Math 407A: Linear Optimization

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Math 407A: Linear Optimization"

Transcription

1 Math 407A: Linear Optimization Lecture 4: LP Standard Form 1 1 Author: James Burke, University of Washington

2 LPs in Standard Form Minimization maximization Linear equations to linear inequalities Lower and upper bounded variables Interval variable bounds Free variable Two Step Process to Standard Form

3 LPs in Standard Form We say that an LP is in standard form if its matrix representation has the form

4 LPs in Standard Form We say that an LP is in standard form if its matrix representation has the form max s.t. c T x Ax b 0 x

5 LPs in Standard Form We say that an LP is in standard form if its matrix representation has the form max c T x It must be a maximization problem. s.t. Ax b 0 x

6 LPs in Standard Form We say that an LP is in standard form if its matrix representation has the form max c T x It must be a maximization problem. s.t. Ax b Only inequalities of the correct direction. 0 x

7 LPs in Standard Form We say that an LP is in standard form if its matrix representation has the form max c T x It must be a maximization problem. s.t. Ax b Only inequalities of the correct direction. 0 x All variables must be non-negative.

8 Every LP can be Transformed to Standard Form

9 Every LP can be Transformed to Standard Form minimization maximization

10 Every LP can be Transformed to Standard Form minimization maximization To transform a minimization problem to a maximization problem multiply the objective function by 1.

11 Every LP can be Transformed to Standard Form minimization maximization To transform a minimization problem to a maximization problem multiply the objective function by 1. linear inequalities

12 Every LP can be Transformed to Standard Form minimization maximization To transform a minimization problem to a maximization problem multiply the objective function by 1. linear inequalities If an LP has an inequality constraint of the form a i1 x 1 + a i2 x a in x n b i, it can be transformed to one in standard form by multiplying the inequality through by 1 to get a i1 x 1 a i2 x 2 a in x n b i.

13 Every LP can be Transformed to Standard Form linear equations

14 Every LP can be Transformed to Standard Form linear equations The linear equation a i1 x i + + a in x n = b i can be written as two linear inequalities a i1 x a in x n b i and a i1 x a in x n b i. or equivalently a i1 x a in x n b i a i1 x 1 a in x n b i

15 Every LP can be Transformed to Standard Form variables with lower bounds

16 Every LP can be Transformed to Standard Form variables with lower bounds If a variable x i has lower bound l i which is not zero (l i x i ), one obtains a non-negative variable w i with the substitution x i = w i + l i. In this case, the bound l i x i is equivalent to the bound 0 w i.

17 Every LP can be Transformed to Standard Form variables with lower bounds If a variable x i has lower bound l i which is not zero (l i x i ), one obtains a non-negative variable w i with the substitution x i = w i + l i. In this case, the bound l i x i is equivalent to the bound 0 w i. variables with upper bounds

18 Every LP can be Transformed to Standard Form variables with lower bounds If a variable x i has lower bound l i which is not zero (l i x i ), one obtains a non-negative variable w i with the substitution x i = w i + l i. In this case, the bound l i x i is equivalent to the bound 0 w i. variables with upper bounds If a variable x i has an upper bound u i (x i u i ) one obtains a non-negative variable w i with the substitution x i = u i w i. In this case, the bound x i u i is equivalent to the bound 0 w i.

19 Every LP can be Transformed to Standard Form variables with interval bounds

20 Every LP can be Transformed to Standard Form variables with interval bounds An interval bound of the form l i x i u i can be transformed into one non-negativity constraint and one linear inequality constraint in standard form by making the substitution x i = w i + l i.

21 Every LP can be Transformed to Standard Form variables with interval bounds An interval bound of the form l i x i u i can be transformed into one non-negativity constraint and one linear inequality constraint in standard form by making the substitution x i = w i + l i. In this case, the bounds l i x i u i are equivalent to the constraints 0 w i and w i u i l i.

