Slide 1 Math 1520, Lecture 10
|
|
|
- Kelly Green
- 5 years ago
- Views:
Transcription
1 Slide 1 Math 1520, Lecture 10 In this lecture, we study the simplex method which is a powerful method for solving maximization/minimization problems developed in the late 1940 s. It has been implemented on computers and can be used on problems with a large number of variables and constraints. The simplex method is used in some of the management courses at RPI. The simplex method is based on the Gauss-Jordan method we studied in chapter 5. The simplex method is more involved and harder than the methods we have encountered thus far in this course.
2 Slide 2 A description of the simplex method Suppose we want to maximize a linear objective function. When there are many vertexes to the feasible set, it is very time consuming to find all the vertexes and test each one to find the maximum as we did in the method of corners. The simplex method shortens this process by iteratively traversing the feasible set in directions so as to make the objective function bigger. It organizies all this in a matrix called the simplex tableau.
3 Slide 3 Standard Maximization Problem The simplex method we describe first applies to standard maximization problems: The standard maximization problem is one in which 1. The goal is to maximize a linear objective function. 2. All the variables involved are non-negative (e.g. x 0 ) 3. All the other linear constraints may be written so that the expression is less than or equal to a non-negative constant. The following problem is not in standard form: Maximize P = 2x + y Subject to the constraints x 0, y 0, x y 4, x + 3y 6 But we can rewrite it to one that is in standard form as: Maximize P = 2x + y Subject to the constraints x 0, y 0, x + y 4, x + 3y 6
4 Slide 4 Slack Variables Consider this problem: Maximize P = x + 2y Subject to the constraints x 0, y 0, x + y 4, x + 3y 6 To solve it usinig the simplex method we first introduce slack variables called u and v to rewrite the last two inequality constraints as equalities to get x + y + u = 4, x + 3y + v = 6, u 0,, v 0 We then convert the objective function P = x + 2y into the form x 2y + P = 0 and put these into a system of linear equations x + y + u = 4 x + 3y + v = 6 x 2y + P = 0 where we want to maximize P subject to x 0, y 0, u 0, v 0.
5 Slide 5 iclicker Question Introduce slack variables for the standard maximization problem:. Maximize P = 3x + 2y Subject to the constraints x 0, y 0, 2x + 3y 12, 2x + y 8 A. B. C. D. E. None of the above 2x + 3y + u = 12 2x + y + v = 8 3x + 2y P = 0 2x + 3y + u = 8 2x + y + v = 12 3x + 2y P = 0 2x + 3y + u = 12 2x + y + v = 8 3x 2y + P = 0 2x + 3y + u = 8 2x + y + v = 12 3x 2y + P = 0
6 Answer to Question Introduce slack variables for the standard maximization problem:. A. Maximize P = 3x + 2y Subject to the constraints x 0, y 0, 2x + 3y 12, 2x + y 8 B. C. D. is the correct answer. E. None of the above 2x + 3y + u = 12 2x + y + v = 8 3x + 2y P = 0 2x + 3y + u = 8 2x + y + v = 12 3x + 2y P = 0 2x + 3y + u = 12 2x + y + v = 8 3x 2y + P = 0 2x + 3y + u = 8 2x + y + v = 12 3x 2y + P = 0
7 Slide 6 Simplex Tableau Once we introduce slack varibles, we take the resulting equations which were x + y + u = 4 x + 3y + v = 6 x 2y + P = 0 and from this set of equations, we write what is called the simplex tableau as In this tableau, the variables associated with unit columns are called basic variables and those with non-unit columns are called non-basic variables. The goal will be to go through a sequence of pivoting steps to make all the entries to the left of the vertical bar non-negative..
