The Linear Programming Problem is by far the most widely used optimization model. Its impact on economic and government modeling is immense. The Simplex Method for solving the Linear Programming (LP) Problem, due to George Dantzig, has been an extremely efficient computational tool for almost four decades. The method has been the subject of intense investigations for many years, but some major aspects of its behavior are not fully understood yet. The purpose of this paper is to survey the body of knowledge on the efficiency of the Simplex Method, from both practical and theoretical points of view. Adopting the number of iterations (pivot steps) as the yardstick for efficiency, we survey four aspects of the issue: 1. Reports on practical experience of the method's performance on real-life LP problems. 2. Results on controlled (Monte-Carlo) experiments solving LP problems which were randomly generated according to some predetermined distributions. 3. Complexity results, including theoretical analyses on both upper and lower bounds for the performance of the Simplex as well as non-Simplex algorithms for LP. 4. Results of recent theoretical studies using probabilistic analysis to derive bounds on the average behavior of the Simplex Method. We discuss the consequences and limitations of the various studies. Special emphasis is given to open questions.

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research paper on simplex method

Linear Programming Problem Solving Simplex Method

10 Pages Posted: 7 Apr 2022

Patna University; INSTITUT de DIPLOMATIE PUBLIQUE, United Kingdom

Date Written: February 25, 2022

Most real-world linear programming problems have more than two variables and thus are too complex for graphical solution. A procedure called the simplex method may be used to find the optimal solution to multivariable problems. The simplex method is actually an algorithm (or a set of instructions) with which we examine corner points in a methodical fashion until we arrive at the best solution—highest profit or lowest cost. Computer programs and spreadsheets are available to handle the simplex calculations for you. But you need to know what is involved behind the scenes in order to best understand their valuable outputs. The Simplex method is an approach to solving linear programming models by hand using slack variables, tableaus, and pivot variables as a means to finding the optimal solution of an optimization problem. A linear program is a method of achieving the best outcome given a maximum or minimum equation with linear constraints. Most linear programs can be solved using an online solver such as MatLab, but the Simplex method is a technique for solving linear programs by hand.

Keywords: Linear Programming, Simplex Method, LPP, Problem Solving

Suggested Citation: Suggested Citation

Ajit Singh (Contact Author)

Patna university ( email ).

Ashok Rajpath Patna, Bihar 800005 India

INSTITUT de DIPLOMATIE PUBLIQUE, United Kingdom ( email )

United Kingdom

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Inductive proof of the simplex method.

by George Bernard Dantzig

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A proof of the existence of an optimal basis in the simplex method based on induction. Specifically, it is shown that a finite chain of feasible basis changes exists, resulting in either an optimal feasible solution or in an infinite class of feasible solutions, such that the objective form tends to minus infinity.

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research paper on simplex method

Simplex Method

A different type of methods for linear programming problems are interior point methods , whose complexity is polynomial for both average and worst case. These methods construct a sequence of strictly feasible points (i.e., lying in the interior of the polytope but never on its boundary) that converges to the solution. Research on interior point methods was spurred by a paper from Karmarkar (1984). In practice, one of the best interior-point methods is the predictor-corrector method of Mehrotra (1992), which is competitive with the simplex method, particularly for large-scale problems.

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optimization problem

simplex method , standard technique in linear programming for solving an optimization problem, typically one involving a function and several constraints expressed as inequalities. The inequalities define a polygonal region, and the solution is typically at one of the vertices. The simplex method is a systematic procedure for testing the vertices as possible solutions.

