6: II 7: III
4.3 Attribute Control Charts in Phase I
4.3.1 Use of a p Chart in Phase I
Theqccpackage includes data from a Phase I study using apchart taken from Montgomery[74]. The study was on a process to produce cardboard cans for frozen orange juice. The plant management requested that the use of control charts be implemented in effort to improve the process. In the Phase I study, 30 subgroups of 50 cans each were initially inspected at half-hour intervals and classified as either conforming or nonconforming. The R code below taken from theqccpackage documentation makes the data available. The first three lines of the data frameorangejuiceare shown below the code.
R>library(qcc) R>data(orangejuice) R>attach(orangejuice) R>orangejuice
sample D size trial
1 1 12 50 TRUE
2 2 15 50 TRUE
3 3 8 50 TRUE
. . .
The column sample in the data frame is the subgroup number, D is the number of nonconforming cans in the subgroup, size is the number of cans inspected in the subgroup, and trial is an indicator variable. Its value is TRUE for each of the 30 subgroups in the initial sample (there are additional subgroups in the data frame). The code below produces the initial p chart shown inFigure 4.6.
R>with(orangejuice,
qcc(D[trial], sizes=size[trial], type="p"))
In this code the[trial](afterDandsize), restricts the data to the initial 30 samples wheretrial=TRUE.
In this Figure, the proportion nonconforming for subgroups 15 and 23 fell above the upper control limit. With only, conforming-nonconforming informa- tion available on each sample of 50 cans, it could be difficult to determine the cause of a high proportion nonconforming. A good way to gain insight about the possible cause of an out of control point is to classify the nonconforming items. For example, if the 50 nonconforming items in sample 23 were classi- fied into 6 types of nonconformites. A Pareto diagram, described later in this chapter, could then be constructed.
After a Pareto diagram was constructed, and it showed that the sealing failure and adhesive failure represented more than 65% of all defects. These defects usually reflect operator error. This sparked an investigation of what operator was on duty at that time. It was found that the half-hour period
FIGURE 4.6: pchart of the number nonconforming
when subgroup 23 was obtained occurred when a temporary and inexperienced operator was assigned to the machine; and that could account for the high fraction nonconforming.
A similar classification of the nonconformities in subgroup 15 led to the discovery that a new batch of cardboard stock was put into production during that half hour period. It was known that introduction of new batches of raw material sometimes caused irregular production performance. Preventative measures were put in place to prevent use of inexperienced operators and untested raw materials in the future.
Subsequently, subgroups 15 and 23 were removed from the data, and the limits were recalculated as p = .2150, U CL = 0.3893, andLCL = 0.0407.
Subgroup 21 (p=.40) now falls above the revisedU CL.
There were no other obvious patterns, and at least with respect to operator controllable problems, it was felt that the process was now stable.
Nevertheless, the average proportion nonconforming cans is 0.215, which is not an acceptable quality level (AQL). The plant management agreed, and asked the engineering staff to analyze the entire process to see if any improve- ments could be made. Using methods like those to be described in Chapter 5, the engineering staff found that several adjustments could be made to the machine that should improve its performance.
After making these adjustments, 24 more subgroups (numbers 31-54) of n=50 were collected. The R code below plots the fraction nonconforming for these additional subgroups on the chart with the revised limits. The result is shown inFigure 4.7.
R>library(qcc)
R># remove out-of-control points (see help(orangejuice) for the reasons)
R>inc <- setdiff(which(trial), c(15,23))
R>qcc(D[inc], sizes=size[inc], type="p", newdata=D[!trial], newsizes=size[!trial])
R>detach(orangejuice)
FIGURE 4.7: Revised limits with additional subgroups
The statement inc <- setdiff(which(trial), c(15,23)) creates a vector (inc) of the subgroup numbers, where trial=TRUE, excluding sub- groups 15 and 23. In the call to theqccfunction, the argumentD[inc]is the vector containing the number of nonconforming cans for all the the subgroups in the original 30 samples minus groups 15 and 23. This is the set that will be used to calculate the control limits. The argument sizes=size[inc] is the vector of subgroup sizes associated with these 28 subgroups. The argument newdata=D[!trial] specifies that the data for nonconformities in the addi- tional 24 subgroups in data frameorangejuice, where trial=FALSEshould be plotted on the control chart with the previously calculated limits. The ar- gumentnewsizes=size[!trial]) is the vector of subgroup sizes associated with the 24 new subgroups. For more information and examples of selecting subsets of data frames (see Wickham and Grolemund[102]).
The immediate impression from the new chart is that the process, after adjustments, is operating at a new lower proportion defective. The run of consecutive points below the center line confirms that impression. The obvious assignable cause for this shift is successful adjustments.
Based on this improved performance, the control limits should be recalcu- lated based on the new data. This was done, and additional data was collected over the next 5 shifts of operation (subgroups 55 to 94). The R code below first loads the data from subgroups 31 to 54, which is in the data frame orangejuice2 in the R package qcc. In this data set trial=TRUE for sub- groups 31 to 54, andtrial=FALSEfor subgroups 55 to 94. Theqccfunction call in the code then plots all the data on the control chart with limits calcu- lated from just subgroups 31 to 54. The result is shown inFigure 4.8.
