Ch03_6e - Chapter 3 Chapter Statistical Process Control...

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Unformatted text preview: Chapter 3 Chapter Statistical Process Control Statistical Process Control Control • Take periodic samples from Take process process • Plot sample points on a Plot control chart control • Determine if process UCL Determine is within limits is • Correct process before Correct LCL defects are produced defects Control Chart Control Out of control Upper control limit Process average Lower control limit 1 2 3 4 5 6 Sample number 7 8 9 10 Variation Variation Common Causes Common Variation inherent in a process Can be eliminated only through improvements in the system Special Causes Variation due to identifiable factors Can be modified through operator or management action Types of Data Types Attribute data Product characteristic evaluated with a discrete choice Good/bad, yes/no Variable data Product characteristic that can be measured Length, size, weight, height, time, velocity Control Charts for Attributes Attributes p Charts Calculate percent defectives in sample Based on binonomial distribution c Charts Count number of defects in an item Based on Poisson distribution Based p-Chart p-Chart UCL = p + zσ p UCL where LCL = p - zσ p LCL z = the number of standard the deviations from the process average deviations p = the sample percent the defective; an estimate of the process average average σp = the standard deviation of the the sample proportion the p(1 - p) (1 σp = n p-Chart Example p-Chart SAMPLE 1 2 3 : : 20 NUMBER OF DEFECTIVES 6 0 4 : : 18 Total 200 PROPORTION DEFECTIVE 6/100 = .06 6/ .00 .04 : : .18 Avg .10 20 samples of 100 pairs of jeans 20 100 p-Chart Calculations p-Chart p= total defectives = 200 / 20(100) = 0.10 total sample observations UCL = p + z p(1 - p) = 0.10 + 3 n 0.10(1 - 0.10) 100 UCL = 0.190 LCL = p - z LCL = 0.010 p(1 - p) = 0.10 - 3 n 0.10(1 - 0.10) 100 0.20 Is the process in control? UCL = 0.190 0.18 Proportion defective 0.16 0.14 0.12 0.10 No! Pts are outside the limits. p = 0.10 0.08 0.06 0.04 0.02 0.02 LCL = 0.010 2 4 6 8 10 12 14 Sample number 16 18 20 c-Chart c-Chart UCL = c + zσ c UCL LCL = c - zσ c LCL σc = c where c = number of defects per sample c-Chart Example c-Chart The number of defects in 15 sample rooms SAMPLE NUMBER OF DEFECTS 1 2 3 12 8 16 : : : : 15 15 190 c-Chart c-Chart 24 UCL = 23.35 Yes Number of defects Is this process in control? 21 18 c = 12.67 15 12 9 6 LCL = 1.99 3 2 4 6 8 10 Sample number 12 14 16 Control Charts for Variables for Mean chart ( X-bar Chart ) uses average of a sample Range chart ( R Chart ) uses amount of dispersion in a sample X-bar Chart X-bar = = x1 + x2 + ... xk x k =+ A R =- A R UCL = x LCL = x UCL LCL 2 2 where = x = average of sample means R = range range A2 = value from 3σ table X-bar Chart Example X-bar OBSERVATIONS (SLIP- RING DIAMETER, CM) SAMPLE k 1 2 3 4 5 x- bar R 1 2 3 4 5 6 7 8 9 10 5.02 5.01 4.99 5.03 4.95 4.97 5.05 5.09 5.14 5.01 5.01 5.03 5.00 4.91 4.92 5.06 5.01 5.10 5.10 4.98 4.94 5.07 4.93 5.01 5.03 5.06 5.10 5.00 4.99 5.08 4.99 4.95 4.92 4.98 5.05 4.96 4.96 4.99 5.08 5.07 4.96 4.96 4.99 4.89 5.01 5.03 4.99 5.08 5.09 4.99 4.98 5.00 4.97 4.96 4.99 5.01 5.02 5.05 5.08 5.03 0.08 0.12 0.08 0.14 0.13 0.10 0.14 0.11 0.15 0.10 50.09 1.15 Total X-bar is the average of each row. R is the range, i.e., the difference between the largest and smallest value in a row. X- bar Chart XCalculations Calculations 50.09 = ∑x x= = = 5.01 cm 10 k X-bar-bar is the average of the x’s = UCL = x + A2R = 5.01 + (0.58)(0.115) = 5.08 = LCL = x - A2R = 5.01 - (0.58)(0.115) = 4.94 Lookup A2 from the table (on next slide) by sample size Look back at the data (on previous slide). There are 10 samples. The size of each sample is 5. Determining Control Limits for x-bar and R-Charts SAMPLE SIZE n FACTOR FOR x-CHART A2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1.88 1.02 0.73 0.58 0.48 0.42 0.37 0.44 0.11 0.99 0.77 0.55 0.44 0.22 0.11 0.00 0.99 0.99 0.88 Fa Fa ct or s FACTORS FOR R-CHART D3 D4 0.00 0.00 0.00 0.00 0.00 0.08 0.14 0.18 0.22 0.26 0.28 0.31 0.33 0.35 0.36 0.38 0.39 0.40 0.41 3.27 2.57 2.28 2.11 2.00 1.92 1.86 1.82 1.78 1.74 1.72 1.69 1.67 1.65 1.64 1.62 1.61 1.61 1.59 5.10 – Is this process in control? 