Section 3.4 Simplex Method

In this section, we will only solve linear programming problems that are so called standard maxi-

mization problems.

A linear programming problem is a standard maximization problem if:

1. The variables are constrained to be nonnegative

2. All of the problem constraint ineqaulities are a nonnegative constant

3. The objective function is to be maximized

The first step in the simplex method is to convert the system of linear inequalities to a system of linear

equalities.

Example 1: Convert the inequality 3x� 5y 12 into an equality.

For the simplex method, we convert all the ineqaulities into equalities and treat the objective function

as a variable. The Simplex Method has 4 steps:

Simplex Method:

Step 1. Set up the initial simplex tableau in the following manner:

Introduce a slack variable for each of the constraints and write each constraint as an equality.

Rewrite the objective function in the form · · ·+ P = 0.

Put this system of linear equations in an augmented matrix with the objective function as

the bottom row.

Example 2: Consider the linear programming problem that is Example 1 of Section 3.3 in the

book:

Maximize P = 60x+ 80y

Subject to 2x+ 4y 80

4x+ 2y 84

2x+ 2y 50

x � 0 , y � 0

(a) Convert the linear programming problem into a system of equalities.

(b) Now write the system of equations as a simplex tableau. This is the initial simplex

tableau.

Step 2. Examine the entries in the bottom row of the augmented matrix. If no negative entries remain,

then a solution has been found. (Reading the solution will be discussed in Step 4)

Step 3. Select and perform the pivot:

Pivot Column: Find the most negative entry in the bottom row of the simplex tableau.

The column this entry is in is the pivot column. If there is a tie for the most negative

entry, use any of the columns with the most negative entry.

Pivot Row: Form the ratio q = a/b where a is the entry in the last column of the simplex

tableau and b is the entry in the pivot column above the horizontal line. Skip this ratio for

negative values of b. The row that results in the smallest nonnegative ratio is the pivot

row.

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Pivot about the entry in the pivot column and pivot row. Note: We will use technology

to do this, and thus it will NOT be testing on the exam, but will be tested in the WebAssign

homework.

After pivoting, repeat Step 2.

Example 3: For the simplex tableau formed in Example 2, use pivots until there are no more

negative entries in the bottom row of the matrix.

x1 x2 s1 s2 s3 P

2 4 1 0 0 0 80

4 2 0 1 0 0 84

2 2 0 0 1 0 50

-60 -80 0 0 0 1 0

Result of the pivot:

x1 x2 s1 s2 s3 P

1/2 1 1/4 0 0 0 20

3 0 -1/2 1 0 0 44

1 0 -1/2 0 1 0 10

-20 0 20 0 0 1 1600

Result of the pivot:

x1 x2 s1 s2 s3 P

0 1 1/2 0 -1/2 0 15

0 0 1 1 -3 0 14

1 0 -1/2 0 1 0 10

0 0 10 0 20 1 1800

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1. Read the solution:

Identify the nonunit columns. These nonbasic variables are set to zero.

Read each row to find the values of the basic variables.

Example 4: Read the solution from the final tableau in Example 3.

x1 x2 s1 s2 s3 P

0 1 1/2 0 -1/2 0 15

0 0 1 1 -3 0 14

1 0 -1/2 0 1 0 10

0 0 10 0 20 1 1800

Example 5: Consider the following simplex tableau. Use the simplex method to find the next tableau.x1 x2 s1 s2 s3 z

1 0 0 1 1 0 5

3 0 1 -2 0 0 3

7 1 0 1 0 0 6

-3 0 0 -4 2 1 10

Result of the pivot:x1 x2 s1 s2 s3 z

1 0 0 1 1 0 5

5 0 1 0 2 0 13

6 1 0 0 -1 0 1

1 0 0 0 6 1 30

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Example 6: Determine if the following simplex tableau is in final form. If it is in final form, find the

solution. If it is not, what is the next pivot?x1 x2 x3 s1 s2 s3 z

1 0 0 0.4 -0.7 0 0 4

0 0 1 -0.2 0.3 -0.1 0 2

0 1 0 1.2 -0.2 0.3 0 5

0 0 0 3.4 2 9.6 1 145

Example 7: Setup the initial simplex tableau needed to solve the following linear programming prob-

lem, and determine the the first pivot. DO NOT PERFORM THE PIVOT.

Maximize R = 5x+ 6y + 4z

Subject to 0.5x+ y + 0.25z 5000

�x+ 3y + 3z 0

�2y + z 0

x � 0 , y � 0 , z � 0

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system : .5× + y+ . Z5z + S ,

= 5000

- X + 3y + 3 Z + Sz= 0

- Zy + Z + S 3= 0

- 5x - by - 4z + R = 0

nsxotsorfsooo- 1

�3�3 6 1 0 0

@6- 2 1 0 0

1 0@=- 5 - 6 -4 O 0 0 1 | 0

Pivot column ? Column 2 since

=

- 6 is the most negative .

Pivot Row ?

q ,= 5000/1=5000

Ez = 013 = °

since qzis the smallest.

%p,°÷nrowzco1um#€

Example 8: You manage an ice cream factory that makes two flavors: Vanilla and Mocha. Each quart

of Vanilla requires 2 eggs and 3 cups of cream. Each quart of Mocha requires 1 egg and 4 cups of cream.

You have in stock 500 eggs and 1200 cups of cream. You make a profit of $3 on each quart of Vanilla

and $5 on each quart of Mocha.

(a) How many quarts of each flavor should you make in order to earn the largest profit?

(b) Are there any leftover resources? Be specific.

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