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10-1 Quality Control Production planning and control Chapter 1 Quality Control William J. Stevenson 9th edition 10-2 Quality Control Quality Control Quality control is a process that measures output relative to standard, and acts when output doesn't meet standards. The purpose of quality control is to assure that processes are performing in an acceptable manner. Companies accomplish quality control by monitoring process output using statistical techniques. 10-3 Quality Control Phases of Quality Assurance Figure 10.1 Inspection before/after production Acceptance sampling The least progressive Inspection and corrective action during production Process control Quality built into the process Continuous improvement The most progressive 10-4 Quality Control Inspection Inspection is an appraisal activity that compares goods or services to a standard. Inspection can occur at three points: - before production: is to make sure that inputs are acceptable. - during production: to make sure that the conversion of inputs into outputs is proceeding in an acceptable manner. - after production: to make a final verification of conformance before passing goods to customers 10-5 Quality Control Inspection Inspection before and after production involves acceptance sampling procedure. Monitoring during the production process is referred as process control Inputs Acceptance sampling Figure 10.2 Transformation Process control Outputs Acceptance sampling 10-6 Quality Control Inspection The purpose of inspection is to provide information on the degree to which items conform to a standard. The basic issues of inspection are: 1 - how much to inspect and how often 2- At what points in the process inspection should occur. 3 - whether to inspect in a centralized or on-site location. 4- whether to inspect attributes (counts) or variables (measures) 10-7 Quality Control How much to inspect and how often The amount of inspection can range from no inspection to inspection of each item many times. Low-cost, high volume items such as paper clips and pencils often require little inspection because: 1. the cost associated with passing defective items is quite low. 2. the process that produce these items are usually highly reliable, so that defects are rare. High-cost, low volume items that have large cost associated with passing defective items often require more intensive inspection such as airplanes and spaceships. The majority of quality control applications ranges between these two extremes. The amount of inspection needed is governed by the cost of inspection and the expected cost of passing defective items. 10-8 Quality Control Inspection Costs Cost Figure 10.3 Total Cost Cost of inspection Cost of passing defectives Optimal Amount of Inspection 10-9 Quality Control Where to Inspect in the Process Inspection always adds to the cost of the product; therefore, it is important to restrict inspection efforts to the points where they can do the most good. In manufacturing, some of the typical inspection points are: Raw materials and purchased parts Finished products Before a costly operation Before an irreversible process Before a covering process 10-10 Quality Control Examples of Inspection Points Table 10.1 Type of business Fast Food Inspection points Cashier Counter area Eating area Building Kitchen Hotel/motel Parking lot Accounting Building Main desk Supermarket Cashiers Deliveries Characteristics Accuracy Appearance, productivity Cleanliness Appearance Health regulations Safe, well lighted Accuracy, timeliness Appearance, safety Waiting times Accuracy, courtesy Quality, quantity 10-11 Quality Control Centralized versus on-site inspection Some situations require that inspections be performed on site such as inspecting the hull of a ship for cracks. Some situations require specialized tests to be performed in a lab such as medical tests, analyzing food samples, testing metals for hardness, running viscosity tests on lubricants. 10-12 Quality Control Statistical process control Quality control is concerned with the quality of conformance of a process: Does the output of a process conform to the intent of design? Managers use Statistical Process Control (SPC) to evaluate the output of a process to determine if it is statistically acceptable. Statistical Process Control: Statistical evaluation of the output of a process during production Quality of Conformance: A product or service conforms to specifications 10-13 Quality Control Control Chart Control Chart: an important tool in SPC Purpose: to monitor process output to see if it is random (in control) or not (out of control). A time ordered plot representative sample statistics obtained from an on going process (e.g. sample means). Upper and lower control limits define the range of acceptable variation. 