Optimality of Monotone procedure
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The University of Electro-CommunicaionsFaculty of Electro-Communications
Department of Systems EngineeringSuzuki Lab
Dinesh Rajapaksha2013-02-19
1
Optimality of Monotone Procedure for a System Working Under
Variable Operating Conditions
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Background
Systems deteriorate over time when used
Companies want the maximum performance from their systems during its life time
Optimal decisions made in a short period of time in operation will give the maximum performance
2
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http://www.intel.com/pressroom/archive/releases/2006/20060125comp.htm(2013/02/15 access)
Plasma Enhanced-CVD[1]
3
WaferChemical Layers
Semiconductor Industry
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Diagram of Plasma Enhanced-CVD (Chemical Vapor depositor)[2]
Plasma
Gases
G
a
S
4
Wafer
Chemical Layers deposited
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Plasma
Gases
Wafer
“System” and “Operating Condition”
5
SYSTEM
Wafer type 1
Wafer type 2
Wafer type 3
Operations needed ioProduct
)1(.ConditionOp
)2(.ConditionOp
)3(.ConditionOp
Operating Conditions
Plasma Enhanced-CVD
Which Operating Condition to use ? When to Use ?
To get the maximum cost performance ofPlasma Enhanced-CVD during it’s life time
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…0 1 i2 nNew
System’s state
Problem in decision making
10nStates of the system :
Available operating conditions : 3h
System
There are possible decisions available.59049310nh
Method: Monotone procedure 6610121310 CCH nnh
6
Decision Combinations (repeated combinations)
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Purpose of the research
Purpose of this research is to providesufficient conditions for optimal decisionmaking to be given by a monotoneprocedure.
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Previous research
8
C. Deman, et.al[3] M. Ohnishi, et.al[4] This research
System’s state Directly observable Partially observable Directly observable
Options toDeicide
Replace, Keep Replace , Inspection, Keep
h Different operatingconditions
…0 1 i2 nNew FailureSystem’s state
ReplaceKeep
…0 1 i2 nNew FailureSystem’s state
Keep ReplaceInspection Keep
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Denotation
9
: State space of system;
: True state of system;
: Operating condition space of system;
: Operating condition;
: Transition probability of system state from i to j when
: Transition probability matrix of system;
: Operation cost per 1-period at when and
: Discount factor 0 < < 1
},...,1,0{ nS
SX
},...,1{ h
O
kO
Sji
k
ij
k p ,}{P
iS
kO
S
X
O
k
ijp
kP
k
iC
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Product 1
SYSTEM
Op. Condition 1 Op. Condition 2 Op. Condition 3
10When do we select the suitable Operating condition ?
Model Description
deterioration
Product 2Product 3
)2()2()2(
)2()2()2(
)2()2()2(
0
0
000
nnn
iniji
j
ppp
ppp
ppp
nj
n
curren
t state
next transition state
1P
2P
3P
)()()(
)()()(
)()()(
0
0
000
hh
nj
h
hhh
h
n
hh
nnn
iniji
j
ppp
ppp
ppp
)1()1()1(
)1()1()1(
)1()1()1(
0
0
000
nnn
iniji
j
ppp
ppp
ppp
nj
n
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),()(
)3,()(
)2,()(
)1,()(
min)(
0
)()(
0
)3()3(
0
)2()2(
0
)1()1(
hivjVpc
ivjVpc
ivjVpc
ivjVpc
iV
n
j
h
ij
h
i
n
j
iji
n
j
iji
n
j
iji
when using Operating condition 1
),()(min)(0
)()( kivjVpciVn
j
k
ij
k
i hk ,...,2,1
Model Formulation
discount factor 10 11
when using Operating condition 2
when using Operating condition 3
Total discounted cost function
when using Operating condition h
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Assumptions
A-1
A-2
A-3
A-4
A-5
)()1()( icc k
i
k
i
hkSIP k ,......2,1)(
)()( ic k
i
SIPP kk )1()(
12
)(
0
)(
0
)1(
0 ...... hk ccc
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Properties of the Optimal Cost function
)0(),0(....)2,0()1,0( ihvvv
)(),( ikiv
)()1,(),( ikivkiv
1)
2)
3)
v(0,1)
v(0,2)
v(0,3)
State i
Cost
Is a non-decreasing function of i
),( kiv v(i,k)
v(i,k+1)
13
0
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14
deterioration
Cost
Op Cond3Op Cond1
Op Cond2
Op Cond3
L0New
Op Cond3
Op Cond2
Cost function for Op condition1
Cost function for Op Conditon3
Cost function for Op condition2
A System with 3 Variable Operating Conditions
Monotone Procedure
Op Cond1 Op Cond2 Op Cond3
Example
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conclusion
• Systems that follow Assumption (A-1) to (A-5),
the optimal decision procedure can be given by a monotone procedure.
