Below is the computer solution to a linear programming problem linear programming:
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X1 |
X2 |
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RHS |
Shadow Price |
| Maximize |
40 |
30 |
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| Constraint 1 |
0.4 |
0.5 |
<= |
20 |
33.3333 |
| Constraint 2 |
0.0 |
0.2 |
<= |
5 |
0 |
| Constraint 3 |
0.6 |
0.3 |
<= |
21 |
44.4444 |
| Solution |
25 |
20 |
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| Sensitivity Analysis |
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| Variable |
Value |
Reduced Cost |
Original Value |
Lower Bound |
Upper Bound |
| X1 |
25 |
0 |
40 |
24 |
60 |
| X2 |
20 |
0 |
30 |
20 |
50 |
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| Constraint |
Shadow Price |
Slack/ Surplus |
Original Value |
Lower Bound |
Upper Bound |
| Constraint 1 |
33.3333 |
0 |
20 |
14.00 |
21.50 |
| Constraint 2 |
0 |
1 |
5 |
4.00 |
Infinity |
| Constraint 3 |
44.4444 |
0 |
21 |
18.75 |
30.00 |
Note: The Reduced Cost is often referred to as the Coefficient Sensitivity.
For the above information, answer the following questions. Provide your answers based on the above information and explain your answers in terms of this information.
a) What are the objective function and the constraints?
b) What are the values of the variables at optimality and what is the value of the objective function at optimality?
c) If there was an opportunity to purchase additional units of each resource (as expressed by the constraints), based on the sensitivity analysis which resource(s) would you consider for purchase? Why? Identify all of the resources that you would consider. If you did not consider a resource, why did you exclude it? Remember that Constraint 1 is equivalent to Resource 1, Constraint 2 is Resource 2, and Constraint 3 is Resource 3.
d) Suppose the contribution of variable X2 in the objective function increases by 10. What are the values of the variables at optimality now and what is the value of the objective function at optimality