MtT Type Risk 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. 

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It is worthwhile to note a subtle difference between the definitions of impact and occurrence in VaR and MtT type risk functions. In the case of VaR type risk function, impact is a probability distribution of loss due to a risk event and occurrence is the probability distribution of the number of risk events during a period. For the MtT type risk function, impact represents the loss due to deviation from a performance target value and occurrence is the distribution of that performance measure. 

The parameters of the impact function fit into case 1 presented in Equation 7.12. Thus the MtT type risk function is given by ... 

These risk measures can be used for both VaR type and MtT type risks. Values of R(a), PLe(r), and PLa r) can be read from the risk table for VaR type risks or calculated using the risk fimction for MtT type risks. When the risk table does not have the desired value, an interpolation method can be used to estimate the real value. Numerical search methods can be used to find out the corresponding a value for a certain risk value from the MtT type risk functions. In order to get ER, extra calculations need to be done based on the VaR type risk table or the MtT type risk function. 

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... 

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. 

So far in our discussion of VaR and MtT type risks, we have used the risk measure as the maximum loss at confidence level a, denoted by R a). We can also derive other risk measures to support managerial decision making in risk management and mitigation. Given next are examples of several risk measures that can be obtained directly as a by-product of the VaR and MtT type disruption risk functions developed earlier. 

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. 

The first objective function (7.23) minimizes the total purchasing cost. Purchasing costs has two components, total variable cost (TVC) a.nd fixed cost (FC). TVC constitutes the first part and PC is the second part of Equation 7.23. Note that FC depends on whether a supplier is selected or not therefore, we use the binary variable Zj. in the expression of FC. The second objective (7.24) represents MtT risk for quality resulting from past transactions with the suppliers. The third objective refers to the minimization of average lead time (725). Note that Dy is a constant and can be removed during the optimization process. The fourth objective in (7.26) minimizes VaR risk of supply disruptions due to hurricanes. We model the quality objective as an S-type MtT risk (see Figure 76). The calculations of exact mathematical expressions for VaR and MtT type risks for the different suppliers are given in Section 714.5. ... 

Suppliers are the source of several MtT type operational risks e.g., late delivery, low service rate, high defective rate, etc. We assume buyers suffer mainly from defective parts hence, we focus on the defective rate (i.e., quality) issue. Naturally, buyers ask for the lowest defective rate possible. For MtT type risks of smaller-the-better type, we proposed using an S-type impact function (see Figure 7.6). We present the general form of the S-type impact function in Equation 7.51... 

Consider the delivery time problems faced by the computer manufacturer XYZ from its supplier ABC in Example 7.2. This was modeled as an MtT-N type risk. Using the disruption risk function given by Equation 7.14, compute the following risk measures for missing delivery time window ... 

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