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Chapter 8 1 Chapter 8 2 Learning Objectives Chapter 8 3 Process Capability • In short: The process must first be brought into statistical control so its performance can be predicted; then its capability to meet specifications can be assessed. Chapter 8 4 Process Control Chapter 8 5 Process Capability Chapter 8 6 Capable V.S. In-Control Process • Capable process– A process that has aC pk greater the 1. this means that the histogram distribution and the normal curve are contained within the actual specification dimensions. Chapter 8 7 Capable V.S. In-Control Process • In-control process– A process that when charted on a SPC chart and all measurements are within the upper and lower control limits. These measurements plots must be random in nature, i.e., no points over or under the control limits and do not violate the Western Electric rules. Chapter 8 8 Capable V.S. In-Control Process • A process may be in-control, but not capable. • A process may be in-control and capable. Chapter 8 9 Process Capability • “Natural Tolerance” (process spread) of a process is the given range that the process can maintain during ordinary operation without the action of any person in making process adjustments. • Normally 3 Limits (6 sigma) Chapter 8 10 How Do We Know a Process is Capable? CP and CPK statistical process indexes • These indexes tell if the process is capable of producing features within the engineering tolerances. • This is essentially a grade for the process where the higher number is better. Chapter 8 11 The Cp Index Cp Spec width Upper spec Lower spec Natural tolerance 6 • Example: suppose the Spec width=10 units and 6 =5 units L.S. 6 Cp U.S. 10 2 units 5 • This implies the spec is twice as wide as the natural spread. Chapter 8 12 CP and CPK • Capability indexes are useful tools in the analysis of capability data. The process must be capable. • CP is a capability index. • Formula: Chapter 8 Engineering Tol. 6 sigma 13 CP and CPK • CPK is a capability index • Formula: USL mean 3 sigma or mean LSL 3 sigma what ever is less USL=Upper SPEC Limit LSL=Lower SPEC Limit Chapter 8 14 CP and CPK • Statistical Process Control– The use of statistical methods or techniques such as control charts to analyze a process or its outputs and to take appropriate actions to achieve and maintain a state of statistical control and to improve the process capability. Chapter 8 15 CP and CPK • Characteristic– A distinguishing feature of a process or its output on which variable data or attribute data can be collected. Chapter 8 16 CP and CPK (Key) • Dominant characteristics– Those characteristics that greatly influence product quality and are most important to product form, fit, or function. • Variable Data– Quantitative data, where measurements are used for analysis. Chapter 8 17 CP and CPK • Attribute Data– Qualitative data that can be counted for recording and analysis, such as the presence of a required label, the installation of all the required fasteners or the absence of errors on an expense report. It can also be characteristics that are inherently measurable but where the results are recorded in a simple yes/no fashion, such as: the acceptability of a shaft diameter when checked on a go/no-go gauge. Chapter 8 18 CP and CPK • Sigma– The standard deviation of a statistical population. Chapter 8 19 CP and CPK • Capability– When the process average plus and minus the 3 standard deviations (sigma) spread of the distribution is contained within the specification tolerance for variable data, or when at least 99.73% of individuals produced by the process meet specification for attribute data, the process is considered to be capable. Capability can only be determined after the process is in statistical control. Chapter 8 20 CP and CPK • Capability Indices– A measure of the capability of a process. Cp is the inherent capability of a process and is defined as the ratio of the tolerance to the process variation. Cpk is a measure of the capability of a process in relation to the process average. It is based on the distance between the process average and the closest specification limit. NORMAL DISTRIBUTION CURVE 68.26% 99.73% Chapter 8 [SIGMA] 4 3 2 1 X 1 2 3 4 21 Process Fallout Table Centered Process Chapter 8 Process capability ratio 0.50 0.75 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 2.00 Parts per million defective 133,600.00 24,400.00 2,700.00 967.00 318.00 96.00 26.00 6.80 1.60 0.34 0.06 0.0018 22 Relationship of Specifications, Control Limits, and Natural Tolerance Limits in the Process Chapter 8 23 Relationship of Specifications, Control