Taguchi's Loss Function

Definition
Simply put, the Taguchi loss function is a way to show how each non-perfect part produced, results in a loss for the company. Deming states that it shows
"a minimal loss at the nominal value, and an ever-increasing loss with departure either way from the nominal value." - W. Edwards Deming Out of the Crisis. p.141
A technical definition is
A parabolic representation that estimates the quality loss, expressed monetarily, that results when quality characteristics deviate from the target values. The cost of this deviation increases quadratically as the characteristic moves farther from the target value. - Ducan, William Total Quality Key Terms. p. 171
Graphically, the loss function is represented as shown above.
Interpreting the chart
This standard representation of the loss function demonstrates a few of the key attributes of loss. For example, the target value and the bottom of the parabolic function intersect, implying that as parts are produced at the nominal value, little or no loss occurs. Also, the curve flattens as it approaches and departs from the target value. (This shows that as products approach the nominal value, the loss incurred is less than when it departs from the target.) Any departure from the nominal value results in a loss!
Loss can be measured per part. Measuring loss encourages a focus on achieving less variation. As we understand how even a little variation from the nominal results in a loss, the tendency would be to try and keep product and process as close to the nominal value as possible. This is what is so beneficial about the Taguchi loss. It always keeps our focus on the need to continually improve.
A business that misuses what it has will continue to misuse what it can get. The point is--cure the misuse. - Ford and Crowther
Application
A company that manufactures parts that require a large amount of machining grew tired of the high costs of tooling. To avoid premature replacement of these expensive tools, the manager suggested that operators set the machine to run at the high-end of the specification limits. As the tool would wear down, the products would end up measuring on the low-end of the specification limits. So, the machine would start by producing parts on the high-end and after a period of time, the machine would produce parts that fell just inside of the specs.
The variation of parts produced on this machine was much greater than it should be, since the strategy was to use the entire spec width allowed rather than produce the highest quality part possible. Products may fall within spec, but will not produce close to the nominal. Several of these "good parts" may not assemble well, may require recall, or may come back under warranty. The Taguchi loss would be very high.
We should consider these vital questions:
* Is the savings of tool life worth the cost of poor products?
* Would it be better to replace the tool twice as often, reduce variation, or look at incoming part quality?
Calculations
Formulas:
Loss at a point: L(x) = k*(x-t)^2where,k = loss coefficientx = measured valuet = target value
Average Loss of a sample set: L = k*(s^2 + (pm - t)^2) where,s = standard deviation of sample pm = process mean
Total Loss = Avg. Loss * number of samples
For example: A medical company produces a part that has a hole measuring 0.5" + 0.050". The tooling used to make the hole is worn and needs replacing, but management doesn't feel it necessary since it still makes "good parts". All parts pass QC, but several parts have been rejected by assembly. Failure costs per part is $0.45. Using the loss function, explain why it may be to the benefit of the company and customer to replace or sharpen the tool more frequently. Use the data below:
measured value0.459 0.478 0.495 0.501 0.511 0.5270.462 0.483 0.495 0.501 0.516 0.5320.467 0.489 0.495 0.502 0.521 0.532 0.474 0.491 0.498 0.505 0.524 0.5330.476 0.492 0.500 0.509 0.527 0.536
Solution:
The average of the points is 0.501 and the standard deviation is about 0.022.
find k,using L(x) = k * (x-t)^2 $0.45 = k * (0.505 - 0.500)^2 k = 18000
next,
using the Average loss equation: L=k * (s^2 + (pm - t)^2)
L = 18000 * (.022^2 + (.501 - .500)^2) = 8.73
So the average loss per part in this set is $8.73.
For the loss of the total 30 parts produced,
= L * number of samples= $8.73 * 30 = $261.90
From the calculations above, one can determine that at 0.500", no loss is experienced. At a measured value of 0.501", the loss is $0.018, and with a value of 0.536", the loss would be as much as $23.00.
Even though all measurements were within specification limits and the average hole size was 0.501", the Taguchi loss shows that the company lost about $261.90 per 30 parts being made. If the batch size was increased to 1000 parts, then the loss would be $8730 per batch. Due to variation being caused by the old tooling, the department is losing a significant amount of money.
From the chart, we can see that deviation from the nominal, could cost as much as $0.30 per part. In addition we would want to investigate whether this kind of deviation would compromise the integrity of the final product after assembly to the point of product failure.