| Summary: | 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.
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