# 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

Save this PDF as:

Size: px
Start display at page:

Download "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"

## Transcription

1 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, if the objective function P is optimized at two adjacent vertices of S, then it is optimized at every point on the line segment joining these two vertices, in which case there are infinitely many solutions to the problem. THEOREM 2 Suppose we are given a linear programming problem with a feasible set S and an objective function P = ax+by. Then, If S is bounded then P has both a maximum and minimum value on S If S is unbounded and both a and b are nonnegative, then P has a minimum value on S provided that the constraints defining S include the inequalities x 0 and y 0. If S is the empty set, then the linear programming problem has no solution; that is, P has neither a maximum nor a minimum value. THE METHOD OF CORNERS Graph the feasible set (region), S. Find the EXACT coordinates of all vertices (corner points) of S. Evaluate the objective function, P, at each vertex The maximum (if it exists) is the largest value of P at a vertex. The minimum is the smallest value of P at a vertex. If the objective function is maximized (or minimized) at two vertices, it is minimized (or maximized) at every point connecting the two vertices. Example: Find the maximum and minimum values of P=3x+2y subject to x + 4y 20 2x + 3y 30 x 0 y 0 1. Graph the feasible region. Start with the line x+4y=20. Find the intercepts by letting x=0 and y=0 : (0,5) and (20,0). Test the origin: (0) + 4(0) 20. This is true, so we keep the half-plane containing the origin. Graph the line 2x+3y=30. Find the intercepts (0,10) and (15,0). Test the origin and find it true. Shade the upper half plane. Non-negativity keeps us in the first quadrant. 2. Find the corner points. We find (0,0) is one corner, (0,5) is the corner from the y-intercept of the first equation, The corner (15,0) is from the second equation. We can find the intersection of the two lines using intersect on the calculator or using rref: Value is (12, 2) 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 4. The region is bounded, therefore a max and a min exist on S. The minimum is at the point (0,0) with a value of P=0. The maximum is at the point (15,0) and the value is P=45.

2 Example: A farmer wants to customize his fertilizer for his current crop. He can buy plant food mix A and plant food mix B. Each cubic yard of food A contains 20 pounds of phosphoric acid, 30 pounds of nitrogen and 5 pounds of potash. Each cubic yard of food B contains 10 pounds of phosphoric acid, 30 pounds of nitrogen and 10 pounds of potash. He requires a minimum of 460 pounds of phosphoric acid, 960 pounds of nitrogen and 220 pounds of potash. If food A costs \$30 per cubic yard and food B costs \$35 per cubic yard, how many cubic yards of each food should the farmer blend to meet the minimum chemical requirements at a minimal cost? What is this cost? Before we can use the method of corners we must set up our system. Start by defining your variables properly and completely: x = no. of cubic yards of plant food mix A y = no. of cubic yards of plant food mix B Next find the objective function. This is the quantity to be minimized or maximized: Minimize C=30x+35y Finally, set up the constraints on the problem: Subject to 20x + 10y 460 phosphoric acid requirement 30x + 30y 960 nitrogen requirement 5x + 10y 220 potash requirement x 0 y 0 Now use the method of corners: 1. Graph the feasible region. Start by putting in the three lines and finding their intercepts. Test the origin in each inequality and find that the origin is false, so we shade the lower half-plane of each. This feasible region is unbounded. That means that there is a minimum, but no maximum. 2. Find the corner points. We have 4 corners of the feasible region. 3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex C=30x+35y (0, 46) 1610 (44, 0) 1320 (20, 12) 1020 minimum (14, 18) The minimum occurs at (20,12). So the farmer should buy 20 cubic yards of plant food mix A and 12 cubic yards of plant food mix B at a cost of \$1020.

3 Example: Find the maximum value of f=10x+8y subject to 2x + 3y 120 5x + 4y 200 x 0 y 0 1. Graph the feasible region. 2x+3y=120 Intercepts are (0,40) and (60,0). Origin is true so shade the upper half plane. 5x+4y=200 Intercepts are (0,50) and (40,0). Origin is true so shade the upper half plane. 2. Find the corner points. We find (0,0) is one corner, (0,40) is the corner from the y-intercept of the first equation. The corner (40,0) is from the second equation. We can find the intersection of the two lines using the intersection or rref: (120/7, 200/7). 3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex F=10x+8y (0, 0) 0 (0, 40) 320 (40, 0) 400 Max (120/7, 200/7) 400 Max 4. We find that the objective function is maximized at two points, therefore it is maximized along the line segment connecting these two points. The value of the objective function is 400 = f = 10x + 8y. Write this in slope-intercept form, y = -1.25x + 50 and 120/7 x 40.

