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Quiz 4. The Simplex Method

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Each question has only one correct answer. The results of the quiz do not affect the final marks.

1: Consider a minimization problem with two extreme points x1 and x2 and two extreme directions d1 and d2. Suppose the objective function z = c x has the following values:

c x1 = 3, c x2 = 3, c d1 = 1, c d2 = 0

Find a complete set of optimal solutions to the problem.

x = a x1 + (1 - a) x2, where 0 <= a <= 1
x = x2 + b d2, where b >= 0
x = a x1 + (1 - a) x2 + b d2, where 0 <= a <= 1 and b >= 0
x = a x1 + (1 - a) x2 + b1 d1 + b2 d2, where 0 <= a <= 1 and b1, b2 >= 0

2: Identify a wrong entry in the simplex tableau:


x1 x2 x3 x4
x2 -1 1 -3 0 0
x4 2 0 -5 1 1
z 1 -1 0 0 -5
Objective function has negative value
Basic variable x2 has zero value
Non-basic variable x3 has zero entry in the objective row
Basic variable x2 has negative entry in the objective row

3-4: Consider the minimization of

z = -2 x1 - 3 x2 + 4 x3

subject to the system

x1 - x2 + 2 x3 + x4 = 3
-x1 + 3 x2 + 5 x3 + x5 = 1

where all variables are non-negative. Find the missing entries A and B in the simplex tableau:


x1 x2 x3 x4 x5
x1 1 -1 2 1 0 3
x5 0 2 7 1 1 A
z 0 -5 8 2 0 B
A = 6
A = 4
A = 2
A = 0
B = -6
B = -2
B = 0
B = 2

5: Identify status of the simplex method from the tableau:


x1 x2 x3 x4
x1 1 -1 0 3 3
x3 0 -2 1 1 4
z 0 0 0 0 -1
One more iteration is required
Simplex method terminates: unique optimal solution
Simplex method terminates: alternative optimal solutions
Simplex method terminates: unbounded optimal solutions


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