Math 482
HW6 Section:
Name:
Due Friday, March 15, 2015
Students in section D13 (three credit hours) need to solve any four of the following ve
problems. Students in section D14 (four credit hours) must solve all ve problems.
1. Interpreting the numbers on
Math 482
HW1
1/24/2014
Name:
Due Friday, January 31, 2014
1. A company produces three types of chemicals: chemical A, chemical B and chemical C.
They sell chemical A for $30 per barrel, chemical B for $20 per barrel, and chemical C
for $10 per barrel. Che
Math 482
HW7
3/28/2014
Name:
Due Friday, April 4, 2014
Students in section B13 (three credit hours) need to solve any four of the following ve
problems. Students in section B14 (four credit hours) must solve all ve problems.
1. THIS PROBLEM IS COUNTED AS
Homework 1
Math 482
Laura Escobar
This homework is due January 27, 2016 at the beginning of the class. In order to obtain
full marks, each solution should be clearly written with explanations for all claims
made. You must list all your sources and the nam
Math 482
HW6
3/08/2014
Name:
Due Friday, March 14, 2014
Students in section B13 (three credit hours) need to solve any four of the following ve
problems. Students in section B14 (four credit hours) must solve all ve problems.
1. Use the complementary slac
Math 482
HW4
2/21/2014
Name:
Due Friday, February 28, 2014
Students in section B13 (three credit hours) need to solve any four of the following ve
problems. Students in section B14 (four credit hours) must solve all ve problems.
1. State the dual to the f
Math 482
HW5
2/28/2014
Name:
Due Friday, March 7, 2014
Students in section B13 (three credit hours) need to solve any four of the following ve
problems. Students in section B14 (four credit hours) must solve all ve problems.
1. Suppose that there exists x
Math 482
HW8
4/06/2014
Name:
Due Friday, April 11, 2014
Students in section B13 (three credit hours) need to solve any two of the following three
problems. Students in section B14 (four credit hours) must solve all three problems.
1. Apply the Ford-Fulker
Math 482
HW9
4/13/2014
Name:
Due Friday, April 18, 2014
Students in section B13 (three credit hours) need to solve any four of the following ve
problems. Students in section B14 (four credit hours) must solve all ve problems.
1. Find the length of a short
Math 482
HW10
4/23/2014
Name:
Due Wednesday, April 30, 2014
Students in section B13 (three credit hours) need to solve any four of the following ve
problems. Students in section B14 (four credit hours) must solve all ve problems.
1. Prove that if A is TUM
Blands algorithm does not cycle
Suppose we are at a step of the simplex procedure represented by tableau T . Let ai,j
be the entry at row i and column j of T . Suppose that there exists 1 i n such
that a0,i < 0, since otherwise we would have found a optim
Second example of simplex method
Suppose we are given the problem
Minimize z = x1 + x2 x3
subject to
2x1 x2 +2x3 +x4
2x1 3x2 +x3
+x5
x1 +x2 2x3
+x6
x1 ,
x2 ,
x3 ,
x4 , x5
x6
=
4
= 5
= 1
0.
(1)
This system is solved with respect to x4 , x5 , and x6 , bu
First example of the simplex method
Suppose we are given the problem
Minimize z = x1 + 2x2 x3
x1 2x2 +x3 +x4
= 10
2x1 3x2 x3
+x5 = 6
x1 , x2 ,
x3 , x4 , x5 0.
subject to
Rewrite the objective function as a new equation with the new variable z and switch t
An example of the dual simplex method
Suppose we are given the problem
Minimize z = 2x1 + 3x2 + 4x3 + 5x4
x1 x2 +x3 x4
x1 2x2 +3x3 4x4
3x1 4x2 +5x3 6x4
x1 , x2 ,
x3 ,
x4
subject to
10,
6,
15
0.
(1)
If we would have inequalities instead of , then the
An example of the revised 2-phase simplex method
Suppose we are given the problem
Minimize z = 19x1 13x2 12x3 17x4
3x1 +2x2 +x3 +2x4
x1 +x2 +x3 +x4
4x1 +3x2 +3x3 +4x4
x1 , x2 ,
x3 ,
x4
subject to
= 225,
= 117,
= 420
0.
(1)
Add to each of the equations it
Math 482
HW3
2/7/2014
Name:
Due Friday, February 14, 2014
Students in section B13 (three credit hours) need to solve any four of the following ve
problems. Students in section B14 (four credit hours) must solve all ve problems.
1. Solve the problem on the
Math 482
HW2
1/31/2014
Name:
Due Friday, February 7, 2014
Students in section B13 (three credit hours) need to solve any four of the following ve
problems. Students in section B14 (four credit hours) must solve all ve problems.
1. A farmer has 100 acres o
Example (2.7 from book)
I
We will use the following pivot rules. Pivot rules are additional
rules for selecting the pivot row and the pivot column.
I
Select the pivot column s so that a0,s = c s c j = a0,j for all
j [n]. (In this example, this always give
Primal-dual example
Suppose we are given the problem P:
Minimize z = x1 + 3x2 + 3x3 + x4
subject to
3x1
3x1
6x1
x1 ,
+4x2
2x2
+4x2
x2 ,
3x3
+6x3
x3 ,
+x4
x4
+x4
x4
= 2,
= 1,
= 4
0.
(1)
The dual D of P is the following:
Maximize w = 21 + 2 + 43
subject
Suppose we are given the problem
Minimize z = 19x1 13x2 12x3 17x4
subject to
3x1
x1
4x1
x1 ,
+2x2
+x2
+3x2
x2 ,
+x3
+x3
+3x3
x3 ,
+2x4
+x4
+4x4
x4
= 225,
= 117,
= 420
0.
(1)
I
There is no obvious bfs, so we use the revised two phase simplex method
I
To st
Suppose we are given the problem
Minimize z = 2x1 + 3x2 + 4x3 + 5x4
subject to
x1
x1
3x1
x1 ,
x2
2x2
4x2
x2 ,
+x3
+3x3
+5x3
x3 ,
x4
4x4
6x4
x4
10,
6,
15
0.
(1)
I
Do we add slack variables or surplus variables?
I
Do we automatically have a basic feasib
Two phase simplex
I
Suppose we are given the problem (LP1).
Minimize z = x1 + x2 x3
subject to
2x1 x2 +2x3 +x4
2x1 3x2 +x3
+x5
x1 +x2 2x3
+x6
x1 ,
x2 ,
x3 ,
x4 ,
x5
x6
=
4
= 5
= 1
0.
(LP1)
I
This system is solved for the basis (4, 5, 6), but this basis i
Theorem 1. Suppose we have the following LP in standard form
(P)
min cT x s.t Ax = b and x 0,
and that B = (j1 , . . . , jm ) is a basis.
(a) The vector x defined by xB = A1
B b and xj = 0 for every nonbasic variable xj is
the unique basic solution associ
Homework 5
Math 482
Laura Escobar
This homework is due March 3, 2017 at the beginning of the class. In order to obtain full
marks, each solution should be clearly written with explanations for all claims made. You
must list all your sources and the names
Homework 3
Math 482
Laura Escobar
This homework is due February 10, 2017 at the beginning of the class. In order
to obtain full marks, each solution should be clearly written with explanations for
all claims made. You must list all your sources and the na