Max 30x1 x2 s.t. 2x1 x2 ≤ 4 2x1 2x2 ≤ 6 x1, x2 ≥ 0 (a) Solve graphically and state the optimal solution. (b) Keeping all the other data as is, what per unit profitability should the product, whose current optimal value is zero, have in order that this product enter the optimal solution at a positive level? (c) How many optimal corner solutions exist after making the change described in part (b)? What are they? (d) In the original problem, how much can the right-hand side (RHS) of the second constraint be increased (or decreased) before the optimal solution is changed? (e) Answer part (d) for the RHS of the first constraint. (f) How do you explain the difference between parts (d) and (e)? (g) What will be the impact of adding the constraint 4x1 x2 = 4 to the original model? (h) What is the impact (on the optimal solution) of adding the constraint 3x1 3x2 ≤ 15 to the original model? (i) Fill in the blanks: The difference between parts (g) and (h) is that the original optimal solution already _______ the constraint in (h) but does not _______ the constraint in (g). (j) Solve this LP problem using the Simplex method but change the objective function to Max 30x1 20x2 but retain the constraints. Then give the optimal solution and objective function value. You need not answer (b) – (i)
Max 30x1 x2 s.t. 2x1 x2 ≤ 4 2x1 2x2 ≤ 6 x1, x2 ≥ 0 (a) Solve graphically and state the optimal solution. (b) Keeping all the other data as is, what per unit profitability should the product, whose current optimal value is zero, have in order that this product enter the optimal solution at a positive level? (c) How many optimal corner solutions exist after making the change described in part (b)? What are they? (d) In the original problem, how much can the right-hand side (RHS) of the second constraint be increased (or decreased) before the optimal solution is changed? (e) Answer part (d) for the RHS of the first constraint. (f) How do you explain the difference between parts (d) and (e)? (g) What will be the impact of adding the constraint 4x1 x2 = 4 to the original model? (h) What is the impact (on the optimal solution) of adding the constraint 3x1 3x2 ≤ 15 to the original model? (i) Fill in the blanks: The difference between parts (g) and (h) is that the original optimal solution already _______ the constraint in (h) but does not _______ the constraint in (g). (j) Solve this LP problem using the Simplex method but change the objective function to Max 30x1 20x2 but retain the constraints. Then give the optimal solution and objective function value. You need not answer (b) – (i)
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter7: Nonlinear Optimization Models
Section: Chapter Questions
Problem 49P: If a monopolist produces q units, she can charge 400 4q dollars per unit. The variable cost is 60...
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Max 30x1 x2 s.t. 2x1 x2 ≤ 4 2x1 2x2 ≤ 6 x1, x2 ≥ 0 (a) Solve graphically and state the optimal solution. (b) Keeping all the other data as is, what per unit profitability should the product, whose current optimal value is zero, have in order that this product enter the optimal solution at a positive level? (c) How many optimal corner solutions exist after making the change described in part (b)? What are they? (d) In the original problem, how much can the right-hand side (RHS) of the second constraint be increased (or decreased) before the optimal solution is changed? (e) Answer part (d) for the RHS of the first constraint. (f) How do you explain the difference between parts (d) and (e)? (g) What will be the impact of adding the constraint 4x1 x2 = 4 to the original model? (h) What is the impact (on the optimal solution) of adding the constraint 3x1 3x2 ≤ 15 to the original model? (i) Fill in the blanks: The difference between parts (g) and (h) is that the original optimal solution already _______ the constraint in (h) but does not _______ the constraint in (g). (j) Solve this LP problem using the Simplex method but change the objective function to Max 30x1 20x2 but retain the constraints. Then give the optimal solution and objective function value. You need not answer (b) – (i)
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