1. An approach to the SN ratios based on
the proportional models and its application
The Institute of Statistical Mathematics, Tokyo, JAPAN
KAWAMURA Toshihiko
Yokohama College of Pharmacy
IWASE Kosei

2. Robust Parameter Design
Outline
Taguchi Design of Experiments
Robust Parameter Design
Signal-to-Noise (SN) ratios
The testing problem of the equality for two SN ratios

3. Taguchi Design of Experiments
Robust Parameter Design, also called the Taguchi Method pioneered by Dr. Genichi TAGUCHI, greatly improves engineering productivity.
Comparable in importance to Statistical Process Control, the Deming approach and the Japanese concept of TQC
Robust Parameter Design is a method for designing products and manufacturing process that are robust to uncontrollable variations.
Based on a Design of Experiments (Fisher’s DOE) methodology for determining parameter levels
DOE is an important tool for designing processes and products
A method for quantitatively identifying the right inputs and parameter levels for making a high quality product or service
Taguchi approaches design from a robust design perspective

4. The Taguchi Approach to DOE
Traditional Design of Experiments (Fisher’s DOE) focused on how different design factors affect the average result level
Taguchi’s DOE (robust design)
Variation is more interesting to study than the average
Run experiments where controllable design factors and disturbing signal factors take on 2 or 3 levels.

5. Robust Design (I)
By consciously considering the noise factors and the cost of failure in the Taguchi method helps ensure customer satisfaction.
Environmental variation during the product’s usage
Manufacturing variation, component deterioration
Noise factors (Disturbances) are events that cause the design performance to deviate from its target values
A three step method for achieving robust design
Concept design
Parameter design
Tolerance design
The focus of Taguchi is on Parameter design

6. Robust Design (II) Robust Parameter Design (e.g. Wu and Hamada 2000)
A statistical / engineering methodology that aim at reducing the performance “variation” of a system.
The selection of control factors and their optimal levels.
The input variables are divided into two board categories.
Control factor: the design parameters in product or process design.
Noise factor: factors whoes values are hard-to-control during normal process or use conditions
The “optimal” parameter levels can be determined through experimentation

7. Signal to Noise (SN) Ratios (I)
Performance measure
Taguchi’s SN ratio

8. Signal to Noise (SN) Ratios (II)

9. The Taguchi Quality Loss Function
The traditional model for quality losses
No losses within the specification limits!
Scrap Cost
LSL
USL
Target
Cost
The Taguchi loss function
the quality loss is zero only if we are on target
loss function
risk function
SN ratios

10. A new performance measure (I)
However, if the adopted principles of the signal-response systems are diffent and the physical quantities of the response values are different between the systems, the comparison of the Taguchi’s SN ratios has no sense.
A new performance measure for the systems :
We propose a dimensionless SN ratios (Kawamura et al. 2006).
Proportional model, K loss function, Dynamic SN ratios
The response and the signal factor values are positive real values.

11. A new performance measure (II)
The response and the signal factor values are positive real values.
Consider two-parameter statistical models for positive continuous observation.
Log normal distribution
Gamma distribution
Inverse Gaussian distribution etc.
Error distribution

12. A new performance measure (III)
K loss function
K risk function

13. A new performance measure (III)
Calculation !

14. A test of the Equality for two SN ratios (I)
We consider the testing problem of the equality for two SN ratios
SN ratios for the systems with Dynamic Characteristics
Performance comparison of the systems

15. A test of the Equality for two SN ratios (II)
Data 2
Data 1
SN ratio A2
SN ratio A1
Which performance is good ?
Testing homogeneity of SN ratios

16. A test of the Equality for two SN ratios (III)
Null hypothesis
A Variance Stabilizing Transformation
Approximation Test

17. A numerical example (I)

18. A numerical example (II)
Significant level 1%
Not significant !
In this example, the significant difference of the SN ratios between A1and A2 is not shown.