Six Sigma Black Belt Study Guides 1 www.pmtutor.org Powered by POeT Solvers Limited.
Causes of variation No process can be variation free. There will be some sort of variation in a process. Major causes of variation in a process can be classified into two categories. They are: Common (chance) causes of variation Inherent or natural in a process Small in magnitude Difficult to identify or eliminate from the process Special causes of variation Variation due to some special causes Large in magnitude Easy to identify and eliminate from the process Stable process A process is said to be a stable process if there exists no special causes of variation. Hence, a stable process runs under common causes only. 2 www.pmtutor.org Powered by POeT Solvers Limited.
Process Capability study: Process capability study gives numerical assessment of the capability of the process. It compares the voice of the customer with the voice of the process The different steps involved in the capability study are Step 1: Check whether the process is stable Stability of the process can be checked by using control charts. Step 2: Check whether the process data is taken from a normal distribution This can be done by constructing a histogram using the original reading from the control chart. Different techniques should be adopted to find the process capability in the case of normal and non-normal data 3 www.pmtutor.org Powered by POeT Solvers Limited.
REJECTS REJECTS REJECTS REJECTS Process Capability Analysis Allowed Variation Actual Variation Determined by the Customer Determined by the Process Voice of Customer Voice of Process LSL USL LCL TARGET UCL -3σ +3σ VOP = UCL LCL = 6σ VOC = USL - LSL LSL Lower Spec Limit USL Upper Spec Limit LSL GOOD CAPABILITY USL LCL Lower Control Limit UCL Upper Control Limit LCL TARGET UCL -3σ +3σ VOP = UCL LCL = 6σ VOC = USL - LSL BAD CAPABILITY 4 www.pmtutor.org Powered by POeT Solvers Limited.
Process capability indices: Process capability indices are used to evaluate the capability of a process in a single number. Three important process capability indices are Cp, Cpk and CR. The important requirements that these measures to be effective are Process should be stable. Data should be normal. Potential capability Cp: Cp is a measure, which measures the ability of the process to meet the customer specifications. Customer requirements can be expressed as lower specification limit (LSL) and upper specification limit (USL). 5 www.pmtutor.org Powered by POeT Solvers Limited.
Potential capability C p : Process Capability Analysis In the formula for C p, σ refers to the point estimate for the process standard deviation and it is given by the formula σ = R d 2 here, R is the average range and d 2 is a constant depends on the subgroup size. The different values of d 2 corresponding to the subgroup size is given in the following table Subgroup 2 3 4 5 6 7 8 9 10 Size d 2 1.128 1.693 2.059 2.326 2.534 2.704 2.847 2.970 3.078 Allowed variation. 6 SD < (USL LSL) C p > 1. Hence, C p should be greater than 1 for a process in order to meet the customer requirements. Limitations of C p : It checks only the potentiality of the process to meet the customer requirements. But it never checks whether the process is actually meeting customer requirements. 6 www.pmtutor.org Powered by POeT Solvers Limited.
Cpk: Cpk gives an idea about the position of the mean comparing with the nearest specification limit It checks centering of the process. The formula for Cpk is given by x = is the process average LSL and USL are the lower and upper specification limits respectively. Please note: If C p = Cpk, then process mean is at the center of the population CR: Capability Ratio (CR) is the reciprocal of C p 7 www.pmtutor.org Powered by POeT Solvers Limited.
Centered Process Off Centered Process Target = Process mean Target Process mean LSL USL LSL USL C p = C pk C p C pk 8 www.pmtutor.org Powered by POeT Solvers Limited.
Process performance indices: This is used to analyze and compare the current process with improvement efforts Like capability measures, performance measures are also effective when The process is stable The data is normal Three important performance indices are P p, P pk, C pm The only difference in the formula of P p, P pk and the corresponding capability indices is, in the performance indices long term sample standard deviation(s) is used instead of short term process spread (σ). The formulae are given by Here s refers to the sample standard deviation. 9 www.pmtutor.org Powered by POeT Solvers Limited.
Cpm: Cpm is considered as the most accurate index than other indices and it is commonly known as Taguchi index. The formula for finding this index is Here µ is the process average and T is the target value. 10 www.pmtutor.org Powered by POeT Solvers Limited.
Short-term and Long-term capability: The validity of the capability analysis increases with the time span of the collected data increases. Process Capability Study is helpful to determine the short term stability and capability of the process, whereas the Process Performance Study is helpful to determine the long term stability and capability of a process Short Term Performance Long Term Performance Off centered Process C p P p Centered Process C pk P pk 11 www.pmtutor.org Powered by POeT Solvers Limited.
This process is short-term capable but not long-term capable 12 www.pmtutor.org Powered by POeT Solvers Limited.
Process Capability for non-normal data: In process capability study, it is assumed that the process data is normally distributed. In the case of non- normal data the different indices of capability study will not give valid results. For a non- normal data the capability studies are performed by Finding another distribution such as Exponential, chi-square and F distribution, that fits the given data. Using nonlinear regression to fit a curve to the data. Using transformations like Box- Cox and Johnson s transformations to transform the data to another variable which is normally distributed. 13 www.pmtutor.org Powered by POeT Solvers Limited.
