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Do the following problems showing all steps and calculations. Please hand in this question sheet with your solution set. Thank you.

1. Solve the following problem

Max 2X2 - X1, subject to X1 + X2  5, X1 + 2X2  6, and X1  0, X2  0,

by the graphical method indicate the optimal solution for both X1 and X2, as well as the optimal value for the objective function. (10%)

If the objective function changes to Max 2X2 + X1 what is the optimal solution and optimal value for the new problem? (Why?). (10%)

2.Joe's Garage specializes in oil changes and tune-ups. Profit per oil change is $7 and $15 per tune-up. Joe's has a fleet account customer, which guarantees 30 oil changes per week. Each oil change requires 20 minutes labor and $8 in supplies. A tune-up takes one hour and costs Joe's $15 in supplies. Mechanics are paid $10 per hour and Joe's currently employs two mechanics working 40 hours each per week. Every week Joe's orders $1,750 in supplies. Joe's wishes to maximize profit, formulate Joe’s problem as a linear program. (10%)

  1. Solve the following problem:

Maximize X1 + 2X2 + 3X3

Subject to: 4X1 + 2X2 + 3X3  24, all variables Xi's  0 (10%)

4. Which of the following statement is correct and why?

A) Zero divided by any number is zero.

B) Any number divided by itself is 1. (10%)

5. Consider the following LP problem

Max 7X1+10X2

Subject to:

5X1 + 6X2  3600 (Raw material)

X1 + 2X2  960 (Labor)

X1  500 (Production limit)

X2  500 (Production limit)

X1, X2  0 (Non-negativity)(50%)

Answer the following questions:

1.What is the optimal solution and optimal value for the problem?

2. Which of the constraints are binding?

3. What is the impact on the optimal solution and optimal value if we decrease the cost coefficient c (1) = 7 to 6.1? Why?

4. What is the shadow price for the RHS # 1.? How do you interpret it?

5. What are the optimal solution and the optimal value for the dual problem? Why?

6. What is the impact on the optimal value if we decrease the right-hand side of constraint # 1 by 500? Why?

7. What is the impact on the optimal value if we increase the right-hand side of constraint # 4 by 50? Why?

8. Is it profitable to produce a new product with net profit of $16 requiring 6 units of resource #1 and 3 units of resource # 2? Why?

9. What is the impact of deleting the second constraint on the optimal solution? Why?

10. What is the impact of adding the following new constraint X2 + 5X1  2100 on the optimal solution? Why?