The War on Error – Vol. IX: Taguchi Loss Function

     Choosing effective strategies for waging war against error in manufacturing and service operations requires an understanding of “the enemy.”  The types of error to be combatted, the sources of these errors, and the amount of error that will be tolerated are important components of a functional definition (see Vol. I for an introduction).
     The traditional view is that the amount of error to be accepted is defined by the specification limits of each characteristic of interest.  Exceeding the specified tolerance of any characteristic immediately transforms the process output from “good” to “bad.”  This is a very restrictive and misleading point of view.  Much greater insight is provided regarding product performance and customer satisfaction by loss functions.

     Named for its developer, the renowned Genichi Taguchi, the Taguchi Loss Function treats quality as a variable output.  In contrast to the “goal post” philosophy described above, which uses a digital step function, the Taguchi Loss Function quantifies quality on an analog scale.
     According to goal-post philosophy, characteristic values that fall inside the tolerance band by the slightest amount are acceptable (no quality loss) and those that fall outside the tolerance band by the slightest amount are unacceptable (total loss).  This is shown conceptually in Exhibit 1.  Recognizing that the difference in quality between these two conditions was quite small, Taguchi realized that quality at any point in the characteristic range could be expressed relative to its target value.  Furthermore, he concluded that any deviation from a characteristic’s target value represents a corresponding reduction in quality with a commensurate “loss to society.”  The Taguchi Loss Function is shown conceptually in Exhibit 2.

Sources of Loss
     Many struggle to understand Taguchi’s characterization of variable quality as loss to society.  It is, however, a clear connection once given proper consideration.  Resources consumed to compensate for reduced quality are unavailable for more productive use.  Whether the effects on consumers and providers are direct or indirect, each incurs a loss.
     Sources of loss are varied and may not be immediately obvious, in large part due to the paradigm shift required to transition from goal post thinking.  The most salient loss to providers comes in the form of scrap and rework.  Material, labor, and time losses are easily identifiable, even by goal-post thinkers.  Somewhat more difficult to capture fully, warranty costs are also salient to most providers.  Lost productive capacity, reverse logistics, and product replacement costs can be substantial.  Worse, a damaged reputation and subsequent loss of good will in the marketplace can result in lost sales.  Some existing customers may remain loyal, but attracting new customers becomes ever more difficult.  The difficulty calculating this loss accurately further compounds the problem; misattribution of the cause of slumping sales will make a turnaround nearly impossible.
     Consumers experience losses due to reduced product or service performance.  If the time required to receive service or to use a product increases, this is a direct cost to the consumer.  Reduced performance lowers the value of the product or service and may prompt consumers to seek alternatives.
     A product that requires more frequent maintenance or repair than anticipated incurs losses of time, money, and productivity for consumers.  This could, in turn, lead to warranty costs, reputational damage, and lost sales for the producer.
     The examples cited are common losses and are somewhat generic for broad applicability.  Careful review of any product or service offered is warranted to discover other sources of loss.  Safety and ergonomic issues that could lead to injury and liability claims should be front-of-mind.  Environmental impacts and other regulatory issues must also be considered.  There may be other sources of loss that are unique to a product type or industry; thorough consideration of special characteristics is vital.
 
Nominal is Best
     The most common formulation of the Taguchi Loss Function is the “nominal is best” type (see Exhibit 2).  Typical assumptions of this formulation include:

• Target (“best”) value is at center of tolerance range.
• Loss = 0 at target value.
• Characteristic values are normally distributed.
• Out-of-spec conditions are reliably captured.
• Failures are due to compound effects (i.e. each characteristic is within its specified range, but the combination is unsatisfactory).

     For product manufacturing, the scrap value may be used to calculate the loss function.  If out-of-tolerance conditions can escape, or other conditions exist that would cause greater losses, the calculation can be modified accordingly.  Also, service providers can define the maximum loss as is most appropriate for them, such as repair cost, loss of future sales, etc.  Whatever definition is chosen, it should be applied consistently to allow for comparisons across time, products, or service offerings.  Using the scrap value of a product provides consistency that may be difficult to achieve with other definitions of maximum loss.
     The nominal is best Taguchi Loss Function is a quadratic function of the form:
            L(x) = k (x – t)^2, where:

• L(x) is the loss experienced at characteristic value x;
• k is the loss coefficient, a proportionality constant;
• x is the measured (actual) characteristic value;
• t is the target characteristic value.

