Linear Programming Notes V Problem Transformations

Save this PDF as:

Size: px
Start display at page:

Transcription

1 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 constraints are all of the form: linear expression constant (a i x b i ), and all variables are constrained to be non-negative. In symbols, this form is: max c x subject to Ax b, x 0. In the second form, the objective is to maximize, the material constraints are all of the form: linear expression = constant (a i x = b i ), and all variables are constrained to be non-negative. In symbols, this form is: max c x subject to Ax = b, x 0. In this formulation (but not the first) we can take b 0. Note: The c, A, b in the first problem are not the same as the c, A, b in the second problem. In order to rewrite the problem, I need to introduce a small number of transformations. I ll explain them in these notes. 2 Equivalent Representations When I say that I can rewrite a linear programming problem, I mean that I can find a representation that contains exactly the same information. For example, the expression 2x = 8 the expression x = 4. They both describe the same fact about x. In general, an equivalent representation of a linear programming problem will be one that contains exactly the same information as the original problem. Solving one will immediately give you the solution to the other. When I claim that I can write any linear programming problem in a standard form, I need to demonstrate that I can make several kinds of transformation: change a minimization problem to a maximization problem; replace a constraint of the form (a i x b i ) by an equation or equations; replace a constraint of the form (a i x b i ) by an equation or equations; replace a constraint of the form (a i x = b i ) by an inequality or inequalities; replace a variable that is not explicitly constrained to be non-negative by a variable or variables so constrained. If I can do all that, then I can write any problem in either of the desired forms. 1

2 3 Rewriting the Objective Function The objective will be either to maximize or to minimize. If you start with a maximization problem, then there is nothing to change. If you start with a minimization problem, say min f(x) subject to x S, then an equivalent maximization problem is max f(x) subject to x S. That is, minimizing f is the same as maximizing f. This trick is completely general (that is, it is not limited to LPs). Any solution to the minimization problem will be a solution to the maximization problem and conversely. (Note that the value of the maximization problem will be 1 times the value of the minimization problem.) In summary: to change a max problem to a min problem, just multiply the objective function by 1. 4 Rewriting a constraint of the form (a i x b i ) To transform this constraint into an equation, add a non-negative slack variable: a i x b i a i x + s i = b i and s i 0. We have seen this trick before. If x satisfies the inequality, then s i = b i a i x 0. Conversely, if x and s i satisfy the expressions in the second line, then the first line must be true. Hence the two expressions are equivalent. Note that by multiplying both sides of the expression a i x + s i = b i by 1 we can guarantee that the right-hand side is non-negative. 5 Rewriting a constraint of the form (a i x b i ) To transform this constraint into an equation, subtract a non-negative surplus variable: a i x b i a i x s i = b i and s i 0. The reasoning is exactly like the case of the slack variable. To transform this constraint into an inequality pointed in the other direction, multiply both sides by 1. a i x b i a i x + s i b i. 2

3 6 Rewriting a constraint of the form (a i x = b i ) To transform an equation into inequalities, note that w = z is exactly that same as w z and w z. That is, the one way for two numbers to be equal is for one to be both less than or equal to and greater than or equal to the other. It follows that a i x = b i a i x b i and a i x b i. By the last section, the second line : a i x b i and a i x b i. 7 Guaranteeing that All Variables are Explicitly Constrained to be Non-Negative Most of the problems that we look at requires the variables to be non-negative. The constraint arises naturally in many applications, but it is not essential. The standard way of writing linear programming problems imposes this condition. The section shows that there is no loss in generality in imposing the restriction. That is, if you are thinking about a linear programming problem, then I can think of a mathematically equivalent problem in which all of the variables must be non-negative. The transformation uses a simple trick. You replace an unconstrained variable x j by two variables u j and v j. Whenever you see x j in the problem, you replace it with u j v j. Furthermore, you impose the constraint that u j, v j 0. When you carry out the substitution, you replace x j by non-negative variables. You don t change the problem. Any value that x j can take, can be expressed as a difference (in fact, there are infinitely many ways to express it). Specifically, if x j 0, then you can let u j = x j and v j = 0; if x j < 0, then you can let u j = 0 and v j = x j. 8 What Is the Point? The previous sections simply introduce accounting tricks. There is no substance to the transformations. If you put the tricks together, they support the claim that I made in the beginning of the notes. Who cares? The form of the problem with equality constraints and non-negative variables is the form that the simplex algorithm uses. The inequality constraint form (with non-negative variables) is the form used in for the duality theorem. 3

