Explanation Given Information: The given function is, f = 88 x 1 + 86 x 2 + 100 x 3 + 100 x 4 And the given inequalities are, 3 x 1 + 2 x 2 + 2 x 3 + 5 x 4 ≤ 200 2 x 1 + 2 x 2 + 4 x 3 + 5 x 4 ≤ 100 x 1 + x 2 + x 3 + x 4 ≤ 200 x 1 ≤ 40 Formula used: The following method is used, Step 1: Use different slack variables to write each constraint inequality as equation, with positive constants on the right side. Step 2: Create the simplex matrix. Step 3: Find the pivot entry. Step 4: 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 5: If there is negative indicator then return to step 3. If all indicators are positive or zero, an optimum value has obtained. Step 6: 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 = 88 x 1 + 86 x 2 + 100 x 3 + 100 x 4 subject to: 3 x 1 + 2 x 2 + 2 x 3 + 5 x 4 ≤ 200 2 x 1 + 2 x 2 + 4 x 3 + 5 x 4 ≤ 100 x 1 + x 2 + x 3 + x 4 ≤ 200 x 1 ≤ 40 Introducing the slack variables and rewriting the objective function, 3 x 1 + 2 x 2 + 2 x 3 + 5 x 4 + s 1 = 200 2 x 1 + 2 x 2 + 4 x 3 + 5 x 4 + s 2 = 100 x 1 + x 2 + x 3 + x 4 + s 3 = 200 x 1 + s 4 = 40 − 88 x 1 − 86 x 2 − 100 x 3 − 100 x 4 + f = 0 Write this in a matrix with the objective function in the last row, x 1 x 2 x 3 x 4 s 1 s 2 s 3 s 4 f [ 3 2 2 5 1 0 0 0 0 200 2 2 ( 4 ) 5 0 1 0 0 0 100 1 1 1 1 0 0 1 0 0 200 1 0 0 0 0 0 0 1 0 40 − 88 − 86 − 100 − 100 0 0 0 0 1 0 ] Most negative value in the last row is -100 which is in two columns. So, let x 3 is the pivot column. Now divide the constants of pivot column from the last row as, 200 2 = 100 100 4 = 25 200 1 = 200 40 0 = ∞ Here 25 is the smallest quotient. So, 4 is the pivot entry. The pivot entry is shown in the matrix inside brackets. Now, multiply the Row 2 with 1 4 to get pivot entry as 1, [ 3 2 2 5 1 0 0 0 0 200 1 2 1 2 ( 1 ) 5 4 0 1 4 0 0 0 25 1 1 1 1 0 0 1 0 0 200 1 0 0 0 0 0 0 1 0 40 − 88 − 86 − 100 − 100 0 0 0 0 1 0 ] Now use operations with other rows to make all other entries in the pivot column equal to zero. So, use the following operations, Add − 2 times row 2 to row 1, add − 1 times row 2 to row 3 and add 100 times row 2 to row 5, [ 2 1 0 2.1 1 − 0

Mathematical Applications for the Management, Life, and Social Sciences

12th Edition

ISBN: 9781337625340

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 4, Problem 33RE

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To calculate: The maximum value of function f=88x1+86x2+100x3+100x4 subjected to constraints 3x1+2x2+2x3+5x4≤200,2x1+2x2+4x3+5x4≤100,x1+x2+x3+x4≤200 and x1≤40.

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Chapter 4, Problem 32RE

Chapter 4, Problem 34RE

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The maximum value of function f = 88 x 1 + 86 x 2 + 100 x 3 + 100 x 4 subjected to constraints 3 x 1 + 2 x 2 + 2 x 3 + 5 x 4 ≤ 200 , 2 x 1 + 2 x 2 + 4 x 3 + 5 x 4 ≤ 100 , x 1 + x 2 + x 3 + x 4 ≤ 200 and x 1 ≤ 40 .

Solution Summary: The author explains how to calculate the maximum value of the function f, which is subjected to constraints.

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To calculate: The maximum value of function f=88x1+86x2+100x3+100x4 subjected to constraints 3x1+2x2+2x3+5x4≤200,2x1+2x2+4x3+5x4≤100,x1+x2+x3+x4≤200 and x1≤40.

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Chapter 4 Solutions