Hansen, Geraldine by wfq74180


									                            Hansen, Geraldine
        Destructive Testing Versus Nondestructive Evaluation:
                      A Statistical Comparison
                           Faculty Mentor: C. Shane Reese, Statistics

Researchers and developers in manufacturing and industry are often confronted with the problem
of testing component functionality. Two important testing methods used on equipment are
destructive testing (DT) or nondestructive evaluation (NDE). DT implies that the part is
destroyed by disassembly, or it is destroyed by utilizing the equipment to determine if the part
was defective. In the case we examined, DT is like exploding a weapon. DT is very expensive
and reduces valuable assets, but it is consistently very accurate. This type of testing is often
referred to as obtaining a “snapshot in time” because we know at that instant (and at no other
point in time) whether or not the equipment is working.

The other type of testing, NDE, is most often associated with techniques such as radiography,
where an internal picture of the equipment is taken and then evaluated to predict if the equipment
will work. This generally has lower cost and usually no reduction of assets, but the accuracy is
often sacrificed, and unfortunately, often unknown. NDE is more informative than DT over a
period of time because the functionality of parts can be evaluated again and again as the
equipment ages. The benefits of NDE indicate the importance of considering it as a testing
technique; however, the uncertainty with regard to NDE needs to be examined.

The proposal that we submitted indicated that we would examine a complex situation where a
success weapon is evaluated on a continuous scale. However, in the past year, our research
focus has been to perfect and more completely understand the case where an explosive is
evaluated to be either a failure (mathematically denoted as 0) or success (denoted as 1). During
the last few months, we have begun the more complex case, although at this time it is still in the
developmental stages. The research we did allowed us to develop a statistical framework for
comparison that can be generalized according to parameters that are given by researchers. This
information was shared at the College of Physical and Mathematical Science’s Spring Research
Conference, where it was recognized with a reward of merit.

This summer we produced a report that analyzes how nuclear weapon testing has changed and
been affected by legislation in the United States, focusing on current rumors that nuclear testing
may resume. This library-based research allowed us to connect our research with current issues.

An outline of our procedure for creating the statistical framework follows, in which we make the
comparison based on a simulation to estimate the probability that NDE detects a good part
correctly. We employ the distributions of the two types of testing, and misclassification
probabilities to create that simulation.

We assume DT data ( X 1, X 2 ,..., X n ) accurately reflects the probability of a good part.
Furthermore, each test has an independent Bernoulli distribution (a useful distribution for
modeling success/failure outcomes) with parameter p where p = Pr( X i = 1) , or the probability
that a part functions as designed.

NDE data (Z1 , Z 2 ,..., Z n ) also has a Bernoulli distribution, but instead of success parameter p,
NDE has success parameter p* where p* = Pr(Z i = 1) , or the probability of the test indicating a
functional part. There are two misclassifications, or errors, that are associated with NDE and
each of these is assigned a probability of occurring. Misclassification probabilities can be
defined as the errors when NDE indicates a component is good when it actually is bad
 Pg |b =Pr(Zi=1|Xi=0), or that a component is bad when it is, in fact, good Pb | g =Pr(Zi=0|Xi=1). The
NDE parameter p* can be shown to be equal to p* = p(1 − Pb| g ) + (1 − p ) Pg |b . Because p*
incorporates the misclassification probabilities as well as what we know from the true
probability, a comparison between the methods can then be achieved.

The parameter p (proportion of successes based on the DT data) can be shown to have a
distribution which follows a Beta distribution (by simple application of Bayes Theorem). All
distribution characteristics, such as means, standard deviations, and quantiles, can then be
obtained analytically. However, the distribution of p* which is based on NDE data must be
simulated through Markov Chain Monte Carlo (MCMC) methods since the form is unknown.
This process results in a generated distribution of p*. After generating thousands of
distributions of the parameters p and p*, we compare the two different distributions by using the
90 percentile of each, so that the center and spread of the distributions are both accounted for.

The practical application of this work is that the probability of success based on DT and the
probability of success based on NDE can be compared. Using this comparison the number of
NDE tests that must be done to reach the same uncertainty level (similar to “confidence level” in
classical statistics) that DT gives you. Figure 1 shows the different number of tests for each
method that are equivalent. The approximation of these points allows us to estimate that for
however many DT tests that are required, we will need twice as many NDE tests to obtain the
            Comparison of Sample Size                    same level of confidence. This simulation
            (Accounting for Uncertainty)                 was created as if the known probability
                                                         was p =0.4167, and the misclassification
                                                         probabilities were Pg |b =0.35 and Pb | g =.005,
                      80 100

                                                         where all of these values were based on
Number of NDE Tests

                                                         information from an actual NDE
                                                         measurement process.

                                                                       By slight modifications of code, we can
                                                                       examine the relationship between DT and

                                                                       NDE for a variety of choices of Pg |b , Pb | g ,

                                                                       and p. Each situation would result in a
                                                                       new “equivalence” curve (like the one
                                                                       shown in Figure 1). These curves can be

                               0   10   20     30     40     50   60   used by decision makers to decide if a
                                        Number of DT Tests             shift to NDE is a worthwhile option.

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