Bayesian Control Chart for Number of Defects in Production Quality Control

Abstract

This study investigates the extension of the c-chart control chart to Bayesian methodology, utilizing the gamma distribution to establish control limits. By comparing the performance of the Bayesian approach with that of two existing methods (the traditional frequentist method and the Bayesian with Jeffreys method), we assess its effectiveness in terms of the average run lengths (ARLs) and false alarm rates (FARs). Simulation results indicate that the proposed Bayesian method consistently outperforms the existing techniques, offering larger ARLs and smaller FARs that closely approximate the expected nominal values. While the Bayesian approach excels in most scenarios, challenges may arise with large values of the λ parameter, necessitating adjustments to the hyperparameters of the gamma prior. Specifically, smaller values of the rate parameter are recommended for optimal performance. Overall, our findings suggest that the Bayesian extension of the c-chart provides a promising alternative for enhanced process monitoring and control.

1. Introduction

The control chart, a fundamental tool in quality control, is utilized in production management to assess the stability of manufacturing or business processes. Often referred to as Shewhart charts, they were pioneered by Walter A. Shewhart in the early 1920s and remain widely employed today. These charts serve as graphical devices for statistical process monitoring, with the aim of enhancing or maintaining process quality by reducing the variability in the product or service characteristics. Traditionally, they are primarily used to monitor process parameters when the underlying process distributions are well understood. The versatility of control charts is evident in their ability to handle various process characteristics, including numeric data, under the assumption of a normal distribution or non-Gaussian distributions such as binomial or Poisson data.

The c-chart is recognized as a foundational tool within the realm of control charts for the tracking of nonconformities. It finds widespread application across various industries, serving to monitor and uphold the process integrity by tallying the occurrence of defects or nonconformities within a sample unit of production. Through the systematic recording and analysis of these data over time, manufacturing operators can effectively identify trends and deviations from the expected performance and take proactive measures to enhance process reliability and product quality.

The primary objective when employing the c-chart is to distinguish between common cause variation arising from inherent fluctuations within the process and special cause variation arising from identifiable factors outside the norm. By establishing control limits on the c-chart, operators and quality control personnel can discriminate between random fluctuations and significant changes in process performance. This differentiation is critical to making informed decisions, implementing corrective actions, and ultimately ensuring that products meet or exceed customer expectations.

The establishment of control limits on the c-chart is typically based on the estimation of the average number of nonconformities in an initial sample $\lambda$. It is denoted by $\overline{\lambda }$. The control limits can be defined as follows [1]:

$$\begin{array}{l}UCL=\overline{\lambda }+3\sqrt{\overline{\lambda }},\\ \hspace{0.17em}CL=\overline{\lambda },\\ LCL\hspace{0.17em}=\overline{\lambda }-3\sqrt{\overline{\lambda }}.\end{array}$$

(1)

The statistical boundaries mentioned above help to determine whether a process is in a state of statistical control, even when the value of λ is unknown, by identifying when the number of defects per unit is significantly lower or higher than expected. Deviations beyond these control limits suggest the presence of special causes of variation, which may require investigation and corrective action. By effectively utilizing the c-chart, organizations can maintain a high level of process control, swiftly detect issues, and make data-driven improvements, ensuring consistent product quality and operational efficiency.

The aforementioned control limits of control charts were proposed based on their performance through the evaluation of the average run length (ARL) values. The ARL is a statistical measurement commonly used in quality control and process monitoring; it considers the number of observations taken before an out-of-control signal is detected. A higher ARL is desired when a process is in control, as it leads to fewer false alarm signals.

However, even when a process operates within the expected parameters, there is an approximately 0.27% probability that a data point will exceed the 3-sigma control limits. The run lengths follow a geometric distribution [2]. Consequently, even a well-monitored process depicted on a well-constructed control chart may trigger a warning regarding the potential existence of an unusual occurrence, even if no event has occurred. For the Shewhart control chart using 3-sigma limits, this false alarm rate occurs, on average, once every 1/0.0027 or 370.4 observations. Therefore, the average run length (ARL) of a Shewhart chart under normal conditions is 370.4.

Many researchers have utilized the average run lengths (ARLs) to compare the performance of control charts, as the average typically follows a geometric distribution for both the classical and Bayesian approaches. Chakraborti and Human [3] proposed the false alarm rate (FAR) and ARL of the c-chart through the classical estimation of the true unknown average number of nonconformities in an inspection unit, denoted by ‘c’. Their results revealed significant differences in the performance of the chart in terms of both the FAR and the in-control ARL. Particularly in cases where ‘c’ is small, the actual FAR and in-control ARL can deviate substantially from the nominal expected values, such as 0.0027 for FAR or 370.4 for ARL. Subsequently, Raubenheimer and Merwe [4] extended the conventional operation of the c-chart by introducing a Bayesian approach. They used a noninformative Jeffreys prior to derive the predictive density in order to obtain the upper and lower control limits of the chart. A simulation study compared the unconditional ARLs and FARs using their proposed Bayesian method with the frequentist approach [3]. The upper and lower control limits were calculated for given values of inspection unit m and parameter λ, consistent with those used by Chakraborti and Human. The results indicated that the Bayesian approach yielded larger values of the unconditional ARL and smaller values of the unconditional FAR than the classical method for most variations in m and λ, with the exception of λ values of 8, 10, and 20.

Bayesian methods have been recommended due to their effectiveness over classical methods in predictive tasks and uncertainty management, their flexibility in accommodating complex models and sequential situations, and the ease of integrating prior information, as described by Bayrri and Garcia-Donato [5]. Several studies by Calabrese [6], Taylor [7], and Taylor [8] also support the superiority of Bayesian methods over classical methods regarding action decisions, the sampling size, and the frequency, based on posterior probability determination during out-of-control states. Additionally, Bayesian approaches have been applied to control charts for the mean of a normal distribution, utilizing predictive distributions to derive rejection regions and construct control charts, as proposed by Menzefricke [9,10]. This approach has been extended to the development of combined exponentially weighted moving average charts for the mean and variance of a normal distribution, using Bayesian methods for chart construction, as proposed by Menzefricke [11]. Saghir [12,13] investigated the X-bar chart for normality and the S2-chart for variance, using a Bayesian framework for the characterization of uncertainty, and compared it with the frequentist design structure. The proposed Bayesian design structure demonstrated superior performance in detecting shifts in the process parameters. Consequently, the Bayesian method was applied to construct control charts using a normal prior [14], exponential prior [15], and uniform prior [16]. An exponentially weighted moving average (EWMA) control chart was utilized for the monitoring of the variance of a distribution-free process [17], as well as to implement a ranked set sampling procedure with measurement errors in industrial engineering [18,19]. Recently, Alshahrani et al. [20] used the Bayesian framework to identify posterior and predictive densities for the construction of control limits for the monitoring of the Maxwell scale parameter. Their proposed method was compared with existing control charts and performed well in monitoring the Maxwell scale parameter.

As demonstrated by the aforementioned studies, there is growing interest in enhancing the efficiency of the c-chart for the monitoring of nonconformities using Bayesian approaches. Of particular interest is the development of Bayesian methods based on informative gamma priors, which are well suited for the modeling of the Poisson distribution [21,22]. These methods leverage predictive distribution techniques to achieve superior performance. This study’s findings will be compared with those obtained using frequentist approaches and the methods proposed by Raubenheimer and Merwe. Specifically, variations in the parameters of the gamma prior will be explored in terms of their impact on the average run lengths (ARLs) and false alarm rates (FARs).

2. Predictive Density of C-Chart

If individual inspection units are randomly chosen at evenly spaced intervals of time, the number of nonconformities in the i^th inspection will adhere to a Poisson distribution with parameter and is given by

$$f\left(x_i|\lambda \right)=\frac{e^{-\lambda }{\lambda }^{x_i}}{x_i!},\hspace{0.17em}{x}_i=0,1,2,\dots \hspace{0.17em},i=1,\dots ,n;\hspace{0.17em}\lambda >0.$$

(2)

In this study, the utilization of an informative conjugate prior is deemed suitable for the Bayesian approach. Let us assume that a random variable, denoted as $X$, follows a gamma distribution with parameters a and b, noted as X~$Gamma(a,b)$. Supharakonsakun [23] and Song and Kim [24] propose the following informative prior for the Poisson distribution:

$$g\left(\lambda |a,b\right)\hspace{0.17em}=\hspace{0.17em}\frac{b^a}{\Gamma \left(a\right)}{\lambda }^{a-1}{e}^{-b\lambda }\hspace{0.17em};\hspace{0.17em}a,b>0.$$

(3)

The posterior distribution can be derived as follows:

$$h\left(\lambda |\underset{\_}{X}\right)\hspace{0.17em}=\hspace{0.17em}\frac{L\left(\lambda \right)g\left(\lambda |a,b\right)}{{\displaystyle \int L\left(\lambda \right)g\left(\lambda |a,b\right)d\lambda }},$$

(4)

where $L\left(\lambda \right)$ represents the likelihood function of the Poisson mass probability function.

Therefore, the posterior distribution can be presented as follows:

$$h\left(\lambda |\underset{\_}{X}\right)\hspace{0.17em}=\hspace{0.17em}\frac{{\left(n+b\right)}^{{\displaystyle \sum _{i=1}^n{x}_i+a}}}{\Gamma \left({\displaystyle \sum _{i=1}^n{x}_i+a}\right)}{\lambda }^{{\displaystyle \sum _{i=1}^n{x}_i+a-1}}{e}^{-\left(n+b\right)\lambda }.$$

(5)

The posterior distribution of the parameter $\lambda$ is a gamma distribution, denoted as $Gamma({\displaystyle \sum _{i=1}^n{X}_i}+a,n+b)$. It can be expressed in the following form:

$${\pi }_j(\lambda |data)=\hspace{0.17em}\frac{{\left(n+b\right)}^{{\displaystyle \sum _{i=1}^n{x}_i+a}}}{\Gamma \left({\displaystyle \sum _{i=1}^n{x}_i+a}\right)}{e}^{-\left(n+b\right)\lambda }{\lambda }^{{\displaystyle \sum _{i=1}^n{x}_i+\alpha -1}}.$$

(6)

The unconditional predictive density can be obtained as follows [25]:

$$f\left(x_f|data\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}f\left(x_f|\lambda \right)}\hspace{0.17em}{\pi }_j(\lambda |data)d\lambda ,$$

(7)

where $X_f$ is the number of nonconformities in a future inspection unit.

