- max Z = 3x1 + 5x2 + 4x3 subject to 2x1 + 3x2 2x2 + 5x3 3x1 + 2x2 + 4x3 and x1,x2,x3 >= 0
- max Z = 5x1 + 10x2 + 8x3 subject to 3x1 + 5x2 + 2x3 4x1 + 4x2 + 4x3 2x1 + 4x2 + 5x3 and x1,x2,x3 >= 0
- max Z = 4x1 + 3x2 subject to 2x1 + x2 x1 + x2 x1 x2 and x1,x2 >= 0
- =,>=`4,7`');">min Z = x1 + x2 subject to 2x1 + 4x2 >= 4 x1 + 7x2 >= 7 and x1,x2 >= 0
- =,>=`80,60`');">min Z = 600x1 + 500x2 subject to 2x1 + x2 >= 80 x1 + 2x2 >= 60 and x1,x2 >= 0
- =`12,10,10`');">min Z = 5x1 + 3x2 subject to 2x1 + 4x2 2x1 + 2x2 = 10 5x1 + 2x2 >= 10 and x1,x2 >= 0
- max Z = x1 + 2x2 + 3x3 - x4 subject to x1 + 2x2 + 3x3 = 15 2x1 + x2 + 5x3 = 20 x1 + 2x2 + x3 + x4 = 10 and x1,x2,x3,x4 >= 0
- max Z = 3x1 + 9x2 subject to x1 + 4x2 x1 + 2x2 and x1,x2 >= 0
- max Z = 3x1 + 2x2 + x3 subject to 2x1 + 5x2 + x3 = 12 3x1 + 4x2 = 11 and x2,x3 >= 0 and x1 unrestricted in sign
- max Z = 3x1 + 3x2 + 2x3 + x4 subject to 2x1 + 2x2 + 5x3 + x4 = 12 3x1 + 3x2 + 4x3 = 11 and x1,x2,x3,x4 >= 0
- =`30,24,3`');">max Z = 6x1 + 4x2 subject to 2x1 + 3x2 3x1 + 2x2 x1 + x2 >= 3 and x1,x2 >= 0
- =`6,10,1`');">max Z = 3x1 + 5x2 subject to x1 - 2x2 x1 x2 >= 1 and x1,x2 >= 0
- =`5,8`');">max Z = 6x1 + 4x2 subject to x1 + x2 x2 >= 8 and x1,x2 >= 0
- =,>=`-5,8`');">max Z = 6x1 + 4x2 subject to -x1 - x2 >= -5 x2 >= 8 and x1,x2 >= 0

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## 4.2: Maximization By The Simplex Method

## Learning Objectives

- Identify and set up a linear program in standard maximization form
- Convert inequality constraints to equations using slack variables
- Set up the initial simplex tableau using the objective function and slack equations
- Find the optimal simplex tableau by performing pivoting operations.
- Identify the optimal solution from the optimal simplex tableau.

## THE SIMPLEX METHOD

- Set up the problem. That is, write the objective function and the inequality constraints.
- Convert the inequalities into equations. This is done by adding one slack variable for each inequality.
- Construct the initial simplex tableau. Write the objective function as the bottom row.
- The most negative entry in the bottom row identifies the pivot column.
- Calculate the quotients. The smallest quotient identifies a row. The element in the intersection of the column identified in step 4 and the row identified in this step is identified as the pivot element. The quotients are computed by dividing the far right column by the identified column in step 4. A quotient that is a zero, or a negative number, or that has a zero in the denominator, is ignored.
- Perform pivoting to make all other entries in this column zero. This is done the same way as we did with the Gauss-Jordan method.
- When there are no more negative entries in the bottom row, we are finished; otherwise, we start again from step 4.
- Read off your answers. Get the variables using the columns with 1 and 0s. All other variables are zero. The maximum value you are looking for appears in the bottom right hand corner.

Now, we use the simplex method to solve Example 3.1.1 solved geometrically in section 3.1.

## Example \(\PageIndex{1}\)

In solving this problem, we will follow the algorithm listed above.

STEP 1. Set up the problem. Write the objective function and the constraints.

- \(x_1\) = The number of hours per week Niki will work at Job I and
- \(x_2\) = The number of hours per week Niki will work at Job II.

It is customary to choose the variable that is to be maximized as \(Z\).

The problem is formulated the same way as we did in the last chapter.

\[x_1 + x_2 + y_1 = 12 \nonumber \]

We rewrite the objective function \(Z = 40x_1 + 30x_2\) as \(- 40x_1 - 30x_2 + Z = 0\).

After adding the slack variables, our problem reads

\[y_1 = 12 \quad y_2 = 16 \quad Z = 0 \nonumber \]

STEP 4. The most negative entry in the bottom row identifies the pivot column.

The most negative entry in the bottom row is -40; therefore the column 1 is identified.

Question Why do we choose the most negative entry in the bottom row?

Question Why do we find quotients, and why does the smallest quotient identify a row?

Question Why do we identify the pivot element?

STEP 6. Perform pivoting to make all other entries in this column zero.

We make the pivot element 1 by multiplying row 1 by 2, and we get

We no longer have negative entries in the bottom row, therefore we are finished.

Question Why are we finished when there are no negative entries in the bottom row?

Answer The answer lies in the bottom row. The bottom row corresponds to the equation:

STEP 8. Read off your answers.

The matrix reads \(x_1 = 4\), \(x_2= 8\) and \(z = 400\).

## Solve Linear Programming Problem Using Simplex Method

## Linear Programming Simplex Algorithm Calculation

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## Calculators and Converters

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## Online Calculator: Simplex Method

Solve linear programming tasks offline!

## Solution example

Transfer to the table the basic elements that we identified in the preliminary stage:

Q 1 = P 1 / x 1,2 = 600 / 1 = 600;

Q 2 = P 2 / x 2,2 = 225 / 0 = ∞;

Q 3 = P 3 / x 3,2 = 1000 / 4 = 250;

Q 4 = P 4 / x 4,2 = 150 / 2 = 75;

Q 5 = P 5 / x 5,2 = 0 / 0 = ∞;

We deduce from the basis the variable with the least positive value of Q.

P 4 = P 4 / x 4,2 = 150 / 2 = 75;

x 4,1 = x 4,1 / x 4,2 = 0 / 2 = 0;

x 4,2 = x 4,2 / x 4,2 = 2 / 2 = 1;

x 4,3 = x 4,3 / x 4,2 = 0 / 2 = 0;

x 4,4 = x 4,4 / x 4,2 = 0 / 2 = 0;

x 4,5 = x 4,5 / x 4,2 = 0 / 2 = 0;

x 4,6 = x 4,6 / x 4,2 = -1 / 2 = -0.5;

x 4,7 = x 4,7 / x 4,2 = 0 / 2 = 0;

x 4,8 = x 4,8 / x 4,2 = 1 / 2 = 0.5;

x 4,9 = x 4,9 / x 4,2 = 0 / 2 = 0;

P 1 = (P 1 * x 4,2 ) - (x 1,2 * P 4 ) / x 4,2 = ((600 * 2) - (1 * 150)) / 2 = 525;

P 2 = (P 2 * x 4,2 ) - (x 2,2 * P 4 ) / x 4,2 = ((225 * 2) - (0 * 150)) / 2 = 225;

P 3 = (P 3 * x 4,2 ) - (x 3,2 * P 4 ) / x 4,2 = ((1000 * 2) - (4 * 150)) / 2 = 700;

P 5 = (P 5 * x 4,2 ) - (x 5,2 * P 4 ) / x 4,2 = ((0 * 2) - (0 * 150)) / 2 = 0;

x 1,1 = ((x 1,1 * x 4,2 ) - (x 1,2 * x 4,1 )) / x 4,2 = ((2 * 2) - (1 * 0)) / 2 = 2;

x 1,2 = ((x 1,2 * x 4,2 ) - (x 1,2 * x 4,2 )) / x 4,2 = ((1 * 2) - (1 * 2)) / 2 = 0;

x 1,4 = ((x 1,4 * x 4,2 ) - (x 1,2 * x 4,4 )) / x 4,2 = ((0 * 2) - (1 * 0)) / 2 = 0;

x 1,5 = ((x 1,5 * x 4,2 ) - (x 1,2 * x 4,5 )) / x 4,2 = ((0 * 2) - (1 * 0)) / 2 = 0;

x 1,6 = ((x 1,6 * x 4,2 ) - (x 1,2 * x 4,6 )) / x 4,2 = ((0 * 2) - (1 * -1)) / 2 = 0.5;

x 1,7 = ((x 1,7 * x 4,2 ) - (x 1,2 * x 4,7 )) / x 4,2 = ((0 * 2) - (1 * 0)) / 2 = 0;

x 1,8 = ((x 1,8 * x 4,2 ) - (x 1,2 * x 4,8 )) / x 4,2 = ((0 * 2) - (1 * 1)) / 2 = -0.5;

x 1,9 = ((x 1,9 * x 4,2 ) - (x 1,2 * x 4,9 )) / x 4,2 = ((0 * 2) - (1 * 0)) / 2 = 0;

x 2,1 = ((x 2,1 * x 4,2 ) - (x 2,2 * x 4,1 )) / x 4,2 = ((0 * 2) - (0 * 0)) / 2 = 0;

x 2,2 = ((x 2,2 * x 4,2 ) - (x 2,2 * x 4,2 )) / x 4,2 = ((0 * 2) - (0 * 2)) / 2 = 0;

x 2,4 = ((x 2,4 * x 4,2 ) - (x 2,2 * x 4,4 )) / x 4,2 = ((1 * 2) - (0 * 0)) / 2 = 1;

x 2,5 = ((x 2,5 * x 4,2 ) - (x 2,2 * x 4,5 )) / x 4,2 = ((0 * 2) - (0 * 0)) / 2 = 0;

x 2,6 = ((x 2,6 * x 4,2 ) - (x 2,2 * x 4,6 )) / x 4,2 = ((0 * 2) - (0 * -1)) / 2 = 0;

x 2,7 = ((x 2,7 * x 4,2 ) - (x 2,2 * x 4,7 )) / x 4,2 = ((0 * 2) - (0 * 0)) / 2 = 0;

x 2,8 = ((x 2,8 * x 4,2 ) - (x 2,2 * x 4,8 )) / x 4,2 = ((0 * 2) - (0 * 1)) / 2 = 0;

x 2,9 = ((x 2,9 * x 4,2 ) - (x 2,2 * x 4,9 )) / x 4,2 = ((0 * 2) - (0 * 0)) / 2 = 0;

x 3,1 = ((x 3,1 * x 4,2 ) - (x 3,2 * x 4,1 )) / x 4,2 = ((5 * 2) - (4 * 0)) / 2 = 5;

x 3,2 = ((x 3,2 * x 4,2 ) - (x 3,2 * x 4,2 )) / x 4,2 = ((4 * 2) - (4 * 2)) / 2 = 0;

x 3,4 = ((x 3,4 * x 4,2 ) - (x 3,2 * x 4,4 )) / x 4,2 = ((0 * 2) - (4 * 0)) / 2 = 0;

x 3,5 = ((x 3,5 * x 4,2 ) - (x 3,2 * x 4,5 )) / x 4,2 = ((1 * 2) - (4 * 0)) / 2 = 1;

x 3,6 = ((x 3,6 * x 4,2 ) - (x 3,2 * x 4,6 )) / x 4,2 = ((0 * 2) - (4 * -1)) / 2 = 2;

x 3,7 = ((x 3,7 * x 4,2 ) - (x 3,2 * x 4,7 )) / x 4,2 = ((0 * 2) - (4 * 0)) / 2 = 0;

x 3,8 = ((x 3,8 * x 4,2 ) - (x 3,2 * x 4,8 )) / x 4,2 = ((0 * 2) - (4 * 1)) / 2 = -2;

x 3,9 = ((x 3,9 * x 4,2 ) - (x 3,2 * x 4,9 )) / x 4,2 = ((0 * 2) - (4 * 0)) / 2 = 0;

x 5,1 = ((x 5,1 * x 4,2 ) - (x 5,2 * x 4,1 )) / x 4,2 = ((0 * 2) - (0 * 0)) / 2 = 0;

x 5,2 = ((x 5,2 * x 4,2 ) - (x 5,2 * x 4,2 )) / x 4,2 = ((0 * 2) - (0 * 2)) / 2 = 0;

x 5,4 = ((x 5,4 * x 4,2 ) - (x 5,2 * x 4,4 )) / x 4,2 = ((0 * 2) - (0 * 0)) / 2 = 0;

x 5,5 = ((x 5,5 * x 4,2 ) - (x 5,2 * x 4,5 )) / x 4,2 = ((0 * 2) - (0 * 0)) / 2 = 0;

x 5,6 = ((x 5,6 * x 4,2 ) - (x 5,2 * x 4,6 )) / x 4,2 = ((0 * 2) - (0 * -1)) / 2 = 0;

x 5,7 = ((x 5,7 * x 4,2 ) - (x 5,2 * x 4,7 )) / x 4,2 = ((-1 * 2) - (0 * 0)) / 2 = -1;

x 5,8 = ((x 5,8 * x 4,2 ) - (x 5,2 * x 4,8 )) / x 4,2 = ((0 * 2) - (0 * 1)) / 2 = 0;

x 5,9 = ((x 5,9 * x 4,2 ) - (x 5,2 * x 4,9 )) / x 4,2 = ((1 * 2) - (0 * 0)) / 2 = 1;

Q 1 = P 1 / x 1,1 = 525 / 2 = 262.5;

Q 2 = P 2 / x 2,1 = 225 / 0 = ∞;

Q 3 = P 3 / x 3,1 = 700 / 5 = 140;

Q 4 = P 4 / x 4,1 = 75 / 0 = ∞;

Q 5 = P 5 / x 5,1 = 0 / 0 = ∞;

P 3 = P 3 / x 3,1 = 700 / 5 = 140;

x 3,1 = x 3,1 / x 3,1 = 5 / 5 = 1;

x 3,2 = x 3,2 / x 3,1 = 0 / 5 = 0;

x 3,3 = x 3,3 / x 3,1 = 0 / 5 = 0;

x 3,4 = x 3,4 / x 3,1 = 0 / 5 = 0;

x 3,5 = x 3,5 / x 3,1 = 1 / 5 = 0.2;

x 3,6 = x 3,6 / x 3,1 = 2 / 5 = 0.4;

x 3,7 = x 3,7 / x 3,1 = 0 / 5 = 0;

x 3,8 = x 3,8 / x 3,1 = -2 / 5 = -0.4;

x 3,9 = x 3,9 / x 3,1 = 0 / 5 = 0;

P 1 = (P 1 * x 3,1 ) - (x 1,1 * P 3 ) / x 3,1 = ((525 * 5) - (2 * 700)) / 5 = 245;

P 2 = (P 2 * x 3,1 ) - (x 2,1 * P 3 ) / x 3,1 = ((225 * 5) - (0 * 700)) / 5 = 225;

P 4 = (P 4 * x 3,1 ) - (x 4,1 * P 3 ) / x 3,1 = ((75 * 5) - (0 * 700)) / 5 = 75;

P 5 = (P 5 * x 3,1 ) - (x 5,1 * P 3 ) / x 3,1 = ((0 * 5) - (0 * 700)) / 5 = 0;

x 1,1 = ((x 1,1 * x 3,1 ) - (x 1,1 * x 3,1 )) / x 3,1 = ((2 * 5) - (2 * 5)) / 5 = 0;

x 1,3 = ((x 1,3 * x 3,1 ) - (x 1,1 * x 3,3 )) / x 3,1 = ((1 * 5) - (2 * 0)) / 5 = 1;

x 1,4 = ((x 1,4 * x 3,1 ) - (x 1,1 * x 3,4 )) / x 3,1 = ((0 * 5) - (2 * 0)) / 5 = 0;

x 1,5 = ((x 1,5 * x 3,1 ) - (x 1,1 * x 3,5 )) / x 3,1 = ((0 * 5) - (2 * 1)) / 5 = -0.4;

x 1,6 = ((x 1,6 * x 3,1 ) - (x 1,1 * x 3,6 )) / x 3,1 = ((0.5 * 5) - (2 * 2)) / 5 = -0.3;

x 1,7 = ((x 1,7 * x 3,1 ) - (x 1,1 * x 3,7 )) / x 3,1 = ((0 * 5) - (2 * 0)) / 5 = 0;

x 1,8 = ((x 1,8 * x 3,1 ) - (x 1,1 * x 3,8 )) / x 3,1 = ((-0.5 * 5) - (2 * -2)) / 5 = 0.3;

x 1,9 = ((x 1,9 * x 3,1 ) - (x 1,1 * x 3,9 )) / x 3,1 = ((0 * 5) - (2 * 0)) / 5 = 0;

x 2,1 = ((x 2,1 * x 3,1 ) - (x 2,1 * x 3,1 )) / x 3,1 = ((0 * 5) - (0 * 5)) / 5 = 0;

x 2,3 = ((x 2,3 * x 3,1 ) - (x 2,1 * x 3,3 )) / x 3,1 = ((0 * 5) - (0 * 0)) / 5 = 0;

x 2,4 = ((x 2,4 * x 3,1 ) - (x 2,1 * x 3,4 )) / x 3,1 = ((1 * 5) - (0 * 0)) / 5 = 1;

x 2,5 = ((x 2,5 * x 3,1 ) - (x 2,1 * x 3,5 )) / x 3,1 = ((0 * 5) - (0 * 1)) / 5 = 0;

x 2,6 = ((x 2,6 * x 3,1 ) - (x 2,1 * x 3,6 )) / x 3,1 = ((0 * 5) - (0 * 2)) / 5 = 0;

x 2,7 = ((x 2,7 * x 3,1 ) - (x 2,1 * x 3,7 )) / x 3,1 = ((0 * 5) - (0 * 0)) / 5 = 0;

x 2,8 = ((x 2,8 * x 3,1 ) - (x 2,1 * x 3,8 )) / x 3,1 = ((0 * 5) - (0 * -2)) / 5 = 0;

x 2,9 = ((x 2,9 * x 3,1 ) - (x 2,1 * x 3,9 )) / x 3,1 = ((0 * 5) - (0 * 0)) / 5 = 0;

x 4,1 = ((x 4,1 * x 3,1 ) - (x 4,1 * x 3,1 )) / x 3,1 = ((0 * 5) - (0 * 5)) / 5 = 0;

x 4,3 = ((x 4,3 * x 3,1 ) - (x 4,1 * x 3,3 )) / x 3,1 = ((0 * 5) - (0 * 0)) / 5 = 0;

x 4,4 = ((x 4,4 * x 3,1 ) - (x 4,1 * x 3,4 )) / x 3,1 = ((0 * 5) - (0 * 0)) / 5 = 0;

x 4,5 = ((x 4,5 * x 3,1 ) - (x 4,1 * x 3,5 )) / x 3,1 = ((0 * 5) - (0 * 1)) / 5 = 0;

x 4,6 = ((x 4,6 * x 3,1 ) - (x 4,1 * x 3,6 )) / x 3,1 = ((-0.5 * 5) - (0 * 2)) / 5 = -0.5;

x 4,7 = ((x 4,7 * x 3,1 ) - (x 4,1 * x 3,7 )) / x 3,1 = ((0 * 5) - (0 * 0)) / 5 = 0;

x 4,8 = ((x 4,8 * x 3,1 ) - (x 4,1 * x 3,8 )) / x 3,1 = ((0.5 * 5) - (0 * -2)) / 5 = 0.5;

x 4,9 = ((x 4,9 * x 3,1 ) - (x 4,1 * x 3,9 )) / x 3,1 = ((0 * 5) - (0 * 0)) / 5 = 0;

x 5,1 = ((x 5,1 * x 3,1 ) - (x 5,1 * x 3,1 )) / x 3,1 = ((0 * 5) - (0 * 5)) / 5 = 0;

x 5,3 = ((x 5,3 * x 3,1 ) - (x 5,1 * x 3,3 )) / x 3,1 = ((0 * 5) - (0 * 0)) / 5 = 0;

x 5,4 = ((x 5,4 * x 3,1 ) - (x 5,1 * x 3,4 )) / x 3,1 = ((0 * 5) - (0 * 0)) / 5 = 0;

x 5,5 = ((x 5,5 * x 3,1 ) - (x 5,1 * x 3,5 )) / x 3,1 = ((0 * 5) - (0 * 1)) / 5 = 0;

x 5,6 = ((x 5,6 * x 3,1 ) - (x 5,1 * x 3,6 )) / x 3,1 = ((0 * 5) - (0 * 2)) / 5 = 0;

x 5,7 = ((x 5,7 * x 3,1 ) - (x 5,1 * x 3,7 )) / x 3,1 = ((-1 * 5) - (0 * 0)) / 5 = -1;

x 5,8 = ((x 5,8 * x 3,1 ) - (x 5,1 * x 3,8 )) / x 3,1 = ((0 * 5) - (0 * -2)) / 5 = 0;

x 5,9 = ((x 5,9 * x 3,1 ) - (x 5,1 * x 3,9 )) / x 3,1 = ((1 * 5) - (0 * 0)) / 5 = 1;

Q 1 = P 1 / x 1,6 = 245 / -0.3 = -816.67;

Q 2 = P 2 / x 2,6 = 225 / 0 = ∞;

Q 3 = P 3 / x 3,6 = 140 / 0.4 = 350;

Q 4 = P 4 / x 4,6 = 75 / -0.5 = -150;

Q 5 = P 5 / x 5,6 = 0 / 0 = ∞;

P 3 = P 3 / x 3,6 = 140 / 0.4 = 350;

x 3,1 = x 3,1 / x 3,6 = 1 / 0.4 = 2.5;

x 3,2 = x 3,2 / x 3,6 = 0 / 0.4 = 0;

x 3,3 = x 3,3 / x 3,6 = 0 / 0.4 = 0;

x 3,4 = x 3,4 / x 3,6 = 0 / 0.4 = 0;

x 3,5 = x 3,5 / x 3,6 = 0.2 / 0.4 = 0.5;

x 3,6 = x 3,6 / x 3,6 = 0.4 / 0.4 = 1;

x 3,7 = x 3,7 / x 3,6 = 0 / 0.4 = 0;

x 3,8 = x 3,8 / x 3,6 = -0.4 / 0.4 = -1;

x 3,9 = x 3,9 / x 3,6 = 0 / 0.4 = 0;

P 1 = (P 1 * x 3,6 ) - (x 1,6 * P 3 ) / x 3,6 = ((245 * 0.4) - (-0.3 * 140)) / 0.4 = 350;

P 2 = (P 2 * x 3,6 ) - (x 2,6 * P 3 ) / x 3,6 = ((225 * 0.4) - (0 * 140)) / 0.4 = 225;

P 4 = (P 4 * x 3,6 ) - (x 4,6 * P 3 ) / x 3,6 = ((75 * 0.4) - (-0.5 * 140)) / 0.4 = 250;

P 5 = (P 5 * x 3,6 ) - (x 5,6 * P 3 ) / x 3,6 = ((0 * 0.4) - (0 * 140)) / 0.4 = 0;

x 1,1 = ((x 1,1 * x 3,6 ) - (x 1,6 * x 3,1 )) / x 3,6 = ((0 * 0.4) - (-0.3 * 1)) / 0.4 = 0.75;

x 1,2 = ((x 1,2 * x 3,6 ) - (x 1,6 * x 3,2 )) / x 3,6 = ((0 * 0.4) - (-0.3 * 0)) / 0.4 = 0;

x 1,3 = ((x 1,3 * x 3,6 ) - (x 1,6 * x 3,3 )) / x 3,6 = ((1 * 0.4) - (-0.3 * 0)) / 0.4 = 1;

x 1,4 = ((x 1,4 * x 3,6 ) - (x 1,6 * x 3,4 )) / x 3,6 = ((0 * 0.4) - (-0.3 * 0)) / 0.4 = 0;

x 1,5 = ((x 1,5 * x 3,6 ) - (x 1,6 * x 3,5 )) / x 3,6 = ((-0.4 * 0.4) - (-0.3 * 0.2)) / 0.4 = -0.25;

x 1,6 = ((x 1,6 * x 3,6 ) - (x 1,6 * x 3,6 )) / x 3,6 = ((-0.3 * 0.4) - (-0.3 * 0.4)) / 0.4 = 0;

x 1,8 = ((x 1,8 * x 3,6 ) - (x 1,6 * x 3,8 )) / x 3,6 = ((0.3 * 0.4) - (-0.3 * -0.4)) / 0.4 = 0;

