Label each of the following statements as true or false. If you label a statemen
ID: 1720015 • Letter: L
Question
Label each of the following statements as true or false. If you label a statement as
false, then justify your answer.
(a) The simplex method’s rule for choosing the entering basic variable is used because it
always leads to the best adjacent BFS (largest z).
(b) The simplex method’s minimum ratio rule for choosing the leaving basic variable is used because making another choice with a larger ratio would yield a basic solution that is not feasible.
(c) When the simplex method solves for the next BFS, elementary algebraic operations (or elementary row operations) are used to eliminate each nonbasic variable from all but one equation (its equation) and to give it a coefficient of +1 in that one equation.
(d) In a particular iteration of the simplex method, if there is a tie for which variable should be the leaving basic variable, then the next BFS must have at least one basic variable equal to zero.
(e) If there is no leaving basic variable at some iteration, then the problem has no feasible solutions.
(f) If at least one of the basic variables has a coefficient of zero in row (0) of the final tableau, then the problem has multiple optimal solutions.
(g) If the problem has multiple optimal solutions, then the problem must have a bounded feasible region.
Explanation / Answer
a) False.
Sharpest increase in slope, not largest increase in z.
b)True
c)True
d) true
e)False
Has unbounded optimal solution.
f)True
Example: The following problem has many optimal solutions:
Max 6X1 + 4X2
subject to:
X1 + 2X2 <=16
3X1 + 2X2 <= 24
all decision variables >= 0.
If you run the above problem on say WinQSB or Lindo you will find four zeros. However, you must notice that this is only a Necessary condition and not a Sufficient one, as for the above numerical example. Unfortunately, the QSB uses this necessary condition. Therefore, sometimes it gives Wrong messages.
g)True
If the problem has multiple optimal solutions, at least two must be CPF solutions. Hence the region is ensured to be bounded.
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