Here,

$$\begin{array}{ll}f\left(x_f|data\right)& ={\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{e^{-\lambda }{\lambda }^{X_f}}{X_f!}}\frac{{\left(n+\beta \right)}^{{\displaystyle \sum _{i=1}^n{X}_i+\alpha }}}{\Gamma \left({\displaystyle \sum _{i=1}^n{X}_i+\alpha }\right)}{e}^{-\left(n+\beta \right)\lambda }{\lambda }^{{\displaystyle \sum _{i=1}^n{X}_i}+\alpha -1}d\lambda \\ & =\frac{{\left(n+b\right)}^{{\displaystyle \sum _{i=1}^n{X}_i+a}}}{X_f!\Gamma \left({\displaystyle \sum _{i=1}^m{X}_i+a}\right)}{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\left(n+b+1\right)\lambda }{\lambda }^{{\displaystyle \sum _{i=1}^n{X}_i+X_f+a-1}}}d\lambda \\ & =\frac{{\left(n+b\right)}^{{\displaystyle \sum _{i=1}^n{X}_i+\alpha }}\Gamma \left({\displaystyle \sum _{i=1}^n{X}_i+X_f+a}\right)}{X_f!\Gamma \left({\displaystyle \sum _{i=1}^n{X}_i+a}\right){\left(n+b+1\right)}^{{\displaystyle \sum _{i=1}^n{X}_i+X_f}+a}}{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{{\left(n+b+1\right)}^{{\displaystyle \sum _{i=1}^n{X}_i+X_f}+\alpha }{e}^{-\left(n+b+1\right)\lambda }\cdot {\lambda }^{{\displaystyle \sum _{i=1}^n{X}_i+X_f+a-1}}}{\Gamma \left({\displaystyle \sum _{i=1}^n{X}_i+X_f+a}\right)}}\hspace{0.17em}d\lambda .\end{array}$$

Hence,

$$f\left(x_f|data\right)=\frac{{\left(n+b\right)}^{{\displaystyle \sum _{i=1}^n{X}_i+\alpha }}\Gamma \left({\displaystyle \sum _{i=1}^n{X}_i+X_f+a}\right)}{X_f!\Gamma \left({\displaystyle \sum _{i=1}^n{X}_i+a}\right){\left(n+b+1\right)}^{{\displaystyle \sum _{i=1}^n{X}_i+X_f}+a}}.$$

(8)

The aforementioned expression can be reformulated in the form of a predictive density, as shown below:

$$f\left(x_f|data\right)=\frac{\left({\displaystyle \sum _{i=1}^n{X}_i+X_f+a-1}\right)!}{\left({\displaystyle \sum _{i=1}^n{X}_i+a-1}\right)!\left(X_f\right)!}{\left(\frac{n+b}{n+b+1}\right)}^{{\displaystyle \sum _{i=1}^n{X}_i+a}}{\left(\frac{1}{n+b+1}\right)}^{X_f}.$$

(9)

Specifically,

$$f\left(x_f|data\right)=\left(\begin{array}{c}{\displaystyle \sum _{i=1}^n{X}_i+X_f+a-1}\\ X_f\end{array}\right){\left(\frac{n+b}{n+b+1}\right)}^{{\displaystyle \sum _{i=1}^n{X}_i+a}}{\left(\frac{1}{n+b+1}\right)}^{X_f}.$$

(10)

Therefore, the predictive density follows a negative binomial distribution with parameters ${\displaystyle \sum _{i=1}^n{X}_i}+a$ and $\frac{n+b}{n+b+1}$. It can be denoted as

$$X_f~NB\left({\displaystyle \sum _{i=1}^n{X}_i+a,\frac{n+b}{n+b+1}}\right)$$

(11)

To establish the c-chart, the predictive density is utilized to determine the upper and lower control limits through simulation studies. The parameters of $\lambda$ and inspection unit n will be varied in this process.

3. Results

This study aims to compare the unconditional average run lengths (ARLs) and unconditional false alarm rates (FARs) using the classical method, the Bayesian approach with the Jeffreys prior, and the proposed method. The variations in inspection unit n and parameter λ when calculating the lower and upper control limits follow the methodology outlined by Raubenheimer and Merwe (B_J). The control limits are derived from the predictive density provided by the Jeffreys and proposed Bayesian approaches. Meanwhile, the frequentist method, as described by Chakraborti and Human (F), is employed to obtain the control limits.

