COMP SCI / SFWR ENG 4O03 Midterm Solution
1a. True; max 0x1 +0x2 and no constraint on x1 and x2 (both variables are free variable)
1b. True; max 1 x2 with x1 1, x2 1, x2 1, x1 0, x2 0
1c. True; we can without loss of generality that the LP is a maximizati
COMP SCI / SFWR ENG 4O03 / 6O03: McMaster University Final Exam - Day Class
December 17, 2013 (Instructor: Antoine Deza)
SOLUTIONS
Problem 1
Prove or disprove each of the following claims:
(a) Let G be a simple connected directed graph with distinct edge
COMP SCI / SFWR ENG 4O03 / 6O03: McMaster University Final Exam - Day Class
April 13, 2012 (Instructor: Antoine Deza)
SOLUTIONS
Problem 1
Consider the following knapsack problem:
max
4x1 + 5x2 + 3x3 + 4x4
such that
5x1 + 4x2 + 2x3 + 3x4 9
(a) Solve the pr
COMP SCI / SFWR ENG 4O03 / 6O03: McMaster University Final Exam - Day Class
December 15, 2014 (Instructor: Antoine Deza)
DURATION OF THE EXAM: 3 hours - 40 points
Name:
Student Number:
This examination paper includes 2 pages and 6 problems. You are respon
COMP SCI / SFWR ENG 4O03 / 6O03: McMaster University Final Exam - Day Class
April 13, 2012 (Instructor: Antoine Deza)
DURATION OF THE EXAM: 3 hours - 40 points
Name:
Student Number:
This examination paper includes 2 pages and 6 problems. You are responsib
COMP SCI / SFWR ENG 4O03 Operations Research
Problem Set 3 18 points
Handed out: Oct 28, 2013
Due date: Nov 26, 2013
* justify your answers *
1. Find the shortest path from vertex 1 to vertex 3 in the following graph using Dijkstras
algorithm.
4p
1
7
2
10
COMP SCI / SFWR ENG 4O03 Operations Research
Problem Set 1 10 points
Handed out: September 6, 2013
Due date: October 4, 2013
* justify your answers *
1. Consider the following problem: Find (, ) such that the plane T x = (i.e. 1 x1 +
2 x2 + 3 x3 = ) separ
COMP SCI / SFWR ENG 4O03: McMaster University MIDTERM
October 11, 2013 (Instructor: Antoine Deza)
DURATION OF THE EXAM: 2 hours - 30 points
Name:
Student Number:
This examination paper includes 2 pages and 5 problems. You are responsible for ensuring that
COMP SCI / SFWR ENG 4O03: McMaster University MIDTERM
October 16, 2012 (Instructor: Antoine Deza)
DURATION OF THE EXAM: 2 hours - 30 points
Name:
Student Number:
This examination paper includes 2 pages and 5 problems. You are responsible for ensuring that
COMP SCI / SFWR ENG 4O03 / 6O03: McMaster University Final Exam - Day Class
December 17, 2013 (Instructor: Antoine Deza)
DURATION OF THE EXAM: 3 hours - 40 points
Name:
Student Number:
This examination paper includes 2 pages and 6 problems. You are respon
COMP SCI / SFWR ENG 4O03 Operations Research
Problem Set 2 26 points
Handed out: September 30, 2013
Due date: October 29, 2013
* justify your answers *
1. While solving an LP maximization problem we obtain the following tableau. The basic
variables are x1
COMP SCI / SFWR ENG 4O03 Operations Research
Problem Set 1 Solution (2014)
1. We can normalize the right hand side and represent a plane as a1 x1 + a2 x2 + a3 x3 = 1.
a1 , a2 and a3 are the variables of the LP problem. Since it is a feasibility problem, a
COMP SCI / SFWR ENG 4O03 Operations Research
Problem Set 3 Solution
1. The shortest path is 1 4 6 7 3 with length 13. Details ignored.
2.1. We can construct a directed weighted graph in the following way: (1) Create a
vertex i for each variable xi ; (2) F
COMP SCI / SFWR ENG 4O03 / 6O03: McMaster University Final Exam - Day Class
December 17, 2013 (Instructor: Antoine Deza)
SOLUTIONS
Problem 1
Prove or disprove each of the following claims:
(a) Let G be a simple connected directed graph with distinct edge
COMP SCI / SFWR ENG 4O03 Operations Research
Problem Set 2 Solution
1.1. a 0
1.2. a < 0, d > max( 3b , 0)
7
1.3. a < 0; b, d 0.
2.1. Introduce the slack variables x5 and x6 , the standard form is:
min
such that
3x1 x2 + x3
x1 + x2 + x3 + x4
2x1 x2 x5
x1 2
COMP SCI / SFWR ENG 4O03 Midterm Solution
1a. False; counterexample: max x1 + x2 such that x1 + x2 2 and x1 + x2 1.
1b. False; counterexample: max x1 + x2 such that x1 + x2 2.
1c. True; example: max cx1 such that x1 1 and x1 1, .
1d. False; counterexample
COMP SCI / SFWR ENG 4O03 / 6O03: McMaster University Final Exam - Day Class
December 15, 2014 (Instructor: Antoine Deza)
SOLUTIONS
Problem 1
Prove or disprove each of the following claims:
(a) If all edges in an undirected graph have dierent positive inte