طراحي اقتصادي نمودار كنترل EWMA-3 تعديل شده براي پايش پروفايل‌هاي خطي ساده با رويكرد زنجيره‌ي ماركوف

Economic Design of Modified EWMA-3 Control Chart for Monitoring Simple Linear Profiles Using the Markov Chain Approach

در بعضي از مسايل كنترل فرايند آماري، كيفيت به‌وسيله‌ي يك پروفايل خطي ساده توصيف مي‏شود. يكي از متداول‌ترين روش‌ها براي پايش پروفايل‌هاي خطي ساده در فاز 2، نمودار كنترل EWMA-3 است. در اين مقاله ابتدا نمودار كنترل EWMA-3 با اندكي تغيير به‌گونه‌يي طراحي مي‌شود كه بتوان به‌جاي يك محصول در هر سطح از مقادير x، از چندين محصول استفاده كرد. سپس روشي براي محاسبه‌ي متوسط طول دنباله‌ي نمودار كنترل EWMA-3 تعديل شده براي پايش پروفايل‌هاي خطي ساده براساس رويكرد زنجيره‌ي ماركوف ارايه مي‏شود. در ادامه، طراحي اقتصادي نمودار كنترل EWMA-3 تعديل شده مورد بررسي قرار مي‏گيرد. از تابع لورنز و ونس به‌عنوان تابع هزينه استفاده شده كه در آن پارامترهاي هزينه‌يي توليد محصول نامنطبق براساس تابع هزينه‌ي تاگوچي محاسبه مي‏شود. براي حل مدل طراحي اقتصادي نمودار EWMA-3 تعديل شده از الگوريتم فراابتكاري ژنتيك استفاده شده است. در انتها آناليز حساسيت روي پارامترهاي مدل و اندازه‌ي جمعيت الگوريتم ژنتيك انجام و نتايج تحليل شده است.

In some statistical process control applications, the quality of a process or a product is characterized by the relationship between a response variable and one or more explanatory variables referred to as a profile by researchers. Simple linear profiles are a different type of profile which has many applications, especially in calibration. One of the most powerful methods in phase II monitoring of simple linear profiles is the EWMA-3 method proposed by Kim et al. (2003). In phase II, the aim is detecting assignable causes as soon as possible. The power of control charts in detecting assignable causes is usually measured by an average run length criterion. Kim et al. (2003) proposed a method in cases where there is only one response value at each level of explanatory variable. However, there are some situations in which there is more than one response value at each level of explanatory variable. In this paper, first, we modify the EWMA-3 method by Kim et al. (2003) for cases with more than one response value at each level of explanatory variable. Then, we present a Markov chain model to measure the ARL criterion of the modified EWMA-3 method. After that, we design the modified EWMA-3 control chart economically to account for the economic properties of the control chart. For this purpose, we use the Lorenzen and Vance cost model and optimize parameter sample size, sampling interval, smoothing parameters and the coefficients of control limits for three EWMA control charts, including the EWMA control charts for monitoring the intercept, the slope and the standard deviation. In addition, we use the idea of the Taguchi cost function in our proposed model. Finally, a genetic algorithm is used to solve the proposed model. The performance of the proposed economic model is evaluated through a numerical example. Then, a sensitivity analysis is undertaken to identify the effect of different parameters and the population size of the genetic algorithm on the economic and statistical properties of the modified EWMA-3 control charts.