A = Chapter 6. Linear Programming: The Simplex Method. + 21x 3 x x 2. C = 16x 1. + x x x 1. + x 3. 16,x 2.
|
|
- Martha Parks
- 6 years ago
- Views:
Transcription
1 Chapter 6 Linear rogramming: The Simple Method Section The Dual roblem: Minimization with roblem Constraints of the Form Learning Objectives for Section 6. Dual roblem: Minimization with roblem Constraints of the Form > The student will be able to formulate the dual problem. The student will be able to solve minimization problems. The student will be able to solve applications of the dual such as the transportation problem. The student will be able to summarize problem types and solution methods. Dual roblem: Minimization With roblem Constraints of the Form > Initial Matri Associated with each minimization problem with > constraints is a maimization problem called the dual problem. The dual problem will be illustrated through an eample. We wish to minimize the objective function C subject to certain constraints: C = ,, 0 We start with an initial matri A which corresponds to the problem constraints: A =
2 Transpose of Matri A To find the transpose of matri A, interchange the rows and columns so that the first row of A is now the first column of A transpose. The transpose is used in formulating the dual problem to follow. A T = Dual of the Minimization roblem The dual of the minimization problem is the following maimization problem: Maimize under the constraints: = y + y 16 + y 9 + y 1, y Formation of the Dual roblem Theorem 1: Fundamental rinciple of Duality Given a minimization problem with problem constraints, 1. Use the coefficients and constants in the problem constraints and the objective function to form a matri A with the coefficients of the objective function in the last row.. Interchange the rows and columns of matri A to form the matri A T, the transpose of A.. Use the rows of A T to form a maimization problem with problem constraints. A minimization problem has a solution if and only if its dual problem has a solution. If a solution eists, then the optimal value of the minimization problem is the same as the optimum value of the dual problem. 7 8
3 Forming the Dual roblem Form the Simple Tableau for the Dual roblem We transform the inequalities into equalities by adding the slack variables 1,, : + y + 1 = 16 + y + = 9 + y + = y + p = 0 The first pivot element is (in red) because it is located in the column with the smallest negative number at the bottom (-16), and when divided into the rightmost constants yields the smallest quotient (16/=8) y1 y Simple rocess Divide row 1 by the pivot element () and change the entering variable to y Simple rocess erform row operations to get zeros in the column containing the pivot element. Result is shown below. Identify the net pivot element (0.5) (in red) y y 1 1 y New pivot element ivot element located in this column because of negative indicator y y 1 1 y
4 Simple rocess Simple rocess Variable becomes new entering variable Divide row by 0.5 to obtain a 1 in the pivot position (circled.) y1 y 1 y Get zeros in the column containing the new pivot element. y 1 y We have now obtained the optimal solution since none of the bottom row indicators are negative Solution of the Linear rogramming roblem Solution of a Minimization roblem Solution: An optimal solution to a minimization problem can always be obtained from the bottom row of the final simple tableau for the dual problem. Minimum of is 16, which is also the maimum of the dual problem. It occurs at 1 = 4, = 8, = 0 y y 1 y y Given a minimization problem with non-negative coefficients in the objective function, 1. Write all problem constraints as inequalities. (This may introduce negative numbers on the right side of some problem constraints.). Form the dual problem.. Write the initial iti system of fthe dual problem, using the variables from the minimization problem as slack variables. 16
5 Solution of a Minimization roblem Application: 4. Use the simple method to solve the dual problem. 5. Read the solution of the minimization problem from the bottom row of the final simple tableau in step 4. Note: If the dual problem has no optimal solution, the minimization problem has no optimal solution. One of the first applications of linear programming was to the problem of minimizing the cost of transporting materials. roblems of this type are referred to as transportation problems. Eample: A computer manufacturing company has two assembly plants, plant A and plant B, and two distribution outlets, outlet I and outlet II. lant A can assemble at most 700 computers a month, and plant B can assemble at most 900 computers a month. Outlet I must have at least 500 computers a month, and outlet II must have at least 1,000 computers a month Transportation costs for shipping one computer from each plant to each outlet are as follows: $6 from plant A to outlet I; $5 from plant A to outlet II: $4 from plant B to outlet I; $8 from plant B to outlet II. Find a shipping schedule that will minimize the total cost of shipping the computers from the assembly plants to the distribution outlets. What is the minimum cost? Solution: To form a shipping schedule, we must decide how many computers to ship from either plant to either outlet. This will involve 4 decision variables: 1 = number of computers shipped from plant A to outlet I = number of computers shipped from plant A to outlet II = number of computers shipped from plant B to outlet I 4 = number of computers shipped from plant B to outlet II 19 0
