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
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 no
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 dict
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
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) an
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
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 sh
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)
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
LINEAR PROGRAMMING
Foundations and Extensions
Third Edition
Recent titles in the INTERNATIONAL SERIES IN
OPERATIONS RESEARCH & MANAGEMENT SCIENCE
Frederick S. Hillier, Series Editor, Stanford Universi
MATH 340 Homework 6 Solutions
Throughout these solutions we use the notation from the lecture notes on
game theory from Z. Bradshaws website for section 201:
https:/www.math.ubc.ca/~ zbradshaw/2015W2.
MATH 340 Introduction to Linear Programming Section 201 TTh .
https:/sites.google.com/site/nguyenlammath/teaching/math-340-in.
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Mathematics 340
Homework 2: Solutions
2010/2011
1. Initial dictionary
(a)
s1 = 5
x1 x2 3x3 +x4
s2 = 3
x1
2x3 x4
z = 0 +10x1 +2x2 +18x3 2x4
x1 enters and s2 leaves
sh is
ar stu
ed d
vi y re
aC s
o
ou u
Math 340 Assignment 2 Solutions
Problem 1. Suppose we have an LP in standard form with
x1 + 2x2 3x3 x4 2
being the first constraint listed. Assume that youve found an optimal solution to the LP with x
Math 340Assignment #1 Due Friday January 26, 2016 (two pages) at the beginning of class.
1. Show that the three inequalities
x y 2
x + 2y 5
x y 6
have no solution x, y with x, y 0 by using our two p
MATH 340
Practice for Quiz # 3
October 7, 2010
These practice problems give the spirit of the questions that can be asked. Obviously
question 1 could be altered so that you are given a dual optimal so
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 varia
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