Quality Loss Function Example
An example of the quality problems that occur in the fabrication of printed circuit boards (PCBs) is the fit of a PCB edge male connector into the product housing female connector or “card cage”. If the variability of the edge connector size is large, the fit is difficult or impossible to achieve, which could result in scrapping the PCB.
Assume that the tolerance for acceptable fit is 土6 mm，the cost of removing a defect in the PCB at the fabrication shop is $100, and the cost of removing a defect at the customer site after the PCB has been assembled is $500. A typical lot of 18 PCBs from the PCB fabricator was measured. The following shows the calculations of the loss function due to the variability of the edge connector and estimation of the savings incurred by either adjusting the average to target or reducing variability of the PCB edge connector.
Assuming actual deviations from the target value of a set of 18 PCBs at fabrication shop: 0, 0, -3, 0, 0, 1，0, -5, -2, -2, 3, -5, -1, 0, -4, 3, 0, 1. Then
There are two ways to improve quality: set the average to target, or reduce variability. It can be readily seen that the second alternative results in the greatest quality cost improvement:
The importance of the loss function is that it gives a monetary value to the state of the output of the process, both in terms of the process average not meeting the specification nominal and the process deviation. In the example outlined above, the average for all 18 measurement was -0.78 mm and the standard deviation was 2.274. Note that in this case theσ, which is 2.274, is different than theσn-1, which is 2.34. The maximum loss function for an assembled PCB that causes customer dissatisfaction is set at $500, and if it does not cause dissatisfaction, there is no loss. Using the formula, the loss due to the process average not being equal to target is calculated to be $8.40, whereas the loss due to variability around the average is $71.84. Taguchi used this technique to compare two Sony television factories in Tokyo and San Diego, CA in 1973.
The quality loss function can also be used to find an optimum level of quality at which the target factory quality can be balanced by the customer dissatisfaction of escaping potential defects. This would imply balancing the product shipping tolerance at $100 per defect removal at the factory versus the advertised specifications (±6mm) with a defect removal of $500 at the customer site. This can be shown mathematically as follows:
The above calculations indicate that the factory should set the tolerance of the manufacturing process at ±2.68 mm with a $100 cost per defect in order to balance the customer tolerance of ±6 mm and $500 cost per defect.
It can be seen that this methodology can provide an alternate approach to six sigma is setting product specifications based on the trade-offs of removing defects at various points in the product life cycle. This analysis is similar to the one performed for testing strategy in Chapter 4. Obviously, the quality loss function methodology is difficult to quantify, especially since the customer defect cost, as expressed in terms of loss to society, is difficult to ascertain.