Midterm Exam
Math 308
Simon Fraser University
Fall, 2013
D. Funk
Pertinent info:
1. To receive full credit for a particular question you must provide a complete and well presented
solution.
2. This exam has 8 questions on 4 pages. Please check to make sur
Homework Solutions
Math 308-3 Fall 2005 Luis Goddyn
1. Assignment 1 Page 23 #1: a: Any triangle in the plane will do, for example. b: The whole of R2 will do, for example. e: Any disc with a circular boundary will do, for example. i: The positive quadrant
SCH/Li Ié for! K
Second Midterm Exam MATH 308-3 Family Name: __________________________ H- j
November 9, 2010 Mathematics, SFU Given Name: __________________________ __
Prof. Luis Goddyn Student Number: __________________________ _,
Instructions: Do a
Simon - Examination Number of booklets used
.' Fraser BOO This is bookiet number
$§Wiii U mVe rSIty / ' '\ Place additional exam books and
notes inside first exam booklet.
Student number
our examination \.
C as of R #/
me (Ptease print in block le
MATH 308 - Assignment 1
1. Find necessary and sufficient conditions for the numbers s and t to make the LP problem
Maximize: f (x1 , x2 ) = x1 + x2
subject to sx1 + tx2 1
x1 , x2 0
a) have an optimal solution.
b) be infeasible.
c) be unbounded.
Prove your
Assignment 2 Solutions
Math 308 Linear Optimization
Fall 2013
D. Funk
Exercises from Chapter 2 (page 82):
2.1 A set of vectors A is a basis for Rn if and only if they form a minimal spanning set
of Rn if and only if |A| = n and the vectors are linearly in
MATH 308 HW3 ANSWER KEY
1.
LP problem in an equational form is
maximize f (x1 , x2 , x3 , x4 ) = c| xs
subject to As xs = b, xs > 0
with xs = (x1 , x2 , x3 , x4 ), As =
1
4
4
1
1 0
, cs = (1, 1, 0, 0), b =
0 1
(18, 18).
Obviously last two columns of the
First Midterm Exam
MATH 308-3
Solution Key
Instructions: Do all ve questions, writing each answer in the space provided (use back if necessary).
Their point values are as indicated. Total = 50 points. Duration = 50 minutes. No calculators!
(1) (a) Convert
Review - Midterm
Math 308 Linear Optimization
Fall, 2013
D. Funk
Sections covered in the text: 1.1-1.4, 2.1-2.7, 3.1-3.8.
Ive listed vocabulary and concepts you should know. Dont try to memorize everything
on this list as a collection of unrelated facts.
MATH 308 - Assignment 2
1. A post office requires different numbers of full-time employees on different days of the
week. The number of full-time employees required on each day is as follows: Day1 (Monday)
17, Day2 (Tuesday) 13, Day3 (Wednesday) 15, Day4
MATH 308 - Assignment 4
1. A company sells bags of grapes and cartons of grape juice. The company grades grapes
on the scale from 1 (poor) to 10 (excellent). At present, the company has on hand 150,000
lb of grade 9 grapes and 200,000 lb of grade 6 grapes
MATH 308 HW5 ANSWER KEY
1. If a = 0, all cases follow easily. Therefore, we assume a 6= 0. By symmetry of
the equations, we may also assume without loss of generality that a > 0.
a) Using the SA, the current tableau is maximum basic feasible, but not opti
MATH 308 HW4 ANSWER KEY
1. Let x1 and x2 be the pounds of grade 9 grapes used in bags and juice, respectively; let y1 and y2 be the pounds of grade 6 grapes used in bags and juice,
respectively.
There are 150,000 lb of grade 9 grapes, so x1 + x2 6 150000.
MATH308 D100, Spring 2016
9. Simplex algorithm for maximum tableau
(based on notes from Dr. J. Hales)
Ladislav Stacho
SFU Burnaby
1/8
SA for Maximum Tableaux
Algorithm (SA for MT)
a11
a21
.
.
.
am1
c1
(ind vars)
a12
.
a22
.
.
.
.
.
.
am2
.
c2
.
a1n
a2n
.
MATH308 D100, Spring 2016
7. Tucker tableau and pivot transformation
(based on notes from Dr. J. Hales)
Ladislav Stacho
SFU Burnaby
1/9
Tucker Tableaux
Given system of linear equations As x s = b and corresponding augmented matrix
a11
a21
a12
a22
.
.
.
.
MATH308 D100, Spring 2016
8. Simplex algorithm for maximum basic feasible tableau
(based on notes from Dr. J. Hales)
Ladislav Stacho
SFU Burnaby
1/9
So Far We Know. . . .
How to describe a problem as a maximization LP problem.