Some simple optimization problems can be solved by drawing the constraints on a graph. However, this method is useful only for systems of inequalities involving two variables. In practice, problems often involve hundreds of equations with thousands of variables, which can result in an astronomical number of extreme points. In 1947 George Dantzig , a mathematical adviser for the U.S. Air Force, devised the simplex method to restrict the number of extreme points that have to be examined. The simplex method is one of the most useful and efficient algorithms ever invented, and it is still the standard method employed on computers to solve optimization problems.

optimization problem

First, the method assumes that an extreme point is known. (If no extreme point is given, a variant of the simplex method, called Phase I, is used to find one or to determine that there are no feasible solutions.) Next, using an algebraic specification of the problem, a test determines whether that extreme point is optimal. If the test for optimality is not passed, an adjacent extreme point is sought along an edge in the direction for which the value of the objective function increases at the fastest rate. Sometimes one can move along an edge and make the objective function value increase without bound. If this occurs, the procedure terminates with a prescription of the edge along which the objective goes to positive infinity . If not, a new extreme point is reached having at least as high an objective function value as its predecessor. The sequence described is then repeated. Termination occurs when an optimal extreme point is found or the unbounded case occurs. Although in principle the necessary steps may grow exponentially with the number of extreme points, in practice the method typically converges on the optimal solution in a number of steps that is only a small multiple of the number of extreme points.

To illustrate the simplex method, consider the example of a factory producing two products, x 1 and x 2 . If the profit on the second type is twice that on the first, then x 1 + 2 x 2 represents the total profit. The function x 1 + 2 x 2 is known as the objective function.

Clearly, the profit will be highest if the factory devotes its entire production capacity to making the second type of commodity. In a practical situation, however, this may not be possible; a set of constraints is introduced by such factors as availability of machine time, labour, and raw materials. For example, if the second type of commodity requires a raw material that is limited so that no more than five can be made in any batch, then x 2 must be less than or equal to five; i.e., x 2 ≤ 5. If the first commodity requires another type of material limiting it to eight per batch, then x 1 ≤ 8. If x 1 and x 2 take equal time to make and the machine time available allows a maximum of 10 to be made in a batch, then x 1 + x 2 must be less than or equal to 10; i.e., x 1 + x 2 ≤ 10.

Two other constraints are that x 1 and x 2 must each be greater than or equal to zero, because it is impossible to make a negative number of either; i.e., x 1 ≥ 0 and x 2 ≥ 0. The problem is to find the values of x 1 and x 2 for which the profit is a maximum. Any solution can be denoted by a pair of numbers ( x 1 , x 2 ); for example, if x 1 = 3 and x 2 = 6, the solution is (3, 6). These numbers can be represented by points plotted on two axes, as shown in the figure . On this graph the distance along the horizontal axis represents x 1 and that along the vertical represents x 2 . Because of the constraints given above, the feasible solutions must lie within a certain well-defined region of the graph. For example, the constraint x 1 ≥ 0 means that points representing feasible solutions lie on or to the right of the x 2 axis. Similarly, the constraint x 2 ≥ 0 means that they also lie on or above the x 1 axis . Application of the entire set of constraints gives the feasible solution set, which is bounded by a polygon formed by the intersection of the lines x 1 = 0, x 2 = 0, x 1 = 8, x 2 = 5, and x 1 + x 2 = 10. For example, production of three items of commodity x 1 and four of x 2 is a feasible solution since the point (3, 4) lies in this region. To find the best solution, however, the objective function x 1 + 2 x 2 = k is plotted on the graph for some value of k , say k = 4. This value is indicated by the broken line in the figure. As k is increased, a family of parallel lines are produced, and the line for k = 15 just touches the constraint set at the point (5, 5). If k is increased further, the values of x 1 and x 2 will lie outside the set of feasible solutions. Thus, the best solution is that in which equal quantities of each commodity are made. It is no coincidence that an optimal solution occurs at a vertex, or “extreme point,” of the region. This will always be true for linear problems, although an optimal solution may not be unique. Thus, the solution of such problems reduces to finding which extreme point (or points) yields the largest value for the objective function.