4 Warnings: These transformations really are transformations. If you start with a problem in which x 1 is not constrained to be non-negative, but act as if it is so constrained, then you may not get the right answer (you ll be wrong if the solution requires that x 1 take a negative value). If you treat an equality constraint like an inequality constraint, then you ll get the wrong answer (unless the constraint binds at the solution). Similarly, you can t treat as inequality constraint as an equation in general. The transformations involve creating a new variable or constraint to compensate for the changing inequalities to equations, equations to inequalities, or whatever it is you do. 9 Example You know all the ideas. Let me show you how they work. Start with the problem: min 4x 1 + x 2 subject to 2x 1 + x 2 6 x 2 + x 3 = 4 x 1 4 x 2 x 3 0. and let s write it in either of the two standard forms. First, to get it into the form: max c x subject to Ax b, x 0 change the objective to maximize by multiplying by 1: max 4x 1 x 2. Next, change the constraints. Multiply the first constraint by 1: 2x 1 x 2 6; replace the second constraint by two inequalities: x 2 + x 3 4 and x 2 x 3 4; and replace the third constraint by the inequality: x 1 4. Finally, replace the unconstrained variable x 1 everywhere by u 1 v 1 and add the constraints that u 1, v 1 0. Putting these together leads to the reformulated problem 4

5 max 4u 1 + 4v 1 x 2 subject to 2u 1 2v 1 x 2 6 x 2 + x 3 4 x 2 x 3 4 u 1 + v 1 4 u 1 v 1 x 2 x 3 0. In notation, this problem is in the form: max c x subject to Ax b, x 0 with c = ( 4, 4, 1, 0), b = ( 6, 4, 4, 4) and A = Next, to put the problem into the form: max c x subject to Ax = b, x 0, change the objective function to max as above; replace x 1 by u 1 v 1 as above; replace the first constraint with 2u 1 + 2v 1 + x 2 s 1 = 6 and s 1 0; leave the second constraint alone; and replace the third constraint with The problem then becomes u 1 + v 1 + s 3 = 4. max 4u 1 + 4v 1 x 2 subject to 2u 1 + 2v 1 + x 2 s 1 = 6 x 2 + x 3 = 4 u 1 + v 1 + s 3 = 4 u 1 v 1 x 2 x 3 s 1 s 3 0. In notation, this problem is in the form: max c x subject to Ax = b, x 0 with c = ( 4, 4, 1, 0, 0, 0), b = (6, 4, 4) and A = In the two different transformations, the A, b, and c used in the representation differ. Indeed, the two descriptions of the problem have different numbers of variables and different numbers of constraints. It does not matter. If you solve either problem, you can substitute back to find values for the original variables, x 1,..., x 4, and the original objective function. 5

6 Computer programs that solve linear programming problems (like Excel) are smart enough to perform these transformations automatically. That is, you need not perform any transformations of this sort in order to enter an LP into Excel. The program asks you whether you are minimizing or maximizing, whether each constraint is an inequality or an equation, and whether the variables are constrained to be nonnegative. 6

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

Math 407A: Linear Optimization

Math 407A: Linear Optimization Lecture 4: LP Standard Form 1 1 Author: James Burke, University of Washington LPs in Standard Form Minimization maximization Linear equations to linear inequalities Lower

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

Readings. D Chapter 1. Lecture 2: Constrained Optimization. Cecilia Fieler. Example: Input Demand Functions. Consumer Problem

Economics 245 January 17, 2012 : Example Readings D Chapter 1 : Example The FOCs are max p ( x 1 + x 2 ) w 1 x 1 w 2 x 2. x 1,x 2 0 p 2 x i w i = 0 for i = 1, 2. These are two equations in two unknowns,

Lecture 4 Linear Programming Models: Standard Form. August 31, 2009

Linear Programming Models: Standard Form August 31, 2009 Outline: Lecture 4 Standard form LP Transforming the LP problem to standard form Basic solutions of standard LP problem Operations Research Methods

4.4 The Simplex Method: Non-Standard Form

4.4 The Simplex Method: Non-Standard Form Standard form and what can be relaxed What were the conditions for standard form we have been adhering to? Standard form and what can be relaxed What were the

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

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

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.

