An np control chartfor monitoringthe mean of a variable

based onan attribute inspection

Zhang Wu* (corresponding author)

School of Mechanical and Aerospace Engineering

Nanyang Technological University, Singapore 639798

Tel: (65) 67904445, Fax: (65) 67911859, Email:

B. C. MichaelKhoo

School of Mathematical Science

University Sains Malaysia, Penang11800, Malaysia

Lianjie Shu

Faculty of Business Administration, University of Macau, Macau

Wei Jiang

Department of Systems Engineering and Engineering Management

Stevens Institute of Technology, Hoboken, NJ07030, USA

Abstract

This article proposes a new np control chart, called the npx chart, thatemploysan attribute inspection (inspecting whether a unit is conforming or nonconforming) to monitor the mean value of a variablex.The distinctive featureof the npxchart is using the statistical warning limits to replace the specification limits for the classification of conforming or nonconforming units. By optimizing the warning limits, the npx chart usually outperforms the chart considerably on the basis of same inspection cost. In addition, the npx chart often works as a leading indicator of trouble and allows operators to take corrective action before any defectives are actually produced.

Keywords:Quality control; Statistical process control; Attribute and variable control charts; Attribute inspection; Average Time to Signal; Loss function.

Zhang Wu, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Tel: (65) 67904445, Fax: (65) 67911859, Email:

1.Introduction

InStatistical Process Control (SPC), control charts for attributes(such as the np, p, c and u charts) detect the out-of-control conditions of a process by checking the number of nonconforming units or nonconformities in a sample.The attribute control charts are widely usedlargely owing toitssimplicity in implementation. By using attribute charts, “expensive and time-consuming measurements may be avoided by attributes inspection” (Montgomery 2005).The sample sizes of the attribute charts are usually much larger than that of the variable charts. This alsoimplies that the attribute inspection is much simpler and less time-consuming than the variable inspection.

Nevertheless, it is commonly believed that attribute charts are unable or inefficient to deal with a quality characteristic that is of a variable type. The effectiveness of a control chart is usually measured by the Average Time to Signal (ATS) which is the average time required to signal a process shift (e.g., a mean shift ) after it occurs. A small out-of-control ATS means that the out-of-control conditions will be signalled promptly and the amount of defectives produced during the out-of-control status would be reduced. On the other hand, the in-control ATS0 should be large so that the false alarm rate is low.

The Shewhart chart is widely applied to monitoring the mean of a variable. It inspects the sample mean .

(1)

where xi is the ith observation in a sample of size . The more advanced CUSUM chart monitors the process mean by checking the cumulative sum Ct.The implementation of both and CUSUM charts relies on a variable inspection which measures the actual value of x and requires the calculation of and/or Ct.

Wu and Jiao (2007) proposed an attribute chart(namely the MON chart) for monitoring the mean of a variable. This chart checks the run length between two consecutive nonconforming samples.

Among all of the attribute control charts, the np chart may be the simplest one for understanding and implementation.The np chart is equivalent to the p chart when the sample size is constant, but the former is easier for non-statistically trained personnel tounderstand and handle(Montgomery 2005).The np chart counts the number, d, of the nonconforming units in a sample of size nnp. If d falls beyond the lower control limit LCLnp or upper control limitUCLnp of the np chart, the process is thought out of control.

Whether the np chart is ableto detect the mean shift of a variable with satisfactory efficiency?Montgomery gave an example in his text (2005), in which an chart and an np chart are compared for detecting the mean shift of a quality characteristic x with a normal distribution ~N(50, 22). The chart uses the3-sigma control limits and a sample size of nine. Its power for detecting a mean shift of 2 (= 1σ) is equal to 0.50. In contrast,to detect the same mean shift,thesample size nnpof the np chart must be at least equal to 60 in order to achieve the same detection power and to maintain the same false alarm frequency. The ratio between the sample sizes (i.e., nnp/) is as large as 6.667. This huge difference will scare away many Quality Assurance (QA) practitioners from considering the np chart for monitoring the mean of a variable.

In this article, a new type of np control chart, the npxchart, is proposed to compete the chart for detecting mean shift δ. It is found that even when using the same sample size and sampling frequency, the npx chart is less effective than the chart only by 30% to 40 %. In fact, since the npx chart employs the simple attribute inspection and eliminates the need for any computation, it may use a greater sample size and/or sampling frequency than the chart in many SPC applications on the basis of same inspection cost.A greater sample sizeand/or sampling frequency will, in turn, make the npx chart more effective than the chart, measured by both Average Time to Signal (ATS) and Extra Quadratic Loss (EQL)(Reynolds and Stoumbos 2004).

