Estimation of the Six Sigma Quality Index
The measurement of the process capability is a key part of quantitative quality control, and process capability indices are statistical measures of the process capability. Six Sigma level represents the maximum achievable process capability, and many enterprises have implemented Six Sigma improvemen...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/19/3458 |
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| author | Chun-Chieh Tseng Kuo-Ching Chiou Kuen-Suan Chen |
| author_facet | Chun-Chieh Tseng Kuo-Ching Chiou Kuen-Suan Chen |
| author_sort | Chun-Chieh Tseng |
| collection | DOAJ |
| description | The measurement of the process capability is a key part of quantitative quality control, and process capability indices are statistical measures of the process capability. Six Sigma level represents the maximum achievable process capability, and many enterprises have implemented Six Sigma improvement strategies. In recent years, many studies have investigated Six Sigma quality indices, including <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mrow><mi>p</mi><mi>k</mi></mrow></msub></mrow></semantics></math></inline-formula>. However, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mrow><mi>p</mi><mi>k</mi></mrow></msub></mrow></semantics></math></inline-formula> contains two unknown parameters, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>, which are difficult to use in process control. Therefore, whether a process quality reaches the <i>k</i> sigma level must be statistically inferred. Moreover, the statistical method of sampling distribution is challenging for the upper confidence limits of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mrow><mi>p</mi><mi>k</mi></mrow></msub></mrow></semantics></math></inline-formula>. We address these two difficulties in the present study and propose a methodology to solve them. Boole’s inequality, Demorgan’s theorem, and linear programming were integrated to derive the confidence intervals of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mrow><mi>p</mi><mi>k</mi></mrow></msub></mrow></semantics></math></inline-formula>, and then the upper confidence limits were used to perform hypothesis testing. This study involved a case study of the semiconductor assembly process in order to verify the feasibility of the proposed method. |
| first_indexed | 2024-03-09T21:28:35Z |
| format | Article |
| id | doaj.art-6e29035f4304488b8318875920b133a0 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T21:28:35Z |
| publishDate | 2022-09-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-6e29035f4304488b8318875920b133a02023-11-23T21:01:51ZengMDPI AGMathematics2227-73902022-09-011019345810.3390/math10193458Estimation of the Six Sigma Quality IndexChun-Chieh Tseng0Kuo-Ching Chiou1Kuen-Suan Chen2School of Internet Economics and Business, Fujian University of Technology, Fuzhou 350014, ChinaDepartment of Finance, Chaoyang University of Technology, Taichung 413310, TaiwanDepartment of Industrial Engineering and Management, National Chin-Yi University of Technology, Taichung 411030, TaiwanThe measurement of the process capability is a key part of quantitative quality control, and process capability indices are statistical measures of the process capability. Six Sigma level represents the maximum achievable process capability, and many enterprises have implemented Six Sigma improvement strategies. In recent years, many studies have investigated Six Sigma quality indices, including <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mrow><mi>p</mi><mi>k</mi></mrow></msub></mrow></semantics></math></inline-formula>. However, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mrow><mi>p</mi><mi>k</mi></mrow></msub></mrow></semantics></math></inline-formula> contains two unknown parameters, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>, which are difficult to use in process control. Therefore, whether a process quality reaches the <i>k</i> sigma level must be statistically inferred. Moreover, the statistical method of sampling distribution is challenging for the upper confidence limits of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mrow><mi>p</mi><mi>k</mi></mrow></msub></mrow></semantics></math></inline-formula>. We address these two difficulties in the present study and propose a methodology to solve them. Boole’s inequality, Demorgan’s theorem, and linear programming were integrated to derive the confidence intervals of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mrow><mi>p</mi><mi>k</mi></mrow></msub></mrow></semantics></math></inline-formula>, and then the upper confidence limits were used to perform hypothesis testing. This study involved a case study of the semiconductor assembly process in order to verify the feasibility of the proposed method.https://www.mdpi.com/2227-7390/10/19/3458Six Sigma quality indexlinear programmingestimationsupper confidence limitstatistic hypothesis testing |
| spellingShingle | Chun-Chieh Tseng Kuo-Ching Chiou Kuen-Suan Chen Estimation of the Six Sigma Quality Index Mathematics Six Sigma quality index linear programming estimations upper confidence limit statistic hypothesis testing |
| title | Estimation of the Six Sigma Quality Index |
| title_full | Estimation of the Six Sigma Quality Index |
| title_fullStr | Estimation of the Six Sigma Quality Index |
| title_full_unstemmed | Estimation of the Six Sigma Quality Index |
| title_short | Estimation of the Six Sigma Quality Index |
| title_sort | estimation of the six sigma quality index |
| topic | Six Sigma quality index linear programming estimations upper confidence limit statistic hypothesis testing |
| url | https://www.mdpi.com/2227-7390/10/19/3458 |
| work_keys_str_mv | AT chunchiehtseng estimationofthesixsigmaqualityindex AT kuochingchiou estimationofthesixsigmaqualityindex AT kuensuanchen estimationofthesixsigmaqualityindex |