1. INTRODUCTION Solu:on to Errors: Increase sample size Varia8on
• Quality of design = degree to which quality characteris:cs are embedded into the product specifica:ons Sampling plans • Chance causes (= common causes) = small, unavoidable, random changes in the process; can only be re-
(Am vs Jap fridge, …) • Sampling by variables: numerical scale (con:nuous): more info, more complex moved by changing the exis:ng process; Assignable causes (= special causes) = Varia:ons in the process
• Quality of conformance = degree to which a product is manufactured according to the specifica:ons (tar- • Sampling by ayributes: present or not (yes/no scale): cheaper, requires larger sample size that have a specific cause; special circumstances àIf only common causes of variability, then the process
gets & balances determined in design) -> directly measurable, not directly associated with customers per- ATTRIBUTES (𝑟 = rejec:on rate) is ‘in (sta:s:cal) control’, or ‘stable’ ó external variability: that doesn’t have a common cause
cep:on of quality • Single sampling plan: Decision based on 1 sample of size n; acceptance number c / rejec:on number (c+1) • Over :me: 1. Process in control; 2. Sustained shi• = shi•ed once and stayed; 3. Dri•ing average = slowly
Cost of quality = cost of producing quality + cost of not producing quality (equilibrium: combined minimum) = r -> simple but expensive; distribu:ons of quality characteris:cs must be known the average is changing in one direc:on; 4. Vola:le average = constantly changing (some:mes L/R)
• COPQ = preven:on cost + appraisal cost (expenses of checking if product was produced right) • Double sampling plan: First sample: size n1 ; if necessary second sampling size n2 ; combine both samples Process control vs capability (P.Con.: gelijke verdeling gemiddelde, P.Cap.: dikte staart verdeling à dun J)
• CONPQ = internal failure cost (in plant) + external failure cost (defec:ve at customer) to decide ; acceptance nr c1 and c2 rejec:on nr r1 (c1<r1) • Capability of process to meet the requirements -> make sure that quality parameters within specifica:ons
3. MEASURING Take 1st sample (size n1): if D1 (= # defecGves) < c1 à accept, other: if D1>r1 à reject, other: if D1<r1: take a (in lines) => Process needs to be in control to be able to measure capability
Scales 2nd sample of size n2 and inspect, let D2 be the # defecGves: if D1+D2< c2 , then accept lot other: decline • Sta:s:cal process control -> goal: achieve process stability and improve process capability
• Nominal scale: categorize objects; Ordinal scale: Order btw objects, says nothing about distance • Mul:ple sampling plan: Con:nue taking samples un:l enough evidence collected for acceptance/rejec:on Sta8s8cal Process Control
Interval scale: comparable, no ra:o (T with °C scale); Ra:o scale: Comparable, meaningful 0 (lengths, of the lot ; generaliza:on of double sampling with smaller sample sizes • Monitor process à detect defect à im-
mass) • Sequen:al sampling plan (=con:nuous sampling): Item-by-item sampling ; number of inspected products prove quality
Human aspects vs instrumental aspects not known beforehand; may lead to 100% inspec:on (if acceptance/rejec:on line never exceeded) 𝑿, 𝑹 CONTROL CHART
• Human: high complexity -> more mistakes; low defect rate: more defects overlooked OC-Curve • R -chart (= range chart): The difference
• Instrumental: calibra:on= verify & adjust the performance of measuring device compared to traceable • Opera:ng Characteris:c curve -> shows probability that a lot with a certain frac:on defec:ve p will be ac- between the largest and smallest obser-