A distinctive feature of the npx chart is the use ofthe statistical warning limitswL and wU to replacethe specification limits for the classification of conforming and nonconforming units. While the specification limits are fixed by design engineers, the warning limits can be set at an optimal level by the QA engineers so that the npx chart has the highest detection power. Moreover, sincewL and wUare continuous variables, they can be adjusted so that the in-control ATS0 of the npx chart is exactly equal to a specified value τ. It ensures that the npx chart can meet the requirement on false alarm rate, and meanwhile has the tightest possible control limit and the highest possible detection power.

Finally, since the warning limits are often much closer to the target value than the specification limits,a nonconforming unit that falls beyond one warning limit of an npxchart may still reside within the specifications. Or in other words, a nonconforming unit classified bythe npx chart may not be a defective. It means that the npx chart often provides an indication of impending trouble and allows operators to take corrective action before any defectives are actually produced. It is an advantage thatonly the variable charts have in the past (Montgomery 2005).

The rationality of the warning limits lies in the fact thatthe major objective of SPC is to quickly detect the occurrence of assignable causes of process shiftsusing the control chart as an on-line process-monitoring tool (Montgomery 2005), rather than evaluating the quality level of the units in a few samples.

As aforementioned, while the chart adopts variable inspections, the npx chart employs the simple attribute inspections.In an attribute inspection, the only concern is whether a unit is conforming (e.g., whether x falls within the warning limits) rather than the actual value of x. A typical example of the attribute inspection is to use a “Go/No Go” ring gage to check whether the diameter x of a shaft exceeds a limit (Kennedy etal. 1987). Suppose the calibrated dimension of the ring gage is made equal to the upper warning limit wU, a shaft is deemed to be oversized if it cannot pass through the gage. Figure 1 shows (a) a double-end gage and (b) a progressive gage. They are able to check both oversizing and undersizing in one run (the calibrated dimension in the left hand side equals wU and that in the right hand side equals wL). Other examples include using a snap gage to check whether the thickness of a component passes a specification, and using a horizontal line (bar) to decide whether a bus passenger is taller than a standard. A special spring scale can be designed to check the weights of objects. The scale will ring or blink if an object is heavier than a predetermined limit.Montgomery (2005) observed that “variable-type inspection is usually much more expensive and time consuming on a per unit basis than attribute inspection.” It thereby may be more appropriate and fair to compare the effectiveness of the charts based on identical inspection costper unit time (W), rather than based onthe equal sample size (n) and sampling interval(h).

,(2)

where c is the inspection cost per unit. If the cost for instrument is neglected, c is simply proportional to the timet spent on the inspection of a single unit. The following condition must be satisfied for a fair comparison between an chart and an npx chart.

.(3)

In Montgomery’s example aforementioned (2005), as both the and np charts use the same sampling interval h, accordingly,

.(4)

That is, the sample size should be inversely proportional to the inspection cost per unit for identical h. On the other hand, if two charts use the same sample size but different sample intervals, Equation (2) leads to:

.(5)

Since an npxchart simply checks whether the units are conforming (within the warning limits) or nonconforming (beyond a warning limit) and does not require any calculation, the inspection costcnp required by thischart must be substantially lower than the inspection cost required by an chart for many SPC applications. According to Equation (2), the sample size nnp of the npx chart may be allowed to be larger than of the chart, or the sampling interval hnp be smaller than .

The actual value of the ratio (/cnp) (ornnp/, or /hnp) in a particular application depends on the extent of simplicity of the attribute inspection with respect to the variable inspection. For example, if the charts are implemented manually, the operator of an charthas to conduct the time consuming variable inspection and also calculate sample mean . Under such circumstances, the inspection costmay be much higher thancnp. Based on same W value (inspection cost per unit time), the npx chart may use a much larger sample size and/or a much smaller sampling interval. In example 1 in the latter section 4, the ratio of (/cnp)in a typical experiment in mechanical engineering is equal to 10.082 ( includes the time for measuring x and for computing by a calculator). It means that the inspection cost for an chart using a sample size of and a sampling interval is nearly the same as the inspection cost for an npx chart using (nnp = 10, hnp = ) or (nnp = , hnp= /10).

On the other hand, if a computer-aided system is available for the SPC implementation as in many modern industries, the operators are released from calculating the sample mean . They only have to measure the value of x and enter the reading into a computer through the keyboard. However, the variable inspection used by the chart is still intrinsically more or much more difficult than the attribute inspection.In fact, in many applications, just keying in a reading (for example, the measured diameters 74.030, 74.002, 74.019, 73.992, 74.008 … of the forged piston rings (Montgomery 2005)) through a keyboard may take longer time than carrying out an attribute inspection. In example 1 in section 4, when an on-site computer is in place, the ratio of (/cnp) is equal to 4.482, where the cost includes the time spent on measuring x and keying the reading into a computer. This suggests that the sample size or sampling frequency of the npx chart may be 4 to 5 times higher than that of the chart even for computer-aided SPC.