Op Cond2 Op Cond2 Op Cond3
State i
cost
Op Cond1
Op Cond2
Op Cond3
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• Reference
[1] 谷口研二(2004)「LSI設計者のためのCMOSアナログ回路入門」 pp.22-29.
[2]https://www.crystec.com/trioxide.htm (2013/01/20 access)
[3] C.Derman(1963): “On optimal replacement rules when changes of state
are Markovian”, Mathematical optimization techniques,University of California Press.
[4] M.Ohnishi, H.Kawai and H.Mine, (1986): “An optimal inspection and replacement policy under incomplete state information” European Journal of Operations Research , vol.27, pp.117-128.
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Thank You
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)()()(
)()()(
)()()(
0
0
000
aa
nj
a
aaa
a
n
aa
a
nnn
iniji
j
ppp
ppp
ppp
P
18
A-3A-4A-5
i
k
iC
iC k
i
)(
)(),,1,0(1 iinkppSISIn
kj
s
ji
n
kj
s
ij
skk PPP
A-2
)( SI EaaP
≧≧
A-1
increasing Stochastic:SI
k
iC
1k
iC
iCC k
i
k
i
)1()(
)(
0
)1(
0
)0(
0 .. hCCC
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Suppose that we have a set of Operating Condition Options
Op. Cond(1) < Op. Cond(2) < ……. < Option(h)
Option(x)Option(x)
Option(x)Option(x)
Option(x)Option(x)
operating conditions set
)1(.)(. kConditionOpkConditionOp ),...2,1( Nk
cost
Op. Condition number k
19
W1W4
W4W3
W3W2
W2
Ordering Operating Conditions
Temp.
Speed
Chemicals
Op. Condition(x)
Example
W1
SETTING
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Mass Production
Ordering the productions with MULTIPLE Plasma Enhanced-CVDs
CVD
W1W1
W1W1
W1W1
W1
CVD
W1W1
W1W1
W1W1
W2
W3
CVD
W3W3
W3W3
W3W3 Op condition1
Op condition2 Op condition 3
State i
cost
Op condition 1
Op condition 2
Op condition 3
W3
CVD
W3W3
W3W3
W3W3
20
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Cost function of system working underOperating condition k
SYSTEM
Op. Condition 1
Productions
n
j
k
ij
k
i jVpckiv0
)()( )(),(
),( kiv
State i
Cost
),( kiv
0
Cost function of the system when operating condition k and state is i
21
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Numerical Example
When Parameters follow assumptions made
7.02.01.0
2.03.05.0
1.02.07.0
4.03.03.0
2.02.06.0
1.01.08.0
2.03.05.0
1.02.07.0
05.005.09.0
9.0
2N
k
ic101
40
20
1
2
1
1
1
0
c
c
c
103
42
23
2
2
2
1
2
0
c
c
c
105
45
28
3
2
3
1
3
0
c
c
c
SIP )1( SIP )2( SIP )3(
Discount factor
number of periods considered
Transition probability matrix for each Option
One period operating cost for each Option
22
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result
48.8
57.8
92
52.43
61.16
76.91
57.43
63.19
74.78
0
10
20
30
40
50
60
70
80
90
100
Op. Condition1 Op. Condition2 Op. Condition 3
Cost
State
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Numerical Example
When Parameters NOT follow assumption A-4
7.02.01.0
2.03.05.0
1.02.07.0
4.03.03.0
2.02.06.0
1.01.08.0
2.03.05.0
1.02.07.0
05.005.09.0
9.0
2N
k
ic100
40
30
1
2
1
1
1
0
c
c
c
95
35
25
2
2
2
1
2
0
c
c
c
85
25
15
3
2
3
1
3
0
c
c
c
SIP )1( SIP )2( SIP )3(
Discount factor
Number of periods considered
Transition probability matrix for each Option
One period operating cost for each Option
24
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Cost
State
result
65.1
82.3
172.9
54.7
71.9
135.4
32.1
46.6
113.9
0
20
40
60
80
100
120
140
160
180
200
Op. Condition1 Op. Condition2 Op. Condition3