Taguchi quality loss function and specification tolerance design

Title: Taguchi Quality Loss Function and Specification Tolerance Design

Authors: Erin Knight, Matt Russell, Dipti Sawalka, Spencer Yendell

Date Presented: November 28, 2006 Date Revised: December 8, 2006
 * First round reviews for this page
 * Rebuttal for this page

Introduction
Genichi Taguchi provided a whole new way to evaluate the quality of a product. Traditionally, product quality has been a correlation between loss and market size for the product. Actual quality of the product was thought of as an adherance to product specifications. Loss due to quality has usually only been thought of as additional costs in manufacturing (i.e. materials, re-tooling, etc.) to the producer up to the time of shipment or sale of the product. It was believed that after sale of the product, the consumer was the one to bear costs due to quality loss either in repairs or the purchase of a new product. It has actually been proven in most cases that in the end the manufacturer is the one to bear the costs of quality loss due to things like negative feedback from customers. Taguchi changed the perspective of quality by correlating quality with cost and loss in dollars not only at the manufacuring level, but also to the customer and society in general.

Taguchi Quality Loss Function
You will most likely encounter Taguchi methods in a manufacturing context. They are statistical methods developed by Genichi Taguchi to improve the quality of products. Where as statisticians before him focused on improving the mean outcome of a process, Taguchi recognized that in an industrial process it is vital to produce a product on target, and that the variation around the mean caused poor manufactured quality. For example, car windshields that have the target average mean are useless if they each vary significantly from the target specifications.

What are the losses to society from poor quality?
Taguchi's key argument was that the cost of poor quality goes beyond direct costs to the manufacturer such as reworking or waste costs. Traditionally manufacturers have considered only the costs of quality up to the point of shipping out the product. Taguchi aims to quantify costs over the lifetime of the product. Long term costs to the manufacturer would include brand reputation and loss of customer satisfaction leading to declining market share. Other costs to the consumer would include costs from low durability, difficulty interfacing with other parts, or the need to build in safety margins.

Great, so what is the actual loss function?
Think for a moment about how the costs of quality would vary with the products deviation on either side of the mean. Now if you were to plot the costs versus the diameter of a nut, for example, you would have a quadratic function, with a minimum of zero at the target diameter. We expect therefore that the loss ($$L$$) will be a quadratic function of the variance ($$ \sigma $$, or standard deviation) from the target ($$m$$). The squared-error loss function has been in use since the 1930's, but Taguchi modified the function to represent total losses. Next we will walk though the derivation of the Taguchi Loss Function.

Loss function for one piece of product:

$$L = k(y-m)^2\qquad$$ Where: $$L$$ = Loss in Dollars $$y$$ = Quality Characteristic (diameter, concentration, etc) $$m$$ = Target Value for y $$k$$ = Constant (defined below)

The cost of the counter measure, or action taken by the customer to account for a defective product at either end of the specification range, Ao, is found by substituting $$y = m + \Delta_0$$ into the loss function:

$$Ao = k(y-m)^2 = k(( m + \Delta_0) - m) ^2\qquad$$

Now we can solve for the constant k,

$$k = \frac{Ao} {\Delta_0 ^2} \qquad$$

Since we are not usually concerned with only one piece of product, the loss function for multiple units is

$$L = k \Delta_0 ^2$$

The process capability index (Cp) is used to forecast the quality level of non-defective products that will be shipped out. The Cp has been used in traditional quality control and is defined rather abstractly as:

$$Cp = \frac{ 2 \Delta_0 } {6  \sigma } $$

where $$2 \Delta_0 \qquad$$ represents the tolerance of the product.

Substituting k into the loss function and then rearranging in terms of Cp,

Taguchi Loss Function:

$$L = \frac{Ao} {\Delta_0 ^2} \sigma ^2 = \frac{ Ao} {9Cp ^2 }$$

The Taguchi quality loss function is a way to assess economic loss from a deviation in quality without having to develop the unique function for each quality characteristic. As a function of the traditionally used process capability index, it also puts this unitless value into monetary units.

