becomes a meaningful and standardized unit of probability such as errors/year, if estimating warranty. Depending on the size and com-
plexity of an FTA, the final spread- sheet may be the result of multiple revisions and editing. Te initial effort is expected to have a very high probability for the “top event”; that is the result of estimations made in numerous places, all of them likely conservatively overestimating the chance of error. As improvement efforts are made, the change from the previous probability to the new one quantifies the change in risk, and this movement is a very real measure of the improvement in the frequency of the “top event.”
Using an FTA With Warranty
Te primary way to utilize the FTA to make improvement in an organized and deliberate manner is to perform a contribution analysis. Tis becomes the check part of a closed PDCA (Plan-Do-Check-Act) loop to drive a reduction in the probability of the “top event,” the unwanted warranty defect. Essentially, this is a mathematical
procedure for calculating the percent contribution of intermediate and basic events to the “top event” (see Fig. 5). Simple arithmetic reveals the rela-
tive contributions to the “top event” by the three main causal branches, as given in Table 1. Te probabilities of the events A, B and C simply add to the “top event” so these contributions are simply the ratio (e.g., PA/PTOP) of the two probabilities. The basic events D and E in turn
contribute to Event A. Their contri- bution can be evaluated as shown in Table 2. This contribution analysis is performed exactly as the one in Table 1. The contribution to the “top event” can be calculated using the same technique. With “AND” relationships, the method of calculation of contribution must be modified somewhat. Table 3 shows the contribution to event B. Here, the ratio of event F to event B does not provide a correct net contri- bution to event B, since the events F and G probabilities are multiplied to yield event B’s probability. Instead, the
38 | MODERN CASTING September 2014
• Adding redundancy: If two systems must fail for a fault to occur, the probability of their occurring simul- taneously is much smaller than any one check system.
• Evaluating the factors that lead to basic events to further break down (and expose to correction) underly- ing causes.
• Evaluating and improving gage error through MSA (Measurement System Analysis) methods.
Fig. 7. The diagram illustrates a PDCA loop as applied to the use of FTA.
relative contribution of event F can be given by this equation: 100[PF/(PF+PG)] = % PF
Contribution Te net impact of event PF to the
“top event” is then: 100[(%PF)(PB)/PTOP] = % PF
Contribution Te remainder of Figure 5 can be
evaluated for contribution using meth- ods already described and as shown in this figure. In Tables 2, 3 and 4 the basic
events have been shaded to distinguish them from the intermediate events. By reviewing the contributions of these basic events to the “top event,” a Pareto analysis can be performed. Tis provides insight into the multiplicity of contributors where action should be taken first (and what it might yield in terms of improvement in the “top event”). Figure 6 illustrates the Pareto of contribution for the FTA illustra- tion in Figure 5.
The Improvement Process Using contribution analysis of the
FTA provides a prioritized direc- tion for improvement and a useful metric for quantifying effective- ness. This activity is intended to be cyclical and continual in a closed PDCA loop, as shown in Figure 7. Improvements may come from sev- eral perspectives, but the following may prove most useful in a warranty evaluation where error probabilities are relatively low. • Error proofing: Human error rates will tend to be the highest probabil- ity events on the FTA.
• Refining probability evaluations via specific capability studies or mainte- nance record mining.
Reporting and Communication
Interaction with the warranty engineer or quality engineer and the customer regarding this type of analysis may require some education regarding the meaning of the results and the approach used. Te following is recommended: • The cross functional team involved in the analysis should be identified.
• The “top event” should be clearly communicated and decisions about its scope should be explained.
• The overall structure and logic of the FTA should be described, branch by branch, so it makes sense to the customer.
• The meaning of the probability numbers should be explained, including the units chosen.
• The translation from probability of occurrence to actual expected parts in the field should be explicit (and con- stant over the revisions of the FTA).
• The provisional and conservative nature of the “top event” prob- ability should be described. The probability is more useful as a measurement of improvement than as an absolute prediction of rate of occurrence of a defect.
• A comparison of actual warranty to predicted warranty within the FTA should be made, especially over a period of years.
• Where suppliers have impact on the results, they should be involved in reviewing or improving on the FTA logic and estimated probabilities.
• Reports of FTA results, especially at the initial use, should be made in person where the appropriate
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