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The loss function of Taguchi is the second part of philosophy. It makes the link between the technique and the cost of quality, allowing to define the tolerances of a system..

## Introduction

The loss function is the initial idea of Taguchi’s robust design. Generally, a tolerance interval is used to define whether a product is good or not. For Taguchi, the challenge is not only to be in the meantime, but to be closer to the target. It considers that the cost generated by the gap with the optimal increases continuously.

#### It is based on the assumption that the loss (good / not good) does not occur suddenly as if we fall into the water at the end of a pontoon.

To illustrate this phenomenon, take the figure opposite :

• Qwhat is the difference between part 2 and 3 ? None of them conform.
• What is the difference between part 1 and part 2 Exhibit 1 is not compliant. Yet, technically speaking, there is more difference between 2 and 3 than between 1 and 2. Especially since the tolerance could be slightly offset on one side or the other to tip the balance.

Taguchi then proposes the following solution.

“Any deviation from the goal results in a financial loss equal to the square of the deviation from the goal”.  G. taguchi

Concretely, the cost is not considered as a direct cost of non-quality, but as a consequence. Taguchi considers that over time, the dispersion of our results generates indirect costs :

• Customer dissatisfaction
• Higher warranty costs
• Higher service costs

The interest of this approach is to :

• Compare multiple vendors with similar capabilities.
• Link technical notions to economic concepts.
• Identify tolerance intervals on customer perception.

## 1 – The coefficient of the loss function

The loss function is dependent on a constant of the system being studied. This coefficient is determined by the ratio between the cost generated by an intervention and the known tolerance interval. We pose:

• A : Sum of the costs generated by a repair, replacement …
• Δ : Tolerance of our system. We find 4 cases that we detail in the table below :

 The target is the best and symmetrical Lthe smallest is the best The biggest is the best The target is the best but asymmetric Formula to use k = A  / Δ2 k = A  / Δ2 k = A  * Δ2 k+ = A0+ / Δ+2 k- = A0- / Δ-2 Value of A A A A In this case, the cost of going up or down may be different. So we have : A0+ = 200 € for example A0- = 100 € for example Value of the tolerance interval Target ± 0,3 Cible +0, +1,2 but we are looking for 0 Cible +10, +… but we are looking for +… In the same way, the tolerance can be different depending on whether one is below or above the target. We can have for example : Target -0,2, +0,8 Value of Δ 0,3 1,2 10 Δ- = -0,2 Δ+ = 0,8

It is assumed that beyond these tolerances, we generate customer dissatisfaction, replacement, scrap … Attention, we must take here all the costs without worrying about who pays (the customer, the customer). business, insurance…).

Exemple :

It is assumed that the functional limit for proper operation is ± 0.3mm. It has been calculated that the cost of replacing the part when it exceeds the tolerances is 20 €. We obtain a constant k of 20/0,32 = 222.

## 2 – The calculation of the loss function

Taguchi translates the loss function by taking into account the constant and the distance between our value obtained and the target sought.

This economic approach to tolerance limits and dispersion is a culture change. For Taguchi, it is not enough for the parts to be within tolerance limits. It also requires a distribution focused on the objective with a dispersion as low as possible.

To calculate it, we will use the formulas described below.

### Case where the target is the best  “symmetric »

We are in the most common case where we will seek to be closest to the target. The formula for the loss function is as follows :

For an element

Cost = k * (Y1 – Y )2

For a set of elements

Cost = k * (σ2 + (μ –Y )2)

k : system constant

Y1 : the value we have of our process

Y : the target value that we seek to achieve and that we know how to satisfy the customer

σ2 : estimated variance from the sample

μ : average of the results obtained by our sample ### Case where the smallest is the best

It is the specific case when one studies a rate of defects or consumptions of energies … One will seek to be the smallest possible (ideally 0) compared to the target. The formula is :

For an element

Cost = k * Y12

For a set of elements

Cost = k * (σ2 + μ2)

k : system constant

Y1 : the value we have of our process

Y : the target value that we seek to achieve and that we know how to satisfy the customer

σ2 : estimated variance from the sample

μ : average of the results obtained by our sample ### Where the biggest is the best

Reverse case of the previous one, it is the case for example when one studies the resistance of a weld. The stronger the resistance, the better. We will try to be as big as possible in relation to the target. The formula is :

For an element

Cost = k * (1 / Y1)2

For a set of elements

Cost = k * (1/n * Σ ((1/Yi)2))

For a set of elements

k : system constant

k : system constant

Y1 : the value we have of our process

Y : the target value that we seek to achieve and that we know how to satisfy the customer

