Soft Constraints
Goal Programming
Flow Models
IE426: Optimization Models and Applications:
Lecture 11
Jeff Linderoth
Department of Industrial and Systems Engineering
Lehigh University
October 5, 2006
Jeff Linderoth
IE426:Lecture 11
Soft Constraints
Goal Programming
Flow Models
Review
Got MILP?
Mixed Integer Linear Program
minimize
c
T
x
subject to
Ax
≥
b
x
≥
0
x
j
∈
Z
∀
j
∈
I
⊆
N.
Jeff Linderoth
IE426:Lecture 11
Soft Constraints
Goal Programming
Flow Models
Review
IP Stuff
Many people would just call this “integer programming” (IP).
(Assumed linear unless expressly stated otherwise)
Mixed integer programming
Only some of the variables have integer restrictions (
I
⊂
N
)
Pure integer programming
(
I
=
N
)
All variables have integer restrictions
01 integer programming
All integer restricted variables also have bounds of
≥
0
and
≤
1
.
B
n
def
=
The set of all
n
dimensional (0,1) (binary) vectors. (Or
maybe
{
0
,
1
}
n
)
Jeff Linderoth
IE426:Lecture 11
Soft Constraints
Goal Programming
Flow Models
Review
Why Integer Programming?
1
Indivisible quantities
If
x
represents the number of airplanes to build
Best to use
IP
if
x
will be small.
It is not worth the extra burden of IP if
x
is the number of
chickens in Arkansas.
2
Decision variable
x
∈ {
0
,
1
}
represents a “yes” or “no”
decision.
We can impose logical connections between these decisions
(often using linear constraints).
We’ll do lots of examples of this
Jeff Linderoth
IE426:Lecture 11
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Soft Constraints
Goal Programming
Flow Models
SmallVille
Multiple Objectives
Weighted Goal Programming
Hierarchical Goal Programming
How We Solve Integer Programs
IP models can be
very
much more difficult to solve than LP
models.
I’m not kidding
IP models can be
very
much more difficult to solve than LP mod
els.
It is important that you have a handle on...
1
How to build a problem that is likely to be solved – Proper
formulation is important!
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 Spring '08
 Linderoth
 Operations Research, Linear Programming, Optimization, Systems Engineering, Jeff Linderoth, Programming Flow Models, Soft Constraints

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