Uncapacitated Lot Sizing
Activity Selection
IE170: Algorithms in Systems Engineering:
Lecture 14
Jeff Linderoth
Department of Industrial and Systems Engineering
Lehigh University
February 23, 2007
Jeff Linderoth
IE170:Lecture 14
Uncapacitated Lot Sizing
Activity Selection
Taking Stock
Last Time
Lot Sizing and Java Code
This Time
Lot Sizing—WagnerWhitin
Greedy Algorithm
Jeff Linderoth
IE170:Lecture 14
Uncapacitated Lot Sizing
Activity Selection
Uncapacitated Lot Sizing
Lot sizing is
the
canonical production planning problem
Given a planning horizon
T
=
{
1
,
2
, . . . , T
}
You must meet given demands
d
t
for
t
∈ T
You can meet the demand from a combination of production
(
x
t
) and inventory (
s
t

1
)
Production cost:
c
(
x
t
) =
K
+
cx
t
if
x
t
>
0
0
if
x
t
= 0
Inventory cost:
I
(
s
t
) =
h
t
s
t
Jeff Linderoth
IE170:Lecture 14
Uncapacitated Lot Sizing
Activity Selection
In General
A General Recursive Relationship
f
t
(
s
) =
min
x
∈
0
,
1
,
2
,...
{
c
t
(
x
) +
h
t
(
s
+
x

d
t
) +
f
t
+1
(
s
+
x

d
t
)
}
.
What if
K
= 250
, d
= [220
,
280
,
360
,
140
,
270]
, c
t
= 2
, h
t
= 1
This might be a problem, as you need to consider producing
every
possible amount between 0 and 1270
Instead, as is often the case in dynamic programming, we look
for
structural properties
of an optimal solution that will make
the algorithm more efficient.
Jeff Linderoth
IE170:Lecture 14
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Uncapacitated Lot Sizing
Activity Selection
I Love Lemmas
Lemma (Fact) 1
Let
x
*
be an optimal policy (production schedule).
If
x
*
t
>
0
,
then
x
*
t
=
∑
T

t
j
=0
d
t
+
j
for some
j
∈ {
0
,
1
, . . . T

t
}
Why? Oh Why?
If Lemma 1 was false, then there would be some period
t
and some
subsequent period
t
+
j
such that production
x
*
t
only partially
satisfied the demand in
t
+
j
. Say this is a quantity
0
< p < d
t
+
j
.
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 Spring '07
 Ralphs
 Dynamic Programming, Systems Engineering, sij, Jeff Linderoth, Uncapacitated Lot Sizing

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