Department of Industrial Engineering. Chap. 8: Process Capability Presented by Dr. Eng. Abed Schokry

Transcription

1 Department of Industrial Engineering Chap. 8: Process Capability Presented by Dr. Eng. Abed Schokry Learning Outcomes: After careful study of this chapter, you should be able to do the following: Investigate and analyze process capability using control charts, Understand the difference between process capability and process potential, (explain the difference between process capability and Process capability index), Calculate and properly interpret process capability ratios, Differentiate between Control limits and specification limits, Understand the role of the normal distribution in interpreting most process capability ratios, Estimate the components of variability in a measurement system, Set specifications on components in a system involving interaction components to ensure that overall system requirements are met, Estimate the natural limits of a process from a sample of data from that process, 1

2 Process Capability Process capability refers to the ability of a process to produce products or provide services capable of meeting the specifications set by the customer or designer. Specification limits are set by management in response to customers expectations The upper specification limit (USL) is the largest value that can be obtained and still conform to customers expectations The lower specification limit (LSL) is the smallest value that is still conforming Process Capability Tolerances or specifications Range of acceptable values established by engineering design or customer requirements Process variability Natural variability in a process Process capability Process variability relative to specification 2

3 Tolerance Limits vs. Process Capability Specification Width Actual Process Width Specification Width Actual Process Width Process Capability PROCESS SPREAD SPECIFIED TOLERANCES ` SPECIFIED TOLERANCES ` PROCESS SPREAD A capable process An incapable process 3

4 Process Capability A process capability index is an aggregate measure of a process s ability to meet specification limits The larger the value, the more capable a process is of meeting requirements Specification limits Specification limits, the allowable spread of the individuals, are compared with the spread of the process to determine how capable the process is of meeting the specifications. Three different situations can exist when specifications and are compared: (I) The process spread can be less than the spread of the specification limits; (II) The process spread can be equal to the spread of the specification limits; (III) The process spread can be greater than the spread of the specification limits. 4

5 (I)The spread of the individuals is less than the spread of the specifications The control limits have been placed on the diagram, as well as the spread of the process averages (dotted line). The spread of the process individuals is shown by the solid line. As expected, the spread of the individual values is greater than the spread of the averages; however, the values are still within the specification limits. This allows for more room for process shifts while staying within the specifications. Notice that even if the process drifts out of control, the change must be dramatic before the parts are considered out of specification. Case I situation 5

6 (II) The process spread can be equal to the spread of the specification limits In this situation, is equal to the tolerance As long as the process remains in control and centered, with no change in process variation, the parts produced will be within specification. A shift in the process mean will result in the production of parts that are out of specification. An increase in the variation present in the process also creates an out-of-specification situation. Case II situation 6

7 (III) The spread is greater than the tolerance spread Case III: The spread is greater than the tolerance spread. Even though the process is exhibiting only natural patterns of variation, it is incapable of meeting the specifications set by the customer. To correct this problem, management intervention will be necessary in order to change the process to decrease the variation. The capability of the process cannot be improved without changing the existing process. Case 3 situation 7

8 Process Capability Process Capability Lower Specification Upper Specification A. Process variability matches specifications Lower Specification Upper Specification B. Process variability well within specifications Lower Specification Upper Specification C. Process variability exceeds specifications 3 Sigma and 6 Sigma Quality Lower specification Upper specification 1350 ppm 1350 ppm 1.7 ppm 1.7 ppm Process mean +/- 3 Sigma +/- 6 Sigma 8

9 The capability index C p The capability index C p is the ratio of tolerance (USL LSL) and 6 C p USL LSL 6 Capability ratio C r Capability ratio C r C r 6 USL LSL 9

10 Capability ratio C pk C pk is the ratio that reflects how the process is performing in terms of a nominal, center, or target value: C pk Z(min) 3 where Z(min) is the smaller of Z(USL) USL X or Z(LSL) X LSL The relationships between C p and C pk 1. When C p has a value of 1.0 or greater, the process is producing product capable of meeting specifications. 2. The C p value does not reflect process centering. 3. When the process is centered, C p = C pk. 4. C pk is always less than or equal to C p. 5. When C p is greater than or equal to 1.0 and C pk has a value of 1.00 or more, it indicates the process is producing product that conforms to specifications. 10

11 The relationships between C p and C pk (Cont.) 6. When C pk has a value less than 1.00, it indicates the process is producing product that does not conform to specifications. 7. A C p value of less than 1.00 indicates that the process is not capable. 8. A C pk value of zero indicates the process average is equal to one of the specification limits. 9. A negative C pk value indicates that the average is outside the specification limits. Meanings of C pk Measures C pk = negative number C pk = zero C pk = between 0 and 1 C pk = 1 C pk > 1 11