10-14 Quality Control Control Chart Figure 10.4 Abnormal variation due to assignable sources Out of control UCL Mean Normal variation due to chance LCL Abnormal variation due to assignable sources 0 1 2 3 4 5 6 7 8 9 Sample number 10 11 12 13 14 15 10-15 Quality Control The Statistical Process Control essence of statistical process control is to assure that the output of a process is random so that future output will be random. 10-16 Quality Control Statistical Process Control The Control Process include Define what is to be controlled. Measure the attribute or the variable to be controlled Compare with the standard Evaluate if the process in control or out of control Correct when a process is judged out of control Monitor results to ensure that corrective action is effective. 10-17 Quality Control Statistical Process Control Variations and Control Random variation: Common natural variations in the output of a process, created by countless minor factors. It would be negligible. Assignable variation: A special variation whose source can be identified (it can be assigned to a specific cause) 10-18 Quality Control Sampling Distribution The variability of a sample statistic can be described by its sampling distribution. The sampling distribution is a theoretical distribution that describe the random variability of a sample statistic. The goal of the sampling distribution is to determine whether nonrandom-and thus, correctable-source of variation are present in the output of a process. How? 10-19 Quality Control Sampling distribution Suppose there is a process for filling bottles with soft drink. If the amount of soft drink in a large number of bottles (e.g., 100) is measured accurately, we would discover slight differences among the bottles. If these amounts were arranged in a graph, the frequency distribution would reflect the process variability. The values would be clustered close to the process average, but some values would vary somewhat from the mean. 10-20 Quality Control Sampling distribution (cont.) If we return back to the process and take samples of 10 bottles each and compute the mean amount of soft drink in each sample, we would discover that these values also vary, just as the individual values varied. They, too, would have a distribution of values. The following figure shows the process and the sampling distribution. 10-21 Quality Control Sampling Distribution Figure 10.5 Sampling distribution Process distribution Mean 10-22 Quality Control Sampling distribution Properties The sampling distribution exhibits much less variability than the process distribution. The sampling distribution has the same mean as the process distribution. The sampling distribution is a normal distribution regardless of the shape of the process distribution. (central limit theorem). 10-23 Quality Control Process and sampling distribution Process distribution Mean = Variance = 2 Standard deviation = Where: n = sample size Sampling distribution Mean = Variance = 2 n Standard deviation = n 10-24 Quality Control Normal Distribution Figure 10.6 Standard deviation Mean 95.44% 99.74% 10-25 Quality Control Control limits Control charts have two limits that separate random variation and nonrandom variation. Control limits are based on sampling distribution Theoretically, the normal distribution extends in either direction to infinity. Therefore, any value is theoretically possible. As a practical matter, we know that 99.7% of the values will be within ±3 standard deviation of the mean of the distribution. Therefore, we could decide to set the control limit at the values that represent ±3 standard deviation from the mean 10-26 Quality Control Control Limits Figure 10.7 Sampling distribution Process distribution Mean Lower control limit Upper control limit 10-27 Quality Control SPC hypotheses Null hypothesis H0: the process is in control Alternative hypothesis H1: the process is out of control Actual situation Decision Reject H0 Don’t reject H0 H0 is true Type I error H0 is false Correct Correct Type II error 10-28 Quality Control Type I error SPC Errors Concluding a process is not in control when it actually is. The probability of rejecting H0 when it is actually true. Type II error Concluding a process is in control when it is not. The probability of accepting H0 when it is actually not true. 10-29 Quality Control Type I Error Figure 10.8 /2 /2 Mean Probability of Type I error LCL UCL Using wider limits (e.g., ± 3 sigma limits) reduces the probability of Type I error 10-30 Quality Control Observations from Sample Distribution Figure 10.9 UCL LCL 1 2 Sample number 3 4 10-31 Quality Control There Types of control charts are four types of control charts; two for variables, and two for attributes Attribute: counted data (e.g., number of defective items in a sample, the number of calls per day) Variable: measured data, usually on a continuous scale (e.g., amount of time needed to complete a task, length, width, weight, diameter of a part). 