How to convert the problem t
MATH 308 - Assignment 3
1. Consider the following maximization LP problem in an equational form:
Maximize f (x1 , x2 , x3 , x4 ) = x1 x2 , subject to
x1 + 4x2 + x3 = 18
4x1 x2 + x4 = 18
x1 , x2 , x3 , x4 0
What are the matrix As and vectors cs and b of t
MATH 308 ASSIGNMENT 5 ANSWER KEY
1. If a = 0, all cases follow easily. Therefore, we assume a 6= 0. By symmetry of
the equations, we may also assume without loss of generality that a > 0.
a) Using the SA, the current tableau is maximum basic feasible, but
First Midterm Exam
MATH 308-3
2017
Solution Key
(1) (a) [5 pts] Convert the following linear program (LP) into a canonical maximization LP problem.
min 2x + 3y z
x+y + z =2
y 2z 4
x, z 0
Solution: Negate the objective function, convert = to inequalities,
Homework Solutions
Math 308-3
Fall 2005
Luis Goddyn
1. Assignment 1 Page 23
#1:
b.
a.
c.
d.
(empty set)
e.
i.
f.
f. and g.
h.
j.
Notes: I have drawn simple examples of each. Of course there are many variations for
most of the solutions.
a: Any polygon in
MATH 895-4 Fall 2010
Course Schedule
f ac ulty of sc ience
depar tment of mathem atics
A SSIGNMENT 3 Solutions
MATH 308
Spring 2017
Textbook P. 83
Week
#1 a.
1
2
3
4
5
Date
Oct 5
12
7
19
t1
8
0
2
26
1
9
Nov 2
11
12
Part/ References
Topic/Sections
Notes/Sp
MATH 895-4 Fall 2010
Course Schedule
f ac ulty of sc ience
depar tment of mathem atics
A SSIGNMENT 2
MATH 308
Spring 2017
Due Monday January 27, 2:20pm in Assignment Box 5b, across the hall from AQ 4110.
Week
Date
Sections
Part/ References
1
Sept
(a) 7
I.
MATH 895-4 Fall 2010
Course Schedule
f ac ulty of sc ience
depar tment of mathem atics
A SSIGNMENT 1
MATH 308
Spring 2017
Due Friday January 13, 2:20pm in Assignment Box 5b, across the hall from AQ 4110.
Week
Date
Sections
1
Sept 7
Part/ References
Topic/
MATH 895-4 Fall 2010
Course Schedule
f ac ulty of sc ience
depar tment of mathem atics
A SSIGNMENT 2
MATH 308
Spring 2017
Due Monday January 27, 2:20pm in Assignment Box 5b, across the hall from AQ 4110.
Week
Date
Sections
1
Sept 7
Part/ References
Topic/
MATH 895-4 Fall 2010
Course Schedule
f ac ulty of sc ience
depar tment of mathem atics
A SSIGNMENT 3
MATH 308
Spring 2017
Due Friday March 10, 2:20pm in Assignment Box 5b, across the hall from AQ 4110.
Week
Date
Sections
1
Sept 7
Part/ References
Topic/Se
MATH 308 - Assignment 5
1. Consider the dual canonical tableau.
x1
y1
a
y2 a
1
c
= s1
x2
a
a
c
= s2
1
b
b
0
=g
= t1
= t2
=f
Prove that
a) If b > 0 and c > 0, then the primal problem (the maximization problem) is unbounded
and the dual problem is infeasibl
The Transportation Problem - continued
Department of Mathematics, Simon Fraser University Surrey
Lecture 17 - Math 308 - 2016-11-18
Department of Mathematics, Simon Fraser University
The Transportation Problem - continued
Transportation problem
Minimize
m
SFU Surrey - Mathematics
MATH 308 Fall 2016 Surrey campus
Lecture 6, 2016-09-23, Friday
Fundamental Theorem of Linear Programming
Abraham Punnen
Abraham .P. Punnen
Lecture6-MATH308-2016-09-23-article-3.pdf
Page 1 of 21
SFU Surrey - Mathematics
A linear e
SFU Surrey - Mathematics
MATH 308 Fall 2016 Surrey campus
Lecture 13, 2016-11-2, Wednesday
Post-optimality analysis
Abraham Punnen
Abraham .P. Punnen
Lecture13-MATH308-2016-11-2-article-3.pdf
Page 1 of 15
SFU Surrey - Mathematics
Changes in cost vector c
SFU Surrey - Mathematics
MATH 308 Fall 2016 Surrey campus
Lecture 11, 2016-10-19, Wednesday
The dual simplex method
Abraham Punnen
Abraham .P. Punnen
Lecture11-MATH308-2016-10-19-article-3.pdf
Page 1 of 16
SFU Surrey - Mathematics
Consider the linear pro