Statistical Process Control - PDF Free Download

S6 Statistical Process Control PowerPoint presentation to accompany Heizer and Render Operations Management, 10e Principles of Operations Management, 8e PowerPoint slides by Jeff Heyl S6-1

Statistical Process Control The objective of a process control system is to provide a statistical signal when assignable causes of variation are present S6-2

Statistical Process Control (SPC) Variability is inherent in every process Natural or common causes Special or assignable causes Provides a statistical signal when assignable causes are present Detect and eliminate assignable causes of variation S6-3

Natural Variations Also called common causes Affect virtually all production processes Expected amount of variation Output measures follow a probability distribution For any distribution there is a measure of central tendency and dispersion If the distribution of outputs falls within acceptable limits, the process is said to be in control S6-4

Assignable Variations Also called special causes of variation Generally this is some change in the process Variations that can be traced to a specific reason The objective is to discover when assignable causes are present Eliminate the bad causes Incorporate the good causes S6-5

Samples To measure the process, we take samples and analyze the sample statistics following these steps (a) Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight Frequency # # # # # # # # # # # # # # # # # # # # # # # # # # Each of these represents one sample of five boxes of cereal Figure S6.1 Weight S6-6

Samples To measure the process, we take samples and analyze the sample statistics following these steps (b) After enough samples are taken from a stable process, they form a pattern called a distribution Frequency The solid line represents the distribution Figure S6.1 Weight S6-7

Samples To measure the process, we take samples and analyze the sample statistics following these steps (c) There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape Frequency Central tendency Weight Weight Variation Weight Shape Figure S6.1 S6-8

Samples To measure the process, we take samples and analyze the sample statistics following these steps (d) If only natural causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable Frequency Weight Prediction Figure S6.1 S6-9

Samples To measure the process, we take samples and analyze the sample statistics following these steps (e) If assignable causes are present, the process output is not stable over time and is not predicable Frequency Weight????????????????? Prediction Figure S6.1 S6-10

Control Charts Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes S6-11

Process Control Lower control limit Frequency (a) In statistical control and capable of producing within control limits Upper control limit (b) In statistical control but not capable of producing within control limits (c) Out of control Size (weight, length, speed, etc.) Figure S6.2 S6-12

Types of Data Variables Characteristics that can take any real value May be in whole or in fractional numbers Continuous random variables Attributes Defect-related characteristics Classify products as either good or bad or count defects Categorical or discrete random variables S6-13

Central Limit Theorem Regardless of the distribution of the population, the distribution of sample means drawn from the population will tend to follow a normal curve 1. The mean of the sampling distribution (x) will be the same as the population mean µ 2. The standard deviation of the sampling distribution (σ x ) will equal the population standard deviation (σ) divided by the square root of the sample size, n x = µ σ x = σ n S6-14

Population and Sampling Distributions Three population distributions Beta Normal Distribution of sample means Mean of sample means = x Standard deviation of the sample means = σ x = σ n Uniform -3σ x -2σ x -1σ x x +1σ x +2σ x +3σ x 95.45% fall within ± 2σ x 99.73% of all x fall within ± 3σ x Figure S6.3 S6-15

Sampling Distribution Sampling distribution of means Process distribution of means x = µ (mean) Figure S6.4 S6-16

Control Charts for Variables For variables that have continuous dimensions Weight, speed, length, strength, etc. x-charts are to control the central tendency of the process R-charts are to control the dispersion of the process These two charts must be used together S6-17

Setting Chart Limits For x-charts when we know σ Upper control limit (UCL) = x + zσ x Lower control limit (LCL) = x - zσ x where x = mean of the sample means or a target value set for the process z = number of normal standard deviations σ x = standard deviation of the sample means = σ/ n σ = population standard deviation n = sample size S6-18

Setting Control Limits n = 9 Hour 1 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 σ = 1 Hour Mean Hour Mean 1 16.1 7 15.2 2 16.8 8 16.4 3 15.5 9 16.3 4 16.5 10 14.8 5 16.5 11 14.2 6 16.4 12 17.3 For 99.73% control limits, z = 3 UCL x = x + zσ x = 16 + 3(1/3) = 17 ozs LCL x = x - zσ x = 16-3(1/3) = 15 ozs S6-19

Setting Control Limits Control Chart for sample of 9 boxes Out of control Variation due to assignable causes 17 = UCL 16 = Mean Variation due to natural causes 15 = LCL 1 2 3 4 5 6 7 8 9 10 11 12 Sample number Out of control Variation due to assignable causes S6-20

Setting Chart Limits For x-charts when we don t know σ Upper control limit (UCL) = x + A 2 R Lower control limit (LCL) = x - A 2 R where R = average range of the samples A 2 = control chart factor found in Table S6.1 x = mean of the sample means S6-21

Control Chart Factors Sample Size Mean Factor Upper Range Lower Range n A 2 D 4 D 3 2 1.880 3.268 0 3 1.023 2.574 0 4.729 2.282 0 5.577 2.115 0 6.483 2.004 0 7.419 1.924 0.076 8.373 1.864 0.136 9.337 1.816 0.184 10.308 1.777 0.223 12.266 1.716 0.284 Table S6.1 S6-22

Setting Control Limits Process average x = 12 ounces Average range R =.25 Sample size n = 5 S6-23

