以複式模擬法構建非對稱規格區間下兩個製程能力指標 C"pmk值差異之信賴區間

Abstract

在評估供應商之優劣時，一般多以製程能力指標值的大小來決定其製程能力的好壞，大的製程能力指標值表示製造過程穩定且產品能符合規格要求。Cp 及Cpk 即為早期發展出來的製程能力指標，也是目前產業界最常使用的兩個製程能力指標。但是 Cp 及 Cpk 兩個指標並未考慮製程平均值偏離目標值，因此發展出指標 Cpm。Cpm 是將田口損失函數期望值的概念納入 Cp 指標中，但此指標值之大小對製程良率的反應會不一致。針對這個缺點，學者提出了Cpmk 指標。 指標可同時考慮製程平均值偏離目標值的製程期望損失及製程良率。 一般常見之製程皆使用對稱之規格區間，但在某些情況下會使用非對稱規格區間，譬如當製品偏離不同方向的規格界限，所造成的損失不相同時﹔或是將非常態的製程分布資料作常態轉換時，也會產生非對稱規格區間的情形。近年來陸續有許多研究發展了一些非對稱規格下之製程能力指標，如C"pk 、C"pm 和 C"pmk 等，其中C"pmk 最能反映出製程平均值偏離目標值的影響。然而由於C"pmk 之估計式之機率分配的推導過於複雜，迄今學界也僅能推導出其近似的機率分配，故與C"pmk 相關的假說檢定與信賴區間尚未被發展出，因而導致C"pmk 之用途有限。 因此，本研究之主要目的即是以複式模擬法(bootstrap simulation)來構建兩個常態製程 C"pmk 指標值之差異，即( C"pmk1 - C"pmk2 )的100(1-α)%信賴區間；此信賴區間可以用來有效地評估兩個製程或供應商製程能力之優劣。研究最後並將所發展之信賴區間構建過程寫成完整的流程，並以一實例來說明如何應用本流程，以供業界沒有太多統計背景之工程人員可以簡單快速地構建出此兩個 C"pmk 值差異之信賴區間，以正確的比較兩個廠商或製程能力之優劣。

The process capability indices are utilized to evaluate a supplier’s general process capability. A larger process index value usually leads to a more capable production process, that is, more products will meet the specifications. Among many developed process capability indices, Cp and Cpk are two most popular indices adopted by industry. However, these two indices do not take into account the effect of mean process deviates from its target value. Therefore, the process index Cpmk was developed to reflect the impact of mean process deviates from its target value on the process capability. The conventional process capability indices are utilized in cases where the specification interval is symmetry about its target value. In the cases where the specification interval is asymmetry, the cost loss varies according to whether the product mean is close to the lower or upper specification limits. Many process capability indices such as C"pk , C"pm and C"pmk were developed recently to reflect the impact of the asymmetry specification limits. Among these indices, C"pmk is superior. However, the exact probability distribution of C"pmk is too complicated to be derived, only the approximate probability distribution of C"pmk hat , which is the estimator of C"pmk , can be derived. Consequently, the related hypotheses testing and confidence interval can not be developed. For this reason, the application of C"pmk is limited. Hence, the main objective of this study is to utilize Bootstrap method to construct a 100(1-α)% confidence interval for the difference between two normal process capability indices, C"pmk1 - C"pmk2 , in asymmetry specification interval . The proposed Bootstrap interval can be effectively employed to determine which one of the two production processes or manufacturers has a better production capability. A detailed procedure of constructing the confidence interval for C"pmk1 - C"pmk2 is also provided by this study for engineers without much statistics background.