4 When you can t find the corners of the feasible region graphically (or don t want to!), we can use the simplex method to find the corners algebraically. The section we cover is for STANDARD MAXIMIZATION PROBLEMS. That is, the linear programming problem meets the following conditions: The objective function is to be maximized. All the variables are non-negative Each constraint can be written so the expression involving the variables is less than or equal to a non-negative constant. THE SIMPLEX METHOD: 1. Set up the initial tableau Take the system of linear inequalities and add a slack variable to each inequality to make it an equation. Be sure to line up variables to the left of the ='s and constants to the right. Rewrite the objective function in the form -c 1 x 1 - c 2 x c n x n +P=0. Note the variables are all on the same side of the equation as the objective function and the objective function gets the positive sign. Put this linear equations into an augmented matrix. This is the initial simplex tableau. 2. Look at the bottom row to the left of the vertical line. If all the entries are NON-NEGATIVE, simplex is done. go to step 4. If there are negative entries, the pivot element must be selected. 3. Select the pivot element Find the most negative entry in the bottom row to the left of the vertical line. This is the pivot column. Mark it with an arrow. find the quotients of the constant column value divided by the pivot column number, q=a/b. Only positive numbers are allowed for b. Choose the row with the smallest non-negative quotient. This is the pivot row. Perform the pivot Go to step 2 4. Solution - write out the equations row by row. Remember, if a column is messy it is a NB variable (parameter) and has a value of 0. The simplex algorithm moves us from corner to corner in the feasible region (as long as the pivot element is chosen correctly). In this first example, I will show how the augmented matrix corresponds to a corner of the feasible region. In problems with more than two variables, you just must trust it to work.

5 Example: Maximize f=3x+2y x+y 4-2x+y 2 x 0 y 0 subject to Introduce a slack variable into each inequality to make an equation. Rewrite the objective function into an equation: x + y + u = 4-2x + y + v = 2-3x + 2y + f = 0 Set up the initial simplex tableau: x y u v f What are the values of our variables here? Remember, a column that is not a unit column (one 1 and the rest zero) is a non-basic variable and is equal to 0. We find the following: x=0, y=0, u=4, v=2 and f=0. Look now at the feasible region. This point is at the origin. Can we increase f? Yes, and changing x will increase it fastest. So we want to pivot on the x column. Look at the quotients. Only one is possible. So we will pivot on the first row. Use the calculator program SMPLX. What we find is x y u v f What are the values of the variables now? x=4, y=0, u=0, v=10, and f=12. What has happened on the graph? We have moved to the corner point (4,0). We couldn't have moved to the other corner point on the x -axis as it is out of the feasible region (that's what the negative number would cause). By choosing the smallest non-negative quotient we move to a corner of the feasible region, not just any intersection point. Are we done? There are no more negative numbers in the bottom row and so we have found the maximum value of f.

6 Example: In the following matrices, find the pivot element if simplex is not done. Find the values of all the variables if simplex is done. (a) Pivot on row1 column 1 or row 2 column 1 as in a tie you can choose either (b) NO SOLUTION (nothing to pivot on and there is a negative number in the bottom row) (c) pivot on row 1 column 1 (d) Pivot on row 1 column 2 (e)

7 Example: A company makes three models of desks, an executive model, an office model and a student model. Each desk spends time in the cabinet shop, the finishing shop and the crating shop as shown in the table: Type of desk Cabinet shop Finishing shop Crating shop Profit Executive Office Student Available hours How many of each type of model should be made to maximize profits? Answer: Start by defining your variables: x = number of executive desks made y = number of office desks made z = number of student desks made Maximize P=150x+125y+50z.subject to 2x + y + z 16 cabinet hours x + 2y + z 16 finishing hours x + y +.5z 10 crating hours x 0, y 0, z 0. We will give each of the inequalities a slack variable. Write the initial tableau, x y z u v w P Use the simplex algorithm to find the pivot elements. Choose row 1, column 1 for our first pivot element. The second pivot will be on row 3 column 2. our final tableau looks like x y z u v w P Set the NB variables to zero, z=0, u=0, w=0. Now read across each row to find x=6, y=4, v=2 and P=1400. This is a word problem and needs to be explained: Make 6 executive desks, 4 office desks and no student desks. All of the hours in the cabinet shop and crating shop will be used. There will be 2 hours of finishing shop unused. The profit will be \$1400.