The numerical simulation study considers parameter values of n = 5, 10, 15, 20, 25, 30, 50, 100, 200 and λ = 1, 2, 3, 4, 5, 8, 10, 15, 20, 50. The hyperparameters of the gamma prior for the proposed Bayesian method are set to (a, b) = (5, 0.25) and (5, 0.5). These parameter values are applied in the proposed Bayesian method to determine the unconditional average run lengths (ARLs) and false alarm rates (FARs). The simulations are repeated 20,000 times for robustness.

The results of all three methods are presented in

Table 1. Unconditional ARLs given (a, b) = (5, 0.25).

λ	F	BJ	BG	F	BJ	BG	F	BJ	BG
		n = 5			n = 10			n = 15
1	2.5213	2.4565	2.6787	2.5927	2.5489	2.6723	2.6169	2.5894	2.6679
2	6.6119	6.3843	7.0773	6.8370	6.6884	7.0620	6.8810	6.7807	7.0383
3	16.6544	15.9202	18.2051	17.3270	16.8506	18.1016	17.4702	17.1052	18.0182
4	39.2982	38.0653	44.8697	40.8478	39.7670	44.1196	41.4880	40.5102	43.6796
5	87.7513	86.3532	105.0799	87.7049	87.2830	99.4738	87.9545	87.6690	96.5618
8	454.9500	494.6330	599.6453	383.5887	480.5528	556.3826	330.5614	415.8965	475.5315
10	398.9501	382.9649	432.2839	382.6195	409.4388	448.5112	354.0101	397.3419	430.3330
15	323.9318	284.3721	294.5420	343.5854	330.6152	339.2149	340.5875	340.7814	348.3136
20	303.0356	258.3695	258.3695	330.1730	305.3085	304.7872	340.5875	320.0669	315.7853
50	257.3339	209.1611	189.9394	294.7512	257.1939	237.9175	312.0903	280.9855	264.5450
		n = 20			n = 25			n = 30
1	2.6267	2.6061	2.6647	2.6348	2.6189	2.6638	2.6367	2.6240	2.6617
2	6.9377	6.8562	7.0468	6.9360	6.8745	7.0312	6.9612	6.9132	7.0357
3	17.6675	17.3812	18.0483	17.6565	17.4120	18.0166	17.7847	17.5371	18.0392
4	41.8971	41.1831	43.6275	42.2358	41.4934	43.4372	42.1687	41.5009	43.2472
5	87.5315	87.4401	94.6250	87.5459	87.4412	92.9462	88.0608	88.0290	92.9611
8	320.0732	391.8287	436.8171	302.7570	364.3714	402.3618	293.0335	344.1186	374.0170
10	353.4235	388.8555	416.8962	344.5182	379.2704	402.8741	336.8641	367.9865	386.4273
15	338.9864	343.0262	351.6103	338.5060	344.1869	350.5746	334.8327	344.5416	346.9817
20	340.2283	331.0582	332.7987	337.3543	333.3624	331.6497	335.0866	334.1162	332.5582
50	321.2072	295.6685	280.1818	326.9583	304.7529	292.0903	331.7540	311.6192	298.6964
		n = 50			n = 100			n = 200
1	2.6393	2.6329	2.6572	2.6398	2.6401	2.6527	2.6390	2.6394	2.6485
2	6.9886	6.9554	7.0286	7.0307	6.9939	7.0339	7.0820	7.0363	7.0558
3	17.8675	17.7380	18.0204	17.9925	17.8715	18.0304	18.1618	17.9982	18.0830
4	42.4864	42.1223	43.1579	42.6179	42.5305	43.1355	42.5747	42.5554	42.9669
5	88.7302	88.7932	91.9146	88.1357	88.1357	90.4300	86.1348	88.0959	89.0128
8	277.0461	306.9855	324.1538	261.3801	277.1624	284.7046	252.8359	263.1057	266.9809
10	322.4659	344.2188	356.2091	307.6547	322.7460	331.3688	294.3781	308.2915	310.5624
15	332.9095	340.9520	342.4873	324.5143	331.8624	334.5469	314.0639	314.0639	325.2001
20	334.6021	337.9960	336.2559	335.1888	335.9407	336.0102	333.4542	333.2912	332.8310
50	338.8756	327.2756	317.3416	345.4461	338.7309	334.0351	348.5869	344.8088	341.6380

Note: Bold values indicate the maximal ARL for the method.

,

Table 2. Unconditional FARs given (a, b) = (5, 0.25).