8 Slide 7 iclicker Question Write the simplex tableau for the following problem. Maximize P = 3x + 2y Subject to the constraints x 0, y 0, 2x + 3y 12, 2x + y 8. A. B. C. None of the above D. E
9 Answer to Question Write the simplex tableau for the following problem.. A. Maximize P = 3x + 2y Subject to the constraints B. x 0, y 0, 2x + 3y 12, 2x + y 8 is the correct answer. C. None of the above D. E
10 Slide 8 Pivoting For the tableau We proceed by making some of the columns to the left of the vertical bar into unit columns using Gauss-Jordan elimination. But we have to choose the pivots more carefully than we did with Gauss-Jordan elimination. To choose the pivot column, pick the column with the most negative entry to the left of the vertical bar in the last row. To choose the pivot row, divide each positive entry in the pivot column into the corroseonding entry in the column of constants. The pivot row is the row corresponding to the smallest ratio. We have a solution when all the entries to the left of the vertical bar in the last row are non-negative. This is called a simplex tableau in final form. In the above matrix the pivot entry is in the second column and second row. We pivot on this entry to make the second column into a unit column. 2/ / / / / /3 1 4
11 Slide 9 Pivoting (continued) We now have the matrix: 2/ / / / / /3 1 4 The next pivot entry is the first column and the first row. We pivot on this entry to get the first column to be a unit column /2 1/ /2 1/ /2 1/2 1 5 We can then read off the solution from matching the unit columns with the last column to get x = 3, y = 1 and P = 5 and the other variables are set to 0 so u = 0, v = 0.
12 Slide 10 Practice Try the same thing for the simplex tableau you found in the last iclicker question: We proceed by making some of the columns to the left of the vertical bar into unit columns using Gauss-Jordan elimination. But we have to choose the pivots more carefully than we did with Gauss-Jordan elimination. To choose the pivot column, pick the column with the most negative entry to the left of the vertical bar in the last row. To choose the pivot row, divide each positive entry in the privot column into the corroseonding entry in the column of constants. The pivot row is the row corresponding to the smallest ratio. We have a solution when all the entries to the left of the vertical bar in the last row are non-negative. This is called a simplex tableau in final form. We can read off the solution by matching the unit columns with the last column. All other variables associated with non-unit columns are set to zero in the solution.
13 Slide 11 iclicker Question Which of the following tableaus is in final form? A. B. C. D. E x y z u v P Constant
14 Answer to Question Which of the following tableaus is in final form? A. B. C. D. E. is the correct answer x y z u v P Constant
15 Slide 12 iclicker Question What is the solution for the following tableau? x y z u v P Constant A. x = 0, y = 4, z = 0, u = 6, v = 0 for a maximum of 15 B. x = 4, y = 0, z = 6, u = 0, v = 0 for a maximum of 15 C. x = 0, y = 4, z = 6, u = 0, v = 0 for a maximum of 15 D. x = 6, y = 4, z = 0, u = 0, v = 0 for a maximum of 15 E. None of the above
16 Answer to Question What is the solution for the following tableau? x y z u v P Constant A. x = 0, y = 4, z = 0, u = 6, v = 0 for a maximum of 15 is the correct answer. B. x = 4, y = 0, z = 6, u = 0, v = 0 for a maximum of 15 C. x = 0, y = 4, z = 6, u = 0, v = 0 for a maximum of 15 D. x = 6, y = 4, z = 0, u = 0, v = 0 for a maximum of 15 E. None of the above
Part III: A Simplex pivot
MA 3280 Lecture 31 - More on The Simplex Method Friday, April 25, 2014. Objectives: Analyze Simplex examples. We were working on the Simplex tableau The matrix form of this system of equations is called
Professor Alan H. Stein October 31, 2007
Mathematics 05 Professor Alan H. Stein October 3, 2007 SOLUTIONS. For most maximum problems, the contraints are in the form l(x) k, where l(x) is a linear polynomial and k is a positive constant. Explain
Lesson 27 Linear Programming; The Simplex Method
Lesson Linear Programming; The Simplex Method Math 0 April 9, 006 Setup A standard linear programming problem is to maximize the quantity c x + c x +... c n x n = c T x subject to constraints a x + a x
Math Models of OR: Some Definitions
Math Models of OR: Some Definitions John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell Some Definitions 1 / 20 Active constraints Outline 1 Active constraints
Lecture 5 Simplex Method. September 2, 2009
Simplex Method September 2, 2009 Outline: Lecture 5 Re-cap blind search Simplex method in steps Simplex tableau Operations Research Methods 1 Determining an optimal solution by exhaustive search Lecture
The augmented form of this LP is the following linear system of equations:
1 Consider the following LP given in standard form: max z = 5 x_1 + 2 x_2 Subject to 3 x_1 + 2 x_2 2400 x_2 800 2 x_1 1200 x_1, x_2 >= 0 The augmented form of this LP is the following linear system of
IE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 5: The Simplex method, continued Prof. John Gunnar Carlsson September 22, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 22, 2010
4.3 Minimizing & Mixed Constraints
Mathematics : Mattingly, Fall 6 8 4. Minimizing & Mixed Constraints So far, you have seen how to solve one type of problem: Standard Maximum. The objective function is to be maximized.. Constraints use..