10-32 Quality Control Variables Control Charts Mean control charts Used to monitor the central tendency of a process. X-bar charts Range control charts Used to monitor the process dispersion R charts 10-33 Quality Control Mean Chart (X-bar chart) The control limits of the mean chart is calculated as follows: (first approach) Upper Control Limit (UCL) = x z x x z Lower Control Limit (LCL) = x Where: n = sample size z = standard normal deviation (1,2 and 3; 3 is recommended) = process standard deviation x = standard deviation of the sampling distribution of the means x x= average of sample means n 10-34 Quality Control Mean Chart (X-bar chart) Example A quality inspector took five samples, each with four observations, of the length of time for glue to dry. The analyst computed the mean of each sample and then computed the grand mean. All values are in minutes. Use this information to obtain three-sigma (i.e., z = 3) control limits for the means of future time. It is known from previous experience that the standard deviation of the process is 0.02 minute. 10-35 Quality Control Mean chart Sample 1 Observation 2 3 4 5 1 12.11 12.15 12.09 12.12 12.09 2 12.10 12.12 12.09 12.10 12.14 3 12.11 12.10 12.11 12.08 12.13 4 12.08 12.11 12.15 12.10 12.12 x 12.10 12.12 12.11 12.10 12.12 10-36 Quality Control Solution n=4 z=3 = 0.02 12.10 12.12 12.11 12.10 12.12 x 12.11 5 0.02 UCL : 12.11 3 12.14 4 0.02 LCL : 12.11 3 12.08 4 10-37 Quality Control Control chart UCL 12.14 x 12.11 LCL 12.08 1 2 3 Sample 4 5 10-38 Quality Control Mean chart A second approach to calculate the control limits: This approach assumes that the range is in control UCL x A2 R LCL x A2 R Where: A2 = A factor from table 10.2 Page 441 R = Average of sample ranges This approach is recommended when the process standard deviation is not known 10-39 Quality Control Example Twenty samples of n = 8 have been taken from a cleaning operations. The average sample range for the 20 samples was 0.016 minute, and the average mean was 3 minutes. Determine three-sigma control limits for this process. Solution x = 3 min. R, = 0.016, A2 = 0.37 for n = 8 (table 10.2) UCL x A2 R 3 0.37(0.016) 3.006 LCL x A2 R 3 0.37(0.016) 2.994 10-40 Quality Control Range Control Chart (R-chart) The R-charts are used to monitor process dispersion; they are sensitive to changes in process dispersion. Although the underlying sampling distribution of the range is not normal, the concept for use of range charts are much the same as those for use of mean chart. Control limits: UCL D4 R LCL D3 R Where values of D3 and D4 are obtained from table 10.2 page 441 10-41 Quality Control R-chart Example Twenty-five samples of n = 10 observations have been taken from a milling process. The average sample range was 0.01 centimeter. Determine upper and lower control limits for sample ranges. Solution R = 0.01 cm, n = 10 From table 10.2, for n = 10, D4 = 1.78 and D3 = 0.22 UCL = 1.78(0.01) = 0.0178 or 0.018 LCL = 0.22(0.01) = 0.0022 or 0.002 10-42 Quality Control R-Chart Example Small boxes of cereal are labeled “net weight 10 ounces.” Each hour, a random sample of size n = 4 boxes are weighted to check process control. Five hours of observation yielded the following: Time 9 A.M. 10 A.M 11 A.M Noon 1 P.M Box 1 9.8 10.1 9.9 9.7 9.7 Box 2 10.4 10.2 10.5 9.8 10.1 Box 3 9.9 9.9 10.3 10.3 9.9 Box 4 10.3 9.8 10.1 10.2 9.9 Range 0.6 0.4 0.6 0.6 0.4 10-43 Quality Control R-Chart Solution n=4 For n = 4 , D3 = 0 and D4 = 2.28 0.6 0.4 0.6 0.6 0.4 0.52 5 UCL D4 R 2.28(0.52) 1.1865 R LCL D3 R 0(0.52) 0 Since all ranges are between the upper and lower limits, we conclude that the process is in control 10-44 Quality Control Using Mean and Range Charts Mean control charts and range control charts provide different perspectives on a process. The mean charts are sensitive to shifts in process mean, whereas range charts are sensitive to changes in process dispersion. Because of this difference in perspective, both types of charts might be used to monitor the same process. 10-45 Quality Control Mean and Range Charts Figure 10.10A (process mean is shifting upward) Sampling Distribution UCL Detects shift x-Chart LCL UCL R-chart LCL Does not detect shift 10-46 Quality Control Mean and Range Charts Figure 10.10B Sampling Distribution (process variability is increasing) UCL x-Chart LCL Does not reveal increase UCL R-chart Reveals increase LCL 10-47 Quality Control Using the Mean and Range Chart To use the Mean and Range control chart, apply the following procedure: 1. 2. 3. 4. 5. Obtain 20 to 25 samples. Compute the appropriate sample statistics (mean and range) for each sample. Establish preliminary control limits using the formulas. Determine if any points fall outside the control limits. If you find no out-of-control signals, assume that the process is in control. If not, investigate and correct assignable cause of variation. Then resume the process and collect another set of observations upon which control limits can be based. Plot the data on a control chart and check for out-ofcontrol signals. 10-48 Quality Control Control Chart for Attributes Control charts for attributes are used when the process characteristic is counted rather than measured. Two types are available: P-Chart - Control chart used to monitor the proportion of defectives in a process C-Chart - Control chart used to monitor the number of defects per unit Attributes generate data that are counted. 10-49 Quality Control Use of p-Charts Table 10.3 When observations can be placed into two categories. Good or bad Pass or fail Operate or don’t operate When the data consists of multiple samples of several observations each 10-50 Quality Control P-Charts The theoretical basis for the P-chart is the binomial distribution, although for large sample sizes, the normal distribution provides a good approximation to it. A P-chart is constructed and used in much the same way as a mean chart. The center line on a P-chart is the average fraction defective in the population, P. The standard deviation of the sampling distribution when P is known is: p p(1 p) n 10-51 Quality Control P-Chart The Control limits UCL p z p LCL p z p If p is unknown, it can be estimated from the samples. That estimates p, replaces p in the preceding formulas, and ^ p replaces p. p Total number of defectives Total number of observations 10-52 Quality Control P-Chart Example An inspector counted the number of defective monthly billing statements of a company telephone in each of 20 samples. Using the following information, construct a control chart that will describe 99.74 percent of the chance variation in the process when the process is in control. Each sample counted 100 statements. 10-53 Quality Control P-Chart Example (cont.) Sample # of defective Sample # of defective 1 4 11 8 2 10 12 12 3 12 13 9 4 3 14 10 5 9 15 21 6 11 16 10 7 10 17 8 8 22 18 12 9 13 19 10 10 10 20 16 Total 220 10-54 Quality Control P-Chart Solution Z for 99.74 percent is 3 p ^ p 220 0.11 20(100) p (1 p ) n 0.11(1 0.11) 0.03 100 Control limits are ^ UCL p z p 0.11 3(0.03) 0.20 p 0.11 3(0.03) 0.02 ^ LCL p z 10-55 Quality Control P-Chart Solution (cont.) Fraction defective 0.20 UCL 0.11 p 0.02 LCL 1 10 Sample number 20 10-56 Quality Control Use of c-Charts Table 10.3 Use only when the number of occurrences per unit of measure can be counted; nonoccurrences cannot be counted. Scratches, chips, dents, or errors per item Cracks or faults per unit of distance Breaks or Tears per unit of area Bacteria or pollutants per unit of volume Calls, complaints, failures per unit of time 10-57 Quality Control C-Chart When the goal is to control the number of occurrences (e.g., defects) per unit, a C-chart is used. Units might be automobiles, hotel rooms, typed papers, or rolls of carpet. The underlying sampling distribution is the Poisson distribution. Use of Poisson distribution assumes that defects occur over some continuous region and that the probability of more than one defect at any particular point is negligible. The mean number of defects per unit is c and the standard deviation is: c 10-58 Quality Control C-Chart Control Limits UCL c z c LCL c z c If the value of c is unknown, as is generally the case, the sample estimate, c , is used in place of c. where: c = Number of defects ÷ Number of samples 10-59 Quality Control C-Chart Example Rolls of coiled wire are monitored using c-chart. Eighteen rolls have been examined, and the number of defects per roll has been recorded in the following table. Is the process in control? Plot the values on a control chart using three standard deviation control limit. sample # of Sample # of defects defects 1 2 3 4 5 6 7 8 9 3 2 4 5 1 2 4 1 2 10 11 12 13 14 15 16 17 18 1 3 4 2 4 2 1 3 1 45 10-60 Quality Control C-Chart Solution Average number of defects per coil = c = 45/18 =2.5 UCL c 3 c 2.5 3 2.5 7.24 LCL c 3 c 2.5 3 2.5 2.24 0 When the computed lower control limit is negative, the effective lower limit is zero. The calculation sometimes produces a negative lower limit due to the use of normal distribution as an approximation to the Poisson distribution. The control chart is left for the student as a homework 10-61 Managerial consideration concerning control charts Quality Control At what point in the process to use control charts: at the part