“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 relative contribution of event F can be given by this equation: 100[PF/(PF+PG)] = % PF
Contribution Te net impact of event PF to
Fig. 6. The Pareto of contributions made by basic events in the Figure 5 FTA illustration is shown.
Group members will require train- ing in the overall strategy of FTA prior to initiating the work. Later, when the probabilities need to be determined, a second training session can cover the math associated with turning process capability numbers into probabilities or estimates. “We had three failures over the last 5 years of maintenance records,” 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 prob- ability for the “top event”; that is the result of estimations made in numer- ous places, all of them likely conserva- tively overestimating the chance of er- ror. As improvement efforts are made, the change from the previous prob- ability 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
Event H Event K
Event K Probability:
Calculated Probability
Event J Event M Event N
0.000007 0.00003 0.000003
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
Table 3. Contribution Analysis of Event B Event B Probability:
Calculated Probability
Event F Event G
0.015 0.0002
Table 4. Contribution Analysis of Event C Event C Probability:
Calculated Probability
0.000008 0.00004
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.
0.000003
Percent Contribution to Event B
98.7% 1.3%
0.000048
Percent Contribution to Event C
16.7% 83.3%
0.00004
Percent Contribution to Event C
17.5% 75.0% 7.5%
Percent Contribution to the Top Event
8.5%
36.6% 3.7%
Jul/Aug 2014 | METAL CASTING DESIGN & PURCHASING | 45
Percent Contribution to the Top Event
9.8% 48.8%
Percent Contribution to the Top Event
3.6% 0.05%
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