6 Constraints are as follows: 1 + < 700 Available from A + 4 < 900 Available from B 1 + > 500 Required at I + 4 > 1,000 Required at II Total shipping charges are: C = Thus, we must solve the following linear programming problem: subject to 1 + Minimize C = < 700 Available from A + 4 < 900 Available from B + > 500 Required at I > 1,000 Required at II Before we can solve this problem, we must multiply the first two constraints by -1 so that all are of the > type. 1 The problem can now be stated as: Minimize C = subject to > > > > 1,000 1,,, 4 > A = , T A = ,
7 The dual problem is; Maimize = y +500y + 1,000y 4 subject to - + y < y 4 < 5 -y + y < 4 -y + y 4 < 8, y, y, y 4 > 0 Introduce slack variables 1,,, and 4 to form the initial system for the dual: - + y + 1 = y 4 + = 5 -y + y + = 4 -yy + y = y +500y + 1,000y 4 + = Solution Summary of roblem Types and Simple Solution Methods If we form the simple tableau for this initial system and solve, we find that the shipping schedule that minimizes the shipping charges is 0 from plant A to outlet I, 700 from plant A to outlet II, 500 from plant B to outlet I, and 00 from plant B to outlet II. The total shipping cost is $7,900. roblem Constraints Right-Side Coefficients of Method of of Type Constants Objective Solution Function Maimization < Nonnegative Any real number Minimization > Any real Nonnegative number Simple Method Form dual and solve by simple method 7 8
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
More informationCPS 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
More information9.5 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
More informationLesson 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
More information4.4 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
More information6.2: 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
More informationUnderstanding 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
More informationFinite 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
More informationECE 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
More informationThe 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
More informationMath 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
More informationc) 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
More information56:270 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
More informationThe 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
More informationChapter 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
More informationNetwork Flow Problems Luis Goddyn, Math 408
Network Flow Problems Luis Goddyn, Math 48 Let D = (V, A) be a directed graph, and let s, t V (D). For S V we write δ + (S) = {u A : u S, S} and δ (S) = {u A : u S, S} for the in-arcs and out-arcs of S
More informationOPERATIONS 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
More informationLinear & Integer programming
ELL 894 Performance Evaluation on Communication Networks Standard form I Lecture 5 Linear & Integer programming subject to where b is a vector of length m c T A = b (decision variables) and c are vectors
More information56:171 Operations Research Midterm Exam - October 26, 1989 Instructor: D.L. Bricker
56:171 Operations Research Midterm Exam - October 26, 1989 Instructor: D.L. Bricker Answer all of Part One and two (of the four) problems of Part Two Problem: 1 2 3 4 5 6 7 8 TOTAL Possible: 16 12 20 10
More informationSection 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?
More informationExam 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
More informationSystems 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
More informationProfessor 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
More informationOptimum 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,
More informationLinear Programming Duality P&S Chapter 3 Last Revised Nov 1, 2004
Linear Programming Duality P&S Chapter 3 Last Revised Nov 1, 2004 1 In this section we lean about duality, which is another way to approach linear programming. In particular, we will see: How to define
More informationSAMPLE QUESTIONS. b = (30, 20, 40, 10, 50) T, c = (650, 1000, 1350, 1600, 1900) T.
SAMPLE QUESTIONS. (a) We first set up some constant vectors for our constraints. Let b = (30, 0, 40, 0, 0) T, c = (60, 000, 30, 600, 900) T. Then we set up variables x ij, where i, j and i + j 6. By using
More informationThe Dual Simplex Algorithm
p. 1 The Dual Simplex Algorithm Primal optimal (dual feasible) and primal feasible (dual optimal) bases The dual simplex tableau, dual optimality and the dual pivot rules Classical applications of linear
More informationOptimization II. Now lets look at a few examples of the applications of extrema.
Optimization II So far you have learned how to find the relative and absolute etrema of a function. This is an important concept because of how it can be applied to real life situations. In many situations
More informationIE 400 Principles of Engineering Management. The Simplex Algorithm-I: Set 3
IE 4 Principles of Engineering Management The Simple Algorithm-I: Set 3 So far, we have studied how to solve two-variable LP problems graphically. However, most real life problems have more than two variables!
