If we set b = c = 0, A = AT , then (14) and (15) will be the same problem. Thus we can construct
another example as follows:
max 0
s.t.
x1 x2 + 2x3 = 0
(16)
x1 + 2x2 + 3x3 = 0
2x1 + 3x2 + 3x3 = 0
x1 ,
Practice
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practice
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IEOR 4004
Other methodologies for Integer Programming
1
Frequency assignment revisited
In the following picture we show (in blue) the signal frequencies that can be operated at any of the nodes. Each
IEOR 4004
Integer Programming Models
Integer Programming problems are optimization models where some or all variables are required to take integral
values. Frequently, these variables are in fact bina
IEOR 4106, Final Exam, Columbia University
Prof. S. Kou, Dec 21, 2004, from 4:10pm to 7pm.
1. (10 pts). Suppose cars arrive at a gas service station according to a Poisson process with
a rate cars per
Midterm Solution. Fall 2005.
E3106. Prof. S. Kou, Columbia University
1. The state space of the Markov chain is cfw_(i, j)i 1, j 1. Suppose
that the Markov chain is currently in the state (i, j). Let
IEOR 4106, Final Exam, Columbia University
Prof. S. Kou, Dec. 21, 2004
1. (10 pts). (a). Y has a Poisson distribution with parameter t. E[Y ] = t.
(b). X has an exponential distribution with rate . E[
Solving Linear Programs
2
In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the
simplex method, proceeds by moving from one feasible solution to an
Converting a Linear Program to Standard Form
Hi, welcome to a
tutorial on converting
an LP to Standard
Form.
Amit, an MIT Beaver
We hope that you
enjoy it and find it
useful.
Mita, an MIT
Beaver
2
Lin
IEOR 4004
More integer programming models
1
Multilevel location problems
A company is designing a distribution network. In the following figure,
8
25
4
1
100
9
5
2
760
10
6
3
11
7
1000
400
12
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Practice
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6. (Problem 7.22 from text)
a) The objective function is :
max
cij xik xjk .
ijk
b) Constraints:
Every person is assigned to a car pool:
xik = 1, i
k
The number of people in a car pool is between L
4. Let fj (A) be the optimal prot the vendor can make in days j, j + 1, . . . , N , assuming he
starts day j in location A. Similarly, let fj (B) be the optimal prot in days j through N
assuming he st
IEOR 4004
Lecture 3  Intro to Linear Programming Algorithms
1
Motivation
We assume an LP in standard form:
(LP):
Subject to
max wT x
(1a)
Ax = b
x 0.
(1b)
(1c)
Here we are assuming that A has m rows
IEOR 4004
Simplex Method: duality
1
Pricing interpretation
Consider a manufacturing problem with two resources, blocks of wood and cans of paint, and two products,
toy soldiers and toy trains.
Manufac
IEOR 4004
Network Problems, III
1
MinimumCost Flow problem
A delivery company runs a delivery network between major US cities. Selected cities are connected by routes as
shown below. On each route a
IEOR 4004
Simplex Method: duality
1
Pricing interpretation
Consider a manufacturing problem with two resources, blocks of wood and cans of paint, and two products,
toy soldiers and toy trains.
Manufac
IEOR E4004: Deterministic Models
Assignment 3: Due October 26
1. Problem 6 (Page 322).
2. Problem 12 (Page 349).
3. Problem 1a (Page 471). Check your answer by writing the shortest path problem as an
Sensitivity Analysis
3
We have already been introduced to sensitivity analysis in Chapter 1 via the geometry of a simple example.
We saw that the values of the decision variables and those of the slac