Taguchi Robust Design and Loss function was proposed by Genichi Taguchi. In United States his concepts related to robustness for the evaluation and improvement of product development process was referred as “Taguchi Methods”. Taguchi named his concepts as “Quality Engineering”, where as other authors referred it as “Robust design” or “Robust engineering”.

**Taguchi Robust Design Approach**

Robust design is the key development in design process during recent years. Robust approach in design is key aspect as it produces reliable design both during manufacturing and also during product use. The basic concept of robust design is that parameter control that makes design strong enough that does not cause failure due to random “noise”

The Taguchi robustness approach says that process or products are controlled by various factors to get the desired response.

**Signal factors:**Signal factors are the signals used to get the desired response. Ex: pressure valve setting to get the required air pressure in the line**Control factors:**These factors produce the response based on noise in the process and also these are controlled by the designer. For example heat shrinks or thermo couples will respond based on the heat generation in the process. Sometimes these factors adds costs to the design hence these are also called tolerance factors.**Noise factors:**These are random events in the process, only mean and variance can be predicted but are not controllable by designer. Ex: voltage fluctuation, air pressure variation etc.

The noise factors impacts the response and leads to errors hence designers to focus on appropriate control factors to produce maximum response to signal factors. Taguchi defined Noise into three categories

- External Noise: variations in the environment where the product is used
- Deterioration Noise: wear and tear inside the unit
- Internal Noise: deviation from target values

**Taguchi Robust Design Principles**

Taguchi demonstrated robust design from the below three method concept

### Concept Design

Concept design is a process of selecting product or process based on competing technologies, customer, price or any other important considerations. Moreover the most advantage of this method is that organizations can produce high quality products with low production costs.

### Parameter Design

Parameter design refers to the identification of control factors for the process and also to determine the optimal (target) level of each factor. Initially in in this method design is developed using low cost parts and manufacturing methods. Afterwards response is optimized to minimize the noise. Furthermore the ultimate goal of this method is to find the efficient process (functionality robust) and service design.

Steps to perform Parameter Design

- Identify signal factors and noise factors and also respective ranges
- Select as much as possible control factors and also levels of the factors, and then dispense these levels to suitable orthogonal arrays ( to identify main effect with minimal number of runs). Functionality can be improved by adjusting different levels of control factors.
- Determine the S/N ratio

- Where
- r is the measurement of magnitude of input signal
- S
_{β}sum of squares of ideal function - V
_{e }mean square of nonlinearity - V
_{N }error term of linearity and nonlinearity

**Signal to noise (S/N) ratio** is especially used to calculate the systems performance. Interestingly the ratio of S/N depends on the situation, sometime Nominal S/N ratio is best for instance target. Sometimes larger S/N ratio is better. Ex: performance and few instances smaller S/N ratio is better. Ex: variation.

- Identify the optimal conditions from the experimental data, for this maximum S/N ratio to be used for each level of control factors.
- Finally, perform real production runs.

**Experimental design using standard orthogonal array**

Design of experiment is an experimental strategy in which effect of multiple factors are studied simultaneously by running tests at various level of factors. Usually in experiments some have few factors, some have too many factors, while there are others that have mixed levels.

Total number of experiments required for 2 level can be calculated as below

Full factorial always needs more number of experiments. However majority of experiments all factors posses same number of levels. In Taguchi approach a fixed number of orthogonal arrays are used to handle common experimental situations.

Particularly the Design means determine the number of experiments to be preformed and the manner in which they should be carried out (ie. number of factors and levels). Taguchi has constructed a number of orthogonal arrays to accomplish experiment design. Moreover each array can be used to suit a number of experimental situations.

Example: L_{4} (2^{3})
orthogonal array

Standard notation for orthogonal arrays L_{n} (l^{m})

- n- number of experiments
- l- number of levels
- m- number of factors
- Row represents the experimental conditions
- Columns represents the factors
- Each array can be used for many experimental situations

Orthogonal array have the property that every factor setting occurs the same number of times every test setting of all the other factors. This allows to make a balanced comparison among factor levels under various conditions.

**Example of Using Taguchi’s orthogonal array design in DMAIC**

XYZ plastic industry, plastic extrusion was
using in which raw plastic is melted and formed into a continuous profile. The
aim of experiments is to improve the component strength (N/m^{2}). Identified
4 control factors (A,B,C and D) of 3
levels (Low, medium and high ) and 3
noise factors (L,M and N) of 2 levels (Low and High)

Since 4 control factors of 3 level , L_{9} orthogonal method used for control factors and 3 noise factors of 2 levels, L_{8} orthogonal method used for noise factors. Here are the experimental results.