measurement standards cepted (=Pa(P)) and tells us how strict a sampling plan is, every sampling plan has an OC-curve!! va:on in one sample will be larger when
Varia8on • AQL: Acceptance Q level = highest defec:ve rate that is considered acceptable -> if % def units < AQL => there is a lot of varia:on. So this will keep
& & &
• 𝛼 !"#$% = 𝛼'(")*+# +𝛼,-$.*(-/-0# -> Varia:on = measurement error = true value - measured value accept; Producers risk: p(reject | %defec:ves <= AQL) = α (i.e. p(accept | %defec:ves <= AQL) = 1-α) track of varia:on (eg: sample (2,3,6,8,10)
Accuracy vs Precision • RQL: Rejec:on Q level = highest defec:ve rate that is considered tolerable -> if % def units >= RQL => re- ; R-chart: 10-2 = 8); 𝑋-chart: keeps track
• Accuracy = difference between the true value & the observed average (low accuracy = result of systema:c ject aka lot tolerance percent defec:ve (LTPD) -> Consumer’s risk: p(accept | %defec:ves >= RQL) = β of sample averages (29/5 = 5,8)
bias in the measurement) à hoe dicht liggen de punten rond het gemiddelde (midden vd cirkel) Single sampling plan (aMributes) • Central line = process mean ; Control lines = will depend on how much varia:on there is
• Precision = Closeness of repeated measurements (low Precision = result of random varia:ons built into • The effect of n (keeping c propor:onal to n) ; if n then discrim- Designing 𝑿, 𝑹 control chart:
the instrument) à hoe dicht liggen de punten tegen elkaar? (ongeact hoe dicht tegen midden) inatory power . The more n the closer it moves towards the The I𝑿 chart (= mean chart)
Repeatability & Reproducibility (= R&R-analysis) ideal OC-curve
1. Compute 𝑋i mean for subgroup i of size n (for all i = 1, …, k)
• Repeatability = consistency of measuring instrument: EV= equipment varia:on; • The effect of c (keeping n constant) ; if c¯ then discriminatory ∑ 9: :::::::::::::::::::
9 <9 … 9
• Reproducibility = consistency of operator: OV = operator varia:on power », but discrimina:on moves towards lower values of p. 2. Compute process mean 𝜇 = 𝑥3 = = " !# %
; ;
• R&R-analysis = study of varia:on in measurement systems If you want to reduce the probability of an error, increase 3. Compute center line (CL) (F 1) and control limits (UCL, LCL) ( = ac:on lines) (F 2,3), dn and An from table!!
à Goal: how much of the total variability is due to the measurement system? n & the acceptance number (keep them propor:onal) a. Why 3𝜎 limits? Chance for process interrup:on due to false alarm should be small -> with 3𝜎 limits and
1. R&R analysis STEPS: Select m operators & n products normal distribu:on, it is 3/1000
&
2. Calibrate the measuring instrument b. Some:mes ‘warning lines’ at 𝑋3 ± 𝐴& 𝑅*
Each of the m operators ?
3. Let each operator measure each product in random order and will measure each of the R- chart ( = range chart)
repeat, r³2 ( Mijk= kth measurement by operator i on product j) n products, r 4mes 4. Compute Rk for subgroup k of size n (for all k)
4. Compute average measurement for each operator 𝑥1 (Formula 1) 5. Compute mean range 𝑅* = CL (F 4)
& difference btw largest & smallest average 𝑥2 (F 3) 6. Compute UCL and LCL (F 5,6)
5. Compute the range for each part & each operator Rij (F 4) + average range for each operator 𝑅1 (F 5) + a. Distribu:on of sample ranges is posi:vely skewed (rechts aflopend) -> UCL and LCL are asymmetrical
overall range 𝑅& (F 6) (K-values have a table!- OC-curve of 𝑿-chart
6. Compute part averages 𝑥3 (F 2) & the range of part averages Rp (F 7) • Describes how well the control chart discovers assignable causes of various magnitude, how likely it is that
