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**BMM 3633 Industrial Engineering**

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**Learning Objectives: Define the concept and application of SPC chart.**

Construct a control chart for variable and attribute data. Calculate process capability ratio (Cp) and process capability index (Cpk). Learning Objectives:

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**Contents: Statistical Process Control Control Charts**

Development of Control Chart Control Chart Patterns Process Capability Contents:

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**Statistical Process Control**

Take periodic samples from process Plot sample points on control chart Determine if process is within limits Prevent quality problems UCL LCL

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**Statistical Process Control (cont..)**

Variation Common Causes 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

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**Statistical Process Control (cont..)**

Types of Data 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

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**Control Charts Graph establishing process control limits**

Charts for variables Mean (x-bar), Range (R) Charts for attributes p and c

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**Control Charts (cont..) Process Control Chart 1 2 3 4 5 6 7 8 9 10**

Sample number Upper control limit Process average Lower Out of control

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**A Process is In Control if**

Control Charts (cont..) A Process is In Control if No sample points outside limits Most points near process average About equal number of points above & below centerline Points appear randomly distributed

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**Development of Control Chart**

Based on in-control data If non-random causes present discard data Correct control chart limits

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**Development of Control Chart (cont..)**

Control Charts for Attributes p Charts Calculate percent defectives in sample c Charts Count number of defects in item

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**Control Charts for Variables**

Development of Control Chart (cont..) Control Charts for Variables Mean chart ( x-Chart ) Uses average of a sample (control the central tendency of the process) Range chart ( R-Chart ) Uses amount of dispersion in a sample (control the dispersion of the sample)

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**Setting Chart Limits For x-Charts when we know s**

Upper control limit (UCL) = x + zsx Lower control limit (LCL) = x - zsx where x = mean of the sample means or a target value set for the process z = number of normal standard deviations sx = standard deviation of the sample means = s/ n s = population standard deviation n = sample size

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**Example Hour 1 Hour Mean Hour Mean Sample Weight of Number Oat Flakes**

1 17 2 13 3 16 4 18 5 17 6 16 7 15 8 17 9 16 Mean 16.1 s = 1 Hour Mean Hour Mean n = 9 For 99.73% control limits, z = 3 UCLx = x + zsx = (1/3) = 17 ozs LCLx = x - zsx = (1/3) = 15 ozs

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**Variation due to natural causes**

Example (cont..) Control Chart for sample of 9 boxes Variation due to assignable causes Out of control Sample number | | | | | | | | | | | | 17 = UCL 15 = LCL 16 = Mean Variation due to natural causes Out of control

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**Setting Chart Limits (cont..)**

For x-Charts when we don’t know s Upper control limit (UCL) = x + A2R Lower control limit (LCL) = x - A2R where R = average range of the samples A2 = control chart factor found in the table x = mean of the sample means

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**Setting Chart Limits (cont..)**

Control Chart Factors Sample Size Mean Factor Upper Range Lower Range n A2 D4 D3

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**Example Process average x = 12 ounces Average range R = .25**

Sample size n = 5

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**Example (cont..) Process average x = 12 ounces Average range R = .25**

Sample size n = 5 UCLx = x + A2R = 12 + (.577)(.25) = = ounces From Table

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**Example (cont..) Process average x = 12 ounces Average range R = .25**

Sample size n = 5 UCL = Mean = 12 LCL = UCLx = x + A2R = 12 + (.577)(.25) = = ounces LCLx = x - A2R = = ounces

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**Exercise OBSERVATIONS (SLIP-RING DIAMETER, CM) SAMPLE k 1 2 3 4 5 x R**

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**Restaurant Control Limits**

For salmon filets at Darden Restaurants Sample Mean x Bar Chart UCL = x – LCL – | | | | | | | | | 11.5 – 11.0 – 10.5 – Sample Range Range Chart UCL = R = LCL = 0 | | | | | | | | | 0.8 – 0.4 – 0.0 –

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**Restaurant Control Limits (cont..)**

Capability Histogram LSL USL 10.2 10.5 10.8 11.1 11.4 11.7 12.0 Specifications LSL USL 12 Capability Mean = Std.dev = 1.88 Cp = 1.77 Cpk = 1.7

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**R – Chart Type of variables control chart**

Shows sample ranges over time Difference between smallest and largest values in sample Monitors process variability Independent from process mean

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**Setting Chart Limits For R-Charts Upper control limit (UCLR) = D4R**

Lower control limit (LCLR) = D3R where R = average range of the samples D3 and D4 = control chart factors from Table

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**Example Average range R = 5.3 pounds Sample size n = 5**

