Taguchi’s Quality EngineeringLoss FunctionTaguchi has defined quality as the loss imparted to society from the time a product is shipped. Societal losses include failure to meet customer requirement, failure to meet ideal performance, and harmful side effects. Also included the losses due to production, such as raw material, energy, and labor consumed on unusable products or toxic by-products.

Loss to Society

Distribution of Color Density for Sony-USA and Sony-Japan

Nominal-the-Best

The quadratic loss function is described by the equation

The loss coefficient is determined by setting , the deviation from the target. When Δ is at the USL or LSL, the loss to the customer of repairing or discarding the product is $A. Thus,

Average Loss

The loss described previously assumes that the quality characteristic is static. In reality, we cannot always hit the target, τ. It is varying due to noise, and the loss function must reflect the variation of many pieces rather than just one piece.

Noise factors are classified as external and internal, with internal being further classified as unit-to-unit and deterioration.

Nominal-the-best loss function with the distribution of the noise factor:

An equation can be derived by summing the individual loss valuesand dividing by their number to give

Other Loss Functions

Orthogonal ArraysOA are a simplified method of putting together an experiment.

An orthogonal array with 8 rows:

Procedure to determine the appropriate OA:1. Define the number of factors and their levels.2. Determine the degrees of freedom.3. Select an orthogonal array.4. Consider any interactions.

Degrees of Freedom

The number of degrees of freedom is a very important value because it determines the minimum number of treatment conditions. It is equal to the sum of

At least seven treatment conditions are needed for the two-level, and 17 conditions are needed for the three-level.

The maximum degrees of freedom is equal to

Selecting the OA

The number of treatment conditions is equal to the number of rows in the OA and must be equal to or greater than the degrees of freedom.

The available OA, up to OA36:

Thus, if the number of dof is 13, then the next available OA is OA16.

Interaction Table

Confounding is the inability to distinguish among the effects ofone factor from another factor and/or interaction. In order to prevent confounding, we must know which column to use for the factors.

Let’s assume that factor A is assigned to column 1 and factor B to column 2. if there is an interaction between factors A and B,then column 3 is used for the interaction, AB.

Another factor, C, would need to be assigned to column 4. If there is interaction between factor A (column 1) and factor C (column 4), then interaction AC will occur in column 5.

The columns that are reserved for interactions are used so that calculations can be made to determine whether there is a strong interaction. If there are none, then all columns can be used forfactors.

The actual experiment is conducted using columns designated for factors, referred to as design matrix. All the columns are referred to as design space.

Linear Graphs

Taguchi developed a simpler method of working with interactions using linear graph. They make it easier to assign factors and interactions to the various columns of an array. Factors are assigned to the points. If there is an interaction between two factors, then it is assigned to the line segment between the two points.

Linear graphs for OA8:

Interactions

Taguchi’s approach to interactions are:

1. Interactions use dof; therefore, more treatment conditions are needed or fewer factors can be used.

2. OA are used in parameter design to obtain optimal factor/levels for robustness and cost in order to improve product and process performance.

3. Interactions are primarily between control factors and noise factors.

4. Main effect analysis will give the optimal results and good reproducibility as long as interactions are relatively mild.

5. OA12 (two-level) and OA18 (three-level) are recommended so that if interactions are present, they are dispersed among all the factors.

6. Engineers should develop a design that uses main effects only.

7. An unsuccessful confirmation run may indicate an interaction.

Signal-to-Noise (S/N) RatioThe S/N concept:

Signal factors ( ) are set by the designer or operator to obtain the intended value of the response variable. Noise factors ( ) are not controlled or are very expensive or difficult to control. Both the average,a , and the variance, , need to be controlled with single figure of merit.

Squaring and taking the log transformation gives

Adjusting for small sample sizes and changing from Bels to decibles by multiplying by ten yields

Smaller-the-Better

The S/NS ratio for smaller-the-better is used for situations where target value (τ) is zero, such as computer response time, automotive emissions, or corrosion. The equation is

The negative sign is used to ensure that the largest value gives the optimum value for the response variable and robust design. Mean standard deviation (MSD) is given to show the relationship to the loss function.

Larger-the-Better

It is used where the largest value is desired, such as weld stregth, gasoline mileage, or yield. From mathematical view point, the target value is 00. like the loss function, it is thereciprocal of smaller-the-better. The equation is