Total Quality Management (TQM)

Transcription

1 Total Quality Management (TQM) Use of statistical techniques for controlling and improving quality and their integration in the management system Statistical Process Control (SPC) Univariate and multivariate control chart Variable control chart Attribute control chart Customer satisfaction measurement Quality dimensions Servqual Customer satisfaction measurement

2 Statistical process control (SPC) Application of statistical methods to the monitoring and control of a process to ensure that it operates at its full potential to produce conforming product. Requisite: measurable output. Key tools: control charts SPC examines a process and the sources of variation in that process using tools that give weight to objective analysis over subjective opinions and that allow the strength of each source to be determined numerically. Early detection and prevention of problems Variations in the process that may affect the quality of the end product or service can be detected and corrected, thus reducing waste as well as the likelihood that problems will be passed on to the customer. Feedback: process cycle time reductions diminished likelihood that the final product will have to be reworked, using SPC data to identify bottlenecks, waiting times, and other sources of delays

3 Univariate control charts Single quality characteristic: the control chart contains a center line that represents the mean value for the in-control process and two other horizontal lines, called the upper control limit (UCL) and the lower control limit (LCL), chosen so that almost all of the data points will fall within these limits as long as the process remains in-control. 38 Average Thickness (mm), X-bar Sample #

4 Basics X~N ( µ, σ ) µ and σ are known Random sample X 1,X,,Xn µ σ µ µ + σ X X µ σ ( ) ~ N,1 X µ P zα zα = 1 α σ α z α -1 1 z α α X µ σ

5 Control limits determination X ~ N(µ, σ ) X µ α α = α σ P z z 1 ( ) α µ α P z σ X z σ = 1 α ( µ ) α µ + α P z σ X z σ = 1 α Control chart LCL UCL = µ z σ α = µ + z σ α Fixing α we decide the size of the interval [LCL, UCL] as a multiple of σ Same as hypothesis testing framework

6 How to construct the control chart Phase I: Build the Historical Data Set (HSD), that is a reference data set for comparisons. Aim: computing the average (the standard of the process), finding LCL and UCL UCL LCL Sample # Detect outliers, clean data: inclusion of atypical observations will increase the variation and distort the correlation among variables CL, UCL and LCL can be provided by process specifications (considered as the standard), if they are available

7 Phase II: Compare new data with the HDS, in order to detect out-of-control situations. Aim: detect out-of-control Average Thickness (mm), X-bar UCL; CL; LCL; Sample # In-control process: All new points are randomly distributed between the LCL and UCL

8 Out-of-control process One or more points lye above the UCL or below the LCL Average Thickness (mm), X-bar UCL; CL; LCL; Sample #

9 The probability limits If: Then: α/ =.1 probability limits chance causes alone were present the probability of a point falling above the upper limit or below the lower limit would be /1 (so small that we should search for an assignable cause if a point would fall outside these limits). The.1 limits may be said to give practical assurances that, if a point falls outside these limits, the variation was caused be an assignable cause. The decision about probability level depends upon the amount of risk the management of the quality control program is willing to take. In general (in the world of quality control) it is customary to use limits that approximate the. standard. The.1 probability limits will be very close to the 3σ limits. For normal distributions, therefore, the 3σ limits are the practical equivalent of.1 probability limits.

10 Dealing with out-of-control findings If a data point falls outside the control limits, we assume that the process is probably out of control and that an investigation is warranted to find and eliminate the cause or causes. Does this mean that when all points fall within the limits, the process is in control? Not necessarily. If the plot looks non-random, that is, if the points exhibit some form of systematic behavior, there is still something wrong. For example, if the first 5 of 3 points fall above the center line and the last 5 fall below the center line, we would wish to know why this is so. Statistical methods to detect sequences or nonrandom patterns can be applied to the interpretation of control charts. To be sure, "in control" implies that all points are between the control limits and they form a random pattern.

