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ISyE 3232
Stochastic Manufacturing and Service Systems
Spring 2011
H. Ayhan and J. Dai
Solutions to Homework 10
1.
(a)
P
(
T
≤
1) = 1

e

10
/
7
≈
0
.
7603 and
P
(
T
≤
2) = 1

e

20
/
7
≈
0
.
9426.
(b) The tardiness will be
T
0
= max
{
(
T

1)
,
0
}
.
E
(
T
0
) =
Z
∞
1
(
t

1)
λe

λt
dt
=
e

λ
Z
∞
0
sλe

λs
dt
=
e
10
/
7
0
.
7
≈
0
.
1678
(c) Let
g
1
(
t
)
,g
2
(
t
)
,g
3
(
t
) denote the proﬁt for the “$1250”, “$1000”, “$750” contract with
delivery time
t
respectively. Then they can be written as:
g
1
(
t
) =
1250
0
≤
t
≤
1
/
2
2500(1

t
) 1
/
2
< t
≤
1
0
t >
1
g
2
(
t
) =
1000
0
≤
t
≤
1
1000(2

t
) 1
< t
≤
2
0
t >
2
g
3
(
t
) =
750
0
≤
t
≤
7
750(14

t
)
/
7 7
< t
≤
14
0
t >
14
Let
T
be a exponential random variable with mean
δ
, let’s compute
E
[
g
1
(
T
)]:
f
1
(
δ
) =
E
[
g
1
(
T
)] =
Z
0
.
5
0
1250
1
δ
e

t/δ
dt
+
Z
1
0
.
5
2500(1

t
)
1
δ
e

t/δ
dt
= 1250

2500
δ
(
e

1
2
δ

e

1
δ
)
Similarly,
f
2
(
δ
) =
E
[
g
2
(
T
)] = 1000

1000
δ
(
e

1
δ

e

2
δ
)
f
3
(
δ
) =
E
[
g
3
(
T
)] = 750

750
7
δ
(
e

7
δ

e

14
δ
)
Plugin 0
.
7 into each of the expected proﬁt function,
f
1
(0
.
7) = 812
.
69,
f
2
(0
.
7) = 872
.
45
and
f
3
(0
.
7) = 749
.
99. Hence, choosing the “$1000” contract will give us the max expected
proﬁt.
(d) Using a software package like Maple or Mathematica, we can plot
f
i
(
δ
) for
i
= 1
,
2
,
3.
f
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This note was uploaded on 01/16/2012 for the course ISYE 3232 taught by Professor Billings during the Spring '07 term at Georgia Institute of Technology.
 Spring '07
 Billings

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