### 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

### 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

### Section 8-7 Systems of Linear Inequalities

658 8 SYSTEMS; MATRICES Section 8-7 Systems of Linear Inequalities FIGURE Half-planes. Graphing Linear Inequalities in Two Variables Solving Systems of Linear Inequalities Graphically Application Many

### 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

### 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.

### 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

### 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

### 3.1 Graphing Systems of Linear Inequalities

3.1 Graphing Systems of Linear Inequalities An inequality is an expression like x +3y < 4 or 2x 5y 2. They look just like equations except the equal sign is replaced by one of ,,. An inequality does

### 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.

### Standard Form of a Linear Programming Problem

494 CHAPTER 9 LINEAR PROGRAMMING 9. THE SIMPLEX METHOD: MAXIMIZATION For linear programming problems involving two variables, the graphical solution method introduced in Section 9. is convenient. However,

### 1. Graphing Linear Inequalities

Notation. CHAPTER 4 Linear Programming 1. Graphing Linear Inequalities x apple y means x is less than or equal to y. x y means x is greater than or equal to y. x < y means x is less than y. x > y means

### 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

### Chapter 5.1 Systems of linear inequalities in two variables.

Chapter 5. Systems of linear inequalities in two variables. In this section, we will learn how to graph linear inequalities in two variables and then apply this procedure to practical application problems.

### 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

### The revenue function, R(x), is the total revenue realized from the sale of x units of the product.

Linear Cost, Revenue and Profit Functions: If x is the number of units of a product manufactured or sold at a firm then, The cost function, C(x), is the total cost of manufacturing x units of the product.

### 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

### 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

### Using the Simplex Method in Mixed Integer Linear Programming

Integer Using the Simplex Method in Mixed Integer UTFSM Nancy, 17 december 2015 Using the Simplex Method in Mixed Integer Outline Mathematical Programming Integer 1 Mathematical Programming Optimisation

### The Simplex Method. yyye

Yinyu Ye, MS&E, Stanford MS&E310 Lecture Note #05 1 The Simplex Method Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/

### 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

### 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

### 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

### 3y 1 + 5y 2. y 1 + y 2 20 y 1 0, y 2 0.

1 Linear Programming A linear programming problem is the problem of maximizing (or minimizing) a linear function subject to linear constraints. The constraints may be equalities or inequalities. 1.1 Example

### 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

10.9 Systems Of Inequalities Copyright Cengage Learning. All rights reserved. Objectives Graphing an Inequality Systems of Inequalities Systems of Linear Inequalities Application: Feasible Regions 2 Graphing

### 0.1 Linear Programming

0.1 Linear Programming 0.1.1 Objectives By the end of this unit you will be able to: formulate simple linear programming problems in terms of an objective function to be maximized or minimized subject

### 4.3 The Simplex Method and the Standard Maximization Problem

. The Simplex Method and the Standard Maximiation Problem Question : What is a standard maximiation problem? Question : What are slack variables? Question : How do you find a basic feasible solution? Question

### 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

### LINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA Hyderabad

LINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA 98481 85073 Hyderabad Page 1 of 19 Question: Explain LPP. Answer: Linear programming is a mathematical technique for determining the optimal allocation of resources

### Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.

Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value

### 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

### 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.

### (a) Let x and y be the number of pounds of seed and corn that the chicken rancher must buy. Give the inequalities that x and y must satisfy.

MA 4-4 Practice Exam Justify your answers and show all relevant work. The exam paper will not be graded, put all your work in the blue book provided. Problem A chicken rancher concludes that his flock

### Section 1.4 Graphs of Linear Inequalities

Section 1.4 Graphs of Linear Inequalities A Linear Inequality and its Graph A linear inequality has the same form as a linear equation, except that the equal symbol is replaced with any one of,,

### 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

### Study Resources For Algebra I. Unit 1D Systems of Equations and Inequalities

Study Resources For Algebra I Unit 1D Systems of Equations and Inequalities This unit explores systems of linear functions and the various methods used to determine the solution for the system. Information

### 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

### Module1. x 1000. y 800.

Module1 1 Welcome to the first module of the course. It is indeed an exciting event to share with you the subject that has lot to offer both from theoretical side and practical aspects. To begin with,

### 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

### Geometry of Linear Programming

Chapter 2 Geometry of Linear Programming The intent of this chapter is to provide a geometric interpretation of linear programming problems. To conceive fundamental concepts and validity of different algorithms

### 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 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

### In this section, we ll review plotting points, slope of a line and different forms of an equation of a line.

Math 1313 Section 1.2: Straight Lines In this section, we ll review plotting points, slope of a line and different forms of an equation of a line. Graphing Points and Regions Here s the coordinate plane:

### 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

### 4.1 Solving a System of Linear Inequalities

4.1 Solving a System of Linear Inequalities Question 1: How do you graph a linear inequality? Question : How do you graph a system of linear inequalities? In Chapter, we were concerned with systems of

### 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

### 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

### 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

### The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6

Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means

### Lines and Linear Equations. Slopes

Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means

### 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

### 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),

### Simplex method summary

Simplex method summary Problem: optimize a linear objective, subject to linear constraints 1. Step 1: Convert to standard form: variables on right-hand side, positive constant on left slack variables for