11 Warning situations: flags for instability 1. Trend in the point position: something in the process is not due to chance Average Thickness (mm), X-bar UCL; CL; LCL; Sample #. Points lye closer to one of the CL than the other: the standard could be changed Average Thickness (mm), X-bar UCL; CL; LCL; Sample #

12 3. Sequences of increasing and/or decreasing points: something systematic affects the process Average Thickness (mm), X-bar UCL; CL; LCL; Sample #

13 Plus or minus "3 sigma" limits are typical In the U.S., whether X is normally distributed or not, it is an acceptable practice to base the control limits upon a multiple of the standard deviation. Usually this multiple is 3 and thus the limits are called 3-sigma limits. This term is used whether the standard deviation is the universe or population parameter, or some estimate thereof, or simply a "standard value" for control chart purposes. In the U.K. statisticians generally prefer to adhere to probability limits (i.e. to consider the distribution). If the underlying distribution is skewed (e.g. for the presence of outliers in the HDS), say in the positive direction, the 3-sigma limit will fall short of the upper.1 limit, while the lower 3-sigma limit will fall below the.1 limit. This situation means that the risk of looking for assignable causes of positive variation when none exists will be greater than one out of a thousand. But the risk of searching for an assignable cause of negative variation, when none exists, will be reduced. The net result, however, will be an increase in the risk of a chance variation beyond the control limits. How much this risk will be increased will depend on the degree of skewness.

14 Shewhart Control Charts for variables X = characteristic of interest (e.g., thickness) Natural Tolerance Limits (NTL) X ~ N(µ, σ ) µ, σ known from: historical data process standards UNTL = µ w + kσ w Center Line = µ w LNTL = µ w - kσ w k = distance of the control limits from the centre line, expressed in terms of standard deviation units. k = 3 3-sigma control charts accepted standard in industry µ and σ are usually unknown, so they are replaced by their sample estimates: Mean estimate Variance estimate x = i i n R = x x X x-bar chart ( xi x) s j j max j min = i n 1 x-bar and R chart x-bar and S chart

15 x-bar and R control chart X ~ N(µ, σ ) µ, σ unknown m samples of size n: x x = R = x x i = 1,,n; j = 1,,m n i i j j j max j min X = j x m j R j j R = j = 1,,m m x-bar chart: UCL = X + AR CL = X LCL = X A R R chart: UCL = D R CL = R 3 4 LCL = D R A, D 3, D 4 are tabulated for n values

16 Problema: Se la variabile X ha distribuzione simmetrica, sia la media campionaria che la mediana campionaria sono stimatori non distorti per la media della popolazione: Quale stimatore scegliere? La proprietà della correttezza non è sufficiente a raccogliere tutte le caratteristiche rilevanti di uno stimatore.. Efficienza relativa Un secondo criterio consente di discriminare tra stimatori corretti per uno stesso parametro considerando la dispersione delle distribuzioni campionarie degli stimatori attorno al valore centrale. Dati due stimatori, T 1 e T, entrambi corretti per il parametro θ, lo stimatore T 1 sarà più efficiente di T se risulta: Var T Var T ( 1) ( ) < 1 T 1 T 1 più efficiente di T T E ( Tn ) = θ T n

17 Why using R as an estimation of σ? In general n is small (4 or 5) n Rel. eff. 1. ( ) ( ) Var S Var R From a computational point of view it s easier to compute R than s. As n increases, R looses more and more information (single observations)

18 Control chart for process specific Average Thickness (mm), X-bar UCL; CL; LCL; Sample # Control chart for process stability Range Sample # UCL; 9.19 CL; 4.36 LCL;.

