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Research article Β· 2023 Β· Full text

International Journal of Statistics and Applied Mathematics

Bayesian approach to control charts for process precision

Sharada V. BhatChetan Malagavi

Volume 8(6) Β· Pages 30–40 Β· DOI: 10.22271/maths.2023.v8.i6a.1400 β†—

Research areasAdvanced Statistical Process MonitoringFault Detection and Control SystemsAdvanced Control Systems Optimization

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Abstract

Author-written

Monitoring process variation is essential to maintain quality of certain product in manufacturing industry. Coefficient of variance (CV) is a relative measure of process variation and inverse coefficient of variance (ICV) measures relative precision. In this paper, control charts for ICV under normal model using Bayesian approach is proposed. The control charts are constructed under noninformative and gamma priors. Their performance is investigated. The proposed control charts are illustrated through examples.

Summary

Source-checked

Plain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.

  1. Why it matters

    • Monitoring process variation is critical for maintaining product quality within the manufacturing industry. View evidence β†—

      Why it matters

      Monitoring process variation is critical for maintaining product quality within the manufacturing industry.

      In the paper's own words

      β€œMonitoring process variation is essential to maintain quality of certain product in manufacturing industry.”

      Page 1

      Found in: Introduction

      Read it in the full textOpen PDF at page 1 β†—
    • The inverse coefficient of variance (ICV) is used as a measure of process precision. View evidence β†—

      Why it matters

      The inverse coefficient of variance (ICV) is used as a measure of process precision.

      In the paper's own words

      β€œThe reciprocal of CV is ICV and it accounts for the measure of precision.”

      Page 1

      Found in: Introduction

      Read it in the full textOpen PDF at page 1 β†—
    • ICV control charts are useful for quality maintenance in industries producing items like surgical instruments, footwear, apparel, and jewelry. View evidence β†—

      Why it matters

      ICV control charts are useful for quality maintenance in industries producing items like surgical instruments, footwear, apparel, and jewelry.

      In the paper's own words

      β€œThe control chart for ICV is helpful in maintaining quality of certain production processes such as manufacturing of surgical instruments, footwear, gems and jewelry, apparel, etc.”

      Page 1

      Found in: Introduction

      Read it in the full textOpen PDF at page 1 β†—

  2. The question

    • The study aims to develop Bayesian control charts for the inverse coefficient of variance (ICV) under a normal model using noninformative (NI) and gamma priors. View evidence β†—

      The question

      The study aims to develop Bayesian control charts for the inverse coefficient of variance (ICV) under a normal model using noninformative (NI) and gamma priors.

      In the paper's own words

      β€œIn this paper, we focus on the development of Bayesian control charts for ICV under normal model. We use NI and gamma priors.”

      Page 2

      Found in: Introduction

      Read it in the full textOpen PDF at page 2 β†—

  3. What they did

    • The researchers derived the expected value and variance for both standard ICV and range-based ICV to construct the control charts. View evidence β†—

      What they did

      The researchers derived the expected value and variance for both standard ICV and range-based ICV to construct the control charts.

      In the paper's own words

      β€œIn section 2, we obtain 𝐸(𝜏), 𝑉(𝜏), 𝐸(πœπ‘Ÿ), 𝑉(πœπ‘Ÿ) and construct control charts for ICV.”

      Page 2

      Found in: Introduction

      Read it in the full textOpen PDF at page 2 β†—
    • The performance of the proposed control charts was evaluated based on power and average run length (ARL). View evidence β†—

      What they did

      The performance of the proposed control charts was evaluated based on power and average run length (ARL).

      In the paper's own words

      β€œWe evaluate the performance of the proposed control charts using power and average run length (ARL) in section 3”

      Page 2

      Found in: Introduction

      Read it in the full textOpen PDF at page 2 β†—
    • The analysis tested sample sizes of n=15, 20, and 25 to determine the power and ARL of the T control chart under a noninformative prior. View evidence β†—

      What they did

      The analysis tested sample sizes of n=15, 20, and 25 to determine the power and ARL of the T control chart under a noninformative prior.

      In the paper's own words

      β€œFor n=15, 20, 25, power and ARL of T control chart under NI prior are given in table 2 and plotted in figure 2.”

      Page 4

      Found in: Performance of the proposed control charts

      Read it in the full textOpen PDF at page 4 β†—

  4. What they found

    • For a fixed sample size, the power of the control chart increases as the shift in the mean (π‘₯β€²) increases and decreases as the shift in ICV (πœΜ‚β€²) increases. View evidence β†—

      What they found

      For a fixed sample size, the power of the control chart increases as the shift in the mean (π‘₯β€²) increases and decreases as the shift in ICV (πœΜ‚β€²) increases.

      In the paper's own words

      β€œFrom figure 2 and table 2, it is seen that, for a given value of n, power increases as π‘₯β€² increases and decreases as πœΜ‚β€² increases.”

