Explanation Given Information: The given function is, f = 4 x + 3 y And the given inequalities are, − x + y ≥ 1 x + y ≤ 45 10 x + y ≥ 45 Formula used: The following method is used, Step 1: If the problem is to minimize f , then maximize – f . Step 2: Express all the constraints as ≤ constraints by multiplying the inequality with ≥ constraint to − 1 . Use slack variables and form the simplex matrix. Step 3: In the simplex matrix, find the upper portion of the last column for any negative entries. If there are no negative entries, apply the simplex method. If there are negative entries then go to step 4. Step 4: When there is a negative value in the upper portion of the last column, proceed as follows, Select any negative entry in the same row and use its column as the pivot column. In the pivot column, compute all quotients for the entries above the line and determine the pivot from the smallest positive quotient. After completing the pivot operations, return to step 3. Step 5: When all values are positive in the upper portion of the last column, then simplex method can be applied. 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 function f = 4 x + 3 y subject to: − x + y ≥ 1 x + y ≤ 45 10 x + y ≥ 45 Because of the mixed constraints, maximize − f = − 4 x − 3 y subject to, x − y ≤ − 1 x + y ≤ 45 − 10 x − y ≤ − 45 The simplex matrix for this problem is, x y s 1 s 2 s 3 f [ 1 ( − 1 ) 1 0 0 0 − 1 1 1 0 1 0 0 45 − 10 − 1 0 0 1 0 − 45 4 3 0 0 0 1 0 ] In this case, there are two negative entries in the upper portion of the last column. So, start with row 1. In row 1, there is a negative entry in the column 2. So, choose pivot column as column 2. Now calculate the pivot entry by dividing the constants of pivot column from the last row as, − 1 − 1 = 1 45 1 = 45 − 45 − 1 = 45 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 1 with − 1 to get pivot entry as 1, [ − 1 ( 1 ) − 1 0 0 0 1 1 1 0 1 0 0 45 − 10 − 1 0 0 1 0 − 45 4 3 0 0 0 1 0 ] Now use operations with other rows to make other entries in the pivot column equal to zero

Mathematical Applications for the Management, Life, and Social Sciences

11th Edition

ISBN: 9781305108042

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 4, Problem 32RE

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To calculate: The minimum value of function f=4x+3y subjected to constraints −x+y≥1,.x+y≤45,10x+y≥45.

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The minimum value of function f = 4 x + 3 y subjected to constraints − x + y ≥ 1 , . x + y ≤ 45 , 10 x + y ≥ 45 .

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To calculate: The minimum value of function f=4x+3y subjected to constraints −x+y≥1,.x+y≤45,10x+y≥45.

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