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION,, 1 n A linear equation in the variables equation that can be written in the form a a a b 1 1 2 2 n n a a is an where
More informationDuality 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
More informationF 1 F 2 Daily Requirement Cost N N N
Chapter 5 DUALITY 5. The Dual Problems Every linear programming problem has associated with it another linear programming problem and that the two problems have such a close relationship that whenever
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Stud Guide for Test II Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the linear inequalit. 1) 3 + -6 1) - - - - A) B) - - - - - - - -
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 132 Eam 2 Review (.1 -.5, 7.1-7.5) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine whether the given ordered set of numbers is a solution
More informationPaper Name: Linear Programming & Theory of Games. Lesson Name: Duality in Linear Programing Problem
Paper Name: Linear Programming & Theory of Games Lesson Name: Duality in Linear Programing Problem Lesson Developers: DR. VAJALA RAVI, Dr. Manoj Kumar Varshney College/Department: Department of Statistics,
More informationM 340L CS Homework Set 1
M 340L CS Homework Set 1 Solve each system in Problems 1 6 by using elementary row operations on the equations or on the augmented matri. Follow the systematic elimination procedure described in Lay, Section
More informationPart 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
More informationSELECT TWO PROBLEMS (OF A POSSIBLE FOUR) FROM PART ONE, AND FOUR PROBLEMS (OF A POSSIBLE FIVE) FROM PART TWO. PART ONE: TOTAL GRAND
1 56:270 LINEAR PROGRAMMING FINAL EXAMINATION - MAY 17, 1985 SELECT TWO PROBLEMS (OF A POSSIBLE FOUR) FROM PART ONE, AND FOUR PROBLEMS (OF A POSSIBLE FIVE) FROM PART TWO. PART ONE: 1 2 3 4 TOTAL GRAND
More information1. 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
More informationMATH 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
More informationMA 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
More informationStandard 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
More information4.3 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..
More informationUNIVERSITY OF KWA-ZULU NATAL
UNIVERSITY OF KWA-ZULU NATAL EXAMINATIONS: June 006 Solutions Subject, course and code: Mathematics 34 MATH34P Multiple Choice Answers. B. B 3. E 4. E 5. C 6. A 7. A 8. C 9. A 0. D. C. A 3. D 4. E 5. B
More informationMATH 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
More informationFarkas 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
More informationA Review of Linear Programming
A Review of Linear Programming Instructor: Farid Alizadeh IEOR 4600y Spring 2001 February 14, 2001 1 Overview In this note we review the basic properties of linear programming including the primal simplex
More informationUNIT-4 Chapter6 Linear Programming
UNIT-4 Chapter6 Linear Programming Linear Programming 6.1 Introduction Operations Research is a scientific approach to problem solving for executive management. It came into existence in England during
More informationAM 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
More informationOptimization Methods: Optimization using Calculus - Equality constraints 1. Module 2 Lecture Notes 4
Optimization Methods: Optimization using Calculus - Equality constraints Module Lecture Notes 4 Optimization of Functions of Multiple Variables subect to Equality Constraints Introduction In the previous
More informationAn introductory example
CS1 Lecture 9 An introductory example Suppose that a company that produces three products wishes to decide the level of production of each so as to maximize profits. Let x 1 be the amount of Product 1
More informationΩ 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
More informationLinear Programming. H. R. Alvarez A., Ph. D. 1
Linear Programming H. R. Alvarez A., Ph. D. 1 Introduction It is a mathematical technique that allows the selection of the best course of action defining a program of feasible actions. The objective of
More informationSummary 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:
More informationCSC 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
More informationAnswer 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
More informationLINEAR 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
More informationSpecial 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
More informationWeek_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 (=)
More informationExam 2 Review Math1324. Solve the system of two equations in two variables. 1) 8x + 7y = 36 3x - 4y = -13 A) (1, 5) B) (0, 5) C) No solution D) (1, 4)
Eam Review Math3 Solve the sstem of two equations in two variables. ) + 7 = 3 3 - = -3 (, 5) B) (0, 5) C) No solution D) (, ) ) 3 + 5 = + 30 = -, B) No solution 3 C) - 5 3 + 3, for an real number D) 3,
More informationChap6 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
More informationMarch 13, Duality 3
15.53 March 13, 27 Duality 3 There are concepts much more difficult to grasp than duality in linear programming. -- Jim Orlin The concept [of nonduality], often described in English as "nondualism," is
More informationFoundations of Operations Research
Solved exercises for the course of Foundations of Operations Research Roberto Cordone Gomory cuts Given the ILP problem maxf = 4x 1 +3x 2 2x 1 +x 2 11 x 1 +2x 2 6 x 1,x 2 N solve it with the Gomory cutting
More informationLinear 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
More informationMVE165/MMG631 Linear and integer optimization with applications Lecture 5 Linear programming duality and sensitivity analysis
MVE165/MMG631 Linear and integer optimization with applications Lecture 5 Linear programming duality and sensitivity analysis Ann-Brith Strömberg 2017 03 29 Lecture 4 Linear and integer optimization with
More informationThe 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:.