      Page 4

      Found in: Performance of the proposed control charts

      Read it in the full textOpen PDF at page 4 β†—
    • For a specific shift in ICV, the power of the control chart increases as the sample size (n) increases. View evidence β†—

      What they found

      For a specific shift in ICV, the power of the control chart increases as the sample size (n) increases.

      In the paper's own words

      β€œAlso, for a specified value of πœΜ‚β€², power increases as n increases.”

      Page 4

      Found in: Performance of the proposed control charts

      Read it in the full textOpen PDF at page 4 β†—
    • Under both noninformative and gamma priors, increasing the sample size (n) leads to increased power and decreased average run length (ARL) for the T control chart. View evidence β†—

      What they found

      Under both noninformative and gamma priors, increasing the sample size (n) leads to increased power and decreased average run length (ARL) for the T control chart.

      In the paper's own words

      β€œFrom tables 2 and 3 it is observed that, power of T control chart increases and ARL decreases with increasing values of n under both priors.”

      Page 4

      Found in: Performance of the proposed control charts

      Read it in the full textOpen PDF at page 4 β†—
    • The range-based TR control chart exhibits higher power than the T control chart under both NI and gamma priors. View evidence β†—

      What they found

      The range-based TR control chart exhibits higher power than the T control chart under both NI and gamma priors.

      In the paper's own words

      β€œOn comparing T and TR control charts under NI and gamma priors, power of TR control chart is higher than power of T control chart.”

      Page 4

      Found in: Performance of the proposed control charts

      Read it in the full textOpen PDF at page 4 β†—
    • The power and ARL of T and TR control charts are nearly identical when the sample size (n) is small. View evidence β†—

      What they found

      The power and ARL of T and TR control charts are nearly identical when the sample size (n) is small.

      In the paper's own words

      β€œHowever, for smaller values of n, power and ARL of both T and TR control charts are almost equal.”

      Page 4

      Found in: Performance of the proposed control charts

      Read it in the full textOpen PDF at page 4 β†—
    • In the hard bake process example, control limits for both T and TR charts were lower under the noninformative prior compared to the gamma prior. View evidence β†—

      What they found

      In the hard bake process example, control limits for both T and TR charts were lower under the noninformative prior compared to the gamma prior.

      In the paper's own words

      β€œFrom figure 4 and table 5, it is observed that, the control limits of T and TR control charts under NI prior are lower than that of gamma prior.”

      Page 7

      Found in: Example 1: Hard bake process

      Read it in the full textOpen PDF at page 7 β†—

  5. What it means

    • The authors conclude that the TR control chart is superior to the T control chart for moderate and large sample sizes. View evidence β†—

      What it means

      The authors conclude that the TR control chart is superior to the T control chart for moderate and large sample sizes.

      In the paper's own words

      β€œTR control chart outperforms T control chart for moderate and large values of n.”

      Page 7

      Found in: Conclusions

      Read it in the full textOpen PDF at page 7 β†—
    • The T control chart is recommended when prior information on mean and variance is available, while the TR control chart is recommended when only extreme observations are available. View evidence β†—

      What it means

      The T control chart is recommended when prior information on mean and variance is available, while the TR control chart is recommended when only extreme observations are available.

      In the paper's own words

      β€œT control chart is useful when process precision is of prime concern and practitioners have prior information about the process mean and variance, whereas, TR control chart is helpful when sample information is available in terms of extreme observations.”

      Page 7

      Found in: Conclusions

      Read it in the full textOpen PDF at page 7 β†—

Visual summary

3 visuals

Illustrations explain the paper; charts are replotted from the paper's own table values.

AI-generated illustration
AI-generated conceptual illustration (Industrial semiconductor wafers on a cleanroom conveyor belt). It explains the idea and does not show study data.β†— Enlarge
Decision Logic for ICV Control Chart Selection
AI-generated conceptual illustration (Industrial semiconductor wafers on a cleanroom conveyor belt). It explains the idea and does not show study data.

AI-generated conceptual illustration (Industrial semiconductor wafers on a cleanroom conveyor belt). It explains the idea and does not show study data.

Open full-size image β†—

Decision Logic for ICV Control Chart Selection

AI-generated conceptual illustration (Industrial semiconductor wafers on a cleanroom conveyor belt). It explains the idea and does not show study data.

View source evidence β†—
Replotted from source data
Replotted from Table 2 (page 8) of the original paper. Every bar is the paper's own table value.β†— Enlarge
Power / n=15 / 2.46221E-05 and n=20 / 2.82939E-05 and more: 0.5000 vs 0.3750 vs 0.3000
Replotted from Table 2 (page 8) of the original paper. Every bar is the paper's own table value.

Replotted from Table 2 (page 8) of the original paper. Every bar is the paper's own table value.

Open full-size image β†—

Power / n=15 / 2.46221E-05 and n=20 / 2.82939E-05 and more: 0.5000 vs 0.3750 vs 0.3000

Replotted from Table 2 (page 8) of the original paper. Every bar is the paper's own table value.

View source evidence β†—
Replotted from source data
Replotted from Table 4 (page 10) of the original paper. Every bar is the paper's own table value.β†— Enlarge
0.6667: Power / n=15 vs n=20 vs n=25
Replotted from Table 4 (page 10) of the original paper. Every bar is the paper's own table value.