CHAPTER 17. Linear Programming: Simplex Method

CHAPTER 17 Linear Programming: Simplex Method CONTENTS 17.1 AN ALGEBRAIC OVERVIEW OF THE SIMPLEX METHOD Algebraic Properties of the Simplex Method Determining a Basic Solution Basic Feasible Solution 17.2

SUPPLEMENT TO CHAPTER

SUPPLEMENT TO CHAPTER 6 Linear Programming SUPPLEMENT OUTLINE Introduction and Linear Programming Model, 2 Graphical Solution Method, 5 Computer Solutions, 14 Sensitivity Analysis, 17 Key Terms, 22 Solved

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

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

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

Section Notes 3. The Simplex Algorithm. Applied Math 121. Week of February 14, 2011

Section Notes 3 The Simplex Algorithm Applied Math 121 Week of February 14, 2011 Goals for the week understand how to get from an LP to a simplex tableau. be familiar with reduced costs, optimal solutions,

Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

The Simplex Method. What happens when we need more decision variables and more problem constraints?

The Simplex Method The geometric method of solving linear programming problems presented before. The graphical method is useful only for problems involving two decision variables and relatively few problem

Simplex Method. Introduction:

Introduction: In the previous chapter, we discussed about the graphical method for solving linear programming problems. Although the graphical method is an invaluable aid to understand the properties of

max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

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

Outline. Linear Programming (LP): Simplex Search. Simplex: An Extreme-Point Search Algorithm. Basic Solutions

Outline Linear Programming (LP): Simplex Search Benoît Chachuat McMaster University Department of Chemical Engineering ChE 4G03: Optimization in Chemical Engineering 1 Basic Solutions

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

value is determined is called constraints. It can be either a linear equation or inequality.

Contents 1. Index 2. Introduction 3. Syntax 4. Explanation of Simplex Algorithm Index Objective function the function to be maximized or minimized is called objective function. E.g. max z = 2x + 3y Constraints

Duality in Linear Programming

Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow

Algebraic Simplex Method - Introduction Previous

Algebraic Simplex Method - Introduction To demonstrate the simplex method, consider the following linear programming model: This is the model for Leo Coco's problem presented in the demo, Graphical Method.

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

Stanford University CS261: Optimization Handout 6 Luca Trevisan January 20, In which we introduce the theory of duality in linear programming.

Stanford University CS261: Optimization Handout 6 Luca Trevisan January 20, 2011 Lecture 6 In which we introduce the theory of duality in linear programming 1 The Dual of Linear Program Suppose that we

Fundamentals of Operations Research Prof.G. Srinivasan Department of Management Studies Lecture No. # 03 Indian Institute of Technology, Madras

Fundamentals of Operations Research Prof.G. Srinivasan Department of Management Studies Lecture No. # 03 Indian Institute of Technology, Madras Linear Programming solutions - Graphical and Algebraic Methods

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.

1. (a) Multiply by negative one to make the problem into a min: 170A 170B 172A 172B 172C Antonovics Foster Groves 80 88

Econ 172A, W2001: Final Examination, Possible Answers There were 480 possible points. The first question was worth 80 points (40,30,10); the second question was worth 60 points (10 points for the first

3.5 Reduction to Standard Form

3.5 Reduction to Standard Form 77 Exercise 3-4-4. Read the following statements carefully to see whether it is true or false. Justify your answer by constructing a simple illustrative example (for false),

3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

Solving Linear Programs

Solving Linear Programs 2 In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the simplex method, proceeds by moving from one feasible solution to another,

LINEAR PROGRAMMING THE SIMPLEX METHOD

LINEAR PROGRAMMING THE SIMPLE METHOD () Problems involving both slack and surplus variables A linear programming model has to be extended to comply with the requirements of the simplex procedure, that