12 Limitations of Capability Indexes 1. Process may not be stable 2. Process output may not be normally distributed 3. Process not centered but C p is used Please visit this links: Estimating Process Capability Must first have an in-control process Estimate the percentage of product or service within specification Assume the population of X values is approximately normally distributed with mean estimated by X and standard deviation estimated by R / d 2 12

13 Process Capability Ratio If the process is centered use Cp Process capability ratio, Cp = specification width process width Cp = upper specification lower specification 6 If the process is not centered use Cpk C pk = min X LTL UTL - X or 3 3 C p Index A measure of potential process performance is the C p index USL LSL Cp 6(R / d ) 2 specification spread process spread C p > 1 implies a process has the potential of having more than 99.73% of outcomes within specifications 13

14 CPL and CPU To measure capability in terms of actual process performance: X LSL CPL 3(R / d ) USL X CPU 3(R / d ) CPL (CPU) > 1 implies that the process mean is more than 3 standard deviation away from the lower (upper) specification limit 2 2 CPL and CPU Used for one-sided specification limits Use CPU when a characteristic only has a USL Use CPL when a characteristic only has an LSL 14

15 C pk Index The most commonly used capability index is the C pk index Measures actual process performance for characteristics with two-sided specification limits C pk = min(cpl, CPU) C pk = 1 indicates that the process average is 3 standard deviation away from the closest specification limit Larger C pk indicates greater capability of meeting the requirements Process Capability & Tolerance When spec. established without knowing whether process capable of meeting it or not serious situations can result Process capable or not actually looking at process spread, which is called process capability (6 ) Let s define specification limit as tolerance (T) : T = USL LSL 3 types of situation can result the value of 6 < USL-LSL the value of 6 = USL - LSL the value of 6 > USL - LSL 15

16 Das Bild kann zurzeit nicht angezeigt werden. Das Bild kann zurzeit nicht angezeigt werden. Both Cp and Cpk are identical because process mean is at the center of the specification spread As the process mean starts to drift away from the center of the specification spread, value of Cpk starts getting smaller (although Cp does not change) Process Capability (6 ) And Tolerance Cp - Capability Index T = U-L Cp = 1 Case II 6 = T Cp > 1 Case I 6 < T Cp < 1 Case III 6 > T Usually Cp = 1.33 (de facto std.) Measure of process performance Shortfall of Cp - measure not in terms of nominal or target value >>> must use Cpk Formulas Cp = (T)/6 Cpk = Z (USL) = Z(min) 3 16

17 Example Determine Cp and Cpk for a process with average 6.45, = 0.030, having USL = 6.50, LSL = T = 0.2 L T U x = Solution Cp= T/6 = 0.2/6(0.03)=1.11 Cpk = Z(min)/3 Z(U) = (USL - x)/ = )/0.03 = 1.67 Z(L) = ( x LSL)/ = )/0.03 = 5.00 Cpk = 1.67/3 = 0.56 Process NOT capable since not centered. Cp > 1 doesn t mean capable. Have to check Cpk Interpreting the Process Capability Index C pk < 1 C pk > 1 C pk > 1.33 C pk > 1.67 C pk > 2 Not Capable Capable at 3 Capable at 4 Capable at 5 Capable at 6 17

18 Process Capability Example You are the manager of a 500-room hotel. You have instituted a policy that all luggage deliveries must be completed within ten minutes or less. For seven days, you collect data on five deliveries per day. Compute an appropriate capability index for the delivery process. Process Capability: Hotel Example Solution n 5 X R d USL X CPU 3(R / d ) (3.894 / 2.326) Since there is only the upper specification limit, we need only to compute CPU. The capability index for the luggage delivery process is.8337, which is less than 1. The upper specification limit is less than 3 standard deviation above the mean. 18

19 Comments On Cp, Cpk Cp does not change when process center (avg.) changes Cp = Cpk when process is centered Cpk Cp always this situation Cpk = 1.00 de facto standard Cpk < 1.00 process producing rejects Cp < 1.00 process not capable Cpk = 0 process center is at one of spec. limit (U or L) Cpk < 0 i.e. value, avg. outside of limits Process Capability: The Control Chart Method for Variables Data 1. Construct the control chart and remove all special causes. NOTE: special causes are special only in that they come and go, not because their impact is either good or bad. 2. Estimate the standard deviation. The approach used depends on whether a Rbar or S chart is used to monitor process variability. = Rbar / d2 = S / c4 Several capability indices are provided on the following slide. 19

