Chapter 9 Time-Weighted Control Charts. Statistical Quality Control (D. C. Montgomery)

Transcription

1 Chapter 9 Time-Weighted Control Charts 許湘伶 Statistical Quality Control (D. C. Montgomery)

2 Introduction I Shewhart control chart: Chap. 5 7: basic SPC methods Useful in phase I implementation( 完成 ) of SPC

3 Introduction II Disadvantage: use only the information about the process contained in the last sample observation ignores any information given by the entire sequence of points insensitive( 不敏感 ) to small process shift (ex: µ 1 µ 0 1.5σ) less useful in phase II monitoring problems

4 Introduction III Two effective approaches: small process shift cumulative sum (CUSUM) control chart ( 累積和管制圖 ) exponentially weighted moving average (EWMA) control chart ( 指數加權移動平均管制圖 ) Sometimes called time-weighted control chart

5 The Cumulative Sum Control Chart I Monitoring the process mean: x 1,..., x 20 N (10, 1 2 ) x 21,..., x 30 N (11, 1 2 ) µ 0 = 10, µ 1 = 11, σ = 1

6 The Cumulative Sum Control Chart II Table 9.1: small shift; no strong evidence that the process is out of control Shewhart control chart has failed to detect the shift relative small magnitude( 量 ) of the shift Shewhart chart: effective if the shift is 1.5σ to 2σ or larger

7 The Cumulative Sum Control Chart III CUSUM chart: incorporate( 包含 ) all the information in the sequence of sample values plotting the cumulative sums of the deviations of the sample values from a target value n 1 µ 0 : the target for the process mean C i : the cumulatice sum up to and including the ith sample i i 1 C i = ( x j µ 0 ) = ( x i µ 0 )+ ( x j µ 0 ) = ( x i µ 0 )+C i 1 j=1 j=1 combing information from several samples

8 The Cumulative Sum Control Chart IV More effective than Shewhart charts for detecting small process shifts; particular with n = 1 In control at µ 0 : C i is a random walk with mean zero µ 0 µ 1 : µ 0 < µ 1 C i 向上趨勢 µ 0 µ 1 : µ 0 > µ 1 C i 向下趨勢

9 The Cumulative Sum Control Chart V n = 1 x i = x i ; µ 0 = 10 not a control chart; lack of limits

10 The Cumulative Sum Control Chart VI Two ways to represent CUSUMs: 1. tabular (or algorithmic( 規則系統的 )) CUSUM: preferable 2. V-mask form of the CUSUM: not the best representation of a CUSUM

11 Tabular CUSUM for µ I CUSUMs: for individual observations; for the average of rational subgroups x i : the ith observation on the process In control: x i N (µ 0, σ 2 ) (σ: known or a reliable estimate is available) Upper-sided CUSUM: C + i = max [ 0, x i (µ 0 + K) + C i 1 + ] Lower-sided CUSUM: C i = max [ 0, (µ 0 K) x i + Ci 1 ] (Minitab: Ci = min(0, x i µ 0 + k + Ci 1 ) C i 0) C + 0 = C0 = 0 µ 1 = µ 0 + δσ Reference value(slack value 寬鬆變數值 ): K = δ 2 σ = µ 1 µ 0 2

12 Tabular CUSUM for µ II C + i (or C i ) > H the process is out of control H : decision interval K, H : two important parameters for CUSUM 一般取 : K = µ 1 µ 0, H = 5σ 2 Using ARL to choose (K, H )

13 Tabular CUSUM for µ III

14 Tabular CUSUM for µ IV require(qcc) data = read.table("table9_1.csv",sep=",",header=t) q = cusum(data,sizes=1,center=10,std.dev=1) summary(q) plot(q, chart.all=false)

15 Tabular CUSUM for µ V

16 Tabular CUSUM for µ VI vertical bar: the value of C i +, Ci solid dot: x i =the value of the observations

17 Tabular CUSUM for µ VII Sample 29: C + 29 = 5.28 > H = 5 且 N + = 7 (the first period) the shift occurred between periods (29-7=22)

18 Tabular CUSUM for µ VIII helpful in determining when the assignable cause has occurred count backward from the out-of-control signal to the time period process shift: adjustment back to the target value µ 0 ˆµ = µ 0 + K + C + i, if C + N + i > H µ 0 K C i N, if C i > H Table 9.2: ˆµ = µ 0 + K + C + i N + = = 11.25

