# Presentation: 481972

## Operating Window Experiments

- C. F. Jeff Wu  and V. Roshan Joseph

- Department of Statistics
- University of Michigan
-  Ann Arbor, MI, USA

- To appear in *Journal of Quality Technology, 2002*
- Available in * http://www.stat.lsa.umich.edu/~jeffwu/*

## Outline

-  Paper Feeder Example
-  SN Ratio
-  Probability of Failure Modeling
-  Performance Measure
-  Modeling and Optimization
-  Generalized Performance Measure
-  Sequential Design/Estimation
-  Application to Real Data
-  Summary

## Paper Feeder Example

- Two failure modes
-  Misfeed  : fails to feed a sheet
-  Multifeed: Feeds more than one sheet

## Objective: Minimize both misfeed and multifeed 

- 	      through parameter design optimization.
**Approach 1**
-  Feed, say, 1000 sheets at different design settings
-  Observe # of misfeeds and # of multifeeds
-  Choose a design setting to minimize both
- Problems:
-  Large sample size required to get reasonable power for
-    statistical tests
-  Conflicting levels: design settings that minimizes misfeeds
-    tend to increase multifeeds
-  Difficult to distinguish between two design settings

## Approach 2

- Failures are extremities of certain *functional characteristics*
-  Identify functional characteristics (continuous variables) that
-    relates to misfeed and multifeed
-  Do nominal-the-best optimization on these characteristics

**Example**
- Misfeed   :  Roller-to-paper friction force
- Multifeed : Paper-to-paper friction force

## Advantages (Approach 2):

-  More information contained in continuous variables than
-    in categorical variables
-  Better understanding about the functioning of the system
-  Optimization is easier

- Problems:
-  Sometimes not easy to identify the right functional
-   characteristics
-  Sometimes difficult to measure due to inadequate or
-   expensive measurement system

**An Alternative Approach**: *Operating Window*

## Operating Window 

*Operating window* is defined as the boundaries of a critical
- parameter at which certain failure modes are excited.

-  Clausing (1994), Taguchi (1993).

**Approach:**
-  Identify a critical parameter: low values of which lead to
-   one failure mode and high values lead to the other failure mode
-  Measure the operating window at different design settings
-  Choose a design to maximize the operating window

## Paper feeder example continued

*Stack force* is a critical parameter and is easy to measure.
- A small force leads to misfeed and a large force leads to multifeed.

- misfeed

- operating window

- multifeed

- 0               * **l*                               *u *                      stack force

- (*l, u*): operating window
- Stack force: *operating window factor*

## Examples

## Notes: 

-  There are no clear boundaries separating the failure modes
-  The boundaries of operating window are random
-  They are defined with respect to a threshold failure rate
-  For example:
-         * **l *= force at which 50% misfeed occurs
-         * **u *= force at which 50% multifeed occurs

## Effect of Noise Factors

- N1:

- 0

- 0

- 0

- N2:

- N3:

- Operating window

-  The two failure modes can be minimized by keeping the
-    OW factor at the center of the operating window
-  A large operating window then corresponds to a robust system
-  Achieving robust design          maximizing operating window

## Optimization

-  Control factors: wrap angle, belt tension, radius of retard roll,...
-  Noise factors   : paper weight and type, alignment errors,...
-  Determine *l* and *u* for several control and noise settings
-  Perform two-step optimization

-  Find a control factor setting to maximize the
-        signal-to-noise ratio

- 2.    Adjust OW factor to the middle of the operating window

## SN Ratio

-  Developed at Fuji Xerox
-  misfeed is eliminated if  *l*= 0
-  multifeed is eliminated if * u* =

*l* is STB:

*u* is LTB:

-  Then

## Probability of failure modeling

***X ***= set of control factors
***N ***= set of noise factors
*M* = operating window factor

- Probability of failure due to defect type 1:

- Probability of failure due to defect type 2:

- = failure threshold (such as 50%)

## Assumed model:

- Reparameterized as

## Performance Measure

-  We can set M to minimize the expected loss

-  Loss:

-  Expected loss:

-  If *p* is the fraction rejection, then the number of extra units to
-   produce to get a non-defective product has a geometric
-   distribution with mean              .

-  Substituting (*),

## EL at M=M*,

- Performance measure:

- PM can be estimated by its sample analog, which is the
- same as the SN Ratio.

## Note:

- Thus *      *                                  is a composite measure for doing
-  the following jobs:
-  Minimize *E(l)*
-  Minimize *Var(l)*
-  Maximize *E(u)*
-  Minimize *Var(u)*

- Thus operating window optimization is much more than just
- widening the operating window

## Modeling and Optimization

- Fit

-  Find
- 2.    Adjust

-  Special case: single noise level,

-  Maximize
- 2.    Adjust

## Generalized Performance Measure

-                              is the operating window

-  Noise factor compounding

-              is the width of operating window in log scale

-  As                  , it becomes

## Sequential Design/Estimation

-  Fixed design : Use two logistic regressions
-  Sequential design:  Use logit-MLE method of Wu (1985, JASA)

- Suppose after *k* experiments we have the data

- Then choose

- Ying and Wu (1997, Statistica Sinica)

- where *y *is the number of failures (say misfeed) in a sample of
- size *m*.

## A Real Example

-  Peace (1993)
-  Wave soldering process
-  Two failure modes: solder voids and solder bridges
-  OW factor: top side board temperature

*l*:  topside board temperature at which solder voids stop
*u*: topside board temperature at which solder bridges begin

-  Five noise levels: board carriers
- 15 control factors:           design

## Slide 23

## Slide 24

## Half-normal Plots

**Half-normal Plots**

## Slide 26

##          : failed to identify any significant effects

-          : failed to identify any significant effects
-                                                                    are similar , which
-   means the performance measures are not significant to               .

## Summary

- A rigorous statistical foundation
- Operating window SN ratio is justified using PerMIA theory under some restrictive modeling assumptions
- New and more effective modeling and optimization strategy
- Generalized performance measure (GPM) under more general modeling assumptions
- Logit-MLE method for sequential design
- Superiority of the proposed strategy is illustrated with an example