Replotted from Table 4 (page 10) of the original paper. Every bar is the paper's own table value.

Open full-size image β†—

0.6667: Power / n=15 vs n=20 vs n=25

Replotted from Table 4 (page 10) of the original paper. Every bar is the paper's own table value.

View source evidence β†—

Figures & tables

From the original paper
πœΜ‚β€² for various values of π‘₯β€²and 𝑠′Figure 1Page 4Power and ARL for T control chart under NI priorFigure 2Page 5Power and ARL for T control chart under gamma prior for πœ‚, 𝛿= 3Figure 3Page 5T and TR control charts for hard bake process dataFigure 4Page 6T and TR control charts for automobile engine piston ring dataFigure 5Page 6The values of πœΜ‚β€²for various values of π‘₯β€² and 𝑠′Table 1Page 8Power and ARL of T control chart for various n under NI priorTable 2Page 8Power and ARL of T control chart for various hyper parameters under gamma priorTable 3Page 9Power and ARL of TR control chart under NI and gamma priorTable 4Page 10Power and ARL of TR control chart under NI and gamma priorTable 4Page 10Control limits and width of T and TR control chartsTable 5Page 11
FULL TEXT Original article Β· reflowed for reading

Introduction

Process variation plays a prime role in process monitoring. Hence monitoring process variation using control charts is inevitable in manufacturing. Sometimes, quality practitioners are interested in calibrating relative change in process variance with respect to process mean known as CV which is unit free. The reciprocal of CV is ICV and it accounts for the measure of precision. The population ICV is given by 𝜏 = πœ‡ 𝜎 (1) where πœ‡ is population mean and 𝜎 is population standard deviation (sd). The sample ICV is given by πœΜ‚= π‘₯ 𝑠 (2) where π‘₯ = is sample sd. The control chart for ICV is helpful in maintaining quality of certain production processes such as manufacturing of surgical instruments, footwear, gems and jewelry, apparel, etc. Singh (1993) [9] studied the behavior of population ICV when a random sample of size n is drawn from various distributions including life testing distributions. Sharma and Krishna (1994) [8] constructed asymptotic distribution of sample ICV. Van Zyl and Van der Merwe (2017) [11] developed Bayesian procedure of obtaining control limits for CV using reference and probability matching priors. Kalkur and Rao (2017) [6] developed Bayes estimators for CV and ICV under normal model using various priors. Using Bayesian approach, the control charts were developed under conjugate priors for process average and process variance respectively by Bhat and Gokhale (2014a) and (2014b) [1][2] . These authors developed control chart for process sd using noninformative (NI), conjugate and non-conjugate priors in 2016. Bhat and Malagavi (2021a) [4] and (2021b) [5] respectively developed range based posterior control charts for process average and process variation.

In this paper, we focus on the development of Bayesian control charts for ICV under normal model. We use NI and gamma priors. In section 2, we obtain 𝐸(𝜏), 𝑉(𝜏), 𝐸(𝜏 π‘Ÿ ), 𝑉(𝜏 π‘Ÿ ) and construct control charts for ICV. We evaluate the performance of the proposed control charts using power and average run length (ARL) in section 3 and illustrate them with examples in section 4. Conclusions are given in section 5. Tables supporting our study are given in appendix.

Bayesian control charts for ICV

In this section, we derive posterior joint distribution of (πœ‡, 1 𝜎 2 ) and propose posterior control charts (T and T R control charts) for ICV. We find 𝐸(𝜏), 𝑉(𝜏), 𝐸(𝜏 π‘Ÿ ) and 𝑉(𝜏 π‘Ÿ ) using joint posterior distribution of πœ‡ and 1 𝜎 2 under NI and gamma priors. Let X 1 , X 2 ,… X n be a random sample of size n from normal distribution with mean πœ‡ and variance 𝜎 2 . Then the likelihood function is given by

Assuming the prior distribution for πœ‡ as πœ‹(πœ‡) = 1, the posterior distribution of πœ‡ is given by

𝑛

Hence, the probability density function of

We derive the posterior distribution of (

The posterior distribution of ( 1 𝜎 2 |𝑠 2 ) under NI prior is given by

Also, the posterior distribution of ( 1 𝜎 2 |𝑠 2 ) under gamma prior is given by

Since πœ‡ and 𝜎 2 are independent, the posterior joint distribution of (πœ‡, 1 𝜎 2 ) is written as

Using (12), under NI prior,

Also, under gamma prior, using (13), we get

The proposed T control chart for ICV is given by 𝐿𝐢𝐿 𝑇 = 𝐸(𝜏) -3βˆšπ‘‰(𝜏), 𝐢𝐿 𝑇 = 𝐸(𝜏) and π‘ˆπΆπΏ 𝑇 = 𝐸(𝜏) + 3βˆšπ‘‰(𝜏).

where 𝐿𝐢𝐿 𝑇 is lower control limit, π‘ˆπΆπΏ 𝑇 is upper control limit and 𝐢𝐿 𝑇 is central line of T control chart. Under NI prior, the control chart is given by substituting ( 14), (15) in (18) and under gamma prior by substituting ( 16), (17) in (18). Further the width, W T of the T control chart is given by

When sample information is available in terms of extreme observations, the range is π‘Ÿ = π‘₯ (𝑛) -π‘₯ (1) where π‘₯ (𝑛) and π‘₯ (1) are respectively the highest and lowest values of the data. According to Tippett (1925) in appropriate equations. Also, we obtain range based T R control chart and its width using 𝐸(𝜏 π‘Ÿ ) and 𝑉(𝜏 π‘Ÿ ).

