This article discusses the importance of measuring hidden quality costs that are associated with a product. Quality costs are defined as “costs incurred for ensuring conformance to quality standards or compensating for nonconformance to quality standards” (page 8). Conventional accounting systems are not able to measure intangible quality costs, so these costs become hidden quality costs. Some examples of hidden quality costs are customer dissatisfaction with a product or defects in a product that causes a loss in sales. These hidden quality costs can be the major factor in the total cost of quality for a product. 

Taguchi’s Quality Loss Function

To estimate these hidden quality costs, a Taguchi quality loss function (QLF) has been proposed. Taguchi’s approach is different than the traditional approach of quality costs. In the traditional approach, if you have two products and one is within the specified limits and the other is just outside of the specified limits, the difference is small. Although the difference is small the product within the limits is considered a good product while the outside one is considered a bad product.  Taguchi disagrees with this approach. Taguchi believes that when a product moves from its target value, that move causes a loss no matter if the move falls inside or outside the specified limits. For this reason, Taguchi developed the QLF to measure the loss associated with hidden quality costs.  This loss happens when a variation causes the product to move away from its target value.

            L(y) =k(y-T)²   

Where:
 k = a proportionality constant dependent upon the organization’s failure cost structure,
 y = actual value of quality characteristic,
 T = target value of quality characteristic.

The value of k must first be determined before the loss can be estimated.

To determine the value of k:

                        k= c/d²

            Where:
            c = loss associated with the specification limit, and
              d = deviation of the specification from the target value. 

The value of k determines the slope of the QLF, the larger the value of k the steeper the parabola.  This is a symmetric QLF because it is assumed that there is a constant k for the whole loss function.  The value of c is a major component in the loss function. This value represents the intangible quality costs of a product. 

Asymmetric Quality Loss Function

The term asymmetric implies that variations can have different sensitivities to loss. If a variation happens on one side of the loss function, that loss may be more or less sensitive than if the same amount of variation happened on the other side of the target value. This involves having to add to the previous formula, there will now be two k’s.  k will now represent the different sensitivities that happen when a variation moves on either side of the target value. This loss function now becomes an asymmetric QLF because there can be different values for k. 

            The unit loss function becomes:

                        L(y) = k₁[(y-T)⁺]²  + k₂[(T-y)⁺]²

            Where: k₁ > or < k₂
                             x⁺ = Max (x,0)

Again k must be determined before the loss can be estimated. 

                        k₁ = c₁ / (U-T)²
                              k₂ = c₂ / (U-T)²

Where: U = upper specification limit of characteristic,
                          L = lower specification limit of characteristic,
                         c₁ = loss associated with U, and
                         c₂ = loss associated with L.

Example Using the Asymmetric Quality Loss Function

Given these values:

First the value of k must be determined.

                        k₁=   80 / (10.4-10)² =$500
                        k₂=   48 / (10-9.6)² =$300

Now the estimation of loss can be determined. Assume that a variation of .2mm happened on both sides of the target value. The product with an actual value of 10.2mm, moved to the right of the target value

                        L(y =10.2) = $500(10.2 -10)² + $300(0)²
                                         = $20

            Or actual value of 9.8mm, moved to the left of the target value

                        L(y =9.8) = $500(0)² + $300(10-9.8)²
                                                 = $12

Since c₁ > c₂, then k₁ > k_2.  This implies that the right side of the loss function is more sensitive than the left side. In the example above the product moved .2mm in both directions and the move to the right causes the greater loss.  

Insensitive Region of Quality Loss Function

This deals with different levels of loss sensitivity on either side of the target value. Insensitive means that there could be different sections in the loss function that are more or less sensitive than other sections. These sensitivity differences cause larger losses in some sections in relation to  other sections of the loss function. For these situations, a separate L(y) formula is needed for each different section in the loss function. 

Examples

Summary

	Web maaw.info