IEOR 4004 - Intro to OR - Deterministic models - Lecture 8
October 1, 2013
1. Duality
1.1. Pricing interpretation
Back to our usual manufacturing LP problem. For the sake of illustration, we drop the 3rd constraint, and
consider the items as blocks of woo
IEOR 4004 - Intro to OR - Deterministic models - Lecture 7-1
September 26, 2013
1. Sensitivity Analysis
changing the objective function
changing the right-hand side (rhs)
i = shadow price of the i-th constraint with rhs bi
the amount by which the valu
Columbia University
IEOR E4004
Recitation #1
Instructor: Dawn Strickland
Teaching Assistant: Gonzalo Muoz
n
P1. (Problem 1.2 from BHM) Consider the following linear program:
max
s.t
2x1 + x2
12x1 + 3x2 6
3x1 + x2 7
x2 10
x1 , x2 0
(a) Draw a graph of the
IEOR 4004 Homework 2 solutions
Note: The solutions below are only sample solutions. Any solution so long as it is correct (and so long as
it uses dictionaries) will be accepted.
Problem #1: Solve the following LPs using the dictionary Simplex method.
a) m
Homework
Assignment
#2
(due
9/27)
I EOR E4004: Introduction to Operations Research: D eterministic Models
Instructors: Juraj Stacho & Dawn Strickland
TAs: Itai Feigenbaum & Peter Maceli
Problem #1: Solve the following LPs using the dictionary Simplex
IEOR 4004 - Intro to OR - Deterministic models - Lecture 6
September 24, 2013
1. Sensitivity Analysis
Also called post-optimality analysis: determining what eect (if any) do changes to the input problem have
on the optimality (feasibility) of a solution t
9.4
Some Characteristics Of Integer ProgramsA Sample Problem
287
subject to:
J
D
j =1 J j =1 I i =1 I
di j xi j 0 xi j = 1 xi j y j I pi xi j = 0 f j (s j ) B ,
(i = 1, 2, . . . , I ), (i = 1, 2, . . . , I ), ( j = 1, 2, . . . , J ), ( j = 1, 2, . . . , J
Homework Assignment #2 (due 9/27)
IEOR E4004: Introduction to Operations Research: Deterministic Models
Instructors: Juraj Stacho & Dawn Strickland
TAs: Itai Feigenbaum & Peter Maceli
Problem #1: Solve the following LPs using the dictionary Simplex method
Homework Assignment #1 (due 9/20)
I EOR E4004: Introduction to Operations Research: D eterministic Models
Instructors: Juraj Stacho & Dawn Strickland
TAs: Itai Feigenbaum & Peter Maceli
Problem #1. The Silicon Solutions software company needs to rent lapt
Homework Assignment #4 (due 10/25)
IEOR E4004: Introduction to Operations Research: Deterministic Models
Instructors: Juraj Stacho & Dawn Strickland
TAs: Itai Feigenbaum & Peter Maceli
Problem #1: Construct the Dual problems to the following LPs (do not s
IEOR 4004
Fall 2014
Homework 4
1. max = 41 + 2
1 + 22 = 6
1 2 3
21 + 2 10
1 , 2 0
a. Solve the LP.
b. Find the dual and solve it as well.
c. Find the range of values of b3 (rhs value of 3rd constraint) for which the current basis remains
optimal. What is
Homework Assignment #7 (due 12/6)
IEOR E4004: Introduction to Operations Research: Deterministic Models
Instructors: Juraj Stacho & Dawn Strickland
TAs: Itai Feigenbaum & Peter Maceli
Problem #1: A company is considering the options available for its next
Homework Assignment #8 (practice only)
IEOR E4004: Introduction to Operations Research: Deterministic Models
Instructors: Juraj Stacho & Dawn Strickland
TAs: Itai Feigenbaum & Peter Maceli
Problem #1: Consider the binary knapsack problem with five items.
IEOR E4004: Introduction to Operations Research: Deterministic Models
Jay Sethuraman & Dawn Strickland
HW 3 (due 10/05)
Problems not written out explicitly are from the text: Applied Mathematical Programming, by Bradley, Hax and Magnanti.
1. While solving
Network Flow Programming Models
8
Notes: Exercise 6b. Add Do as many jobs as possible.
6.
A company has three workers. On a particular day, six jobs are scheduled to be
completed. The cost for each worker to do each job is shown in the matrix below.
Using
Homework Assignment #8 (practice only)
IEOR E4004: Introduction to Operations Research: Deterministic Models
Instructors: Juraj Stacho & Dawn Strickland
TAs: Itai Feigenbaum & Peter Maceli
Problem #1: Consider the binary
Algorithms
Lecture 17: Applications of Maximum Flow
For a long time it puzzled me how something so expensive, so leading edge, could be so useless, and then it occurred to me that a computer is a stupid machine with the ability to do incredibly smart thin
IEOR 4004 Optimization Models and Methods
Quiz 1 Solution
Problem 1
(a) The dual problem is
min
Subject to
20y1 + 10y2
5y1 + y2 6
y1 + 2y2 8
y 0.
(b) (Plot the corresponding graphs). Using the graphical method, x = (10/3, 10/3) and
y = (4/9, 34/9) are opt
IEOR E4004 Optimization Models and Methods
Quiz 2 solution
Fall 2016
November 11, 2016
Problem 1
(a) The negative augmenting cycle is 1 2 5 4 1. It is easy to see that we can
augment 5 units of flow on this cycle.
(b) The corresponding network is shown in
IEOR 4004
Quiz 2, November 14 2016
1. In this question we deal with the minimum-cost flow problem. As usual for each arc (i, j) we
list, first, its per-unit cost cij and second, its capacity uij .
50
250
500,10
1
5,45
2
10,300
15,20
5,250
4
5
10,250
15,29
Solving a regression problem by linear programming
A very common and important problem in statistics is linear regression, the problem of
fitting a straight line to statistical data. The most commonly employed technique is the
method of least squares, but
IEOR 4004: Optimization Models and Methods
Homework 3
Instructor: Shipra Agrawal
Due on: 09/29/2016
Each assignment must be submitted either on paper at the beginning of the class it is due, or online on
courseworks before the class. NO EMAIL SUBMISSIONS
IEOR 4004: Introduction to OR: Deterministic Models
Homework 1
Instructor: Shipra Agrawal
Due on: Thursday, September 15, 2016
Each assignment must be submitted either on paper at the beginning of the class it is due, or online on
courseworks before the
IEOR 4004: Introduction to OR: Deterministic Models
Homework 2
Instructor: Shipra Agrawal
Due on: Thursday, 09/22/2016
Each assignment must be submitted either on paper at the beginning of the class it is due, or online on
courseworks before the class. N
IEOR 4004: Optimization Models and Methods
Homework 7
Instructor: Shipra Agrawal
Due on: 11/03/2016
Each assignment must be submitted either on paper at the beginning of the class it is due, or online on
courseworks before the class. NO EMAIL SUBMISSIONS
IEOR 4004: Introduction to OR: Deterministic Models
Homework 4
Instructor: Shipra Agrawal
Due on: 10/05/2016
Each assignment must be submitted either on paper at the beginning of the class it is due, or online on
courseworks before the class. NO EMAIL SU