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

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

1 Gaussian Elimination

Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

Linear Inequalities and Linear Programming. Systems of Linear Inequalities in Two Variables

Linear Inequalities and Linear Programming 5.1 Systems of Linear Inequalities 5.2 Linear Programming Geometric Approach 5.3 Geometric Introduction to Simplex Method 5.4 Maximization with constraints 5.5

Chap 4 The Simplex Method

The Essence of the Simplex Method Recall the Wyndor problem Max Z = 3x 1 + 5x 2 S.T. x 1 4 2x 2 12 3x 1 + 2x 2 18 x 1, x 2 0 Chap 4 The Simplex Method 8 corner point solutions. 5 out of them are CPF solutions.

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

I. Solving Linear Programs by the Simplex Method

Optimization Methods Draft of August 26, 2005 I. Solving Linear Programs by the Simplex Method Robert Fourer Department of Industrial Engineering and Management Sciences Northwestern University Evanston,

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,

Linear Programming: Chapter 5 Duality

Linear Programming: Chapter 5 Duality Robert J. Vanderbei October 17, 2007 Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/ rvdb Resource

Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

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

2.1 Systems of Linear Equations

. Systems of Linear Equations Question : What is a system of linear equations? Question : Where do systems of equations come from? In Chapter, we looked at several applications of linear functions. One

Theory of Linear Programming

Theory of Linear Programming Debasis Mishra April 6, 2011 1 Introduction Optimization of a function f over a set S involves finding the maximum (minimum) value of f (objective function) in the set S (feasible

Solution to homework problem # and 7.1

Let us first review some basic concepts. Solution to homewk problem # 4.107 and 7.1 1. Derive the dual problem from the Lagrangian duality. It wks f convex problems, including all linear programming problems.

AM 221: Advanced Optimization Spring Prof. Yaron Singer Lecture 7 February 19th, 2014

AM 22: Advanced Optimization Spring 204 Prof Yaron Singer Lecture 7 February 9th, 204 Overview In our previous lecture we saw the application of the strong duality theorem to game theory, and then saw

Graphical method. plane. (for max) and down (for min) until it touches the set of feasible solutions. Graphical method

The graphical method of solving linear programming problems can be applied to models with two decision variables. This method consists of two steps (see also the first lecture): 1 Draw the set of feasible

7. Solving Linear Inequalities and Compound Inequalities

7. Solving Linear Inequalities and Compound Inequalities Steps for solving linear inequalities are very similar to the steps for solving linear equations. The big differences are multiplying and dividing

Lecture notes 1: Introduction to linear and (mixed) integer programs

Lecture notes 1: Introduction to linear and (mixed) integer programs Vincent Conitzer 1 An example We will start with a simple example. Suppose we are in the business of selling reproductions of two different

3 Does the Simplex Algorithm Work?

Does the Simplex Algorithm Work? In this section we carefully examine the simplex algorithm introduced in the previous chapter. Our goal is to either prove that it works, or to determine those circumstances

Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs

Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs These questions are from previous years and should you give you some idea of what to expect on Quiz 1. 1. Consider the

Lecture 1: Linear Programming Models. Readings: Chapter 1; Chapter 2, Sections 1&2

Lecture 1: Linear Programming Models Readings: Chapter 1; Chapter 2, Sections 1&2 1 Optimization Problems Managers, planners, scientists, etc., are repeatedly faced with complex and dynamic systems which

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

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

Linear Programming Notes VII Sensitivity Analysis

Linear Programming Notes VII Sensitivity Analysis 1 Introduction When you use a mathematical model to describe reality you must make approximations. The world is more complicated than the kinds of optimization

Operation Research. Module 1. Module 2. Unit 1. Unit 2. Unit 3. Unit 1

Operation Research Module 1 Unit 1 1.1 Origin of Operations Research 1.2 Concept and Definition of OR 1.3 Characteristics of OR 1.4 Applications of OR 1.5 Phases of OR Unit 2 2.1 Introduction to Linear

By W.E. Diewert. July, Linear programming problems are important for a number of reasons:

APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)

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

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

LINEAR PROGRAMMING PROBLEM: A GEOMETRIC APPROACH

59 LINEAR PRGRAMMING PRBLEM: A GEMETRIC APPRACH 59.1 INTRDUCTIN Let us consider a simple problem in two variables x and y. Find x and y which satisfy the following equations x + y = 4 3x + 4y = 14 Solving

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

The Graphical Method: An Example

The Graphical Method: An Example Consider the following linear program: Maximize 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 0, where, for ease of reference,

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

Fundamentals of Operations Research. Prof. G. Srinivasan. Department of Management Studies. Indian Institute of Technology, Madras. Lecture No.

