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11th Edition

Ronald J. Harshbarger + 1 other

Publisher: Cengage Learning

ISBN: 9781305108042

Chapter 4, Problem 30RE

Textbook Problem

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In Problems 28-35, use the simplex method. Assume that all variables are nonnegative.

To determine

**To calculate:** The minimization of function

**Given Information:**

The provided function is,

**Formula used:**

The following steps to solve the minimization of a function with the help of simplex method.

Step 1. Form a dual matrix by writing the matrix *A* that does not contain any slack variable and the last row have only positive coefficients.

Step 2. Transpose matrix *A* to yield matrix *B.*

Step 3. Rename the variable and function of a dual maximization.

Step 4. Solve this problem with simplex method by introducing slack variables.

Step 5. Use different slack variables to write each constraint inequality as an equation, with positive constants on the right side.

Step 6. Set up the simplex matrix.

Step 7. Find the pivot entry.

Step 8. Use row operations with only the row containing the pivot entry to make the pivot entry 1 and all other entries in the pivot column zero.

Step 9. For the following conditions:

(a) Return back to step 7 when there is negative quantity.

(b) The values are now optimum value when all the quantity is positive or zero.

Step 10. Set the non-basic variables equal to zero.

**Calculation:**

Consider the function,

The duality of a minimization function is a maximization function.

Make the dual form of the provided minimization function.

Now, transpose matrix *A* to yield matrix *B* as,

Now write the dual maximization function, rename the variables and the function.

Maximize

Now solve the above function with simplex method by introduce the slack variables and rewrite the objective function,

Write the above function in a matrix with the objective function in the last row,

Here, the most negative value in the last row is

Now divide the constants of pivot column from the last row as,

Mathematical Applications for the Management, Life, and Social Sciences

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