Limits, and Natural Tolerance Limits in the Process Chapter 8 24 Relationship of Specifications, Control Limits, and Natural Tolerance Limits in the Process LSL USL Target Cp Cp spread Cp +3σ 2σ σ -σ -2σ -3σ C p is the ratio of spec. width to the natural variability present in the process; i.e. Chapter 8 Cp USL LSL 6 25 Relationship of Specifications, Control Limits, and Natural Tolerance Limits in the Process LSL USL Target Center Line (Avg) C pk is the ratio of the distance between the process center and the nearest spec. Limit to one half of process 3); i.e. C min X LSL , USL X variability ( Chapter 8 pk 3 3 26 Chapter 8 27 Chapter 8 28 Chapter 8 29 Process Capability When? • Prior to taking delivery of new process equipment • Before approving newly-installed process equipment for production use • As production begins, to establish capability of the equipment-tooling-material-operator combination. • On an ongoing basis to verify continuing capability. • When out-of-specification conditions are found. Chapter 8 30 Process Capability The need for capability studies • Products must meet specifications. • It is more efficient for cost and timing to produce to specification than to sort to specification. • Consistent performance requires inherent capability. • Process capability equals machine capability plus process control. • To verify capability, conduct capability studies. Chapter 8 31 Process Capability Study Flowchart 決定關鍵品質特性 從製程中隨機抽30個樣 本 （短期製程能力分析） 計算平均數，標準差， Cp/Cpk指標 是 是否通過常態性檢定 且Cpk1.23 決定SPC管制計畫中的4個 W 否 檢查並排除影響量測儀器及製程產 生變異之原因，考慮進行長程製程 能力分析 開始使用計量值或前置管制 圖 （長期製程能力分析） 使用管制圖時，每次取兩個 以上之樣本，至少收集25筆 （天）以上之資料 是 是否通過常態性檢定 且Cpk>1 檢查全距管制圖是否穩定，去掉脫 離管制界限之異常點 確認資料之正確性並使用目標值作為製程 平均，再一次估計Cp/Cpk指標 檢查平均管制圖是否穩定， 去掉異常點後再計算Cp及 Cpk之指標 決定SPC管制計畫，開始使 用 管制圖 Chapter 8 決定SPC管制計畫，開始 使用管制圖 否 是 是否通過常態性檢 定並接受Cpk>1 否 考慮以實驗設計的方法減少製程變 異，或更新設備 32 Procedures for Determining Process Capability 1) Define the process Is it a line, machine, operator, portion of manufacturing, etc.? Chapter 8 33 Procedures for Determining Process Capability 2) Determine if Specifications are now being met – conduct “Short Term” Process Capability Study a. Get current data b. Calculate X and s (usually from grouped data) to estimate the mean and standard deviation of the process. ( n= 30) Chapter 8 34 Procedures for Determining Process Capability c. Assume normality and estimate percent meeting specifications. Determine from frequency distribution if the assumption is valid. d. If specs are being met, generally go to another problem. If specs are not being met, proceed below. Chapter 8 35 Procedures for Determining Process Capability 3) Determine Inherent Variability, Using XR/Pre-Control Charts a. Get consecutive samples, two (2) or more at a time in a sample group. Get at least twenty (20) groups over a shift or other short production run during which operation appears stable or without unusual problems. Chapter 8 36 Procedures for Determining Process Capability b. Calculate average range R , and control limits for R. c. Discard any ranges outside control limits. Assume that assignable cause would have been identified and removed. If assumption cannot be made, retain the range. Chapter 8 37 Procedures for Determining Process Capability d. Recalculate average range and control limits. e. Repeat steps c and d (removal of excessive ranges and the recalculation of control limits) until all ranges in control. Chapter 8 38 Procedures for Determining Process Capability f. Estimate the standard deviation from the last average range: Estimate of σ= R /d2 g. Use midpoint of specification limits as the process mean, assume normality, and estimate percent meeting specs. Chapter 8 39 Procedures for Determining Process Capability h. If specs can be met, investigate process to determine why specs are not being met. Chapter 8 40 Procedures for Determining Process Capability i. If specs cannot be met, consider management alternatives: Change specs Change process Make best of it Drop product j. Set up X -R charts for future control. Chapter 8 41 Chapter 8 42 SPC Control Plan Sheet (1) Supplier (2) Supplier No. Address (3) Supplier Representative City/State (4) Phone No. (5) of . (6) Control Characteristics A. F. B. G. C. H. D. I. E. J. Chapter 8 43 SPC Control Plan (7) Spec Limit (8) (9) (10) (11) Station/ Location Inspection Methods Sample Size Inspection Frequency (12) (13) Analysis Cpk Method Index (14) Reaction to Out of Control A. B. C. D. E. F. G. H. I. J. (15) Submitted by . Title . Chapter 8 (16) Approved By . Date . 