Performance of the proposed control charts

In this section, we investigate the performance of the proposed T and T R control charts for ICV in terms of power and ARL. The power and ARL of any control chart is given by

where 𝛽 is probability of not detecting the shift in πœΜ‚β€², where πœΜ‚β€² = π‘₯ β€² 𝑠 β€² , π‘₯ β€² = π‘₯ + π‘Ž and 𝑠 β€² = 𝑠 + 𝑏, π‘Ž and 𝑏 are non-negative constants.

Under NI and gamma prior, 𝛽 of T control chart is respectively given by

Using (20) , we compute power and ARL of T control chart for various values of πœΜ‚β€²and n. In table 1, we provide values of πœΜ‚β€² which are computed for various values of π‘₯ β€² and 𝑠 β€² . Here, π‘₯ = 1, π‘Ž = 0 (0.25) 2, 𝑠 = 1 and 𝑏 = 0 (0.5) 4. The values of πœΜ‚β€² for different shifts are plotted in figure 1 using table 1. For n=15, 20, 25, power and ARL of T control chart under NI prior are given in table 2 and plotted in figure 2. Also, under gamma prior, for hyper parameters πœ‚ = 𝛿 = 0,3,5 the corresponding power, ARL values are given in tables 3. Using table 3 power and ARL of T control chart are plotted in figure 3 for πœ‚, 𝛿 = 3. Table 4 consists of power and ARL of T R control chart under NI prior and gamma prior with πœ‚, 𝛿 = 3. In these tables 10𝐸 -𝑙 represents 𝑒 -𝑙 , where 𝑙 is an integer.

Table 4. Power and ARL of TR control chart under NI and gamma prior
NI priorβ€”β€”β€”β€”β€”β€”β€”β€”
π‘₯β€²π‘ β€²πœΜ‚β€²PowerARL
n=15n=20n=25n=15n=20n=25
1.01.01.00008.03E-042.43E-071.34E-041245.4132254115537.1927257490.360917
1.50.66671.19E-061.05E-046.82E-05841814.4257849482.80459214669.241479
2.00.50000.0000120.0013020.00128182863.164954768.062929780.706406
3.00.33330.0006400.0026960.0026991563.323966370.917632370.542859
4.00.25000.0022450.0027000.002700445.350905370.398347370.398347
5.00.20000.0026830.0027000.002700372.761656370.398347370.398347
1.51.01.50000.0427600.0000200.01525723.38608949947.91362965.542902
1.51.00000.0000630.0086890.00779115807.401854115.087359128.360258
2.00.75000.0006430.1072780.1463831555.9858659.3215396.831415
3.00.50000.0340650.2221430.30841729.3557464.5016143.242361
4.00.37500.1195790.2224540.3085388.3626994.4953123.241097
5.00.30000.1428650.2224540.3085386.9996344.4953123.241097
2.01.02.00000.2405050.0000840.0483254.15792211953.75630420.693220
1.51.33330.0003560.0363070.0246762810.47195227.54321740.525931
2.01.00000.0036150.4482550.463646276.6460092.2308722.156816
3.00.66670.1915970.9282070.9768695.2192951.0773461.023679
4.00.50000.6725660.9295080.9772501.4868431.0758381.023280
5.00.40000.8035370.9295080.9772501.2444981.0758381.023280
3.01.03.00000.2974100.0000900.0494503.36236511111.11112720.222447
1.52.00000.0004400.0390600.0252502272.72963125.60163939.603960
2.01.50000.0044700.4822500.474440223.7138792.0736132.107748
3.01.00000.2369300.9986000.9996104.2206601.0014021.000390
4.00.75000.8316991.0000001.0000001.2023581.0000001.000000
5.00.60000.9936591.0000001.0000001.0063811.0000001.000000
Gamma Prior with πœ‚, 𝛿= 3
π‘₯β€²π‘ β€²πœΜ‚β€²PowerARL
n=15n=20n=25n=15n=20n=25
1.01.01.00001.08E-071.13E-063.91E-069259958.683615881900.827012255447.136100
1.50.66672.01E-051.05E-046.54E-0449651.2529959568.5442351528.361243
2.00.50000.0003290.0012340.0024813035.803191810.074244403.137112
3.00.33330.0023800.0026960.002700420.248187370.928775370.398347
4.00.25000.0026990.0027000.002700370.546566370.398347370.398347
5.00.20000.0027000.0027000.002700370.398347370.398347370.398347
1.51.01.50000.0000060.0000930.000447173881.42038910703.1243492235.238969
1.51.00000.0010730.0086110.074774932.340056116.12793113.373619
2.00.75000.0175420.1017150.28348157.0056299.8314063.527571
3.00.50000.1267210.2221360.3085387.8913264.5017503.241097
4.00.37500.1437190.2224540.3085386.9580404.4953123.241097
5.00.30000.1437760.2224540.3085386.9552574.4953123.241097
2.01.02.00000.0000320.0003900.00141730915.1914742561.519208705.710172
1.51.33330.0060330.0359810.236837165.76510227.7922524.222322
2.01.00000.0986650.4250080.89788710.1352982.3528961.113725
3.00.66670.7127400.9281790.9772501.4030361.0773791.023280
4.00.50000.8083400.9295080.9772501.2371021.0758381.023280
5.00.40000.8086640.9295080.9772501.2366081.0758381.023280
3.01.03.00000.0000400.0004200.00145025000.0259392380.952384689.655172
1.52.00000.0074600.0387100.242350134.04839625.8331184.126264
2.01.50000.1220100.4572400.9187908.1960582.1870351.088388
3.01.00000.8813790.9985701.0000001.1345861.0014321.000000
4.00.75000.9995991.0000001.0000001.0004011.0000001.000000
5.00.60000.9999991.0000001.0000001.0000011.0000001.000000

Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.

View original table image
Power and ARL of TR control chart under NI and gamma prior↗ Enlarge
Table 4
Power and ARL of TR control chart under NI and gamma prior

Power and ARL of TR control chart under NI and gamma prior

Open full-size image β†—
Original paper tableSource PDF Β· page 10 β†—
Table 4. Power and ARL of TR control chart under NI and gamma prior
β€”5.00.30000.1437760.2224540.3085386.9552574.4953123.241097
2.01.02.00000.0060570.0091180.008649165.101156109.667489115.624830
1.51.33330.0033640.0284240.093591297.26145635.18110110.684763
2.01.00000.0754160.3923550.74229013.2597862.5487151.347183
3.00.66670.7015080.9278530.9772501.4255011.0777561.023280
4.00.50000.8083320.9295080.9772501.2371151.0758381.023280
5.00.40000.8086640.9295080.9772501.2366081.0758381.023280
3.01.03.00000.0074900.0098100.008850133.511487101.936799112.994350
1.52.00000.0041600.0305800.095770240.38486532.70111210.441683
2.01.50000.0932600.4221100.75957010.7227222.3690511.316534
3.01.00000.8674890.9982201.0000001.1527521.0017831.000000
4.00.75000.9995891.0000001.0000001.0004111.0000001.000000
5.00.60000.9999991.0000001.0000001.0000011.0000001.000000

Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.

View original table image
Power and ARL of TR control chart under NI and gamma prior↗ Enlarge
Table 4
Power and ARL of TR control chart under NI and gamma prior