19 Tabular CUSUM for µ IX successive values of C i +, Ci are not independent: sensitizing rules; zone rules; run tests Choosing of K, H : ARL Define K = kσ, H = hσ Using h = 4 or h = 5, k = 1/2: generally provide a CUSUM that has good ARL properties against a shift about 1σ in the process mean

20 Tabular CUSUM for µ X a 1σ shift: require samples to detect the shift for a Shewhart charts

21 Tabular CUSUM for µ XI h = 4.77 ARL 0 = 370 samples (matches for a Shewhart charts with 3σ limits)

22 Tabular CUSUM for µ XII Siegmund s approximation: (one-sided CUSUM) ARL = exp( 2 b) + 2 b Upper one-sided CUSUM: = δ k Lower one-sided CUSUM: = δ k δ = (µ 1 µ 0 )/σ (shift in the mean, in the units of σ) b = h if = 0 ARL = b 2 δ = 0 approximate ARL 0 δ 0 approximate ARL 1

23 Tabular CUSUM for µ XIII ARL +, ARL : ARL of the two one-sided statistics ARL of the two-sided CUSUM: 1 ARL = 1 ARL ARL Ex: k = 1/2, h = 5, δ = 0; = δ k = 1/2; b = h = ARL + 0 exp [ 2( 1/2)(6.166)] + 2( 1/2) ( 1/2) 2 = ARL + 0 = ARL 0 (symmetric) 1 ARL 0 = 1 ARL ARL 0 (In Table 9.3: ARL 0 = 465) ARL 0 = 469.1

24 Tabular CUSUM for µ XIV Ex: the mean shift by 2σ δ = 2 = 1.5(upper), = 2.5(lower) ARL 1 = 3.89

25 Standardized CUSUM I standardize x i : y i = x i µ 0 σ Standardized Two-Sided CUSUM [ ] C i + = max 0, y i k + C i 1 + [ ] Ci = max 0, k y i + Ci 1 h, k: the choices do not depend on σ Standardized CUSUM: for controlling variability

26 Improving CUSUM for Large Shifts I CUSUM: effective in detecting small shift; not as effective as the Shewhart chart in detecting large shift Approach for improving the ability to detect large process shifts: combined CUSUM-Shewhart procedure for on-line control Shewhart control limits: ±3.5σ from CL or the target µ0 An out-of-control signal on either (or both) charts: an action signal

27 Improving CUSUM for Large Shifts II FIR(Fast Initial Response 快速起始反應 or Headstart( 有利的開端 ) Feature): improve the sensitivity of a CUSUM at process start-up C + 0, C 0 : some nonzero value: 一般取 H /2 (50% headstart)

28 Improving CUSUM for Large Shifts III decline( 下降 ) rapidly to zero from the starting value Period 2 for C i + headstart or Period 3 for C i : unaffected by the

29 Improving CUSUM for Large Shifts IV Benefit: quickly drop to zero if in control (little effect by the headstart) detect the shift more quickly shorter out-of-control ARL values

30 Improving CUSUM for Large Shifts V If the process is in control when the CUSUM is reset but shifts out of control later, the more appropriate ARL is Column (a) in Table 9.5

31 CUSUM for process variability I Hawkins (1981, 1993a): a new standardized quantity v i = yi (H.W.), y i = x i µ 0 σ sensitive to variance changes rather than mean changes approx. In control: v i N (0, 1) (in control distribution) CUSUM for v i : (S + 0 = S 0 = 0 unless a FIR feature is used) S + i = max[0, v i k + S + i 1 ] S i = max[0, k v i + S i 1 ] choose h, k and interpretation is similar to the CUSUM of the mean

32 Self-Starting CUSUM I Hawkins (1987): easy to implement( 執行 ) x n : the average of the first n observations w n : the sum of squared deviations from x n w n = Recursive form n (x i x n ) 2 i=1 x n = x n 1 + x n x n 1 n w n = w n 1 + (n 1)(x n x n 1 ) 2 n

33 Self-Starting CUSUM II s 2 n = w n /(n 1): sample variance of the first n observations Standardize new process observation: (n 3) T n = x n x n 1 s n 1 n 1 n T n T n 1 F( ): the cumulative t distribution with n 1 d.f. ( ) n 1 n 1 n 1 P(T n t) = P( n T n t n ) = F t n Φ 1 : the inverse normal cumulative distribution n 1 U n = Φ 1 [F(a n T n )], a n = n plot all values of U n, n 3 on a N(0,1) CUSUM

34 Self-Starting CUSUM III The users of a self-starting CUSUM should take investigative and corrective action immediately following an out-of-control signal. If the process is not adjusted and the CUSUM reset, the self-starting CUSUM will turn back downward( 往後推移地 ).

