Multiplication principle
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乘數原理 (multiplication principle). Fundamentals of Probability and Statistics. 假設一試驗 E1 有 n1 種可能結果,對此 n1 個可能結果進行另一試驗 E2 ,又有 n2 可能結果,則此組合試驗 E1-E2 ,共有 n1  n2 可能結果。. 抽出不放回. 抽出放回. 考慮順序. 不考慮順序. n 個事物中取 r 個之四種情況. Fundamentals of Probability and Statistics. (permutation). (combination). 問題 1.

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乘數原理 (multiplication principle)

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Multiplication principle

乘數原理(multiplication principle)

Fundamentals of Probability and Statistics

  • 假設一試驗E1有n1種可能結果,對此n1個可能結果進行另一試驗E2,又有n2可能結果,則此組合試驗E1-E2,共有n1n2可能結果。

Quality Control


Multiplication principle

抽出不放回

抽出放回

考慮順序

不考慮順序

n個事物中取r個之四種情況

Fundamentals of Probability and Statistics

(permutation)

(combination)

Quality Control


Multiplication principle

問題1

Fundamentals of Probability and Statistics

  • 某西餐廳推出系列套餐組合,第一道湯品共有5種,第二道沙拉共有4種,第三道主菜共有10種,最後附餐則有5種。試問此系列套餐共有幾種搭配方式?

  • 某公司推出兒童電子琴玩具,其中有兩個可撥0~9的數字鍵,此兩鍵任意撥一號碼即發一種特殊音效,試問此玩具共可發出多少種音效?

Quality Control


Multiplication principle

問題2

Fundamentals of Probability and Statistics

  • 汽車牌照號碼前兩碼可為英文字母或數字,後四碼可為數字。試問此設計共可容納多少車輛?

  • 六個學生要坐一排有10張椅子的座位上,有多少種不同坐法?

  • 自生產線上抽出10個產品檢驗,其中檢查出三個不良品。然而此三個不良品可能是第1、2、3個,也有可能是第2、5、8個…。試問共有多少種可能情況?

Quality Control


Probability distributions

Probability Distributions

Fundamentals of Probability and Statistics

  • Definitions

    • Sample A collection of measurements selected from some larger source or population.

    • Probability Distribution A mathematical model that relates the value of the variable with the probability of occurrence of that value in the population.

    • Random Variable variable that can take on different values in the population according to some “random” mechanism.

Quality Control


Two types of probability distributions

Two Types of Probability Distributions

Fundamentals of Probability and Statistics

  • Continuous When a variable being measured is expressed on a continuous scale, its probability distribution is called a continuous distribution. The probability distribution of piston-ring diameter is continuous.

  • Discrete When the parameter being measured can only take on certain values, such as the integers 0, 1, 2, …, the probability distribution is called a discrete distribution. The distribution of the number of nonconformities would be a discrete distribution.

Quality Control


Important discrete distributions

Important Discrete Distributions

Fundamentals of Probability and Statistics

  • The Hypergeometric Distribution

  • The Binomial Distribution

  • The Poisson Distribution

  • The Pascal and Related Distributions

Quality Control


Hypergeometric distribution

Hypergeometric Distribution

Fundamentals of Probability and Statistics

The Hypergeometric distributionis

N:母體數; D:某特定類別之物品數; n:樣本數

x:抽出n個物品中正好有x個是屬於D類之個數

Quality Control


Example

Example

Fundamentals of Probability and Statistics

  • 有100支保險絲,隨機抽5支,若在某電壓下均燒斷,則接受。若已知該批有20支不良保險絲,則允收機率為何?

Quality Control


Bernoulli trial

Bernoulli trial

Fundamentals of Probability and Statistics

An experiment with two, and only two, possible outcomes. A random variable X has a Bernoulli(p) distribution if

Quality Control


The binomial distribution 1

Fundamentals of Probability and Statistics

The Binomial Distribution1

A quality characteristic follows a binomial

distribution if:

1. All trials are independent.

2. Each outcome is either a “success” or “failure”.

  • The probability of success on any trial is given as p. The probability of a failure is 1- p.