Standard Form An LP is in standard form when: All variables are non-negativenegative All constraints are equalities Putting an LP formulation into sta
Chapter 4 Linear Programming: The Simplex Method An Overview of the Simplex Method Standard Form Tableau Form Setting Up the Initial Simplex Tableau Improving the Solution Calculating the Next Tableau
Non-Standard Constraints. Setting up Phase 1 Phase 2
Non-Standard Constraints Setting up Phase 1 Phase 2 Maximizing with Mixed Constraints Some maximization problems contain mixed constraints, like this: maximize 3x 1 + 2x 2 subject to 2x 1 + x 2 50 (standard)
The simplex algorithm
The simplex algorithm The simplex algorithm is the classical method for solving linear programs. Its running time is not polynomial in the worst case. It does yield insight into linear programs, however,
Week_4: simplex method II
Week_4: simplex method II 1 1.introduction LPs in which all the constraints are ( ) with nonnegative right-hand sides offer a convenient all-slack starting basic feasible solution. Models involving (=)
9.5 THE SIMPLEX METHOD: MIXED CONSTRAINTS
SECTION 9.5 THE SIMPLEX METHOD: MIXED CONSTRAINTS 557 9.5 THE SIMPLEX METHOD: MIXED CONSTRAINTS In Sections 9. and 9., you looked at linear programming problems that occurred in standard form. The constraints
Lecture 11 Linear programming : The Revised Simplex Method
Lecture 11 Linear programming : The Revised Simplex Method 11.1 The Revised Simplex Method While solving linear programming problem on a digital computer by regular simplex method, it requires storing
Prelude to the Simplex Algorithm. The Algebraic Approach The search for extreme point solutions.
Prelude to the Simplex Algorithm The Algebraic Approach The search for extreme point solutions. 1 Linear Programming-1 x 2 12 8 (4,8) Max z = 6x 1 + 4x 2 Subj. to: x 1 + x 2
1. Introduce slack variables for each inequaility to make them equations and rewrite the objective function in the form ax by cz... + P = 0.
3.4 Simplex Method If a linear programming problem has more than 2 variables, solving graphically is not the way to go. Instead, we ll use a more methodical, numeric process called the Simplex Method.
Systems Analysis in Construction
Systems Analysis in Construction CB312 Construction & Building Engineering Department- AASTMT by A h m e d E l h a k e e m & M o h a m e d S a i e d 3. Linear Programming Optimization Simplex Method 135
Math 210 Finite Mathematics Chapter 4.2 Linear Programming Problems Minimization - The Dual Problem
Math 2 Finite Mathematics Chapter 4.2 Linear Programming Problems Minimization - The Dual Problem Richard Blecksmith Dept. of Mathematical Sciences Northern Illinois University Math 2 Website: http://math.niu.edu/courses/math2.
Linear Programming, Lecture 4
Linear Programming, Lecture 4 Corbett Redden October 3, 2016 Simplex Form Conventions Examples Simplex Method To run the simplex method, we start from a Linear Program (LP) in the following standard simplex
Today s class. Constrained optimization Linear programming. Prof. Jinbo Bi CSE, UConn. Numerical Methods, Fall 2011 Lecture 12
Today s class Constrained optimization Linear programming 1 Midterm Exam 1 Count: 26 Average: 73.2 Median: 72.5 Maximum: 100.0 Minimum: 45.0 Standard Deviation: 17.13 Numerical Methods Fall 2011 2 Optimization
4.4 The Simplex Method and the Standard Minimization Problem
. The Simplex Method and the Standard Minimization Problem Question : What is a standard minimization problem? Question : How is the standard minimization problem related to the dual standard maximization
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Math324 - Test Review 2 - Fall 206 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the vertex of the parabola. ) f(x) = x 2-0x + 33 ) (0,
Review Solutions, Exam 2, Operations Research
Review Solutions, Exam 2, Operations Research 1. Prove the weak duality theorem: For any x feasible for the primal and y feasible for the dual, then... HINT: Consider the quantity y T Ax. SOLUTION: To
Linear Programming: Simplex Method CHAPTER The Simplex Tableau; Pivoting
CHAPTER 5 Linear Programming: 5.1. The Simplex Tableau; Pivoting Simplex Method In this section we will learn how to prepare a linear programming problem in order to solve it by pivoting using a matrix
Math Models of OR: Sensitivity Analysis
Math Models of OR: Sensitivity Analysis John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 8 USA October 8 Mitchell Sensitivity Analysis / 9 Optimal tableau and pivot matrix Outline Optimal
Understanding the Simplex algorithm. Standard Optimization Problems.
