The simplex method can be used to solve any standard maximization problem. Which of the following minimization problems have dual problems that are standard maximization problems? (Do not solve the problems.) A Minimize C = 2 x 1 + 3 x 2 subject to 2 x 1 − 5 x 2 ≥ 4 x 1 − 3 x 2 ≥ − 6 x 1 , x 2 ≥ 0 B Minimize C = 2 x 1 − 3 x 2 subject to − 2 x 1 + 5 x 2 ≥ 4 − x 1 + 3 x 2 ≥ 6 x 1 , x 2 ≥ 0 What conditions must a minimization problem satisfy so that its dual problem is a standard maximization problem?
The simplex method can be used to solve any standard maximization problem. Which of the following minimization problems have dual problems that are standard maximization problems? (Do not solve the problems.) A Minimize C = 2 x 1 + 3 x 2 subject to 2 x 1 − 5 x 2 ≥ 4 x 1 − 3 x 2 ≥ − 6 x 1 , x 2 ≥ 0 B Minimize C = 2 x 1 − 3 x 2 subject to − 2 x 1 + 5 x 2 ≥ 4 − x 1 + 3 x 2 ≥ 6 x 1 , x 2 ≥ 0 What conditions must a minimization problem satisfy so that its dual problem is a standard maximization problem?
Solution Summary: The author explains that the dual problem is a standard maximization problem.
The simplex method can be used to solve any standard maximization problem. Which of the following minimization problems have dual problems that are standard maximization problems? (Do not solve the problems.)
A
Minimize
C
=
2
x
1
+
3
x
2
subject to
2
x
1
−
5
x
2
≥
4
x
1
−
3
x
2
≥
−
6
x
1
,
x
2
≥
0
B
Minimize
C
=
2
x
1
−
3
x
2
subject to
−
2
x
1
+
5
x
2
≥
4
−
x
1
+
3
x
2
≥
6
x
1
,
x
2
≥
0
What conditions must a minimization problem satisfy so that its dual problem is a standard maximization problem?
Give an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.
3. [10 marks]
Let Go (Vo, Eo) and G₁
=
(V1, E1) be two graphs that
⚫ have at least 2 vertices each,
⚫are disjoint (i.e., Von V₁ = 0),
⚫ and are both Eulerian.
Consider connecting Go and G₁ by adding a set of new edges F, where each new edge
has one end in Vo and the other end in V₁.
(a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so
that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian?
(b) If so, what is the size of the smallest possible F?
Prove that your answers are correct.
Let T be a tree. Prove that if T has a vertex of degree k, then T has at least k leaves.
Chapter 6 Solutions
Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
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