In the simplex method, the problem is first put into canonical form by converting the linear inequalities into equalities by introducing “slack variables” x 3 ≥ 0 (so that x 1 + x 3 = 8), x 4 ≥ 0 (so that x 2 + x 4 = 5), x 5 ≥ 0 (so that x 1 + x 2 + x 5 = 10), and the variable x 0 for the value of the objective function (so that x 1 + 2 x 2 − x 0 = 0). The problem may then be restated as that of finding nonnegative quantities x 1 , …, x 5 and the largest possible x 0 satisfying the resulting equations. One obvious solution is to set the objective variables x 1 = x 2 = 0, which corresponds to the extreme point at the origin. If one of the objective variables is increased from zero while the other one is fixed at zero, the objective value x 0 will increase as desired (subject to the slack variables satisfying the equality constraints). The variable x 2 produces the largest increase of x 0 per unit change; so it is used first. Its increase is limited by the nonnegativity requirement on the variables. In particular, if x 2 is increased beyond 5, x 4 becomes negative.

At x 2 = 5, this situation produces a new solution—( x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ) = (10, 0, 5, 8, 0, 5)—that corresponds to the extreme point (0, 5) in the figure. The system of equations is put into an equivalent form by solving for the nonzero variables x 0 , x 2 , x 3 , x 5 in terms of those variables now at zero; i.e., x 1 and x 4 . Thus, the new objective function is x 1 − 2 x 4 = −10, while the constraints are x 1 + x 3 = 8, x 2 + x 4 = 5, and x 1 − x 4 + x 5 = 5. It is now apparent that an increase of x 1 while holding x 4 equal to zero will produce a further increase in x 0 . The nonnegativity restriction on x 3 prevents x 1 from going beyond 5. The new solution—( x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ) = (15, 5, 5, 3, 0, 0)—corresponds to the extreme point (5, 5) in the figure. Finally, since solving for x 0 in terms of the variables x 4 and x 5 (which are currently at zero value) yields x 0 = 15 − x 4 − x 5 , it can be seen that any further change in these slack variables will decrease the objective value. Hence, an optimal solution exists at the extreme point (5, 5). - Free Term Papers, Essays and Research Documents

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Simplex Method in Operations Research

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Linear Programming: Chapter 2 The Simplex Method Operations Research and Financial Engineering Simplex Method An Example. maximize subject to −x1 + 3x2 − 3x3 3x1 − x2 − 2x3 ≤ 7 −2x1 − 4x2 + 4x3 ≤ 3 x1 − 2x3 ≤ 4 −2x1 + 2x2 + x3 ≤ 8 3x1 ≤ 5 x1, x2, x3 ≥ 0. Rewrite with slack variables maximize −x1 + 3x2 − 3x3 ζ = subject to w1 w2 w3 w4 w5 = = = = = 7 3 4 8 5 − + − + − 3x1 + x2 + 2x3 2x1 + 4x2 − 4x3 x1 + 2x3 2x1 − 2x2 − x3 3x1 x1 , x 2 , x 3 , w 1 , w 2 , w 3 , w 4 , w 5 ≥ 0. Notes: • This layout is called a dictionary. • Setting x1 , x2 , and x3 to 0, we can read off the values for the other variables: w1 = 7, w2 = 3, etc. This specific solution is called a dictionary solution. • Dependent variables, on the left, are called basic variables. • Independent variables, on the right, are called nonbasic variables. Dictionary Solution is Feasible maximize −x1 + 3x2 − 3x3 ζ = subject to w1 w2 w3 w4 w5 = = = = = 7 3 4 8 5 − + − + − 3x1 + x2 + 2x3 2x1 + 4x2 − 4x3 x1 + 2x3 2x1 − 2x2 − x3 3x1 x1, x2, x3, w1, w2, w3 w4 w5 ≥ 0. Notes: • All the variables in the current dictionary solution are nonnegative. • Such a solution is called feasible. • The initial dictionary solution need not be feasible—we were just lucky above. Simplex Method—First Iteration • If x2 increases, obj goes up. • How much can x2 increase? Until w4 decreases to zero. • Do it. End result: x2 > 0 whereas w4 = 0. • That is, x2 must become basic and w4 must become nonbasic. • Algebraically rearrange equations to, in the words of Jean-Luc Picard, ”Make it so.” • This is a pivot. A Pivot: x2 ↔ w4 becomes Simplex Method—Second Pivot Here’s the dictionary after the first pivot: • Now, let x1 increase. • Of the basic variables, w5 hits zero first....

