ANSWERS to Midterm 2 questions
1. (20 points.) Prove that the intersection of two convex set is also convex. (Hint: use definition of convex
set and formal definition of intersection of two sets: ( ) )
Take two arbitrary points x and y in ( ).
Since x and

Chapter 2
Linear programming
1
Introduction
Many management decisions involve trying to make the
most effective use of an organizations resources.
Resources typically include machinery, labor, money, time,
warehouse space, or raw materials.
Resources m

ANSWERS TO MIDTERM EXAM 1 QUESTIONS
1. (30 pts.)
x1 = the number of soldiers produced per week
x2 = the number of trains produced per week
si+ = amount by which the ith goal level is exceeded.
si- = amount by which the ith goal level is underachieved
The

College of Management, NCTU
Operation Research I
Fall, 2008
Chap 4 The Simplex Method
The Essence of the Simplex Method
Recall the Wyndor problem
Max Z = 3x1 + 5x2
4
S.T. x1
2x2 12
3x1 + 2x2 18
x1, x2 0
8 corner point solutions. 5 out of
them are CPF sol

CHAPTER
LINEAR
PROGRAMMING
AND APPLICATIONS
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.to
Basic Concepts Linear Programming
in
Degenerate
LP's-Graphical Solution
Natural Occurrence Linear Constraints
of
The Simplex Method of SolvingLinear ProgrammingProblems.

104 CHAPTER 2 Systems of Linear Equations and Matrices
where xi , x2 x3 x4 represent the amount, in millions of dollars, that must be produced to satisfy internal and external
demands of the four sectors; N is the total workforce required
for a particular

A Tutorial on Convex Optimization
Haitham Hindi
Palo Alto Research Center (PARC), Palo Alto, California
email: hhindi@parc.com
Abstract In recent years, convex optimization has become a computational tool of central importance in engineering, thanks to it

Example
A company is planning the manufacture of a product for
March, April, May and June of next year. The demand
quantities are 520, 720, 520 and 620 units, respectively.
The company has 10 employees at the beginning and can
meet fluctuating production

NAME
1.
MATH 304
Examination 2
Page 1
[18 points]
(a) Find the following determinant. However, use only properties of determinants,
without calculating directly (that is without expanding along a column or row or
otherwise). Explain your answer.
1 3 2 1
5

What is Operation Research?
What is Operation Research?
What is Operation Research?
During World War II, British military leaders asked
scientists and mathematicians to analyze several
military problems, involving
the deployment of radar
the management

Linear programming modeling and examples
Example 1
Each gallon of milk, pound of cheese, and pound of
apple produces a known number of milligrams of
protein and vitamins A, B, and C, as given in following.
The minimum weekly requirements of the nutritiona

OPIM 915
Final Examination Spring 2015
Read at least twice before looking at the exam questions.
Sign the NO CHEATING FORM and indicate the time at which you started the exam. An
exam without this form signed and dated will not be graded.
If you want to g

Integer Programming
Introduction to Integer Programming (IP)
Difficulties of LP relaxation
IP Formulations
Branch and Bound Algorithms
Reference: Chapter 9 in W. L. Winstons book.
Integer Programming Model
An Integer Programming model is a linear pro

Starting Solutions and Convergence
CONTENTS
The Initial Basic Feasible Solutions
The Two-Phase Method
The Big-M Method
Degeneracy, Cycling, and Stalling
Reference: Chapter 4 in BJS book.
Starting Solutions
The simplex method assumes the existence of

INTRODUCTION TO LINEAR PROGRAMMING
CONTENTS
Introduction to Linear Programming
Applications of Linear Programming
Reference: Chapter 1 in BJS book.
A Typical Linear Programming Problem
Linear Programming Formulation:
Minimize c1x1 + c2x2 + c3x3 + . + cn

BASIC FEASIBLE SOLUTIONS
CONTENTS
Polyhedral Sets and Polytopes
Basic Feasible Solutions and Polytopes
Reference: Chapters 2 and 3 in BJS book.
Hyperplanes
A hyperplane in En generalizes the notion of a straight line in
E2 and the notion of a plane in E

PRELIMINARIES
CONTENTS
Linear Algebra
Convex Analysis
Reference: Chapter 2 in BJS book.
Vectors
Row or Column Vector:
An n vector is a row or column array of n numbers. Each n
vector can be represented by a point or by a line from the
origin to the poin

ESI 6417
Linear Programming
and
Network Optimization
Fall 2003
Ravindra K. Ahuja
370 Weil Hall, Dept. of ISE
ahuja@ufl.edu
352-392-3615
Course Objectives
Engineers and managers are constantly attempting to
optimize, particularly in the design, analysis, a

Interm1
30th of October, Thursday
At 4:30
Duration: 2hrs
Interm2
First week of December
Final
The week starting with 22nd of December
(most probably)
Linear Algebra and Convex Analysis
In this lecture, we will review the elementary linear
algebra and co

Linear Programming: Chapter 14
Network Flows: Applications
Robert J. Vanderbei
October 22, 2007
Operations Research and Financial Engineering
Princeton University
Princeton, NJ 08544
http:/www.princeton.edu/rvdb
Transportation Problem
Each node is one of

Representation Theorem for the General Case
Let = cfw_: , be a nonempty polyhedral set.
The Simplex Method
When an optimal solution of a linear programming problem exists, an optimal
extreme point also exists.
Consider the following linear programming pr

Geometric Solution
The geometric solution of a two-dimensional LP problem is quite
useful to discuss the main ideas of the solution methods. Lets
start with an illustrative example before explaining the geometric
method in detail.
Example
Consider the fol