Power and ARL of TR control chart under NI and gamma prior

Open full-size image β†—
Original paper tableSource PDF Β· page 10 β†—
Table 3. Power and ARL of T control chart for various hyper parameters under gamma prior
𝜼, 𝜹= πŸŽβ€”β€”β€”β€”β€”β€”β€”β€”
𝒙′𝒔′𝝉̂′PowerARL
n=15n=20n=25n=15n=20n=25
1.001.01.00002.47E-052.33E-050.00002040480.69369942919.85484949504.950495
1.50.66673.132E-063.132E-060.000076319308.920125319308.92012513157.894737
2.00.50006.558E-050.0001030.00121015249.0056549688.682902826.763321
3.00.33330.0011920.0018890.002697838.707396529.480877370.780289
4.00.25000.0025120.0026860.002700398.110843372.338230370.398356
5.00.20000.0026960.0027000.002700370.936205370.405755370.398356
1.501.01.50000.0013160.0019200.002305760.137357520.893653433.881644
1.51.00000.0001670.0002580.0086825995.9110483875.269161115.177557
2.00.75000.0034920.0085040.138228286.342397117.5859867.234429
3.00.50000.0634960.1556180.30822015.7490596.4260063.244438
4.00.37500.1337680.2212950.3085387.4756364.5188553.241097
5.00.30000.1435680.2224500.3085386.9653574.4954023.241097
2.001.02.00000.0073990.0080220.007300135.148378124.662580136.985232
1.51.33330.0009380.0010780.0275001066.041085927.44661036.363886
2.01.00000.0196420.0355350.43781850.91015528.1412002.284056
3.00.66670.3571300.6502370.9762432.8000991.5379001.024335
4.00.50000.7523730.9246650.9772501.3291281.0814731.023280
5.00.40000.8074910.9294890.9772501.2384031.0758601.023280
3.001.03.00000.0091500.0086300.007470109.289731115.874855133.868809
1.52.00000.0011600.0011600.028140862.069860862.06896735.536603
2.01.50000.0242900.0382300.44801041.16924826.1574682.232093
3.01.00000.4416300.6995500.9989702.2643411.4294901.001031
4.00.75000.9303890.9947901.0000001.0748191.0052371.000000
5.00.60000.9985490.9999801.0000001.0014531.0000201.000000
πœ‚, 𝛿= 3
π‘₯β€²π‘ β€²πœΜ‚β€²PowerARL
n=15n=20n=25n=15n=20n=25
1.01.01.00003.13E-053.12E-052.18E-0531958.44239432069.12098245955.129944
1.50.66672.48E-064.16E-051.97E-04402606.89928824036.2327935087.190597
2.00.50000.0000570.0007740.00185717397.7617351292.297632538.534069
3.00.33330.0014640.0026740.002700683.026328373.924455370.409460
4.00.25000.0026410.0027000.002700378.610407370.398347370.398347
5.00.20000.0027000.0027000.002700370.420573370.398347370.398347
1.51.01.50000.0016660.0025690.002487600.108440389.204522402.121154
1.51.00000.0001320.0034280.0224657560.061756291.71396744.514442
2.00.75000.0030610.0637600.212209326.69125515.6838754.712334
3.00.50000.0779680.2203560.30852812.8257154.5381073.241194
4.00.37500.1406580.2224540.3085387.1094614.4953123.241097
5.00.30000.1437680.2224540.3085386.9556744.4953123.241097
2.01.02.00000.0093720.0107360.007877106.69608893.146153126.957785
1.51.33330.0007440.0143240.0711541344.13876069.81428114.054110
2.01.00000.0172160.2664160.67214358.0839673.7535351.487779
3.00.66670.4385300.9207430.9772212.2803441.0860801.023310
4.00.50000.7911240.9295080.9772501.2640241.0758381.023280
5.00.40000.8086150.9295080.9772501.2366821.0758381.023280
3.01.03.00000.0115900.0115500.00806086.28136686.580087124.069479
1.52.00000.0009200.0154100.0728101086.95765064.89292713.734377
2.01.50000.0212900.2866200.68779046.9704573.4889401.453932
3.01.00000.5422890.9905700.9999701.8440341.0095201.000030
4.00.75000.9783091.0000001.0000001.0221721.0000001.000000
5.00.60000.9999391.0000001.0000001.0000611.0000001.000000
πœ‚, 𝛿= 5
π‘₯β€²π‘ β€²πœΜ‚β€²PowerARL
n=15n=20n=25n=15n=20n=25
1.01.01.00002.02E-052.65E-052.39E-0549452.38282337757.22195241852.920604
1.50.66671.12E-058.26E-052.59E-0489038.06426612112.4377813867.582201
2.00.50000.0002520.0011400.0020513971.674323877.492472487.642149
3.00.33330.0023420.0026950.002700426.977080371.058832370.398347
4.00.25000.0026990.0027000.002700370.550273370.398347370.398347
5.00.20000.0027000.0027000.002700370.398347370.398347370.398347
1.51.01.50000.0010770.0021820.002731928.605716458.237740366.225594
1.51.00000.0005980.0068030.0295491671.936735147.00170833.842503
2.00.75000.0134090.0939000.23435674.57920710.6496234.267015
3.00.50000.1247240.2220580.3085388.0176804.5033283.241097
4.00.37500.1437170.2224540.3085386.9581104.4953123.241097

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Power and ARL of T control chart for various hyper parameters under gamma prior↗ Enlarge
Table 3
Power and ARL of T control chart for various hyper parameters under gamma prior

Power and ARL of T control chart for various hyper parameters under gamma prior

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Table 2. Power and ARL of T control chart for various n under NI prior
𝒙′𝒔′𝝉̂′Powerβ€”β€”ARLβ€”β€”
n=15n=20n=25n=15n=20n=25
1.01.01.00002.46221E-052.82939E-052.04645E-0540613.85387635343.35375448865.217328
1.50.66672.96978E-071.81966E-050.0001565343367257.70313454955.2444136388.381293
2.00.50001.15551E-050.0005030.00178286541.6699401988.395680561.311674
3.00.33330.0006420.0026190.0027001557.080660381.798861370.405755
4.00.25000.0022430.0027000.002700445.865550370.398347370.398347
5.00.20000.0026820.0027000.002700372.836699370.398347370.398347
1.51.01.50000.0013110.0023310.002339762.637809428.942006427.585291
1.51.00000.0000160.0014990.01788963229.607414666.96027155.900250
2.00.75000.0006150.0414390.2035981625.06000424.1320184.911645
3.00.50000.0342010.2158110.30853129.2385104.6336743.241161
4.00.37500.1194410.2224540.3085388.3723634.4953123.241097
5.00.30000.1428360.2224540.3085387.0010444.4953123.241097
2.01.02.00000.0073750.0097410.007408135.592945102.656304134.997328
1.51.33330.0000890.0062650.05666111241.887809159.61989117.648840
2.01.00000.0034610.1731490.644868288.9270235.7753821.550706
3.00.66670.1923650.9017530.9772305.1984521.1089511.023300
4.00.50000.6717890.9295080.9772501.4885621.0758381.023280
5.00.40000.8033750.9295080.9772501.2447481.0758381.023280
3.01.03.00000.0091200.0104800.007580109.64923795.419847131.926121
1.52.00000.0001100.0067400.0579809090.918523148.36795317.247327
2.01.50000.0042800.1862800.659880233.6451025.3682631.515427
3.01.00000.2378800.9701400.9999804.2038051.0307791.000020
4.00.75000.8307391.0000001.0000001.2037471.0000001.000000
5.00.60000.9934591.0000001.0000001.0065841.0000001.000000

