The Big M Method. Modify the LP
|
|
|
- Bruno Jones
- 5 years ago
- Views:
Transcription
1 The Big M Method Modify the LP 1. If any functional constraints have negative constants on the right side, multiply both sides by 1 to obtain a constraint with a positive constant. Big M Simplex: 1
2 The Big M Method Modify the LP 1. If any functional constraints have negative constants on the right side, multiply both sides by 1 to obtain a constraint with a positive constant. 2. Introduce a slack variable s i 0 for each constraint. Big M Simplex: 1
3 The Big M Method Modify the LP 1. If any functional constraints have negative constants on the right side, multiply both sides by 1 to obtain a constraint with a positive constant. 2. Introduce a slack variable s i 0 for each constraint. 3. Introduce a surplus variable s j 0 and an artificial variable x i 0 for each constraint. Big M Simplex: 1
4 The Big M Method Modify the LP 1. If any functional constraints have negative constants on the right side, multiply both sides by 1 to obtain a constraint with a positive constant. 2. Introduce a slack variable s i 0 for each constraint. 3. Introduce a surplus variable s j 0 and an artificial variable x i 0 for each constraint. 4. Introduce an artificial variable x j 0 in each = constraint. Big M Simplex: 1
5 The Big M Method Modify the LP 1. If any functional constraints have negative constants on the right side, multiply both sides by 1 to obtain a constraint with a positive constant. 2. Introduce a slack variable s i 0 for each constraint. 3. Introduce a surplus variable s j 0 and an artificial variable x i 0 for each constraint. 4. Introduce an artificial variable x j 0 in each = constraint. 5. For each artificial variable x i, add a penalty term M x i to the objective function. Use the same constant M for all the artificial variables. (In numerical software, use a very large number for M.) Big M Simplex: 1
6 The Big M Method: Example Example (Big M in Action) Maximize P = 2x 1 + x 2 subject to x 1 + x 2 10 x 1 + x 2 2 with x 1, x 2 0. Big M Simplex: 2
7 The Big M Method: Example Example (Big M in Action) Maximize P = 2x 1 + x 2 subject to x 1 + x 2 10 x 1 + x 2 2 with x 1, x 2 0. The Big M Simplex Tableau Eq Z x 1 x 2 s 1 s 2 x 1 b (0) M 0 (1) (2) Use Maple Big M Simplex: 2
8 Exercise (O-Jay) The Big M Method: Exercise O-Jay is a mixture of orange juice and orange soda. We need to restrict the amount of sugar to 4gm/bottle and maintain at least 20mg/bottle of vitamin C. What is the least cost mixture? Let: x 1 = number of ounces of orange soda in a bottle of O-Jay x 2 = number of ounces of orange juice in a bottle of O-Jay The LP is: Minimize z = 2x 1 + 3x 2 subject to 0.5x x 2 4 (sugar constraint) with x 1, x 2 0 x 1 + 3x 2 20 x 1 + x 2 = 10 (Vitamin C constraint) (10 oz in per bottle) Big M Simplex: 3
9 The Big M Method: Summary Summary Big M Simplex: 4
10 Summary The Big M Method: Summary 1. If the problem is minimize Z, change to maximize (-Z). Big M Simplex: 4
11 The Big M Method: Summary Summary 1. If the problem is minimize Z, change to maximize (-Z). 2. Add a slack variable s i to change to =. Big M Simplex: 4
12 Summary The Big M Method: Summary 1. If the problem is minimize Z, change to maximize (-Z). 2. Add a slack variable s i to change to =. 3. Subtract a surplus variable s j and add an artificial variable x j to change to =. Big M Simplex: 4
13 Summary The Big M Method: Summary 1. If the problem is minimize Z, change to maximize (-Z). 2. Add a slack variable s i to change to =. 3. Subtract a surplus variable s j and add an artificial variable x j to change to =. 4. Add an artificial variable x k to each = constraint. Big M Simplex: 4
