MthSc 810: Mathematical Programming
Lecture 2
Pietro Belotti
Dept. of Mathematical Sciences
Clemson University
August 30, 2011
Reading for today: Chapters 1.1, 1.2, 1.3
Reading for Sep. 1: Chapters 1.4, 2.1, 2.2
Linear programming
The optimization problem, with
n
variables and
m
constraints:
min
∑
n
i
=
1
c
i
x
i
s
.
t
.
∑
n
i
=
1
a
ji
x
i
=
b
j
∀
j
=
1
,
2
. . . ,
m
x
i
≥
0
∀
i
=
1
,
2
. . . ,
n
is called a
LinearProgramming
(from now on LP) problem.
They are often written in matricial form:
min
c
⊤
x
s
.
t
.
A
x
=
b
x
≥
0
A
is the (
m
×
n
)
coefficient matrix
,
b
is the
righthand side vector
,
and
c
is the
objective coefficient vector
.
Standard form of an LP
◮
The problem in the previous slide is said to be written in
standard
form.
◮
An LP in the previous slide is the intersection between an
affine subspace (
A
x
=
b
) and the first orthant (
x
≥
0
).
◮
Any LP
not
in standard form can be rewritten with
equations only:
◮
for any inequality
a
⊤
x
≤
b
, introduce a new
slack
variable
s
and rewrite the inequality as
a
⊤
x
+
s
=
b
◮
s
must be nonnegative, of course:
s
≥
0
◮
If a variable
x
i
is
unrestricted in sign
(u.r.s.), that is, it is not
constrained to be nonnegative or nonpositive, we can
replace it with
x
+
i

x
−
i
with
x
+
i
≥
0 and
x
−
i
≥
0, thus
obtaining an LP in standard form.
Maximization problems
They are not so different from their minimization counterpart.
max
{
f
(
x
) :
x
∈
F
}
=

min
{
f
(
x
) :
x
∈
F
}
⇒
we take the opposite of the objective function
only
.
Example:
max
{
2
x

3
:
x
∈
[
4
,
5
]
}
=

min
{
2
x
+
3
:
x
∈
[
4
,
5
]
}
7
=

(

7
)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Example: Transportation problem
◮
A large manufacturing company produces liquid nytrogen
in
five
plants spread out in East Georgia
◮
Each plant has a monthly production capacity
Plant
i
1
2
3
4
5
Capacity
p
i
120
95
150
120
140
◮
There are
seven
retailers in the same area
◮
Each retailer has a monthly demand to be satisfied
Retailer
j
1
2
3
4
5
6
7
Demand
d
j
55
72
80
110
85
30
78
◮
Transportation between any plant
i
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Staff
 Math, Linear Programming, Optimization, Yi, Dept. of Mathematical Sciences Clemson University, j=1 akj xj

Click to edit the document details