Math 340
Assignment #1
Due Thursday January 26, 2012 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 phase method (not using LINDO; you
need the practice! Fractions a
Math 340 Assignment #1 Solutions
class.
Due Wednesday January 27, 2016 at the beginning of
1. The three inequalities x + 2y 2, 2x + y 1, 3x + y 4 have no solution x, y with
x, y 0:
We rst transform the inequalities to standard form x+2y 2, 2xy 1, +3xy 4.
M340(921) SolutionsProblem Set 1
(c) 2013, Philip D. Loewen
1. Consider the linear system Ax = b,
2 6
1 3
A=
1 3
where
0
0
1
0 3 1 0
1 9 2 1 ,
0 1 0
1
10
b = 5 .
5
Which of the ve vectors below, if any, are basic solutions of Ax = b? Explain your decision
MATH 340 Homework 1
Due January 15 (in class)
1. Question 1: Convert the following optimization problem into an LP
problem stating it in standard form.
minimize 3x 2(y + z)
subject to
|x y| 3
xyz =3
z2
x, y, z 0.
2. Question 2: Decide if the problem above
MATH 340 Homework 1
1. Question 1:
Standard form of a linear programming problem in n decision variables
having m constraints:
maximize:
n
=
cj x j
j=1
subject to:
n
aij xj bi
i = 1, 2, . . . , m
j=1
xj 0
j = 1, 2, . . . , n.
You need to transform each li
Math 340 Lecture 9
The simplex method in matrix formalism. Now we want to go back to thinking about things in
terms of matrices, and see how we can rephrase what were doing in the simplex method in terms of them.
Remember that a while back we said that if
Math 340 Lecture 5
Summary of the simplex method. Lets go back and look at what we did in our first example of the
simplex method last time. We started with an LP in standard form:
Maximize 6x1 + 8x2 + 5x3 + 9x4 subject to:
2x1 + x2 + x3 + 3x4 5
x1 + 3
Math 340 Lecture 2
We had a couple of examples last class, so now we have an idea of what a linear program is. General setup:
going back to linear algebra, a linear function on n variables (usually well denote them x1 , . . . , xn ) is any
function of the
MATH 340
Sample Revised Simplex Computations for Quiz 4.
In each of the following questions you are given Aaug , b, caug , the current basis and
1
AB . Determine, using our revised simplex methods, the next entering variable (if there
is one), next leavin
Question 1.
The picture below represents a partial matching (the red edges) in a bipartite graph. Find the maximum
matching using the Ford-Fulkerson algorithm. Formulate the question as a maximum flow problem in a
network. Draw the network and consider th
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.html
Problem 1. This game is symmetric so we automatica
MATH 340 Homework 5
Due February 12 (in class)
1. Question 1:
True or False? Indicate the correct answer and justify it
in a short sentence. We can suppose that the LP problem
is formulated in the standard form. When we refer to the
value of a given feasi
MATH 340 Homework 3
Due February 5 (in class)
1. Question 1:
Solve the following linear programming problems using the simplex
method. Use the simplex method (tableau or dictionary form) and
clearly indicate the steps of the algorithm. No other solution m
MATH 340 Homework 3
Due January 29 (in class)
1. Question 1: Solve the problem from the previous assignment using
the simplex method. Find the largest value of 2x + y under the following conditions;
2x y 2
2y x 2
x+y 7
x, y 0.
Copy the picture of the regi
MATH 340 Homework 7
Due March 26 (in class)
1. Question 1: Below is the payo matrix of a zero-sum game. In this
matrix the entries represent the payo of the row player. For example if
row player plays A and column player plays b then row player receives
$
max x1 + 5x2
subject to
4x1 + 2x2 4
x1 + x2 6
x1 , x2 0
Solution: The standard form is
4x1 + 2x2 4
x1 x2 6
x1 , x2 0
Introduce x0 and the slack variables x3 , x4 . The rst dictionary of the auxiliary problem is
x3 = 4 4x1 2x2 + x0
x4 = 6 + x1 + x2 + x0
w
MATH 340 Homework 2
Due January 22 (in class)
1. Question 1: Sketch the region on the plane with coordinates x, y
where
2x y 2
2y x 2
x+y 7
x, y 0
Mark the point in this region where the value of 2x + y is the largest.
2. Question 2:
Find the solutions to
Math 340 Lecture 4
Slack variables. Lets actually try solving a system of linear equations using the simplex method. Chvtals textbook has an example it starts with but I figured Id let you read that and try a different one. So I
looked in Vanderbeis textb
Math 340 Lecture 1
The subject of linear programming might be better called linear optimization: the goal is to maximize a
linear function subject to linear constraints. Rather than go into the mathematical formalism right away,
its probably better to loo
Math 340 Lecture 7
Infeasible initial dictionaries. Started discussing this last time, with the example
Maximize 2x1 x2 subject to
x1 + x2 1
x1 2x2 2
x2 1
x1 , x2 0.
Setting up the slack variables gives us an infeasible dictionary, i.e. one where put
MATH 340
Practice for Quiz # 1 on Wednesday Jan 13.
1. Put the following LP in Standard Inequality Form.
Minimize 2x1
x1
+4x2
+x3
+x3
x2
+6
4
=1
x2 , x3 0;
x1 free
2. Using the following dictionary and Anstees Rule, perform two pivots to get to an optimal
MATH 340
Sample Revised Simplex Computations for Quiz 4.
In each of the following questions you are given A, b, c, current basis and B 1 . Determine, using
our revised simplex methods, the next entering variable (if there is one), next leaving variable (i
MATH 340
Practice for Quiz # 3
Wednesday Feb. 3, 2016
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 solution. And in question 2, a different
error/inc
Math 340: Linear Programming
HW# 3
February-March 2017
Problem 1.
Consider an LP in standard form which has as its first constraint
x1 + x2 x3 x4 1.
(1)
Assume that you know (based on other constraints) that the optimal solution has x2 = 0.
Show that the
The University of British Columbia
Midterm Examination - March 6-7, 2017
Mathematics 304
Sections 201-202
Closed book examination
Last Name
Time: 50 minutes
First
Signature
Student Number
Special Instructions:
No books, notes, or calculators are allowed.
Math 340: Linear Programming
HW# 5
March-April 2017
Problem 1.
Players A and B each hide a nickel or a dime. If the hidden coins match, player A gets both;
if they dont match, then B gets both. Find the optimal strategies. Which player has the
advantage,
Math 340 Lecture 10
Avoiding degeneracy - the perturbation method. Last week we left off with one problem: the simplex
method (with the standard rule) could cycle. Its rare enough to not really be an issue in practice, but is
still kind of worrying theore
Math 340 Lecture 3
Matrix formalism for linear programs. The book doesnt go into this until Chapter 7, but I want to
mention it here because thinking in terms of matrices will be important as we go along. We can write a
linear program in standard form ver
Math 340 Lecture 8
Degeneracy. So the one problem were let with is: how do we know that this process actually terminates,
and we dont get stuck in an infinite loop? Well, this is kind of a subtle question. But one way we can get
stuck in an infinite loop
Math 340 Lecture 6
The standard rule, and examples by computer. Last time I set down the following standard rule
for choosing which variables enter and leave using the simplex method.
Choose the entering variable to be the non-basic one with the largest