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Power and ARL of T control chart for various n under NI prior↗ Enlarge
Table 2
Power and ARL of T control chart for various n under NI prior

Power and ARL of T control chart for various n under NI prior

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Table 1. The values of πœΜ‚β€²for various values of π‘₯β€² and 𝑠′
π‘₯β€²π‘ β€²πœΜ‚β€²π‘₯β€²π‘ β€²πœΜ‚β€²π‘₯β€²π‘ β€²πœΜ‚β€²
1.001.01.00001.751.01.75002.501.02.5000
1.50.66671.51.16671.51.6667
2.00.50002.00.87502.01.2500
2.50.40002.50.70002.51.0000
3.00.33333.00.58333.00.8333
3.50.28573.50.50003.50.7143
4.00.25004.00.43754.00.6250
4.50.22224.50.38894.50.5556
5.00.20005.00.35005.00.5000
1.251.01.25002.001.02.00002.751.02.7500
1.50.83331.51.33331.51.8333
2.00.62502.01.00002.01.3750
2.50.50002.50.80002.51.1000
3.00.41673.00.66673.00.9167
3.50.35713.50.57143.50.7857
4.00.31254.00.50004.00.6875
4.50.27784.50.44444.50.6111
5.00.25005.00.40005.00.5500
1.501.01.50002.251.02.25003.001.03.0000
1.51.00001.51.50001.52.0000
2.00.75002.01.12502.01.5000
2.50.60002.50.90002.51.2000
3.00.50003.00.75003.01.0000
3.50.42863.50.64293.50.8571
4.00.37504.00.56254.00.7500
4.50.33334.50.50004.50.6667
5.00.30005.00.45005.00.6000

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The values of πœΜ‚β€²for various values of π‘₯β€² and 𝑠′↗ Enlarge
Table 1
The values of πœΜ‚β€²for various values of π‘₯β€² and 𝑠′

The values of πœΜ‚β€²for various values of π‘₯β€² and 𝑠′

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Figure 3. Power and ARL for T control chart under gamma prior for πœ‚, 𝛿= 3
Power and ARL for T control chart under gamma prior for πœ‚, 𝛿= 3β†— Enlarge
Figure 3
Power and ARL for T control chart under gamma prior for πœ‚, 𝛿= 3

Power and ARL for T control chart under gamma prior for πœ‚, 𝛿= 3

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Figure 2. Power and ARL for T control chart under NI prior
Power and ARL for T control chart under NI prior↗ Enlarge
Figure 2
Power and ARL for T control chart under NI prior

Power and ARL for T control chart under NI prior

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Figure 1. πœΜ‚β€² for various values of π‘₯β€²and 𝑠′
πœΜ‚β€² for various values of π‘₯β€²and 𝑠′↗ Enlarge
Figure 1
πœΜ‚β€² for various values of π‘₯β€²and 𝑠′

πœΜ‚β€² for various values of π‘₯β€²and 𝑠′

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From figure 2 and table 2, it is seen that, for a given value of n, power increases as π‘₯ β€² increases and decreases as πœΜ‚β€² increases.

Also, for a specified value of πœΜ‚β€², power increases as n increases. For all the values of π‘₯ β€² , for n=15 and increasing πœΜ‚β€², ARL increases up to some value of πœΜ‚β€² and then decreases. But for n=20, 25, ARL increases as πœΜ‚β€² increases for different values of π‘₯ β€² .

From figure 3 and table 3, it is noticed that, for a given value of n, power increases as π‘₯ β€² increases and decreases as πœΜ‚β€² increases.

Also, for a specified value of πœΜ‚β€², power increases as n increases. For all the values of π‘₯ β€² , for n=15, 20 and increasing πœΜ‚β€², ARL increases up to some value of πœΜ‚β€² and then decreases. But for n=25, ARL increases as πœΜ‚β€² increases.

