MATH 140, Chapter 3 19

SECTION 3.4: The Simplex Method Linear programming problems can quickly become complicated to solve. In this section, we will learn the ___________________________________ to more efficiently

solve certain ____________________________________ problems. These certain problems are

called standard maximization problems.

Standard Maximization Problems Definition: Standard Maximization Problem A linear programming problem is a standard maximization problem if

1. Decision variables are __________________________________, i.e. 𝑥,𝑦 ≥ 0.

2. All other constraints are of the form: ________________________________

3. Objective function is to be ________________________________________

Example 1: Identifying standard maximization problems. Is the following linear programming problem is a standard maximization problem? Maximize 𝑃 = 60𝑥 + 80𝑦 subject to: 2𝑥 + 4𝑦 ≤ 80 2𝑥 + 2𝑦 ≤ 50 4𝑥 + 2𝑦 ≤ 84 𝑥,𝑦 ≥ 0 The Simplex Method You can use the simplex method to solve any standard maximization problem Overview

1. Translate system into a simplex tableau

2. Pivot on specific elements until the last row is nonnegative

3. Read off solution and maximum value of objective function

MATH 140, Chapter 3 20

Setting Up the Simplex Tableau A _____________________________ is a special kind of augmented ___________________ which will help solve the standard maximization problem. Steps for creating the Simplex Tableau:

1. Introduce a new variable called a ____________________ variable into each of the

problem constraints of the system.

• Each inequality gets its own slack variable, usually denoted _________, and

all of the inequality signs get changed to equal signs.

• Example: If your constraints are: 2𝑥 + 4𝑦 ≤ 80 2𝑥 + 2𝑦 ≤ 50 4𝑥 + 2𝑦 ≤ 84 𝑥,𝑦 ≥ 0 Then system of inequalities becomes: 2𝑥 + 4𝑦 + _____ _____ 80 2𝑥 + 2𝑦 + _____ _____ 50 4𝑥 + 2𝑦 + _____ _____ 84

2. Rewrite the ___________________________ function

• Set your objective function equal to zero by subtracting the

____________________variable from both sides.

• Example: If your objective function is: 𝑃 = 60𝑥 + 80𝑦, then you modify the objective function to be:

3. Create your ______________________________ matrix, called the simplex tableau,

using the rewritten constraints and the rewritten objective function.

• Label your variables across the top, including the original variables, the slack variables, and the dependent variable of the objective function

• Put your modified constraint equations into the matrix • The last line is your modified objective function • Example: Using the constraints and objective function from above, the

simplex tableau for this problem is:

MATH 140, Chapter 3 21

Selecting the Pivot Element The simplex method finds the solution to the standard maximization problem by pivoting on carefully chosen matrix entries. Here are the steps to finding the matrix entry on which to pivot:

1. Find the __________________ ______________________

• The column of the pivot element is the column of the most _______________

entry in the bottom row. If there is a tie for most negative entry, you can pick

any column with the most negative entry.

• Example: Using the simplex tableau above, the pivot column would be:

2. Find the __________________ ______________________

• For all but the last row, divide the element in the last column by the

corresponding element in the pivot column, provided the element in the pivot

column is _____________________________. This will give you ratios for

each row.

• The row with the smallest ratio is your pivot row.

• Example: From the simplex tableau and pivot column from above, we can

find the pivot row by the following:

Therefore for this example, the pivot element is:

MATH 140, Chapter 3 22

Reading off the Solution Once all of the pivoting steps are complete, one can read off the solutions by the following:

1. Identify the ___________ ___________________. These are the columns with one 1 and

zeroes elsewhere.

2. If a column is _____________ a unit column, then it is a nonunit column. The variable

associated with a nonunit column is a _________________________ variable.

3. Convert the constraint rows back into equations.

4. Set the nonbasic variables equal to 0. This will give the values for the decision variables.

5. The maximum value that the objective function obtains will be the number in the bottom

right-hand corner.

Steps for the Simplex Method A standard maximization problem can be solved using the simplex method by the following:

1. Set up the simplex tableau

• Follow the steps in the “Setting Up the Simplex Tableau” section above.

2. Check the bottom row.

• If no negative entries are in the bottom row, then a solution has been found

and the simplex tableau is in final form. Go to Step 4 to read the solution.

• Otherwise, if negative entries do exist in the bottom row, go to Step 3.

3. Select the pivot element.

• Follow the steps in the “Selecting the Pivot Element” section above to find the

pivot column and the pivot row.

• Pivot on the element in the pivot column and the pivot row. On homework

and take home quizzes you can use the following online pivot tool:

http://www.zweigmedia.com/RealWorld/tutorialsf1/scriptpivot2.html

Simply enter the numbers from your matrix, click on the element you want to

pivot on, then click “Pivot on Selection”.

• Go to Step 2.

4. Read off the solution.

• Follow the steps in the “Reading off the Solution” section above.

MATH 140, Chapter 3 23

Example 1: Using the Simplex Method on a Standard Maximization Problem A wooden boat manufacturer creates canoes (x) and rowboats (y). The manufacture of each boat must go through three operations: cutting of the wood, painting of the completed boat, and assembly of the pieces. Each canoe requires 2 hours of cutting, 2 hours of painting, and 4 hours of assembling. Each rowboat requires 4 hours of cutting, 2 hours of painting, and 2 hours of assembling. The total time available per week in the cutting section is 80 hours, in the painting section 50 hours, and 84 hours in the assembly section. There is a profit of $60 for every canoe made and $80 for every rowboat made. How many of each type of boat should be made in order to maximize profits? This problem translates to: Maximize 𝑃 = 60𝑥 + 80𝑦 subject to: 2𝑥 + 4𝑦 ≤ 80 2𝑥 + 2𝑦 ≤ 50 4𝑥 + 2𝑦 ≤ 84 𝑥,𝑦 ≥ 0

MATH 140, Chapter 3 24

(Example continued)

MATH 140, Chapter 3 25

Excess Resources After reading the solution to a simplex tableau, any nonzero slack variable represents an

_______________ of ________________________.

For instance, in the boat example, we obtained that 𝑠! = 14. Recall that 𝑠! was the slack variable

relating to the assembly constraint. This means that there were _______ hours allocated to

assembling boat parts that were not needed. We would say that there were 14 assembly hours

leftover.

MATH 140, Chapter 3 26

Example 2: Using the Simplex Method A firm manufactures 3 kinds of bulk trail mix, each of which uses peanuts, chocolate chips, and dried fruit. A bag of trail mix A requires 2 pounds of peanuts, 1 pound of chocolate chips, and 1 pound of dried fruit. A bag of trail mix B requires 1 pound of peanuts, 1 pound of chocolate chips, and 3 pounds of dried fruit. A bag of trail mix C requires 1 pound of peanuts, 4 pounds of chocolate chips, and 3 pounds of dried fruit. There are 20 pounds of peanuts, 30 pounds of chocolate chips, and 40 pounds of dried fruit available. The profit on each bag of trail mix A is $26, on each bag of trail mix B is $24, and on each bag of trail mix C is $30. Find the number of bags of each trail mix that should be made in order to maximize profits and find the maximum profits.