22 Every LP can be Transformed to Standard Form free variables

23 Every LP can be Transformed to Standard Form free variables Sometimes a variable is given without any bounds. Such variables are called free variables. To obtain standard form every free variable must be replaced by the difference of two non-negative variables. That is, if x i is free, then we get with 0 u i and 0 v i. x i = u i v i

24 Transformation to Standard Form Put the following LP into standard form. minimize 3x 1 x 2 subject to x 1 + 6x 2 x 3 + x 4 3 7x 2 + x 4 = 5 x 3 + x x 2, x 3 5, 2 x 4 2.

25 Transformation to Standard Form The hardest part of the translation to standard form, or at least the part most susceptible to error, is the replacement of existing variables with non-negative variables.

26 Transformation to Standard Form The hardest part of the translation to standard form, or at least the part most susceptible to error, is the replacement of existing variables with non-negative variables. Reduce errors by doing the transformation in two steps.

27 Transformation to Standard Form The hardest part of the translation to standard form, or at least the part most susceptible to error, is the replacement of existing variables with non-negative variables. Reduce errors by doing the transformation in two steps. Step 1: Make all of the changes that do not involve a variable substitution.

28 Transformation to Standard Form The hardest part of the translation to standard form, or at least the part most susceptible to error, is the replacement of existing variables with non-negative variables. Reduce errors by doing the transformation in two steps. Step 1: Make all of the changes that do not involve a variable substitution. Step 2: Make all of the variable substitutions.

29 Must be a maximization problem minimize 3x 1 x 2 subject to x 1 + 6x 2 x 3 + x 4 3 7x 2 + x 4 = 5 x 3 + x x 2, x 3 5, 2 x 4 2.

30 Must be a maximization problem minimize 3x 1 x 2 subject to x 1 + 6x 2 x 3 + x 4 3 7x 2 + x 4 = 5 x 3 + x x 2, x 3 5, 2 x 4 2. Minimization = maximization

31 Must be a maximization problem minimize 3x 1 x 2 subject to x 1 + 6x 2 x 3 + x 4 3 7x 2 + x 4 = 5 x 3 + x x 2, x 3 5, 2 x 4 2. Minimization = maximization maximize 3x 1 + x 2.

32 Inequalities must go the right way.

33 Inequalities must go the right way. x 1 + 6x 2 x 3 + x 4 3

34 Inequalities must go the right way. x 1 + 6x 2 x 3 + x 4 3 becomes

35 Inequalities must go the right way. x 1 + 6x 2 x 3 + x 4 3 becomes x 1 6x 2 + x 3 x 4 3.

36 Equalities are replaced by 2 inequalities. 7x 2 + x 4 = 5

37 Equalities are replaced by 2 inequalities. becomes 7x 2 + x 4 = 5

38 Equalities are replaced by 2 inequalities. becomes 7x 2 + x 4 = 5 7x 2 + x 4 5

39 Equalities are replaced by 2 inequalities. becomes 7x 2 + x 4 = 5 7x 2 + x 4 5 and

40 Equalities are replaced by 2 inequalities. becomes 7x 2 + x 4 = 5 7x 2 + x 4 5 and 7x 2 x 4 5

41 Grouping upper bounds with the linear inequalities. The double bound 2 x 4 2 indicates that we should group the upper bound with the linear inequalities.

42 Grouping upper bounds with the linear inequalities. The double bound 2 x 4 2 indicates that we should group the upper bound with the linear inequalities. x 4 2

43 Step 1: Transformation to Standard Form Combining all of these changes gives the LP maximize 3x 1 + x 2 subject to x 1 6x 2 + x 3 x 4 3 7x 2 + x 4 5 7x 2 x 4 5 x 3 + x 4 2 x x 2, x 3 5, 2 x 4.

