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.
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
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
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
Problem 1 (10 Marks)
Solve the following initial value problems.
(a)p%=(p21)(q2+1) g(1)=0
=p ii. \ 1 a _1
12+: = EH 0? (1 P
_ Am
J 3;: JP VP Problem 2 (15 Marks)
Compute the following integrals.
0 d9: "
(a) f +~ z j *3 ax.
00 8 61:; + l
" DO
,2 I
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
Cardiac Muscle
Found only in heart
Striated
Each cell usually has one nucleus
Has intercalated disks and gap junctions
Autorhythmic cells
Action potentials of longer duration and
longer refractory period
Ca2+ regulates contraction
Cardiac Muscle
Elongated
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
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 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 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
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 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
$
October 2015
MATH 340
Name
Page 2 of 6 pages
Marks
[8]
1.
Consider the problem: maximize x1 subject to x1 6, x1 10, x1 7, x1 0. Write
this as a linear program in standard form. Use the two-phase method, adding an
auxilliary variable x0 to EVERY slack vari
Math 340
Midterm
Wishesday, Octember 1, 2016
Explain your work. Name LP theorems as you use them.
1.[30pts] Solve the following LP using our two phase method with Anstees rule. You will need a
fake pivot to feasibility and two more pivots in Phase One. In
Math 340
Assignment #2
Due Monday Oct 24, 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 Pivots Preserve the Set of All Solutions
It is crucial that we know that the pivot process preserves the set of solutions to the
dictionary. We will not concern ourselves with positivity. You should be able to see that
the result applies to all p
MATH 340 Example of Infeasibility.
Richard Anstee
We give an example of an LP where we are unable to drive the artificial variable x0 to 0. Or
equivalently we find that when we maximize w = x0 we are ubable to get ot to zero so we cannot
proceed to Phase
Example of Cycling (from Chvatal)
This example has the virtue of suffering from no roundoff errors when
run on a computer. Cycling in LPs remains rare and so many implementations do not implement an anti-cycling rule. We use Anstees pivot rules
(which are
MATH 340
Simplex Algorithm
One of the interesting features of MATH 340 is that it is dominated by an algorithm. In fact we
prove the important Duality theorems using our algorithm. This approach of proving a theorem
using an algorithm may be unusual to yo
MATH 340 Standard Inequality Form
Richard Anstee
Any Linear Programming problem can be put in standard inequality form which is the maximization of a linear objective function subject to inequalities each of which is a linear function
of the variables les
Math 340 Dual Simplex resulting in infeasibility
Consider a primal
max c x
Ax b .
x0
If we have a dictionary with all the coefficients in the z row are negative (namely cN cB B 1 AN
0T then we can call this dual feasible since cTB B 1 would be a feasible
MATH 340 Example of pivoting to optimal solution(s).
Richard Anstee
We considered the following LP in standard inequality form
max 4x1 +3x2 +x3 +x4
x1 +2x2
x4 3
2x1 +x2 x3 +x4 2
x2 +x3
2
x1 , x2 , x3 , x4 0
We add slack variables x5 , x6 , x7 correspondi
MATH 340 Example of Degenreracy in pivoting process.
Richard Anstee
We give an example of an LP where we are forced to do what we call degenerate pivots. We
still obtain an optimal solution(s). Degenerate pivots can result in cycling and you should read
C
MATH 340
A Sensitivity Analysis Example from lectures
The following examples have been sometimes given in lectures and so the fractions are rather
unpleasant for testing purposes. Note that each question is imagined to be independent; the changes
are not
Math 340
Practice Midterm Solutions
Explain your work. Name LP theorems as you use them.
October 29, 2016
1.[30pts] Solve the following LP using our two phase method with Anstees rule. You will need a
fake pivot to feasibility and two more pivots in Phase
MATH 340
Practice for Quiz # 5
This quiz on sensitivity analysis will have a few questions about determining ranges in which
the basis stays optimal and one question where you need to do either usual or dual simplex method
pivoting. In either case you nee
Math 340
Midterm
Wednesday February 22, 2006
Explain your work. Name LP theorems as you use them.
1.[30pts] Solve the following LP using our two phase method with Anstees rule. You will
need a fake pivot to feasibility and two more pivots in Phase One. In
Math 340
Assignment #3
Due November 4, 2016
1. Consider an LP in standard form which has as its first constraint
x1 + x2 x3 x4 1.
Assume that you know (based on other constraints) that x2 = 0. Show that the dual variable
y1 associated with the first const
Assignment #3
Math 340
Due Monday November 7, 2016
1. Consider an LP which has as its first constraint
x1 + x2 x3 x4 1.
Assume that you know (based on other constraints) that x2 = 0. Show that the dual variable
y1 associated with the first constraint is