method for Solving Eigenvalue Problems Described by Coupled Second-Order Differential Equations

A systematic iteration scheme is presented to numerically solve the eigenvalue problem described by a coupled set of second-order differential equations under the boundary conditions that the solution is localized near the origin. The method consists of the application of Newton's iteration scheme to the shooting method. A useful analytical expression for the correction to the trial eigenvalue is obtained. Application is made for two examples, simple harmonic oscillator and electro-magnetic drift-wave in a finite-β plasma in a sheared magnetic field, to demonstrate a remarkable stability, efficiency and accuracy of the method. In the latter example, reduction of the CPU time by factor 20–30 is obtained as compared with the standard simplex method.