σ2 : estimated variance from the sample

μ : average of the results obtained by our sample ### Case where the target is best “asymmetric”

This is the specific case of the nominal is best but the tolerance is asymmetrical. This is the case for example when we study the temperature variation of a fridge: too cold, we will freeze food, too hot, we will have a loss of food and especially the risk of disease. consequences are therefore not the same and the cost generated below or above the target is not the same. We will use an asymmetric, formula to evaluate the loss function :

For an element

Cost+ = k+ * (Y1 – Y )2 avec Y1 > Y

Coût- = k- * (Y1 – Y )2 avec Y1 ≤ Y

For a set of elements

Cost+ = k+ * (σ2 + (μ –Y )2)

Cost- = k- * (σ2 + (μ –Y )2)

k+ et k- : system constant above and below the target

Y 1 : the value we have of our process below or above the target

σ 2 : variance estimated from the sample

μ : average of the results obtained by our sample ### An example

A well-known example is that of Sony in the 1970s. At that time, a study was conducted on the color density of televisions. The results were compared between the Japanese and American factories of Sony. The desired density was then defined with a tolerance of ± 5 with respect to the target. The distributions of the results are as follows. Source : M. S. Phadke (1989) – Quality engineering using robust design

We notice that the Japanese factory produces a production that follows a normal law and it induces about 0.3% of default. Conversely, the American plant does not produce a defect and the distribution of the measurements is homogeneous around the average.
The consumer study has actually shown that customers have a preference for televisions at the Japanese factory. The reason is that customers preferred TVs that had the target density. Outside the Japanese factory manufactured much more than the American factory.
By calculating the proportionality coefficient k of the loss function, we see that it is \$ 8.35 for the Japanese plant and \$ 25 for the American plant (Source: Asahi Japan Journal, 1979).

## 3 – Determine the tolerance

The loss function allows us to determine the tolerances of an element or a subset. Depending on the situation, we will be faced with two cases that we detail below via an example.

### Case 1 : the tolerance of an element

Take the example of the voltage of a circuit. We know that at the factory, we can calibrate the products for a cost of € 2. The nominal voltage of the devices is 230 Volts with a tolerance of + or – 20. We also know that replacing the device costs us 100 €.

We want to know when it is “profitable” for us to calibrate.

.

#### 3.1.1 – Determine the function k

We determined k by performingt :

k = 100 / (250 – 230)2 = 0,25 € / Volt

#### 3.1.2 – Deduce tolerance

We will determine the level of profitability of the calibration. We perform the following calculation :

Δ = ± √ (Cost / k) = ± 2,83 Volts

We conclude that below a deviation of 2.83 volts at the factory, it is not necessary to perform a recalibration, as this would not be profitable.

### Case 2 :Tolerance of a subset

We are in the classic case of the designer. We are very often subjected to the following situation: our company designs color printers. We know that the quality of the image is dependent on a parameter (spacing of the anchor tube and the drum) which itself is dependent on the straightness of the anchor tube, the position of the terminal block and the position drum.

#### 3.2.1 – Calculate the tolerance

The general equation for calculating tolerance is:

Δi = √ (Costi/Cost ) * Δ  / βi

Avec :

• Δi : Component tolerance i
• Δ  : tolerance of the system as a whole
• Costi : Cost of repairing and replacing the item i
• Cost : cost result of failure to respect the system’s tolerance
• βi :Sensitivity of the system response to component variationi

In our example, we know that the overall cost generated by exceeding the tolerance is 500 €, and that the tolerance of the system is 0.045. For components, we have the following data with which we deduce the tolerances of each components :

 Cost in \$ βi Calculated tolerance Δi Straightness of the anchor tube 22 7 ± 0,0013 Position of the terminal block 100 6,85 ± 0,003 Drum position 44 6,5 ± 0,002

#### 3.2.2 – Validate the capability

Finally, the last step of the calculation, we will validate the capability of our production means thanks to the calculated tolerances. Using standard capability formulas (Customer Voice / Process Voice).

Using the previous example, we obtain the following table

:

 Variability in manufacturing (3σ) short term Capability Cp Calculated tolerance Δi Straightness of the anchor tube 0,0012 1,083 ± 0,0013 Position of the terminal block 0,0033 0,909 ± 0,003 Drum position 0,0009 2,22 ± 0,002

Only the manufacturing capability of the drum is very good. For the 2 others, it is necessary to put them under control and to carry out actions of progress.

## Source

W. Y. Fowlkes, C. M. Creveling (2000) – L’ingénierie robuste, méthodes Taguchi en conception

J. M. Buforn (2008) – Plans d’expériences, outils de modélisation et d’optimisation