44 DATA COLLECTION FOR CAPABILITY ANALYSIS Char measured Part No & Name Operation No & Dese. Date SAMPLE DATA: No Value No Value No Value No Value No 1 21 41 61 81 2 22 42 62 82 3 23 43 63 83 4 24 44 64 84 5 25 45 65 85 6 26 46 66 86 7 27 47 67 87 8 28 48 68 88 9 29 49 69 89 10 30 50 70 90 11 31 51 71 91 12 32 52 72 92 13 33 53 73 93 14 34 54 74 94 15 35 55 75 95 16 36 56 76 96 17 37 57 77 97 18 38 58 78 98 19 39 59 79 99 20 40 60 80 100 Value Remarks TALLY SHEET: VALUE TALLY FREQUENCY Chapter 8 45 Process Capability Studies 1) Train engineers to learn the process capability study that is to measure the inherent variability of the process so that the performance potential can be detected under normal, in control conditions. 2) Train engineers to use the control chart method and the frequency distribution method for measuring process capability at 99.7% confidence level. Chapter 8 46 Process Capability Studies 3) The applications include the following: a. Information to facilitate the design of the product b. Acceptance of new or reconditioned piece of equipment. c. Scheduling work to machine. Chapter 8 47 Process Capability Studies d. Setting up the machine for a production run. e. Selection of operators. f. Introduction of a new product or process. Chapter 8 48 CPK • A key characteristic will be considered capable if the supplier can demonstrate with 90% confidence that the true Cpk exceeds 1.0. (In some cases, an alternate Cpk requirement will be defined in the contract.) When computing Cpk (see section 7-3.2), the number of measurements collected must be taken into account. Table 2.3.2 must be used to determine the minimum calculated Cpk to demonstrate capability Chapter 8 49 TABLE 2.3.2 CPK Chapter 8 50 CPK • The values in the above table are the calculated Cpk values required to be 90% confident that the actual Cpk is greater than or equal to the Cpk value at the top of the respective column. The values listed in the column titled “Number of measurements taken” are the actual number of measurements, not the number of plot points. The table assumes that the underlying distribution of the individual measurements are normally distributed with a fixed mean and standard deviation. Chapter 8 51 CPK • Examples: If 30 parts are measured and the required Cpk is 1.0, the calculated Cpk from the 30 parts needs to be at least 1.23. If 20 parts are measured and the calculated Cpk is 1.93, we have sufficient confidence that the actual Cpk is 1.50 or better. Chapter 8 52 Cp, Cpk Index / Process Capability Studies (1)The test specification limits for incoming diodes is .008 ohms (upper) and .001 (lower) and the standard deviation for this population is .002 ohms. What is Cp index for this process? Does this number indicate the process is within the specification limits? Chapter 8 53 Cp, Cpk Index / Process Capability Studies Why or why not? a. Suppose action was taken on this incoming test process and the standard deviation is now .001 ohms. What is the Cpk index? Is it within the limits? Given that x =0.04 Chapter 8 54 Cp, Cpk Index / Process Capability Studies (2) Measurements of the solder thicknesses on print circuit boards going through the wave solder machine are naturally between .006 and .012. Given that .001 ,whereas the specification limits are at .004, .002 from the mean of .009. What is the Cp/Cpk index for this process? Chapter 8 55 Cp, Cpk Index / Process Capability Studies (3) When should a process capability study be conducted? Chapter 8 56 Process Capability Natural tolerance limits are defined as follows: Chapter 8 57 Uses of process capability data: Chapter 8 58 Reasons for Poor Process Capability Process may have good potential capability Chapter 8 59 Chapter 8 60 Chapter 8 61 Chapter 8 62 Probability Plotting Chapter 8 63 • The distribution may not be normal; other types of probability plots can be useful in determining the appropriate distribution. Chapter 8 64 Chapter 8 65 Chapter 8 66 For the hard bake process: Chapter 8 67 One-Sided PCR Chapter 8 68 Interpretation of the PCR Chapter 8 69 Assumptions for Interpretation of Numbers in Table 8.2 • Violation of these assumptions can lead to big trouble in using the data in Table 8.2. Chapter 8 70 Chapter 8 71 • Cp does not take process centering into account • It is a measure of potential capability, not actual capability Chapter 8 72 A Measure of Actual Capability Chapter 8 73 Normality and Process Capability Ratios • The assumption of normality is critical to the usual interpretation of these ratios (such as Table 8.2) For non-normal data, options are • 1. Transform non-normal data to normal 2. Extend the usual definitions of PCRs to handle non-normal data 3. Modify the definitions of PCRs for general