14 Summary The Big M Method: Summary 1. If the problem is minimize Z, change to maximize (-Z). 2. Add a slack variable s i to change to =. 3. Subtract a surplus variable s j and add an artificial variable x j to change to =. 4. Add an artificial variable x k to each = constraint. 5. Add M x j to the objective function for each artificial variable x j. Big M Simplex: 4
15 Summary The Big M Method: Summary 1. If the problem is minimize Z, change to maximize (-Z). 2. Add a slack variable s i to change to =. 3. Subtract a surplus variable s j and add an artificial variable x j to change to =. 4. Add an artificial variable x k to each = constraint. 5. Add M x j to the objective function for each artificial variable x j. 6. Use a row operation with each artificial variable row to eliminate M from the objective function in x j columns. 1 1 Just operate on Row (0), the objective function, to reduce arithmetic. Big M Simplex: 4
16 Summary The Big M Method: Summary 1. If the problem is minimize Z, change to maximize (-Z). 2. Add a slack variable s i to change to =. 3. Subtract a surplus variable s j and add an artificial variable x j to change to =. 4. Add an artificial variable x k to each = constraint. 5. Add M x j to the objective function for each artificial variable x j. 6. Use a row operation with each artificial variable row to eliminate M from the objective function in x j columns Run the simplex algorithm. 1 Just operate on Row (0), the objective function, to reduce arithmetic. Big M Simplex: 4
17 The Big M Method: Big Example Example (Big Big M ) Maximize Z = 2x 1 + 5x 2 + 3x 3 subject to x 1 + 2x 2 x 3 7 with x 1, x 2, x 3 0. x 1 + x 2 2x 3 5 x 1 + 4x 2 + 3x 3 1 2x 1 x 2 + 4x 3 = 6 Big M Simplex: 5
18 The Big M Method: Big Example Example (Big Big M ) Maximize Z = 2x 1 + 5x 2 + 3x 3 subject to x 1 + 2x 2 x 3 7 with x 1, x 2, x 3 0. x 1 + x 2 2x 3 5 x 1 + 4x 2 + 3x 3 1 2x 1 x 2 + 4x 3 = 6 Variables There are: Big M Simplex: 5
19 The Big M Method: Big Example Example (Big Big M ) Maximize Z = 2x 1 + 5x 2 + 3x 3 subject to x 1 + 2x 2 x 3 7 with x 1, x 2, x 3 0. x 1 + x 2 2x 3 5 x 1 + 4x 2 + 3x 3 1 2x 1 x 2 + 4x 3 = 6 Variables There are: 3 decision variables x i Big M Simplex: 5
20 The Big M Method: Big Example Example (Big Big M ) Maximize Z = 2x 1 + 5x 2 + 3x 3 subject to x 1 + 2x 2 x 3 7 with x 1, x 2, x 3 0. x 1 + x 2 2x 3 5 x 1 + 4x 2 + 3x 3 1 2x 1 x 2 + 4x 3 = 6 Variables There are: 3 decision variables x i 1 slack variable s 1 Big M Simplex: 5
21 The Big M Method: Big Example Example (Big Big M ) Maximize Z = 2x 1 + 5x 2 + 3x 3 subject to x 1 + 2x 2 x 3 7 with x 1, x 2, x 3 0. x 1 + x 2 2x 3 5 x 1 + 4x 2 + 3x 3 1 2x 1 x 2 + 4x 3 = 6 Variables There are: 3 decision variables x i 2 surplus variables s j 1 slack variable s 1 Big M Simplex: 5
22 The Big M Method: Big Example Example (Big Big M ) Maximize Z = 2x 1 + 5x 2 + 3x 3 subject to x 1 + 2x 2 x 3 7 with x 1, x 2, x 3 0. x 1 + x 2 2x 3 5 x 1 + 4x 2 + 3x 3 1 2x 1 x 2 + 4x 3 = 6 Variables There are: 3 decision variables x i 1 slack variable s 1 2 surplus variables s j 3 artifical variables x j Big M Simplex: 5
23 The Big M Method: Big Example Initial Artificial Problem Tableau BV Eq Z x 1 x 2 x 3 s 1 s 2 s 3 x 1 x 2 x 3 b (0) M M M 0 s 1 (1) x 1 (2) x 2 (3) x 3 (4) Big M Simplex: 6
24 The Big M Method: Big Example Initial Artificial Problem Tableau BV Eq Z x 1 x 2 x 3 s 1 s 2 s 3 x 1 x 2 x 3 b (0) M M M 0 s 1 (1) x 1 (2) x 2 (3) x 3 (4) Beginning Simplex Tableau Eq x 1 x 2 x 3 s 1 s 2 s 3 x 1 x 2 x 3 b (0) 2 4M 5 2M 3 9M 0 M M M (1) (2) (3) (4) Use Maple Big M Simplex: 6
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
Chapter 4 The Simplex Algorithm Part II
Chapter 4 The Simple Algorithm Part II Based on Introduction to Mathematical Programming: Operations Research, Volume 4th edition, by Wayne L Winston and Munirpallam Venkataramanan Lewis Ntaimo L Ntaimo