In this article, the observations of aquality characteristic x are assumed to be independent and have an identical normal distribution with known in-control mean μ0 and standard deviation σ0. It is also assumed that the process variance remains unchanged. When process shift occurs, the mean value  will change, i.e.,

,(6)

where is the mean shift in terms of 0.

The remainder of the article proceeds as follows. The implementation and design of the npxchart is introduced in Section 2. This chart is compared with the chartin Section 3. Subsequently, two practicalexamplesare illustrated in Section 4. Finally, the conclusions and discussions are drawn in the last Section.

2.Implementation and design of the npx chart

Implementation

The npx chart has five parameters: the sample size nnp, the sampling interval hnp, the lower and upper warning limitswL and wU, and the upper control limit UCLnp. Unlike a conventional np chart, the npx chart does not need a lower control limit. The operation of an npx chart is as simple as that of the conventional np charts except the specification limits being replaced by the warning limits.During the implementation, nnp units as a sample are inspected at the end of each sampling intervalhnp. If the number, d, of nonconforming units is larger than UCLnp, the process is signalled as out of control; otherwise the process is thought in control. The whole procedure does not need any calculation.It is noted that, with only an upper control limit UCLnp, an npx chart is able to detect both increasing ( > 0) and decreasing ( < 0) mean shifts, depending on whether thexvalues of the dnonconforming units fall above theupper warning limit wU or below the lower warning limitwL. The upper control limit UCLnpis used to check d and, therefore, is an integer. However, the warning limits wL and wU are variables.They are symmetrical about the in-control mean μ0.

,(7)

where, kw is called the warning limit coefficient.

Design specifications

To design an npx chart, the following five specifications need to be decided:

nnpthe sample size

hnpthe sampling interval

the allowed minimum in-control Average Time to SignalATS0

μ0the in-control process mean

σ0the in-control process standard deviation

The sample size nnpand sampling interval hnp are also the charting parameters and decided based on the available inspection resources (e.g., manpower and instrument). The value of  is decided with regard to the tolerable false alarm rate. The values of the in-control mean 0 and standard deviation 0 are usually estimated from the data observed during the pilot runs in phase I operation. For a process,the control charts used in phase I and phase II operations may not be the same. For example,Montgomery (2005) suggested using a simple chart in phase I operation and a more advanced EWMA or CUSUM chart in phase II operation, as it is much simpler to use an chart to collect the sample data for estimating process parameters. With the same reason, one may employ an chart in phase I operation to estimate 0 and 0 in order to build the npx chart. Then he can use the npx chart to monitor the process mean in phase II operation.

Constraint function

The actual in-control ATS0 should be no smaller than τ in order to satisfy the requirement on false alarm rate. However, ATS0 cannot be too large, otherwise the out-of-control ATS will also be quite large and the detection effectiveness is low. It is most ideal thatATS0be equal to τ.

,(8)

where, α is the probability of type I error (the probability that the control chart produces an out-of-control signal when the process is in fact in control). On the other hand, if the number dof nonconforming units in a sample follows a binomial distribution, the value of α produced by an npx chart is calculated by

,(9)

where, p0 is the probability that a unit is nonconforming (falling beyond one of the warning limits) when the process is in control. Referring to Equation (7),

,(10)

where, Ф() is the cumulative probability function of the standard normal distribution.Combining Equations (8), (9) and (10), we have

.(11)

When the upper control limit UCLnp is given, the warning limit coefficient kw is the only unknown in the above equation and can be solved by any numerical method. The resultant value of kw will ensure the satisfaction of the constraint(ATS0= τ).

Objective function

In many control chart designs, the out-of-control ATS at a specifiedmean shift level is used as the objective function to be minimized. However, since the goal of the design is to make the control chart efficient at signalling a wide range of mean shifts, it is therefore more desirable that the objective function measures the holistic performance of the charts cross a process shift range rather than just at a specific point. Researchers usually compare the performance of two charts by examining the corresponding out-of-control ATS values of the two charts at some discrete points of process shift δ cross a range. For most of the cases, one chart is unable to excel another chart at all the points. However, as long as one chart has smaller ATS at more points and/or to a larger degree, this chart is thought more effective than the other. Moreover, since it is usually assumed that all mean shifts within a range are equally important (Sparks 2000), a uniform distribution for δ is implied. Such comparison scenario may be formulated as follows:

,(12)

where,RATS indicates the average of the ratio of the ATS values of two charts cross the range (δmin ≤ δ ≤ δmax). In Equation (12),ATS() is produced by one chart at  and ATSbenchmark() is generated by another chart that acts as the benchmark. Obviously, if the RATS value of a chart is larger than one, this chart is generally less effective than the benchmark, and vice versa.