Question

1. Given the following linear programming model, answer the questions that follow. You are given the result of a computer program. The results are

Maximize 9 X₁ + 12 X₂ + 10 X₃

Subject to:

Machine Constraint:   3 X₁ + 4 X₂ + 3 X₃ < 160

Labor Constraint:        6 X₁ + 10 X₂ + 4 X₃ < 288

Materials Constraint: 2 X₁ + 2 X₂ + 7 X₃ < 200

Product 2 Constraint: X₁ < 16

OPTIMAL SOLUTION

Objective Function Value =         483.097

      Variable             Value             Reduced Costs  

   --------------     ---------------      ------------------

         X1                    16.000                   0.000

         X2                    10.839                   0.000

         X3                    20.903                   0.000

  

     Constraint        Slack/Surplus           Dual Prices   

   --------------     ---------------      ------------------

         1                      5.935                   0.000

         2                      0.000                   1.032

         3                      0.000                   0.839

         4                      0.000                   1.129

OBJECTIVE COEFFICIENT RANGES

    Variable       Lower Limit       Current Value     Upper Limit

------------   ---------------    --------------- ---------------

      X1                  7.871              9.000   No Upper Limit

      X2                  2.857             12.000           14.059

      X3                  4.800             10.000           18.750

RIGHT HAND SIDE RANGES

  Constraint      Lower Limit       Current Value     Upper Limit

------------   ---------------    --------------- ---------------

       1                154.065            160.000   No Upper Limit

       2                192.000            288.000          304.727

       3                 70.400            200.000          226.286

       4                  0.000             16.000           30.154

1. What is the optimal solution?

1. What is the profit?

2. If the profit of Product 3 was changed to $20 and the profit of Product 1 was changed $20, would that change the solution?   Provide proof.

3. If the number of minutes of Machine time was decreased to 155 minutes and the amount of materials were decreased to 170 pounds, would this change the solution? Provide proof.

4. The Dual Price for Constraint 1 (Machine time) is 4.2. In terms of this problem what does that mean?

Answer 1

(Since there are more than 4 parts i will answer first 4)

a.

The optimal solution is X1 = 0.00, X2 = 4.00 and X3 = 48.00

b.

The optimal profit is 528

c.

Given, the profit of products 1 & 3 are changed to 20 $

The lower and upper limit for objective coefficient ranges of variable X1 are (no limit , 22.2)

coefficient (Profit) for X1 is 20$ which lies in the range (no limit, 22.2)

The lower and upper limit for objective coefficient ranges of variable X3 are (13.5, 36.0)

New coefficient (Profit) for X3 is 20 $ which lies in the range (13.5, 36.0)

we see that both new values lie in the objective coefficient range, new values of Profit would not change the optimal solution values for X1 and X2

d.

Given, number of minutes of machines reduced to 155 min and number of materials decreased to 170 pounds

The lower and upper limit for right hand side ranges of minutes of machines (constraint 1) is (150, 177.5)

New value of 155 min lies in the range (150, 177.5)

The lower and upper limit for right hand side ranges of materials (constraint 3) is (160, 213.33)

New value of 170 pounds lies in the range (160, 213.33)

As, both new values lie in the right hand side range, new values would not change the solution.

P.S. (please upvote if you find the answer satisfactory)