20 Process Capability Indices: Variables Data CP = (engineering tolerance)/6 = (USL LSL) / 6 This index is generally used to evaluate machine capability. tolerance to the engineering requirements. Assuming that the process is (approximately) normally distributed and that the process average is centered between the specifications, an index value of 1 is considered to represent a minimally capable process. HOWEVER allowing for a drift, a minimum value of 1.33 is ordinarily sought bigger is better. A true Six Sigma process that allows for a 1.5 shift will have Cp = 2. Process Capability Indices: Variables Data CR = 100*6 / (Engineering Tolerance) = 100* 6 /(USL LSL) This is called the capability ration. Effectively this is the reciprocal of Cp so that a value of less than 75% is generally needed and a Six Sigma process (with a 1.5 shift) will lead to a CR of 50%. 20

21 Process Capability Indices: Variables Data CM = (engineering tolerance)/8 = (USL LSL) / 8 This index is generally used to evaluate machine capability. Note this is only MACHINE capability and NOT the capability of the full process. Given that there will be additional sources of variation (tooling, fixtures, materials, etc.) CM uses an 8 spread, rather than 6. For a machine to be used on a Six Sigma process, a 10 spread would be used. Process Capability Indices: Variables Data ZU = (USL X) / ZL = (X LSL) / Zmin = Minimum (ZL, ZU) Cpk = Zmin / 3 This index DOES take into account how well or how poorly centered a process is. A value of at least +1 is required with a value of at least being preferred. Cp and Cpk are closely related. In some sense Cpk represents the current capability of the process whereas Cp represents the potential gain to be had from perfectly centering the process between specifications. 21

22 Limitations of Capability Indexes 1. Process may not be stable 2. Process output may not be normally distributed 3. Process not centered but Cp is used Process Capability: Example 1. Assume that we have conducted a capability analysis using X- bar and R charts with subgroups of size n = 5. Also assume the process is in statistical control with an average of and an average range of A table of d2 values gives d2 = (for n = 5). Suppose LSL = and USL = = R bar / d2 = /2.326 = Cp = ( ) / 6(.00948) = CR = 100*(6* ) / ( ) = 142.2% CM = ( ) / (8*( )) = T,ZL = ( )/(.00948) = 1.9 T,ZU = ( )/(.00948) = 2.3 so that L, Zmin = 1.9 C pk = Z min / 3 = 1.9 / 3 =

23 Process Capability: Interpretation Cp = since this is less than 1, the process is not regarded as being capable. CR = 142.2% implies that the natural tolerance consumes 142% of the specifications (not a good situation at all). CM = = Being less than 1.33, this implies that if we were dealing with a machine, that it would be incapable of meeting requirements. ZL = 1.9 This should be at least +3 and this value indicates that approximately 2.9% of product will be undersized. ZU = 2.3 should be at least +3 and this value indicates that approximately 1.1% of product will be oversized. Cpk = 0.63 since this is only slightly less that the value of Cp the indication is that there is little to be gained by centering and that the need is to reduce process Capability indices: Cr & Cm The Cr capability ratio is used to summarize the estimated spread of the system compared to the spread of the specification limits (upper and lower). The lower the Cr value, the smaller the output spread. Cr does not consider process centering. When the Cr value is multiplied by 100, the result shows the percent of the specifications that are being used by the variation in the process. Cr is calculated using an estimated sigma and is the reciprocal of Cp. In other words, Cr = 1/Cp. Cm (capability machine) The Cm index describes machine capability; it is the number of times the spread of the machine fits into the tolerance width. The higher the value of Cm, the better the machine. Example: if Cm = 2.5, the spread fits 2½ times into the tolerance width, while Cm = 1 means that the spread is equal to the tolerance width. 23

24 Process Performance Pp and Ppk How do you know if your process is capable? Process Capability Pp measures the process spread vs the specification spread. In other words, how distributed the outcome of your process is vs what the requirements are. Pp = (USL LSL) / 6* s Process Mean close to USL (Process Mean close to LSL) If your Process Mean (central tendency) is closer to the USL, use: Ppk = [ USL x(bar) ] / 3 s, where x(bar) is the Process Mean. Interpreting Ppk Scores A Ppk of 1 means that there is half of a bell curve between the center of the process and the nearest specification limit. That means your process is completely centered. The Cereal Box Example Consumer Reports has just published an article that shows that we frequently have less than 15 grams of cereal in a box. Let s assume that the government says that we must be within ± 5 percent of the weight advertised on the box. Upper Tolerance Limit = (16) = 16.8 grams Lower Tolerance Limit = (16) = 15.2 grams We go out and buy 1,000 boxes of cereal and find that they weight an average of grams with a standard deviation of grams. 24

