Taguchi Loss Function
Created by the late Dr. Genichi Taguchi, the Taguchi loss function is a response to the adherence to specification (spec) concept of quality. Adherence to spec is a binary model of loss governed by a square loss function (see Figure 1). Here, anything inside of the specification limits is good and anything outside of them is bad. While this model of loss was a step forward when it was first conceptualized, its flaws make it a liability today. The fundamental truth of manufacturing is that no two parts, assemblies, or products produced by the same process will ever be the same. Even when parts are in spec, differences will exist because variation always asserts an influence.
Figure 1. Square loss function governing adherence to specification concept of quality.
Recognizing the limitations of adherence to spec, Dr. Taguchi developed the Taguchi loss function. As shown in Figure 2, loss under the Taguchi loss function increases quadratically from the target with product of the highest quality being produced at the parabolic vertex (target). As a quality or performance characteristic deviates from the target, loss increases until maximum loss is realized at the specification limits. In instances where rework can be performed, a portion of the loss can be recovered. In instances where parts must be scrapped, the full loss due to poor quality is realized.
Figure 2. Quadratic loss function governing the continual improvement concept of quality.
The Taguchi loss function is remarkable for how it reflects the realities of manufacturing. It recognizes that variation, in its ubiquity, influences process outcomes regardless of the specification limits. This promotes a new way of thinking about quality and new goals. Rather than chase “good enough” with adherence to spec, it promotes efforts to produce product that is virtually the same. This inevitably led Dr. Taguchi to his definition of world class quality:
On-target with minimum variance.
Operating a process on-target requires process knowledge that enables the process mean to be set as close to the target as possible. Operating a process with minimum variance requires it to be operated predictably (i.e., in a state of statistical control). The only tool capable of helping achieve both of these tasks is the process behavior charts.
An economic model of loss
The Taguchi loss function is not only a conceptually useful model of loss, but also an economic one. That is, it can be used to calculate the financial costs of producing parts off-target with variance. Doing this requires an understanding of the formula that governs the function and its constituent parts.
For an individual quality or performance characteristic (i.e., one measurement from one part), the Taguchi loss function is written as:
Here, L(x) is the loss function, K is a numeric constant expressed in dollars per unit squared, x is the measured quality or performance characteristic, and T is the target value.
To calculate the value of K, the cost of scrapping a part must be associated with the deviation from the target. This is achieved by connecting the deviation from the target at the specification limits with the cost of scrapping a part, C_scrap, at the spec limits. If we let x_scrap represent the measured value of a quality or performance characteristic at the specification limits, the cost of scrap can be written as:
Here, C_scrap is a function of K and the squared deviation of x_scrap from the target. Rewriting this formula in terms of K yields:
Thus, the numeric constant K is a function of the cost of scrapping a part and the squared deviation of x_scrap from the target.
Calculating loss for one off-target measurement
A manufacturing process produces parts with a diameter of 30 units ± 3 units. This makes the target 30 units, the Upper Specification Limit (USL) 33 units, the Lower Specification Limit (LSL) 27 units, and the tolerance 6 units. The cost of scrap for each part, C_scrap, is $5. Using the Taguchi loss function, what is the loss due to poor quality when the measured diameter of a part is 29 units?
To calculate the loss associated with a part diameter of 29 units, substitute and solve the Taguchi loss function using the requisite values for K, x, and T. In this example, x = 29 and T = 30 but K is unknown. Thus, before the loss associated with the part diameter of 29 units can be calculated, the value of K must be determined.
K is a function of the cost of scrap, C_scrap, and the squared deviation of x_scrap from the target. Here, C_scrap = $5, x_scrap = 33, and T = 30. Substituting these values into the formula for K yields a value $0.56.
With the value of K in hand, the loss associated with a part diameter of 29 units is calculated as follows:
Thus, the loss associated with a part diameter of 29 units is approximately $0.56.
Figure 3. Quadratic loss function with loss associated with a part diameter of 29 units.
Calculating loss for a distribution of measurements
Although L(x) can be used to estimate the loss in dollars for a particular value of x, it does not take into account the fact that no manufacturing process produces a single part. The purpose and point of manufacturing is to produce multiples of the same part that are, for all intents and purposes, the same. To determine the likelihood that a particular value for a quality or performance characteristic will occur, the distribution of the values of X must be considered.
Assume the manufacturing process from above produced 100 parts. Each of these parts will have a unique quality characteristic, x, and an associated loss that can be described as L(x). By summing the individual losses for the 100 parts, an average loss per unit of production can be obtained. This is expressed by the equation:
Here, E{L(x)} is the expected average loss per unit, K is again a numeric constant expressed in dollars per unit squared, X is the mean of the distribution, T is the target, and σ is the standard deviation statistic.
For the run of 100 units, K is still $0.56 and the target is still 30 units. The mean of the distribution is 31.2 units and the standard deviation is 0.7 units. Substituting these values into the expected loss formula yields an average expected loss per unit of $1.08. This means that if the process continues to operate with a mean of 31.2 units and standard deviation of 0.7 units, the average loss per unit will be $1.08.
To reduce the average loss per unit, the mean deviation from target and the variation around the mean must be minimized. That is, the process must be operated on-target with minimum variance. While opinions abound on how to achieve this task, the only tool capable of realizing it is the process behavior chart.
Figure 4. Quadratic loss function with a product distribution composed of 100 measurements.
Summary
A contrast to the adherence to specification concept of quality, the Taguchi loss function is a high-fidelity model of loss. It trades the binary world of good and bad for one that is based on a quadratic continuum. In doing so, the Taguchi loss function reflects the realities of manufacturing. It recognizes that, whether we like it or not, the measured quality and performance characteristics of parts, assemblies, and products will never be the same. Variation always asserts its influence.
The model of loss reflected in the Taguchi loss function is not just theoretical but economic. Given a single measurement or a distribution of measurements, an economic cost can be associated with loss due to poor quality. This connects the world and work of engineers to the world and work of business. It assigns a financial cost to inaction and a financial benefit to operating on-target with minimum variance. It promotes a new way of thinking about processes and systems and encourages efforts to continually improve instead of settling for “good enough.”