     To calculate k, using the assumptions stated above, set x equal to the maximum allowable value of the bilateral tolerance (USL) and L(x) equal to the maximum loss (i.e. scrap cost).  Solve for k by rearranging the loss function expression:  k = L(x)/(x – t)^2.
     With a known k value, the loss incurred by the deviation from the target characteristic value can be calculated for each occurrence.  Total and average loss/part can then be calculated and used to analyze process performance.
     For a visual comparison of the goal post model and the loss function, use the interactive model from GeoGebra.
 
Other Loss Function Formulations
     There are two special cases that warrant particular attention:  Smaller is Better and Bigger is Better.  When lower characteristic values are desirable – i.e. zero is ideal – the loss function can be simplified by setting t = 0.  The resulting smaller is better formulation is L(x) = kx^2, where k is calculated using the value of x at which a product would be scrapped or a process ceased.  Example characteristics that utilize this formulation include noise levels, pollutant emissions, and response time of a system.  The smaller is better loss function is shown conceptually in Exhibit 3.

     When larger characteristic values provide greater performance or customer value, the bigger is better formulation of the loss function is used, in which the inverse of the characteristic value is used to calculate losses.  It takes the form L(x) = k (1/x)^2.  At x = 0, the loss would be infinite – an unrealistic result.  More likely, there is a minimum anticipated value of the characteristic; this value should be used to calculate k.  This value also defines the maximum expected loss per unit.  The bigger is better loss function is shown conceptually in Exhibit 4.

     This formulation also suggests that zero loss cannot be achieved.  Doing so requires the characteristic to reach an infinite value – another unrealistic result.  In practical terms, there may be a value beyond which the relative benefits are imperceptible, resulting in an effectively zero loss.  This idiosyncrasy does not diminish its comparative value.
 
Relationship to Other Programs
     Six Sigma initiatives attempt to reduce variation and center process means in their distributions.  Though the objectives are often defined in goal post terminology, Six Sigma is highly compatible with the paradigm of the loss function.  Both pursue consistent output, though they value that output differently.
     The Taguchi Loss Function fosters a continuous improvement mindset.  Until all losses are zero, there are potential improvements to be made.  The loss function formulations presented provide a method to determine the cost-effectiveness of proposed improvement projects.  First, a baseline is established; then anticipated gains (reduced losses) can be calculated.  If the anticipated gains exceed planned expenditures, the project is justified.
     Project objectives may even be defined by a loss function.  That is, a target loss value may be defined; then solutions are sought to achieve the target level.  This application is not in common use; the difficulty in achieving the necessary paradigm shift ensures it.
 
Summary
     A mindset adjustment is required to transition from traditional (goal post) quality evaluations to Taguchi Loss Functions.  If this is achieved, they can be used to effect by following a simple procedure:

• Define the reference loss (e.g. scrap cost).
• Define the value of the characteristic at which this loss is incurred.
• Calculate the loss coefficient, k.
• Calculate the loss incurred by each deviation from target values.

     At this point, the process branches according to the objectives sought.  Totals and averages can be calculated, comparisons made, or other analyses conducted.  The loss function provides a useful reference and encourages an expanded view of perceived quality.
 
     For assistance introducing loss functions to your organization or pursuing other operational improvement efforts, contact JayWink Solutions for a consultation.

     For a directory of “The War on Error” volumes on “The Third Degree,” see “Vol. I:  Welcome to the Army.”
 
References
[Link] “Taguchi loss function.”  Wikipedia.
[Link] “Taguchi Loss Function.”  WhatIsSixSigma.net
[Link] “Taguchi Loss Function.”  Lean Six Sigma Definition.
[Link] “Taguchi loss function.”  Six Sigma Ninja, November, 11, 2019.
[Link] “The Taguchi loss function.”  Thomas Lofthouse; Work Study, November 1, 1999.
[Link] “Taguchi’s Loss Function.”  Elsmar.com.
[Link] “Principles of Robust Design.”  Dr. Nicolo Belavendram; International Conference on Industrial Engineering and Operations Management, July 2012.
[Link] “Robust Design Seminar Report.”  Shyam Mohan; November 2002.
 
Jody W. Phelps, MSc, PMP®, MBA
Principal Consultant
JayWink Solutions, LLC
jody@jaywink.com