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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
Publisher: Cengage Learning
ISBN: 9781305108042

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BuyFindarrow_forward

Mathematical Applications for the ...

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.

30.  Minimize  g = 25 y 1 +   10 y 2 +   4 y 3  subject to 7.5 y 1 +   4.5 y 2 +       2 y 3     650 6.5 y 1 +     3 y 2 +   1.5 y 3 400 y 1 + 1.5 y 2 + 0.5 y 3 200

To determine

To calculate: The minimization of function g=25y1+10y2+4y3 by simplex method assume all variables are non-negative and the function is subjected to, 7.5y1+4.5y2+2y36506.5y1+3y2+1.5y3400y1+1.5y2+0.5y3200.

Explanation of Solution

Given Information:

The provided function is, g=25y1+10y2+4y3 and the inequalities are, 7.5y1+4.5y2+2y36506.5y1+3y2+1.5y3400y1+1.5y2+0.5y3200.

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, g=25y1+10y2+4y3 and inequalities 7.5y1+4.5y2+2y36506.5y1+3y2+1.5y3400y1+1.5y2+0.5y3200.

The duality of a minimization function is a maximization function.

Make the dual form of the provided minimization function.

A=[7.54.526506.531.540011.50.520025104g]

Now, transpose matrix A to yield matrix B as,

B=[7.56.51254.531.51021.50.54650400200g]

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

Maximize f=650x1+400x2+200x3 subject to:

7.5x1+6.5x2+x3254.5x1+3x2+1.5x3102x1+1.5x2+0.5x34

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

7.5x1+6.5x2+x3+s1=254.5x1+3x2+1.5x3+s2=102x1+1.5x2+0.5x3+s3=4650x1400x2200x3+f=0

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

    x1          x2         x3     s1  s2   s3   f[7.56.511000254.531.5010010(2)1.50.50010465040020000010]

Here, the most negative value in the last row is 650. So, first column is the pivot column.

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

257.5=3.33104.5=2

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Chapter 4 Solutions

Mathematical Applications for the Management, Life, and Social Sciences
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