λ	F	BJ	BG	F	BJ	BG	F	BJ	BG
		n = 5			n = 10			n = 15
1	0.39662270	0.40708550	0.40708550	0.38569400	0.39232380	0.37421000	0.38212840	0.38618380	0.37482830
2	0.15124300	0.15663420	0.14129660	0.14626240	0.14951160	0.14160280	0.14532670	0.14747760	0.14208050
3	0.06004426	0.06281317	0.05492979	0.05771337	0.05934518	0.05524374	0.05724033	0.05846155	0.05549938
4	0.02544645	0.02627064	0.02228677	0.02448115	0.02514650	0.02266568	0.02410337	0.02468512	0.02289396
5	0.01139585	0.01158035	0.00951657	0.01140187	0.01145698	0.01005290	0.01136951	0.01136951	0.01035606
8	0.00219804	0.00202170	0.00166765	0.00260696	0.00208094	0.00179732	0.00302516	0.00240444	0.00210291
10	0.00250658	0.00261121	0.00231329	0.00261356	0.00244237	0.00222960	0.00282478	0.00251673	0.00232378
15	0.00308707	0.00351652	0.00339510	0.00291048	0.00302467	0.00294798	0.00293610	0.00293443	0.00287098
20	0.00329994	0.00387043	0.00388933	0.00302872	0.00327538	0.00328098	0.00300033	0.00312435	0.00316671
50	0.00388600	0.00478100	0.00526484	0.00339269	0.00388812	0.00420314	0.00320420	0.00355890	0.00378008
		n = 20			n = 25			n = 30
1	0.38070290	0.38371150	0.37527970	0.37953070	0.38184120	0.37540080	0.37926550	0.38109870	0.37569620
2	0.14414100	0.14585320	0.14190820	0.14417550	0.14546450	0.14222320	0.14365370	0.14464900	0.14213290
3	0.05660097	0.05753329	0.05540701	0.05663632	0.05743179	0.05550451	0.05622827	0.05543484	0.05543484
4	0.02386803	0.02428183	0.02292132	0.02367660	0.02410020	0.02302176	0.02371426	0.02409589	0.02312288
5	0.01142446	0.01143640	0.01056803	0.01142258	0.01143625	0.01075891	0.01135580	0.01135989	0.01075718
8	0.00312429	0.00255214	0.00228929	0.00330298	0.00274445	0.00248533	0.00341258	0.00290598	0.00267368
10	0.00282947	0.00257165	0.00239868	0.00290260	0.00263664	0.00248217	0.00296856	0.00271749	0.00258781
15	0.00294997	0.00291523	0.00284406	0.00295416	0.00290540	0.00285246	0.00298657	0.00290241	0.00258781
20	0.00293920	0.00302062	0.00300482	0.00296424	0.00299974	0.00301523	0.00298430	0.00299297	0.00300699
50	0.00311326	0.00338217	0.00356911	0.00305849	0.00328135	0.00342360	0.00301428	0.00320905	0.00334788
		n = 50			n = 100			n = 200
1	0.37889310	0.37980500	0.37633970	0.37882120	0.37877600	0.37698020	0.37892750	0.37886810	0.37757490
2	0.14309070	0.14377280	0.14227490	0.14223350	0.14298200	0.14216920	0.14120300	0.14212070	0.14172810
3	0.05596745	0.05637619	0.05549266	0.05557858	0.05595490	0.05546201	0.05506071	0.05556127	0.05530066
4	0.02353694	0.02374041	0.02317074	0.02346430	0.02351255	0.02318279	0.02348811	0.02349877	0.02327374
5	0.01127012	0.01126212	0.01087967	0.01134614	0.01127120	0.01105828	0.01160971	0.01135127	0.01123434
8	0.00360951	0.00325748	0.00308496	0.00382585	0.00360799	0.00351241	0.00395513	0.00380075	0.00374559
10	0.00310110	0.00290513	0.00280734	0.00325040	0.00309841	0.00301779	0.00339699	0.00324368	0.00321997
15	0.00300382	0.00293296	0.00291982	0.00308153	0.00301330	0.00298912	0.00318407	0.00307384	0.00307503
20	0.00298863	0.00295861	0.00297393	0.00298339	0.00297672	0.00297610	0.00299891	0.00300038	0.00300453
50	0.00295094	0.00305553	0.00315118	0.00289481	0.00295220	0.00299370	0.00286873	0.00290016	0.00292708

Note: Bold values indicate the minimal ARL for the method.

,

Table 3. Unconditional ARLs given (a, b) = (5, 0.5).