44 Step 2: Variable Replacement The variable x 1 is free, so we replace it by

45 Step 2: Variable Replacement The variable x 1 is free, so we replace it by x 1 = z + 1 z 1 with 0 z+ 1, 0 z 1.

46 Step 2: Variable Replacement The variable x 1 is free, so we replace it by x 1 = z + 1 z 1 with 0 z+ 1, 0 z 1. x 2 has a non-zero lower bound ( 1 x 2 ) so we replace it by

47 Step 2: Variable Replacement The variable x 1 is free, so we replace it by x 1 = z + 1 z 1 with 0 z+ 1, 0 z 1. x 2 has a non-zero lower bound ( 1 x 2 ) so we replace it by z 2 = x or x 2 = z 2 1 with 0 z 2.

48 Step 2: Variable Replacement The variable x 1 is free, so we replace it by x 1 = z + 1 z 1 with 0 z+ 1, 0 z 1. x 2 has a non-zero lower bound ( 1 x 2 ) so we replace it by z 2 = x or x 2 = z 2 1 with 0 z 2. x 3 is bounded above (x 3 5), so we replace it by

49 Step 2: Variable Replacement The variable x 1 is free, so we replace it by x 1 = z + 1 z 1 with 0 z+ 1, 0 z 1. x 2 has a non-zero lower bound ( 1 x 2 ) so we replace it by z 2 = x or x 2 = z 2 1 with 0 z 2. x 3 is bounded above (x 3 5), so we replace it by z 3 = 5 x 3 or x 3 = 5 z 3 with 0 z 3.

50 Step 2: Variable Replacement The variable x 1 is free, so we replace it by x 1 = z + 1 z 1 with 0 z+ 1, 0 z 1. x 2 has a non-zero lower bound ( 1 x 2 ) so we replace it by z 2 = x or x 2 = z 2 1 with 0 z 2. x 3 is bounded above (x 3 5), so we replace it by z 3 = 5 x 3 or x 3 = 5 z 3 with 0 z 3. x 4 is bounded below ( 2 x 4 ), so we replace it by

51 Step 2: Variable Replacement The variable x 1 is free, so we replace it by x 1 = z + 1 z 1 with 0 z+ 1, 0 z 1. x 2 has a non-zero lower bound ( 1 x 2 ) so we replace it by z 2 = x or x 2 = z 2 1 with 0 z 2. x 3 is bounded above (x 3 5), so we replace it by z 3 = 5 x 3 or x 3 = 5 z 3 with 0 z 3. x 4 is bounded below ( 2 x 4 ), so we replace it by z 4 = x or x 4 = z 4 2 with 0 z 4.

52 Step 2: Transformation to Standard Form Substituting x 1 = z + 1 z 1 into gives maximize 3x 1 + x 2 subject to x 1 6x 2 + x 3 x 4 3 7x 2 + x 4 5 7x 2 x 4 5 x 3 + x 4 2 x x 2, x 3 5, 2 x 4.

53 Step 2: Transformation to Standard Form maximize 3z z1 + x 2 subject to z 1 + z1 6x 2 + x 3 x 4 3 7x 2 + x 4 5 7x 2 x 4 5 x 3 + x 4 2 x z + 1, 0 z 1, 1 x 2, x 3 5, 2 x 4.

54 Step 2: Transformation to Standard Form Substituting x 2 = z 2 1 into maximize 3z z1 + x 2 subject to z 1 + z1 6x 2 + x 3 x 4 3 7x 2 + x 4 5 7x 2 x 4 5 x 3 + x 4 2 x 4 2 gives 0 z + 1, 0 z 1, 1 x 2, x 3 5, 2 x 4.

55 Step 2: Transformation to Standard Form maximize 3z z1 + z 2 1 subject to z 1 + z1 6z 2 + x 3 x = 3 7z 2 + x = 12 7z 2 x = 12 x 3 + x 4 2 x z + 1, 0 z 1, 0 z 2, x 3 5, 2 x 4.

56 Step 2: Transformation to Standard Form maximize 3z z1 + z 2 subject to z 1 + z1 6z 2 + x 3 x 4 3 7z 2 + x z 2 x 4 12 x 3 + x 4 2 x z + 1, 0 z 1, 0 z 2, x 3 5, 2 x 4.

