Mathematics 340
Homework 9: Solutions
2010/2011
1. Since the water must ow from the source to the sink (and not in circles), the maximum
amount of water owing from the source will be the maximum ow. Therefore the objective
function is
n
Maximize
x1k .
k=1
Mathematics 340
Homework 10: Solutions
2010/2011
1. For brevity just the various results are listed, however your solution should resemble that
given in class.
x1
x2
s1
1 0
1 2 3 1
First iteration: xB =
, xN = , B =
, AN =
, cB =
x3
s2
0 1
1 1 2 3
x
Mathematics 340
Homework 11: Solutions
2010/2011
(a) Give B and label its columns appropriately. Calculate B 1 :
x1
4
B= 1
2
x2 s 3
6
0
3.5 0
4
1
7
16
1
8
3
8
B 1 =
3
4
1
2
1
2
0
0
1
(b) Determine the range for c1 so that the basis cfw_x1 , x2 , s3 rem
Mathematics 340
Homework 0: Solutions
2010/2011
1. We have a box whose side lengths are x1 , x2 and x3 . It has faces with circumferences
10, 14 and 12.
(a) The linear system of equations on the variables x1 , x2 , x3 expressing this
situation:
2x1 + 2x2
December 1999
MATH 340102
Name
Page 2 of 4 pages
Marks
[10]
1.
Consider the problem
subject to x1 , x2 , x3 0,
maximize 3x1 + 2x2 + 4x3
x1
2x1
2x1
+x2
+x2
+2x3
+3x3
+3x3
4,
5,
7.
A friend thinks she solved it using the simplex method, with a nal dictio
Math 340 Final 99: Very Brief Solutions
Problem 1. (a) x =
21
2
5 3
, ,0
2 2
satises the constraints.
(b) Twice the rst constraint plus half the second yields 3x1 +2x2 + 11 x3
2
so clearly 3x1 + 2x2 + 4x3 21 .
2
Problem 2. (a) x0 enters, s1 leaves. Then
An Example of Degeneracy in Linear Programming
An LP is degenerate if in a basic feasible solution, one of the basic variables takes on a zero
value. Degeneracy is caused by redundant constraint(s) and could cost simplex method extra
iterations, as demons
Mathematics 340
Homework 8: Solutions
2010/2011
1. Computing matrix products eciently:
(a)
Compute B = AT A - 1002 = 10000 multiplications
Compute v = Bx - 10 10 = 100 multiplications
Compute w = By - 10 10 = 100 multiplications
Compute u = v + w - ju
Mathematics 340
Homework 6: Solutions
2010/2011
1. For each of the following LPs, use complementary slackness to check if the suggested
solution is optimal:
(a) The solution x = 1 x = 2, x = 1 z = 12 for the problem:
3
2
1
maximize
z = 5x1 + 2x2 + 3x3
sub
Mathematics 340
Homework 5: Solutions
2010/2011
1. Answer: This happens if and only if every pivot between your current dictionary and the
nal dictionary is degenerate.
Explanation: Say we have a dictionary D such that the last line of the dictionary read
Mathematics 340
Homework Problems
2010/2011
Homework 0 - practice problems, do not hand in
1. We have a box whose side lengths are x1 , x2 and x3 . It has faces with circumferences
10, 14 and 12.
(a) Write down a linear system of equations on the variable
Mathematics 340
Homework 3: Solutions
2010/2011
1. False.
Consider the LP problem:
Maximize x1 x2
subject to x1 x2 0
x1 , x 2
0
Drawing the region of feasible solutions we see it is unbounded:
x2
x1
However, using the simplex method we have the initial d
Math 340
Assignment #2
Due Friday Oct 21, 2016 at the beginning of class.
1. Give an example of a dictionary for which the current basic feasible solution is optimal and
yet the coefficients of the non-basic variables in the z row are not all negative.
2.
Math 340
Assignment #2
Due Wednesday Feb 10, 2016 at the beginning of class.
1. Give an example of a dictionary for which the current basic feasible solution is optimal and
yet the coefficients of the non-basic variables in the z row are not all negative.
March 2009
MATH 340202
Name
Page 2 of 8 pages
Marks
[8]
1.
Each of the dictionaries below has x1 entering variable. Below each dictionary, write
which variable leaves (if any); also write unbounded pivot (meaning that you know
the LP is unbounded at that