行政院國家科學委員會專題研究計畫 成果報告
廣義田口能力指標估計式在大樣本下的分配與統計推論(I)
研究成果報告(精簡版)
計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 98-2221-E-151-034- 執 行 期 間 ： 98 年 08 月 01 日至 99 年 07 月 31 日 執 行 單 位 ： 國立高雄應用科技大學工業工程管理系 計 畫 主 持 人 ： 林谷鴻 處 理 方 式 ： 本計畫涉及專利或其他智慧財產權，1 年後可公開查詢中 華 民 國 99 年 09 月 29 日
行政院國家科學委員會補助專題研究計畫
; 成 果 報 告
□期中進度報告
廣義田口能力指標估計式在大樣本下的分配與統計推論(I)
計畫類別：;個別型計畫 □整合型計畫
計畫編號：
NSC
98-2221-E-151-034
執行期間：2009 年 8 月 1 日至 2010 年 7 月 31 日
執行機構及系所：國立高雄應用科技大學工業工程與管理系
計畫主持人：林谷鴻
共同主持人：
計畫參與人員：吳榮展、黃宇晨
成果報告類型(依經費核定清單規定繳交)：;精簡報告 □完整報告
本計畫除繳交成果報告外，另須繳交以下出國心得報告：
□赴國外出差或研習心得報告
□赴大陸地區出差或研習心得報告
□出席國際學術會議心得報告
□國際合作研究計畫國外研究報告
處理方式：
除列管計畫及下列情形者外，得立即公開查詢
□涉及專利或其他智慧財產權，;一年□二年後可公開查詢
附件一ABSTRACT
The generalized Taguchi capability index "
pm
C has been proposed to measure the manufacturing
performance with asymmetric tolerances. Investigations on the generalized Taguchi capability index are
almost focused on sample samples with a normality assumption. In this project, a natural estimator of "
pm
C is considered. The limiting distribution and related large sample properties of the considered estimator are studied under general populations having fourth central moment exists. A decision-making procedure based
on an approximate (1 - α)% lower bound of "
pm
C is also constructed. A demonstrate example is provided
to illustrate how the proposed procedure may be applied for judging whether the process runs under the
desirable quality requirement.
Keywords: approximate lower bound; generalized Taguchi capability index.
摘要
廣義的田口能力指標 " pm C 被用來量測俱有非對稱允差規格的製程績效。關於廣義田口能力指標的研究 幾乎皆是以常態分配為前題，採取單一樣本進行估計。本研究計劃考慮C 的一個自然估計式，針對"pm 任何俱有四階中心動差的母體，推導該估計式的極限分配並研究其在大樣本下的相關性質。本研究計 劃建構一個基於 " pm C 的一個信賴下限，提出一個近似(1 - α)%信心水準的決策程序。本研究計劃最後提 供一個實務案例說明如何以本研究的成果應用在實務上作成決策。 關鍵詞：近似下界，廣義的田口能力指標1. Introduction
Process capability index Cpm (Hsiang and Taguchi (1985), Chan et al. (1988)) has been widely used in
the manufacturing industry to provide numerical measures of process potential and performance. As noted
by many quality control researchers and practitioners, Cpm is not originally designed to provide an exact
measure on the number of non-conforming items. But, Cpm includes the process departure ( −μ T)2 (rather
than 6σ alone) in the denominator of the definition to reflect the degree of process targeting. The index Cpm
is defined as the following:
2 2 2 2 pm T) (μ σ 3 d T) (μ σ 6 LSL USL C − + = − + − = , (1)
where USL and LSL are the upper and lower specification limits preset by the customers, the product
designers, μ is the process mean, σ is the process standard deviation, T is the preset target value (a known
constant) and d = (USL - LSL)/2 is half length of the specification interval.
The index Cpm takes the process targeting issue into consideration, however, it fails to account for
process with asymmetric tolerances. A process is said to have asymmetric tolerances if the upper tolerance
dU = USL – T, is unequal to the lower tolerance, dL = T - LSL. To handle processes with asymmetric
tolerances, Chen et al. (1999) considered a generalization of the Taguchi capability index Cpm. The
generalization, referred as " pm C , is defined as follows: 2 2 * " pm A σ 3 d C + = , (2)
where d* = min{dU, dL}, A = max{d(μ – T)/dU, d(T – μ)/dL}. Clearly, if the preset target value T = m = (USL
+ LSL)/2 (symmetric case), then d* = d = (USL - LSL)/2, A = |μ – T|, and the generalization "
pm
C reduces to
the original index Cpm. The factor A in the definition ensures that the generalization C"pm obtains its
maximal value at = T (process is on -target) regardless of whether the tolerances are symmetric (T = m) or
asymmetric (T ≠ m).