λ	F	BJ	BG	F	BJ	BG	F	BJ	BG
		n = 5			n = 10			n = 15
1	2.5233	2.4572	2.6748	2.5902	2.5443	2.6675	2.6174	2.5896	2.6643
2	6.6229	6.4023	7.0214	6.8260	6.6835	7.0203	6.8738	6.7870	7.0060
3	16.6449	16.6449	17.8280	17.3424	16.8308	17.9019	17.4687	17.1058	17.8753
4	39.3661	37.8868	43.1480	40.7582	39.8308	42.8859	41.6155	40.8729	43.0164
5	87.5318	86.5160	98.0466	87.5600	87.0566	94.7797	87.6750	87.7005	93.0653
8	456.9219	493.1376	518.5925	375.8899	468.6201	480.5386	332.8215	419.2790	426.7985
10	402.4407	387.8796	381.4358	378.7237	409.0835	396.0685	358.4815	398.2550	388.9642
15	326.0085	284.4770	258.4064	340.9400	329.3362	298.6948	338.5840	338.4772	315.4739
20	303.9213	259.1802	221.6345	329.7544	303.4682	269.3523	332.0111	318.7007	282.2210
50	256.3661	210.5426	140.2344	296.0692	258.2642	200.4727	313.9835	283.3336	233.0509
		n = 20			n = 25			n = 30
1	2.6288	2.6068	2.6631	2.6336	2.6167	2.6599	2.6366	2.6248	2.6601
2	6.9355	6.8503	7.0232	6.9377	6.8787	7.0113	6.9631	6.9119	7.0211
3	17.6310	17.3377	17.9002	17.6617	17.3994	17.8630	17.7455	17.5314	17.9095
4	41.8540	41.0597	42.8223	42.1195	41.4906	42.8501	42.1345	41.5007	42.7617
5	87.8871	87.7707	92.3624	87.7346	87.7406	91.1625	87.6568	88.0440	90.7840
8	393.3209	393.3209	397.9287	302.4668	361.1734	367.9019	290.8101	342.4454	344.9910
10	356.7776	394.4178	384.6940	344.6354	380.6124	373.1827	332.6781	360.8223	359.1106
15	338.4139	343.7880	320.2326	339.4823	345.9631	327.5580	333.3335	340.9038	324.1026
20	337.3600	331.5635	301.3750	336.1248	332.8756	305.8301	334.6695	333.6483	310.6800
50	320.6388	294.8311	252.0554	326.8175	305.9600	267.2054	331.6683	312.3347	278.4506
		n = 50			n = 100			n = 200
1	2.6381	2.6318	2.6549	2.6398	2.6397	2.6512	2.6390	2.6393	2.6482
2	6.9853	6.9522	7.0217	7.0317	6.9953	7.0317	7.0826	7.0395	7.0572
3	17.8699	17.7442	17.9680	17.9892	17.8775	17.9930	18.1734	18.0104	18.0739
4	42.5169	42.1612	42.8746	42.6079	42.4718	42.8995	42.6216	42.6172	42.8391
5	88.8421	88.5996	90.6512	88.1742	88.8398	89.8989	86.1159	87.9510	88.4350
8	276.7123	307.2505	311.8946	262.5758	278.5654	279.9313	251.6594	262.0512	263.2078
10	323.3059	345.7096	342.3620	309.8530	325.9426	323.4057	294.9882	308.7353	307.5628
15	333.8996	340.5693	329.5273	324.8547	332.6512	325.3336	314.2074	324.1764	321.4006
20	336.0491	337.7370	321.4377	334.4073	333.7946	326.4139	333.9665	334.1646	328.1677
50	338.8835	326.4732	302.0031	345.7355	339.9024	323.9124	349.7454	346.0734	337.4349

Note: Bold values indicate the maximal ARL for the method.

and

Table 4. Unconditional FARs given (a, b) = (5, 0.5).

λ	F	BJ	BG	F	BJ	BG	F	BJ	BG
		n = 5			n = 10			n = 15
1	0.39630460	0.40697220	0.37386390	0.38606500	0.39303320	0.37487780	0.38205250	0.38615420	0.37532930
2	0.15099010	0.15619350	0.14242240	0.14649910	0.14962180	0.14244380	0.14547910	0.14734040	0.14273460
3	0.06007845	0.06274271	0.05609142	0.05766229	0.05941489	0.05586000	0.05724533	0.05724533	0.05594302
4	0.02540255	0.02639445	0.02317604	0.02453494	0.02510619	0.02331767	0.02402952	0.02446612	0.02324695
5	0.01142442	0.01155856	0.01019923	0.01142074	0.01148678	0.01055079	0.01140576	0.01140244	0.01074515
8	0.00218856	0.00202783	0.00192830	0.00266035	0.00213393	0.00208100	0.00300461	0.00238505	0.00234303
10	0.00248484	0.00257812	0.00262167	0.00264045	0.00244449	0.00252482	0.00278955	0.00251095	0.00257093
15	0.00306741	0.00351522	0.00386987	0.00293307	0.00303641	0.00334790	0.00295348	0.00295441	0.00316983
20	0.00329033	0.00385832	0.00451193	0.00303256	0.00329524	0.00371261	0.00301195	0.00313774	0.00354332
50	0.00390067	0.00474963	0.00713092	0.00337759	0.00387201	0.00498821	0.00318488	0.00352941	0.00429091
		n = 20			n = 25			n = 30
1	0.38039740	0.38361130	0.37550780	0.37970210	0.38215570	0.37596040	0.37928360	0.38097830	0.37592760
2	0.14418640	0.14597810	0.14238510	0.14413970	0.14537660	0.14262610	0.14361330	0.14467830	0.14242800
3	0.05671837	0.05767788	0.05586524	0.05661960	0.05747328	0.05598164	0.05635218	0.05704039	0.05583628
4	0.02389257	0.02435477	0.02335234	0.02374195	0.02410183	0.02333716	0.02373350	0.02409597	0.02409597
5	0.01137823	0.01139332	0.01082692	0.01139802	0.01139723	0.01096943	0.01140813	0.01135796	0.01101516
8	0.00313724	0.00254245	0.00251301	0.00330615	0.00276875	0.00271812	0.00343867	0.00292017	0.00289863
10	0.00280287	0.00253538	0.00259947	0.00290162	0.00262735	0.00267965	0.00300591	0.00277145	0.00278466
15	0.00295496	0.00290877	0.00312273	0.00294566	0.00289048	0.00305289	0.00300000	0.00293338	0.00308544
20	0.00296419	0.00301601	0.00331813	0.00297509	0.00300413	0.00326979	0.00298802	0.00299717	0.00321869
50	0.00311877	0.00339177	0.00396738	0.00305981	0.00326840	0.00374244	0.00301506	0.00320169	0.00359130
		n = 50			n = 100			n = 200
1	0.37906670	0.37996640	0.37666900	0.37881600	0.37883430	0.37718550	0.37893600	0.37889080	0.37760990
2	0.14315800	0.14384040	0.14241610	0.14221330	0.14295410	0.14221220	0.14119100	0.14205480	0.14170010
3	0.05595995	0.05635656	0.05565437	0.05558894	0.05593614	0.05557725	0.05502545	0.05552335	0.05552335
4	0.02352006	0.02352006	0.02332383	0.02346981	0.02354502	0.02331030	0.02346227	0.02346469	0.02334315
5	0.01125592	0.01128673	0.01103129	0.01134119	0.01125622	0.01112361	0.01161226	0.01136997	0.01130774
8	0.00361386	0.00325467	0.00320621	0.00380842	0.00358982	0.00357231	0.00397363	0.00381605	0.00379928
10	0.00309305	0.00289260	0.00292089	0.00322734	0.00306803	0.00309209	0.00338997	0.00323902	0.00325137
15	0.00299491	0.00293626	0.00303465	0.00307830	0.00300615	0.00307377	0.00318261	0.00308474	0.00311138
20	0.00297576	0.00296088	0.00311102	0.00299037	0.00299585	0.00306360	0.00299431	0.00299254	0.00304722
50	0.00295087	0.00306304	0.00331122	0.00289239	0.00294202	0.00308726	0.00285922	0.00288956	0.00296354