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...Op"erations Research This page intentionally left blank Copyright © 2007, 2005 New Age International (P) Ltd., Publishers Published by New Age International (P) Ltd., Publishers All rights reserved. No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the publisher. All inquiries should be emailed to [email protected] ISBN (13) : 978-81-224-2944-2 PUBLISHING FOR ONE WORLD NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS 4835/24, Ansari Road, Daryaganj, New Delhi - 110002 Visit us at PREFACE I started my teaching career in the year 1964. I was teaching Production Engineering subjects till 1972. In the year 1972 I have registered my name for the Industrial Engineering examination at National Institution of Industrial Engineering, Bombay. Since then, I have shifted my field for interest to Industrial Engineering subjects and started teaching related subjects. One such subject is OPERATIONS RESEARCH. After teaching these subjects till my retirement in the year 2002, it is my responsibility to help the students with a book on Operations research. The first volume of the book is LINEAR PORGRAMMING MODELS. This was published in the year 2003. Now I am giving this book OPERATIONS RESEARCH, with other chapters to students, with a hope that it will help them to......

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...Solutions 56:171 Operations Research Homework #3 Solutions – Fall 2002 1. Revised Simplex Method Consider the LP problem Maximize subject to z = 3 x1 − x2 + 2 x3 x1 + x2 + x3 ≤ 15 2 x1 − x2 + x3 ≤ 2 − x1 + x2 + x3 ≤ 4 x j ≥ 0, j = 1, 2,3 a. Let x4 , x5 , &, x6 denote the slack variables for the three constraints, and write the LP with equality constraints. Answer: Maximize z = 3 x1 − x2 + 2 x3 subject to x1 + x2 + x3 + x4 = 15 2 x1 − x2 + x3 + x5 = 2 − x1 + x2 + x3 + x6 = 4 x j ≥ 0, j = 1, 2,3, 4,5, 6 After several iterations of the revised simplex method, 1 0  the basis B={4,3,2} and the basis inverse matrix is ( AB ) −1 =  0 1 2  0 − 1   2 −1   1 . 2 1   2 b. Proceed with one iteration of the revised simplex method, by i. Computing the simplex multiplier vector π Answer: 1 0 −1    B −1 0 1 1  =  0, 3 , 1  π = CB ( A ) = [0 2 −1] 2 2  2 2  0 − 1 1    2 2  = [ 0, 1.5, 0.5] ii. “pricing”, i.e., computing the “relative profits”, of the non-basic columns. Answer: 56:171 O.R. -- HW #3 Solutions Fall 2002 page 1 of 8 Solutions  1 0 0 C = [3 0 0 ] , A =  2 1 0     −1 0 1    N N N −3 −1  C = C −π A =  1 2 2  2 The relative profits for non-basic variables are C1 = 0.5 , C5 = −1.5 , C6 = −0.5 . iii. Selecting the column to enter the basis. Answer: Only the relative profit of X 1 is positive and the problem is Max problem, and so X 1 should enter the basic. iv. Computing the substitution rates of the entering column. Answer: The......

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  6. The Simplex Method.

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    Paperback 9 pages. $15.00. $12.00 20% Web Discount. A proof of the existence of an optimal basis in the simplex method based on induction. Specifically, it is shown that a finite chain of feasible basis changes exists, resulting in either an optimal feasible solution or in an infinite class of feasible solutions, such that the objective form ...

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