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
21. Solve the LP given in Exercise 19 using the big-m method discussed in Exercise 20.
Extra Problems for Chapter 3. Linear Programming Methods 20. (Big-M Method) An alternative to the two-phase method of finding an initial basic feasible solution by minimizing the sum of the artificial
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
The Simplex Algorithm
The Simplex Algorithm How to Convert an LP to Standard Form Before the simplex algorithm can be used to solve an LP, the LP must be converted into a problem where all the constraints are equations and
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 (=)
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
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
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
Metode Kuantitatif Bisnis. Week 4 Linear Programming Simplex Method - Minimize
Metode Kuantitatif Bisnis Week 4 Linear Programming Simplex Method - Minimize Outlines Solve Linear Programming Model Using Graphic Solution Solve Linear Programming Model Using Simplex Method (Maximize)
The Simplex Method. Formulate Constrained Maximization or Minimization Problem. Convert to Standard Form. Convert to Canonical Form
The Simplex Method 1 The Simplex Method Formulate Constrained Maximization or Minimization Problem Convert to Standard Form Convert to Canonical Form Set Up the Tableau and the Initial Basic Feasible Solution
Simplex Method for LP (II)
Simplex Method for LP (II) Xiaoxi Li Wuhan University Sept. 27, 2017 (week 4) Operations Research (Li, X.) Simplex Method for LP (II) Sept. 27, 2017 (week 4) 1 / 31 Organization of this lecture Contents:
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
Theory of Linear Programming
SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT BF360 Operations Research Unit Two Theory of Linear Programming Moses Mwale e-mail: moses.mwale@ictar.ac.zm BF360 Operations Research Contents Unit 2: Theory
Variants of Simplex Method
Variants of Simplex Method All the examples we have used in the previous chapter to illustrate simple algorithm have the following common form of constraints; i.e. a i x + a i x + + a in x n b i, i =,,,m
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
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
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
Using IOR Tutorial; Artificial Variables and Initial Basic Feasible Solutions
Using IOR Tutorial; Artificial Variables and Initial Basic Feasible Solutions Class 11 October 4, 2017 Announcements Guest Speaker Thursday at 3 pm Warner 208 Refreshments The Sleeping Beauty Controversy:
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
OPERATIONS RESEARCH. Michał Kulej. Business Information Systems
OPERATIONS RESEARCH Michał Kulej Business Information Systems The development of the potential and academic programmes of Wrocław University of Technology Project co-financed by European Union within European
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.
