15.053
Overview of Techniques for Solving
Integer Programs
Tuesday, April 9
Enumeration Techniques
Branch and Bound
Complete Enumeration
list all solutions and choose the best
Branch and Bound
Implicitly search all solutions, but cleverly eliminate th
15.053
A 2-Variable Integer program
Thursday, April 4
Introduction to Integer Programming
maximize 3x + 4y
Integer programming models
subject to
Handouts: Lecture Notes
5x + 8y 24
x, y 0 and integer
What is the optimal solution?
1
The Feasible Region
Why
15.053
The Minimum Cost Flow Problem
Tuesday, April 2
The Shortest Path Problem
Dijkstras Algorithm for Solving the Shortest
Path Problem
Handouts:
Lecture Notes
1
Formulation
Directed Graph G = (N, A).
Node set N, arc set A;
Capacities uij on arc (i,j)
l
15.053
Network Models
Thursday, March 14
Linear Programming models that exhibit a
very special structure
Can use this structure to dramatically
reduce computational complexity
First widespread application of LP to
problems of industrial logistics
Addresse
15.053
Bounds
Tuesday, March 5
Duality
One of the great contributions of
optimization theory (and math
programming) is the providing of upper
bounds for maximization problems
The art of obtaining bounds
weak and strong duality
We can prove that solution
15.053
February 26, 2002
Glass Example
x1 = # of cases of 6-oz juice glasses (in 100s)
x2 = # of cases of 10-oz cocktail glasses (in 100s)
x3 = # of cases of champagne glasses (in 100s)
Sensitivity Analysis
presented as FAQs
max 5 x1
s.t
6 x1
10 x1
x1
x1
15.053
Todays Lecture
February 21, 2002
Simplex Method Continued
Review of the simplex algorithm.
Formalizing the approach
Degeneracy and Alternative Optimal Solutions
Is the simplex algorithm finite? (Answer, yes,
but only if we are careful)
Handouts: Le
15.053
Review of Linear Algebra
February 7, 2002
Some elementary facts about vectors and
matrices.
The Gauss-Jordan method for solving
systems of equations.
Bases and basic solutions and pivoting.
A brief review of Linear Algebra
Linear Programming Models
15.053
An Airline Revenue Management
Problem
To accompany lecture
on February 7
Some additional Linear Programs (not covered
in lecture)
Background: Deregulation occurred in 1978
Prior to Deregulation
Carriers only allowed to fly certain routes. Hence ai
15.053
February 5, 2002
Overview
Course Description
Course Administration and Logistics
What is Management Science?
Linear Programming Examples
Introduction to Optimization
Handouts: Lecture Notes
MSR Marketing
GTC
Handouts:
Syllabus and General Info.
L
Practice Final Exam.
15.053 May, 2002
Note. This exam does not cover every topic mentioned in class on the last day, each of
which may appear on the final exam.
1. Consider the following quadratic programming problem.
Min x12 + 2x22 + 2x1 4x2
s.t.
3x1 + 2
15.053 Midterm
Tuesday, March 12, 2001
(closed book )
1.
Answer all questions in exam books provided.
2.
Budget your time. If a problem (or a part of a problem) is taking too long, you
may want to go on to the next one.
3.
If you think that there is an am
Homework 8
15.053 Introduction to Optimization
Due at the beginning of class Thursday, May 2, 2002.
1. BH&M, Exercise 1b, p. 608
2. BH&M, Exercise 4, p.609-610
3. Consider the following optimization problem:
Min
s.t.
e
x1+ x2
0 x1 4
0 x2 4
a) Express the
15.053
Example: Fire company location.
Thursday, April 11
Some more applications of integer
programming
Consider locating fire companies in different
districts.
Objective: place fire companies so that each
district either has a fire company in it, or one
15.053
Thursday, April 25
Difficulties of NLP Models
Linear
Program:
Nonlinear Programming Theory
Separable programming
Nonlinear
Programs:
Handouts: Lecture Notes
1
Graphical Analysis of Non-linear programs
in two dimensions: An example
Minimize
( x 14)
15.053 Tuesday, April 30
Dynamic Programming
Transforms a complex optimization problem
into a sequence of simpler ones.
Usually begins at the end and works
backwards
Can handle a wide range of problems
Relies on recursion, and on the principle of
optimali
15.053
April 5, 2016
Duality in Linear Programming
Quote of the Day
Just as we have two eyes and two feet, duality
is a part of life.
- Carlos Santana
2
Comments on the upcoming midterm
4 to 5 PM. 4/13. E51-345.
Review session. April 12 at 7 PM?
if ther
Midterm 1 Information.
Day: March 16, 2016
Time: 7:30 PM to 9:30 PM
Room: E51-315
Note time change
Crib notes:
Each student is permitted crib notes written on one side of one page of paper
Midte
5/2/16
15.053
Comments on the upcoming midterm
May 3, 2016
4 to 5 PM. 5/4.
E51-395 (Last name begins with A-M)
E51-361 (Last name begins with N-Z)
Review session, course overview
C
A note on reduced costs.
Problem 1e from Recitation 5 is a problem on reduced costs. It asks the following:
1e. The contribution of each unit of Desks to Revenue is 90. Producing one unit of Desks
requires 2 hours of Cutti
Helpful Tips for Using Microsoft Excel
Throughout this course, we will be using Microsoft Excel extensively. The purpose of this note is to
provide some troubleshooting tips for the often-quirky Solver function and to introduce some built-in
functions tha
Recitation 12
15.053 Introduction to Optimization
May 10, 2002
1.
Suppose that there are two piles of matches on a table. One pile has 5 matches
and the other pile has 10 matches. The person who picks up the last match wins.
At each alternating turn, my o
Recitation 11
15.053 Introduction to Optimization
Friday May 3, 2002.
1. Shortest path on an acyclic network: Given an acyclic network with topological ordering,
write the stages, recursion, initial condition and the final goal for solving the shortest pa
Recitation 9 Problems
More Practice Problems for Midterm II
These problems are intended to aid in your understanding of the topics. Please make sure
you also understand everything from the practice midterm.
15.053 Introduction to Optimization
April 19,200
Recitation 5 Midterm 1 Review
Our thoughts on topics covered so far (with help from BHM)
This is not intended to be comprehensive
March 8, 2002
First tip: ANSWER THE QUESTION ASKED!
Second tip: TRY NOT TO LEAVE ANY QUESTION BLANK. PUT SOMETHING
DOWN! Oth
Recitation 3 Problems
15.053 Introduction to Optimization
February 22, 2002
1. Fernwood Labor (More practice on formulations)
Fernwood Lumber produces plywood. The cost to produce 1000 board feet of plywood
varies from month to month because of the variat