    4. The probability of a success is constant.

Quality Control


The binomial distribution 2

Fundamentals of Probability and Statistics

The Binomial Distribution2

The binomial distribution with parameters

n 0 and 0 < p < 1, is

The mean and variance of the binomial distribution are

Quality Control


Example1

Example

Fundamentals of Probability and Statistics

  • 經長時間觀察,在一項製程中所生產的產品,平均10項產品中會有1個不良品。今自生產線上抽出5項檢驗,則最多有一項不良品之機率為何?

Quality Control


Poisson distribution 1

Fundamentals of Probability and Statistics

Poisson Distribution1

  • 在一定連續區間內計算變化之次數

  • 期本假設

    • 在極短區(h)內正好發生一次的變化機率接近λh

    • 發生和其他不重疊的區間之變化次數無關

    • 基本上在極短時間內發生兩次或兩次以上變化的機率幾乎為零

Quality Control


Poisson distribution 2

Fundamentals of Probability and Statistics

Poisson Distribution2

The Poisson distribution is

Where the parameter  > 0. The mean and variance of the Poisson distribution are

Quality Control


The poisson distribution 3

Fundamentals of Probability and Statistics

The Poisson Distribution3

  • The Poisson distribution is useful in quality engineering

    • Typical model of the number of defects or nonconformities that occur in a unit of product.

    • Any random phenomenon that occurs on a “per unit” basis is often well approximated by the Poisson distribution.

Quality Control


Example2

Example

Fundamentals of Probability and Statistics

  • 某品牌錄音帶之瑕疵數平均1200呎有一個,則4800呎裏有0個瑕疵之機率為何?

Quality Control


Important continuous distributions

Fundamentals of Probability and Statistics

Important Continuous Distributions

  • The Normal Distribution

  • The Exponential Distribution

  • The Gamma Distribution

  • The Weibull Distribution

Quality Control


The normal distribution 1

Fundamentals of Probability and Statistics

The Normal Distribution1

The normal distribution

is an important

continuous distribution.

  • Symmetric, bell-shaped

  • Mean, 

  • Standard deviation, 

Quality Control


The normal distribution 2

Fundamentals of Probability and Statistics

The Normal Distribution2

For a population that is

normally distributed:

  • approx. 68% of the data will lie within 1 standard deviation of the mean;

  • approx. 95% of the data will lie within 2 standard deviations of the mean, and

  • approx. 99.7% of the data will lie within 3 standard deviations of the mean.

Quality Control


Standard normal distribution 1

Fundamentals of Probability and Statistics

Standard normal distribution1

  • Many situations will involve data that is normally distributed. We will often want to find probabilities of events occuring or percentages of nonconformities, etc.. A standardized normal random variable is:

Quality Control


Standard normal distribution 2

Fundamentals of Probability and Statistics

Standard normal distribution2

  • Z is normally distributed with mean 0 and standard deviation, 1.

  • Use the standard normal distribution to find probabilities when the original population or sample of interest is normally distributed.

  • Tables, calculators are useful.

Quality Control


Example3

Fundamentals of Probability and Statistics

Example

  • The tensile strength of paper is modeled by a normal distribution with a mean of 35 lbs/in2 and a standard deviation of 2 lbs/in2.

  • What is the probability that the tensile strength of a sample is less than 40 lbs/in2?

  • If the specifications require the tensile strength to exceed 30 lbs/in2, what proportion of the samples is scrapped?

Quality Control


Central limit theorem

Fundamentals of Probability and Statistics

Central Limit Theorem

  • 若 是由具有平均數μ及變異數σ2的分配所抽出之一組樣本數的平均數,則當n時

Quality Control


The exponential distribution 1

The Exponential Distribution1

Fundamentals of Probability and Statistics

  • The exponential distribution is widely used in the field of reliability engineering.

  • The exponential distribution is

    The mean and variance are

Quality Control


The exponential distribution 2

The Exponential Distribution2

Fundamentals of Probability and Statistics

  • The relationship between the exponential distribution and Poisson distributionx=0 implies that there are no occurrences of the event in (0,t], and

  • P(0) is the probability that the interval to the first occurrence is greater than t 

Quality Control


Some useful approximations

Fundamentals of Probability and Statistics

Some Useful Approximations

  • In certain quality control problems, it is sometimes useful to approximate one probability distribution with another. This is particularly useful if the original distribution is difficult to manipulate analytically.

  • Some approximations:

    • Binomial approximation to the hypergeometric

    • Poisson approximation to the binomial

    • Normal approximation to the binomial

Quality Control


Multiplication principle

生產前後

之檢驗

生產期間之

矯正措施

將品質設計

於製程中

允收抽樣

製程管制

連續改進

進步最少

進步最多

品質管制的各種層面

Quality Control


Multiplication principle

投入

轉換

產出

允收抽樣

允收抽樣

製程管制

檢驗

  • 檢驗多少量?檢驗多少次

  • 要在什麼地點進行製程檢驗

  • 應進行集中檢驗或現場檢驗

Quality Control


Multiplication principle

檢驗成本

成本

總成本

檢驗成本

不良品通

過的成本

最佳檢驗數

Quality Control


Multiplication principle

在製程中何處進行檢驗

  • 進料

  • 最終產品

  • 在昂貴作業前

  • 在不可變更的製程之前

  • 在包裝製程之前

Quality Control


Multiplication principle

統計的製程管制

  • 管制程序

    • 定義

    • 衡量

    • 與標準比較

    • 評估

    • 採取矯正措施

    • 評估矯正措施

Quality Control


Chance and assignable causes of quality variation

Chance and Assignable Causes of Quality Variation

  • A process that is operating with only chance causes of variation present is said to be in statistical control.

  • A process that is operating in the presence of assignable causesis said to be out of control.

  • The eventual goal of SPC is reduction or elimination of variability in the process by identification of assignable causes.

Quality Control


Multiplication principle

管制圖的目的

  • 發現製程有無引起變異的非機遇原因,進而針對原因予以消除,維持製程的穩定,防止異常原因之再次發生。

Quality Control


Multiplication principle

管制圖的種類 —數據的性質分類1

  • 計量值管制圖

    • These charts are applied to data that follow a continuous distribution (measurement data).

  • 計數值管制圖

    • These charts are applied to data that follow a discrete distribution.

Quality Control


Multiplication principle

計量值質管制圖

平均數與全距管制圖

中位數與全距管制圖

平均數與標準差管制圖

個別值平均數與全距管制圖

複式管制圖

機率管制圖

趨勢管制圖…

計數值管制圖

不良率管制圖(p管制圖)

不良數管制圖(np管制圖)

缺點數管制圖(c管制圖)

平均缺點數管圖(u管制圖)

管制圖的種類 —依數據的性質分類2

Quality Control


Multiplication principle

計量值與計數值管圖之優缺點

Quality Control


Multiplication principle

製程分析用管制圖

為決定方針用

為工程解析用

為工程能力研究用

為製程管制之準備用

管制用管制圖

用以控制製程的品質,於繪製完成後將管制界限延長,就每日管制特性的數據計算統計量數,並予以點繪於圖上,以管制製程,如不在管制狀態即採取下列措施:

追查異常原因

迅速消除此種原因

研究此種原因使其不再發生

管制圖的種類 —依管制圖的用途分類

Quality Control


Multiplication principle

編製管制圖的步驟

1. 決定品質特性

2. 決定合理之樣組數

3. 抽驗並收集樣本觀察值

4. 計算試驗用之管制界限與中心線

5. 建立修訂後之管制界限與中心線

Quality Control


X r 3

計量值管制圖—X-R管制圖(管制界限為±3個標準差)

UCL:管制上限(upper control limit)

CL:管制中心線(center line)

LCL:管制下限(lower control limit)

Quality Control


Control charts for and r 1

Control Charts for and R (1)

Notation for variables control charts

  • n - size of the sample (sometimes called a subgroup) chosen at a point in time

  • m - number of samples selected

  • = average of the observations in the ith sample (where i = 1, 2, ..., m)

  • = grand average or “average of the averages (this value is used as the center line of the control chart)

Quality Control


Control charts for and r 2

Control Charts for and R(2)

Notation and values

  • Ri = range of the values in the ith sample

    Ri = xmax - xmin

  • = average range for all m samples

  •  is the trueprocess mean

  • is the trueprocess standard deviation

Quality Control


Control charts for and r 3

Control Charts for and R(3)

Statistical Basis of the Charts

  • Assume the quality characteristic of interest is normally distributed with mean , and standard deviation, .

  • If x1, x2, …, xn is a sample of size n, then he average of this sample is

  • is normally distributed with mean, , and standard deviation,

Quality Control


Control charts for and r 4

Control Charts for and R (4)

Statistical Basis of the Charts

  • The probability is 1 -  that any sample mean will fall between

  • The above can be used as upper and lower control limits on a control chart for sample means, if the process parameters are known.

Quality Control


Control charts for and r 5

Control Charts for and R (5)

Estimating the Process Standard Deviation

  • The process standard deviation can be estimated using a function of the sample average range.

  • This is an unbiased estimator of 

Relative range: W=R/σ

E(W)=E(R/σ)=d2

Quality Control


Control charts for and r 6

Control Charts for and R (6)

Control Limits for the chart

Quality Control


Control charts for and r 7

^

σR=d3(R/d2)

Control Charts for and R (7)

R=Wσ

Var(W)=d32

σR=d3σ

Quality Control


Control charts for and r 8

Control Charts for and R (8)

Control Limits for the R chart

  • D3 and D4 are found in Appendix VI for various values of n.

Quality Control


Interpretation of and r charts

Interpretation of and R Charts

  • In interpreting patterns on the X bar chart, we must first determinewhether or not theR chartis in control.

  • Patterns of the plotted points will provide useful diagnostic information on the process, and this information can be used to make process modifications that reduce variability.

    • Cyclic Patterns

    • Mixture

    • Shift in process level

    • Trend

    • Stratification

Quality Control


Multiplication principle

Quality Control


Multiplication principle

平均數-全距管制圖三個標準差管制界限相關因子表

Quality Control


The shewhart control chart for individual measurements 1

The Shewhart Control Chart for Individual Measurements(1)

  • What if you could not get a sample size greater than 1 (n =1)? Examples include

    • Automated inspection and measurement technology is used, and every unit manufactured is analyzed. There is no basis for rational subgrouping.

    • The production rate is very slow, and it is inconvenient to allow samples sizes of N > 1 to accumulate before analysis

    • Repeat measurements on the process differ only because of laboratory or analysis error, as in many chemical processes.

  • The X and MR charts are useful for samples of sizes

    n = 1.

Quality Control


Multiplication principle

個別值與移動全距管制圖(2)

  • 使用理用

    • 產品需經很久的時間才能製造完成

    • 分析或測定一件產品的品質較為麻煩且費時

    • 產品係非常貴重的物品

    • 產品品質頗為均勻(自動化生產)

    • 屬破壞性試驗

    • 在於管制製造條件如溫度、壓力、濕度…等

Quality Control


The shewhart control chart for individual measurements 3

The Shewhart Control Chart for Individual Measurements(3)

Moving Range Chart

  • The moving range (MR) is defined as the absolute difference between two successive observations:

    MRi = |xi - xi-1|

    which will indicate possible shifts or changes in the process from one observation to the next.

Quality Control


The shewhart control chart for individual measurements 4

The Shewhart Control Chart for Individual Measurements(4)

X and Moving Range Charts

  • The X chart is the plot of the individual observations. The control limits are

where

Quality Control


The shewhart control chart for individual measurements 5

The Shewhart Control Chart for Individual Measurements(5)

X and Moving Range Charts

  • The control limits on the moving range chart are:

Quality Control


The shewhart control chart for individual measurements 51

The Shewhart Control Chart for Individual Measurements(5)

Example

Ten successive heats of a steel alloy are tested for hardness. The resulting data are

Heat HardnessHeat Hardness

152 6 52

251 750

354 851

455 958

550 1051

Quality Control


The shewhart control chart for individual measurements 6

The Shewhart Control Chart for Individual Measurements(6)

Example

Quality Control


The shewhart control chart for individual measurements 7

The Shewhart Control Chart for Individual Measurements(7)

Interpretation of the Charts

  • X Charts can be interpreted similar to charts. MR chartscannotbe interpreted the same as or R charts.

  • Since the MR chart plots data that are “correlated” with one another, then looking for patterns on the chart does not make sense.

  • MR chartcannotreally supply useful information about process variability.

  • More emphasis should be placed on interpretation of the X chart.

Quality Control


The shewhart control chart for individual measurements 8

The Shewhart Control Chart for Individual Measurements(8)

Average Run Lengths (ARL=1/(1-β))

  • The ability of the individuals control chart to detect small shifts is very poor.

Quality Control


The shewhart control chart for individual measurements 9

The Shewhart Control Chart for Individual Measurements(9)

  • The normality assumption is often taken for granted.

  • When using the individuals chart, the normality assumption is very important to chart performance.

Quality Control


Multiplication principle

  • 計數值管制圖1

Quality Control


Multiplication principle

計數值管制圖2

Quality Control


Multiplication principle

管制圖的判讀1

1) One point plots outside the 3-sigma control limits.

2) Two out of three consecutive points plot beyond the 2-sigma warning limits.

3) Four out of five consecutive points plot at a distance of 1-sigma or beyond from the center line.

4) Eight consecutive points plot on one side of the center line.

5) Six points in a row steadily increasing or decreasing.

Quality Control


Multiplication principle

管制圖的判讀2

6) Fifteen points in a row in zone C (both above and below the center line).

7) Fourteen points in a row steadily increasing or decreasing.

8) Eight points in a row on both sides of the center line with none in zone C.

9) An unusual or nonrandom pattern in the data.

10) One or more points near a warning or control limit.

Quality Control


Multiplication principle

連串檢定方法

  • 連串(run):指擁有某種特性的一連串觀測值。

  • 連串檢定方法:

    • 檢查連串的向上與向下的連串數目

    • 計算中位數上方與下方的連串數目

第10章 品質管制


Multiplication principle

連串檢定方法之步驟

  • 將數字資料轉換並計算個別的連串數。

    • 轉換成一系列的U 與D(即代表「上」與「下」)

    • 轉換為一系列的A 與B(即「大於」中位數與「小於」中位數)。

  • 計算連串數期望值與標準差。

  • 區分出連串數的觀測值與期望值不同後,計算標準差數字z。

  • 將數字與完全隨機系列的期望連串數比較。

  • 將z 值與±2(95.5% 的z值)或其他期望值(例如,95% 的±1.96 或98% 的±2.33)互相比較。

    • 若z 值超過期望界限,則表示存在非隨機模式。

第10章 品質管制


Multiplication principle

連串數的抽樣分配可用來區分機遇變異及其他模式

第10章 品質管制


Multiplication principle

連串檢定公式彙整

其中

N =總觀測數

r = A 與B 或U 與D 的觀察連串數,取決於所涉及的檢定。

第10章 品質管制


Multiplication principle

例題6—連串檢定之計算

第10章 品質管制


Multiplication principle

例題6—解答

第10章 品質管制


Choice of control limits

Choice of Control Limits

“99.73% of the Data”

  • If approximately 99.73% of the data lies within 3 of the mean (i.e., 99.73% of the data should lie within the control limits), then 1 - 0.9973 = 0.0027 or 0.27% of the data can fall outside 3 (or 0.27% of the data lies outside the control limits).

  • 0.0027 is the probability of a Type I error or a false alarm in this situation.

Quality Control


Types of error 1

Types of Error1

  • Type I error - rejecting the null hypothesis when it is true.

  • The risk of a point falling beyond the control limits, indicating an out-of-control condition when no assignable cause is present.

  • Pr(Type I error) = . Sometimes called the producer’s risk.

  • 可能造成不必要的預防成本與鑑定成本

Quality Control


Types of error 2

Types of Error2

  • Type II error - not rejecting the null hypothesis when it is false.

  • The risk of a point falling between the control limits when the process is really out of control.