Note: Bold values indicate the minimal ARL for the method.

. The average run length (ARL) is a well-known measure used to evaluate the performance of control charts. It represents the expected number of inspection units sampled before the initial signal appears on the chart and is preferred to be as large as possible. The hyperparameters of the gamma prior are varied within (a, b) = (5, 0.25) for Table 1 and Table 2 and (a, b) = (5, 0.5) for Table 3 and Table 4 to assess their impacts on the unconditional ARL and FAR values.

Table 1 displays the unconditional average run lengths (ARLs) for various sample sizes (n = 5, 10, 15, 20, 25, 30, 50, 100 and 200). For n = 5 and 10, the Bayesian method with the gamma prior achieves larger ARLs compared to other methods across parameter values $\lambda$ = 1 to 10. In contrast, the frequentist method exhibits the maximum ARLs when parameter λ ranges from 15 to 50. For n = 15, the Bayesian method with the gamma prior outperforms the others for λ = 1 to 15, while the classical approach excels for $\lambda$ = 20 to 50. For n = 20, 25, and 30, the results indicate the excellent performance of the Bayesian method with the gamma prior for λ = 1 to 15, while the frequentist method demonstrates efficiency for λ = 20 to 50 across all sample sizes.

As the sample sizes increase to n = 50, 100, and 200, the Bayesian method with the gamma prior maintains strong performance for $\lambda$ = 1 to 15 at n = 50, with the Bayesian method using Jeffrey’s prior providing the largest ARLs for $\lambda$ = 20. Conversely, the frequentist method performs well for $\lambda$ = 50. For n = 100, the proposed method yields the largest ARLs for $\lambda$ = 1 to 20, while the frequentist method shows effectiveness for $\lambda$ = 50. Finally, at n = 200, the proposed Bayesian method achieves the maximum ARLs for $\lambda$ = 1, 4, 5, 8, 10, and 15, while the frequentist method yields the largest ARLs for $\lambda$ = 2, 3, 20, and 50.

Table 2 presents the unconditional false alarm rates (FARs) for the proposed method with hyperparameters (a, b) = (5, 0.25). The simulation results demonstrate that the proposed method offers smaller FARs compared to the existing methods for various combinations of inspection unit (n) and parameter λ. Specifically, for $\lambda$ = 8, 10, the proposed Bayesian method exhibits significantly smaller FARs, approaching the nominal value of FAR = 0.0027. However, for λ = 1 to 5, the three methods yield slightly different FAR values.

Additionally, as the inspection unit decreases, the proposed method demonstrates resilience to increasing $\lambda$ values.

Table 3 presents the unconditional average run lengths (ARLs) for varying sample sizes (n = 5, 10, 15, 20, 25, 30, 50, 100, and 200) and parameter values $\lambda$. For n = 5 and 10, the proposed Bayesian method achieves larger ARLs across parameter values $\lambda$ = 1 to 8, while the frequentist method performs best for $\lambda$ ranging from 10 to 50. Similarly, for n = 15, the proposed method excels for $\lambda$ between 1 and 8, with the Raubenheimer and Der Merwe method outperforming the others for $\lambda$ = 10, and the classical approach is superior for $\lambda$ = 15 to 50.