57 Step 2: Transformation to Standard Form Substituting x 3 = 5 z 3 into maximize 3z z1 + z 2 subject to z 1 + z1 6z 2 + x 3 x 4 3 7z 2 + x z 2 x 4 12 x 3 + x 4 2 x 4 2 gives 0 z + 1, 0 z 1, 0 z 2, x 3 5, 2 x 4.

58 Step 2: Transformation to Standard Form maximize 3z z1 + z 2 subject to z 1 + z1 6z 2 z 3 x = 8 7z 2 + x z 2 x 4 12 z 3 + x = 3 x z + 1, 0 z 1, 0 z 2, 0 z 3, 2 x 4.

59 Step 2: Transformation to Standard Form Substituting x 4 = z 4 2 into maximize 3z z1 + z 2 subject to z 1 + z1 6z 2 z 3 x 4 8 7z 2 + x z 2 x 4 12 z 3 + x 4 3 x 4 2 gives 0 z + 1, 0 z 1, 0 z 2, 0 z 3, 2 x 4.

60 Step 2: Transformation to Standard Form maximize 3z z1 + z 2 subject to z 1 + z1 6z 2 z 3 z = 10 7z 2 + z = 14 7z 2 z = 14 z 3 + z = 1 z = 4 0 z + 1, 0 z 1, 0 z 2, 0 z 3, 0 z 4. which is in standard form.

61 Step 2: Transformation to Standard Form After making these substitutions, we get the following LP in standard form:

62 Step 2: Transformation to Standard Form After making these substitutions, we get the following LP in standard form: maximize 3z z1 + z 2 subject to z 1 + z1 6z 2 z 3 z z 2 + z z 2 z 4 14 z 3 + z 4 1 z z + 1, z 1, z 2, z 3, z 4.

63 Transformation to Standard Form: Practice Transform the following LP to an LP in standard form. minimize x 1 12x 2 + 2x 3 subject to 5x 1 x 2 + 3x 3 = 15 2x 1 + x 2 20x x 2, 1 x 3 4

Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

More information

Linear Equations and Inequalities

Linear Equations and Inequalities Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109 - Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................

More information

Chapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints

Chapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and

More information

Mathematical finance and linear programming (optimization)

Mathematical finance and linear programming (optimization) Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may

More information

Linear Programming I

Linear Programming I Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

More information

Math 240: Linear Systems and Rank of a Matrix

Math 240: Linear Systems and Rank of a Matrix Math 240: Linear Systems and Rank of a Matrix Ryan Blair University of Pennsylvania Thursday January 20, 2011 Ryan Blair (U Penn) Math 240: Linear Systems and Rank of a Matrix Thursday January 20, 2011

More information

Definition of a Linear Program

Definition of a Linear Program Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

More information

4.6 Linear Programming duality

4.6 Linear Programming duality 4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal

More information

7.4 Linear Programming: The Simplex Method

7.4 Linear Programming: The Simplex Method 7.4 Linear Programming: The Simplex Method For linear programming problems with more than two variables, the graphical method is usually impossible, so the simplex method is used. Because the simplex method

More information

Basic Components of an LP:

Basic Components of an LP: 1 Linear Programming Optimization is an important and fascinating area of management science and operations research. It helps to do less work, but gain more. Linear programming (LP) is a central topic

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

More information

Absolute Value Equations and Inequalities

Absolute Value Equations and Inequalities . Absolute Value Equations and Inequalities. OBJECTIVES 1. Solve an absolute value equation in one variable. Solve an absolute value inequality in one variable NOTE Technically we mean the distance between

More information

Study Guide 2 Solutions MATH 111

Study Guide 2 Solutions MATH 111 Study Guide 2 Solutions MATH 111 Having read through the sample test, I wanted to warn everyone, that I might consider asking questions involving inequalities, the absolute value function (as in the suggested

More information

Question 2: How do you solve a matrix equation using the matrix inverse?