More information1. 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.
More informationWeek 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
More informationLinear 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
More informationChapter 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
More informationIntroduction 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
More informationWater Resources Systems: Modeling Techniques and Analysis
INDIAN INSTITUTE OF SCIENCE Water Resources Systems: Modeling Techniques and Analysis Lecture - 9 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc. Summary of the previous lecture
More informationThe 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
More informationNon-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)
More information1 Kernel methods & optimization
Machine Learning Class Notes 9-26-13 Prof. David Sontag 1 Kernel methods & optimization One eample of a kernel that is frequently used in practice and which allows for highly non-linear discriminant functions
More informationM340(921) Solutions Problem Set 6 (c) 2013, Philip D Loewen. g = 35y y y 3.
M340(92) Solutions Problem Set 6 (c) 203, Philip D Loewen. (a) If each pig is fed y kilograms of corn, y 2 kilos of tankage, and y 3 kilos of alfalfa, the cost per pig is g = 35y + 30y 2 + 25y 3. The nutritional
More informationExample 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
More informationAlgebraic 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
More informationChapter 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
More informationLinear Programming. Our market gardener example had the form: min x. subject to: where: [ acres cabbages acres tomatoes T
Our market gardener eample had the form: min - 900 1500 [ ] suject to: Ñ Ò Ó 1.5 2 â 20 60 á ã Ñ Ò Ó 3 â 60 á ã where: [ acres caages acres tomatoes T ]. We need a more systematic approach to solving these
More informationSAMPLE QUESTION PAPER MATHEMATICS (041) CLASS XII
SAMPLE QUESTION PAPER MATHEMATICS (01) CLASS XII 017-18 Time allowed: hours Maimum Marks: 100 General Instructions: (i) All questions are compulsory. (ii) This question paper contains 9 questions. (iii)
More information2 3 x = 6 4. (x 1) 6
Solutions to Math 201 Final Eam from spring 2007 p. 1 of 16 (some of these problem solutions are out of order, in the interest of saving paper) 1. given equation: 1 2 ( 1) 1 3 = 4 both sides 6: 6 1 1 (
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Department of Mathematics & Statistics Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MS4303 SEMESTER: Spring 2018 MODULE TITLE:
More information(includes both Phases I & II)
(includes both Phases I & II) Dennis ricker Dept of Mechanical & Industrial Engineering The University of Iowa Revised Simplex Method 09/23/04 page 1 of 22 Minimize z=3x + 5x + 4x + 7x + 5x + 4x subject
More informationDEPARTMENT 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
More information21. 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
More informationIn the original knapsack problem, the value of the contents of the knapsack is maximized subject to a single capacity constraint, for example weight.
In the original knapsack problem, the value of the contents of the knapsack is maximized subject to a single capacity constraint, for example weight. In the multi-dimensional knapsack problem, additional
More informationSummary 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
More informationReview 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
More informationAbe Mirza Graphing f ( x )
Abe Mirza Graphing f ( ) Steps to graph f ( ) 1. Set f ( ) = 0 and solve for critical values.. Substitute the critical values into f ( ) to find critical points.. Set f ( ) = 0 and solve for critical values.
More informationPrelude 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
More informationChapter 1: Linear Programming
Chapter 1: Linear Programming Math 368 c Copyright 2013 R Clark Robinson May 22, 2013 Chapter 1: Linear Programming 1 Max and Min For f : D R n R, f (D) = {f (x) : x D } is set of attainable values of
More informationMath 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
More informationSlide 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
More informationLecture 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
More informationLinear Programming: The Simplex Method
7206 CH09 GGS /0/05 :5 PM Page 09 9 C H A P T E R Linear Programming: The Simplex Method TEACHING SUGGESTIONS Teaching Suggestion 9.: Meaning of Slack Variables. Slack variables have an important physical
More informationAnother max flow application: baseball
CS124 Lecture 16 Spring 2018 Another max flow application: baseball Suppose there are n baseball teams, and team 1 is our favorite. It is the middle of baseball season, and some games have been played
More informationExam. Name. Use the indicated region of feasible solutions to find the maximum and minimum values of the given objective function.
Exam Name Use the indicated region of feasible solutions to find the maximum and minimum values of the given objective function. 1) z = 12x - 22y y (0, 6) (1.2, 5) Solve the 3) The Acme Class Ring Company
More informationThe use of shadow price is an example of sensitivity analysis. Duality theory can be applied to do other kind of sensitivity analysis:
Sensitivity analysis The use of shadow price is an example of sensitivity analysis. Duality theory can be applied to do other kind of sensitivity analysis: Changing the coefficient of a nonbasic variable
More informationMath 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.
More information