28-05-2013, 12:58 PM
ARL Comparisons Between Neural Network Models and ¯x-Control Charts for Quality Characteristics that are Nonnormally Distributed
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Abstract:
One widely used control chart, the ¯x-chart, is based on the assumption that
means of samples drawn from the process are normally distributed. When the normality
assumption is not valid, control chart users may choose from several different courses of
action. These include using Box-Cox power transformations on the original data to yield an
approximate normal distribution, increasing the size of the samples drawn from the process
until the distribution of the sample means is considered normal, and modifying the ¯x-chart
to employ asymmetric control limits instead of limits that are equidistant from the process
target mean. Since none of the remedies for handling nonnormal processes is completely
satisfactory, we build on previous neural network research by developing a neural network to
control nonnormal processes.
Introduction
The control chart was introduced in 1925 by Shewhart in the Journal of the American
Statistical Association. Since then control charts have become a powerful tool to monitor
process means and variances [12]. When used appropriately, control charts help determine
if processes are “in control” or “out of control” A process found to be out-of-control can
be corrected, thereby reducing the variability of the process means [1].
One of the most widely used control charts, the ¯x-chart, is based on the assumption that
the distribution of the means of samples drawn from the process is sufficiently close to a
normal distribution. However, in many industrial processes, this assumption is invalid.
Neural Network Description
In this study we used a fully-connected feed-forward network with five input neurons,
five hidden neurons, and three output neurons. Choosing five input neurons was logical
once we decided upon using samples of size five because each neuron would represent an
observation in the sample. The simplicity of this neural network architecture guaranteed
a realistic training time. We implemented the back-propagation learning algorithm for
our neural network. Although the slowness of learning is usually a major concern with
back-propagation networks, the learning speed of the training sets was fast enough for
use in our study.
In addition to the architectural features of number of hidden layers, number of hidden layer
neurons, and number of output neurons, factors such as the learning rate, momentum,
learning rule, and type of transfer function are crucial to the performance of a trained
neural network We used a learning rate of 0.1 and a momentum of 0.6 after testing more
than 30 different combinations of the parameters. A sigmoid transfer function was used
because of its desirable features.
Training of the Neural Network
For the neural network to satisfactorily monitor a process, the range of shifts in the process
mean utilized in the training procedure should be representative of the whole spectrum
of the shifts of interest. However, the determination of the range of mean shifts used
in training is subject to the required recognition capability, and the expected training
time. The magnitude of shifts in the process mean was measured in units of the standard
deviation of the mean (σ¯x). When small magnitudes of process changes were included
in the training set the neural network performed poorly, registering a high false alarm
rate (an increase in commitment of Type I errors). This poor performance for small
shifts in the process mean was probably due to the high overlap between the in-control
observations and out-of-control observations. As a result, data with small shifts (.25, .5,
and .75 σ¯x) were not included in the training sets.
Data Collection and Performance Measures
Following the convention established in previous research, comparison of the neural network
and ¯x-control charts was based on average run length (ARL) which is defined as
the average number of subgroups (samples) that are observed from a process before an
out-of-control signal is detected. For a given process, this average was based on 10,000
simulated replications. When the process is in-control (not shifted), it is desirable to have
a large ARL. However, if the mean of the process is out-of-control (shifted), it is desirable
to have a small ARL. In other words, the monitoring process should generate signals as
quickly as possible (i.e., give shorter ARLs) if the production process is out of control,
and as late as possible (i.e., give longer ARLs) if the production process is in control.
Discussion and Conclusion
In this paper the performance of the neural network was evaluated by estimating the
ARLs. The neural network approach presented here offers a competitive alternative to
existing control schemes. Because the performance of the neural network depends on
the selection of training samples, the results reported in this research are not necessarily
optimal. However, we have demonstrated the feasibility of applying a neural network
model to monitoring a process.
The comparisons showed that use of the neural network model appeared to be a better
control procedure for detecting sudden changes in the process mean than the ¯x-control
chart. The neural network developed in this study was primarily designed to detect a
sudden shift in the process mean. Nevertheless there are other characteristics of a process,
such as trends, cycles, systematic and stratification patterns and mixtures, that may cause
one to regard the process as out-of-control. In future research these characteristics ought
to be considered and might need to be redefined under nonnormality.