and to assess the contribution of the basic and intermedi- ate events in numerical terms. If one were to establish the probability of occurrence of the basic events, then by using the mathematics of probability of independent events, a prob- ability can be calculated for the “top event.” For two independent events
that must occur together, the probability of the result is the product of the events neces- sary for its occurrence. When events are connected by an “AND,” the “top event” prob- ability is given as P1 x P2 = P. Where three events are so re- lated (and independent), each having probabilities P1, P2 and P3, then the “top event” likeli- hood would be given as P1 x P2 x P3 = P. Figure 3 illustrates the calculation
for the purchase of spare parts associated with the event. Knowing the spare part ordering frequency over a year or two can give the engineer an idea of the frequency of a given failure. Process Capability: If the
Fig. 3. This is an illustration of the calculation of probabilities when events are connected by an “OR” relationship is shown.
of probability when two events are connected by an “OR” relationship. Where either one of two independent events can cause the “top event,” the probabilities P1 and P2 combine as: P = P1 + P2 – P1P2 Te subtracted portion represents
that overlapping probability where both events occur at the same time. It is not correct to double count these occurrences. Where three events are related by an “OR” connection: P = P1 + P2 + P3 – P1P2 – P2P3 – P1P3 + P1P2P3 To summarize the math, the prob-
abilities for “AND” connections are multiplied and the probabilities for “OR” logical connections are added. Figure 4 provides an illustration of a section of an FTA where the calcula- tion rules are applied.
Assigning Probability Te basic events in such chains of
cause and effect generally will fall into three categories, each with its own source of probability information. Equipment Failure: Many basic
events can be the failure or fault of a piece of equipment or a component within a larger system of equipment, such as a plumbing leak or an electri- cal failure. Establishing a frequency
Fig. 7. The diagram illustrates a PDCA loop as applied to the use of FTA.
for these events relies on maintenance records. Even if maintenance does not keep such records directly, the pur- chasing department may have records
probability of a dimensional defect (say, a certain amount of mismatch on a parting line or a mislocated boss) is needed, a capability study can determine the frequency of a specific defect at a given severity. Capability studies may not be waiting in the file to be used for this purpose, but they can be performed with this calculation in mind. Process characteristics, not just product characteristics, can be evaluated in this man-
ner. If, for example, the frequency of too high concentration of coolant in a machining line is needed, this data can be obtained from a capability study. It
Fig. 4. This illustrates the calculation of probabilities in a more complex FTA section. Jul/Aug 2014 | METAL CASTING DESIGN & PURCHASING | 43
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