From Table, D4 = 2.115, D3 = 0 UCL = 11.2 Mean = 5.3 LCL = 0 UCLR = D4R = (2.115)(5.3) = 11.2 pounds LCLR = D3R = (0)(5.3) = 0 pounds

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**Exercise OBSERVATIONS (SLIP-RING DIAMETER, CM) SAMPLE k 1 2 3 4 5 x R**

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**Mean and Range Charts (a)**

These sampling distributions result in the charts below (Sampling mean is shifting upward but range is consistent) x-chart (x-chart detects shift in central tendency) UCL LCL R-chart (R-chart does not detect change in mean) UCL LCL

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**Mean and Range Charts (cont..)**

(b) These sampling distributions result in the charts below (Sampling mean is constant but dispersion is increasing) x-chart (x-chart does not detect the increase in dispersion) UCL LCL R-chart (R-chart detects increase in dispersion) UCL LCL

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**Using x- and R-Charts Together**

Mean and Range Charts (cont..) Using x- and R-Charts Together Each measures the process differently Both process average and variability must be in control CHAPTER 9

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**The Normal Distribution**

=0 1 2 3 -1 -2 -3 95% 99.74%

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**Steps In Creating Control Charts**

Take samples from the population and compute the appropriate sample statistic Use the sample statistic to calculate control limits and draw the control chart Plot sample results on the control chart and determine the state of the process (in or out of control) Investigate possible assignable causes and take any indicated actions Continue sampling from the process and reset the control limits when necessary

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**Control Charts for Attributes**

For variables that are categorical Good/bad, yes/no, acceptable/unacceptable Measurement is typically counting defectives Charts may measure Percent defective (p-chart) Number of defects (c-chart)

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**Control Limits for p-Charts**

Population will be a binomial distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics UCLp = p + zsp ^ p(1 - p) n ðp = ^ LCLp = p - zsp ^ where p = mean fraction defective in the sample z = number of standard deviations ðp = standard deviation of the sampling distribution n = sample size ^

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**Control Limits for p-Charts (cont..)**

Control Chart Z Values Smaller Z values make more sensitive charts Z = 3.00 is standard Compromise between sensitivity and errors CHAPTER 9

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**Example Samples of work of 20 clerks; each clerk entered 100 records**

Sample Number Fraction Sample Number Fraction Number of Errors Defective Number of Errors Defective Total = 80 p = = 0.04 80 (100)(20) (0.04)( ) 100 sp = = 0.02 ^

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**Example (cont..) UCLp = p + zsp = 0.04 + 3(0.02) = 0.10**

^ LCLp = p - zsp = (0.02) = 0 ^ .11 – .10 – .09 – .08 – .07 – .06 – .05 – .04 – .03 – .02 – .01 – .00 – Sample number Fraction defective | | | | | | | | | | UCLp = 0.10 LCLp = 0.00 p = 0.04

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**Possible assignable causes present**

Example (cont..) UCLp = p + zsp = (0.02) = 0.10 ^ Possible assignable causes present LCLp = p - zsp = (0.02) = 0 ^ .11 – .10 – .09 – .08 – .07 – .06 – .05 – .04 – .03 – .02 – .01 – .00 – Sample number Fraction defective | | | | | | | | | | UCLp = 0.10 LCLp = 0.00 p = 0.04

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**Exercise 20 samples of 100 pairs of jeans 1 6 .06 11 9 .09**

Sample Number Fraction Sample Number Fraction Number of Errors Defective Number of Errors Defective Total = 200

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**Control Limits for c-Charts**

Population will be a Poisson distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics UCLc = c + 3 c LCLc = c - 3 c where c = mean number defective in the sample

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**Example c = 54 complaints/9 days = 6 complaints/day UCLc = c + 3 c**

Number of complaints: 3, 0, 8, 9, 6, 7, 4, 9, 8 over 9-day period c = 54 complaints/9 days = 6 complaints/day UCLc = c + 3 c = = 13.35 | 1 2 3 4 5 6 7 8 9 Day Number defective 14 – 12 – 10 – 8 – 6 – 4 – 2 – 0 – UCLc = 13.35 LCLc = 0 c = 6 LCLc = c - 3 c = = 0

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**Exercise The number of defects in 15 sample rooms**

Sample Number Sample Number Sample Number Number of Defects Number of Defects Number of Defects Total = 190

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**Which Control Chart to Use**

Variables Data Using an x-Chart and R-Chart Observations are variables Collect samples of n = 4, or n = 5, or more, each from a stable process and compute the mean for the x-chart and range for the R-chart Track samples of n observations each.

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**Which Control Chart to Use (cont..)**

Attribute Data Using the p-Chart Observations are attributes that can be categorized as good or bad (or pass–fail, or functional–broken), that is, in two states. We deal with fraction, proportion, or percent defectives. There are several samples, with many observations in each. For example, 20 samples of n = 100 observations in each.

