Using Spreadsheets for Linear Programming with The Simplex Method A sample problem By: Jeffrey Bivin Lake Zurich High School Last Updated: October 11, 2005

A landscaper can buy three types of 100-pound bags of fertilizer, type A, type B, and type C. Each 100- pound bag of type A fertilizer costs $20 and contains 40 pounds of nitrogen, 30 pounds of phosphoric acid, and 10 pounds of potash. Each 100-pound bag of type B fertilizer costs $30 and contains 20 pounds of nitrogen, 20 pounds of phosphoric acid, and 55 pounds of potash. Each 100-pound bag of type C fertilizer costs $20 and contains no nitrogen, 30 pounds of phosphoric acid, and 40 pounds of potash. The landscaper requires 4000 pounds of nitrogen pounds of phosphoric acid, and 2000 pounds of potash. How many bags of each type of fertilizer should the landscaper buy in order to minimize the cost? Also, find the minimum cost. Jeff Bivin -- LZHS

Analyzing the problem Nitrogen Phos. Acidpotash Type Ax1x1 $ Type Bx2x2 $ Type Cx3x3 $ needs Jeff Bivin -- LZHS

Define the Constraints 40x x 2 > x x x 3 > x x x 3 > 2000 x1 > 0 x2 > 0 x3 > 0 Jeff Bivin -- LZHS

Function to minimize z = 20x x x 3 Jeff Bivin -- LZHS

Initial Tableau x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Jeff Bivin -- LZHS

Initial Tableau Analysis Jeff Bivin -- LZHS

A negative basic variable in s 1 leads us to use column x 1 as the pivot column. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Jeff Bivin -- LZHS

A negative basic variable in s 1 leads us to use column x 1 as the pivot column. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues s 1 is a basic variable because it has only one non-zero element in the column Jeff Bivin -- LZHS

A negative basic variable in s 1 leads us to use column x 1 as the pivot column. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues We use Column x 1 as the pivot column because the 40 in column x 1 is the left most positive number in the row with the -1 value (negative basic variable). Jeff Bivin -- LZHS

Divide the value column by the pivot column. This adds an additional column at the right. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Jeff Bivin -- LZHS

is the lowest non-negative quotient. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Jeff Bivin -- LZHS

is the lowest non-negative quotient. Therefore, 30 is determined to be the pivot point. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Jeff Bivin -- LZHS

Determine the row operations: row1 = 3 x row1 – 4 x row2 row3 = 3 x row3 – row2 row4 = 3 x row4 – 2 x row2 x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Jeff Bivin -- LZHS

Perform the row operations to determine Tableau 2 x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues r1=3r1-4r r r3=3r3-r r4=3r4-2r Jeff Bivin -- LZHS

2 nd Tableau Analysis Jeff Bivin -- LZHS

A negative basic variable in s 1 leads us to use column s 2 as the pivot column. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Jeff Bivin -- LZHS

Divide the value column by the pivot column. This adds an additional column at the right. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Jeff Bivin -- LZHS

1000 is the lowest non-negative quotient. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Jeff Bivin -- LZHS

1000 is the lowest non-negative quotient. Therefore, 4 is determined to be the pivot point. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Jeff Bivin -- LZHS

Determine the row operations: row2 = 4 x row2 + row1 row3 = 4 x row3 – row1 row4 = 2 x row4 – row1 x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Jeff Bivin -- LZHS

Perform the row operations to determine Tableau 3 x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues r r2=4r2+r r3=4r3-r r4=2r4-r Jeff Bivin -- LZHS

3 rd Tableau Analysis Jeff Bivin -- LZHS

A negative basic variable in s 3 leads us to use column x 2 as the pivot column. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues r r2=4r2+r r3=4r3-r r4=2r4-r Jeff Bivin -- LZHS

Divide the value column by the pivot column. This adds an additional column at the right. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues r r2=4r2+r r3=4r3-r r4=2r4-r Jeff Bivin -- LZHS

is the lowest non-negative quotient. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues r r2=4r2+r r3=4r3-r r4=2r4-r Jeff Bivin -- LZHS

is the lowest non-negative quotient. Therefore, 30 is determined to be the pivot point. x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues r r2=4r2+r r3=4r3-r r4=2r4-r Jeff Bivin -- LZHS

Determine the row operations: row1 = 30 x row1 + row3 row2 = 10 x row2 – row3 row4 = 5 x row4 – row3 x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Jeff Bivin -- LZHS

Perform the row operations to determine Tableau 4 x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues r1=30r1+r r2=10r2-r r r4=52r4-r Jeff Bivin -- LZHS

4 th Tableau Analysis Jeff Bivin -- LZHS

Now we can determine the solutions x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Basic Variables x 1, x 2, and s 2 are all positive and the bottom row has all positive variable coefficients Jeff Bivin -- LZHS

Determining the solutions: from row 1: s 2 = / 120 = 1100 from row 2: x 1 = / 1200 = 90 from row 3: x 2 = / 600 = 20 from row 4: z = -( / 30) = 2400 also: x 3 = s 1 = s 3 = 0 x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues r1=30r1+r r2=10r2-r r r4=52r4-r Jeff Bivin -- LZHS

Results x 1 = 90 x 2 = 20 x 3 = 0 s 1 = 0 s 2 = 1100 s 3 = 0 z = 2400 Jeff Bivin -- LZHS

Process Complete Answer the Question The minimum cost is $2400 when you order 90 bags of type A fertilizer 20 bags of type B fertilizer 0 bags of type C fertilizer Jeff Bivin -- LZHS

A Quick Recap Jeff Bivin -- LZHS

Initial Tableau x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues Jeff Bivin -- LZHS

Perform the row operations to determine Tableau 2 x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues r1=3r1-4r r r3=3r3-r r4=3r4-2r Jeff Bivin -- LZHS

Perform the row operations to determine Tableau 3 x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues r r2=4r2+r r3=4r3-r r4=2r4-r Jeff Bivin -- LZHS

Perform the row operations to determine Tableau 4 x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues r1=30r1+r r2=10r2-r r r4=52r4-r Jeff Bivin -- LZHS

x1x1 x2x2 x3x3 s1s1 s2s2 s3s3 zValues r1=30r1+r r2=10r2-r r r4=52r4-r Determine the solutions from the 4 th Tableau from row 1: s 2 = / 120 = 1100 from row 2: x 1 = / 1200 = 90 from row 3: x 2 = / 600 = 20 from row 4: z = -( / 30) = 2400 also: x 3 = s 1 = s 3 = 0 Jeff Bivin -- LZHS

Process Complete Answer the Question The minimum cost is $2400 when you order 90 bags of type A fertilizer 20 bags of type B fertilizer 0 bags of type C fertilizer Jeff Bivin -- LZHS