5.08 – UCL = 5.08 5.06 – Yes, pt 9 is on the line, not over it. Mean 5.04 – = x = 5.01 5.02 – 5.00 – 4.98 – 4.96 – X- bar Chart Chart LCL = 4.94 4.94 – 4.92 – | 1 | 2 | 3 | | | | 4 5 6 7 Sample number | 8 | 9 | 10 R- Chart RUCL = D4R UCL LCL = D3R LCL ∑R R= k where R = range of each sample k = number of samples D3, D4 = values from table R-Chart Example R-Chart OBSERVATIONS (SLIP-RING DIAMETER, CM) SAMPLE k SAMPLE 1 2 3 4 5 x R 1 2 3 4 5 6 7 8 9 10 5.02 5.01 4.99 5.03 4.95 4.97 5.05 5.09 5.14 5.01 5.01 5.03 5.00 4.91 4.92 5.06 5.01 5.10 5.10 4.98 4.94 5.07 4.93 5.01 5.03 5.06 5.10 5.00 4.99 5.08 4.99 4.95 4.92 4.98 5.05 4.96 4.96 4.99 5.08 5.07 4.96 4.96 4.99 4.89 5.01 5.03 4.99 5.08 5.09 4.99 4.98 5.00 4.97 4.96 4.99 5.01 5.02 5.05 5.08 5.03 0.08 0.12 0.08 0.14 0.13 0.10 0.14 0.11 0.15 0.10 50.09 1.15 Total Example 15.3 R-Chart Calculations R-Chart ∑R 1.15 R= = = 0.115 k 10 UCL = D4R = 2.11(0.115) = 0.243 LCL = D3R = 0(0.115) = 0 Lookup Values D3 and D4 in table in Determining Control Limits for x-bar and R-Charts SAMPLE SIZE n FACTOR FOR x-CHART A2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1.88 1.02 0.73 0.58 0.48 0.42 0.37 0.44 0.11 0.99 0.77 0.55 0.44 0.22 0.11 0.00 0.99 0.99 0.88 Fa Fa ct or s FACTORS FOR R-CHART D3 D4 0.00 0.00 0.00 0.00 0.00 0.08 0.14 0.18 0.22 0.26 0.28 0.31 0.33 0.35 0.36 0.38 0.39 0.40 0.41 3.27 2.57 2.28 2.11 2.00 1.92 1.86 1.82 1.78 1.74 1.72 1.69 1.67 1.65 1.64 1.62 1.61 1.61 1.59 R-Chart R-Chart 0.28 – 0.24 – UCL = 0.243 Range 0.20 – 0.16 – R = 0.115 0.12 – 0.08 – 0.04 – 0– LCL = 0 | | | 1 2 3 | | | | 4 5 6 7 Sample number | 8 | 9 | 10 Use X- bar and R-Charts Together R-Charts Process average and process variability must be in control. It is possible for samples to have very narrow ranges, but their averages is beyond control limits. It is possible for sample averages to be in control, but ranges might be very large. A Process is In Control if Process 1. No sample points are outside the No control limits control 2. Most points are near the process Most average average 3. About an equal number of points About are above & below the centerline are 4. Points appear to be randomly Points distributed distributed Control Chart Patterns Control 1. 8 consecutive points on one side of the consecutive center line. center 2. 8 consecutive points up or down across consecutive zones. zones. 3. 14 points alternating up and down. 4. 2 out of 3 consecutive points in Zone A out but still inside the control limits. but 5. 4 out of 5 consecutive points in Zone A out or B. or You don’t have to memorize these. They will be provided on quiz. Zones for Pattern Tests Zones Figure 15.4 Rule 1: Eight consecutive points on one side of the center line UCL UCL Mean Mean LCL LCL Rule 2: Eight consecutive points up or down across zones UCL UCL LCL LCL Rule 3: Fourteen points alternating up or down alternating UCL LCL Rule 4: Two of three consecutive points in Zone A consecutive UCL UCL Zone A Zone B Zone C Zone C LCL Zone A Zone B Zone C Zone C Zone B Zone A Zone B Zone A LCL Rule 5: 4 out of 5 consecutive points in Zone B or beyond points UCL UCL Zone C Zone A Zone B Zone C Zone C Zone B Zone C Zone B Zone A Zone A Zone A Zone B LCL LCL Process Capability Process The range of natural variability in a The process process A process cannot meet specifications process if its natural variability exceeds the design tolerances design 3-sigma quality Specifications equal the process control limits. 6-sigma quality Specifications are twice as large as control Specifications limits limits Process Capability Process Design Design Specifications Specifications Cp < 1; Process is not capable (a) Natural variation (a) exceeds design specifications; process is not capable of meeting specifications all the time. all Process Design Design Specifications Specifications (b) Design specifications b) and natural variation the same; process is capable of meeting specifications most the time. most Cp = 1; Process is just capable Process Process Capability Process Cp > 1; process is capable Design Design Specifications Specifications (c) Design specifications c) greater than natural variation; process is capable of always conforming to specifications. specifications. Process Design Design Specifications Specifications (d) Specifications greater d) than natural variation, but process off center; capable but some output will not meet upper specification. specification. Cpk measures if process is off-center Process If Cp = 2, Six sigma quality Process Capability Process Process Capability Ratio tolerance range Cp = process range = upper specification limit upper lower specification limit lower 6σ where 6σ = UCL – LCL Process Capability Index, Cpk Process Cpk = minimum = x - lower specification limit , 3σ = upper specification limit - x 3σ Recall, for x-bar chart, A2(R-bar) = 3σ ...
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