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**Which Control Chart to Use (cont..)**

Attribute Data Using a c-Chart Observations are attributes whose defects per unit of output can be counted. We deal with the number counted, which is a small part of the possible occurrences. Defects may be: number of blemishes on a desk; complaints in a day; crimes in a year; broken seats in a stadium; typos in a chapter of this text; or flaws in a bolt of cloth.

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**Patterns in Control Charts**

Upper control limit Target Lower control limit Normal behavior. Process is “in control.”

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**Patterns in Control Charts (cont..)**

Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. One plot out above (or below). Investigate for cause. Process is “out of control.”

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**Patterns in Control Charts (cont..)**

Upper control limit Target Lower control limit Trends in either direction, 5 plots. Investigate for cause of progressive change.

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**Patterns in Control Charts (cont..)**

Upper control limit Target Lower control limit Two plots very near lower (or upper) control. Investigate for cause.

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**Patterns in Control Charts (cont..)**

Upper control limit Target Lower control limit Run of 5 above (or below) central line. Investigate for cause.

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**Patterns in Control Charts (cont..)**

Upper control limit Target Lower control limit Erratic behavior. Investigate.

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**Sample Size Determination**

Attribute control charts 50 to 100 parts in a sample Variable control charts 2 to 10 parts in a sample CHAPTER 9

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Process Capability The natural variation of a process should be small enough to produce products that meet the standards required A process in statistical control does not necessarily meet the design specifications Process capability is a measure of the relationship between the natural variation of the process and the design specifications

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**Process Capability Ratio**

Cp = Upper Specification - Lower Specification 6σ A capable process must have a Cp of at least 1.0 Does not look at how well the process is centered in the specification range Often a target value of Cp = 1.33 is used to allow for off-center processes Six Sigma quality requires a Cp = 2.0

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**Upper Specification - Lower Specification**

Example Insurance claims process Process mean x = minutes Process standard deviation s = minutes Design specification = 210 ± 3 minutes Cp = Upper Specification - Lower Specification 6σ

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**Upper Specification - Lower Specification**

Example (cont..) Insurance claims process Process mean x = minutes Process standard deviation s = minutes Design specification = 210 ± 3 minutes Cp = Upper Specification - Lower Specification 6σ = = 1.938 6(0.516)

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**Upper Specification - Lower Specification**

Example (cont..) Insurance claims process Process mean x = minutes Process standard deviation s = minutes Design specification = 210 ± 3 minutes Cp = Upper Specification - Lower Specification 6σ = = 1.938 6(0.516) Process is capable

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**Exercise Net weight specification = 9.0 oz 0.5 oz**

Process mean = 8.80 oz Process standard deviation = 0.12 oz CHAPTER 9

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Exercise In a GE insurance claims process, x = minutes, and = minutes. The design specification to meet customer expectations is 210 ± 3 minutes. Determine the process capability ratio. Is the process capable? CHAPTER 9

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**Process Capability Index**

Cpk = minimum of , Upper Specification - x Limit 3σ Lower x - Specification Limit A capable process must have a Cpk of at least 1.0 A capable process is not necessarily in the center of the specification, but it falls within the specification limit at both extremes

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**Example New Cutting Machine New process mean x = 0.250 inches**

Process standard deviation s = inches Upper Specification Limit = inches Lower Specification Limit = inches

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**Example (cont..) New Cutting Machine New process mean x = 0.250 inches**

Process standard deviation s = inches Upper Specification Limit = inches Lower Specification Limit = inches Cpk = minimum of , (0.251) (3)0.0005

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**New machine is NOT capable**

Example (cont..) New Cutting Machine New process mean x = inches Process standard deviation s = inches Upper Specification Limit = inches Lower Specification Limit = inches Cpk = minimum of , (0.251) (3)0.0005 (0.249) Both calculations result in New machine is NOT capable Cpk = = 0.67 0.001 0.0015

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**Exercise Net weight specification = 9.0 oz 0.5 oz**

Process mean = 8.80 oz Process standard deviation = 0.12 oz CHAPTER 9

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**Interpreting Cpk Cpk = negative number Cpk = zero**

Cpk = between 0 and 1 Cpk = 1 Cpk > 1

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Any Questions???

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Thank You

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Copyright 2006 John Wiley & Sons, Inc. Beni Asllani University of Tennessee at Chattanooga Statistical Process Control Operations Management - 5 th Edition.

Copyright 2006 John Wiley & Sons, Inc. Beni Asllani University of Tennessee at Chattanooga Statistical Process Control Operations Management - 5 th Edition.

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