Question 2: How do you solve a matrix equation using the matrix inverse? Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients

More information

Maximizing volume given a surface area constraint

Maximizing volume given a surface area constraint Maximizing volume given a surface area constraint Math 8 Department of Mathematics Dartmouth College Maximizing volume given a surface area constraint p.1/9 Maximizing wih a constraint We wish to solve

More information

Section 1.7 22 Continued

Section 1.7 22 Continued Section 1.5 23 A homogeneous equation is always consistent. TRUE - The trivial solution is always a solution. The equation Ax = 0 gives an explicit descriptions of its solution set. FALSE - The equation

More information

Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010

Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010 Section 2.1: Linear Equations Definition of equation An equation is a statement that equates two algebraic expressions. Solving an equation involving a variable means finding all values of the variable

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

More information

Solving Systems of Linear Equations; Row Reduction

Solving Systems of Linear Equations; Row Reduction Harvey Mudd College Math Tutorial: Solving Systems of Linear Equations; Row Reduction Systems of linear equations arise in all sorts of applications in many different fields of study The method reviewed

More information

Section 1.7 Inequalities

Section 1.7 Inequalities Section 1.7 Inequalities Linear Inequalities An inequality is linear if each term is constant or a multiple of the variable. EXAMPLE: Solve the inequality x < 9x+4 and sketch the solution set. x < 9x+4

More information

Nonlinear Programming Methods.S2 Quadratic Programming

Nonlinear Programming Methods.S2 Quadratic Programming Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective

More information

Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).

Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication). MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix

More information

Linear Equations in One Variable

Linear Equations in One Variable Linear Equations in One Variable MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this section we will learn how to: Recognize and combine like terms. Solve

More information

Converting a Linear Program to Standard Form

Converting a Linear Program to Standard Form Converting a Linear Program to Standard Form Hi, welcome to a tutorial on converting an LP to Standard Form. We hope that you enjoy it and find it useful. Amit, an MIT Beaver Mita, an MIT Beaver 2 Linear

More information

Solving Linear Programs in Excel

Solving Linear Programs in Excel Notes for AGEC 622 Bruce McCarl Regents Professor of Agricultural Economics Texas A&M University Thanks to Michael Lau for his efforts to prepare the earlier copies of this. 1 http://ageco.tamu.edu/faculty/mccarl/622class/

More information

Special Situations in the Simplex Algorithm

Special Situations in the Simplex Algorithm Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the

More information

Solving Linear Equations in One Variable. Worked Examples

Solving Linear Equations in One Variable. Worked Examples Solving Linear Equations in One Variable Worked Examples Solve the equation 30 x 1 22x Solve the equation 30 x 1 22x Our goal is to isolate the x on one side. We ll do that by adding (or subtracting) quantities

More information

Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

More information

SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

More information

1 Solving LPs: The Simplex Algorithm of George Dantzig

1 Solving LPs: The Simplex Algorithm of George Dantzig Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

More information

Solving Systems of Linear Equations Using Matrices

Solving Systems of Linear Equations Using Matrices Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.

More information

3.1 Solving Systems Using Tables and Graphs

3.1 Solving Systems Using Tables and Graphs Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system

More information

Lecture 7 : Inequalities 2.5

Lecture 7 : Inequalities 2.5 3 Lecture 7 : Inequalities.5 Sometimes a problem may require us to find all numbers which satisfy an inequality. An inequality is written like an equation, except the equals sign is replaced by one of

More information

Chapter 3: Section 3-3 Solutions of Linear Programming Problems

Chapter 3: Section 3-3 Solutions of Linear Programming Problems Chapter 3: Section 3-3 Solutions of Linear Programming Problems D. S. Malik Creighton University, Omaha, NE D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming

More information

Linear Programming Refining Transportation

Linear Programming Refining Transportation Solving a Refinery Problem with Excel Solver Type of Crude or Process Product A B C 1 C 2 D Demand Profits on Crudes 10 20 15 25 7 Products Product Slate for Crude or Process G 0.6 0.5 0.4 0.4 0.3 170

More information

8.2. Solution by Inverse Matrix Method. Introduction. Prerequisites. Learning Outcomes

8.2. Solution by Inverse Matrix Method. Introduction. Prerequisites. Learning Outcomes Solution by Inverse Matrix Method 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix algebra allows us

More information

Lecture 2: August 29. Linear Programming (part I)

Lecture 2: August 29. Linear Programming (part I) 10-725: Convex Optimization Fall 2013 Lecture 2: August 29 Lecturer: Barnabás Póczos Scribes: Samrachana Adhikari, Mattia Ciollaro, Fabrizio Lecci Note: LaTeX template courtesy of UC Berkeley EECS dept.