Fundamentals of Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture No. # 22 Inventory Models - Discount Models, Constrained Inventory

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

Linear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:

Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrix-vector notation as 4 {{ A x x x {{ x = 6 {{ b General

1.1 Solving a Linear Equation ax + b = 0

1.1 Solving a Linear Equation ax + b = 0 To solve an equation ax + b = 0 : (i) move b to the other side (subtract b from both sides) (ii) divide both sides by a Example: Solve x = 0 (i) x- = 0 x = (ii)

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

Chapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS 3.2 TERMINOLOGY

Chapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS Once the problem is formulated by setting appropriate objective function and constraints, the next step is to solve it. Solving LPP

The Transportation Problem: LP Formulations

The Transportation Problem: LP Formulations An LP Formulation Suppose a company has m warehouses and n retail outlets A single product is to be shipped from the warehouses to the outlets Each warehouse

LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,

LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are

Math 018 Review Sheet v.3

Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1 - Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.

Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.

Solving Inequalities Examples

Solving Inequalities Examples 1. Joe and Katie are dancers. Suppose you compare their weights. You can make only one of the following statements. Joe s weight is less than Kate s weight. Joe s weight is

No Solution Equations Let s look at the following equation: 2 +3=2 +7

5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

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

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if

1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c

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:

Rational Polynomial Functions

Rational Polynomial Functions Rational Polynomial Functions and Their Domains Today we discuss rational polynomial functions. A function f(x) is a rational polynomial function if it is the quotient of

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,

The Graphical Simplex Method: An Example

The Graphical Simplex Method: An Example Consider the following linear program: Max 4x 1 +3x Subject to: x 1 +3x 6 (1) 3x 1 +x 3 () x 5 (3) x 1 +x 4 (4) x 1, x 0. Goal: produce a pair of x 1 and x that

3.5. Solving Inequalities. Introduction. Prerequisites. Learning Outcomes

Solving Inequalities 3.5 Introduction An inequality is an expression involving one of the symbols,, > or

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

A couple of Max Min Examples. 3.6 # 2) Find the maximum possible area of a rectangle of perimeter 200m.

A couple of Max Min Examples 3.6 # 2) Find the maximum possible area of a rectangle of perimeter 200m. Solution : As is the case with all of these problems, we need to find a function to minimize or maximize

LINEAR PROGRAMMING MODULE Part 2 - Solution Algorithm

LINEAR PROGRAMMING MODULE Part 2 - Solution Algorithm Name: THE SIMPLEX SOLUTION METHOD It should be obvious to the reader by now that the graphical method for solving linear programs is limited to models

Sect Solving and graphing inequalities

81 Sect 2.7 - Solving and graphing inequalities Concepts #1 & 2 Graphing Linear Inequalities Definition of a Linear Inequality in One Variable Let a and b be real numbers such that a 0. A Linear Inequality

Solutions of Linear Equations in One Variable

2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

Solutions to Homework Set 3 (Solutions to Homework Problems from Chapter 2)

Solutions to Homework Set 3 (Solutions to Homework Problems from Chapter 2) Problems from 21 211 Prove that a b (mod n) if and only if a and b leave the same remainder when divided by n Proof Suppose a

Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood

PERFORMANCE EXCELLENCE IN THE WOOD PRODUCTS INDUSTRY EM 8720-E October 1998 \$3.00 Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood A key problem faced

The Simplex Solution Method

Module A The Simplex Solution Method A-1 A-2 Module A The Simplex Solution Method The simplex method is a general mathematical solution technique for solving linear programming problems. In the simplex

3.4. Solving simultaneous linear equations. Introduction. Prerequisites. Learning Outcomes

Solving simultaneous linear equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential