This preview shows pages 1–3. Sign up to view the full content.
Lecture 3: Introduction to integer linear programming
Contents
1 Lumpy linear programs with fxed charges
1
2 Piecewise linear costs
1
3 Knapsack and Capital Budgeting problems
2
4 Set packing, covering and partitioning models
3
5 Assignment and Matching problems
4
6 Facility location and network design problems
5
7 Traveling salesman models
7
8 Processor scheduling and sequencing problems
7
9 Solution methods
8
9
.
1 S
o
lv
in
gIP
sa
sLP
s ..........................................
10
9.2
Rounding LP solutions .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1
Lumpy linear programs with fxed charges
In this lecture we will review optimization models where the constraints and the objective are linear but the
domain
D
has a discrete component in the sense that some or all the variables are restricted to take only a
discrete set of values. This restriction leads to interesting ramiFcations for modeling. We will breiﬂy review
the basic integer linear programming models in the lecture.
Example 1
Suppose we want to invest in three mutual funds. Each of the three mutual funds speciFes a
minimum amount
l
i
that has to be invested in case one decides to invest in that fund. Each fund charges a
frontend load consisting of a Fxed cost
c
i
plus a proportional cost at the rate
α
i
,
i
=1
,...,
3. Suppose the
investor has a capital
W
to invest. Describe the set of all feasible investments.
Problem is crying out for decision variable
y
i
,
i
,
2
,
3 deFned as follows.
y
i
=
±
1
,
invest nonzero amount in fund
i,
0
,
otherwise
Let
x
i
denote the amount invested in the mutual fund
i
3. Using
y
i
we want to ensure that if
y
i
=0
then
x
i
=0andif
y
i
=1then
l
i
≤
x
i
≤
W
, i.e.
l
i
y
i
≤
x
i
≤
Wy
i
The budget constraint is as follows
m
X
i
=1
(
c
i
y
i
+(1+
α
i
)
x
i
)
≤
W.
Thus, the set of all feasible investment decisions is given by
X
=
²
(
x
,
y
):
y
i
∈{
0
,
1
}
,l
i
y
i
≤
x
i
≤
i
,i
,
2
,
3
,
m
X
i
=1
(
c
i
y
i
α
i
)
x
i
)
≤
W
³
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentInteger programming
2
Consider the following modiFcation.
Suppose the mutual funds also stipulate that, after the minimum
amount
l
i
has been invested, all additional investments must be in multiples of
d
i
,
i
=1
,
2
,
3. How does
this change the set of feasible investments ? All this new constraint is saying is that
x
i
=
l
i
+
d
i
z
i
for some
z
i
≥
0 and integer. We will denote the set of integers by
Z
and the set of nonnegative integers by
Z
+
.Thu
s
,
z
i
∈
Z
+
. Incorporating this constraint we get
X
=
±
(
x
,
z
,
y
):
y
i
∈{
0
,
1
}
,z
i
∈
Z
+
,x
=
y
i
+
d
i
z
i
,
0
≤
d
i
z
i
≤
(
W
−
l
i
)
y
i
,i
,
2
,
3
,
∑
m
i
=1
(
c
i
y
i
+(1+
α
i
)
x
i
)
≤
W
²
2
Piecewise linear costs
In the linear programming part of this course we discussed how to handle piecewise linear convex “
≤
”
constraints and piecewise linear concave “
≥
” constraints. In this section we will discuss methods to handle
the other two types of constaints and also piecewise linear functions with Fxed costs.
This is the end of the preview. Sign up
to
access the rest of the document.
 Summer '09
 OptimizationModelsandMethods

Click to edit the document details