of the process that (1) have tendency to go out of control, (2) are critical to the successful operation of the product or service. What size samples to take: there is a positive relation between sample size and the cost of sampling. What type of control chart to use: Variables: gives more information than attributes Attributes: less cost and time than variables 10-62 Quality Control Run Tests Run test – a test for randomness Control charts test for points that are too extreme to be considered random. However, even if all points are within the control limits, the data may still not reflect a random process. Any sort of pattern in the data would suggest a nonrandom process. The presence of patterns, such as trends, cycles, or bias in the output indicates that assignable, or nonrandom, cause of variation exist. Analyst often supplement control charts with a run test, which is another kind of test for randomness. 10-63 Quality Control Nonrandom Patterns in Control charts Figure 10.11 Trend: sustained upward or downward movement. Cycles: a wave pattern Bias: too many observations on one side of the center line Mean shift: A shift in the average Too much dispersion: the values are too spread out 10-64 Quality Control Run Test A run is defined as a sequence of observations with a certain characteristic, followed by one or more observations with a different characteristic. The characteristic can be anything that is observable. For example, in a series AAAB, there are two runs; a run of three A’s followed by a run of one B. The series AABBBA , indicates three runs; a run of two A’s followed by a run of three B’s, followed by a run of one A. 10-65 Quality Control Run test There are two types of run test: 1. 2. Runs up and down Runs above and below the median In order to count these runs, the data are transformed into a series of U’s and D’s (for up and down) and into a series of A’s and B’s (for above and below the median). There are three U/D and four A/B runs for the data: 25 29 42 40 35 38 U U D D U B B A A B A Where the median is 36.5 10-66 Quality Control Figure 10.12 Counting Runs Counting Above/Below Median Runs B A Figure 10.13 A B A B B B A Counting Up/Down Runs U U D U (7 runs) A B (8 runs) D U D U U D 10-67 1. 2. Quality Control Run test procedure To determine whether any patterns are present in control charts, one must do the following: Transform the data into both A’s and B’s and U’s and D’s, and then count the number of runs in each case. Compare the number of runs with the expected number of runs in a completely random series, which is calculated as follows: N 1 2 2N 1 3 E ( r ) med E (r ) u / d Where: N is the number of observations or data points, and E(r) is the expected number of runs 10-68 Quality Control Run test procedure (cont.) 3. Calculate the standard deviations of the runs as: med N 1 4 u/d 16 N 29 90 4. Calculate the test statistic (Ztest) as following: Z test observed number of runs – expected number of runs standard deviation of number of runs N 1) 2 N 1 4 2N 1 r ( ) 3 16 N 29 90 r ( Z t est Z t est For the median Up and down If the Ztest is within ± 2 or ± 3; then the process is random; otherwise, it is not random 10-69 Quality Control Run test Example Twenty sample means have been taken from a process. The means are shown in the following table. Use median and up/down run test with z = 2 to determine if assignable causes of variation are present. Assume the median is 11. sample 1 2 3 4 5 6 7 8 9 10 mean sample Mean 10 11 10.7 10.4 12 11.3 10.2 13 10.8 11.5 14 11.8 10.8 15 11.2 11.6 16 11.6 11.1 17 11.2 11.2 18 10.6 10.6 19 10.7 10.9 20 11.9 10-70 Quality Control Run test Solution sample mean A/B U/D Sample Mean A/B U/D 1 10 B - 11 10.7 B D 2 10.4 B U 12 11.3 A U 3 10.2 B D 13 10.8 B D 4 11.5 A U 14 11.8 A U 5 10.8 B D 15 11.2 A D 6 11.6 A U 16 11.6 A U 7 11.1 A D 17 11.2 A D 8 11.2 A U 18 10.6 B D 9 10.6 B D 19 10.7 B U 10 10.9 B U 20 11.9 A U 10-71 Quality Control Run test Solution (cont.) 1. A/B: 10 runs and U/D: 17 runs 2. Expected number of runs for each test is: N 20 1 1 11 2 2 2 N 1 2(20) 1 13 3 3 E (r ) med E (r ) u / d 3. The standard deviations are: med N 1 4 20 1 2.18 4 u/d 16 N 29 90 16( 20) 29 1.8 90 4. The ztest values are: 10 11 0.46 2.18 17 13 2.22 1.8 Z med Zu / d Although the median test doesn’t reveal any pattern, because its Ztest value is within ±2, the up/down test does; its value exceed +2. consequently, nonrandom variations are probably present in the data and, hence, the process is not in control 10-72 Quality Control Tolerances or specifications Range of acceptable values established by engineering design or customer requirements Process variability Process Capability Natural variability in a process Process capability Process variability relative to specification 