x 1,9 = ((x 1,9 * x 3,6 ) - (x 1,6 * x 3,9 )) / x 3,6 = ((0 * 0.4) - (-0.3 * 0)) / 0.4 = 0;

x 2,1 = ((x 2,1 * x 3,6 ) - (x 2,6 * x 3,1 )) / x 3,6 = ((0 * 0.4) - (0 * 1)) / 0.4 = 0;

x 2,2 = ((x 2,2 * x 3,6 ) - (x 2,6 * x 3,2 )) / x 3,6 = ((0 * 0.4) - (0 * 0)) / 0.4 = 0;

x 2,3 = ((x 2,3 * x 3,6 ) - (x 2,6 * x 3,3 )) / x 3,6 = ((0 * 0.4) - (0 * 0)) / 0.4 = 0;

x 2,4 = ((x 2,4 * x 3,6 ) - (x 2,6 * x 3,4 )) / x 3,6 = ((1 * 0.4) - (0 * 0)) / 0.4 = 1;

x 2,5 = ((x 2,5 * x 3,6 ) - (x 2,6 * x 3,5 )) / x 3,6 = ((0 * 0.4) - (0 * 0.2)) / 0.4 = 0;

x 2,6 = ((x 2,6 * x 3,6 ) - (x 2,6 * x 3,6 )) / x 3,6 = ((0 * 0.4) - (0 * 0.4)) / 0.4 = 0;

x 2,8 = ((x 2,8 * x 3,6 ) - (x 2,6 * x 3,8 )) / x 3,6 = ((0 * 0.4) - (0 * -0.4)) / 0.4 = 0;

x 2,9 = ((x 2,9 * x 3,6 ) - (x 2,6 * x 3,9 )) / x 3,6 = ((0 * 0.4) - (0 * 0)) / 0.4 = 0;

x 4,1 = ((x 4,1 * x 3,6 ) - (x 4,6 * x 3,1 )) / x 3,6 = ((0 * 0.4) - (-0.5 * 1)) / 0.4 = 1.25;

x 4,2 = ((x 4,2 * x 3,6 ) - (x 4,6 * x 3,2 )) / x 3,6 = ((1 * 0.4) - (-0.5 * 0)) / 0.4 = 1;

x 4,3 = ((x 4,3 * x 3,6 ) - (x 4,6 * x 3,3 )) / x 3,6 = ((0 * 0.4) - (-0.5 * 0)) / 0.4 = 0;

x 4,4 = ((x 4,4 * x 3,6 ) - (x 4,6 * x 3,4 )) / x 3,6 = ((0 * 0.4) - (-0.5 * 0)) / 0.4 = 0;

x 4,5 = ((x 4,5 * x 3,6 ) - (x 4,6 * x 3,5 )) / x 3,6 = ((0 * 0.4) - (-0.5 * 0.2)) / 0.4 = 0.25;

x 4,6 = ((x 4,6 * x 3,6 ) - (x 4,6 * x 3,6 )) / x 3,6 = ((-0.5 * 0.4) - (-0.5 * 0.4)) / 0.4 = 0;

x 4,8 = ((x 4,8 * x 3,6 ) - (x 4,6 * x 3,8 )) / x 3,6 = ((0.5 * 0.4) - (-0.5 * -0.4)) / 0.4 = 0;

x 4,9 = ((x 4,9 * x 3,6 ) - (x 4,6 * x 3,9 )) / x 3,6 = ((0 * 0.4) - (-0.5 * 0)) / 0.4 = 0;

x 5,1 = ((x 5,1 * x 3,6 ) - (x 5,6 * x 3,1 )) / x 3,6 = ((0 * 0.4) - (0 * 1)) / 0.4 = 0;

x 5,2 = ((x 5,2 * x 3,6 ) - (x 5,6 * x 3,2 )) / x 3,6 = ((0 * 0.4) - (0 * 0)) / 0.4 = 0;

x 5,3 = ((x 5,3 * x 3,6 ) - (x 5,6 * x 3,3 )) / x 3,6 = ((0 * 0.4) - (0 * 0)) / 0.4 = 0;