Understanding the Simplex algorithm. Ma 162 Spring 2011 Ma 162 Spring 2011 February 28, 2011 Standard Optimization Problems. A standard maximization problem can be conveniently described in matrix form
Math Models of OR: Handling Upper Bounds in Simplex
Math Models of OR: Handling Upper Bounds in Simplex John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 280 USA September 208 Mitchell Handling Upper Bounds in Simplex / 8 Introduction Outline
Dual Basic Solutions. Observation 5.7. Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP:
Dual Basic Solutions Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP: Observation 5.7. AbasisB yields min c T x max p T b s.t. A x = b s.t. p T A apple c T x 0 aprimalbasicsolutiongivenbyx
1. (7pts) Find the points of intersection, if any, of the following planes. 3x + 9y + 6z = 3 2x 6y 4z = 2 x + 3y + 2z = 1
Math 125 Exam 1 Version 1 February 20, 2006 1. (a) (7pts) Find the points of intersection, if any, of the following planes. Solution: augmented R 1 R 3 3x + 9y + 6z = 3 2x 6y 4z = 2 x + 3y + 2z = 1 3 9
MA 162: Finite Mathematics - Section 3.3/4.1
MA 162: Finite Mathematics - Section 3.3/4.1 Fall 2014 Ray Kremer University of Kentucky October 6, 2014 Announcements: Homework 3.3 due Tuesday at 6pm. Homework 4.1 due Friday at 6pm. Exam scores were
Introduction to the Simplex Algorithm Active Learning Module 3
Introduction to the Simplex Algorithm Active Learning Module 3 J. René Villalobos and Gary L. Hogg Arizona State University Paul M. Griffin Georgia Institute of Technology Background Material Almost any
Section Notes 9. IP: Cutting Planes. Applied Math 121. Week of April 12, 2010
Section Notes 9 IP: Cutting Planes Applied Math 121 Week of April 12, 2010 Goals for the week understand what a strong formulations is. be familiar with the cutting planes algorithm and the types of cuts
AM 121: Intro to Optimization Models and Methods Fall 2018
AM 121: Intro to Optimization Models and Methods Fall 2018 Lecture 5: The Simplex Method Yiling Chen Harvard SEAS Lesson Plan This lecture: Moving towards an algorithm for solving LPs Tableau. Adjacent
The Simplex Algorithm
8.433 Combinatorial Optimization The Simplex Algorithm October 6, 8 Lecturer: Santosh Vempala We proved the following: Lemma (Farkas). Let A R m n, b R m. Exactly one of the following conditions is true:.
Math x + 3y 5z = 14 3x 2y + 3z = 17 4x + 3y 2z = 1
Math 210 1. Solve the system: x + y + z = 1 2x + 3y + 4z = 5 (a z = 2, y = 1 and x = 0 (b z =any value, y = 3 2z and x = z 2 (c z =any value, y = 3 2z and x = z + 2 (d z =any value, y = 3 + 2z and x =
Exam 3 Review Math 118 Sections 1 and 2
Exam 3 Review Math 118 Sections 1 and 2 This exam will cover sections 5.3-5.6, 6.1-6.3 and 7.1-7.3 of the textbook. No books, notes, calculators or other aids are allowed on this exam. There is no time
MATH 210 EXAM 3 FORM A November 24, 2014
MATH 210 EXAM 3 FORM A November 24, 2014 Name (printed) Name (signature) ZID No. INSTRUCTIONS: (1) Use a No. 2 pencil. (2) Work on this test. No scratch paper is allowed. (3) Write your name and ZID number
Math Homework 3: solutions. 1. Consider the region defined by the following constraints: x 1 + x 2 2 x 1 + 2x 2 6
Math 7502 Homework 3: solutions 1. Consider the region defined by the following constraints: x 1 + x 2 2 x 1 + 2x 2 6 x 1, x 2 0. (i) Maximize 4x 1 + x 2 subject to the constraints above. (ii) Minimize
Ω R n is called the constraint set or feasible set. x 1
1 Chapter 5 Linear Programming (LP) General constrained optimization problem: minimize subject to f(x) x Ω Ω R n is called the constraint set or feasible set. any point x Ω is called a feasible point We
TIM 206 Lecture 3: The Simplex Method
TIM 206 Lecture 3: The Simplex Method Kevin Ross. Scribe: Shane Brennan (2006) September 29, 2011 1 Basic Feasible Solutions Have equation Ax = b contain more columns (variables) than rows (constraints),
ORF 307: Lecture 2. Linear Programming: Chapter 2 Simplex Methods
ORF 307: Lecture 2 Linear Programming: Chapter 2 Simplex Methods Robert Vanderbei February 8, 2018 Slides last edited on February 8, 2018 http://www.princeton.edu/ rvdb Simplex Method for LP An Example.
Introduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras
Introduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Module - 03 Simplex Algorithm Lecture 15 Infeasibility In this class, we
Linear Programming: Chapter 5 Duality
Linear Programming: Chapter 5 Duality Robert J. Vanderbei September 30, 2010 Slides last edited on October 5, 2010 Operations Research and Financial Engineering Princeton University Princeton, NJ 08544
The Simplex Method. Lecture 5 Standard and Canonical Forms and Setting up the Tableau. Lecture 5 Slide 1. FOMGT 353 Introduction to Management Science
The Simplex Method Lecture 5 Standard and Canonical Forms and Setting up the Tableau Lecture 5 Slide 1 The Simplex Method Formulate Constrained Maximization or Minimization Problem Convert to Standard
Yinyu Ye, MS&E, Stanford MS&E310 Lecture Note #06. The Simplex Method
The Simplex Method Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye (LY, Chapters 2.3-2.5, 3.1-3.4) 1 Geometry of Linear
AM 121: Intro to Optimization
AM 121: Intro to Optimization Models and Methods Lecture 6: Phase I, degeneracy, smallest subscript rule. Yiling Chen SEAS Lesson Plan Phase 1 (initialization) Degeneracy and cycling Smallest subscript
Math 354 Summer 2004 Solutions to review problems for Midterm #1
Solutions to review problems for Midterm #1 First: Midterm #1 covers Chapter 1 and 2. In particular, this means that it does not explicitly cover linear algebra. Also, I promise there will not be any proofs.
Introduce the idea of a nondegenerate tableau and its analogy with nondenegerate vertices.
2 JORDAN EXCHANGE REVIEW 1 Lecture Outline The following lecture covers Section 3.5 of the textbook [?] Review a labeled Jordan exchange with pivoting. Introduce the idea of a nondegenerate tableau and
Chapter 5 Linear Programming (LP)
Chapter 5 Linear Programming (LP) General constrained optimization problem: minimize f(x) subject to x R n is called the constraint set or feasible set. any point x is called a feasible point We consider
Lecture 2: The Simplex method
Lecture 2 1 Linear and Combinatorial Optimization Lecture 2: The Simplex method Basic solution. The Simplex method (standardform, b>0). 1. Repetition of basic solution. 2. One step in the Simplex algorithm.