families of distributions Chapter 8 74 Other Types of Process Capability Ratios • • • • First generation Second generation Third generation Lots of research has been done to develop ratios that overcome some of the problems with the basic ones • Not much evidence that these ratios are used to any significant extent in practice Chapter 8 75 Chapter 8 76 Chapter 8 77 Chapter 8 78 Chapter 8 79 Chapter 8 80 Chapter 8 81 Process Capability Analysis using Control Charts Chapter 8 82 Since LSL = 200 Chapter 8 83 Chapter 8 84 Chapter 8 85 Measurement Accuracy The closeness of measurement to the true value Bias : Systematic Error Precision : Random Error Chapter 8 86 Accuracy and Precision We have focused only on precision Chapter 8 87 Chapter 8 88 Gauge R&R Studies Chapter 8 89 Chapter 8 90 Chapter 8 91 7.8 Gauge and Measurement Systems Capability Studies • Determine how much of the observed variability is due to the gauge or measurement system • Isolate the components of variability in the measurement system • Assess whether the gauge is capable (suitable for the intended application) Chapter 8 92 Chapter 8 93 Chapter 8 94 Chapter 8 95 The P/T ratio: Chapter 8 96 Chapter 8 97 Estimating the Variance Components Chapter 8 98 Chapter 8 99 The gauge is not capable by this criterion Chapter 8 100 Discrimination Ratio Chapter 8 101 Gauge R&R Studies Are Usually Conducted with a Factorial Experiment Chapter 8 102 This is a two-factor factorial experiment ANOVA methods are used to analyze the data and yo estimate the variance components Chapter 8 103 Chapter 8 104 Chapter 8 105 Chapter 8 106 Chapter 8 107 • Negative estimates of a variance component would lead to filling a reduced model, such as, for example: Chapter 8 108 Chapter 8 109 For this Example Chapter 8 110 Other Topics in Gauge R&R Studies • Section 8.7.3 provides a description of methods to obtain confidence intervals on the variance components and measures of gauge R&R • Section 8.7.4 presents a new measure of gauge capability, the probabilities of misclassification of parts – Rejecting good units (producer’s risk) – Passing bad units (consumer’s risk) – Methods for calculating these two probabilities are given Chapter 8 111 Statistical Tolerance • The more realistic statistical approach is based on the relationship between the variances of a number of independent causes(A,B,C,…) and the variance of the overall result. This relationship is : 2ass' y A2 B2 C 2 ass' y A2 B 2 C 2 Chapter 8 112 Statistical Tolerance • From a process capability of 6=T and by substitution (T=tolerance) Tass ' y TA TB TC 2 2 2 • The approach to statistical tolerance is illustrated by s simple mechanical assembly of three parts as shown below : A .500 .005 Chapter 8 B C 1.000 2.000 .010 .020 113 Statistical Tolerance • Dimensions of A,B, and C determine the overall assembly length. A conventional specification on the assembly length would be : Chapter 8 A: .500 .005 B: 1.000 .010 C: 2.000 .020 Assy : 3.500 .035 114 Statistical Tolerance • This is illustrated by again using the previous assembly. Assume that capability studies indicate that if the .005 tolerance on component a could be example to .010 , a secondary operation could be eliminated. What effect will this have on the total length tolerance? Tassy 0.0202 0.0202 0.0402 .01 4.0 4.0 16.0 Tassy .049 .0245 Chapter 8 115 Statistical Tolerance • This is an increase of only .0015 for a component increase of .005. This illustrates an important characteristic of the statistical combination of component variances. The Effect of a component with small variance is vary small; the component with the largest variance has the greatest effect on overall variance. Chapter 8 116 Statistical Tolerance • Exercise • Compare the tolerance found using the conventional engineering approach to the statistically computed assembly tolerance. • Consider the four blocks:AB, BC, CD and DE shown below. Each of these is independent. Determine the specification for the total assembly AE. A Chapter 8 .750〞B .002 〞 .320 〞C .001 〞 .475 〞 D .100 〞 E .003 〞 .002 〞 117 Statistical Tolerance • Assume that each part component was studied for capability, and we found that each required process creating the respective component linear length was stabilized (that is, in control). Each process had respective dimensional characteristics as shown below: Chapter 8 118 Component X AB .750 〞 .00067 〞 BC .320 〞 .000333 CD .475 〞 .001 〞 DE .100 〞 .00067 〞 Chapter 8 119 Statistical Tolerance • From this data, it is apparent that if we assemble these four components, that the grand average of these parts would be 1.645 〞. This is determined