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
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
Summary of the simplex method
MVE165/MMG630, The simplex method; degeneracy; unbounded solutions; infeasibility; starting solutions; duality; interpretation Ann-Brith Strömberg 2012 03 16 Summary of the simplex method Optimality condition:
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
Dr. Maddah ENMG 500 Engineering Management I 10/21/07
Dr. Maddah ENMG 500 Engineering Management I 10/21/07 Computational Procedure of the Simplex Method The optimal solution of a general LP problem is obtained in the following steps: Step 1. Express the
Linear Programming. Xi Chen. Department of Management Science and Engineering International Business School Beijing Foreign Studies University
Linear Programming Xi Chen Department of Management Science and Engineering International Business School Beijing Foreign Studies University Xi Chen (chenxi0109@bfsu.edu.cn) Linear Programming 1 / 148
Linear Programming. Dr. Xiaosong DING
Linear Programming Dr. Xiaosong DING Department of Management Science and Engineering International Business School Beijing Foreign Studies University Dr. DING (xiaosong.ding@hotmail.com) Linear Programming
OPERATIONS RESEARCH. Linear Programming Problem
OPERATIONS RESEARCH Chapter 1 Linear Programming Problem Prof. Bibhas C. Giri Department of Mathematics Jadavpur University Kolkata, India Email: bcgiri.jumath@gmail.com MODULE - 2: Simplex Method for
AM 121 Introduction to Optimization: Models and Methods Example Questions for Midterm 1
AM 121 Introduction to Optimization: Models and Methods Example Questions for Midterm 1 Prof. Yiling Chen Fall 2018 Here are some practice questions to help to prepare for the midterm. The midterm will
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)
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
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?
IP Cut Homework from J and B Chapter 9: 14, 15, 16, 23, 24, You wish to solve the IP below with a cutting plane technique.
IP Cut Homework from J and B Chapter 9: 14, 15, 16, 23, 24, 31 14. You wish to solve the IP below with a cutting plane technique. Maximize 4x 1 + 2x 2 + x 3 subject to 14x 1 + 10x 2 + 11x 3 32 10x 1 +
Algebraic Simplex Active Learning Module 4
Algebraic Simplex Active Learning Module 4 J. René Villalobos and Gary L. Hogg Arizona State University Paul M. Griffin Georgia Institute of Technology Time required for the module: 50 Min. Reading Most
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
Chap6 Duality Theory and Sensitivity Analysis
Chap6 Duality Theory and Sensitivity Analysis The rationale of duality theory Max 4x 1 + x 2 + 5x 3 + 3x 4 S.T. x 1 x 2 x 3 + 3x 4 1 5x 1 + x 2 + 3x 3 + 8x 4 55 x 1 + 2x 2 + 3x 3 5x 4 3 x 1 ~x 4 0 If we
min 4x 1 5x 2 + 3x 3 s.t. x 1 + 2x 2 + x 3 = 10 x 1 x 2 6 x 1 + 3x 2 + x 3 14
The exam is three hours long and consists of 4 exercises. The exam is graded on a scale 0-25 points, and the points assigned to each question are indicated in parenthesis within the text. If necessary,
The Simplex Method of Linear Programming
The Simplex Method of Linear Programming Online Tutorial 3 Tutorial Outline CONVERTING THE CONSTRAINTS TO EQUATIONS SETTING UP THE FIRST SIMPLEX TABLEAU SIMPLEX SOLUTION PROCEDURES SUMMARY OF SIMPLEX STEPS
Dr. S. Bourazza Math-473 Jazan University Department of Mathematics
Dr. Said Bourazza Department of Mathematics Jazan University 1 P a g e Contents: Chapter 0: Modelization 3 Chapter1: Graphical Methods 7 Chapter2: Simplex method 13 Chapter3: Duality 36 Chapter4: Transportation
OPTIMISATION 3: NOTES ON THE SIMPLEX ALGORITHM
OPTIMISATION 3: NOTES ON THE SIMPLEX ALGORITHM Abstract These notes give a summary of the essential ideas and results It is not a complete account; see Winston Chapters 4, 5 and 6 The conventions and notation
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
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),
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..
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
...(iii), x 2 Example 7: Geetha Perfume Company produces both perfumes and body spray from two flower extracts F 1. The following data is provided:
The LP formulation is Linear Programming: Graphical Method Maximize, Z = 2x + 7x 2 Subject to constraints, 2x + x 2 200...(i) x 75...(ii) x 2 00...(iii) where x, x 2 ³ 0 Example 7: Geetha Perfume Company
AM 121: Intro to Optimization Models and Methods
AM 121: Intro to Optimization Models and Methods Fall 2017 Lecture 2: Intro to LP, Linear algebra review. Yiling Chen SEAS Lecture 2: Lesson Plan What is an LP? Graphical and algebraic correspondence Problems
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
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.
