

New method accounts for uncertainties without unjustified assumptions
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FUZZY COINS  Arlin Cooper (12331) uses bent coins to illustrate fuzzy algebra. (Photo by Randy Montoya) Download 150dpi JPEG image, 'fuzzy_pix.jpg', 1 Mb 
"I became intrigued," Arlin says. "This method of computing subjective uncertainty seemed more tied to reality than traditional statistics. It took in all the uncertainties and made no unwarranted assumptions."
Arlin says that it is one thing to make predictions about safety  whether it is nuclear weapons or jetliners  using standard statistics if you know for certain that there is a physical basis. It's another when you have to factor in uncertainties that can only be expressed subjectively.
Fuzzy algebra
That's where fuzzy algebra comes in.
In the traditional mathematical model, probabilities are estimated by clearcut equations or by precisely defined probability distributions. Data are dumped into computer routines and point numbers or probability distributions are calculated.
The problem, Arlin says, is that typically only the analyst doing the computations knows which data are assumed and which are physically justified.
Traditional statistics vs. fuzzy algebra
"The resulting probability analyses go to decision makers who may be unaware of the nature of the uncertainties and make decisions based on the figures given to them," he says.
In explaining the difference between traditional statistics and fuzzy algebra, he uses the example of the probability of throwing four heads in four coin tosses. Traditional analysis says that the probability is 1/16, and this might be viewed as an acceptably small risk for a gambler. But what happens if the consequences of losing are raised beyond what is acceptable  say to loss of life?
Would the gambler then be interested in knowing more about the coins and how they were tossed? How does the probability change if the coins have been subjected to an "abnormal environment" (stresses beyond the operating environment) and are significantly warped or deformed and where there are limited physical data or tests? Knowing more about the coins and how they can fail to perform in a predictable way is analogous to the safety analyst wanting to know how safety devices in highconsequence systems can fail.
A traditional analyst might select one or more distribution equations to determine the probability of the coins falling heads up  a guessing game of sorts. The distribution equations then shape the outcome of the probability analysis, resulting in a phenomenon called "tail (extremes) suppression."
A fuzzy algebra approach, however, takes into account qualitative judgments by "experts" who in the example might view the warped coins and come up with their own separate range estimates about how frequently the coins will land heads up. These predictions are plugged into a fuzzy function, a twodimensional portrayal of range that represents a continuum from the lowest to the highest levels of presumption.
The true consensus on how often the coins would land on heads lies somewhere within the fuzzy function. Instead of an exact point or probability distribution like in traditional analysis, fuzzy algebra provides a subjectivebased range. The process avoids tail suppression, a potential source of overoptimism.
"It may not be a satisfying answer because it is not precise, but it's an honest answer," Arlin says.
As more and more information is received  for example if exact measurements about the way the warped coins fall are determined  the fuzzy math approach can transition toward the probabilistic approach.
"You will start out with as low as 10 percent hard data and 90 percent subjective data," Arlin says. "And as you strive to get better information, the numbers may eventually turn completely around to 90 percent hard data and 10 percent subjective."
Approach suits safety analyses
Arlin says this hybrid approach perfectly suits safety probability analyses because while there may be some welldefined physical responses, there are also a lot of highly uncertain events  ranging from human failure to lightning strikes  that can threaten a nuclear weapon. By having experts analyze what those problems might be and processing them in a hybrid analysis, it is possible to come up with realistic estimates.
Arlin's manager, Perry D'Antonio, is enthusiastic about Arlin's endeavors to find an improved method of doing probability analyses, agreeing that current procedures don't properly take into account subjective assumptions.
"Arlin looked at what others were doing and found a possible solution to a major problem  coming up with a real probability description that doesn't overstate safety," Perry says. "In the business of safety you have to be cautious about what you might have forgotten to include, and you can't assume anything unjustified."
Last modified: February 12, 1999
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