# The Quality Loss Function (QLF)

The quality loss function was defined by Genishi Taguchi, its major author, as “the financial loss to society imparted by the product due to deviation of the product’s functional characteristic from its desired target value.” It is a negative definition of quality, which totals up the quality loss after the product is shipped. This loss is not widely used by product designers since the data required to calculate it are not readily available in the early part of the design of the product. The loss could be tangible as in-service and warranty costs that companies have to pay to repair the product. There are other costs that cannot be measured quantitatively: loss of market share, customer dissatisfaction, and lost future sales.

Quality loss function is a quadratic expression estimating the cost of a product quality characteristic not meeting its target. This deviation from target can be measured by the average shift from target and by the standard deviation of the quality characteristic. Even when a product leaves the factory within its specifications, it carries with^ the inherent loss due to not exactly meeting its target.

The loss function L indicates a monetary measure for the product characteristic average versus its target value and the distribution the average. Generally, it is expressed in terms of the cost of each failure divided by the square of the deviation from the average at which the failure occurs:

where

L = loss function

y = design characteristic

m = target value or specification nominal

A = cost of repair or replacement of the product

△ = functional limit of the product, where customer dissatisfaction occurs. This could be wider than the product specifications.

Rewriting the formula by using the fact that (y - m)2 is similar to the expression for mean square deviation (MSD) or the variance for the product characteristics:

The loss formula can be translated into familiar statistical terms of actual product characteristic average and the standard deviationσ. Theσ term is based on the n divisor of the standard deviation formula and not n - 1 for the sample deviation: