Ch 6 Control Charts for Attributes Attribute control
Ch 6 Control Charts for Attributes 本章介紹三種Attribute control charts 1. Control chart for fraction nonconforming:P chart 2. Control chart for nonconforming:c chart 3. Control chart for nonconformities per unit:u chart 1
使用 3 -sigma limit: 當process的fraction nonconforming p為未知,可以下 法估計之:假設每次取n個樣本,共取m次,則 以此 取代上述 p chart中的 p值, 所得的control chart即稱為trial control limits based on m initial samples. 3
若p是已知或是給定,無需建立trial control limits。 通常process的true p值為未知,但 廠可能給定一個 standard的p值,此時若有out of control的signal出現, 則需注意是否process是out of control at the target p but in control at some other value of p。 5
Example 1 Cardboard cans for frozen orange juice concentrate. Nonconforming的原因:e. g. 從side seam或bottom joint漏出 Sample:n=50 cans,m=30 取樣時間為每天的三次交換班時間,歷時半小時,此時機器 仍在運轉。 1. 2. 3. 4. Data 見附表。 Trial control chart如附圖。 Sample 15及23 outside control limits. 調查發現,於sample 15時,正好一批新的cardboard 在此交換班的半小時中,加入生產過程,而造成此不 規則的情況;而sample 23則由於臨時指定一位相當 沒經驗的operator來操作此架機器。 故可將sample 15及23消去,重建control chart。 6
5. 重建之control chart limits與control chart如附圖。 6. Sample 21 exceed新的control limit,然而我們對該點 無法找出任何的assignable causes,故決定仍保留該 點,並將此新的control chart用來作做future samples 的控制,(我們並已檢定該control chart的maximum run=5,且無任何nonrandom的pattern),故目前之 結論為:該process是in control at the level p=0. 2150, 且該control chart可用來monitor current procedure。 7. 雖然此process目前是in control,但其p=0. 2150則是 偏高,但此已無法從workforce的層次來改進,而必 須由management的階層來improve! 8. Management指示engineering staff進行分析該過程, 由分析結果決定對機器進行數方面的調整。 7
12. 再持續取樣(如附表),其C. C. (如附圖) Process is in control. 然而 仍偏高,可利用statistically designed experiments來決定如何調整機器, 及其調整量,以期達到defect-free的 manufacturing。 9
Fraction nonconforming control chart 的設計 三個主要參數: 1. Sample size(n) 2. Frequency of sampling 3. Width of the control limits(k) (亦可加入economic的考量) 10
Sampling frequency:通常建立不良率的control chart是 採取 100% inspection of all process output over some convenient period of time。在此情況下,sample size( n) 也會被同時決定,以下為兩個考量sampling frequency 1. 的因素。 Production rate 2. Rational subgroup e. g. 若一日有三班,若我們懷疑此三班已有shift發生, 則我們會取一個shift的output為subgroup,而不是以 整日的output為subgroup。 11
建議 II:Duncan(1974) n要取得夠大,當process shift為某一特定量時,我們能 detect 此shift 的機率大約50%。 e. g. p=0. 01 shift 到p=0. 05 ,檢測此shift的機率≧ 0. 5, 即要求 可利用Normal Approximation to Binomial分佈 即選擇n 使其upper control limit等於out of control 時的不良率。 14
在P chart中最常使用的是 3 -sigma limit。 注意:fraction nonconforming control chart僅適用於符合二 項分佈假設的fraction nonconforming data。 i. e. 1. P(nonconforming unit)=constant 2. Successive units of production are independent. 當nonconforming units是cluster在一起(即存在correlation) 或是彼此間是相關的,上述之P chart則不再適用。 17
注意二:在variable sample size的C. C. 上,runs或其他的 nonrandom patterns並無太大的意義。 e. g. 假設 p=0. 2 雖然 ,但 21
但由於一般操作員較不易了解standardize control chart的意義,故可採用variable control limits( 如 方法一)給操作員使用,而同時提供standardize C. C. 給quality engineer’s use。 當length of the production run很短時,亦建議採 用standardize control chart。 Fraction nonconforming的C. C. 在非製造業有廣泛 的應用。 e. g. 1. 計算一個pay period中錯誤或遲到的員 薪 水袋總數。 23 2. 貨商沒有準時交貨的次數。
通常在非製造業fraction nonconforming C. C. 常必 須面臨variable sample size的問題。 e. g. 在一個會計年度中的check requests的總數不是 一個常數,其中 e. g. 在上例中,資料來自於一個大的aerospace company的採購部門(每星期負責向公司的 原料廠商issues purchase order),其造成 purchase order錯誤的原因有多種, 如:incorrect part numbers, incorrect delivery dates, incorrect prices or terms, wrong supplier numbers。 24
O. C. curve及ARL: O. C. curve即計算prob. of Type II error(當 接受 的機率)。 為真, (如附表及附圖) 1. 可由Binomial dist. 的c. d. f. 計算。 2. 當p小時(e. g. p<0. 1),n大,可用Poisson approximation。 3. 當p大時,n大,可用normal approximation。 25
Nonconformities(Defects)的C. C. : a. 一個nonconforming item可能擁有數個 nonconformities(defects)。 b. 具有數個nonconformities的item卻未必 被視為nonconforming item(必須視其 defect的嚴重情形,e. g. PC的外殼刮傷 部分)。 1. c chart:total number of nonconformities in a unit. 2. u chart:average number of nonconformities per unit. 在下列例子中,c or u chart均較p chart更實際。 e. g. 1. 在 100公尺的油管中,焊接不好的總數。 2. 在機翼上卯釘斷掉的個數。 27
c chart(control chart for nonconformities) 基本假設:在一個constant size的sample中的nonconformities 的個數為一個Poisson分佈。 1. The number of opportunities or potential location for nonconformities are infinite large. 2. The probability of occurrence of a nonconformities at any location be small and constant. ( 見Poisson Postulates) 3. 對每一個sample皆有相同的inspection unit. ※ nonconformities可以是不同type,只要每一個class的 nonconformities是滿足上述的條件。 ( Rmk:∵independent的Poisson,其和亦為Poisson。) 28
Inspection unit的定義取決於記錄或檢查時的方便性。 e. g. single unit of product、5 units (or 10 units)of product. 令 x=nonconformities 的個數~Poisson ( c ) (若LCL<0,則取LCL=0) 若c未知,則以 估計之。 = observed average # of nonconformities in a preliminary sample of inspection units. Trial control limits 29
Example 2 Inspection unit: 100 printed circuit boards. 其取 26 successive samples of 100 printed circuit boards. (data 見附表)。 (如附圖) 30
1. Sample 6及20在limits之外, Sample 6 new inspector examined the board,不清楚可 能在board中有數種不同的nonconformities。 Sample 20 在焊接機器上,溫度控制的問題。 2. 故將此兩點除外,再造C. C. revised,並加入新的sample 20 筆(如附圖)。雖然process是in control,但每片board的 nonconformities的個數仍偏高。 必須採取management action才能improve此生產過程。 31
一般而言,c chart較p chart更具資訊(因為通常 nonconformities有數種型態)。 e. g. 上例中,for defect data 500 boards的data以 Pareto chart分析之,見附圖,可發現超過60% 的defect數目與焊接不完全及solder cold joints 相關,由此可見,若能isolate及eliminate wave soldering process的問題,則對process yield必 有很大的改進。 32
1. 在此大部分的defects attributable to a few(在此為 two)defect types,此現象我們稱該nonconformities follow a Pareto distribution。 2. 見附表,對printed circuit board的不同type nonconformities的分析。 全部 40個焊接不完全及20個solder cold joints均 發生在part 0001285號上。 此種board在焊接上常會出狀況。 3. 以cause effect diagram分析:如附圖,在上述分析 知主要問題來自solder process,可由此來協助選取 designed experiment的variables以optimize wave soldering的過程。 33
Sample size:n inspection units (e. g. n=2. 5) 1. Revised chart is the observed mean number of nonconformities in the original inspection unit. 34
2. 用u chart If we find c total nonconformities in a sample of n inspection units, then the average number of nonconformities per inspection unit is 每次抽樣中,inspection unit的個數。 Note that c is a Poisson random variable. represents the observed average number of nonconformities per unit in a preliminary set of data. 35
Example 3 PC製造商 1個PC = 1個inspection unit Sample size = 5個inspection units 共有20個samples(如附表)。 雖無lack of statistical control,但u太過大(如附圖)。 結論:Management must take action to improve the process. 36
Alternative Probability Models for Count Data (可用來描述Count data的其他機率模型) E. g. nonconformities是以cluster出現 參考文獻:Jackson(1972), Leavenworth(1976), Gardiner(1987)。 37
100%檢驗(e. g. 布疋、紙捲)。 c chart的中央線及C. L. 均會改變,u chart只有C. L. 隨n改變。 當樣本數是個變數 僅可用u chart,不可用c chart。 另兩種可行之control chart: 1. Average sample size: 2. Standardized(runs and pattern-recognition): 38
Example 4 紡織公司 檢驗染布 inspection unit = 每 50平方公尺 Data如下表,control chart如附圖。 = 500/50 = 400/50 39
不同型式的defects。 將defects依其severity分類並給予weights。 Demerit scheme 基本假設: Class A Defects—Very Serious 1. Completely unfit for service 2. Cannot be easily corrected in the field 3. Cause personal injury or property damage Class B Defects—Serious 1. Suffer a Class A operating failure 2. Will certainly have reduced life or increase maintenance cost 40
Class C Defects—Moderately Serious 1. Fail in service 2. Possibly have reduced life or increased maintenance costs 3. A major defect in finish appearance, or quality of work Class D Defects—Minor defects in finish, appearance, or quality of work 41
定義number of demerits in the inspection unit: weight The demerit weights of Class A-100, Class B-50, Class C-10, and Class D-1 are used fairly widely in practice. 42
A sample of n inspection units is used. Then the number of demerits per unit is (獨立Poisson r. v. ’s 的線性組合) D is the total number of demerits in all n inspection units. Control chart 43
其他可行的方法: 1. Two-class系統 2. 對每一種defect class 採用各自的C. C. For the c chart, the OC curve plots the probability of type II error against the true mean number of defects c. The expression for is where x is a Poisson random variable with parameter c. 44
We will generate the O. C. curve for the c chart in Example 2 Inspection unit: 100 printed circuit boards. 共取 26 successive samples of 100 printed circuit boards. OC curve 45