7. Calculate a control limit (CL) on the individual ranges D4𝑅& (F 8) -> if range > CL -> look for assignable cause ?@
an observa:on, in the 𝑋-chart falls inside CL’s. Magnitude = shi• in process mean (k). 𝜇 → . Chance
If assignable cause & you have to delete that data and a5er that recompute steps 4-7 with 1 less n N < 10*n -> hypergeometric distribu:on; N ³ 10*n -> Bino- √0
mial distribu:on; n ³ 20 and p £ 5% the binomial distr. can 𝑃(𝐷 ≤ 𝑐 𝕀 𝑝 = 𝑝B ) = 1 − 𝛼 shi• remains undetected = 𝛽 = 𝑃(𝐿𝐶𝐿 ≤ 𝑋* ≤ 𝑈𝐶𝐿 ½𝜇 = 𝜇 + 𝑘𝜎)
8. Compute EV, OV, RR (nega:ve OV -> make 0) (F 9,10,12) EV = Repeatability, OV = Reproducibility
𝑃(𝐷 ≤ 𝑐 𝕀 𝑝 = 𝑝& ) = 𝛽 OC-curve of an R-chart
9. Compute Part Varia:on PV and Total Varia:on TV (F 11,13) (Rp = range of the part averages) be approximated by the Poisson distribu:on ( l = n*p)
+ • Describes how well the control chart discovers assignable causes of various magnitude. Magnitude =
10. Express EV, OV, RR as % of TV (EV/TV, OV/TV,…) & evaluate OR as % of To (tolerance) (OR/To) • How to design single sampling plan? Given AQL (=p1) and α, 𝑛 6
<10% acceptable R S T 𝑝 (1 − 𝑝B )0F6 = 1 − 𝛼 Change in the process standard devia:on (𝜆). 𝜎 → 𝜎B , 𝑤𝑖𝑡ℎ 𝜆 = 𝜎B /𝜎. Chance shi• remains undetected =
and RQL (=p2) and β, determine n and c (upper formula) 𝑥 B
10-30% acceptable, depending on cost & importance • If D is binomially distributed use: (lower formula) 6DE
+
𝛽 = 𝑃(𝐿𝐶𝐿 ≤ 𝑅 ≤ 𝑈𝐶𝐿 ½𝜎 = 𝜎B )
>30% not acceptable Nomograph: AQL and RQL connect with Pa(AQL), Pa(RQL) 𝑛 Interpre8ng 𝑿, 𝑹 control chart
R S T 𝑝&6 (1 − 𝑝& )0F6 = 𝛽
2 2 2 2 45 ! Double sampling plan (aMributes) 𝑥 • Never use 𝑋-chart before R chart is in control
OR EV , OV & RR as % of TV à = 100* ! ( Otherwise EV + OV + PV =/= 100) -> variance ra:os 6DE
!5
• Compute OC-curve: Find Pa (AQL) • For both R and 𝑋:
B
4.ACCEPTANCE SAMPLING = P(accept in 1st sample) + P(accept in 2nd sample) 8 points in a row on the same side of CL = mean of process has probably shi•ed (2 ∗ ( )C = 0,78% chance)
&
= P(D1≤1) + P(D1=2 and D2≤1) = P(D1≤1) + P(D1=2)*P(D2≤1) 6 points in a row in a run upward or downward = detect trend, a slow dri• is happening
Inspec8on: goal =/= es:ma:ng Q but decide acceptance/rejec:on
e.g.: n1 = 50, c1 = 1, r1 = 3, n2 = 100, and c2 = 3, with AQL=0.01 and RQL=0.08 14 points in a row in oscilla:on (op en neer) = cyclic behaviour
• 100% inspec:on: expensive, impossible when destruc:ve, cri:cal parts, only 80% detected
• Acceptance sampling: decide on a lot of N products, based on quality of a sample of n products taken • For only 𝑋-chart (because R-chart is not normal distr): divided into 3 zones: C (green): 68,26% (CL +/- 1 𝜎 );
from it. Problem: sample & lot may not have same % of non-conf. units. -> less expensive, less info, re- B (yellow): 27,20% (CL +/- 2 𝜎 ); A (red): 4,27%% (CL +/- 3𝜎 ); outside control limits: 0,27%
quires planning, risk making wrong decisions 2 of the 3 consecu:ve points in zone A (same side of CL)
• No inspec:on: defec:ve units almost never encountered, Q assurance through cer:fica:on 4 of the 5 consecu:ve points in zone A or B (same side of CL)
Quality control: tradi:onal approach vs. modern approach 8 consecu:ve points outside zone C
• Tradi:onal: Industrial revolu:on setup; Separate inspec:on at line's end; Goal: Ship only good, but costly 15 consecu:ve points in zone C
• Modern: Integrated produc:on and QC; Direct feedback for improvement; Worker owns quality; Emphasis Next: Find Pa (RQL) = same but other probability!
