Explanation Given Information: The provided function is, f = 3 x + 5 y and the function is subjected to x + y ≥ 19 − x + y ≥ 1 − x + 10 y ≥ 190 . Formula used: The following steps to solve the maximization of a function with the help of simplex method. Step 1. Express all the constraints as ≤ constraints by multiply the inequality with ≥ constraint to − 1 . Step 2. Write every inequalities as an equation where positive constant is on the other side with distinct slack variables. Step 3. Set up the simplex matrix. Step 4. Find the pivot entry. Step 5. 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 6. For the following conditions: (a) Return back to step 4 when there is negative quantity. (b) The values are now optimum value when all the quantity is positive or zero. Step 7. Set the non-basic variables equal to zero, note the values of the basic variables and the objective function from the rows of the matrix. Calculation: Consider the provided function, f = 3 x + 5 y and inequalities x + y ≥ 19 − x + y ≥ 1 − x + 10 y ≥ 190 . Firstly, express all the constraints as ≤ constraints. So, multiply the inequality with ≥ constraint to − 1 as, − x − y ≤ − 19 x − y ≤ − 1 x − 10 y ≤ − 190 Now with the help of simplex method introduce the slack variables and rewriting the objective function, − x − y + s 1 = − 19 x − y + s 2 = − 1 x − 10 y + s 3 = − 190 − 3 x − 5 y + f = 0 Write the above equation as a matrix with the objective function in the last row, x y s 1 s 2 s 3 f [ − 1 − 1 1 0 0 0 − 19 1 ( − 1 ) 0 1 0 0 − 1 1 − 10 0 0 1 0 − 190 − 3 − 5 0 0 0 1 0 ] Now, the most negative value in the last row is − 5 . So, second column is the pivot column. Now divide the constants of pivot column from the last row as, − 19 − 1 = 19 − 1 − 1 = 1 − 190 − 10 = 19 Here 1 is the smallest non-negative quotient. So, − 1 is the pivot entry. The pivot entry is shown in the matrix inside brackets. Multiply the row 2 with − 1 to get pivot entry as 1, [ − 1 − 1 1 0 0 0 − 19 − 1 ( 1 ) 0 − 1 0 0 1 1 − 10 0 0 1 0 − 190 − 3 − 5 0 0 0 1 0 ] Now use operations with other rows to make other entries in the pivot column equal to zero. So, use the following operations, R 2 + R 1 → R 1 − 10 R 2 + R 3 → R 3 5 R 2 + R 4 → R 4 The resultant matrix is obtained as, [ ( − 2 ) 0 1 − 1 0 0 − 18 − 1 1 0 − 1 0 0 1 9 0 0 10 1 0 180 − 8 0 0 − 5 0 1 5 ] Now, the most negative value in the last row is 8 . So, first column is the pivot column. Now divide the constants of pivot column from the last row as, − 18 − 2 = 9 1 − 1 = − 1 180 9 = 20 Here 9 is the smallest non-negative quotient

Mathematical Applications for the Management, Life, and Social Sciences

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ISBN: 9781337625340

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 4, Problem 26RE

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To calculate: The maximization of function f=3x+5y by simplex method assume all variables are non-negative and the function is subjected to, x+y≥19−x+y≥1−x+10y≥190.

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Chapter 4, Problem 25RE

Chapter 4, Problem 27RE

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The maximization of function f = 3 x + 5 y by simplex method assume all variables are non-negative and the function is subjected to, x + y ≥ 19 − x + y ≥ 1 − x + 10 y ≥ 190 .

Solution Summary: The author explains how to solve the maximization of a function with the help of simplex method.

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To calculate: The maximization of function f=3x+5y by simplex method assume all variables are non-negative and the function is subjected to, x+y≥19−x+y≥1−x+10y≥190.

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