Summary by Kellee Mundy
Master
of Accountancy Program
University of South Florida, Summer 2003
Deming Main Page | Quality Related Main Page
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.
The QLF is a “U” shaped parabola. The horizontal axis is tangent with the parabola at the target value. This is a quadratic loss function because it assumes that when a product is at its target value (T) the loss is zero.
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.
k= c/d^{2 }
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) = k_{1}[(y-T)^{+}]^{2 } + k_{2}[(T-y)^{+}]^{2 }
Where: k_{1} > or < k_{2 } x^{+ }= Max (x,0)
Again k must be determined before the loss can be estimated.
k_{1} = c_{1 }/ (U-T)^{2 } k_{2} = c_{2 }/ (U-T)^{2 }
Where: U = upper
specification limit of characteristic,
L = lower specification limit of characteristic,
c_{1
}= loss
associated with U, and_{
}
c_{2 }= loss associated
with L.
Example
Using the Asymmetric Quality Loss Function
Given these values:
c_{1 }= $80 | c_{2 }= $48 | U = 10.4mm | L = 9.6mm | T = 10mm |
First the value of k must be determined.
k_{1}=
80 / (10.4-10)^{2 }=$500
k_{2}=
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 c_{1 }> c_{2}, then
k_{1 }> 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
Case | Product, service or characteristic | Effect or explanation |
Symmetric with insensitive regions. | Soft drinks, juice and medicine. | A small deviation from the target value does not create a significant loss, but is not negligible. But too much or too little of an ingredient can cause a large loss. |
Asymmetric with insensitive regions. | Product delivery time and customer service. | Early delivery causes small loss, but late delivery causes larger loss. |
Asymmetric with insensitive regions. | Air pressure in auto tires. | Too little air may cause some loss in performance, but too much air can cause a large loss from a blow out. |
Asymmetric with zero loss in insensitive regions. | Blood pressure and blood cell count. | Within a range of variation from the target there is no loss. Any further deviation from the target value may cause significant loss. |
Summary