Sample Linear Programming Problem
A furniture manufacturer makes two types of furniture chairs and sofas. The
production of the sofas and chairs requires three operations carpentry, finishing, and
upholstery. Manufacturing a chair requires 3 hours of carp
10
Blending of Input Materials
10.1 Introduction
In a blending problem, there are:
1) Two or more input raw material commodities;
2) One or more qualities associated with each input commodity;
3) One or more output products to be produced by blending the
Welcome to the Course Page
of
MATH 3171 3.00 A
Linear Optimization
Department of Mathematics and Statistics
Faculty of Science
York University
4700 Keele Street
Toronto, Ontario M3J 1P3
Course Information:
Course:
M/W 18:00 21:00, TEL 1004
Name:
Dr. Iould
4
The Model Formulation
Process
Count what is countable, measure what is measurable, and
what is not measurable, make measurable.
Galileo Galilei(1564-1642)
4.1 The Overall Process
In using any kind of analytical or modeling approach for attacking a probl
3
Analyzing Solutions
3.1 Economic Analysis of Solution Reports
A substantial amount of interesting economic information can be gleaned from the solution report of a
model. In addition, optional reports, such as range analysis, can provide further informa
6
Product Mix Problems
6.1 Introduction
Product mix problems are conceptually the easiest constrained optimization problems to comprehend.
The Astro/Cosmo problem considered earlier is an example. Although product mix problems are
seldom encountered in th
1
What Is Optimization?
1.1 Introduction
Optimization, or constrained optimization, or mathematical programming, is a mathematical procedure
for determining optimal allocation of scarce resources. Optimization, and its most popular special
form, Linear Pr
9
Multi-period Planning
Problems
9.1 Introduction
One of the most important uses of optimization is in multi-period planning. Most of the problems we
have considered thus far have been essentially one-period problems. The formulations acted as if
decision
Name:
February 27, 2008
Some Simplex Method Examples
Example 1: (from class) Maximize: P = 3x + 4y subject to:
x+y 4
2x + y 5
x 0, y 0
Our rst step is to classify the problem. Clearly, we are going to maximize our objective function, all are variables ar
Y/i
fg
*
Math 3170.06
Test 3
MME
March 5,2073
(print):
(Family)
(Given)
STUDENT NUMBER:
Question 1 [5 rnarks]
- ,./
e d Y
Factories A, B and C produce up to 100,200, and 50 units.
Units may sent to destinations D, E, or F. Destinations D, E and F must rec
Solutions to selected Homework Problems
from Sections 3.8 and 3.9 of the textbook.
Section 3.8
# 10. Let Mi = tons of coal shipped from Mine i and
Xij = tons of coal shipped from Mine i to Customer j.
Here is the appropriate formulation of the problem in
Solutions to selected Homework Problems
from Sections 3.2, 3.8, 3.10 and 3.11 of the textbook.
Section 3.2:
#4.
4a) We want to make x1 larger and x2 smaller so we move down and to the right.
4b) We want to make x1 smaller and x2 larger so we move up and t
Solutions to Homework Problems
from Sections 4.4 of the textbook.
# 1.
From Figure 2 of Chapter 3 we see that the extreme points of the feasible region and
corresponding Basic Feasible Solutions are:
H = (0, 0) - s1 = 100, s2 = 80, s3 = 40, x1 = x2 = x3 =
#4.
A LINGO model that can be used to maximize the profit in this situation is:
MODEL:
SETS:
PRODUCTS/1.3/:MADE,PROFIT;
RESOURCES/1.3/:AVAIL;
RESPRO(RESOURCES,PRODUCTS):USAGE;
ENDSETS
MAX=@SUM(PRODUCTS(I):PROFIT(I)*MADE(I);
@FOR(RESOURCES(I):@SUM(PRODUCTS