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

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

http://www.intel.com/pressroom/archive/releases/2006/20060125comp.htm(2013/02/15 access)

Plasma Enhanced-CVD[1]

3

WaferChemical Layers

Semiconductor Industry

Diagram of Plasma Enhanced-CVD (Chemical Vapor depositor)[2]

Plasma

Gases

G

a

S

4

Wafer

Chemical Layers deposited

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

…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)

Purpose of the research

Purpose of this research is to providesufficient conditions for optimal decisionmaking to be given by a monotoneprocedure.

7

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

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

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

),()(

)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

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

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

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

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

• 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.

16

Thank You

)()()(

)()()(

)()()(

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

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

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

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

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

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

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

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