** LITERATURE REVIEW**

**2.3 CCC Type Control Charts**

Many enterprises today can pursue their goal of achieving a high yield process
with nearly zero defect due to the rapid growth in automated technologies. The
traditional approach monitors the defective rate (fraction nonconforming) for attribute
data by using the p-chart. However, the use of the p-chart is unsuitable for a high yield
process due to its very low defective rate. In order to overcome the problem
encountered in using the p-chart, the CCC chart was developed, where the latter plots
the cumulative count of conforming items between two consecutive nonconforming
items. Furthermore, the CCC chart has been adjudged to be very useful in monitoring
high yield processes. A high yield process refers to a process with a very low in-control
fraction of nonconforming (*p*_{0}), say at most 0.001 or 1000 parts-per-million (ppm)
(Chang and Gan, 2001). Control charts for monitoring processes with low defective
rates include those by McCool and Joyner-Motley (1998), Wang (2009), Acosta-Mejia
(2012) and Abbas et al. (2020), to mention a few. Table 2.1 provides a summary of the
existing CCC charts in the literature.

Table 2.1. A summary of existing CCC charts and their descriptions

Author(s) Title and journal information Description

1 Calvin Title: Quality control technique

for zero defect

Journal: IEEE Transactions on
*Components, * *Hybrid * *and *
*Manufacturing * *Technology, *
1983, 6(3), 323-328

It was pointed out that if attribute control charts are used in achieving zero defect, then the standard p and u charts are not suitable. A control chart that plots the number of good items between two consecutive defects on a logarithmic scale to accommodate a large number of good items can be employed. This chart is known as the cumulative count of conforming (CCC) control chart for high yield processes and it is based on the geometric distribution.

2 Goh Title: A control chart for very
high yield processes
Journal: *Quality Assurance, *
1987, 13(1), 18-22

This study investigated the properties of a geometric chart (also called the CCC chart) developed by Calvin (1983).

3 Xie and Goh Title: The use of probability
limits for process control based
on geometric distribution
Journal: *International Journal *
*of Quality and Reliability *
*Management, 1997, 14(1), *
64-73

It was demonstrated why the use of probability limits is preferred over the traditional k-sigma control limits, as far as the geometric distribution is concerned.

4

Xie et al. Title: A quality monitoring and decision-making scheme for automated production processes

Journal: *International Journal *
*of Quality and Reliability *
*Management, 1999, 16(2), *
148-157

A control scheme was presented for monitoring the cumulative count of items inspected. This procedure limits the consecutive number of nonconforming items to a small value when the process has suddenly deteriorated.

5 Ohta et al. Title: A CCC-r chart for high yield- processes

Journal: Quality and Reliability
*Engineering * *International, *
2001, 17(6), 439-446

The chart is based on the number of items inspected until r nonconforming items are observed. The authors demonstrated that as r increases, the CCC-r chart becomes more sensitive to small changes in upward shifts in the fraction nonconforming.

6 Chang and Gan Title: Cumulative sum chart for high yield processes Journal: Statistica Sinica, 2001, 11(3), 791-805

The CUSUM chart was proposed for monitoring high yield processes with a sole priority of detecting small and moderate parameter changes based on geometric, binomial and Bernoulli counts.

7 Ranjan et al. Title: Optimal control limits for CCC charts in the presence of inspection error

Journal: Quality and Reliability
*Engineering * *International, *
2003, 19(2), 149-160

The effect of inspection error on the Shewhart CCC chart and how the chart’s control limits can be computed are discussed.

8 Liu et al. Title: Cumulative count of conforming with variable sampling intervals

Journal: *International Journal *
*of * *Production * *Economics, *
2006, 101(2), 286-297

The CCC chart with the variable sampling interval scheme is developed. The average time to signal criterion is used in evaluating the efficiency of the chart.

9 Yeh et al. Title: EWMA control charts for monitoring high yield processes based on non-transformed observations

Journal: *International Journal *
*of Production Research, 2008, *
46(20), 5679-5699

EWMA control charts were developed based on non-transformed geometric, binomial and Bernoulli counts.

These charts were designed using one-sided control limits and the average number of inspected samples criterion was used in evaluating the chart’s efficiency.

Table 2.1 (continued)

An estimator for a period of time in which a step change has occurred was suggested. The performance of the model was investigated using several numerical examples.