### 1. Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded:

Final Study Guide MATH 111 Sample Problems on Algebra, Functions, Exponents, & Logarithms Math 111 Part 1: No calculator or study sheet. Remember to get full credit, you must show your work. 1. Determine

### 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.

### 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

### Algebra 1 Topic 10: Systems of linear equations and inequalities Student Activity Sheet 1; use with Overview

Algebra 1 Topic 10: Student Activity Sheet 1; use with Overview 1. Consider the equation y = 3x + 4. [OV, page 1] a. What can you know about the graph of the equation? Numerous answers are possible, such

### Section 3.3 Linear Inequalities in Two Variables

Acc. Alg. II Ch. 3 Notes Name Section 3.3 Linear Inequalities in Two Variables Graph the linear inequalities below. 1. 2x y 3 2. -5x 2y > 4 3. x > -1 Acc. Alg. II Ch. 3 Notes Page 2 4. x < 3 5. Write an

### Linear Programming I: Maximization 2009 Samuel L. Baker Assignment 10 is on the last page.

LINEAR PROGRAMMING I 1 Learning objectives: Linear Programming I: Maximization 2009 Samuel L. Baker Assignment 10 is on the last page. 1. Recognize problems that linear programming can handle. 2. Know

### 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,

### Module 3 Lecture Notes 2. Graphical Method

Optimization Methods: Linear Programming- Graphical Method Module Lecture Notes Graphical Method Graphical method to solve Linear Programming problem (LPP) helps to visualize the procedure explicitly.

### 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

### 10 Linear Programming

10 Linear Programming Learning Objectives: After studying this unit you will be able to: Formulate the Linear Programming Problem Solve the Linear Programming Problem with two variables using graphical

### 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

### Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.

Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the

### Linear Equations in Linear Algebra

1 Linear Equations in Linear Algebra 1.2 Row Reduction and Echelon Forms ECHELON FORM A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties: 1. All nonzero

### 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

### Reduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:

Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in

### Portable Assisted Study Sequence ALGEBRA IIA

SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of

### 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.

### Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010

Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables { ax + by = c dx + ey = f A solution of such a system is an

### 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

### 5 Systems of Equations

Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate

### 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

### 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

### 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

### 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

### Section 7.1 Solving Linear Systems by Graphing. System of Linear Equations: Two or more equations in the same variables, also called a.

Algebra 1 Chapter 7 Notes Name Section 7.1 Solving Linear Systems by Graphing System of Linear Equations: Two or more equations in the same variables, also called a. Solution of a System of Linear Equations:

### Chapter 4: The Mechanics of the Simplex Method

Chapter 4: The Mechanics of the Simplex Method The simplex method is a remarkably simple and elegant algorithmic engine for solving linear programs. In this chapter we will examine the internal mechanics

### 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

### Linear Programming II: Minimization 2006 Samuel L. Baker Assignment 11 is on page 16.

LINEAR PROGRAMMING II 1 Linear Programming II: Minimization 2006 Samuel L. Baker Assignment 11 is on page 16. Introduction A minimization problem minimizes the value of the objective function rather than

### is redundant. If you satisfy the other four constraints, then you automatically satisfy x 1 0. (In fact, if the first two constraints hold, then x 1 m

Linear Programming Notes II: Graphical Solutions 1 Graphing Linear Inequalities in the Plane You can solve linear programming problems involving just two variables by drawing a picture. The method works

### 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

### 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

### 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

### LINEAR PROGRAMMING: MODEL FORMULATION AND 2-1

LINEAR PROGRAMMING: MODEL FORMULATION AND GRAPHICAL SOLUTION 2-1 Chapter Topics Model Formulation A Maximization Model Example Graphical Solutions of Linear Programming Models A Minimization Model Example

### not to be republishe NCERT LINEAR PROGRAMMING Chapter Introduction

504 MATHEMATICS Chapter 12 LINEAR PROGRAMMING The mathematical experience of the student is incomplete if he never had the opportunity to solve a problem invented by himself. G. POLYA 12.1 Introduction

### 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

### 9.4 THE SIMPLEX METHOD: MINIMIZATION

SECTION 9 THE SIMPLEX METHOD: MINIMIZATION 59 The accounting firm in Exercise raises its charge for an audit to \$5 What number of audits and tax returns will bring in a maximum revenue? In the simplex

### Solve the linear programming problem graphically: Minimize w 4. subject to. on the vertical axis.

Do a similar example with checks along the wa to insure student can find each corner point, fill out the table, and pick the optimal value. Example 3 Solve the Linear Programming Problem Graphicall Solve

### 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.

### 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

### 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)

### Solving Equations Involving Parallel and Perpendicular Lines Examples

Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines

### 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

### Row Echelon Form and Reduced Row Echelon Form

These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

### 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

### 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,