Question

The Williams Trunk Company makes trunks for the military, for commercial use, and for decorative pieces. Each military trunk requires 4 hours for assembly, 1 hour for finishing, and 0.1 hour for packaging. Each commercial trunk requires 3 hours for assembly, 2 hours for finishing, and 0.2 hour for packaging. Each decorative trunk requires 2 hours for assembly, 4 hours for finishing, and 0.3 hour for packaging. The profit on each trunk is $18 for military, $21 for commercial, and $27 for decorative. If 4900 hours are available for assembly work, 2200 for finishing, and 210 for packaging, how many of each type of trunk should be made to maximize profit?

Answer 1

Dear student, please refer to the question below. Question:

The Williams Trunk Company makes trunks for the military, for commercial use, and for decorative pieces. Each military trunk requires 4 hours for assembly, 1 hour for finishing, and 0.1 hour for packaging. Each commercial trunk requires 3 hours for assembly, 2 hours for finishing, and 0.2 hour for packaging. Each decorative trunk requires 2 hours for assembly, 4 hours for finishing, and 0.3 hour for packaging. The profit on each trunk is $6 for military, $7 for commercial, and $9 for decorative. If 4900 hours are available for assembly work, 2200 for finishing, and 210 for packaging, how many of each type of trunk should be made to maximize profit?

Ans:

Consider that a company produces trunks for the military, trunks for commercial use and trunks for decorative pieces.

Refer to the information in problem 42 on page 291, the maximization problem can be stated as follows

Subject to

Introduce a slack variable for each equation.

Also, include the objective function with the constraints.

Therefore,

Suppose, the slack variables are

Therefore, the corresponding system of equations is

Therefore, the initial tableau is

The initial basic feasible solution is

Since there are negative entries in the last row, the solution is not optimal.

To find the pivot element, choose the column with the most negative entry.

Therefore, choose the third column as the pivot column.

To select the row, divide each constant above the line in the last column by the corresponding entries in the pivot column.

Write the ratios to the right of the tableau

Since the ratio 550 is least, the element 2200 determines the pivot row.

Therefore, the element 4 in the second row is the pivot element.

Use row operations given below to convert column 3 to unit column.

The transformed tableau is

Since there are negative entries in the last row, the solution is not optimal.

To find the pivot element, choose the column with the most negative entry.

Therefore, choose the column 1 as the pivot column.

To select the row, divide each constant above the line in the last column by the corresponding entries in the pivot column.

Write the ratios to the right of the tableau

Since the ratio 1085.7 is least positive, the element 3800 determines the pivot row.

Therefore, 3.5 is the pivot element.

Use row operations given below to convert column 1 to unit column.

The transformed tableau is

Since there are negative entries in the last row, the solution is not optimal.

To find the pivot element, choose the column with the most negative entry.

Therefore, choose the second column as the pivot column.

To select the row, divide each constant above the line in the last column by the corresponding entries in the pivot column.

Write the ratios to the right of the tableau

Since the ratio 500 is least, the element 17.8571 determines the pivot row.

Therefore, the element 0.03571 in the third row is the pivot element.

Use row operations given below to convert column 2 to unit column.

The transformed tableau is

Since there are all positive entries in the last row, therefore the solution is optimal.

Hence, the basic feasible solution is

And the maximum profit is

Hence, the company should produce about trunks for the military, trunks for the commercial use and trunks for the decorative pieces and the maximum profit earned is.