Grey Relation

The grey relation coefficient, y xo(k),x,ik) k= 1,2, 3), is calculated using Equation (7.3) for each of the failure events identiEed in the FMEA. In the example used in Table 7.5, the grey relation coefficient can be calculated as shown here, assuming that = 0.5  

Similarly, the grey relations for the other two linguistic variables (Severity (y,) and Detectability (y )), can be calculated as follows  

The grey relation coefficients for events B and C are calculated in the same way. The results of these calculations are summarised as seen in Table 7.6. 

The degree of grey relation is calculated using Equation (7.5) for each failure event incorporating the weighted variables. For example, assume that the values of fif, fi, and fid are 0.4, 0.4 and 0.2, respectively, the degree of grey relation in the example shown in Table 7.5 can be calculated as seen here  

Using Equation (7.5), the grey relation for event A can be calculated as  

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Tosun, N. (2006) Determination of optimum parameters for multi-performance characteristics in drilling by using grey relational analysis. International Journal of Advanced Manufacturing Technology, 28 450-5. 

The marine industry is recognising the need for powerful techniques that can be used to perform risk analysis of marine systems. One technique that has been applied in both national and international marine regulations and operations is Failure Mode and Effects Analysis (FMEA). This risk analysis tool assumes that a failure mode occurs in a system/component through some failure mechanism. The effect of this failure is then evaluated. A risk ranking is produced in order to prioritise the attention for each of the failure modes identified. The traditional method utilises the Risk Priority Number (RPN) ranking system. This method determines the RPN by finding the multiplication of factor scores. The three factors considered are probability of failure, severity and detectability. Traditional FMEA has been criticised to have several weaknesses. These weaknesses are addressed in this Chapter. A new approach, which utilises the fuzzy rules base and grey relation theory, is presented. 

The standard series is an objective series that reflects the ideal or desired level of all the decision factors and can be expressed asxo = Ml) X(t(2)... xe(K)]. This could be assumed to be the lowest level of the linguistic terms describing the decision factors. The difference between the two series (comparative and standard series) is calculated. The grey relation coefficient is obtained using Equation (7.3) ... 

At this stage, the standard series for the variables is generated by determining the optimal level of all three variables for the events in the FMEA. The difference between the standard and comparative series is obtained and the results are used to determine the grey relation coefhcient. 

The identified failure events in the FMEA are ranked according to the ascending order of the degree of relation. This entails that the failure mode with the smallest degree of grey relation gets the highest priority for attention. For the example in Table 7.7, failure event B would be at the top of the list for priority for attention, this will be followed by events C and A. The suimnaiy of the results for this example is shown in Table 7.8. 

Consider the first event where Sf, S and Sd are assigned Remote, High and High for the probability of occurrence, severity and detectability, respectively. The grey relation coefficients y/, y and yd are calculated as shown below ... 

Table 7.8 Example of Ranking for Failure Events Using the Degree of Grey Relation...

Failure events Probability of occurrence Severity Detectability Degree of grey relation Ranking (priority for attention)... 

Description Component Failure Mode s, s "U Se Yd Grey Relation... 

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