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ISyE 3232
Stochastic Manufacturing and Service Systems
Fall 2011
H.Ayhan
Solutions to Homework 9
1. Let
{
X
n
:
n
≥
0
}
be the stock level at the evening of day
n
. The problem states an
(
s,S
) inventory policy, i.e., if the stock level is
s
or less than
s
at the review time,
we make orders to bring the stock level up to
S
; we do not order otherwise. We can
establish the relation between
X
n
+1
and
X
n
as follows:
X
n
+1
=
±
S

D
if
X
n
≤
s
X
n

D
if
X
n
> s
.
(1)
By Equation (1),
{
X
n
}
is a discretetime Markov chain. Note that we allow negative
stock levels, which indicate unﬁlled orders at review times.
(
a
) In this case,
s
= 2 and
S
= 6. We have the state space
S
=
{
1
,
0
,
1
,
2
,
3
,
4
,
5
,
6
}
.
The transition probability matrix can be written as,
P
=
0
0
0
.
01
.
04
.
3
.
15
.
5
0
0
0
.
01
.
04
.
3
.
15
.
5
0
0
0
.
01
.
04
.
3
.
15
.
5
0
0
0
.
01
.
04
.
3
.
15
.
5
.
01
.
04
.
3
.
15
.
5
0
0
0
0
.
01
.
04
.
3
.
15
.
5
0
0
0
0
.
01
.
04
.
3
.
15
.
5
0
0
0
0
.
01
.
04
.
3
.
15
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 Fall '07
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