R>library(qcc) R>data(orangejuice2) R>attach(orangejuice2) R>names(D) <- sample
R>qcc(D[trial], sizes=size[trial], type="p", newdata=D[!trial], newsizes=size[!trial]) R>detach(orangejuice2)
FIGURE 4.8: Revised limits with additional subgroups
In this chart there is no indication of assignable causes in the new data.
The proportion nonconforming varies between 0.02 to 0.22 with an average proportion nonconforming of 0.1108. This is not an acceptable quality level.
Due to the fact that no assignable causes are identified inFigure 4.7, thep chart alone gives no insight on how to improve the process and reduce the number of nonconforming items. This is one of the shortcomings of attribute charts in Phase I studies. Another shortcoming, pointed out by Borror, and Champ[10], is that the false alarm rate for p and np charts increases with
the number of subgroups and could lead to wasted time trying to identify phantom assignable causes.
To try to improve the process to an acceptable level of nonconformities, the next step might be to use some of the seven tools to be described inSection 4.4. For example, classify the 351 nonconforming items found in subgroups 31 to 94 and display them in a Pareto Chart. Seeing the most prevalent types of nonconformities may motivate some ideas among those familiar with the process as to their cause. If there is other recorded information about the processing conditions during the time subgroups were collected, then scatter diagrams and correlation coefficients with the proportion nonconforming may also stimulate ideas about the cause for differences in the proportions. These ideas can be documented in cause-and-effect diagrams, and may lead to some Plan-Do-Check-Act investigations. More detail on these ideas will be shown in Section 4.4. Even more efficient methods of investigating ways to improve a process using the Experimental Designs are described in theChapter 5.
Continual investigations of this type may lead to additional discoveries of ways to lower the proportion nonconforming. The control chart (like Figure 4.7) serves as part of a logbook to document the timing of interventions and their subsequent effects. Assignable causes discovered along the way are also documented in the OCAP for future use in Phase II.
The use of the control charts in this example, along with the other types of analysis suggested in the last two paragraphs, would all be classified as part of the Phase I study. Compared to the simple example from Mitra[72] presented in the last section, this example gives a clearer indication of what might be involved in a Phase I study. In Phase I, the control charts are not used to monitor the process but rather as tools to help identify ways to improve and to record and display progress.
Phase I would be complete when process conditions are found where the performance is stable (in control) at a level of nonconformities for attribute charts, or a level of process variability for variable charts that is acceptable to the customer or next process step. An additional outcome of the Phase I study is an estimate of the current process average and standard deviation.
4.3.2 Constructing other types of Attribute Charts with qcc When the number of items inspected in each subgroup is constant,npcharts for nonconformites are preferable. The actual number of nonconformites for each subgroup are plotted on the np chart rather than the fraction noncon- forming. The control limits, which are based on the Binomial distribution, are calculated with the following formulas:
U CL=np+ 3p
np(1−p)
Center line=np (4.2)
LCL=np−3p
np(1−p)
where, n is constant so that the control limits remain constant for all sub- groups. The control limits for apchart, on the other hand, can vary depend- ing upon the subgroup size. The R code below illustrates the commands to produce an np chart with qcc using the data from Figure 14.2 in [21]. In this code, sizes=1000 is specified as a single constant rather than a vector containing the number of items for each subgroup.
R>library(qcc)
R>d<-c(9,12,13,12,11,9,7,0,12,8,9,7,11,10) R>qcc(d,sizes=1000,type="np")
When charting nonconformities, rather than the number of nonconforming items in a subgroup, a c-chart is more appropriate. When counting noncon- forming (or defective) items, each item in a subgroup is either conforming or nonconforming. Therefore, the number of nonconforming in a subgroup must be between zero and the size of the subgroup. On the other hand, there can be more than one nonconformity in a single inspection unit. In that case, the Poisson distribution is better for modeling the situation. The control limits for c-chart are based on the Poisson distribution and are given by the following formulas:
U CL=c+ 3√ c
Center line=c (4.3)
LCL=c−3√ c
where,c is the average number of defects per inspection unit.
Montgomery[74] describes an example where ac-chart was used to study the number of defects found in groups of 100 printed circuit boards. More than one defect could be found in a circuit board so that the upper limit for defects is greater than 100. This data is also included in the qcc package.
When the inspection unit size is constant (i.e. 100 circuit boards) as it is in
this example, thesizes=argument in the call to qccis unnecessary. The R code below, from theqccdocumentation, illustrates how to create a c-chart.
R>library(qcc) R>data(circuit) R>attach(circuit)
R>qcc(circuit$x[trial], sizes=circuit$size[trial], type="c") u-charts are used for charting defects when the inspection unit does not have a constant size. The control limits for theu-chart are given by the fol- lowing formulas:
U CL=u+ 3p u/k
Center line=u (4.4)
LCL=u+ 3p u/k
where,uis the average number of defects per a standardized inspection unit and k is the size of each individual inspection unit. The R code to create anu-chart using the data in Figure 14.3 in [21] is shown below. The qcc() function used in that code uses the formulas for the control limits shown in Equation 4.4 and are not constant. [21] replaces k, in Equation 4.4, withk, the average inspection unit size. Therefore, the control limits in that book are incorrectly shown constant. If you make the chart using the code below, you will see that the control limits are not constant.
R>library(qcc)
R>d<-c(6,7,8,8,6,7,7,6,3,1,2,3,3,4) R>qcc(d,sizes=k,type="u")