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
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
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
LINEAR PROGRAMMING. Relation to the Text (cont.) Relation to Material in Text. Relation to the Text. Relation to the Text (cont.
LINEAR PROGRAMMING Relation to Material in Text After a brief introduction to linear programming on p. 3, Cornuejols and Tϋtϋncϋ give a theoretical discussion including duality, and the simplex solution
Gomory Cuts. Chapter 5. Linear Arithmetic. Decision Procedures. An Algorithmic Point of View. Revision 1.0
Chapter 5 Linear Arithmetic Decision Procedures An Algorithmic Point of View D.Kroening O.Strichman Revision 1.0 Cutting planes Recall that in Branch & Bound we first solve a relaxed problem (i.e., no
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
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
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
Special cases of linear programming
Special cases of linear programming Infeasible solution Multiple solution (infinitely many solution) Unbounded solution Degenerated solution Notes on the Simplex tableau 1. The intersection of any basic
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
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.
Simplex Algorithm Using Canonical Tableaus
41 Simplex Algorithm Using Canonical Tableaus Consider LP in standard form: Min z = cx + α subject to Ax = b where A m n has rank m and α is a constant In tableau form we record it as below Original Tableau
6-3 Solving Systems by Elimination
Another method for solving systems of equations is elimination. Like substitution, the goal of elimination is to get one equation that has only one variable. To do this by elimination, you add the two
Brief summary of linear programming and duality: Consider the linear program in standard form. (P ) min z = cx. x 0. (D) max yb. z = c B x B + c N x N
Brief summary of linear programming and duality: Consider the linear program in standard form (P ) min z = cx s.t. Ax = b x 0 where A R m n, c R 1 n, x R n 1, b R m 1,and its dual (D) max yb s.t. ya c.
THE STEPS OF THE SIMPLEX ALGORITHM
THE STEPS OF THE SIMPLEX ALGORITHM Contents 1. Introduction... 2 2. Slack and surplus variables... 2 3. Basic and nonbasic variables... 2 4. Admissible solutions... 3 5. Solution of a linear program (LP)...
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
(P ) Minimize 4x 1 + 6x 2 + 5x 3 s.t. 2x 1 3x 3 3 3x 2 2x 3 6
The exam is three hours long and consists of 4 exercises. The exam is graded on a scale 0-25 points, and the points assigned to each question are indicated in parenthesis within the text. Problem 1 Consider
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
The Simplex Method for Solving a Linear Program Prof. Stephen Graves
The Simplex Method for Solving a Linear Program Prof. Stephen Graves Observations from Geometry feasible region is a convex polyhedron an optimum occurs at a corner point possible algorithm - search over
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
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
SEN301 OPERATIONS RESEARCH I LECTURE NOTES
SEN30 OPERATIONS RESEARCH I LECTURE NOTES SECTION II (208-209) Y. İlker Topcu, Ph.D. & Özgür Kabak, Ph.D. Acknowledgements: We would like to acknowledge Prof. W.L. Winston's "Operations Research: Applications
Contents. 4.5 The(Primal)SimplexMethod NumericalExamplesoftheSimplexMethod
Contents 4 The Simplex Method for Solving LPs 149 4.1 Transformations to be Carried Out On an LP Model Before Applying the Simplex Method On It... 151 4.2 Definitions of Various Types of Basic Vectors
56:270 Final Exam - May
@ @ 56:270 Linear Programming @ @ Final Exam - May 4, 1989 @ @ @ @ @ @ @ @ @ @ @ @ @ @ Select any 7 of the 9 problems below: (1.) ANALYSIS OF MPSX OUTPUT: Please refer to the attached materials on the
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
MATH 4211/6211 Optimization Linear Programming
MATH 4211/6211 Optimization Linear Programming Xiaojing Ye Department of Mathematics & Statistics Georgia State University Xiaojing Ye, Math & Stat, Georgia State University 0 The standard form of a Linear
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
1 Simplex and Matrices
1 Simplex and Matrices We will begin with a review of matrix multiplication. A matrix is simply an array of numbers. If a given array has m rows and n columns, then it is called an m n (or m-by-n) matrix.