Plot factor contributions and select parameter levels which maximize S/N ratio

From the above graph the maximum S/N ratio for A at high level, B at low level, C at high level and D at low level

The robust design is attained which will maximize the response (strength) under the influence of noise factors when

- A factor set at high level
- B factor set at low level
- C factor set at high level
- D factor set at low level

An ANOVA can also be used apart from S/N ratio, but Taguchi prefers the graphical technique to visualize the significant factors. Moreover the advantage of Taguchi design method is to determine the optimum combination of factors and levels for the analysis. Finally, confirmation productions run to be carried out to check the results.

### Additional Helpful Robust Design Resources

### Tolerance Design

Tolerance is an acceptable limit of deviation in a parameter’s dimension or value and this must be determined for all system components. In other words Tolerance design maintain the balance between quality and design cost. Thus, Tolerance design creates metrics allowing designer to compute the tolerances that can be adjusted to meet customer needs and expectations while delivering a cost effective product.

Tolerance design is a step beyond parameter design as it impacts the budget such as material change, equipment etc. Apart from economics, it is also considers the other important factors like constraints due to safety factor, material property and design choice etc.,

The relationship between tolerance specification, the functional limit and the safety factor are

### Statistical Tolerance

Statistical tolerance is the method to determine the tolerance when two or more components are assembled. Statistical tolerance indicates the amount of process variance. This is the relationship between independent cause variance and the variance of the overall result.

Statistical tolerance uses square root of sum of variances to determine the tolerance required.

**Example: **Find the overall thickness of the stack consists of 4 plates of various thicknesses and variation

- Nominal thickness = 12+15+20+12=59
- First plate 6σ
_{1 }= 0.5; σ_{1 }= 0.083 - Second plate 6σ
_{2 }= 0.4; σ_{2 }= 0.068 - Third plate 6σ
_{3 }= 0.4; σ_{3 }= 0.068 - Fourth plate 6σ
_{4 }= 0.5; σ_{4 }= 0.083

Assuming that the measured dimensions (thickness) of a plate fall within a bell curve and also assuming the standard deviation 3σ is necessary for assembly

σ_{Total} ^{2} = σ_{1}^{2}+ σ_{2}^{2}+ σ_{3}^{2}+σ_{4}^{2}

σ_{Total} =√(0.083^{2}+0.068^{2}+0.068^{2}+0.083^{2})
=0.151

3σ_{Total} = 0.452

Hence the total thickness of the stack = 59±0.452

## Taguchi’s Quality Imperatives

- The basic quality losses are due to poor design
- For any new products, plan experiments to get the parameter targets
- Consider customer environment conditions to develop robust products
- Robustness is a function of product design
- Quality losses are a loss to society
- Robust products have a strong signal to noise ratio
- Manufacture the products that are consistent by reducing variation.
- Tolerances should be set prior to manufacturing and also Quality loss function can be measured

## What is Taguchi’s Loss Function

Taguchi loss function was developed by the Genichi Taguchi. According to Taguchi every time the process deviates from the target, even if it stays within the specifications, there is a loss to the society. In other words it states that increase in process variation leads to customer dissatisfaction even if the process is within specification limit

### When would you use Taguchi’s loss function?

Whenever an organization works on process improvements or optimizing the process they especially focus on lowering the variation form the nominal value (i,e target). As a matter of fact for any business profit is one of the basic and very important metric. Thus Taguchi loss function is very valuable that transform the variation from nominal value with financial depiction. In other words it gives the relation between lowering the variation from target to the financial benefit.

From the above orange example it is clear that customer may pay same amount on day 3, 5 or 7 days, but in reality on day 5 only it’s a value for money for the customer. Which may further leads to customer dissatisfaction or future business impact. Hence Taguchi Loss function is widely using the organizations.

The loss function is to determine the financial loss that will occur when a quality characteristic x deviates from the nominal value t

The loss function depicted as L(x) = k(x-t)^{2}

- k=
loss coefficient = cost of a defective product /(tolerance)
^{2} - t= nominal value
- σ
^{2}= mean value of (x-t)^{2}

### Example of Using Taguchi’s loss function in DMAIC

XYZ is a washing machine manufacturing unit produces various models. For instance, among all the parts compressor is one of the critical part. What is the loss to the society if the voltage is 105V?

- Nominal voltage = 110V
- Mean tolerance = 15V
- Cost of defective washing machine = $5000

k= loss coefficient = cost of a defective product /(tolerance)^{2} = 5000/(15)^{2} = $22.2

L(x) = k(x-t)^{2} = 22.2(105-110)^{2 }= 22.2*25= $555.5

Therefore, if the compressor output voltage is 110V then the loss to the society is zero, where as if the output of the compressor is 105V then Quality loss to the society is $555.5