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Sub: Statistics                                                                    Topic: Operation Research

Design a LPP Model

Question :

Consider the following LP problem:

        Maximize profit = 1X1 + 1X2

        Subject to 2X1 + 1X2 ≤ 100

                     1X1 + 2X2 ≤ 100

   a) What is the optimal solution to this problem? Solve it graphically.

   b) If a technical breakthrough occurred that raised the profit per unit of X1 to $3, would this
        affect the optimal solution?

   c) Instead of an increase in the profit coefficient X1 to $3, suppose the profit was overestimated
        and should have only been $1.25. Does this change the optimal solution?




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 *The Homework solutions from Classof1 are intended to help students understand the approach to solving the problem and not for
                              submitting the same in lieu of their academic submissions for grades.
Sub: Statistics                                                                    Topic: Operation Research

Solution :



a. Replace the variable x1 by x and x2 by y.

              2X1 + 1X2 ≤ 100

              1X1 + 2X2 ≤ 100




Rewrite the inequality in explicit form.

                 y ≤ 100 - 2x

               y ≤ (100 - x)/2


Enter the above inequality in the Graphing Calculator.

Solving the set of linear inequality constraints graphically, we obtain the feasible region.




                             www.classof1.com/homework-help/statistics
 *The Homework solutions from Classof1 are intended to help students understand the approach to solving the problem and not for
                              submitting the same in lieu of their academic submissions for grades.
Sub: Statistics                                                                    Topic: Operation Research




                                           x1                 x2                 Max

                                            0                  0                            0

                                            0                 50                          50

                                         33.33              33.33                    66.66

                                           50                  0                          50


So we conclude that the profit is maximum at x1 = 33.33 and x2 = 33.33

So the optimal solution is (33.33, 33.33).

The profit at optimal point is $66.66.

b. Objective Function is z = 3X1 + 1X2.



                             www.classof1.com/homework-help/statistics
 *The Homework solutions from Classof1 are intended to help students understand the approach to solving the problem and not for
                              submitting the same in lieu of their academic submissions for grades.
Sub: Statistics                                                                    Topic: Operation Research
              x1                x2              Max

               0                 0                        0

               0                50                      50

             33.33            33.33               133.32

              50                 0                    150


If we increase the objective function coefficient of x1 from 1 to 3 then the optimal solution is (50, 0).
Consequently, the profit at optimal point is $150.

c. Objective Function is z = 1.25X1 + 1X2.

               x1               x2               Max

  
								
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