Taguchi’s loss function

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... 

Figure 4.14 Taguchi s loss function to show the trade-off between bias (underfitting) and variance (overfitting) see Ref. [30] for more mathematical details. Here, k would be the optimum dimensionality of the PLS model. 

Quality Assurance, Fig. 2 Taguchi s loss function (Taguchi 1986)... 

Taguchi s loss function can be used to evaluate design decisions on a financial basis to establish whether additional costs in the production process would be worthwhile in the market place. 

Table 7.4 illustrates the differences between VaR and MtT type risks. VaR risks are rare events that can disrupt severely supply chain operations. MtT risks are more common with less severe impact. Yang (2006) used extreme value theory (EVT) to model the impact of VaR type risk event and Taguchi s Loss Functions to model the impact of MtT type risk. Supply disruptions due... 

Plots of Taguchi s loss functions are given in Figure 7.6. 

Taguchi s loss functions represent the impact of the risk. Since the risk is a function of both impact and occurrence, we need the occurrence function of the risk event as well. MtT type occurrence function is actually the distribution of the performance measure from historical data and it can be used as the probability function to predict risk in the future. Firms can use past data to fit an appropriate occurrence function, or use some widely adopted distributions such as Ganuna distribution for S-type occurrence function. Beta distribution for L-type occurrence function, and Generalized Hyperbolic distribution for N-type occurrence function. 

Mathematical expressions for S-type, N-type, and L-type risks can be derived (for a desired a level of confidence) by aggregating the impact function of Section 7.10.1 and the occurrence function of Section 7.10.2. Since Taguchi s loss functions are relatively simple, the aggregation can be done analytically. 

Taguchi proposed that a measure of the loss to society, and to the customer, be a quadratic (square law) function of the difference between the target value and the actual value of the parameter of interest. This has become known as Taguchi s loss function. The goal posts idea just does not address this loss to society it does not penalize a product with a parameter that is not the target value. The loss function has been used as a measure of monetary loss associated with less than optimal performance as well as higher maintenance and repair costs. 

This work can be extended in several directions. The qualitative assessment scores for hazards and vulnerability can be improved by using the more elaborate quantitative models of risk developed by Bilsel and Ravindran (2012) for major disruptive events. For rare events, such as earthquakes and floods, Bilsel and Ravindran have used extreme value distributions to determine the financial impacts of disruptions. For other events, such as transportation failures, they use Taguchi s loss functions. Efforts can be taken to extend the risk assessment to consider multiple decision makers. Fuzzy logic can also be used to handle ambiguity in the scores. 

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. 

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 . 

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 ... 

The quality is accepted by the customer when it is maintained inside a level of tolerance, between an upper and a lower value of the service (Fig. 3). In DNSP, the consiuner tolerance is defined by standards in the sector. So the cost of any deviation, in terms of quality, increases in a quadratic form from the target value as centre of the curve, see Fig. 3. In our case, focus only on intangible costs, the re-establishment time has been considered to reflect the maintenance impact in quality service. The Taguchi s Function Loss (TFL) is ... 

TFLS = Taguchi s Function Loss of a series of samples ... 

Figure 4. Taguchi s Function Loss per customer depending on the reestablishment time.

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 ... 

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)) ... 

Further information on quality loss and signal-to-noise ratios can be found in texts written in (Phadke (1989), Ross (1988), Roy (1990), Suh (1990). Taguchi (1986)). They all provide detailed discussions on how to apply statistical methods and Taguchi s approach in the selection of design parameters for satisfying functional requirements. 

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