25 Cereal Box Process Capability Specification or Tolerance Limits Upper Spec = 16.8 grams, X LTL UTL X C Lower Spec = 15.2 grams pk Min ; 3 3 Observed Weight Mean = grams, Std Dev = grams What does a C pk of mean? Many companies look for a C pk of 1.3 or better 6-Sigma company wants 2.0! Process Capability Tolerance (specification, design) Limits bearing width cm LTL = cm UTL = cm Process Limits The actual distribution from the process Run the process to make 100 bearings, compute the mean and std. dev. (and plot/graph the complete results) Suppose, mean = 1.250, std. dev =

26 The Cereal Box Example Design Specs: Bearing diameter cm s LTL = cm s inches UTL = cm s The actual distribution from the process mean = 1.250, s = s limits (0.002) [1.244, 1.256] The Cereal Box Example Anew process, std. dev. =

27 Process Capability Index, C pk A process has a mean of 45.5 and a standard deviation of 0.9. The product has a specification of 45.0 ± 3.0. Find the Cpk. Process Capability Index, C pk Example problem solution: C pk X LTL UTL-X = min or 3 3 = min { ( )/3(0.9) or ( )/3(0.9) } = min { (3.5/2.7) or (2.5/2.7) } = min { 1.30 or 0.93 } = 0.93 (Not capable!) However, by adjusting the mean, the process can become capable. 27

28 Process Capability Indices Consider our bags of sugar: m 10 kg LSL, USL 9.5, 10.5 kg m 10.1 kg s 0.1 kg C p (0.1) The results look ok, but the results are misleading since Cp is target insensitive Process Capability Example Specification Nominal (target) dimension: 30 mm Tolerance: + 1 mm, mm Process standard deviation: cpu = = 1.33 cpl = = (0.25) 3 (0.25) and therefore Cpk =

29 Process Capability Ratios Centered process (special case): specification width c p = process width Upper Spec Limit Lower Spec Limit = Process Capability Example Specification Nominal (target) dimension: 30 mm Tolerance: ± 1 mm Process standard deviation: c p = = (0.25) 29

30 Process Capability Cp = (design tolerance width)/(process width) = (maxspec min-spec)/ /6 x Example: Plane is on time if it arrives between T 15min and T + 15min. Design tolerance width is therefore 30 minutes x of arrival time is 12 min Cp = 30/6*12 = 30/72 = 0.42 A capable process can still miss target if there is a shift in the mean. Process Capability Motorola Six Sigma is defined as Cp = 2.0 I.e., design tolerance width is +/- 6 x or 12 x 3 3 process width min acceptable Design tolerance width max acceptable 30

31 Process Capability Requirements Process must be normally distributed Process must be in control Process capability result: < 1.00 = not capable < 1.33 = capable, but not acceptable > 1.33 = capable and acceptable (generally) > 2.00 = capable and acceptable (6Σ) > 5 or 10 is overkill, excessive resource use Capability Versus Control Process capability (C p or C pk ) Measure of variability against design specifications Specs set by customer or design engineer Spec width: USL & LSL (or UTL & LTL) Statistical process control (SPC) Measure of variability against control limits Control limits calculated from sample data UCL and LCL 31

32 Capability Versus Control Control Capability In Control Out of Control Capable IDEAL Not Capable Process Control vs. Capability The difference between capability and stability (control) A process is capable if individual products consistently meet specification A process is stable (in control) only if common variation is present in the process 32

33 Example 1 Machine Standard Deviation Machine Capability A /0.78 = 1.03 B /0.48 = 1.67 C /0.96 = 0.83 C p Cp > 1.33 is desirable Cp = 1.00 process is just capable Cp < 1.00 process is not capable Improving Process Capability Simplify Standardize Mistake-proof Upgrade equipment Automate 33

34 When to Use Pp, Ppk, Cp, and Cpk Process Performance Indices Pp and Ppk American National Standards Institute in ANSI Standard Z1 on Process Capability Analysis (1996) states that Pp and Ppk should be used when the process is not in control. Now it is clear that when the process is normally distributed and in control, is essentially and is essentially because for a stable process the difference between s and is minimal. However, please note that if the process is not in control, the indices Pp and Ppk have no meaningful interpretation relative to process capability, because they cannot predict process performance. 34

35 END When to Use Pp, Ppk, Cp, and Cpk Pp, Ppk In Relation to Z Scores Ppk can be determined by diving the Z score by three. A z score is the same as a standard score; the number of standard deviations above the mean Z = x mean of the population / standard deviation. Ppk = ( USL µ) / 3σ = z / 3 35