35 EWMA I Exponentially Weighted Moving Average Control Chart Exponentially weighted moving average: z i = λx i + (1 λ)z i 1, 0 < λ 1 z 0 = µ 0 (target value) Sometimes: z 0 = x z i : a weighted average of all previous sample means

36 EWMA II z i = λx i + (1 λ)[λx i 1 + (1 λ)z i 2 ] = λx i + λ(1 λ)x i 1 + (1 λ) 2 z i 2 i 1 z i = λ (1 λ) j x i j + (1 λ) i z 0 j=0 weights λ(1 λ) j : decreasing geometrically with the age of the sample mean a weighted average of all past and current observations insensitive to the normality assumption

37 EWMA III i 1 λ j=0 (1 λ) j = 1 (1 λ) i EWMA also called a geometric moving average (GMA) If x i are independent r.v. with variance σ 2 ( ) λ [ σz 2 i = σ 2 1 (1 λ) 2i] 2 λ

38 EWMA IV EWMA control chart UCL = µ 0 + Lσ Central Line = µ 0 LCL = µ 0 Lσ ( ) λ [1 (1 λ) 2 λ 2i ] ( ) λ [1 (1 λ) 2 λ 2i ] [1 (1 λ) 2i ] i gets large 1: approach steady-state ( ) λ UCL = µ 0 + Lσ 2 λ ( ) λ LCL = µ 0 Lσ 2 λ

39 EWMA V Example 9.2 λ = 0.10; L = 2.7 µ 0 = 10; σ = 1 z 1 = λx 1 + (1 λ)z 0 = 0.1(9.45) + 0.9(10) = 9.945

40 EWMA VI i = 1 UCL = µ 0 + Lσ = CL = µ 0 = 10 ( LCL = µ 0 Lσ = 9.37 i = 2 UCL = CL = µ 0 = 10 LCL = 9.64 ( ) λ [1 (1 λ) 2i ] 2 λ λ 2 λ ) [1 (1 λ) 2i ]

41 EWMA VII effective against small process shifts L, λ: choose to give ARL performance for the EWMA control chart that closely approximates CUSUM ARL performances for detecting small shifts General: λ [0.05, 0.25] work well in practice ( 通常取 λ = 0.05, 0.10, 0.20) a good rule of thumb( 經驗法則 ): use small values of λ to detect small shifts { L = 3 :works well, when the larger value of λ 2.6 L 2.8 :when λ is small (λ 0.1) λ = 0.1, L = 2.7 ARL 0 500; ARL 1 = 10.3 (detecting a shift of σ, 約與 h = 5, k = 1/2 的 cusum 相同 )

42 EWMA VIII

43 EWMA for Variability I Monitoring Variability: exponentially weighted mean square error(ewms) x i N (µ, σ 2 ) S 2 i = λ(x i µ) 2 + (1 λ)s 2 i 1 E(S 2 i ) = σ 2 (for large i) if x i are independent and normally distributed σ 0 : in-control or target value S 2 i σ 2 approx. χ 2 v, v = (2 λ)/λ

44 EWMA for Variability II EWRMS: exponentially weighted root mean square χ 2 v,α/2 UCL = σ 0 v χ 2 v,1 α/2 LCL = σ 0 v MacGregor and Harris (1993): the EWMS can be sensitive to shifts in both the process mean and the standard deviation Replace µ with ˆµ i at each point in time; a logical estimate of µ: the ordinary EWMA z i S 2 i = λ(x i z i ) 2 + (1 λ)s 2 i 1

45 Moving average control chart I The moving average of span w at time i: M i = x i + x i x i w+1 w V (M i ) = 1 i w 2 V (x j ) = 1 w 2 j=i w+1 The three-sigma control limit for M i UCL = µ 0 + 3σ w LCL = µ 0 3σ 2 i j=i w+1 σ 2 = σ2 w

46 Moving average control chart II Example 9.3: w = 5