Review Exercise 2. 1 a Chemical A 5x+ Chemical B 2x+ 2y12 [ x+ Chemical C [ 4 12]
![Review Exercise 2. 1 a Chemical A 5x+ Chemical B 2x+ 2y12 [ x+ Chemical C [ 4 12]](/thumbs/82/84751684.jpg "Review Exercise 2. 1 a Chemical A 5x+ Chemical B 2x+ 2y12 [ x+ Chemical C [ 4 12]")
Review Exercise a Chemical A 5x+ y 0 Chemical B x+ y [ x+ y 6] b Chemical C 6 [ ] x+ y x+ y x, y 0 c T = x+ y d ( x, y) = (, ) T = Pearson Education Ltd 08. Copying permitted for purchasing institution
Optimization (168) Lecture 7-8-9
Optimization (168) Lecture 7-8-9 Jesús De Loera UC Davis, Mathematics Wednesday, April 2, 2012 1 DEGENERACY IN THE SIMPLEX METHOD 2 DEGENERACY z =2x 1 x 2 + 8x 3 x 4 =1 2x 3 x 5 =3 2x 1 + 4x 2 6x 3 x 6
Lecture 11: Post-Optimal Analysis. September 23, 2009
Lecture : Post-Optimal Analysis September 23, 2009 Today Lecture Dual-Simplex Algorithm Post-Optimal Analysis Chapters 4.4 and 4.5. IE 30/GE 330 Lecture Dual Simplex Method The dual simplex method will
Chapter 4 The Simplex Algorithm Part I
Chapter 4 The Simplex Algorithm Part I Based on Introduction to Mathematical Programming: Operations Research, Volume 1 4th edition, by Wayne L. Winston and Munirpallam Venkataramanan Lewis Ntaimo 1 Modeling
Distributed Real-Time Control Systems. Lecture Distributed Control Linear Programming
Distributed Real-Time Control Systems Lecture 13-14 Distributed Control Linear Programming 1 Linear Programs Optimize a linear function subject to a set of linear (affine) constraints. Many problems can
Slack Variable. Max Z= 3x 1 + 4x 2 + 5X 3. Subject to: X 1 + X 2 + X x 1 + 4x 2 + X X 1 + X 2 + 4X 3 10 X 1 0, X 2 0, X 3 0
Simplex Method Slack Variable Max Z= 3x 1 + 4x 2 + 5X 3 Subject to: X 1 + X 2 + X 3 20 3x 1 + 4x 2 + X 3 15 2X 1 + X 2 + 4X 3 10 X 1 0, X 2 0, X 3 0 Standard Form Max Z= 3x 1 +4x 2 +5X 3 + 0S 1 + 0S 2
Summary of the simplex method
MVE165/MMG631,Linear and integer optimization with applications The simplex method: degeneracy; unbounded solutions; starting solutions; infeasibility; alternative optimal solutions Ann-Brith Strömberg
The Avis-Kalunzy Algorithm is designed to find a basic feasible solution (BFS) of a given set of constraints. Its input: A R m n and b R m such that
Lecture 4 Avis-Kaluzny and the Simplex Method Last time, we discussed some applications of Linear Programming, such as Max-Flow, Matching, and Vertex-Cover. The broad range of applications to Linear Programming
3. Duality: What is duality? Why does it matter? Sensitivity through duality.
1 Overview of lecture (10/5/10) 1. Review Simplex Method 2. Sensitivity Analysis: How does solution change as parameters change? How much is the optimal solution effected by changing A, b, or c? How much
LINEAR PROGRAMMING 2. In many business and policy making situations the following type of problem is encountered:
LINEAR PROGRAMMING 2 In many business and policy making situations the following type of problem is encountered: Maximise an objective subject to (in)equality constraints. Mathematical programming provides
c) Place the Coefficients from all Equations into a Simplex Tableau, labeled above with variables indicating their respective columns
BUILDING A SIMPLEX TABLEAU AND PROPER PIVOT SELECTION Maximize : 15x + 25y + 18 z s. t. 2x+ 3y+ 4z 60 4x+ 4y+ 2z 100 8x+ 5y 80 x 0, y 0, z 0 a) Build Equations out of each of the constraints above by introducing
Matrices, Row Reduction of Matrices
Matrices, Row Reduction of Matrices October 9, 014 1 Row Reduction and Echelon Forms In the previous section, we saw a procedure for solving systems of equations It is simple in that it consists of only
Math Models of OR: The Revised Simplex Method
Math Models of OR: The Revised Simplex Method John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell The Revised Simplex Method 1 / 25 Motivation Outline 1
CHAPTER 2. The Simplex Method
CHAPTER 2 The Simplex Method In this chapter we present the simplex method as it applies to linear programming problems in standard form. 1. An Example We first illustrate how the simplex method works
The Simplex Method. Standard form (max) z c T x = 0 such that Ax = b.