For sample sizes n = 20, 25, and 30, the proposed Bayesian method performs consistently well for $\lambda$ = 1 to 8, with the Bayesian method using the Jeffreys prior showing superior ARLs for $\lambda$ = 10 and 15 and the frequentist method for $\lambda$ = 20 to 50.

As the sample sizes increase to n = 50, 100, and 200, the proposed Bayesian method maintains strong performance for $\lambda$ = 1 to 8. For higher $\lambda$ values (10 to 20 and $\lambda$ = 50), the Bayesian method with the Jeffreys prior and the classical method achieve the largest ARLs, respectively. At n = 100, the proposed method dominates for $\lambda$ = 1 to 8, while the frequentist method and the Bayesian method with the Jeffreys prior perform best for $\lambda$ = 10 to 15 and $\lambda$ = 20 to 50, respectively. Finally, at n = 200, the proposed Bayesian method demonstrates the maximum ARLs for $\lambda$ = 1, 4, 5, and 8, while the Bayesian method with the Jeffreys prior and the classical method excel for $\lambda$ = 10 to 20 and $\lambda$ = 2, 3, 50.

Table 4 reports the unconditional FARs for the proposed method with hyperparameters (a, b) = (5, 0.5). The simulation results show that the proposed method consistently provides smaller FARs compared to the existing methods for various combinations of n and λ values. Specifically, for $\lambda$ = 8 or 10 with n = 25 and 30, the proposed Bayesian method demonstrates significantly smaller FARs, nearing the nominal value of FAR = 0.0027. However, as the sample size increases, the three methods yield slightly different FAR values.

4. Discussion

The extension of the c-chart to the Bayesian methodology, as demonstrated by Raubenheimer and Merwe, offers a novel approach to control limit establishment. By employing the Jeffreys prior, this method presents an alternative to the original frequentist technique introduced by Chakraborti and Human. In this study, we explore the efficacy of an informative prior for the Bayesian method, particularly utilizing the gamma distribution to derive the predictive density for the calculation of control limits.

Our simulation study evaluates the effectiveness of this proposed Bayesian approach alongside two existing methods, with variations in the parameter λ and inspection unit (n). The results consistently demonstrate the superiority of the Bayesian method, providing larger average run lengths (ARLs) and smaller false alarm rates (FARs) that closely approximate the expected nominal value of 0.0027. This superiority can be attributed to the narrower control limits employed by the Bayesian approach, enhancing its ability to detect shifts in the process.

However, it is important to address the computational aspects of Bayesian methods. Bayesian techniques are known for being computationally intensive, and our approach is no exception. In our study, we observed that the time required for the Bayesian computation increases with larger parameter values of λ (20 and 50), where the method may also not perform optimally. Specifically, the computational time for each iteration of the Bayesian control chart was recorded and compared with that of the frequentist method. The results indicated that while the Bayesian approach provided superior statistical performance, it required significantly more computational resources and time, particularly for large datasets.

In our investigation, we employed hyperparameters for the gamma prior with large shape and rate parameters (a, b) = (5, 0.25), (5, 0.5). The variation in the rate parameter b was explored to assess the impact on the performance of the Bayesian method, revealing that smaller values of b are preferable for optimal implementation. Additionally, we found that optimizing these hyperparameters helped to reduce the computational burden, although it was not completely eliminated.

To summarize, while the Bayesian method offers improved detection capabilities and more accurate control limits, it is more time-consuming compared to traditional frequentist methods. Future research could focus on optimizing the computational efficiency of Bayesian algorithms or exploring approximations that maintain the method’s advantages without the extensive computational overhead. Additionally, conducting a sensitivity analysis, if feasible, could provide valuable insights into the robustness of our findings under varying conditions, thus further enhancing the applicability and reliability of Bayesian control charts in practice.

5. Conclusions

Our study investigates the extension of the c-chart control chart to Bayesian methodology, leveraging the gamma distribution to establish control limits. We compare the performance of the Bayesian approach with that of two existing methods: the traditional frequentist approach and the Bayesian method using the Jeffreys prior. Our evaluation, based on metrics such as the average run lengths (ARLs) and false alarm rates (FARs), reveals compelling insights into the efficacy of the Bayesian extension.

Simulation results consistently demonstrate the superiority of the proposed Bayesian method over the existing techniques. We observe that the Bayesian approach achieves larger ARLs and smaller FARs, closely aligning with the expected nominal values. This superiority underscores the effectiveness of Bayesian methodology in enhancing process monitoring and control.

A key aspect of our analysis is the investigation of the impact of parameter values, particularly that of parameter λ values, on the performance of each method. While the Bayesian approach excels in most scenarios, challenges may arise with large values of λ. Through meticulous parameter tuning, we identify strategies to optimize the Bayesian method’s performance, such as recommending smaller values of the rate parameter for improved results.

Overall, our findings underscore the promise of the Bayesian extension of the c-chart as a robust and effective tool for process monitoring and control. By elucidating the nuances of parameter selection and computational considerations, our study provides actionable insights for practitioners seeking to leverage Bayesian methodology in industrial engineering applications.