6.2: The Simplex Method: Maximization (with problem constraints of the form )
6.2: The Simplex Method: Maximization (with problem constraints of the form ) 6.2.1 The graphical method works well for solving optimization problems with only two decision variables and relatively few
Signature: Name (PRINT CLEARLY) and ID number:
AMS 540 / MBA 540 (Fall, 2008) Estie Arkin Linear Programming - Midterm Do all problems. Write your answers on the exam. You are permitted to use the text, your notes and any material handed out in class.
Linear Programming Inverse Projection Theory Chapter 3
1 Linear Programming Inverse Projection Theory Chapter 3 University of Chicago Booth School of Business Kipp Martin September 26, 2017 2 Where We Are Headed We want to solve problems with special structure!
Example Problem. Linear Program (standard form) CSCI5654 (Linear Programming, Fall 2013) Lecture-7. Duality
CSCI5654 (Linear Programming, Fall 013) Lecture-7 Duality Lecture 7 Slide# 1 Lecture 7 Slide# Linear Program (standard form) Example Problem maximize c 1 x 1 + + c n x n s.t. a j1 x 1 + + a jn x n b j
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
Slide 1 Math 1520, Lecture 10
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
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
Practice Final Exam Answers
Practice Final Exam Answers 1. AutoTime, a manufacturer of electronic digital timers, has a monthly fixed cost of $48,000 and a production cost $8 per timer. The timers sell for $14 apiece. (a) (3 pts)
CSCI 1951-G Optimization Methods in Finance Part 01: Linear Programming
CSCI 1951-G Optimization Methods in Finance Part 01: Linear Programming January 26, 2018 1 / 38 Liability/asset cash-flow matching problem Recall the formulation of the problem: max w c 1 + p 1 e 1 = 150
Sensitivity Analysis
Dr. Maddah ENMG 500 /9/07 Sensitivity Analysis Changes in the RHS (b) Consider an optimal LP solution. Suppose that the original RHS (b) is changed from b 0 to b new. In the following, we study the affect
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
In Chapters 3 and 4 we introduced linear programming
SUPPLEMENT The Simplex Method CD3 In Chapters 3 and 4 we introduced linear programming and showed how models with two variables can be solved graphically. We relied on computer programs (WINQSB, Excel,
1. Algebraic and geometric treatments Consider an LP problem in the standard form. x 0. Solutions to the system of linear equations
The Simplex Method Most textbooks in mathematical optimization, especially linear programming, deal with the simplex method. In this note we study the simplex method. It requires basically elementary linear
Answer the following questions: Q1: Choose the correct answer ( 20 Points ):
Benha University Final Exam. (ختلفات) Class: 2 rd Year Students Subject: Operations Research Faculty of Computers & Informatics Date: - / 5 / 2017 Time: 3 hours Examiner: Dr. El-Sayed Badr Answer the following
Introduction. Very efficient solution procedure: simplex method.
LINEAR PROGRAMMING Introduction Development of linear programming was among the most important scientific advances of mid 20th cent. Most common type of applications: allocate limited resources to competing
END3033 Operations Research I Sensitivity Analysis & Duality. to accompany Operations Research: Applications and Algorithms Fatih Cavdur
END3033 Operations Research I Sensitivity Analysis & Duality to accompany Operations Research: Applications and Algorithms Fatih Cavdur Introduction Consider the following problem where x 1 and x 2 corresponds
Generating Gomory's Cuts for linear integer programming problems: the HOW and WHY
Page 1 of 7 Generating Gomory's Cuts for linear integer programming problems: the HOW and WHY A Gomory's Cut is a linear constraint with the property that it is strictly stronger than its Parent, but it
Duality in LPP Every LPP called the primal is associated with another LPP called dual. Either of the problems is primal with the other one as dual. The optimal solution of either problem reveals the information
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
CO350 Linear Programming Chapter 6: The Simplex Method
CO50 Linear Programming Chapter 6: The Simplex Method rd June 2005 Chapter 6: The Simplex Method 1 Recap Suppose A is an m-by-n matrix with rank m. max. c T x (P ) s.t. Ax = b x 0 On Wednesday, we learned