Quality Loss Function for Various Quality Characteristics
There are three characteristics used to define the quality loss function:


 * 1) Nominal–the-Best Characteristic
 * 2) Smaller-the-Better Characteristic
 * 3) Larger-the-Better Characteristic

Each of these characteristic types is defined by a different set of equations, which is different from the general form of the loss function equation.

Nominal–Defined upper and lower boundries
For a nominal characteristic, there is a defined target value for the product which has to be achieved. There is a specified upper and lower limit, with the target specification being the middle point. Quality is in this case is defined in terms of deviation from the target value. An example of this characteristic is the thickness of a windshield in a car.

The equation used to describe the loss function of one unit of product:

$$L = k(y-m)^2\qquad$$ Where: $$L$$ = Loss in Dollars $$y$$ = Output Value $$m$$ = Target Value of Output $$k$$ = Proportionality Constant

The proportionality constant ($$k$$) for nominal-the-best characteristics can be defined as:

$$k = \frac {A_0} {\Delta_0^2} $$ Where: $$ A_0 $$ = Consumer Loss (in Dollars) $$ \Delta_0 $$ = Maximum Deviation from Target Allowed by Consumer

When there is more than one piece of product the following form of the loss function is used:

$$L = k(MSD)\qquad$$

$$MSD = (y-m)^2\qquad$$ = Mean Squared Deviation

A graphical representation of the Nominal Characteristic is shown below. As the output value ($$y$$) deviates from the target value ($$m$$) increasing the mean squared deviation, the loss ($$L$$) increases. There is no loss when the output value is equal to the target value ($$y = m$$).

Smaller-the- Better
In the case of Smaller-the-Better characteristic, the ideal target value is defined as zero. An example of this characteristic is minimization of heat losses in a heat exchanger. Minimizing this characteristic as much as possible would produce a more desirable product.

The equation used to describe the loss function of one unit of product:

$$L = ky^2\qquad$$ Where: $$k$$ = Proportionality Constant $$y$$ = Output Value

The proportionality constant ($$k$$) for the Smaller-the-Better characteristic can be determined as follows:

$$k = \frac {A_0} {y_0^2}\qquad $$ $$A_0$$ = Consumer Loss (in Dollars) $$y_0$$ = Maximum Consumer Tolerated Output Value

A graphical representation of the Smaller-the-Better characteristic is below. The loss is minimized as the output value is minimized.



Larger-the-Better
The Larger–the-Better characteristic is just the opposite of the Smaller-the-Better characteristc. For this characteristic type, it is preferred to maximize the result, and the ideal target value is infinity. An example of this characteristic is maximizing the product yield from a process.

The equation used to describe the loss function of one unit of product:

$$L = \frac{k} {y_0 ^2}\qquad $$ Where: $$k$$ = Proportionality Constant $$y_0$$ = Minimum Consumer Tolerated Output Value

The proportionality constant ($$k$$) for the Larger-the-Better characteristic can be calculated by using the equation given for the Smaller-the-Better proportionality constant. The only difference between the two is the deffinition of $$y_0$$.

A graphical representation of the Larger-the-Better characteristic is shown below. This characteristic is the opposite of the Smaller-the-Better characteristic, as the loss is minimized as the output value is maximized.



Specifying Tolerances for a Process
A manufacturer is responsible for only shipping products that meet certain specifications. Products that do not meet these determined specifications are defective and cannot be shipped for sale, resulting in a loss to the company. In aiming to meet these specifications, manufacturers have a determined level of tolerance for deviation from the desired target specification. The problem that often occurs is products that barely meet specifications are shipped and fail after customer purchase. This causes negative feedback from customers, which results in losses to the manufacturer and ultimately society. The standard to fix this problem is to tighten up the tolerances. More stringent tolerances would result in fewer products failing on customers, reducing losses in the market, but they would also result in increased costs to manufacturers. Before Taguchi, there was no set method for determining optimal tolerances for a given process.