Process Capability for Attributes data: In the case of attributes data, the process capability is defined by using the mean of nonconformity. Example: A electronic manufacturing process has a customer specification of maximum two defectives per batch, produces lots of size 56 and the customer specification is given by 2/56 = 0.035714. The value p for the stable process is given by 0.03. Since the value of p is better than the customer specification, the process is capable. To find the percentage of batches having more than two defectives: The probability of occurrence corresponding to each defective can be calculated by using binomial distribution. In the case of 0 defective, the probability of occurrence is given by P (0) = 56 C 0 (0.03) 0 (1 0.03) 56 = (0.97) 56 = 0.182 Similarly we can find the probability of the other defective cases also using binomial distribution and the values are given in the following table. Defectives 0 1 2 Total Probability of 0.182 0.315 0.268 0.765 Occurrence Hence the percentage of batches have more than 2 defective are 100-76.5 = 23.5% 14 www.pmtutor.org Powered by POeT Solvers Limited.
Calculation of process performance metrics: There are numerous performance indicators used in Six Sigma projects. Percent defective Parts per million (PPM) Defects per million opportunities (DPMO) Defects per unit (DPU) Process Sigma Rolled throughput yield (RTY) Consider the following example A process produces 20,000 books. Five types of defects can occur. The number of occurrences of each defect type are given below: Defect type Frequency Typographical error 345 Missing pages 45 Incorrect ordering of pages 37 Feeding mistake 41 Hazing printing 25 Total number of defects 493 15 www.pmtutor.org Powered by POeT Solvers Limited.
1. Percent defective = Number of defects 100 = 493 100 = 2.4% Number of units 20,000 2. Parts per million (PPM) = Number of defects 1,000,000 = 493 1,000,000 = 24,650 Number of units 20,000 3. Defects per Million Opportunities (DPMO) DPMO = Total number of defects 1,000,000 Total number of opportunities = 493 1,000,000 20000 5 = 4930 4. Defects per unit (DPU) = Number of defects = 493 = 0.02465 Number of units 20,000 16 www.pmtutor.org Powered by POeT Solvers Limited.
5. Process sigma = Process sigma is calculated using excel. 6. Throughput yield = e -DPU = e -0.02465 =0.976 7. Rolled Throughput Yield (RTY): RTY applies to the yield from a series of processes and is calculated by multiplying the individual process yields. If a product goes through five processes whose yields are 0.993 0.987, 0.975, 0.969, and 0.957 then RTY = 0.993 0.987 0.975 0.969 0.957 = 0.886 17 www.pmtutor.org Powered by POeT Solvers Limited.
Example 1: Determine the process capability / performance metrics of a process that had a lower specification of 2 and an upper specification of 16 with responses: 3.4, 8.57, 2.42, 5.59, 9.92, 4.63, 7.48, 8.55, 6.1, 6.42, 4.46, 7.02, 5.86, 4.8, 9.6, 5.92 (data were collected sequentially). Testing of stability: The process is stable (under statistical control) 18 www.pmtutor.org Powered by POeT Solvers Limited.
Testing of normality: Stat > basic stat > Normality test Since the p-value = 0.895 > 0.05, data is normal. 19 www.pmtutor.org Powered by POeT Solvers Limited.
Process capability analysis Select Stat > Quality Tools > Capability Sixpack > Normal Enter the subgroup size as 1, and enter the given LSL and USL 20 www.pmtutor.org Powered by POeT Solvers Limited.
Output 21 www.pmtutor.org Powered by POeT Solvers Limited.
Alternative Minitab capability analysis for normally distributed data: Stat > Quality Tools > Capability Analysis > Normal 22 www.pmtutor.org Powered by POeT Solvers Limited.
Example 2: A chemical process has a specification upper limit of 0.16 and physical lower limit of 0.10 for the level of contaminant. Determine the estimated process capability / performance of the process, which has the following output: 23 www.pmtutor.org Powered by POeT Solvers Limited.
Phase 1: Check for special causes. 5 out of 119 points are out of control limits, which indicate that special causes are present. Many a times we noticed that the LCL value becomes negative, which is impossible in a real life situation. Apparently the individual chart is not the best way to analyze data like this. 24 www.pmtutor.org Powered by POeT Solvers Limited.
As we do not know whether special causes are present, we can t determine the proper distribution for the data. Likewise, if the distribution of data is not known we can t determine whether special causes are present because the control limits may be in the wrong place. The central limit theorem states that stable distributions produce normally distributed averages even when the individual s data are not normally distributed. We create subgroups of size 5 in Minitab and test the stability (X-bar chart). Instead of using individuals chart we use averages. The chart indicates that the process is in statistical control. 25 www.pmtutor.org Powered by POeT Solvers Limited.
Phase 2: Examine the distribution (Test for normality). The p-value indicates that the data is not normal. Hence, we use Box-Cox transformation (to be defined). 26 www.pmtutor.org Powered by POeT Solvers Limited.
The following figure shows a Box-Cox plot. Lambda = -1.69 27 www.pmtutor.org Powered by POeT Solvers Limited.
Testing of normality of the transformed data: The transformed data looks well behaved. 28 www.pmtutor.org Powered by POeT Solvers Limited.
Phase 3: Predicting long term defect rate for the process. Minitab calculates both the process performance indices (Ppk) as well as the process capability indices (Cpk). The denominator for the process performance indices is the overall standard deviation rather than the standard deviation based only on within subgroup variability. This is called the long term process capability, which Minitab labels as Overall Capability. Minitab s analysis indicates that the process is not capable (Ppk < 1). The estimated long-term performance of the process is 193390.19 DPMO, where the observed performance is even worse (201680.67 DPMO). The difference is a reflection of lack of fit. 29 www.pmtutor.org Powered by POeT Solvers Limited.
Conclusion: In this chapter we have learnt about: Process capability studies. Process capability indices. Process performance indices. Short term and long term capability. Process capability for non-normal data. Process capability for attributes data. Process performance vs. specification. Examples using Minitab 30 www.pmtutor.org Powered by POeT Solvers Limited.