Operations Research : Applications and Algorithms
Operations Research : Applications and Algorithms
4th Edition
ISBN: 9780534380588
Author: Wayne L. Winston
Publisher: Brooks Cole
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Chapter 3.2, Problem 6P

Explanation of Solution

Given data:

The farmer Jane owns 45 acres of land and planning to plant with wheat or corn.

On planting wheat she yields $200 profit and corn yields $300 profit.

Given table:

 WheatCorn
Labor3 workers2 workers
Fertilizer2 tons4 tons

Consider x1 be the acres of land planted with wheat and x2 be the acres of land planted with corn.

Objective function:

Maximize z=200x1+300x2

Considering the constraints,

Constraint 1: Total acres of land used

Constraint 2: Maximum number of workers to be used is 100

Constraint 3: Maximum tons of fertilizers are 120 tons.

Expressing the constraint 1 in terms of x1and x2:

x1+x2=45

Expressing the constraint 2 in terms of x1and  x2:

3x1+2x2100

Expressing the constraint 3 in terms of x1, x2and x3 :

2x1+4x2120

Therefore, the mathematical model of given LP is,

Maximizez=200x1+300x2

Subject to the constraints,

x1+x2=453x1+2x21002x1+4x2120x1,x20(Sign restriction)

Converting the inequality constraint without adding any variable:

x1+x2=453x1+2x21002x1+4x2120

The coordinate points for the constraint x1+x2=45 is,

If x1=0 then x2=45, The points are (0,45)

If x2=0 then x1=45, The points are (45,0)

The coordinate points for the constraint 3x1+2x2=100 is,

If x1=0 then 2x2=100, The points are (0,50)

If x2=0 then 3x1=100, The points are (33.33,0)

The coordinate points for the constraint 2x1+4x2=120 is,

If x1=0 then 4x2=120, The points are (0,30)

If x2=0 then 2x1=120, The points are (60,0)

Therefore, the coordinate point for the constraint x1=0 is (3,0)

Graph:

From the above graph, it is known that the vertices of the feasible region lies in the points A(0,30) and C(33.33,0).

Calculating the value of the objective function to find the end points:

6x1+4x2=2002x1+4x2=120                    4x1=80

Therefore, the value of x1=20

Substituting the value of x1=20 in the equation 3x1+2x2=100

3x1+

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Chapter 3 Solutions

Operations Research : Applications and Algorithms

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ISBN:9780534380588
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