Since it is very difficult to quantify the loss to society for a defective product after customer purchase, Taguchi predicts the quality level. The quality loss function is the basis for determining tolerances for a process. In quality engineering, tolerance is defined as the deviation from the target, not the deviation between products. Taguchi's method determines tolerances that aim for a balance between losses to the manufacturer and the customer. To do determine these tolerances, the quality loss function can be used to determine how much it costs the manufacturer to fix the defective product before shippment, and compare that value to the cost that the defective product would have on the customer (society).

Worked out Example 1
Suppose you are manufacturing green paint. To determine a specification for the pigment, you must determine both a functional tolerance and customer loss. The functional tolerance, $$\Delta_0$$ is a value for every product characteristic at which 50% of customers view the product as defective. The customer loss, $$A_0$$, is the average loss occuring at this point. Your target is 200g of pigment in each gallon of paint. The average cost to the consumer, HomePainto, is $10 per gallon from returns or adjusting the pigment. The paint becomes unsatisfactory if it is out of the range $$ 200g \pm 10g$$.

Calculate the loss imparted to society from a gallon of paint with only 185g of pigment.

$$L = k(y-m)^2\qquad$$

$$Ao = $10\qquad$$

$$\Delta_0 = 10g \qquad$$

$$ k = \frac{Ao} {\Delta_0 ^2}  = \frac {$10} {10g ^2} = $0.10 per gram ^2 $$

$$L = $0.10 (y - 200)^2\qquad$$ dollars/ gallon of paint

$$y = 185 g\qquad$$

$$L = $0.10 (185 - 200)^2 = $ 22.5 \qquad$$

This figure is a rough approximation of the cost imparted to society from poor quality.

Worked out Example 2
Expanding on the first paint example, lets decide what the manufacturing tolerance should be. The manufacturing tolerance is the economic break-even point for reworking scrap. Suppose the off-target paint can be adjusted at the end of the line for $1 a gallon. At what pigment level, should the manufacturer spend the $1 to adjust the paint?

The manufacturing tolerance is determined by setting L = $1.

$$ $1 = $0.10 (y - 200)^2\qquad$$ dollars/ gallon of paint

$$y = 200 \pm \sqrt{ \frac{1.00}{0.10} } =  200 \pm 3.16 $$

As long as the paint is within $$200g \pm 3.2 g $$ of pigment, the factory should not spend $1 to adjust the pigment at the end of the line, because the loss without the rework will be less than $1. The manufacturing tolerance represents a break-even point between the manufacturer and the consumer and sets limits for shipping the product. If the paint manufacturer ships the product from example 1 with 185g of pigment, they are saving 1$ of reworking costs but imparting a cost of $22.50 on society. These additional costs will surface through loss of customer satisfaction and thus brand reputation, loss of marketshare and returned products.

Multiple Choice Question 1
How do Taguchi's methods differ from traditional ways of calculating losses due to poor quality?

A) They average losses over a 12 month period of time

B) They include not only losses to the manufacturer up to the point of shipping, but also losses to society

C) They put the losses in Yen instead of Dollars

D) They calculate the cost per product

Multiple Choice Question 2
How do you select a tolerance range?

A) Ask your consumers which products they are disatisfied with

B) Find the break even point for fixing the product and the cost imparted to society from not fixing the product

C) Make the tolerance as small as your equipment will allow

D) Use the standard safety factor of 4

Submitting answers to the multiple choice questions
The deadline for submitting your answers is the start of class on Tuesday, 11/27. You are expected to work on these multiple choice questions under the Honor Code. Please use the following link to submit your answers to the above multiple choice questions: https://lessons.ummu.umich.edu/2k/che_466/TaguchiLoss