  • Pr(Type II error) = . Sometimes called the consumer’s risk.

  • 可能造成不必要的內、外部失敗成本

Quality Control


Multiplication principle

管制界限設定為±3σ的理由

  • 在統計推論時,將發生型I或型II誤差。而在1930年代,所有工業型態之廣大經驗指出,±3σ界限提供此兩種錯誤型態所產生的成本之間的經濟平衡。

Quality Control


Multiplication principle

規格界限與管制上下限

  • 規格界限

    • 乃是顧客要求,而工程設計建立的規範。指示產品必須落在此範圍內才會被接受。

  • 管制界限

    • 乃是統計上的界限。指示樣本平均或樣本全距因隨機變化可跨越的範圍界限。

Quality Control


Multiplication principle

規格界限與管制上下限之比較

  • 規格界限與管制界限並無直接相關,既使製程落在管制界限內,亦可能不合規範要求。

  • 管制界限<規格界限

    • 管制界限優良,製程能滿足標準。

  • 管制界限>規格界限

    • 離散大時,改善製程,減小變異。

    • 偏心時,改善固定性變異,移動平均值至規格中心。

    • 無法改善製程時亦可考慮修改標準。

Quality Control


Multiplication principle

UCL

UCL

UCL

LCL

LCL

製程額外的改進

LCL

製程集中且穩定

製程不集中

且不穩定

使用管制圖追蹤改進過程

Quality Control


Process capability

製程能力(process capability)

  • 意義

    • 對於穩定之製程所持有之特定成果,能夠合理達成之能力界限。

  • 製程能力的表示法

    • 製程準確度(Ca值)

    • 製程精密度(Cp值)製程能力比(process capability ratio,PCR)

    • 製程能力指數(Cpk值)

Quality Control


Ca capability of accuracy

製程平均值-規格中心值

Ca=

規格公差/2

=

X-μ

=

T/2

製程準確度Ca(capability of accuracy)

T:規格公差=規格上限-規格下限

Quality Control


Multiplication principle

製程準確度之評價方法

  • A級

    • 小於等於12.5%理想的狀態,故維持現狀。

  • B級

    • 大於12.5%,小於等於25%儘可能調整改進至A級。

  • C級

    • 大於25%,小於等於50%應立即檢討並加以改善

  • D級

    • 大於50%,小於100%採取緊急措施,並全面檢討,必要時應考停止生產。

Quality Control


Cp capability of precision process capability ratio pcr

雙邊規格:

規格公差

T

=

Cp=

6個估計標準差

單邊規格:

=

規格上限-製程平均值

SU-X

=

Cp=

3個估計標準差

=

製程平均值-規格下限

X-SL

=

Cp=

3個估計標準差

製程精密度Cp (capability of precision) (process capability ratio,PCR)

Quality Control


Multiplication principle

製程精密度評價方式

  • A+級

    • 大於等於1.67可考慮放寛產品變異,尋求管理的簡單化或成本降低方法。

  • A級

    • 小於1.67,大於等於1.33理想的狀態,故維持現狀。

  • B級

    • 小於1.33,大於等於1.00確實進行製程管制,儘可能改善為A級。

  • C級

    • 小於1.00,大於等0.67產生不良品,產品須全數選別,並管理改善製程。

  • D級

    • 小於0.67採取緊急對策,進行品質改善,探求原因並重新檢討規格。

Quality Control


Multiplication principle

{

}

X-SL

SU-X

Cpk=min

=

=

製程能力指數Cpk

Cpk=(1-|Ca|)Cp

當Ca=0時Cpk=Cp

Quality Control


Multiplication principle

製程能力指數評價方式

  • A級

    • 大於等於1.33製程能力足夠

  • B級

    • 小於1.33,大於等於1.00能力尚可應再努力

  • C級

    • 小於1.00應加以改善

Quality Control


Multiplication principle

製程能力指標的限制

  • 如果製程並不穩定,則製程能力指標將沒有意義。

  • 若製程產出並非常態分配,則對製程產出的推論將不正確。

  • 若製程不準確,則使用Cp指標將造成誤導。

Quality Control


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