19 Example sample 1 Xi 74.1 Ri.38 Piston ring production X = diameter X x i i = = = m m = 5 samples n i = j R j.581 R = = =.3 m x-bar chart: UCL = = CL = 74.1 LCL = = R chart: UCL = =.49 CL =.3 LCL =.3 =

20 Average Thickness (mm), X-bar UCL; CL; 74.1 LCL; Sample # Range Sample # UCL;.49 CL;.3 LCL;.

21 Process capability X ~ N(µ, σ ) Control limits: driven by the natural variability of the process Specification limits: determined externally ˆσ = R d can be considered an estimate of σ d tabulated for values of n R.3 ˆσ = = = d Given new specification limits: 74. ±.5 X ~ N(74.1;.99 ) Estimated fraction of out-of-controls: ( ) ( ) ˆp = Pr x < Pr x > 74.5 = = Φ + 1 Φ = ( 5.15) 1 ( 4.4) = Φ + Φ = = =.% Given these specification limits, about.% of the piston ring produced will be not conform

22 Process capability ratio C p = USL LSL 6σ = capacity index SL = specification limits 6σ = control limits (3σ + 3σ) In our example: σ unknown ˆσ = R d Ĉp = = C p > 1 natural control limit (3σ above and below the average) are internal to the specification limits only very few observations will be out-of-control 1 P = 1 C p = percentage of the specification interval used by the process In our example: 1 ˆP = 1 = 59.5% 1.68 The process uses the 6% of the specification interval

23 x-bar and S control chart If: sample size n is large (n > 1 or 1) or: sample size n is not constant s ( xi x) = i n 1 s c Unbiased estimates for σ and σ c 4 is tabulated for values of n 4 X = j x m j S j j S = j = 1,,m m x-bar chart: UCL = X + A3S CL = X LCL = X A S 3 S chart: UCL = B S CL = S 3 4 LCL = B S A 3, B 3, B 4 tabulated

24 In our example: sample Xi Si j S S = j m = x-bar chart: UCL = = CL = 74.1 LCL = = S chart: UCL =.89.9 =.19 CL =.9 LCL =.9 =

25 Average Thickness (mm), X-bar UCL; CL; 74.1 LCL; Sample # Standard Deviation Sample # UCL;. CL;.9 LCL;.

26 Control Charts for attributes If we cannot represent a particular quality characteristic numerically, or if it is impractical to do so, we then often resort to using a quality characteristic to sort or classify an inspected item into one of two categories. Examples of attributes "conforming units" or "nonconforming units" "non defective" and "defective" p chart: Control charts dealing with the proportion or fraction of defective products np chart: Control charts dealing with the number of defects or nonconformities u chart: Control chart handling defects per unit

27 p control chart p = proportion of non-conformities (in the process) Each unit is a Ber(p) (p = probability of non-conformity) In a sample of size n, with D non-conformities: Number of non-conformities n Pr D x p 1 p x x ( = ) = ( ) n x x = 1,,, n E(D) = np Var(D) = np(1-p) Fraction of non-conformities p chart: ( ) p 1 p UCL = p + 3 n CL = p p 1 p LCL = p 3 n ( ) ˆp = D n E p If p is unknown: UCL = p + 3 CL = p LCL = p 3 ( ˆ) p, Var ( pˆ ) ( ) = = p 1 p n ( ) p 1 p n ( ) p 1 p n p = average proportion of m samples of size n Ex. pag. 7

28 np control chart np chart: ( ) UCL = np + 3 np 1 p CL = np LCL ( ) = np 3 np 1 p If p is unknown: ( ) UCL = np + 3 np 1 p CL = np ( ) LCL = np 3 np 1 p

29 Limits of Shewhart charts Shewhart charts are built considering the process in its steady state, so are note sensible to small changes (generally < 1.5 σ) in the process Example m = 3 Xi ~ N(1; 1) i = 1,, Xi ~ N(11; 1) i = 1,, 3 Average Thickness (mm), X-bar Shewhart 3σ chart UCL; 13. CL; 1. LCL; 7. The chart does not detect the change Sample #