x 5,4 = ((x 5,4 * x 3,6 ) - (x 5,6 * x 3,4 )) / x 3,6 = ((0 * 0.4) - (0 * 0)) / 0.4 = 0;

x 5,5 = ((x 5,5 * x 3,6 ) - (x 5,6 * x 3,5 )) / x 3,6 = ((0 * 0.4) - (0 * 0.2)) / 0.4 = 0;

x 5,6 = ((x 5,6 * x 3,6 ) - (x 5,6 * x 3,6 )) / x 3,6 = ((0 * 0.4) - (0 * 0.4)) / 0.4 = 0;

x 5,8 = ((x 5,8 * x 3,6 ) - (x 5,6 * x 3,8 )) / x 3,6 = ((0 * 0.4) - (0 * -0.4)) / 0.4 = 0;

x 5,9 = ((x 5,9 * x 3,6 ) - (x 5,6 * x 3,9 )) / x 3,6 = ((1 * 0.4) - (0 * 0)) / 0.4 = 1;

## Simplex Method Calculator

Solve optimization problems using the simplex method.

The problem in the canonical form can be written as follows:

Add variables (slack or surplus) to turn all the inequalities into equalities:

Write down the simplex tableau:

The leaving variable is $$$ S_{1} $$$ , because it has the smallest ratio.

Divide row $$$ 1 $$$ by $$$ 2 $$$ : $$$ R_{1} = \frac{R_{1}}{2} $$$ .

Add row $$$ 2 $$$ multiplied by $$$ 4 $$$ to row $$$ 1 $$$ : $$$ R_{1} = R_{1} + 4 R_{2} $$$ .

Subtract row $$$ 2 $$$ from row $$$ 3 $$$ : $$$ R_{3} = R_{3} - R_{2} $$$ .

The leaving variable is $$$ S_{2} $$$ , because it has the smallest ratio.

Multiply row $$$ 2 $$$ by $$$ 2 $$$ : $$$ R_{2} = 2 R_{2} $$$ .

Add row $$$ 3 $$$ to row $$$ 1 $$$ : $$$ R_{1} = R_{1} + R_{3} $$$ .

None of the Z-row coefficients are negative.

$$$ Z = 20 $$$ A is achieved at $$$ \left(x_{1}, x_{2}\right) = \left(4, 2\right) $$$ A .

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## Example (part 1): Simplex method

Solve using the Simplex method the following problem:

A change is made to the variable naming, establishing the following correspondences:

Otherwise, the following steps are executed iteratively.

The column of the input base variable is called pivot column (in green color).

In this example: 18/2 [=9] , 42/2 [=21] and 24/3 [=8]

The intersection of pivot column and pivot row marks the pivot value , in this example, 3.

The new coefficients of the tableau are calculated as follows:

- In the pivot row each new value is calculated as: New value = Previous value / Pivot
- In the other rows each new value is calculated as: New value = Previous value - (Previous value in pivot column * New value in pivot row)

Calculations for P 4 row are shown below:

The tableau corresponding to this second iteration is:

- 6.1. The input base variable is X 2 (P 2 ), since it is the variable that corresponds to the column where the coefficient is -1.
- 6.2. To calculate the output base variable, the constant terms P 0 column) are divided by the terms of the new pivot column: 2 / 1/3 [=6] , 26 / 7/3 [=78/7] and 8 / 1/3 [=24]. As the lesser positive quotient is 6, the output base variable is X 3 (P 3 ).
- 6.3. The new pivot is 1/3.
- 6.1. The input base variable is X 5 (P 5 ), since it is the variable that corresponds to the column where the coefficient is -1.
- 6.2. To calculate the output base variable, the constant terms (P 0 ) are divided by the terms of the new pivot column: 6/(-2) [=-3] , 12/4 [=3] , and 6/1 [=6]. In this iteration, the output base variable is X 4 (P 4 ).
- 6.3. The new pivot is 4.

Undoing the name change gives x = 3 and y = 12.

Copyright ©2006-2023. All rights reserved.

Developed by: Daniel Izquierdo Granja Juan José Ruiz Ruiz

English translation by: Luciano Miguel Tobaria

French translation by: Ester Rute Ruiz

Portuguese translation by: Rosane Bujes

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