From tables 2 and 3 it is observed that, power of T control chart increases and ARL decreases with increasing values of n under both priors. Also, power of T control chart increases with increasing values of hyper parameters under gamma prior for a given shift in 𝜏. From table 4, it is found that, power of T R control chart increases with increasing values of n. On comparing T and T R control charts under NI and gamma priors, power of T R control chart is higher than power of T control chart. However, for smaller values of n, power and ARL of both T and T R control charts are almost equal. From figure 1 and table 1, we observe that, for a given value of 𝑠 β€² , πœΜ‚β€² increases as π‘₯ β€² increases and decreases as 𝑠 β€² increases

Illustration

In this section, we illustrate the proposed T and T R control charts in figures 4 and 5 respectively for examples on hard bake process and automobile engine piston ring due to Montgomery (1996). Here, we make use of first twenty five data points of these data and compute πœΜ‚β€² for each subgroup. Then πœΜ‚β€² is plotted for various samples. The control limits and width of the control charts are provided in table 5. When subgroup size is 5, d=2.326 and we consider πœ‚ = 15, 𝛿 = 0.0001.

Table 5. Control limits and width of T and TR control charts
β€”β€”Example 1 Hard bake processβ€”Example 2 Automobile engine piston ringβ€”
Control chartPrior Control limitsNIGamma (𝜼= πŸπŸ“, 𝜹= 𝟎. 𝟎𝟎𝟎𝟏)NIGamma (𝜼= πŸπŸ“, 𝜹= 𝟎. 𝟎𝟎𝟎𝟏)
TUCL16.782621.78610889.713599.03
CL11.875616.86867722.65210547.31
LCL6.968611.95114555.6057495.589
W9.8149.83496334.0946103.44
TRULC15.996220.992411206.813963.73
CL11.316616.07497947.52710830.16
LCL6.637111.15754688.2597696.605
W9.35919.83496518.5366267.121

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Control limits and width of T and TR control charts↗ Enlarge
Table 5
Control limits and width of T and TR control charts

Control limits and width of T and TR control charts

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Example 1: Hard bake process

The data consists of 25 samples with each sample having subgroup of size 5. It is the measurement of flow width (in micron) in hard-bake process for semi-conductor manufacturing. From figure 4 and table 5, it is observed that, the control limits of T and T R control charts under NI prior are lower than that of gamma prior. Also, width under gamma prior is wider than width under NI prior. The width of T R control chart is narrower than width of T control chart under NI prior, whereas widths of both control charts are same under gamma prior.

Figure 4. T and TR control charts for hard bake process data
T and TR control charts for hard bake process data↗ Enlarge
Figure 4
T and TR control charts for hard bake process data

T and TR control charts for hard bake process data

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Example 2: Automobile engine piston ring inside diameter

The data is on measurement of inside diameter (in mm) of piston rings for automobile engine. It contains 25 samples with each of size 5.

Figure 5: T and TR control charts for automobile engine piston ring data

From figure 5 and table 5, it is seen that, for T and T R control charts the control limits under gamma prior are higher than control limits under NI prior, but the width under gamma prior is narrower than width under NI prior. Also, width of T R control chart is wider than width of T control chart under both NI and gamma priors.

Figure 5. T and TR control charts for automobile engine piston ring data
T and TR control charts for automobile engine piston ring data↗ Enlarge
Figure 5
T and TR control charts for automobile engine piston ring data

T and TR control charts for automobile engine piston ring data

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Conclusions

In this section, we record our conclusions about the proposed control charts for process ICV, 𝜏 = πœ‡ 𝜎 , the process precision. Here πœ‡ is process mean and 𝜎 is process sd. ο‚· A posterior control chart, that is, T control chart for process precision using Bayesian approach is proposed. The sample ICV is πœΜ‚= π‘₯ 𝑠 where π‘₯ is sample mean and s is sample sd. ο‚· Under NI and gamma priors, when process measurements follow normal distribution, the T control chart is developed. ο‚· We obtain posterior joint distribution of (πœ‡, 1 𝜎 2 |𝑠 2 ) under NI, gamma priors and derive 𝐸(𝜏), 𝑉(𝜏) under both priors. ο‚· The control limits and width of T control chart are developed using 𝐸(𝜏), 𝑉(𝜏). ο‚· For T control chart, under NI prior, for a given value of n, power increases as shift in π‘₯ increases and decreases as shift in πœΜ‚ increases. For a specified value of shift in 𝜏, power increases with increasing value of n and ARL increases with increasing shifts in πœΜ‚ and higher values of n. ο‚· Under gamma prior, for increasing values of n, power increases with increasing shifts in π‘₯ when πœ‚ and 𝛿 take higher values.

For a given πœ‚, 𝛿 and n, the power decreases as shift in πœΜ‚ increases. ο‚· On substituting 𝑠 = π‘Ÿ 𝑑 , we obtain control limits based on range and develop T R control chart. The behavior of this control chart is similar to that of T control chart. ο‚· T R control chart outperforms T control chart for moderate and large values of n. However, performance of both the control charts are the same for small values of n. ο‚· In hard bake process data, under NI prior width of T control chart is wider than width of T R control chart and under gamma prior width of both control charts are same. Further, in automobile engine piston ring data, the width of T R control chart is wider than width of T control chart under both priors. ο‚· T control chart is useful when process precision is of prime concern and practitioners have prior information about the process mean and variance, whereas, T R control chart is helpful when sample information is available in terms of extreme observations.

Appendix

Appendix

References

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