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Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
Publisher: Cengage Learning
ISBN: 9781337625340

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Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
Publisher: Cengage Learning
ISBN: 9781337625340
Chapter 4.5, Problem 13E
Textbook Problem
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In Problems 11-18, use the simplex method to find the optimal solution. Assume that all variables are nonnegative.

Minimize f = 2 x + 3 y subject to

x 5 y 13 x + y 2

To determine

To calculate: The optimal solution of the LPP with the help of simplex method.

Minimize f=2x+3y subject to

x5y13x+y2

Explanation of Solution

Given Information:

The linear programing problem with mixed constraint is given as:

Minimize f=2x+3y subject to

x5y13x+y2

Formula used:

To solve the linear programming problem by simplex method, follow the steps below:

Step 1: Use slack variables and write the constraint inequalities in equation form.

Step 2: Write the equations in a simplex matrix.

Step 3: Choose the most negative number on the left side of the bottom row and pivot the column.

Step 4: Select the pivot entry which is the smallest of the test ratios ab, where, a is entry in the right most column and b is the corresponding entry in the pivot column.

Step 5: Make the pivot entry as 1 and other entries of pivot column as 0 by the use of row operations.

Step 6: Repeat the above steps till all the entries in the bottom row are non-negative.

Calculation:

Provided the LPP is, Minimize f=2x+3y subject to the constraints

Minimize f=2x+3y subject to

x5y13x+y2

Since, given problem is minimization problem with mixed constraint so first convert it into maximization and also convert all constraint into standard form as by multiplying them by (1) both side.

Maximize f=2x3y subject to

x5y13xy2

Express all constraints as constraints. To form the simplex form convert all the inequalities into equations, add slack variable into the inequalities.

x+s1=5y+s2=13xy+s3=2x0,y0

Where s1,s2 and s3 are two slack variables with s1,s2,s30.

Simplex matrix for the LPP is written below:

  x      y      s1   s2   s3   f[10100050101001311001022300010]

Here, (1) is pivot element for this table.

Solve the given problem by simplex method

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Mathematical Applications for the Management, Life, and Social Sciences
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