The Simplex Method Standard form (max) z c T x = 0 such that Ax = b. The Simplex Method Standard form (max) z c T x = 0 such that Ax = b. Build initial tableau. z c T 0 0 A b The Simplex Method Standard
Part 1. The Review of Linear Programming
In the name of God Part 1. The Review of Linear Programming 1.5. Spring 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Formulation of the Dual Problem Primal-Dual Relationship Economic Interpretation
Motivating examples Introduction to algorithms Simplex algorithm. On a particular example General algorithm. Duality An application to game theory
Instructor: Shengyu Zhang 1 LP Motivating examples Introduction to algorithms Simplex algorithm On a particular example General algorithm Duality An application to game theory 2 Example 1: profit maximization
Homework Assignment 4 Solutions
MTAT.03.86: Advanced Methods in Algorithms Homework Assignment 4 Solutions University of Tartu 1 Probabilistic algorithm Let S = {x 1, x,, x n } be a set of binary variables of size n 1, x i {0, 1}. Consider
9.1 Linear Programs in canonical form
9.1 Linear Programs in canonical form LP in standard form: max (LP) s.t. where b i R, i = 1,..., m z = j c jx j j a ijx j b i i = 1,..., m x j 0 j = 1,..., n But the Simplex method works only on systems
3 The Simplex Method. 3.1 Basic Solutions
3 The Simplex Method 3.1 Basic Solutions In the LP of Example 2.3, the optimal solution happened to lie at an extreme point of the feasible set. This was not a coincidence. Consider an LP in general form,
Gauss-Jordan Elimination for Solving Linear Equations Example: 1. Solve the following equations: (3)
The Simple Method Gauss-Jordan Elimination for Solving Linear Equations Eample: Gauss-Jordan Elimination Solve the following equations: + + + + = 4 = = () () () - In the first step of the procedure, we
Notes on Simplex Algorithm
Notes on Simplex Algorithm CS 9 Staff October 8, 7 Until now, we have represented the problems geometrically, and solved by finding a corner and moving around Now we learn an algorithm to solve this without
Linear programs Optimization Geoff Gordon Ryan Tibshirani
Linear programs 10-725 Optimization Geoff Gordon Ryan Tibshirani Review: LPs LPs: m constraints, n vars A: R m n b: R m c: R n x: R n ineq form [min or max] c T x s.t. Ax b m n std form [min or max] c
Math Models of OR: Extreme Points and Basic Feasible Solutions
Math Models of OR: Extreme Points and Basic Feasible Solutions John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 1180 USA September 018 Mitchell Extreme Points and Basic Feasible Solutions
CS Algorithms and Complexity
CS 50 - Algorithms and Complexity Linear Programming, the Simplex Method, and Hard Problems Sean Anderson 2/15/18 Portland State University Table of contents 1. The Simplex Method 2. The Graph Problem
MATH 445/545 Homework 2: Due March 3rd, 2016
MATH 445/545 Homework 2: Due March 3rd, 216 Answer the following questions. Please include the question with the solution (write or type them out doing this will help you digest the problem). I do not
OPRE 6201 : 3. Special Cases
OPRE 6201 : 3. Special Cases 1 Initialization: The Big-M Formulation Consider the linear program: Minimize 4x 1 +x 2 3x 1 +x 2 = 3 (1) 4x 1 +3x 2 6 (2) x 1 +2x 2 3 (3) x 1, x 2 0. Notice that there are
Lecture Simplex Issues: Number of Pivots. ORIE 6300 Mathematical Programming I October 9, 2014
ORIE 6300 Mathematical Programming I October 9, 2014 Lecturer: David P. Williamson Lecture 14 Scribe: Calvin Wylie 1 Simplex Issues: Number of Pivots Question: How many pivots does the simplex algorithm
IE 400: Principles of Engineering Management. Simplex Method Continued
IE 400: Principles of Engineering Management Simplex Method Continued 1 Agenda Simplex for min problems Alternative optimal solutions Unboundedness Degeneracy Big M method Two phase method 2 Simplex for
1 Review Session. 1.1 Lecture 2
1 Review Session Note: The following lists give an overview of the material that was covered in the lectures and sections. Your TF will go through these lists. If anything is unclear or you have questions
Linear Programming and the Simplex method
Linear Programming and the Simplex method Harald Enzinger, Michael Rath Signal Processing and Speech Communication Laboratory Jan 9, 2012 Harald Enzinger, Michael Rath Jan 9, 2012 page 1/37 Outline Introduction
Optimization 4. GAME THEORY
Optimization GAME THEORY DPK Easter Term Saddle points of two-person zero-sum games We consider a game with two players Player I can choose one of m strategies, indexed by i =,, m and Player II can choose
CSC Design and Analysis of Algorithms. LP Shader Electronics Example
CSC 80- Design and Analysis of Algorithms Lecture (LP) LP Shader Electronics Example The Shader Electronics Company produces two products:.eclipse, a portable touchscreen digital player; it takes hours
Advanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras
Advanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture - 3 Simplex Method for Bounded Variables We discuss the simplex algorithm
Week 3: Simplex Method I
Week 3: Simplex Method I 1 1. Introduction The simplex method computations are particularly tedious and repetitive. It attempts to move from one corner point of the solution space to a better corner point
DEPARTMENT OF STATISTICS AND OPERATIONS RESEARCH OPERATIONS RESEARCH DETERMINISTIC QUALIFYING EXAMINATION. Part I: Short Questions
DEPARTMENT OF STATISTICS AND OPERATIONS RESEARCH OPERATIONS RESEARCH DETERMINISTIC QUALIFYING EXAMINATION Part I: Short Questions August 12, 2008 9:00 am - 12 pm General Instructions This examination is
Supplementary lecture notes on linear programming. We will present an algorithm to solve linear programs of the form. maximize.
