Stochastics and StatisticsThree-triplet np control charts
Introduction
The np control chart is widely used in the industries or service sections to monitor the performance of a process where quality is measured in terms of fraction nonconforming. In attribute statistical process control (SPC), a nonconforming unit is a product or service that fails to meet at least one specified requirement. And the fraction nonconforming p is the ratio of the number of nonconforming units in a process to the total number of units in that process.
The widespread use of the np chart is due to the relative simplicity in handling attribute quality characteristics, the capability of checking multiple quality requirements, and especially, the prevalence of count data in many sections (e.g., the health care industry). The np chart is equivalent to the p chart. While the p chart is more suitable for manipulating the variable sample size, the np chart is easier for personnel with limited statistical knowledge to implement when the sample size is constant (Montgomery, 2001).
The CUSUM type charts are also used for monitoring fraction nonconforming (Lucas, 1985). Even though the CUSUM control chart is more effective than the Shewhart np chart in detecting small and moderate p shifts, the np chart (or the equivalent p chart) is still considered as the most widely used attribute control chart. It is because of an interest in involving production operators in quality improvement (Box et al., 1997). Methods such as the CUSUM chart are more difficult for the operators to use and understand (Juran and Godfrey, 1999).
To run an np chart, four parameters have to be determined, namely, the sample size n, the sampling interval h, the lower control limit LCL and the upper control limit UCL. During the process control, for every h output units (or time units), a sample of n units is inspected. If the number d of nonconforming units found in the sample falls inside the range bracketed by LCL and UCL, the process is thought in control, otherwise an out-of-control condition is signaled.
The lower and upper control limits of a conventional np chart are calculated as follows:where, k is the control limit coefficient, usually set at 3. The control limits calculated by Eq. (1) usually have fractional values. Since d must be an integer, some practitioners prefer to use integer values for control limits on the np chart instead of their fraction counterparts (Montgomery, 2001). If the in-control condition is defined by (LCL⩽d⩽UCL) and the out-of-control condition by (d<LCL or d>UCL), the integral limit LCL can be obtained by rounding up its fractional counterpart, and the integral UCL is obtained by rounding down its fractional counterpart. For example, if the fractional lower and upper control limits are 2.62 and 20.51 respectively, the equivalent integral control limits are LCL=3 and UCL=20.
The power of a control chart can be measured by the run length, which is the number of inspected samples required to signal an out-of-control condition after it has occurred. When a process is out of control, the QA engineers want the control chart to signal quickly, i.e., to have a short out-of-control run length. However, when the process is in control, the QA engineers want the chart to produce fewer false alarms, i.e., to have a long in-control run length. Run length is a random number, the mean value of which is called the average run length (ARL). The out-of-control ARL is commonly used as an indicator of the power of the control chart, and the in-control ARL0 for the false alarm rate. However, a percentage point of the run length distribution may be a more appropriate measure than ARL in some applications (Woodall, 1985). It is because that the use of ARL may bring serious uncertainty to the performance assessment of the charts. For a given process condition, it is quite likely that a particular run length is substantially different from ARL. For example, for an np chart with (n=65, LCL=0, UCL=6), ARL is equal to 4.7 when p=0.075. However, in this case, the probability that a run length is equal to or larger than 8 is almost 20%. Even the probability that a run length is equal to or larger than 10 is over 10%. Furthermore, the run length distribution can be highly skewed. ARL may be quite different from the median run length. Thus, ARL cannot be considered as representing “half the time” (Palm, 1990). Finally, the use of ARL curbs the design flexibility, as the QA engineers have to refrain themselves from specifying a desired performance characteristic for the given application. Obviously, being able to specify a desired probability W for an expected run length RL helps to achieve considerable performance certainty and great design flexibility, and therefore, is more attractive than indiscriminate use of ARL.
In this article, the 3-triplet np chart is proposed. It is so called because the three triplets (p0,RL0,W0), (p−,RL−,W−) and (p+,RL+,W+) can be specified by the QA engineers and fully satisfied by the chart. A design triplet (p,RL,W) is characterized by three parameters: fraction nonconforming p, design run length RL and expected probability W. For an in-control condition (p=p0), the triplet (p0,RL0,W0) stipulates that the probability of a run length being no shorter than RL0 is equal to W0. Similarly, for a downward p− shift, the triplet (p−,RL−,W−) sets down that the probability of a run length (against p− shift) being no longer than RL− is equal to W−. Finally, the triplet (p+,RL+,W+) specifies that the probability of a run length (against p+ shift) being no longer than RL+ is equal to W+. In the design of an np chart, above three triplets (p0,RL0,W0), (p−,RL−,W−) and (p+,RL+,W+) are usually the main concerns in the mind of the QA engineers. It is ideal that the QA engineers are able to specify the parameters of these three triplets based on the design requirements, and then, these requirements are fully satisfied by the resultant chart.
- (1)
The in-control RL0 determines the cost incurred by false alarms (Chiu, 1975; Duncan, 1978). In order to reduce this cost, RL0 should be set sufficiently long (especially if handling false alarms is very difficult and time consuming) with a reasonable probability W0.
- (2)
The out-of-control RL+ determines the cost attributable to the nonconforming units produced under the out-of-control conditions. This cost is very critical to the overall cost in SPC. In order to reduce this cost, RL+ must be set sufficiently small with a high probability W+ when p=p+ (p+ is the smallest upward shift that is considered harmful enough to the product quality).
- (3)
The out-of-control RL− determines the speed of signaling the downward p− shift. The downward shift may have resulted from a real quality improvement, or a malfunction of the instrument, or a mistake made by the operator. If the downward shift is attributable to real quality upgrading, a signaling will deliver useful information for the continuous improvement program. In contrast, if the downward shift is caused by instrumental or personnel errors, a signaling will alert the operators. In any case, it is preferable to signaling the downward p− shift early so that the situation can be addressed.
