Assembly Line Balancing
Uncapacitated Lot Sizing
IE170: Algorithms in Systems Engineering:
Lecture 13
Jeff Linderoth
Department of Industrial and Systems Engineering
Lehigh University
February 21, 2007
Jeff Linderoth
IE170:Lecture 13
Assembly Line Balancing
Uncapacitated Lot Sizing
Taking Stock
Last Time
Assembly Line Balancing
(Intro to) Lot Sizing
Knapsack Problem
This Time: DP + Ilya’s Favorite Algorithm
Lot Sizing
Greedy Algorithm
Jeff Linderoth
IE170:Lecture 13
Assembly Line Balancing
Uncapacitated Lot Sizing
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 13
Assembly Line Balancing
Uncapacitated Lot Sizing
Let’s Solve it with DP
What should our stages be?
Hint: Typically stages have type “from beginning until now”
(like
S
ij
) or from “now until end” (like in capital budgeting)
Stage
Let
f
t
(
s
)
: be the minimum cost of meeting demands from
t, t
+
1
, . . . T
if
s
units are in inventory at the beginning of period
t
Jeff Linderoth
IE170:Lecture 13
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Assembly Line Balancing
Uncapacitated Lot Sizing
Let’s Solve an Example
T
= 3
d
= [2
,
1
,
2]
h
= [1
,
1
,
0]
K
= 2
, c
= 1
Busy Going Backwards
f
3
(0) = 2 + 2(1) = 4
f
3
(1) = 2 + 1(1) = 3
f
3
(2) = 0
Jeff Linderoth
IE170:Lecture 13
Assembly Line Balancing
Uncapacitated Lot Sizing
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
)
}
.
Let’s do a couple by hand.
This gets tedious – so let’s code it up...
Jeff Linderoth
IE170:Lecture 13
Assembly Line Balancing
Uncapacitated Lot Sizing
Oh Dear!
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
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 Spring '07
 Ralphs
 Dynamic Programming, Systems Engineering, sij, Jeff Linderoth, Uncapacitated Lot Sizing

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