N-Type Risk Function

Since N-type impact exists on both sides of the target value, its mathematical form is more complicated than S-type and L-type. There are 11 normal cases and 13 special cases that need to be considered for the final closed-form 

The final forms of the risk functions for the remaining 23 cases are given in Yang (2006). 

Case 2 Cases Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10 Case 11 

Case 12 Case 13 Case 14 Case 15 Case 16 Case 17 Case 18 Case 19 Case 20 Case 21 Case 22 Case 23 Case 24 

Computer manufacturer XYZ in the US wants to evaluate the risks from its supplier ABC in Southern China. Inaccurate delivery time is one of the major problems affecting XYZ s supply chain. This represents an N-type risk for XYZ. After analyzing the historical data, XYZ found that ABC s delivery time fits a generalized hyperbolic distribution with parameters X = -0.5, a = 3 = 0, 8 = 1, and p = 0.1. This actually is a special case of generalized hyperbolic distribution called the Cauchy distribution. The PDF and CDF functions of a Cauchy distribution are as follows  

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In Example 7.2, the performance measure was delivery time which followed a Cauchy Distribution, given by Figure 7.8. The impact function was an N-type Taguchi loss function, representing the cost due to deviation from the target window for delivery time, given by Equation 713. The MtT-N type risk function was given by Equation 714. 

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