11 Albers Title: The optimal choice of negative binomial charts for quality process

Journal: *Journal of Statistical *
*Planning and Inference, 2010, *
140(1), 214-225

The chart computes the failure rates by using a negative binomial distribution and demonstrated how the optimal number of failures is related to the degree of an increase in the fraction nonconforming.

12 Chen et al. Title: Cumulative conformance control chart with variable sampling intervals and control limits

Journal: *Applied Stochastic *
*Models * *in * *Business * *and *
*Industry, 2011, 27(4), 410-420 *

This study incorporates the variable sampling interval and variable control limit features to increase the sensitivity of the basic CCC chart.

13 Amiri and Khosravi Title: Estimating a change point of the cumulative count of conforming under a drift Journal: Scientia Iranica, 2012, 19(3), 856-861

A maximum likelihood estimator for a change point of the nonconforming level of a high quality process with the linear trend was provided. The Monte Carlo simulation was used in evaluating the performance of the estimator.

14 Amiri and Khosravi Title: Identifying time of monotonic change in the fraction nonconforming of high quality process

Journal: *The * *International *
*Journal * *of * *Advanced *
*Manufacturing * *Technology, *
2013, 68(1-4), 547-555

An approach for estimating the time of a change which does not require prior knowledge of the change type was suggested.

15 Mavroudis and Nicolas Title: EWMA control charts for
monitoring high yield process
Journal: *Communications in *
*Statistics * *- * *Theory * *and *
Markov chain procedure, where the geometric distribution
has been transformed to its analogous exponential
distribution.

16 Chen et al. Title: Economic design of VSI GCCC charts for correlated samples from high yield processes

Journal: South African Journal
*of * *Industrial * *Engineering, *
2013, 24(2), 88-101

An economic design model of the VSI GCCC chart by considering a correlation of the production outputs within the same sample was presented. A cost function that examines the cost of sampling and inspection has been developed.

17 Khilare and Shirke Title: The steady-state
performance of cumulative
count of conforming control
chart based on runs rules
Journal: *Communications in *
*Statistics * *- * *Theory * *and *
*Methods, 2014, 43(15), *
3135-3147

The SS properties of the m-of-m runs rules based control chart have been investigated based on the cumulative count of conforming items for high yield processes.

Table 2.1 (continued)

18 Bersimis et al. A compound control chart for monitoring and controlling high quality processes

Journal: *European Journal of *
*Operational Research, 2014, *
233(3), 595-603

A compound rule that counts the number of conforming
units inspected between the (i1)^{th} and the *i*^{th}
nonconforming items, as well as the number of conforming
units observed between the (i2) and the i^{th} nonconforming
items was proposed.

19 Zhang et al. Title: Performance of

cumulative count of
conforming charts of variable
sampling intervals with
estimated control limits
Journal: *International Journal *
*of * *Production * *Economics, *
2014, 150, 114-124

This paper investigated the performance of the CCC chart with variable sampling intervals when the control limits are estimated. The performance of the chart was evaluated using the ATS and SDATS criteria.

20 Lee and Khoo Variable sampling interval cumulative count of conforming items with runs rules

Journal: *Communications in *
*Statistics - Simulation and *
*Computation, * 2015, 44(9),
2410-2430

A combination of runs rules with the VSI feature by considering the lower sided CCC chart was presented. The findings reveal that the sensitivity of the CCC chart can be improved via the addition of runs rules, as well as varying the sampling intervals.

An overview of control charts based on the time between events, in which the study takes into account of the cumulative quantity control and CCC charts was provided.

22 Fallahnezhad and Golbafian

Economic design of cumulative count of conforming control charts based on average number of inspected items

Journal: Scientia Iranica, 2017, 24(1), 330-341

A mathematical model based on the average number of inspected items for the economic design of the CCC-r chart in minimizing the average cost per item was suggested.

Additionally, sensitivity analyses with respect to the Type-I and Type-Type-IType-I errors were conducted.

23 Zhang et al. CCC-r charts’ performance with estimated parameter for high quality process number of observations to signal and standard deviation of the average number of observations to signal criteria was studied.

**2.3.1 Basic CCC Chart **

The CCC chart was originally introduced by Calvin (1983) and was further enhanced by Goh (1987). The CCC chart is based on the time between defects (Albers, 2010).

Let X be the cumulative count of items inspected until a nonconforming item
is observed. Then *X is said to have a geometric distribution with parameter p. The *
probability mass function (pmf) and cumulative distribution function (cdf) of X are

###

^{1}

( ) 1 * ^{x}* , for 1, 2,...

