The ranges of optimality for the profit of product

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Reference no: EM132236383

Maximize 12X1 + 10X2 + 8X3             Total Profit

Subject to      X1 + X2 + X3 > 160        At least a total of 160 units of all three products needed

                 X1 + 3X2 + 2X3 ≤ 450         Resource 1

                 2X1 + X2 + 2X3 ≤ 300         Resource 2

               2X1 + 2X2 + 3X3 ≤ 400         Resource 3

                  And X1, X2, X3 ≥ 0

Where X1, X2, and X3 represent the number of units of Product 1, Product 2, and Product 3 to be manufactured.

The QM for Windows output for this problem is given below.

Solution List:

Variable           Status   Value

X1       Basic    100

X2       Basic    100

X3       NONBasic       0

surplus 1          Basic    40

slack 2 Basic    50

slack 3 NONBasic       0

slack 4 NONBasic       0

Optimal Value (Z)        2200

Linear Programming Results:

            X1       X2       X3                  RHS     Dual

Maximize         12        10        8                                 

Constraint 1     1          1          1          >=        160      0

Constraint 2     1          3          2          <=        450      0

Constraint 3     2          1          2          <=        300      2

Constraint 4     2          2          3          <=        400      4

Solution           100      100      0                      2200   

Ranging Results:

Variable           Value   Reduced Cost Original Val     Lower Bound   Upper Bound

X1       100      0          12        10        20

X2       100      0          10        6          12

X3       0          8          8          -Infinity           16

            Dual Value      Slack/Surplus   Original Val     Lower Bound   Upper Bound

Constraint 1     0          40        160      -Infinity           200

Constraint 2     0          50        450      400      Infinity

Constraint 3     2          0          300      275      400

Constraint 4     4          0          400      320      420

(a) What are the ranges of optimality for the profit of Product 1, Product 2, and Product 3?

(b) Find the dual prices of the four constraints and interpret their meanings. What are the ranges in which each of these dual prices is valid?

(c) If the profit contribution of Product 1 changes from $12 per unit to $15 per unit, what will be the optimal solution? What will be the new total profit? (Note: Answer this question by using the ranging results. Do not solve the problem).

(d) Which resource should be obtained in larger quantity to increase the profit most? (Note: Answer this question using the ranging results given above. Do not solve the problem).

Reference no: EM132236383

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