Linear Programming with Post-Optimality Analyses
Wilson Problem: Wilson Manufacturing produces both baseballs and softballs, which it
wholesales to vendors around the country. Its facilities permit the manufacture of a
maximum of 500 dozen baseballs and a
CHAPTER 3.
LINEAR PROGRAMMING FORMULATIONS
It can be recalled from the Mine example that the conditions for a mathematical model
to be a linear program (LP) were:
all variables continuous (i.e. can take fractional values)
a single objective (minimize or
CHAPTER 2.
AN INTRODUCTION TO LINEAR PROGRAMMING
Consider the following problem.
Example 2.1 Mining Problem A Mine Company owns two different mines that produce an ore which, after being crushed, is graded into three classes: high, medium and
low-grade. T
CHAPTER 1.
OPERATION RESEARCH
1.1
Introduction
Operations Research (OR) is an interdisciplinary branch of applied mathematics which
uses methods like mathematical modeling, statistics, and algorithms to arrive at optimal
or good decisions in complex probl
Linear
Programming
1
Objectives
Requirements for a linear programming model.
Graphical representation of linear models.
Linear programming results:
Unique optimal solution
Alternate optimal solutions
Unbounded models
Infeasible models
Extreme poin
Math 30210 Introduction to Operations Research
Examination 1 (15 points per question, 90 points total)
Solutions
1.
(a) A linear programming problem calls for the minimization of x + 2y 4w. The
problem has two articial variables, R1 and R2 . Which of the
CHAPTER 5.
SIMPLEX METHOD
5.1
Introduction
Graphical method is simple way to solve LP problems. Unfortunately we can get graphical
solutions of LP problem with 2 and 3 decision variables only. Therefore we need to find
an alternative method to get optimal
MT218: LINEAR PROGRAMMING
TUTORIAL 1
1. A soap manufacturing company is planning an advertising campaign in three different media;
namely, Television, Radio and Magazine. The purpose of the advertising program is to reach as
many customers as possible. Re
MT218:
LINER PROGRAMMING
LECTURE NOTES
1. INTRODUCTION TO OR
1.1 TERMINOLOGY
The British/Europeans refer to "operational research", the Americans to "operations
research" - but both are often shortened to just "OR" (which is the term we will use).
Another
MA30087/50087: Optimisation methods of
operational research
Brief notes to accompany the lectures.1
Prepared by A. E. Kyprianou2
Department of Mathematical Sciences
The University of Bath
Claverton Down
BA2 7AY
Suggested further reading: Elementary Linear
Introduction
Terminology
The Simplex Algorithm
SIMPLEX METHOD
E. Mujuni
Simplex Algorithm for Minimization Problems
Introduction
Terminology
The Simplex Algorithm
Simplex Algorithm for Minimization Problems
Introduction
Graphical method is simple way to s
MT218: Linear Programming
Tutorial 6
Q1: Show that the following LP is unbounded
max z = 2x2
s.t
x1 x2 4
x1 +x2 1
x 1 , x2 0
Q2: Use the Big M method to solve the following LPs
min z = 4x1 +4x2 +x3
s.t
2x1 +x2 +x3
2x1 +x2
2x1 +x2 +3x3
x 1 , x2 , x 3
2
3
3
CHAPTER 4.
GRAPHICAL SOLUTION
4.1
Introduction
An LP with two decision varibles can be solved graphically. This is called graphical
method. The medhod consists of two phases
Finding the values of the decision variables for which all the constraints are m
MT218: LINEAR PROGRAMMING
TUTORIAL 2
Formulate the following problem as LP model
9.
Ms. Fidans diet requires that all the food she eats come from one of the four basic food
groups. At present, the following four foods are available for consumption: browni
3 Homework 2 (due Sept. 10) Name: 50 (K/T! 0N5
The purpose of this homework is to examine the standard form. of a linear programming
problem, and the pivot operation that allows one to nd a basic feasible solution to a
problem.
Reading: Sections 3.1 and 3
5 Homework 3 (due Sept. 17) Name: 50 LU [Fro/J6.
The purpose of this homework is to explore the simplex algorithm for solving a linear pro-
gramming problem in standard form.
Reading: Sections 3.3 and 3.4
1. Problem set 3.3, question 1 (page 76)
- part (a
Math 30210 Introduction to Operations Research
Quiz 3 Wednesday September 19, 2007
NAME:
Instructions: This is a closed-book quiz. Please do not use any notes.
Consider the following problem: Maximize x1 + 2x2 subject to x1 , x2 0 and
x1 + x2 1
x1 + x2 4
Math 30210 Introduction to Operations Research
Quiz 2 Wednesday September 12, 2007
NAME:
Instructions: This is a closed-book quiz. Please do not use any notes.
A linear programming problem is in standard form if all variables are constrained to be
non-neg
Math 30210 Introduction to Operations Research
Quiz 1 Wednesday September 5, 2007
NAME:
Instructions: This is a closed-book quiz. Please do not use any notes.
A caterer has ve mixed fruit drinks available to him, and must produce 500 gallons
of punch for
Math 30210 Introduction to Operations Research
Final Examination (120 points total)
Friday December 14, 2007, 4.15pm6.15pm
Instructions: Please present your answers neatly and legibly. Complete all questions for full
credit. You may not use notes or books
Math 30210 Introduction to Operations Research
Examination 1 (15 points per question, 90 points total)
Solutions
1.
(a) A linear programming problem calls for the minimization of x + 2y 4w. The
problem has two articial variables, R1 and R2 . Which of the
Math 30210 Introduction to Operations Research
Assignment 13 (35 points total; each question worth 5 points)
Solutions
1. Taha 13.4A Q1b): Value of game is 5, A should play A1 and B should play B3 .
2. Taha 13.4A Q2b): Any p 7, any q 7.
3. Taha 13.4A Q3b)
Math 30210 Introduction to Operations Research
Assignment 5 (50 points total)
Due before class, Wednesday October 3, 2007
Instructions: Please present your answers neatly and legibly. Include a cover page
with your name, the course number, the assignment
Capital Market Theory:
An Overview
Capital market theory extends portfolio
theory and develops a model for pricing all
risky assets
Capital asset pricing model (CAPM) will
allow you to determine the required rate of
return for any risky asset
Systematic