on automa:on Repeat for other points of the OC-curve 5.2 STATISTICAL PROCESS CONTROL part 2
Sampling techniques Double Sampling Plan is useful in “extreme situaGons”: very low or very high % of defecGves in lot 𝑿 − 𝑺 CONTROL CHART (à uses sample standard devia:on instead of the range with X – R chart)
• Random Sampling: Equal chance selec:on of units for inspec:on; Methods: random numbers, serial VARIABLES • Use when n>10, because range no longer suited to approximate 𝜎; Use when n is variable / the sample
codes, posi:on in container; Stra:fied Sampling: Division of the lot into subgroups (palet, box, loca:on); Single sampling plan (AQL is smallest mean emission level, RQL is highest) size varies; Use when stricter control of varia:on is needed
Random inspec:on of units within each subgroup; Systema:c Sampling: Random selec:on of a star:ng • Give assurance about average µ of quality characteris:cs X ; assump:on 1: X is normally distributed; 2: the 𝑺-chart
point; Inspec:on of every k-th product in the lot; Cluster sampling: pick 1 or more representa:ve sub- lower X, the beyer • For each subgroup i, compute si =
groups & inspect all units from these subgroups If μ < AQL: accept the lot with probability 1-α; If μ > RQL: accept the lot with probability β • The average of the m subgroup standard devia:ons is 𝑠̅ = Center line (CL) (F 4)
Sampling Errors -> Determine n & c using (F 3,4) and Z-table • Compute UCL and LCL: B3 and B4 instead of D3 and D4 (F 5,6)
• Systema:c Errors: Measuring bias (low accuracy measuring system); Non-homogeneous lots (varying pro- -> Measure the quality characteris:c for n units • if subgroup size is not constant, use weighted average approach to
duc:on factors, e.g. not produced by the same machines); Sampling bias (e.g., "convenience sampling") -> If the sample average 𝑥̅ is lower than c, then accept the lot, else reject the lot find grand average and standard devia:on; A3, B3 and B4 depend on
• Non-systema:c Errors (Example: Tennis Balls): Ques:on: Is it possible that 998 out of 1000 balls are within 5.1 STATISTICAL PROCESS CONTROL part 1 subgroup size n, they will now be different for each sample (now
specifica:ons? Any sample can lead to wrong decisions… Solu:on: increase sample size • Specifica8on Limits apply to every single unit & determine whether the customer wants it or not CL becomes a step func:on that changes from sample to sample!)
Hypotheses: H0 : Nul hypothese -> Current situa:on, Ha : what we want to prove is going on. à is determined at the start -> individual averages
𝑿-chart
Decision Rule: If H0 is wrong, choose Ha . If H0 is correct, s:ck with it. • Control Limits apply to groups of units & determine (via sample averages) if a process is stable and so
• Compute CL, UCL and LCL (F 7,8,9)
Type I Error: Reject a good quality lot (Reject H0 while H0 is true) (α: Probability of Type I error (Producer’s whether we should adjust the machine or not à control lines exceeded = adjustment needed
𝑿-Rm CONTROL CHART
risk) Typical value: α=0.05) • When making conclusion based on sample average & adjus:ng every:me -> the variability à take sub-
***0 -> standard devia:on ¯ 𝜎6̅ = 𝜎. • When subgroup size 1 unit (n=1); When data becomes available too slowly for subgrouping (temperature
Type II Error: Accept a poor quality lot (Reject Ha while H0 is false) (β: Probability of Type II error (Con- group of size n & subgroup mean 𝑋
√𝑛 measurements) or when we want to act directly instead for wai:ng 5 errors (high value items)
sumer’s risk) Typical value: 0.05≤β≤0.10)
