CO 342 Spring 2017 Week 5 Summary
1
Cut space
Theorem 1.1. Let G be connected and let T be a spanning tree. The set of all fundamental cuts
in G with respect to T form a basis for C(G).
Corollary 1.2. If G is connected, then dim C(G) = |V | 1.
Theorem 1.3
CO 342 Spring 2017: Assignment 3
Due: Wednesday May 24 at 11:00am.
1. Let k 2. Let G be a k-connected graph with at least 2k vertices. Prove that G
contains a cycle of length at least 2k.
2. Suppose G has at least 3 vertices. Prove that G is 2-connected i
CO 250: Introduction to Optimization
Module 2: Linear Programs (Possible Outcomes)
c
University
of Waterloo
1 / 12
What Does Solving an Optimization Problem Mean?
input
LP or
IP or
NLP
max
2x1 3x2
s.t.
x1 + x2 1
output
algorithm
(software)
solution
x1 = 1
1
Assignment 1 - CO 480 Spring 2017
I.D. Number:
First Name:
Due Date: Thursday May 18th, 2:30pm
Family Name:
List any references that you used beyond the course text and lectures (e.g. discussions, texts or online
resources) at the end of each relevant q
C&O 454 Scheduling Spring 2017
Assignment 1
Due: Thu, May 25 in class before lecture
Bonus problem: Due by Tue, May 30
Write your name and ID# clearly, and underline your last name. Acknowledge all collaborators.
Unless otherwise stated, all algorithms sh
CO 370 Midterm Exam, Winter 2017
Page 1
UNIVERSITY OF WATERLOO
Midterm Exam
Winter 2017
Student Name
Student Identification Number:
Course Number
CO 370
Course Title
Deterministic Operations Research Models
Section(s)
001
Instructor
Laura Sanit`a
Date of
CO 370 - Homework assignment 5
Fall 15
Page 1
CO 370 - Fall 15
Homework assignment #5: (Due on Friday, Nov 27, at the beginning of the class)
Instructions:
You will be graded not only on correctness, but in clarity of exposition.
Allowed sources of help
CO 370 - Homework assignment 4
Winter17
Page 1
CO 370 - Winter17
Homework assignment #4: (Due on Thurs March 16nd, at the beginning of the class - 2:30
pm)
Instructions:
You will be graded not only on correctness, but also on clarity of exposition.
Allo
CO 370 - Homework assignment 3
Winter17
Page 1
CO 370 - Winter17
Homework assignment #3: (Due on Thurs March 2nd, at the beginning of the class - 2:30
pm)
Instructions:
You will be graded not only on correctness, but also on clarity of exposition.
Allow
CO 370 - Homework assignment 4
Fall 15
Page 1
CO 370 - Fall 15
Homework assignment #4: (Due on Monday, Nov 16, at the beginning of the class)
Instructions:
You will be graded not only on correctness, but in clarity of exposition.
Allowed sources of help
CO 370 Midterm Exam, Fall 2015
Page 1
Question 1 (20 points)
Shoe Inc. forecasts the following demand of pairs of shoes for the next six months:
Month
Demand
1
5000
2
6000
3
5000
4
9000
5
6000
6
5000
Each shoemaker takes 0.25h to make a pair of shoes and
Practice midterm for CO 370
October 21, 2015
1. Daniel Jack produces two types of whiskeys. Both whiskeys are produced using corn, rye and barley.
There are 50kgs of corn, 40kgs of rye and 60kgs of barley available. The ingredients are mashed together,
th
Question 1 (20 points)
Consider the following integer program:
max z = x1
s.t.
4x1
3x1
x1 , x2
+5x2
+3x2 6
+2x2 18
0 and integer
(IPo )
The picture below shows the branch-and-bound tree in the middle of solving this problem (only the
nodes that were alre
CO 370 - Practice assignment
Winter17
Page 1
CO 370 - Winter17
Practice assignment
Question 1 (0 points)
Plastic Inc. produces 4 different products: A, B, C and D. Each product requires a certain amount
of polyurethane (PU) and polypropylene (PE). The tab
CO 370 - Homework assignment 4
Winter17
Page 1
CO 370 - Winter17
Homework assignment #4: (Due on Thurs March 16nd, at the beginning of the class - 2:30
pm)
Instructions:
You will be graded not only on correctness, but also on clarity of exposition.
Allo
CO 370 - Homework assignment 5
Winter17
Page 1
CO 370 - Winter17
Homework assignment #5: (Due on Thursday, March 30th, at the beginning of the class)
Instructions:
You will be graded not only on correctness, but also on clarity of exposition.
Allowed so
CO 370 - Homework assignment 5
Winter17
Page 1
CO 370 - Winter17
Homework assignment #5: (Due on Thursday, March 30th, at the beginning of the class)
Instructions:
You will be graded not only on correctness, but also on clarity of exposition.
Allowed so
CO 370: Midterm Information
When?
Monday February 27th, from 4.30 to 6.20 pm
Where?
DC 1350
Exam type?
I
I
I
Closed book exam
No collaboration is allowed
Allowed material: only Faculty of Math approved
calculators
CO250 I NTRODUCTION TO O PTIMIZATION , W INTER 2017
A SSIGNMENT 10 SOLUTIONS
Exercise 1 (25 marks).
un
e
l
Due: Monday, April 3, 2017, at 4:00 p.m.
(a) (10 marks) Consider the following 2-dimensional polyhedron (which in this case is a square) defined by
CO250 I NTRODUCTION TO O PTIMIZATION , W INTER 2017
A SSIGNMENT 6 SOLUTIONS
Due: Friday, February 17, 2017, at 4:00 p.m.
Exercise 1 (40 marks). [Two-phase simplex method]
3 x
6
2 4 2 1
1 0 1 1 x = 1
7
5 1 2 0
1
1 1
1
0
0
subject to
un
e
max
l
Consider
CO250 I NTRODUCTION TO O PTIMIZATION , W INTER 2017
A SSIGNMENT 7 SOLUTIONS
Exercise 1 (30 marks).
(a) (20 marks) Let
un
e
l
Due: Friday, March 10, 2017, at 4:00 p.m.
3 1 2
1
0
3
0 2
2 2
1 , b := 0 , c :=
2
4
1
2 2
8
5
5
3
7
1
ao
,L
A :=
.T
describe the
CO250 I NTRODUCTION TO O PTIMIZATION , W INTER 2017
A SSIGNMENT 8 SOLUTIONS
Due: Friday, March 17, 2017, at 4:00 p.m.
Exercise 1 (20 marks). [Dual Transformations]
max
(P)
x1 + 2x2 + 3x3
subject to
3x1 + 5x2 + 7x3 11
.T
2x1 + 4x2 + 8x3 = 21
un
e
l
Conside
CO250 I NTRODUCTION TO O PTIMIZATION , W INTER 2017
A SSIGNMENT 9 SOLUTIONS
Due: Friday, March 24, 2017, at 4:00 p.m.
Exercise 1. (35 marks)
(IP)
un
e
l
Consider the integer programming problem:
maxcfw_x1 + x2 : x1 + 9x2 10, 2x1 19, x 0, x integer
(Note:
CO 331: Assignment 2
Due date: Thursday February 2 2017, beginning of class.
Please note that assignments are not weighted equally. Each problem on each assignment is worth
10 marks. The total marks received on assignments will be added together at the en