by the following calculation : X AB X BC X CD X DE Chapter 8 Total X AE =.750 〞 =.320 〞 =.475 〞 = .100 〞 =.100 〞(Assembly average)120 Statistical Approach • Assume that each part component was studied for capability, and we found hat each required process creating the respective component linear length was stabilized (that is , in control ). • Each process had respective dimension characteristic as shown below: Chapter 8 121 Component AB BC CD DE Chapter 8 X .750” .320” .475” .100” .00067” .00033” .001” .00067” 122 Form this data, it is apparent that if we assemble these four components, that the grand average of those parts would be 1.645”. This is determined by the following calculation: X AB =.750” X BC =.320” X CD =.475” X DE =.100” Total Chapter 8 X AE = 1.645” (assembly average) 123 If we could determine the standard deviation of AE, then we could be able to find that region within which 99.7% of all linear lengths of AE (the assembly) would lie. This region be defined as: Chapter 8 124 The standard deviation of the assembly AE can be found by calculating the square root of the sum of the square of each component. This can be stated mathematically as follows: AE AB 2 BC 2 CD 2 DE 2 Therefore, Upper tolerance X 3 Lower tolerance X 3 Total statistical tolerance What percent is this of the “ Worst case ” tolerance? Chapter 8 AE AE AE AE 125 Such shafts and bearings fit together? Is the overlap too much satisfactory assembly? Chapter 8 126 An Analysis The situation pictured in Figure 11-6 can be analyzed as follows. Let xs a value for shaft O.D. xb a value for bearing I.D. s 0.9105, b 0.9210 Chapter 8 127 Bearing 0.9105 0.9210 Shaft .9000 .9050 .9100 Shaft xs .9200 .9150 and bearing xh .9250 .9300 dimensions in cm Figure 11-6 Distributions for Shaft and Bearing with Overlapping Variation Chapter 8 128 and d xb xs Hence d b s 0.0105 The standard deviation for the bearing I.D., s , may be approximated as 0.9195 0.9015 s 0.003 6 Similarly, the standard deviation for the bearing I.D., b , may be approximated using 0.9335 0.9085 b 0.00417 6 Chapter 8 129 Hence, the standard deviation for the difference, d may be approximated using the equation with a1 1 and a2 1 as below: d s2 b2 0.0032 0.004172 0.005134 This distribution of the difference, d , will be normal, since xb and xs are normal, with the approximate mean value and standard deviation obtained above. This situation is pictured in Figure 11-7. Chapter 8 130 This area to the left of zero represents the proportion of bearings having smaller I.D.’s than the O.D.’s of the corresponding shafts. It is estimated as follows: Chapter 8 131 0 0.0105 z 2.045 0.005134 0.0105 with 2.045 0.0204 or 2.04% -.005 0 .005 .010 .015 .020 .025 d xb xs in cm Figure 11-7 Distribution for Evaluating Mating Part Tolerances Chapter 8 132 8.7.5 Attribute Gauge Capability • Sometimes the output of a gauge isn’t numerical – it’s just pass/fail • Nominal or ordinal data is also common • Occurs frequently in service businesses • Common situation – do operating personnel consistently make the same decisions regarding the units they are inspecting or analyzing • Example – a bank uses manual underwriting of mortgage loans • The underwriter uses information to classify the applicant into one of four categories; decline or category 1, 2, 3 – categories 2 & 3 are low-risk and 1 is high risk • Compare underwriters performance relative to a “consensus” evaluation determined by a panel of “experts” Chapter 8 133 Thirty applicants, three underwriters Each underwriter evaluates each application twice The applications are “blinded” by removing names, SSNs, addresses, and other identifying information Chapter 8 134 Attribute Gauge Capability • Determine the proportion of time that the underwriter agrees with him/herself – this measures repeatability • Determine the proportion of time that the underwriter agrees with the correct classification – this measures bias • Minitab performs the analysis – using the attribute agreement analysis routine Chapter 8 135 Chapter 8 136 Chapter 8 137 8.8 Setting Specifications on Discrete Components • • • • • Components interact with other components Complex assemblies Tolerance stack-up problems Linear combinations Nonlinear combinations Chapter 8 138 Chapter 8 139 Chapter 8 140 Chapter 8 141 Chapter 8 142 Chapter 8 143 8.9 Estimating the Natural Tolerance Limits of a Process For a normal distribution with unknown mean and variance: • Difference between tolerance limits and confidence limits • Nonparametric tolerance limits can also be calculated Chapter 8 144 Chapter 8 145 Chapter 8 146 Learning Objectives Chapter 8 147