MATH2003J, OPTIMIZATION IN ECONOMICS,
BDIC 2023/2024, SPRING
Problem Sheet 8
Question 1:
Consider the following LP problem:
Maximize z = 5x1+3x2
subject to 2x1 + 4x2 ≤ 32
3x1 + 2x2 ≤ 24
x1, x2 ≥ 0
(I) Solve the original problem with the simplex method.
(II) Formulate the dual problem.
(III) Determine the optimal solution to the dual problem from the optimal tableau of the original problem.
(IV) Solve the dual problem with the simplex method (without using the original LP prob-lem), to verify the previous result.
Question 2:
Cons attained at (ider the following LP problem:
Maximize z = 40x1 + 20x2 + 60x3
subject to 2x1 + 4x2 + 10x3 ≤ 24
5x1 + x2 + 5x3 ≤ 8
x1, x2, x3 ≥ 0.
(I) Formulate the dual problem.
(II) Find the maximum value of z in the given LP problem by solving the dual problem using the graphical method.
Question 3:
Consider the following LP problem:
Maximize z = 9x + 12y
subject to 2x + 4y ≤ 32
2x + 2y ≤ 20
15x + 8y ≤ 120
x ≤ 7
y ≤ 7
x, y ≥ 0.
(I) Formulate the dual problem.
(II) Find the optimal value of objective function in the dual LP problem by solving the give LP problem using the graphical method.
Question 4:
Determine if the following statements are True or False, and briefly justify your answer:
(1) , The set [1, 2) is open.
(2) , The set [1, 2) is closed.
(3) , The set (1, 2] is open.
(4) , The set (1, 2] is closed.
(5) , The set (0,+∞) is neither closed nor open.
(6) , The set {2n − 1 ∣ n ∈ N} ⊂ R is closed.
(7) , The set {1/n ∣ n ∈ N} ⊂ R is closed.
(8) , The set Q ⊂ R is closed.
(9) , The set of irrational numbers R ∖ Q is closed.
(10) , The set:
is closed.
(11) The set:
is closed.
(12) The set [1, 2] × [0, 1] ⊂ R
2
is closed.
(13) The set [1, 2] × (0, 1) ⊂ R
2
is closed.
(14) The set {2024} is closed.
(15) The set {(x, y) ∈ R2
∣ xy = 1} is open.
(16) The set {(x, y) ∈ R2
∣ xy = 1} is closed.
(17) The set {(x, y) ∈ R2
∣ x2 + xy + y2 = 2023} is closed.
(18) The set {(x, y) ∈ R2
∣ y = x/1} is closed.
(19) The set
is closed.
(20) The set
is closed.