2. Estimating
C
"pm based on single sampleAssuming that the measurements of the characteristic investigated, (X1, X2, …, Xn), are chosen randomly
from a stable process which follows a normal distribution N(μ, σ2). An estimator of "
pm
C considered by
2 2 n * " pm A S 3 d C ˆ ˆ + = , (3)
where Aˆ= max{d(X – T)/dU, d(T –X )/dL} and S2n = ∑ni=1(Xi −X)2/n. We note that if the production
tolerance is symmetric, then Aˆ may be simplified as X −T and the estimator Cˆ becomes the MLE of "pm
Cpm discussed by Boyles (1991). Chen et al. (1999) investigated the statistical properties of the estimated
" pm
C . They obtained the exact distribution and the formulae for the r-th moment, expected value, variance,
and the mean-squared error under the normality assumption. The natural estimator Cˆ ′′pm can be rewritten
as: pm Cˆ ′′ = Y K 3 C + , (4) where C = n1/2 d*/σ, K = n 2 n
S /σ2, and Y = [max{(d/dU) Z, (d/dL) Z}]2 with Z = n1/2(X − T)/σ. On the
assumption of normality, the statistic K is distributed as 2
1 n−
χ , Z is distributed as N(δ0, 1), where δ =
n1/2(μ − T)/σ.
Chen et al. (1999) showed that the statistic Z2 follows a non-central chi-square distribution with one
degree of freedom and non-centrality parameter 2
0
δ . The distribution of Y is a weighted non-central
chi-square distribution with one degree of freedom and non-centrality parameter 2
0
δ , under the assumption of normality. The probability density function of Y is expressed as:
) y ( fY = π 2 e−λ/2
(
)
∑ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∑ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ∞ =0 = j 2 i Y 2 1 i 2i j i j 0 f y/d d 1) ( 2 j 1 Γ ! j ) δ 2 ( j , y > 0, (5) where λ = 2 0δ and Yj is distributed as χ . The probability density function of 1+2 j Cˆ ′′pm is derived as:
) x ( f " pm Cˆ = 3 Γ
(
(n 1)/2)
x C 2 n 1) (n n n/2 1 − + − − × π 2 e−λ/2 ∑ ∞ =0 j j! 1 0 j 3x C δ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∑ − = 2 1 i j i 1) ( ×(
− + ∫1 − − − 0 1)/2 (j 3)/2 (n 1) (j i (1 y) y d ⎟⎟ ⎠ ⎞ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧− (1−y+d− y) dy 18x C exp 2 i 2 2 , x > 0. (6)The research work in Chen et al. (1999) is based on the traditional distribution frequency approach. On the other hand, Lin et al. (2005) proposed a Bayesian approach based on for assessing process capability. Under the assumption of a non-informative prior, Lin et al. (2005) obtained a simple Bayesian procedure for process capability assessment, which allows one to proceed with a Bayesian credible interval estimation for
" pm
3. The limiting behaviors of
C
"pm~
Theorem. Let Xi1, Xi2, …, Xini be a random sample of measurements from a process having fourth central
moment μ4 = E(X - μ)4 exists for each i = 1, 2, …, k, j = 1, 2, …, ni, and LSL < μ < USL, then as N = ∑ki=1n i
→ ∞, (a) " pm C~ is consistent. (b) For USL – T ≠ T – LSL, N(C C" ) pm " pm− ~
converges in distribution to a normal distribution with
asymptotic mean 0 and asymptotic variance "2
pm σ , where ξ= [(μ − Τ)/σ] and "2 pm σ = 2 2 2 U " pm ξ ) (d/d 1 C ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 1 σ μ 4 1 σ μ σ T -μ d d σ T -μ d d 4 4 3 3 2 U 2 4 U , for USL – T < T – LSL, (7) "2 pm σ = 2 2 2 L " pm ξ ) (d/d 1 C ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 1 σ μ 4 1 σ μ σ T -μ d d σ T -μ d d 4 4 3 3 2 L 2 4 L , for USL – T > T – LSL. (8) (c) " pm C~ is asymptotically unbiased.