10-73 Quality Control Capability analysis Capability analysis is the determination of whether the variability inherent in the output of a process falls within the acceptable range of variability allowed by the design specification for the process output. If it is within the specifications, the process is said to be “capable.” if it is not, the manager must decide how to correct the situation. We cannot automatically assume that a process that is in control will provide desired output. Instead, we must specifically check whether a process is capable of meeting specifications and not simply set up a control chart to monitor it. A process should be both in control and within specifications before production begins. 10-74 Quality Control Process Capability Figure 10.15 Lower Specification Upper Specification A. Process variability matches specifications Lower Specification Upper Specification B. Process variability Lower Upper well within specifications Specification Specification C. Process variability exceeds specifications 10-75 Quality Control Capability analysis If the product doesn’t meet specifications (not capable) a manager might consider a range of possible solutions such as: 1. Redesign the process. 2. Use an alternative process. 3. Retain the current process but attempt to eliminate unacceptable output using 100% inspection. 4. Examine the specifications to see whether they are necessary or could be relaxed without adversely affecting customer satisfaction. 10-76 Quality Control Process Capability Ratio Calculate the capability and compare it to specification width. If the capability is less than the specification width, the process is capable. Where: Capability = 6; where is the process SD Or calculate Process capability ratio, Cp = Cp = specification width process width Upper specification – lower specification 6 The process is capable if Cp is at least 1.33, this ratio implies only about 30 parts per million can be expected to not be within the specification 10-77 Quality Control Capability analysis Example A manager has the option of using any one of three machines for a job. The machines and their standard deviations are listed below. Determine which machines are capable if the specifications are 10 mm and 10.8 mm. Machine A Standard deviation (mm) 0.13 B 0.08 C 0.16 10-78 Quality Control Capability analysis Solution Capability = 6 Machine A B C Standard deviation (mm) 0.13 0.08 0.16 Machine capability 0.78 0.48 0.96 Capable Yes Yes No It is clear that machine A and machine B are capable, since the capability is less than the specification width (10.8 – 10 = 0.8) 10-79 Quality Control Capability ratio Example Compute the process capability ratio for each machine in the previous example Solution Machine Standard Machine deviation capability (mm) 6 A 0.13 0.78 B C 0.08 0.16 0.48 0.96 Cp Capable 0.8/0.78= 1.03 No 0.8/0.48 = 1.67 0.8/0.96 = 0.83 Yes No Only machine B is capable because its ratio exceed 1.33 10-80 Quality Control 3 Sigma and 6 Sigma Quality Upper specification Lower specification 1.350 ppm 1.350 ppm 1.7 ppm 1.7 ppm Process mean +/- 3 Sigma +/- 6 Sigma 10-81 Quality Control Cpk ratio If a process is not centered (the mean of the process is not in the center of the specification), a more appropriate measure of process capability is the Cpk ratio, because it does take the process mean into account. The Cpk is equal the smaller of Upper specification – process mean 3 And Process mean – lower specification 3 10-82 Quality Control Cpk Ratio Example A process has a mean of 9.2 grams and a standard deviation 0f 0.3 grams. The lower specification limit is 7.5 grams and upper specification limit is 10.5 grams. Compute Cpk Solution 1. Compute the ratio for the lower specification: 9.2 7.5 1.7 1.89 3(.3) 0.9 2. Compute the ratio for the upper specification: 10.5 9.2 1.3 1.44 3(0.3) .9 The smaller of the two ratios is 1.44 (greater than 1.33), so this is the Cpk . Therefore, the process is capable 10-83 Quality Control Improving Process Capability Simplify the process Standardize the process Mistake-proof Upgrade equipment Automate 10-84 Quality Control Improving Process Capability Method Examples Simplify Eliminate steps, reduce number of parts Standardize use standard parts, standard procedure Make Design parts that can only be assembled mistake-proof the correct way; have simple checks to verify a procedure has been performed correctly Upgrade Replace worn-out equipment; take equipment advantage of technological improvements Automate Substitute processing for manual processing 10-85 Quality Control Taguchi Loss Function Figure 10.17 Traditional cost function Cost Taguchi cost function Lower spec Target Upper spec 10-86 Quality Control 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