   Chapter 4, Problem 11T Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Solutions

Chapter
Section Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

Maximize f = 60 x + 48 y + 35 z subject to 8 x + 6 y + z ≤ 108 4 x + 2 y + 1.5 z ≤ 50 2 x + 1.5 y + 0.5 z ≤ 40 z ≤ 16 x ≥ 0 , y ≥ 0 , z ≥ 0

To determine

To calculate: The solution of the linear programming problem

Maximize f=60x+48y+35z

Subject to

8x+6y+z1084x+2y+1.5z502x  +1.5y +0.5z    40                       z        16

x0,y0,z0

Explanation

Given Information:

The provided problem is:

Maximize f=60x+48y+35z

Subject to

8x+6y+z1084x+2y+1.5z502x  +1.5y +0.5z    40                       z        16

x0,y0,z0

Formula used:

Following steps of the simplex method,

Step 1: Use slack variables and write the inequalities in equation form.

Step 2: Write the equations in an augmented matrix.

Step 3: Choose the smallest negative number on the left side of the bottom row and pivot the column.

Step 4: Select the pivot which is the smallest of the test ratios ab, where a is the entry in the right most column and b is the corresponding entry in the pivot column.

Step 5: Clear the pivot column using the pivot and label the pivot row with the label of the pivot column.

Step 6: Repeat the above steps till all the entries in the bottom row are non-negative.

Use row operations to clear the column of the augmented matrix.

Calculation:

Consider the maximisation problem,

Maximize f=60x+48y+35z

Subject to

8x+6y+z1084x+2y+1.5z502x  +1.5y +0.5z    40                       z        16

x0,y0,z0

Step 1: Use slack variables and write the inequalities in equation form.

Add the slack variables s1,s2,s3 and s4 as below,

8x+6y+z        + s1=1084x+2y+1.5z  +s2=502x  +1.5y +0.5z +s3    =   40                       z     +s4   =   16

Step 2: Write the equations in an augmented matrix.

The equations are,

8x+6y+z        + s1=1084x+2y+1.5z  +s2=502x  +1.5y +0.5z +s3    =   40                       z     +s4   =   16

And,

60x48y35z+f=0

Write the equations in augmented matrix,

[xyzs1s2s3s4f86110000108421.5010005021.50.500100400010001016604835000010]

Step 3: Choose the smallest negative number on the left side of the bottom row and pivot the column.

Since, the smallest negative number on the left side of the bottom row is 60.

Thus, choose the left most entry and pivot the x column.

Step 4: Select the pivot which is the smallest of the test ratios ab, where a is the entry in the right most column and b is the corresponding entry in the pivot column.

Since, the test ratios are neither 0 nor negative.

Thus, the test ratios are 1088=13.5, 504=12.5, 402=20 and 160=

Since, 12.5 is smallest.

Thus, the pivot is 4 in column x.

[xyzs1s2s3s4f86110000108421.5010005021.50.500100400010001016604835000010]

Step 5: Clear the pivot column using the pivot and label the pivot row with the label of the pivot column.

Since, the pivot is s2 in column x and replace s2 with x.

Thus, x is the active variable.

Use row operations to clear the pivot column of the augmented matrix.

Use the row operations 14R2R1 then,

8R2+R1R12R2+R3R360R2+R5R5

[xyzs1s2s3s4f02212000810.50.37500

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