Doi: 10.2306/SCIENCEASIA1513-1874.2014.40.248 (original) (raw)
Max-out-in pivot rule with cycling prevention for the simplex method
ScienceAsia, 2014
A max-out-in pivot rule is designed to solve a linear programming (LP) problem with a non-zero righthand side vector. It identifies the maximum of the leaving basic variable before selecting the associated entering nonbasic variable. Our method guarantees convergence after a finite number of iterations. The improvement of our pivot rule over Bland's rule is illustrated by some cycling LP examples. In addition, we report computational results obtained from two sets of LP problems. Among 100 simulated LP problems, the max-out-in pivot rule is significantly better than Bland's rule and Dantzig's rule according to the Wilcoxon signed rank test. Based on these results, we conclude that our method is best suited for degenerate LP problems.
NEW PIVOTING RULES FOR THE SIMPLEX METHOD
In this paper the aim of work is to introduce a new pivoting rules to the simplex method of linear programming problem (LPP). To select the basisentering and leaving vector to the simplex table to get the maximum improvement from the set of basisentering variables to get a optimal basic feasible solution of the objective function. This new technique is illustrated through the problem for the simplex method under an easily described pivoting rules.
Systematic construction of examples for cycling in the simplex method
Computers & Operations Research, 2006
We present systematic procedures to construct examples of linear programs that cycle when the simplex method is applied. Cycling examples are constructed for diverse variants of pivot selection strategies: most negative reducedcost and steepest-edge rule for the entering variable, and smallest ratio rule for the leaving variable (where ties are broken according to the least-index or the largest coefficient criterion, respectively). The results are of theoretical interest since only a limited number of cycling examples have been presented in the literature up to date. Constructed cycling examples may also serve as test problems to evaluate the practical performance of anticycling procedures or new variants of simplex type methods.
Texas A&M University, Department of …, 1994
In this study we implement Primal and Dual Simplex methods with different pivot-selection techniques, namely, the Non-Basic Gradient, Bland's, the Least-Recently Considered (LRC), the Greatest-Increment and the Steepest-Edge. We present the derivation of the Simplex method from an analytic and geometric viewpoint. We provide derivation and interpretation of the five different pivot selection methods as it realtes to the geometry of the convex polytopes. We perform several experiments and evaluate their performance with respect to the number of pivot iterations until optimal solution, and computational effort for each iteration. There is a trade-off between the number of iterations and the cost per iteration in the solution of LP problem instances. The Bland's method requires the shortest time per iteration but it usually requires many more iterations. The Non-Basic Gradient method performs comparably to the LRC, but the later one is more insensitive to degenerate pivots, and thus, it leads to fewer iterations in those cases. The Greatest-Increment increment has the most computational intensive iterations but it usually solves LP instances with fewer pivots than the previously mentioned three ones. The computation cost per iteration in the Steepest-Edge is lower than that of the Greatest-Increment but higher than those of the other three methods. The Steepest-Edge method requires fewer iterations than all other pivot selection methods that we investigated in this study. All the experiments are carried out in the LPBench linear programming environment, a home-grown system designed for the development and evaluation of pivot selection algorithms for the Simplex method.
Absolute Change Pivot Rule for the Simplex Algorithm
2014
The simplex algorithm is a widely used method for solving a linear programming problem (LP) which is first presented by George B. Dantzig. One of the important steps of the simplex algorithm is applying an appropriate pivot rule, the rule to select the entering variable. An effective pivot rule can lead to the optimal solution of LP with the small number of iterations. In a minimization problem, Dantzig’s pivot rule selects an entering variable corresponding to the most negative reduced cost. The concept is to have the maximum improvement in the objective value per unit step of the entering variable. However, in some problems, Dantzig’s rule may visit a large number of extreme points before reaching the optimal solution. In this paper, we propose a pivot rule that could reduce the number of such iterations over the Dantzig’s pivot rule. The idea is to have the maximum improvement in the objective value function by trying to block a leaving variable that makes a little change in the ...
The simplex algorithm with a new primal and dual pivot rule
1994
Abstract We present a simplex-type algorithm for linear programming that works with primal-feasible and dual-feasible points associated with bases that differ by only one column. The algorithm is almost unaffected by degeneracy, and a preliminary implementation compares favorably with the primal simplex method.
RESEARCH ARTICLE A Streamlined Artificial Variable Free Version of Simplex Method
2016
This paper proposes a streamlined form of simplex method which provides some great ben-efits over traditional simplex method. For instance, it does not need any kind of artificial vari-ables or artificial constraints; it could start with any feasible or infeasible basis of an LP. This method follows the same pivoting sequence as of simplex phase 1 without showing any ex-plicit description of artificial variables which also makes it space efficient. Later in this paper, a dual version of the new method has also been presented which provides a way to easily implement the phase 1 of traditional dual simplex method. For a problem having an initial basis which is both primal and dual infeasible, our methods provide full freedom to the user, that whether to start with primal artificial free version or dual artificial free version without making any reformulation to the LP structure. Last but not the least, it provides a teaching aid for the teachers who want to teach feasibility achieveme...
The Average number of pivot steps required by the Simplex-Method is polynomial
Zeitschrift für Operations Research, 1982
The paper deals with the average number of pivot steps required by the Simplex-Method for solving linear programming problems with m inequality-restrictions in n variables. The m hyperplanes bounding the feasible regions of the corresponding inequalities are assumed to be distributed independently, identically and symmetrically under rotations in the n-dimensional Euclidean space. A certain variant of the Simplexalgorithm, the so-called Schatteneckenalgorithmus, is analyzed. This variant can even be used for the calculation of a start vertex. For the expected number of pivot steps required for the solution of the programming problem an explicit upper bound, which is polynomial in m and n, can be derived. This result implies that the average computation-time required for solving the problem is polynomial in m and n, too. Zusammenfassung: Die vorliegende Arbeit befa~t sich mit der durchschnittlichen Zahl yon Pivotschritten, die ben6tigt werden, um lineare Optimierungsprobleme mit m Ungleichungs-Nebenbedingungen in n Variablen mit dem Simplexverfahren zu 16sen. Die m Hyperebenen, die die Zuliissigkeitsbereiche der zugeh6rigen Ungleichungen begrenzen, seien unabh~ingig, identisch und rotationssymmetrisch im n-dimensionalen euklidischen Raum verteilt. Eine bestimmte Variante des Simplexalgorithmus, der sogenannte Schatteneckenalgorithmus, wird untersucht. Diese Variante kann sogar benutzt werden, um eine Startecke zu bestimmen. Fiir die erwartete Anzahl der Pivotschritte, die zur L6sung des Optimierungsproblems erforderlich sind, kann eine explizite obere Schranke, die polynomial ist in mund n, hergeleitet werden. Dieses Resultat garantiert, dat~ die durchschnittliche Recahenzeit zur L~Ssung des Problems ebenfails polynomial ist in m und n.
International Series in Operations Research & Management Science, 2008
In this chapter we present the simplex method as it applies to linear programming problems in standard form.