非指數失效機制與非常態資料之X-bar 管制圖的經濟與經濟-統計設計

統計方法近來已經受到產業界的重視，將其應用於解決製程問題，尤其是在1924年，Shewhart 博士應用統計的原理發展出著名的管制圖(Control chart)方法，並且被廣泛的使用在監控製程的變異，以確保產品的品質能符合顧客的期待。傳統在探討應用經濟模式決定管制圖的參數時，通常假設製程由管制內(In-control)狀態轉變為管制外(Out-of-control)狀態時，其可歸屬原因(Assignable causes)的發生是服從卜瓦松過程(Poisson process)，並且假設由製程中所抽取之樣本組，其組內數據的分佈與樣本統計量均服從常態分配。然而對於電子產品的零組件、機械設備或機台，指數分配就不是一個適合的製程失效機制假設，且由於其危險函數(Hazard function)會隨使用時間增加，因而產生失效的機率便會遞增。此外產業界為考量降低抽樣與檢驗的成本與時間，在應用管制圖監控製程時會設法減少抽樣數量，導致無法應用中央極限定理使樣本組內的數據分佈與樣本統計量逼近常態分配。因此應用傳統的管制圖理論設計實際監控製程之管制圖時，會影響其偵測製程變異的能力。基於以上因素，本研究嘗試以Burr分配取代常態分配為基礎，並分別於Weibull與Gamma分配的失效機制，以固定抽樣區間（FSI）與變動抽樣區間（VSI）的抽樣方式，及經濟性與經濟統計的設計方法建構X bar管制圖，且應用非線性的搜尋方式決定其參數值，以完成管制圖的設計。最後分別以ECT、型 I 誤差及檢定力三項評估指標，比較應用Burr分配與常態分配於不同非指數失效機制、抽樣區間與設計方法的限制下所建構之X bar管制圖的績效。期望透過研究之結論，可以提供一種降低成本並且維持高品質產品之製程監控工具於實務界。

關鍵字

非常態分配；變動抽樣區間；經濟性設計；經濟統計設計；非指數失效機制

並列摘要

Statistical approaches have recently been used to solve industrial process control problems. In particular, the control chart technique, developed using a statistical theory by Dr. Shewhart in 1924, is a well-known statistical method and is widely applied to monitor the variance of a manufacturing process, such that it can ensure that the quality of products is consistent with the expectations of customers. Traditionally, when an economic model is applied to determine the parameters of control charts, it is assumed that the occurrence of an assignable cause follows Poisson distribution when the status of a process changes from in-control into out-of-control condition. Furthermore, it is also assumed that the within-group sampled data and the sampling statistics are also normally distributed. However, the exponential distribution is not an appropriate failure mechanism for some components, mechanical equipment, or machines of electronic products, because their hazard functions will gradually increase with an increase in the time of use. In addition, considering the savings on sampling cost and time, soothe industries usually try to reduce sample size when applying control charts for process monitoring. In these situations, the distribution of subgroup data sets violates the assumption of normal distribution since the central limit theorem cannot be applied. This may reduce the capability of a control chart when applying it for the detection of process variations. This research intends to study the economic and economic-statistical design of x-bar control charts based on the assumption of Burr distribution instead of normal distribution. Meanwhile, fixed sampling interval (FSI) and variable sampling interval (VSI) approaches will be employed under the cost model of Weibull and Gamma distributions. Moreover, the non-linear search approach is applied to determine the parameters of the control chart. To evaluate the performance of different control chart designs, three performance indexes, namely, ECT, Type I error, and power, are employed. It is expected that the research results can provide industries a process monitoring tool with reduced cost at same quality level.

並列關鍵字

Non-normal distribution ； variable sampling interval ； economic design ； economic statistical design ； non-exponential failure mechanism

參考文獻

1. Al-Oraini, H.A.& Rahim, M.A. (2003). Economic statistical design of X control charts for systems with Gamma (λ, 2) in-control times. Journal of Applied Statistics, 30, 397-409.

2. Autin, S. (1990). An introduction to genetic algorithms. AI Expert, 49-53.

3. Banerjee, P.K. & Rahim, M.A. (1988). Economic design of X control charts under Weibull shock models. Technometrics, 30(4), 407-414.

4. Bai, D. S. & Lee, K. T.(1998).An economic design of variable sampling interval control charts. International Journal of Production Economics, 54, 57-64.

5. Baker, K.R. (1971). Two process models in economic design of an X chart. AIIE Transactions, 3, 257-263.

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非指數失效機制與非常態資料之X-bar 管制圖的經濟與經濟-統計設計

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