*g x* *p* *p* ^{} *x* (2.1)

and

###

^{1}

###

where g(x) and G(x) are the pmf and cdf of X, respectively.

Note that the traditional control limits cannot be applied on the basic CCC chart
because the geometric distribution is highly skewed. Instead, the appropriate control
limits to be used are the probability control limits (Xie and Goh, 1997). Since the
geometric distribution is discrete, the control limits will be rounded to integers and any
point that falls on (or beyond) the upper control limit (UCL) or lower control limit
(LCL) will result in an out-of-control signal. Consequently, the *UCL and LCL are *
obtained by solving the following equation:

###

Pr Pr

*X* *UCL* *X* *LCL* 2

, (2.3)

where α denotes an acceptable probability of a false alarm by the CCC chart. From Equation (2.3), the following is obtained:

###

^{1}

###

^{1}

###

^{1}

###

^{1}

^{1}

2

* ^{UCL}*

*G UCL* *p* _{. } (2.4)

By solving Equation (2.4), the UCL of the CCC chart is obtained as (Liu et al. 2006)

###

In a similar way, the LCL of the CCC chart is obtained as follows:

###

^{1}

###

^{1}

###

It is important to note that the decision procedure with respect to an out-of-control condition on the CCC chart is slightly different from that of the traditional control charts. Thus, if a point in the CCC chart is equal to or exceeds the UCL, the process is believed to have improved. Meanwhile, if a point in the CCC chart is equal to or less than the LCL, the process is deemed to have deteriorated (Xie et al., 1999).

The ARL of the basic CCC chart is given as (Liu et al., 2006)

###

###

^{1}

ARL 1

Pr Pr

1 ,

1 1 1 ^{}

* ^{LCL}*

^{UCL}*X* *LCL* *X* *UCL*

*p* *p*

(2.6)

where *UCL and LCL are defined in Equations (2.5a) and (2.5b), respectively. *

Additionally, the ATS of the basic CCC chart is obtained as (Liu et al., 2006)
*ATS* *ARL* *d*

*p* , (2.7)
The in-control and out-of-control ATSs are obtained using Equation (2.7) when
*p* *p*0 and *p* *p*_{1}, respectively, where d is the sampling interval length of the basic
CCC chart. Some CCC type control charts include those proposed by Ohta et al.

(2001), Ranjan et al. (2003), Noorossana et al. (2009), Amiri and Khosravi (2012 and 2013), Bersimis et al. (2014), Ali et al. (2016), Fallahnezhad and Golbafian (2017), and Zhang et al. (2019).

**2.3.2 VSI CCC Chart **

The VSI *X* chart was proposed by Reynolds et al. (1998). They demonstrated
that the VSI *X* chart was more efficient than the Shewhart *X* chart. Similarly, Prabhu
et al. (1993) and Costa (1994) introduced VSS schemes for the*X* chart, where a

smaller sample size is used for taking the next sample if the current *X* value is close
to the center line, while a larger sample size is considered otherwise.

Additionally, the variable sampling techniques can be used to improve the efficiency of attributes control charts (Vaughan, 1993; Epprecht and Costa, 2001).

Epprecht et al. (2003) investigated a general model regarding adaptive c, np, u and p charts, where one, two or three design parameters (sample size, sampling interval and control limit width) can be used. The design allows switching between two values.

General guidelines for selecting effective design schemes were also provided in the
study. Furthermore, Wu and Luo (2004) investigated the optimal design of VSI, VSS
and VSIVSS *np control charts, especially for detecting small or moderate process *
shifts.

Previous studies on the economic design of control charts indicated that the
VSI control chart exhibits a better performance than the fixed sampling interval (FSI)
chart in relation to cost. Bai and Lee (1998) conducted a study with respect to a cost
model involving the cost of a false alarm, the cost of identifying and eliminating an
assignable cause, and the cost of sampling and testing. Their study revealed that with
proper design parameters, the VSI *X* chart yielded lower expected cost per unit of time
compared with the corresponding FSI *X* chart. Also, Chen (2004) provided an
extension to this study by investigating the VSI *X* chart with non-normal data. They
provided an alternative cost model that uses the Burr distribution in the economic
design of the VSI *X* chart.

Liu et al. (2006) proposed a VSI CCC chart to increase the insensitive nature of the basic CCC chart. They considered Equation (2.1) and therefore, suggested that, since the control limits given in Equations (2.5a) and (2.5b) are rounded to integers,

the true-false alarm rate is unlikely to be exactly equal to the given value of , hence,