پايش متغير خروجي دو‌جمله‌يي با به‌كارگيري الگوهاي خطي تعميم‌يافته در فرايندهاي دومرحله‌يي

Monitoring Binomial Response Variable Based on Generalized Linear Models in Two Stage Processes

امروزه بيشتر محصولات و خدمات داراي فرايندهايي با چندين مرحله‌ي متوالي هستند. با توجه به وجود خاصيت آبشاري در اغلب اين فرايندها، پايش جداگانه‌ي مراحل بدون در نظر گرفتن رابطه‌ي بين مشخصه هاي كيفي در مراحل مختلف باعث افزايش خطا در نتايج مي ‌شود. در اين نوشتار، يك نمودار كنترل براساس الگوهاي خطي تعميم‌يافته به‌منظور پايش فرايندهاي دومرحله‌يي با مشخصه‌ي كيفي دو‌جمله‌يي در مرحله‌ي دوم پيشنهاد شده است. در روش پيشنهادي، به‌منظور پايش متغير دو‌جمله‌يي در مرحله‌ي دوم، يك نمودار كنترل انتخاب عامل انحراف با در نظر گرفتن رابطه‌ي بين مشخصه هاي كيفي در دو مرحله طراحي شده است. به‌منظور برقراري رابطه بين مشخصه هاي كيفي مراحل اول و دوم، از تابع رابط لجيت (Logit) كه براي متغير خروجي دو‌جمله‌يي مناسب است استفاده شده است. عملكرد نمودار كنترل پيشنهادي با استفاده از شبيه‌سازي و براساس معيار متوسط طول دنباله ارزيابي شده است.

Nowadays, the process behind many productions and services has several sequential stages. According to the cascade effect in most of these processes, monitoring each stage without considering the relation between quality characteristics at different stages could lead to misleading results. Therefore, monitoring multistage processes under different assumptions is widely developed by many authors. Sometimes, quality characteristics at different stages have a binomial distribution; for example, the number of nonconforming items in one batch of products at different stages. To the best of our knowledge, there is no method to monitor this type of quality characteristic. Note that monitoring each stage using a conventional np control chart is a misleading approach. In this paper, a two stages process is considered, in which the quality characteristic in the second stage follows a binomial distribution. We propose a generalized linear model (GLM) based control chart for monitoring the process. To establish the relationship between the first and second stage quality characteristics, we use a Logit link function that is suitable for a binomial response variable. Then, the deviance residual (DR) control statistic is constructed using a generalized likelihood ratio (GLR) test to monitor the binomial variable at the second stage. This study is investigated in Phase II, therefore, it is assumed that the distribution parameters of quality characteristics in stages, and the parameters of the link function are known, based on Phase I analysis. The performance of the proposed method is evaluated through two numerical examples, in terms of the average run length criterion, and is compared with that of the np-chart. In the first example, the quality characteristic at the first stage has normal distribution. The simulation results indicate that the proposed chart outperforms the np-chart, while the first stage quality characteristic in the second example has binomial distribution. The simulation study shows that for some parameter values in the latter example, the out-of-control average run length (ARL) is larger than the in-control ARL. This problem roughly can be solved by increasing the sample size. However, the proposed method leads to better results.