Lecture 2: Statistical process Control

Transcription

1 Lecture 2: Statistical prcess Cntrl Definitin f statistical prcess cntrl: SPC is a statistical methd t evaluate the tendency f a prcess t g ut f cntrl. SPC is used in the industry t determine if a prduct is within expectatins. This wuld enable t mnitr and measure if the particular prcess is wing within a particular range. This can imprve the quality f the prcess and the quality f the prduct been prduced. In SPC, cntrl charts are created in rder t assess whether a prcess is statistically cntrl r statistically ut f cntrl. There can be 2 types f variatins that ccur in a prcess. Nrmal variatin r abnrmal variatin. In nrmal variatin is present in the prcess, the values that are pltted in a SPC cntrl chart, fall within the charts cntrl limits and they fall in a randm pattern. In this particular case, the prcess is said t be in statistical cntrl. Hwever, when there is abnrmal variatin in the prcess, pltted values may fall utside the cntrl limits and the prcess is said t be ut f statistical cntrl. These pltted pints are referred t as utliers. There situatins which d nt fall clearly within these 2 categries. Fr example, in sme situatins, abnrmal variatins exist in the prcess but the pltted values are within the chart s cntrl limits. This is referred t instability, that is the prcess is unstable. The pltted pints fr an unstable prcess frm a recgnizable nn-randm pattern (like lines). If a prcess is ut f cntrl, there needs t be a respnse in rder t adjust the prcess s that it falls within the chart s cntrl limits r that it becmes stable again. With SPC, the respnse t an ut f cntrl prcess is intended t eliminate whatever is causing the prcess t be ut f cntrl. Althugh SPC infrms when the prcess is ut f cntrl, n cntrl chart can identify the specific prblem that is causing the prcess t g ut f cntrl. Hence, we can cnclude that a prcess is in cntrl if it satisfies the fllwing requirements. These rules are called the Western electric rules: 1. there are n sample pints utside the limits. 2. Mst pints are near the prcess average 3. Abut an equal number f pints are abve and belw the centre line 4. Pints appear randmly distributed. That is, that there shuld be n nticeable patterns (like straight lines) but data shuld be all ver the place.

2 Nte: These rules apply nly t chart. These rules are evaluated fr ne side f the centre line (ne half f the cntrl band) at a time (e.g., first the centerline t the upper cntrl limit, then the centerline t the lwer cntrl limit). Chart: it tests variatins f data based n the average. It plts the average f the values taken frm each sample grup that is cllected frm a prcess. Hence, it cntrls whether the average f each sample is within the cntrl limits. The data t be evaluated may cme frm samples with r withut replicates. The quality f a sample may be cmpared against histrical data f a cmpany ( knwn) r against its dispersin f the data (unknwn ). A chart has: Data with replicates and is knwn: prcess centre line (): this is the verall average f the average f the replicates within the sample. It is = average f means f the samples = average f replicates within ne sample N = ttal number f samples. Nte: this will yield the same result as if we were t just add up all the replicates per sample and devided by the ttal number f replicates per sample. Fr example, if the example abve had 2 replicates, we can als prceed t average the replicates first and then t average the + and values. 1 2/2. Then, average the + and - values, 1 2/2. This is the same as (replicate1 + replicate2)/4 upper cntrl limit (UCL): this is the maximum limit fr the prcess t be in cntrl. It is = average f sample means. k = the cnstant fr the. Always chse k = 3 n = number f replicates within ne sample lwer cntrl limit (LCL): this is the minimum limit fr the prcess t be cntrl. It is

3 Data replicates and is unknwn: if the prcess is unknwn, the must be apprximated. It is fund that the best apprximatin fr is the ne cming frm the range f data, nrmalized by a factr that depends n the number f replicates frm which the range was derived. prcess centre line (): this is the verall average f the average f the replicates within the sample. It is = average f sample means. = average f replicates within ne sample N = ttal number f samples. upper cntrl limit (UCL): this is the maximum limit fr the prcess t be in cntrl. It is = average f the means f the samples. k = the cnstant fr the. Always chse k = 3 n = number f replicates within ne sample = average f the range given R i /N. where,r i value f the replicates within ne sample. d 2 = cnstant whse value depends n the number f replicates. is estimate by /d 2 lwer cntrl limit (LCL): this is the minimum limit fr the prcess t be cntrl. It is Individual measurements and is unknwn: this is a particular case where we nly have 1 replicate per sample, and hence cannt calculate a value fr R. We, hence replace by called the mving range. prcess centre line (): this is the verall average f the average f the replicates within the sample. It is = average f the samples. N = ttal number f samples.

4 upper cntrl limit (UCL): this is the maximum limit fr the prcess t be in cntrl. It is = average f sample means. k = the cnstant fr the. Always chse k = 3 n = number f replicates within ne sample = average f the range given R i /N. where,r i value f the replicates within ne sample. d 2 = cnstant whse value depends n the number f replicates. /d 2 is estimate by /d 2. = x i x i-1 /(N - 1). that is the sum f the difference between each sample reading (since it nly has 1 reading per sample) divided by the number f samples minus 1. lwer cntrl limit (LCL): this is the minimum limit fr the prcess t be cntrl. It is Nte: we have gne frm knwing, t nt knwing and hence estimating it with, t having t estimate by. We then have the fllwing equatin: Despite nly having 1 replicate per ne sample we still use the value fr d 2 f Althugh this value is fr 2 replicates. chart: it tests variatins based n the range f data. It is the difference between the highest value and the lwest value. We use chart because it is pssible that a prcess average is under cntrl, but their variability (range) might be beynd the cntrl limits. Hence, in rder fr a prcess t be under cntrl bth the prcess average and the prcess variability must be in cntrl. It plts the range f the values taken frm each sample grup that is cllected frm a prcess. Hence, it cntrls whether the range f each sample is within the cntrl limits. The chart lks exactly the same fr that f the average chart but nw the lwest value is set t zer.

5 Data with replicates and is knwn: assume that the range f the samples cme frm a ppulatin with mean µ r and standard deviatin r. prcess centre line (): this is the verall average f the average f the replicates within the sample. It is = average f the range given R i /N. where,r i value f the replicates within ne sample. N = ttal number f samples. upper cntrl limit (UCL): this is the maximum limit fr the prcess t be in cntrl. It is μ μ = ppulatin mean. k = the cnstant fr the. Always chse k = 3 = ppulatin standard deviatin lwer cntrl limit (LCL): this is the minimum limit fr the prcess t be cntrl. It is μ Data replicates and is unknwn: µ r and r will nt be knwn fr mst cases. Hence, the values f the µ r and r must be apprximated. It is fund that the best apprximatin fr µ r and r is the ne cming frm the range f data, multiplied by a factr that depends n the number f replicates frm which the range was derived. prcess centre line (): this is the verall average f the average f the replicates within the sample. It is = average f the range given R i /N. where, R i value f the replicates within ne sample. N = ttal number f samples. upper cntrl limit (UCL): this is the maximum limit fr the prcess t be in cntrl. It is μ μ = ppulatin mean. k = the cnstant fr the. Always chse k = 3 = ppulatin standard deviatin lwer cntrl limit (LCL): this is the minimum limit fr the prcess t be cntrl. It is μ