Question

Assume that you go to supermarket to buy apple and orange. Each apple costs 5 $, and each orange costs 7 $. One apple contains 4 grams of fiber and 2 grams of vitamin. One orange contains 1 gram of fiber and 1 gram of vitamin. Your doctor advised you to get at least 12 grams of fiber and 8 grams of vitamin for every day. You love oranges, so you want to eat at least 2 daily. Pretend you eat no other food that contains fiber and vitamin for the day. You like to decide what to purchase to minimize total cost. 

a) Formulate a linear programming model for this problem. Find the optimal solution using the graphical solution approach. 

b) Determine the types of constraints (binding, nonbinding, and redundant).

c) How much extra (i.e., surplus) fiber and vitamin do you take, and how many oranges do you eat at the optimal solution?

d) Determine the sensitivity ranges for the cost of a apple and orange. 

e) Using the sensitivity ranges you found in part (d), determine the new optimal solution and objective function value if the cost of apple is increased from 5 $ to 10 $.

f) Determine the sensitivity ranges for the RHS of constraints (except nonnegativity constraints). Answer parts (g) and (h) considering the sensitivity ranges you found in part (f). Don’t solve the model again.

 g) You decided that 2 oranges is not enough and you need at least 3. What are the new optimal solution and objective function value? (Hint: Use the shadow price of relevant constraint.) h) After seeing your medical test results, the doctor decided to increase the minimum daily fiber requirement to 14 grams. What are the new optimal solution and objective function value? (Keep the minimum orange at 2.) (Hint: Use the shadow price of relevant constraint.)