More information

CHAPTER 11: BASIC LINEAR PROGRAMMING CONCEPTS

CHAPTER 11: BASIC LINEAR PROGRAMMING CONCEPTS Linear programming is a mathematical technique for finding optimal solutions to problems that can be expressed using linear equations and inequalities. If a real-world problem can be represented accurately

More information

Learning Objectives for Section 1.1 Linear Equations and Inequalities

Learning Objectives for Section 1.1 Linear Equations and Inequalities Learning Objectives for Section 1.1 Linear Equations and Inequalities After this lecture and the assigned homework, you should be able to solve linear equations. solve linear inequalities. use interval

More information

Linear Programming. March 14, 2014

Linear Programming. March 14, 2014 Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

More information

Using Excel to Find Numerical Solutions to LP's

Using Excel to Find Numerical Solutions to LP's Using Excel to Find Numerical Solutions to LP's by Anil Arya and Richard A. Young This handout sketches how to solve a linear programming problem using Excel. Suppose you wish to solve the following LP:

More information

IEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2

IEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2 IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3

More information

EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

More information

COLLEGE ALGEBRA 10 TH EDITION LIAL HORNSBY SCHNEIDER 1.1-1

COLLEGE ALGEBRA 10 TH EDITION LIAL HORNSBY SCHNEIDER 1.1-1 10 TH EDITION COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER 1.1-1 1.1 Linear Equations Basic Terminology of Equations Solving Linear Equations Identities 1.1-2 Equations An equation is a statement that two expressions

More information

Solving simultaneous equations using the inverse matrix

Solving simultaneous equations using the inverse matrix Solving simultaneous equations using the inverse matrix 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix

More information

What is Linear Programming?

What is Linear Programming? Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to

More information

3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max

3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,

More information

Grade 6 Math Circles. Exponents

Grade 6 Math Circles. Exponents Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles November 4/5, 2014 Exponents Quick Warm-up Evaluate the following: 1. 4 + 4 + 4 +

More information

Guide to SRW Section 1.7: Solving inequalities

Guide to SRW Section 1.7: Solving inequalities Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is -6-5 -4-3 -2-1 0

More information

Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method

Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming

More information

Linear Programming. April 12, 2005

Linear Programming. April 12, 2005 Linear Programming April 1, 005 Parts of this were adapted from Chapter 9 of i Introduction to Algorithms (Second Edition) /i by Cormen, Leiserson, Rivest and Stein. 1 What is linear programming? The first

More information

1 Review of Least Squares Solutions to Overdetermined Systems

1 Review of Least Squares Solutions to Overdetermined Systems cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares

More information

Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Method To Solve Linear, Polynomial, or Absolute Value Inequalities: Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

More information

Linear Programming. Solving LP Models Using MS Excel, 18

Linear Programming. Solving LP Models Using MS Excel, 18 SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting

More information

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

More information

Duality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725

Duality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725 Duality in General Programs Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T

More information

Linear Programming in Matrix Form

Linear Programming in Matrix Form Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,

More information

Jeffrey Kantor. August 28, 2006

Jeffrey Kantor. August 28, 2006 Mosel August 28, 2006 Announcements Mosel Enrollment and Registration - Last Day for Class Change is August 30th. Homework set 1 was handed out today. A copy is available on the course web page http://jkantor.blogspot.com

More information

Direct Methods for Solving Linear Systems. Matrix Factorization

Direct Methods for Solving Linear Systems. Matrix Factorization Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

