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
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 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
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 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.
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 - 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 - 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
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 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
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
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
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
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
.
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
Assignment 4 Solutions
Math 308 Linear Optimization
D. Funk
Fall 2013
Due Friday, Nov. 29
Exercises from Chapter 4 (page 187), Chapter 6 (page 319), and Chapter 8 (page 435):
When solving a linear programming problem using the two-phase or big-M method, y
Supplementary Notes
Math 308-3
Fall 2010
Luis Goddyn
These notes are a supplement and summary to the textbook for Math 308. Students are
expected to know this material for the nal examination.
1. Convexity of Feasible Regions
Here is the missing proof of
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
Review - Final Exam
Math 308 Linear Optimization
D. Funk
Fall, 2013
Sections covered in the text: 4.1-4.4, some of 4.6, 6.1-6.2, 8.1-8.2, 9.1.
Ive listed vocabulary and concepts you should know. Dont try to memorize everything
on this list as a collection
Assignment 3 Solutions
Math 308 Linear Optimization
Fall 2013
D. Funk
Due Friday, Nov. 1
1. Solve the following linear programming problem by the simplex method.
Maximize 3x1 + 2x2 + x3
subject to 3x1 3x2 + 2x3 3
x1 + 2x2 + x3 6
x1 , x2 , x3 0
At each ite
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.
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
Page 1 of 3
Department of Mathematics
Homework 4
Term: Fall 2016 (September 6 - December 5)
Student ID Information
Last name:
First name:
Student ID #:
sfu.ca userid:
Course ID:
Math 308
Course Title:
Linear Optimization
Instructor:
Abraham P Punnen
Due D
MATH 308 - ASSIGNMENT 1 - ANSWER KEY
1. There are two cases:
Case 1: s, t > 0
Since 1/s > 0 and 1/t > 0, the set of solutions consists of all
points inside the triangle with vertices (0, 0), (1/s, 0), and (0, 1/t).
Since the set is non-empty and bounded,