Setting Control Limits Process average x = 12 ounces Average range R =.25 Sample size n = 5 UCL x = x + A 2 R = 12 + (.577)(.25) = 12 +.144 = 12.144 ounces From Table S6.1 S6-24

Setting Control Limits Process average x = 12 ounces Average range R =.25 Sample size n = 5 UCL x = x + A 2 R = 12 + (.577)(.25) = 12 +.144 = 12.144 ounces LCL x = x - A 2 R = 12 -.144 = 11.857 ounces UCL = 12.144 Mean = 12 LCL = 11.857 S6-25

Restaurant Control Limits For salmon filets at Darden Restaurants Sample Mean Sample Range 11.5 11.0 x Bar Chart 10.5 1 3 5 7 9 11 13 15 17 0.8 0.4 Range Chart 0.0 1 3 5 7 9 11 13 15 17 UCL = 11.524 x 10.959 LCL 10.394 UCL = 0.6943 R = 0.2125 LCL = 0 S6-26

Restaurant Control Limits LSL Capability Histogram USL Specifications LSL 10 USL 12 Capability Mean = 10.959 Std.dev = 1.88 C p = 1.77 C pk = 1.7 S6-27

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 S6-28

Setting Chart Limits For R-Charts Upper control limit (UCL R ) = D 4 R Lower control limit (LCL R ) = D 3 R where R = average range of the samples D 3 and D 4 = control chart factors from Table S6.1 S6-29

Setting Control Limits Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 D 4 = 2.115, D 3 = 0 UCL R = D 4 R = (2.115)(5.3) = 11.2 pounds UCL = 11.2 Mean = 5.3 LCL R = D 3 R = (0)(5.3) = 0 pounds LCL = 0 S6-30

Mean and Range Charts (a) These sampling distributions result in the charts below x-chart UCL LCL (Sampling mean is shifting upward but range is consistent) (x-chart detects shift in central tendency) UCL R-chart LCL Figure S6.5 (R-chart does not detect change in mean) S6-31

Mean and Range Charts (b) These sampling distributions result in the charts below x-chart UCL LCL (Sampling mean is constant but dispersion is increasing) (x-chart does not detect the increase in dispersion) UCL R-chart LCL Figure S6.5 (R-chart detects increase in dispersion) S6-32

Steps In Creating Control Charts 1. Take samples from the population and compute the appropriate sample statistic 2. Use the sample statistic to calculate control limits and draw the control chart 3. Plot sample results on the control chart and determine the state of the process (in or out of control) 4. Investigate possible assignable causes and take any indicated actions 5. Continue sampling from the process and reset the control limits when necessary S6-33

Manual and Automated Control Charts S6-34

Managerial Issues and Control Charts Three major management decisions: Select points in the processes that need SPC Determine the appropriate charting technique Set clear policies and procedures S6-35

Which Control Chart to Use Variables Data Using an x-chart and R-Chart 1. Observations are variables 2. Collect 20-25 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 3. Track samples of n observations each. Table S6.3 S6-36

Patterns in Control Charts Upper control limit Target Lower control limit Normal behavior. Process is in control. Figure S6.7 S6-37

Patterns in Control Charts Upper control limit Target Lower control limit Figure S6.7 One plot out above (or below). Investigate for cause. Process is out of control. S6-38

Patterns in Control Charts Upper control limit Target Lower control limit Figure S6.7 Trends in either direction, 5 plots. Investigate for cause of progressive change. S6-39

Patterns in Control Charts Upper control limit Target Lower control limit Figure S6.7 Two plots very near lower (or upper) control. Investigate for cause. S6-40

Patterns in Control Charts Upper control limit Target Lower control limit Run of 5 above (or below) central line. Investigate for cause. Figure S6.7 S6-41

Patterns in Control Charts Upper control limit Target Lower control limit Erratic behavior. Investigate. Figure S6.7 S6-42

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 S6-43

Process Capability Ratio C p = Upper Specification - Lower Specification 6σ A capable process must have a C p of at least 1.0 Does not look at how well the process is centered in the specification range Often a target value of C p = 1.33 is used to allow for off-center processes Six Sigma quality requires a C p = 2.0 S6-44

Process Capability Ratio Insurance claims process Process mean x = 210.0 minutes Process standard deviation σ =.516 minutes Design specification = 210 ± 3 minutes C p = Upper Specification - Lower Specification 6σ S6-45

Process Capability Index Upper C pk = minimum of Specification - x, Limit 3σ Lower x - Specification Limit 3σ A capable process must have a C pk 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 S6-48

Process Capability Index New Cutting Machine New process mean x =.250 inches Process standard deviation σ =.0005 inches Upper Specification Limit =.251 inches Lower Specification Limit =.249 inches C pk = minimum of, (.251) -.250 (3).0005 S6-50

Interpreting C pk C pk = negative number C pk = zero C pk = between 0 and 1 C pk = 1 C pk > 1 Figure S6.8 S6-52

Automated Inspection Modern technologies allow virtually 100% inspection at minimal costs Not suitable for all situations S6-53

SPC and Process Variability Lower specification limit Upper specification limit (a) Acceptance sampling (Some bad units accepted) (b) Statistical process control (Keep the process in control) (c) C pk >1 (Design a process that is in control) Process mean, µ Figure S6.10 S6-54