For the u chart, we may generate the OC curve from ≦n. UCL的最大整數 ≧nl. CL的最小整數 如附圖。 46
Dealing With Low-Defect levels(PPM range≦ 1000) 此時,u及c chart均ineffective ! 修正法:建立另一種C. C. — the time between successive occurrences of defects c、u chart在非製造業亦被廣泛的應用。 47
Choice Between Attributes and Variables Control Charts Variable C. C. : Attribute C. C. : 當quality characteristic是無法測量時(如:color of the item),則採用attribute C. C. 。 48
Attribute C. C. 的優點為,可將數個quality characteristic 同時考慮為nonconforming的標準,較variable C. C. 考 慮 multivariate C. C. 簡單、經濟、省時。 Variable C. C. 則提供比Attribute C. C. 更有用的訊息,較 易找出造成out of control的原因。 就process-capability分析而言,亦較傾向於採用variable C. C. 。 49
如下圖,當process mean 是 時,很少的不良品會被製 造出,若process mean開始向上shift,則在其未到達 chart會出現strong nonrandom pattern,或數 前, 個out of control的點,然而p chart則會等到mean已完 全 shift到 時,才會反應出來。 chart較p chart更powerful! 50
Example 5 某一quality characteristic的標準值為 50, LSP=44,USP=56(3 -sigma specification limit), 即 ,當 時,希望 Sample size on the chart n=9 USL Shift後的值 52
Sample size on the P chart ∵ Specification limit is 3 -sigma. ∴ p=0. 0027 n=79. 13(≒ 80) 結論: chart is less expensive to operate. 53
Guidelines for Implementing Control Charts 1. Choosing the proper type of control charts. 2. Determining which process characteristics to control. 3. Determining where the charts should be implemented in the process. 4. Taking actions to improve processes as the result of SPC/control chart analysis. 5. Selecting data-collection systems and computer software. Remember, control charts are not just for process surveillance; they should be used as an active, online method for reduction of process variability. 54
Choosing the Proper Type of Control Chart A. and R(or and S)charts A new process is coming on stream, or a new product is being manufactured by an existing process. The process has being in operation for some time, but it is chronically in trouble or unable to hold the specified tolerances. The process is in trouble, and the control charts can be useful or diagnostic purposes(troubleshooting). Destructive testing(or other expensive testing procedures) is required. 55
B. Attributes Charts(p charts, c charts, and u charts) C. Control Charts for Individuals inconvenient or impossible to obtain more than one measurement per sample, or repeat measurements automated testing and inspection available very slowly 56
Determining Which Characteristics to Control and Where to Put the Control Charts At the beginning of a control charts program, control charts should be applied to any product characteristics or manufacturing operations believed to be important. Action Taken to Improve Process improvement is the primary objective of statistical process control. 1. Statistical control 2. Capability (如附圖) 57
Selection of Data-Collection Systems and Computer Software There are several sources of free software. In addition to the packages available on various personal computer bulletin boards, the Journal of Quality Technology has published computer programs in either BASIC or FORTRAN since 1969. (如附表) 58
分析結果(Example 1) Back
The Poisson postulates The Poisson distribution can be derived from a set of basic assumptions, sometimes called the Poisson postulates. These assumptions relate to the physical properties of the process under consideration. While, generally speaking, the assumptions are not very easy to verify, they do provide an experimenter with a set of guidelines for considering whether the Poisson will provide a reasonable model. For a more complete treatment of the Poisson postulates, see the classic text by Feller(1968)or Barr and Zehna(1983). Back
Theorem:For each t≧ 0, let be an integer-valued random variable with the following properties. (Think of as denoting the number of arrivals in the time period from time 0 to time t. ) Back
The postulates may also be interpreted as describing the Behavior of objects spatially(for example, movement of insects), giving the Poisson application in spatial Distributions. Back
- Slides: 63