- (4)
However, choosing a very high value for RL0, a very low value for RL− or RL+ or unduly high value for any of the probability levels will lead to a prohibitively large sample size, and consequently, the intolerably high SPC cost.
To sum up, it is ideal that the QA engineers are able to set each of RL0, RL− and RL+ at an appropriate value with a rational probability, so that the designed np chart can be run as effectively and economically as expected. As mentioned by Vining and Reynolds (1997), in designing a control chart, “three main factors need to be considered: the false alarm rate of the chart; the ability of the chart to detect an out-of-control process; and, the sampling costs of the chart. A good design will achieve a reasonable balance between these three factors”.
RLs and Ws are largely determined by the sample size and the control limits. However, in traditional chart designs, these parameters are rarely determined analytically in the light of performance characteristics. The QA engineers usually select a sample size based on the rational subgrouping or for the administrative convenience (Duncan, 1986). The control limits are calculated by using very primitive formulae (1). Unfortunately, the sample size and control limits determined by this way are unlikely to give satisfactory in-control and out-of-control statistical performance. The conventional 3-σ np chart cannot even ensure the in-control ARL0 being equal or close to a specified value, let alone the three triplets. Even the more sophisticated CUSUM chart cannot enable the QA engineers to specify the RLs and Ws at the three key values p0, p− and p+. As a result, it is very difficult for the QA engineers to predict and control the performance characteristics of the conventional np chart.
Palm (1990) considers the percentile run lengths for the Shewhart charts with supplementary runs rules. Woodall (1985) presents a method for designing the Shewhart and CUSUM charts on the basis of their statistical performance over specified in-control and out-of-control regions of parameter values. Duncan (1986) suggests a method for determining the sample size n of an np chart, in which n is set large enough so that the chance of detecting a fraction nonconforming shift in a single sample is approximately equal to 50% (a special triplet: p=p+, RL+=1, W+=50%). Ryan and Schwertman, 1997, Schwertman and Ryan, 1999 have proposed several algorithms that select the sample size of the np chart based on p0, so that the resultant in-control ARL0 is very close to the specified value.
Another problem of the conventional np chart is that the design scheme (Eq. (1)) results in (LCL=0) for many cases (Glushkovsky, 1994). It means that the np chart is unable to detect any downward shifts in p. Nelson (1997) suggests using supplementary runs rules to detect downward shifts when there is no lower control limit. However, run length properties of the np charts with such rules in place have not been investigated thoroughly.
The 3-triplet np chart proposed in this article is designed by seeking an optimal combination of the sample size and the control limits, so that the np chart’s power against the false alarm rate, the downward p− shift and the upward p+ shift are all maintained in the satisfactory levels, and the sample size (or SPC cost) is minimized in the meantime.
In case the designed sample size for a 3-triplet np chart is larger than practically allowed, a compromised algorithm can be employed to reduce the run length probabilities in a rational manner.
In this article, in order to design the control chart parameters more precisely, probability control limits, rather than the 3-σ limits, are used. The distribution of the number d of nonconforming units in a sample is assumed to be binomial. Since the binomial distribution is a discrete one, the control limits LCL and UCL designed based on this distribution are integers in the first place.
The design procedure of the 3-triplet np chart will be introduced in the next section. It is followed by the description of a compromised design algorithm and the recommended values for the input parameters. Finally, the performance of the 3-triplet np chart is illustrated by a comprehensive example.
Section snippets
Design of the 3-triplet np chart
In designing a 3-triplet np control chart, the QA engineers will specify the following nine input parameters:
- p0, p−, p+
the fraction nonconforming values for the in-control and two out-of-control (downward and upward) conditions.
- RL0, RL−, RL+
the corresponding design run lengths.
- W0, W−, W+
the corresponding probabilities.
The in-control fraction nonconforming p0 is usually estimated from the observed data in the pilot runs. p− and p+ indicate the downward and upward rejectable quality levels and
Compromised design
Sometimes, due to the shortage of inspection instrument or manpower, the maximum allowable sample size nallow may be smaller than the designed sample size for the 3-triplet np chart. Under such circumstance, many different schemes may be used to find a compromised design. One possibility is to reduce the specifications W0, W− and W+ proportionally, so that nallow becomes a feasible sample size. In our current implementation, a more pragmatic scheme is employed. Here, the priority is put first
Default values of the input parameters
In designing a 3-triplet np chart, the QA engineers are allowed to specify any desired value for each of the six input parameters RL0, RL−, RL+, W0, W− and W+. It certainly gives them great freedom to design, control and stipulate the performance characteristics of the resultant chart. It should be considered as a privilege rather than a burden. In fact, if the QA engineers have difficulty to determine part or all of these six input parameters, they may simply use the following recommended
The problem
In an electronic company, a component produced by a process is classified as nonconforming if any of the five types of defectives (missing part, wrong orientation, misalignment, tombstone, and bend lead) is detected. The in-control p0 is estimated as 0.01 from the historic data. Based on the current and expected product quality, the managerial level lays down the following requirements:
- (1)
When the process is in control (p0=0.01), the false alarm should not occur before the 200th sample with a
Conclusions and discussions
This article has proposed the 3-triplet np chart, a highly specifiable and controllable attribute chart for monitoring fraction nonconforming. The chart is designed by wisely adjusting the control limits and the sample size, so that the specified probabilities of the design run lengths at three key points are warranted. This development enables the QA engineers to have more control in the design and operation of the np control charts. Specifically, the false alarm rate is no higher than a
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