More information

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

Algebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations:

Algebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Algebra Chapter 6 Notes Systems of Equations and Inequalities Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Solution of a system of linear equations: Consistent independent system:

More information

Sensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS

Sensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS Sensitivity Analysis 3 We have already been introduced to sensitivity analysis in Chapter via the geometry of a simple example. We saw that the values of the decision variables and those of the slack and

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

Linear Programming Problems

Linear Programming Problems Linear Programming Problems Linear programming problems come up in many applications. In a linear programming problem, we have a function, called the objective function, which depends linearly on a number

More information

Question 2: How do you solve a linear programming problem with a graph?

Question 2: How do you solve a linear programming problem with a graph? Question 2: How do you solve a linear programming problem with a graph? Now that we have several linear programming problems, let s look at how we can solve them using the graph of the system of inequalities.

More information

Solving Quadratic & Higher Degree Inequalities

Solving Quadratic & Higher Degree Inequalities Ch. 8 Solving Quadratic & Higher Degree Inequalities We solve quadratic and higher degree inequalities very much like we solve quadratic and higher degree equations. One method we often use to solve quadratic

More information

by the matrix A results in a vector which is a reflection of the given

by the matrix A results in a vector which is a reflection of the given Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

More information

Determine If An Equation Represents a Function

Determine If An Equation Represents a Function Question : What is a linear function? The term linear function consists of two parts: linear and function. To understand what these terms mean together, we must first understand what a function is. The

More information

OPRE 6201 : 2. Simplex Method

OPRE 6201 : 2. Simplex Method OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2

More information

Arithmetic and Algebra of Matrices

Arithmetic and Algebra of Matrices Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational

More information

is the degree of the polynomial and is the leading coefficient.

is the degree of the polynomial and is the leading coefficient. Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...

More information

2.2/2.3 - Solving Systems of Linear Equations

2.2/2.3 - Solving Systems of Linear Equations c Kathryn Bollinger, August 28, 2011 1 2.2/2.3 - Solving Systems of Linear Equations A Brief Introduction to Matrices Matrices are used to organize data efficiently and will help us to solve systems of

More information

Transportation Polytopes: a Twenty year Update

Transportation Polytopes: a Twenty year Update Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,

More information

Linear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.

Linear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that

More information

Solving Systems of Linear Equations. Substitution

Solving Systems of Linear Equations. Substitution Solving Systems of Linear Equations There are two basic methods we will use to solve systems of linear equations: Substitution Elimination We will describe each for a system of two equations in two unknowns,

More information

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.

More information

CLASS 3, GIVEN ON 9/27/2010, FOR MATH 25, FALL 2010

CLASS 3, GIVEN ON 9/27/2010, FOR MATH 25, FALL 2010 CLASS 3, GIVEN ON 9/27/2010, FOR MATH 25, FALL 2010 1. Greatest common divisor Suppose a, b are two integers. If another integer d satisfies d a, d b, we call d a common divisor of a, b. Notice that as

More information

Learning Goals. b. Calculate the total cost of using Roller Rama. y 5 5(50) 5 250. The total cost of using Roller Rama is $250.

Learning Goals. b. Calculate the total cost of using Roller Rama. y 5 5(50) 5 250. The total cost of using Roller Rama is $250. Making Decisions Using the Best Method to Solve a Linear System Learning Goals In this lesson, you will: Write a linear system of equations to represent a problem context. Choose the best method to solve

More information

Chapter 6: Sensitivity Analysis

Chapter 6: Sensitivity Analysis Chapter 6: Sensitivity Analysis Suppose that you have just completed a linear programming solution which will have a major impact on your company, such as determining how much to increase the overall production

More information

The Point-Slope Form

The Point-Slope Form 7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

More information

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued). MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

More information

Solutions Of Some Non-Linear Programming Problems BIJAN KUMAR PATEL. Master of Science in Mathematics. Prof. ANIL KUMAR

