MASSACHUSETTS INSTITUTE OF TECHNOLOGY
15.053 Optimization Methods in Management Science (Spring 2007) Problem Set 9, Due May 3rd, 2007
You will need 84 points out of 161 to receive a grade of 5.
Problem 1: Dream Team: IP Formulation (20 Points) Coach
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
15.053 Optimization Methods in Management Science (Spring 2007) Virtual Recitation 6, 2007
Welcome to Recitation 6. As always we start with a quiz. You have 15 minutes to complete the quiz. OK. That was a joke. This v
15.053/8
February 12, 2013
Geometry and visualizations of linear programs
Wikipedia User: 4C. License CC BY-SA. This content is excluded from our Creative
Commons license. For more information, see http:/ocw.mit.edu/help/faq-fair-use/.
1
Quotes of the da
15.053/8
April 2, 2013
IP Techniques 1. Branch and Bound
1
Quotes of the Day
The time to relax is when you don't have time for it.
- Attributed to Jim Goodwin and Sydney J. Harris
There is more to life than increasing its speed.
- Mohandas K. Gandhi
2
Ove
15.053/8
March 19, 2013
Integer Programming Formulations 2
references:
IP Formulation Guide (on the website)
Tutorial on IP formulations.
Applied Math Programming
announcement on meetings of teams with staff
1
Quote of the Day
What chiefly characterizes c
15.053/8
April 30, 2013
Decision Trees 1
The example in the first half of todays lecture is a modification
of the example in Bertsimas and Freund: Data, Models, and
Decisions.
1
Quote of the Day
Truly successful decision making relies on a
balance between
15.053/8
March 5, 2013
Game Theory 2
1
Quotes of the Day
New Quotes
2
From Marilyn Vos Savants column.
Say youre in a public library, and a beautiful stranger strikes
up a conversation with you. She says: Lets show pennies to
each other, either heads or t
15.053/8
May 2, 2013
Decision Analysis 2
The Value of Information
1
Quotes of the day
No sensible decision can be made any longer without taking into
account not only the world as it is, but the world as it will be.
Isaac Asimov (1920 - 1992)
A wise man m
Optimization Methods in Management Science
MIT 15.053
Recitation 3
TAs: Giacomo Nannicini, Ebrahim Nasrabadi
At the end of this recitation, students should b e able to:
1. Understand when an LP is: unbounded, degenerate, has multiple optimal solutions.
2.
15.053/8
February 26, 2013
Sensitivity analysis and shadow prices
special thanks to Ella,
Cathy, McGraph, Nooz,
Stan and Tom
1
Quotes of the Day
If the facts don't fit the theory, change the facts.
- Albert Einstein (attributed)
What's the use of a good q
15.053/8
February 7, 2013
More Linear and Non-linear Programming Models
Optimal meal selection at McDonalds.
A (financial) portfolio selection problem.
Introduction to convex functions
Workforce scheduling.
1
Announcements
Optional recitations for 15.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
15.053 Optimization Methods in Management Science (Spring 2007) Recitation 7, April 12th and April 13th, 2007
Problem 1: Ford-Fulkerson Algorithm Consider the following Maximum Flow network.
2
5 7 8 3
3
s
2 4 6
t 1
3
IP Reference guide for integer programming formulations.
by James B. Orlin
for 15.053 and 15.058
This document is intended as a compact (or relatively compact) guide to the formulation of integer
programs. For more detailed explanations, see the PowerPoin
15.053/8
February 5, 2013
Optimization Methods
in Management Science
and Operations Research
Handout: Lecture Notes
1
Class website + more
Class website
please log on as soon as possible
Problem Set 1 will be due next Tuesday
Lots of class information o
15.053
March 7, 2013
Team Projects (plus more)
1
Quotes of the Day
For me, every day is a new thing. I approach each
project with a new insecurity, almost like the first
project I ever did. And I get the sweats. I go in and
start working, I'm not sure whe
15.053/8
February 28, 2013
2-person 0-sum
(or constant sum) game theory
1
Quotes of the Day
My work is a game, a very serious game.
- M. C. Escher (1898 - 1972)
Conceal a flaw, and the world will imagine the worst.
- Marcus Valerius Martialis (40 AD - 103
15.053/8
March 14, 2013
Introduction to Integer Programming
Integer programming models
1
Quotes of the Day
Somebody who thinks logically is a nice
contrast to the real world.
- The Law of Thumb
Take some more tea, the March Hare said to
Alice, very earne
15.053/8
February 21, 2013
Simplex Method Continued
1
Quote of the Day
Everyone designs who devises courses of
action aimed at changing existing situations into
preferred ones.
- Herbert Simon
2
Todays Lecture
Very quick review of the simplex algorithm.
P
15.053
February 14, 2013
Review of Guassian elimination for solving systems
of equations
Introduction to the Simplex Algorithm
1
Quotes for today
Any impatient student of mathematics or
science or engineering who is irked by having
algebraic symbolism thr
Optimization Methods in Management Science
MIT 15.053
Recitation 1
TAs: Giacomo Nannicini, Ebrahim Nasrabadi
Problem 1
You create your own start-up company that caters high-quality organic food directly to a number
of customers. You receive a number of te
Optimization Methods in Management Science
MIT 15.053
Recitation 3
TAs: Giacomo Nannicini, Ebrahim Nasrabadi
At the end of this recitation, students should b e able to:
1. Understand when an LP is: unbounded, degenerate, has multiple optimal solutions.
2.
Optimization Methods in Management Science
MIT 15.053
Recitation 8
TAs: Giacomo Nannicini, Ebrahim Nasrabadi
At the end of this recitation, students should b e able to:
1. Derive Gomory cut from fractional rows of the Simplex tableau.
2. Have an intuititi
Optimization Methods in Management Science
MIT 15.053
Recitation 9
TAs: Giacomo Nannicini, Ebrahim Nasrabadi
Problem 1
We apply Dijkstras algorithm to the network given below to nd the shortest
path between S and T. (Arc labels represent distances.)
6
A
D
Optimization Methods in Management Science
MIT 15.053
Recitation 9
TAs: Giacomo Nannicini, Ebrahim Nasrabadi
Problem 1
We apply Dijkstras algorithm to the network given below to nd the shortest
path between S and T. (Arc labels represent distances.)
6
A
D
Optimization Methods in Management Science
MIT 15.053, Spring 2013
Recitation 10, Friday May 3th, 2013
Problem 1
The Airfare Problem1 . You are trying to get the cheapest airfare that you can. You just
called up and found that the ticket home will cost $4
Optimization Methods in Management Science
MIT 15.053
Recitation 4
TAs: Giacomo Nannicini, Ebrahim Nasrabadi
At the end of this recitation, students should b e able to:
1. Interpret the solution and sensitivity report of a problem to take decisions relat
15.053/8
April 9, 2013
Integer Programming 5
Gomory cuts
Constraint generation
Formulations using set notations
1
Quote of the Day
Mathematicians are like Frenchmen: whatever you
say to them, they translate into their own language,
and forthwith it is som
Optimization Methods in Management Science
MIT 15.053
Recitation 7
TAs: Giacomo Nannicini, Ebrahim Nasrabadi
Problem 1
Suppose branch and bound is being applied to a 01 integer program in which we are maximizing.
By node 11 the rst three variables have be
Optimization Methods in Management Science
MIT 15.053
Recitation 8
TAs: Giacomo Nannicini, Ebrahim Nasrabadi
At the end of this recitation, students should b e able to:
1. Derive Gomory cut from fractional rows of the Simplex tableau.
2. Have an intuititi