30 Data m = 3 Xi ~ N(1; 1) i = 1,, Xi ~ N(11; 1) i = 1,, 3 sample x i x i Cum sample x x Cum

31 CUSUM control chart (CUmulative SUM) It includes all available information i j= 1 ( i ) C = x µ = cumulative sum of the first i deviations from the mean i 1 Cumulative Sum Sample In-control: Ci = Out-of-control: Ci > Ci < positive drift (upward) negative drift (downward)

32 Tabular CUSUM chart ( ) ( ) C = max ;x µ + k + C + + i i i 1 C = max ; µ k x + C i i i 1 H = 5 σ We want that the chart is sensible for a particular value of OOC µ 1 k = allowance (or slack value) = half of the difference µ - µ 1 Only deviations x i - µ > k are cumulated sample x i C (> H) 5.3 N sample x i C N

33 Adjustment of mean After a change of the mean µ, we can deduce e new reference value: + Ci + µ i > µ = N ˆ µ Ci i > k C H k C H N H = 5σ i = first samples where the process goes out-of-control (i = 9) + Ci ˆµ = µ + k + = + N 5.8 = = The new process is N(11.5; 1)

34 Other CUSUM Standardized CUSUM Unilateral CUSUM Scale CUSUM (for monitoring variability) V-mask CUSUM (not tabular)

35 EWMA control chart Exponentially Weighted Moving Average EWMA = statistic for monitoring the process that averages the data in a way that gives less and less weight to data as they are further removed in time. Comparison of Shewhart and EWMA control chart: Shwehart control chart: the decision regarding the state of control of the process at any time, t, depends solely on the most recent measurement from the process and, of course, the degree of 'trueness' of the estimates of the control limits from historical data. EWMA control chart: the decision depends on an exponentially weighted average of all prior data, including the most recent measurement. By the choice of weighting factor, λ, the EWMA control procedure can be made sensitive to a small or gradual drift in the process, whereas the Shewhart control procedure can only react when the last data point is outside a control limit. Definition of EWMA The statistic that is calculated is: EWMA t = λ Y t + ( 1- λ ) EWMA t-1 for t = 1,,..., n. where: EWMA is the mean of historical data (target) Y t is the observation at time t n is the number of observations to be monitored including EWMA

36 EWMA control chart Exponentially Weighted Moving Average EWMA can be defined as: ( ) z = λ x + 1 λ z i i i 1 Weighted average of the first i observations < λ 1 is a constant that determines the depth of memory of the EWMA ( ) ( ) zi = λ xi + 1 λ λ xi λ zi = ( ) ( ) = λ x + 1 λ z + 1 λ z i i 1 i z i-1 for i = 1 z = µ = λ + λ i 1 j i i i ( λ ) i j + ( λ) j= 1 z x 1 x 1 z ( ) z = λ x + 1 λ µ 1 1 λ(1 - λ) j decrease according to a geometric progression

37 λ = rate at which 'older' data enter into the calculation of the EWMA statistic. λ = 1: only the most recent measurement influences the EWMA (degrades to Shewhart chart). Thus, a large value of λ > 1: more weight to recent data and less weight to older data; λ < 1: more weight to older data. Usually. λ.3 If λ =. for an observation, the previous are.16,.18,.14, EWMA charts are insensible to Normality assumption

38 EWMA standard deviation λ i 1 1 zi λ Variance of z i : σ = σ ( λ) UCL = µ + Lσ CL = µ λ i 1 ( 1 λ) λ LCL = µ Lσ λ i 1 ( 1 λ) λ i 1 (1 - λ) i 1 Then: UCL = µ + Lσ CL = µ LCL = µ Lσ λ λ λ λ

39 In our example: µ = 1 σ = 1 λ =.1 L =.7 ( ) z = λ x + 1 λ z i i i 1 ( ) ( ) z = λ x + 1 λ µ = 1 1 = = ( ) ( ) z = λ x + 1 λ z = 1 = = sample x EWMA sample x EWMA