Solutions Of Some Non-Linear Programming Problems BIJAN KUMAR PATEL. Master of Science in Mathematics. Prof. ANIL KUMAR Solutions Of Some Non-Linear Programming Problems A PROJECT REPORT submitted by BIJAN KUMAR PATEL for the partial fulfilment for the award of the degree of Master of Science in Mathematics under the supervision

More information

7. Show that the expectation value function that appears in Lecture 1, namely

7. Show that the expectation value function that appears in Lecture 1, namely Lectures on quantum computation by David Deutsch Lecture 1: The qubit Worked Examples 1. You toss a coin and observe whether it came up heads or tails. (a) Interpret this as a physics experiment that ends

More information

Solving Linear Systems, Continued and The Inverse of a Matrix

Solving Linear Systems, Continued and The Inverse of a Matrix , Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing

More information

EXCEL SOLVER TUTORIAL

EXCEL SOLVER TUTORIAL ENGR62/MS&E111 Autumn 2003 2004 Prof. Ben Van Roy October 1, 2003 EXCEL SOLVER TUTORIAL This tutorial will introduce you to some essential features of Excel and its plug-in, Solver, that we will be using

More information

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

More information

7 Gaussian Elimination and LU Factorization

7 Gaussian Elimination and LU Factorization 7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

Larson, R. and Boswell, L. (2016). Big Ideas Math, Algebra 2. Erie, PA: Big Ideas Learning, LLC. ISBN

Larson, R. and Boswell, L. (2016). Big Ideas Math, Algebra 2. Erie, PA: Big Ideas Learning, LLC. ISBN ALG B Algebra II, Second Semester #PR-0, BK-04 (v.4.0) To the Student: After your registration is complete and your proctor has been approved, you may take the Credit by Examination for ALG B. WHAT TO

More information

1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where. Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

More information

LINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL

LINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL Chapter 6 LINEAR INEQUALITIES 6.1 Introduction Mathematics is the art of saying many things in many different ways. MAXWELL In earlier classes, we have studied equations in one variable and two variables

More information

3. Reaction Diffusion Equations Consider the following ODE model for population growth

3. Reaction Diffusion Equations Consider the following ODE model for population growth 3. Reaction Diffusion Equations Consider the following ODE model for population growth u t a u t u t, u 0 u 0 where u t denotes the population size at time t, and a u plays the role of the population dependent

More information

Chapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach

Chapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach Chapter 5 Linear Programming in Two Dimensions: A Geometric Approach Linear Inequalities and Linear Programming Section 3 Linear Programming gin Two Dimensions: A Geometric Approach In this section, we

More information

A Numerical Study on the Wiretap Network with a Simple Network Topology

A Numerical Study on the Wiretap Network with a Simple Network Topology A Numerical Study on the Wiretap Network with a Simple Network Topology Fan Cheng and Vincent Tan Department of Electrical and Computer Engineering National University of Singapore Mathematical Tools of

More information

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

More information

Chapter 2 Solving Linear Programs

Chapter 2 Solving Linear Programs Chapter 2 Solving Linear Programs Companion slides of Applied Mathematical Programming by Bradley, Hax, and Magnanti (Addison-Wesley, 1977) prepared by José Fernando Oliveira Maria Antónia Carravilla A

More information

Eleonóra STETTNER, Kaposvár Using Microsoft Excel to solve and illustrate mathematical problems

Eleonóra STETTNER, Kaposvár Using Microsoft Excel to solve and illustrate mathematical problems Eleonóra STETTNER, Kaposvár Using Microsoft Excel to solve and illustrate mathematical problems Abstract At the University of Kaposvár in BSc and BA education we introduced a new course for the first year

More information

Chapter 6 Notes. Section 6.1 Solving One-Step Linear Inequalities

Chapter 6 Notes. Section 6.1 Solving One-Step Linear Inequalities Chapter 6 Notes Name Section 6.1 Solving One-Step Linear Inequalities Graph of a linear Inequality- the set of all points on a number line that represent all solutions of the inequality > or < or circle

More information