# [Excel spreadsheet available at http://web.mit.edu/15.053/www/Exer3.16.xls ] The…

[Excel spreadsheet available at http://web.mit.edu/15.053/www/Exer3.16.xls] The linear-programming program:

has an optimal canonical form given by:

Answer the following questions. All parts are independent and refer to changes in the original problem formulation as stated above. That is, do not consider parts (a), (b), and (c) together, but refer to the original problem formulation in each part.

a) Suppose that the objective function changes to

z = 4×4 + 2×5 + 0x6.

Find the new optimal solution. Is the solution to the new problem unique? Explain your answer.

b) Consider the parametric-programming problem with the objective function:

Maximize z = (4 + 2θ )x4 + (2 − 4θ )x5 + (−3 + θ )x6.

For what values θ ≤ θ ≤ θ of θ do the variables x4, x5, and x3 remain in the optimal basis (e.g., if θ = −1 and θ = 2, then the interval is −1 ≤ θ ≤ 2). What are the variable transitions at θ = θ and θ = θ?

c) Consider the parametric programming problem with the righthand sides

2 + 4θ ,

6 − 2θ ,

6 − θ.

For what values θ ≤ θ ≤ θ of θ do the variables x4, x5, and x3 remain in the optimal basis? What are the variable transitions at θ = θ and θ = θ?

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