Cornell University, Fall 2016 Supplementary lecture notes on linear programming CS 6820: Algorithms 26 Sep 28 Sep 1 The Simplex Method We will present an algorithm to solve linear programs of the form
Math 1314 Week #14 Notes
Math 3 Week # Notes Section 5.: A system of equations consists of two or more equations. A solution to a system of equations is a point that satisfies all the equations in the system. In this chapter,
Optimum Solution of Linear Programming Problem by Simplex Method
Optimum Solution of Linear Programming Problem by Simplex Method U S Hegde 1, S Uma 2, Aravind P N 3 1 Associate Professor & HOD, Department of Mathematics, Sir M V I T, Bangalore, India 2 Associate Professor,
ECE 307 Techniques for Engineering Decisions
ECE 7 Techniques for Engineering Decisions Introduction to the Simple Algorithm George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign ECE 7 5 9 George
3. THE SIMPLEX ALGORITHM
Optimization. THE SIMPLEX ALGORITHM DPK Easter Term. Introduction We know that, if a linear programming problem has a finite optimal solution, it has an optimal solution at a basic feasible solution (b.f.s.).
MATH2070 Optimisation
MATH2070 Optimisation Linear Programming Semester 2, 2012 Lecturer: I.W. Guo Lecture slides courtesy of J.R. Wishart Review The standard Linear Programming (LP) Problem Graphical method of solving LP problem
MATH 445/545 Test 1 Spring 2016
MATH 445/545 Test Spring 06 Note the problems are separated into two sections a set for all students and an additional set for those taking the course at the 545 level. Please read and follow all of these
February 17, Simplex Method Continued
15.053 February 17, 2005 Simplex Method Continued 1 Today s Lecture Review of the simplex algorithm. Formalizing the approach Alternative Optimal Solutions Obtaining an initial bfs Is the simplex algorithm
Farkas Lemma, Dual Simplex and Sensitivity Analysis
Summer 2011 Optimization I Lecture 10 Farkas Lemma, Dual Simplex and Sensitivity Analysis 1 Farkas Lemma Theorem 1. Let A R m n, b R m. Then exactly one of the following two alternatives is true: (i) x
Example. 1 Rows 1,..., m of the simplex tableau remain lexicographically positive
3.4 Anticycling Lexicographic order In this section we discuss two pivoting rules that are guaranteed to avoid cycling. These are the lexicographic rule and Bland s rule. Definition A vector u R n is lexicographically
CPS 616 ITERATIVE IMPROVEMENTS 10-1
CPS 66 ITERATIVE IMPROVEMENTS 0 - APPROACH Algorithm design technique for solving optimization problems Start with a feasible solution Repeat the following step until no improvement can be found: change
Finite Mathematics MAT 141: Chapter 4 Notes
Finite Mathematics MAT 141: Chapter 4 Notes The Simplex Method David J. Gisch Slack Variables and the Pivot Simplex Method and Slack Variables We saw in the last chapter that we can use linear programming
Section 4.1 Solving Systems of Linear Inequalities
Section 4.1 Solving Systems of Linear Inequalities Question 1 How do you graph a linear inequality? Question 2 How do you graph a system of linear inequalities? Question 1 How do you graph a linear inequality?
Worked Examples for Chapter 5
Worked Examples for Chapter 5 Example for Section 5.2 Construct the primal-dual table and the dual problem for the following linear programming model fitting our standard form. Maximize Z = 5 x 1 + 4 x