Quality Loss Function

If, besides the quality-related measure, z, one also wishes to include operating costs, in the analysis, because quality loss functions express quality costs on a monetary basis, commensurate with operating costs, the final global performance metric, y, which reflects total manufacturing cost, is simply the sum of both quality and operating costs (Clausing, 1993),... 

Figure 7.4 Quality loss functions (a) classical and (b) as suggested by Taguchi.

Genichi Taguchi s methods have been widely known in industry for decades. The central idea of his methods is the quahty loss function and robust parameter design [42,43]. The quality loss function is used to estimate costs when the product or process characteristics are shifted from the target value. This is represented by the following equation ... 

Basically, Taguchi works in terms of a quality loss function rather than quality and defined quality loss as the loss imparted by the product to society from the time it is shipped. By taking a target value as the best possible value for the quality characteristic under investigation, he used a simple quadratic loss function, with deviations from the target to show that a decrease in loss was associated with an increase in quality (Figure 19.5). It will be noted that a loss will occur even when the product is within the permitted specification, but this loss is minimal when the product is on target. 

The quality loss is crucial in Taguchi s theory. It is based on the assumption that when a functional characteristic y deviates from the specified target value m, the customer and the society in general experiences an economical loss due to poorer product quality. This economic loss is expressed as the loss function L(y). Based on this, Taguchi defines the quality loss for not being on target by means of the quadratic quality loss function (Phadke (1989), Taguchi (1986)) ... 

This starts with the selection of a correct quality loss function to represent the description of loss attributed in the case. This is a purely mathematical analysis and S/N-ratio for each treatment is calculated according to the selected standard S/N-ratio expressions as described in Section 10.2.4.2. The calculated S/N-ratios are then normalised before proceeding to the next step. 

C. Taguchi Loss Functions as Continuous Quality Cost Models.401... 

This Section addresses cases with a continuous performance metric, y. We identify the corresponding problem statements and results, which are compared with conventional formulations and solutions. Then Taguchi loss functions are introduced as quality cost models that allow one to express a quality-related y on a continuous basis. Next we present the learning methodology used to solve the alternative problem statements and uncover a set of final solutions. The section ends with an application case study. 

Both situations with categorical and continuous, real-valued performance metrics will be considered and analyzed. Since Taguchi loss functions provide quality cost models that allow the different objectives to be expressed on a commensurate basis, for continuous performance variables only minor modifications in the problem definition of the approach presented in Section V are needed. On the other hand, if categorical variables are chosen to characterize the system s multiple performance metrics, important modifications and additional components have to be incorporated into the basic learning methodology described in Section IV. 

Since these loss functions express quality costs on a common and commensurate basis, extending the learning methodology of Section V to a situation with P objectives is straightforward. All one has to do is replace the original definition of the y performance metric [Eq. (23)] by the following more general version ... 

Ross, P.J. (1988), Taguchi Techniques for Quality Engineering Loss Function, Orthogonal Experiments, Parameter and Tolerance Design, McGraw-Hill Book Company, New York, NY. 

The loss function, which is the concrete form of Taguchi s definition of quality The quality of a product is the loss caused by the product to society from the time the product is shipped . 

However, the shape of such a loss function depends on the process or product which is considered and is often difficult to establish. In general the goals of quality improvement (or the shapes of the belonging loss-functions) are often simplified to three types, which are ... 

Figure 4.2 Loss functions for A) nominal value, B) smaller the better and C) larger the better quality characteristics...

The quality characteristic of type nominal the best which uses a specific target value combined with one particular loss function, the quadratic, is probably the most commonly used ... 

If the constant in the quadratic equation is chosen properly then the quadratic loss function can be used for direct calculation of the financial loss induced by an off-target quality characteristic. 

Because Taguchi has made no immediate coupling between the concept of loss-functions (the more philosophical part) and the off-line quality control methods (the more concrete part), the loss functions probably serve best as an important background concept rather than a practical working tool. 

When limiting ourselves to one quality characteristic (y), with preferred value T, a common way to define a measure of quality is the quadratic loss function (as used by Taguchi [13]), which is defined as L(y) = k(y-f) where A is a constant, coupling the loss (y-r) to, for example, an economic quantity. 

The problem is complex (as is any trade-off balance) and it can be approached graphically by the well-known Taguchi s loss function, widely used in quality control, slightly modified to account for Faber s discussions [30]. Thus, Figure 4.14 shows that, in general, the overall error decreases sharply when the first factors are introduced into the model. At this point, the lower the number of factors, the larger is the bias and the lower the explained variance. When more factors are included in the model, more spectral variance is used to relate the spectra to the concentration of the standards. Accordingly, the bias decreases but, at the same time, the variance in the predictions... 

Instead of a single quality-related performance variable, z, as in Section V, let s suppose that one has to consider a total of P distinct objectives and the corresponding continuous performance variables, z, / = 1P, which are components of a performance vector z = [zj,..., zPY- In such case, one has to identify the corresponding Taguchi loss functions, L(z,), i = 1,..., P, for each of the performance variables ... 

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