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Lecture Notes For Optimization Theory
&Ana
lys
is
(ESE 304)
Michael A. Carchidi
November 24, 2010
Chapter 9  Integer Programming
The following notes are based on the text entitled:
Introduction to Mathemat
ical Programming
by Wayne L. Winston and Munirpallam Venkataramanan (4th
edition), and these can be viewed at https://courseweb.library.upenn.edu/ after
you log in using your PennKey user name and Password.
9.1 Introduction to Integer Programming
A
Linear Programming
(LP) problem in which
at least one
the decision vari
ables is required to be integer is called an
Integer Programming
(IP) problem.
If all of the decision variables are required to be integer, then the IP problem is
called a
Pure Integer Programming
(PIP) problem, otherwise it is called a
Mixed
(MIP) problem.
Example
: Revised Giapetto’s Woodcarving
Giapetto’s Woodcarving Inc., manufactures two types of wooden toys: soldiers
and trains. A soldier sells for
$27
and uses
$10
worth of raw materials. Each soldier
that is manufactured increases Giapetto’s variable labor and overhead costs by
$14
.At
ra
ins
e
l
l
sfo
r
$21
and uses
$9
worth of raw materials. Each train that is
manufactured increases Giapetto’s variable labor and overhead costs by
$10
.The
manufacture of wooden soldiers and trains requires two types of labor:
f
nishing
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View Full Documentand carpentry. A soldier requires
4
hours of
f
nishing labor (instead of
2
as in the
original statement of the problem in Chapter
3
)and
1
hour of carpentry labor. A
train requires
1
hour of
f
nishing labor and
1
hour of carpentry labor. Each week,
Giapetto can obtain all the needed raw materials but only
100
f
nishing hours and
80
carpentry hours. Demand for trains is unlimited, but at most
40
soldiers are
bought each week. Giapetto wants to maximize weekly pro
f
ts (revenues minus
costs).
We had shown in chapter 3 that a statement of this problem is as follows
max
z
=3
x
1
+2
x
2
(Objective Function)
s.t.
4
x
1
+
x
2
≤
100
(Finishing Constraint)
x
1
+
x
2
≤
80
(Carpentry Constraint)
x
1
≤
40
(Constraint on Demand for Soldiers)
x
1
,x
2
≥
0
(Sign Restrictions)
x
1
2
are integer (Integer Restrictions)
where
x
1
=
the number of soldiers produced each week, and
x
2
=
thenumbero
ftra
insproducedeachweek
.
With the minor change of a soldier requiring
4
hours of
f
nishing labor instead of
2
hours, the optimal solution to the problem without the integer restrictions is
x
max
=
⎡
⎣
6
2
3
73
1
3
⎤
⎦
with
z
max
=$166
2
3
Although we have been ignoring the integer restrictions in the past chapters, we
must technically include these restrictions in the full development of this problem
since we cannot make a fractional number of soldiers or trains. We see then
that the Giapetto problem is really a
Pure Integer Programming
problem. Using
LINDO (which we shall discuss later) we
f
nd that the solution to the pure IP
problem is given by
x
max
=
∙
6
74
¸
with
z
max
.
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 Winter '11
 MichaelA.Carchidi

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