40 UCL = µ + Lσ CL = µ λ i 1 ( 1 λ) λ Control limits must be computed for each i. LCL = µ Lσ λ i 1 ( 1 λ) λ data points = z i EWMA (mm) CL; Sample #

41 Multivariate control chart Hotelling in 1947 introduced a statistic, called T, which uniquely lends itself to plotting multivariate observations. Hotelling's T is a scalar that combines information from the dispersion and mean of several variables. As in the univariate case, when data are grouped, the T chart can be paired with a chart that displays a measure of variability within the subgroups for all the analyzed characteristics. The combined T and σ (dispersion) charts are thus a multivariate counterpart of the univariate x-bar and S (or x-bar and R) charts.

42 Each chart represents 14 consecutive measurements on the means of four variables. The T chart for means indicates an out-of-control state for groups 1, and The T d chart for dispersions indicates that groups 1, 13 and 14 are also out-of-control. The interpretation is that the multivariate system is suspect. To find an assignable cause, one has to resort to the individual univariate control charts or some other univariate procedure that should accompany this multivariate chart.

43 The Hotelling T statistical distance X = X 1,, X p MVN random variables µ = µ 1,, µ p = Mean vector Σ = Covariance matrix = µ 1 T X Σ X µ Equation of an ellipsoidal contour of X ( ) ( ) T statistic probability distribution 1. T = X µ Σ X µ ~ χ ( ) ( ) 1 p Known parameters µ and Σ. ( ) ( ) ( + ) ( ) p ( n p) p n 1 n 1 1 T = X x S X x ~ F ( ) ( ) ( ) n T = X x S X x ~ B n p n p 1, p, n p Unknown parameters µ and Σ Estimated mean and covariance matrices: x and S not independent from X

44 T distance Multivariate counterpart of the Student's-t statistic. It should be mentioned that for independent variables, the covariance matrix is a diagonal matrix and T becomes proportional to the sum of squared standardized variables. In general, the higher the T value, the more distant is the observation from the mean. The formula for computing the T is: The constant c is the sample size from which the covariance matrix was estimated. The T distances lend themselves readily to graphical displays and as a result the T -chart is the most popular among the multivariate control charts.

45 A simple example T_R α =.1 UCL = 51, 1 - α =.5 UCL =6,99 Out of control UCL.1: 14 UCL.5: 58 When out-of-controls are detected, we should look for responsible variables by univarate Shewart control charts.

46 Detect outliers Graphs and scatter plots of the variables, but some of them may not be as evident Multivariate system Possible (numerical) solutions Purging outliers Trimming Robust estimates

47 Purging outliers Mason, Young, Charting the T in phase I, Phase I Data = X Phase II New data = Y x and S ( i ) ( i ) ( i ) T x = x x S 1 x x UCL UCL = outlier ( n 1) n x H and SH B p n p 1 α;, of the HDS ( i ) ( i H ) H ( i H ) Then: 1 T y = y x S y x UCL we repeat the procedure on purged data and we remove all detected outliers until a homogeneous set of observation (HDS) is obtained x H and SH out-of-control UCL = ( + ) ( ) n( n p) p n 1 n 1 F α; p, n p

48 Trimming Alfaro J.L., Ortega J.F., 8 A Robust Alternative to Hotelling s T Control Chart Using Phase I Trimmed Estimators Data = X y Clean set HDS xtrim Mean vector x C and C efficiency coefficients S Trim Covariance matrix Phase II New data = Y no UCL in phase I 1 T y = y x S y x UCL ( ) ( ) ( ) i i Trim Trim i Trim xtrim and STrim are very good approximations of µ and Σ UCL ~ ( n 1) n B p n p 1 α;,

49 Robust estimates Phase I Data = X HDS Robust estimates of unknown µ and Σ x R and SR no UCL in phase I Phase II New data = Y x R and SR 1 T y = y x S y x UCL ( ) ( ) ( ) i i R R i R UCL Empirical distribution: simulation out-of-control