Explanation Given information: The objective function which has to minimize is: M = x + y which is subjected to the constraints: { 7 x + 5 y ≥ 40 x + 4 y ≥ 9 x ≥ 0 y ≥ 0 Formula used: The minimization problem of simplex can be solved as below: Step 1: The minima problem can be converted into maxima problem by multiplying ( − 1 ) on both sides Step 2: Write the initial tableau and find the feasible solutions. Step 3: Make the all elements of last column positive by row operations. Step 4: Find the pivot column, by finding maximum negative number in the last row of the tableau. Step 5: Now divide the all elements which are in the last column by their corresponding elements of pivot column to decide pivot elements. Step 6: Now to make the pivot element unity, divide all the element of pivot row by 1 and make all other elements of pivot column to zero by row operation. Step 7: If there is one or more than one negative number in the last row, then the above tableau is not optimal, it needs iteration. Calculation: Since, this is minima problem, so convert the minima problem into maxima problem by multiplying ( − 1 ) on both sides of objective function which is: ( − 1 ) × M = − x − y . So, the objective function which has to maximizeis M = − x − y . And, revised constraints are: { − 7 x − 5 y ≤ − 40 − x − 4 y ≤ − 9 x ≥ 0 y ≥ 0 Write the above constraints in the form of linear equations by introducing slack variables, u , and v which are: − 7 x − 5 y + u = − 40 − x − 4 y + v = − 9 x + y + M = 0 Step 1: Write the initial tableau and find the feasible solution. In the matrix form it can be written as: x y u v M [ − 7 − 5 1 0 0 − 40 − 1 − 4 0 1 0 − 9 1 1 0 0 1 0 ] So, the above simplex tableau is initial tableau, therefore one feasible solution is: Step 2: Make the all elements of last column positive. The choice to make positive is either − 40 or − 9 . But choose − 40 because it is most negative. Step 3: Pivot element will be that element which can make minimum positive. So, ( − 40 − 7 = 5.714 ) and ( − 40 − 5 = 8 ) , the minimum positive is 5.714. In this case the pivot element is − 7 . u v M [ − 7 − 5 1 0 0 − 40 − 1 − 4 0 1 0 − 9 1 1 0 0 1 0 ] Step 4: Now to make the pivot element unity, divide all the element of pivot row by − 7 and make all other elements of pivot column to zero by row operation which is as below: x y u v M [ 1 5 7 − 1 7 0 0 40 7 − 1 − 4 0 1 0 − 9 1 1 0 0 1 0 ] R 2 + 1 × R 1 R 3 − 1 × R 1 → [ 1 5 7 − 1 7 0 0 40 7 0 − 23 7 − 1 7 1 0

Finite Mathematics & Its Applications (12th Edition)

12th Edition

ISBN: 9780134437767

Author: Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair

Publisher: PEARSON

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Chapter 4, Problem 5RE

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To calculate: The minimum value of objective function: M=x+y which is subjected to the constraints:

{7x+5y≥40x+4y≥9x≥0y≥0

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Chapter 4, Problem 4RE

Chapter 4, Problem 6RE

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The minimum value of objective function: M = x + y which is subjected to the constraints: { 7 x + 5 y ≥ 40 x + 4 y ≥ 9 x ≥ 0 y ≥ 0

Solution Summary: The author explains how the minimization problem of simplex can be solved by multiplying (-1) on both sides of objective function.

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To calculate: The minimum value of objective function: M=x+y which is subjected to the constraints:

{7x+5y≥40x+4y≥9x≥0y≥0

Want to see the full answer?

Chapter 4 Solutions

The minimum value of objective function: M = x + y which is subjected to the constraints: { 7 x + 5 y ≥ 40 x + 4 y ≥ 9 x ≥ 0 y ≥ 0

Solution Summary: The author explains how the minimization problem of simplex can be solved by multiplying (-1) on both sides of objective function.

Videos

To calculate: The minimum value of objective function: M=x+y which is subjected to the constraints: {7x+5y≥40x+4y≥9x≥0y≥0

Want to see the full answer?

Chapter 4 Solutions

To calculate: The minimum value of objective function: M=x+y which is subjected to the constraints:

{7x+5y≥40x+4y≥9x≥0y≥0