Corollary. If a random sample of measurements Xi1, Xi2, …, Xini follows a normal distribution N(μ, σ2) for
each i = 1, 2, …, k, j = 1, 2, …, ni, then C"pm ~
is asymptotically efficient.
An approximate 100(1 – α)% lower confidence bound of "
pm
C based on k subsamples can be expressed as:
⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∞ − , N σ z C " pm " pm ~ ~ α . (9)
A useful decision-making procedure on process capability with large samples is presented in the
following for in-plant applications to judge whether the process runs under the desirable quality requirement.
STEP 1: Decide the confidence level 1 - α (0.900, 0.950, 0.975, or 0.990).
STEP 2: Choose a random sample of k subgroups each of size ni from a stable (under statistical control)
process to calculate " pm C~ .
STEP 3: Construct an approximate 100(1 – α)% lower confidence bound.
STEP 4: Conclude that the process is capable of the time with 100(1 – α)% confidence if the true "
pm
C value
fell into expression (9). Otherwise, we do not have enough evidence to claim that the process is capable under the given confidence level.
References
1 Boyles, R.A., “The Taguchi capability index”, Journal of Quality Technology, 23 (1991) 17-26.
2 Chan, L.K., Cheng, S.W., and Spiring, F.A., “A new measure of process capability: Cpm”, Journal of Quality
Technology, 20 (1988) 162-175.
3 Chen, K.S., Pearn, W.L., and Lin, P.C., “A new generalization of the capability index Cpm for asymmetric
tolerances”, International Journal of Reliability, Quality and Safety Engineering, 6 (1999) 383-398.
4 Hsiang, T.C., and Taguchi, G., “A tutorial on quality control and assurance — the Taguchi methods”,
ASA Annual Meeting Las Vegas, Nevada, USA, 1985.
5 Lin, G.H., Pearn, W.L., and Yang, Y.S., “A Bayesian approach to obtain a lower bound for the Cpm index”,
Quality & Reliability Engineering International, 21 (2005) 655-668.
6 Lin G.H., “A reliable procedure on performance evaluation - a large sample approach based on the
estimated Taguchi capability index”, Yugoslav Journal of Operations Research, (2010), under revision.
國科會補助專題研究計畫成果報告自評表
請就研究內容與原計畫相符程度、達成預期目標情況、研究成果之學術或應用價
值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）
、是否適
合在學術期刊發表或申請專利、主要發現或其他有關價值等，作一綜合評估。
1. 請就研究內容與原計畫相符程度、達成預期目標情況作一綜合評估
; 達成目標
□ 未達成目標（請說明，以 100 字為限）
□ 實驗失敗
□ 因故實驗中斷
其他原因
說明：
2. 研究成果在學術期刊發表或申請專利等情形：
論文：□已發表 □未發表之文稿 □撰寫中 □無
專利：□已獲得 □申請中 ;無
技轉：□已技轉 □洽談中 ;無
其他：已投稿至國際學術期刊，審查中。
3. 請依學術成就、技術創新、社會影響等方面，評估研究成果之學術或應用價
值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）（以
500 字為限）
本研究計劃的成果主要在學術成就。當樣本數夠大時，本研究所採取的估計
式俱備漸近常態性並且達到 CR-lower bound，換言之，本研究提出的估計式
在大樣本下俱備優良估計式的性質。任何品管從業人員可以合理採用本研究
提出的估計式進行製程績效評估。
附件二98 年度專題研究計畫研究成果彙整表
計畫主持人：林谷鴻 計畫編號： 98-2221-E-151-034-計畫名稱：廣義田口能力指標估計式在大樣本下的分配與統計推論(I) 量化 成果項目 實際已達成 數（被接受 或已發表） 預期總達成 數(含實際已 達成數) 本計畫實 際貢獻百 分比 單位 備 註 （ 質 化 說 明：如 數 個 計 畫 共 同 成 果、成 果 列 為 該 期 刊 之 封 面 故 事 ... 等） 期刊論文 0 0 100% 研究報告/技術報告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 0 0 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國內 參與計畫人力 （本國籍） 專任助理 0 0 100% 人次 期刊論文 0 1 100% 研究報告/技術報告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 章/本 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 0 0 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國外 參與計畫人力 （外國籍） 專任助理 0 0 100% 人次其他成果
