© 2006 Prentice Hall, Inc.S6 – 1 Operations Management Supplement 6 – Statistical Process Control © 2006 Prentice Hall, Inc. PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 6e Operations Management, 8e

© 2006 Prentice Hall, Inc.S6 – 2 Outline  Statistical Process Control (SPC)  Control Charts for Variables  The Central Limit Theorem  Setting Mean Chart Limits (x-Charts)  Setting Range Chart Limits (R-Charts)  Using Mean and Range Charts  Control Charts for Attributes  Managerial Issues and Control Charts

© 2006 Prentice Hall, Inc.S6 – 3 Outline – Continued  Process Capability  Process Capability Ratio (C p )  Process Capability Index (C pk )  Acceptance Sampling  Operating Characteristic Curve  Average Outgoing Quality

© 2006 Prentice Hall, Inc.S6 – 4 Learning Objectives When you complete this supplement, you should be able to: Identify or Define:  Natural and assignable causes of variation  Central limit theorem  Attribute and variable inspection  Process control  x-charts and R-charts

© 2006 Prentice Hall, Inc.S6 – 5 Learning Objectives When you complete this supplement, you should be able to: Identify or Define:  LCL and UCL  P-charts and c-charts  C p and C pk  Acceptance sampling  OC curve

© 2006 Prentice Hall, Inc.S6 – 6 Learning Objectives When you complete this supplement, you should be able to: Identify or Define:  AQL and LTPD  AOQ  Producer’s and consumer’s risk

© 2006 Prentice Hall, Inc.S6 – 7 Learning Objectives When you complete this supplement, you should be able to: Describe or Explain:  The role of statistical quality control

© 2006 Prentice Hall, Inc.S6 – 8  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 Statistical Process Control (SPC)

© 2006 Prentice Hall, Inc.S6 – 9 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”

© 2006 Prentice Hall, Inc.S6 – 10 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

© 2006 Prentice Hall, Inc.S6 – 11 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 Weight # ## # ## ## # ### #### ######### # Each of these represents one sample of five boxes of cereal Figure S6.1

© 2006 Prentice Hall, Inc.S6 – 12 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 The solid line represents the distribution Frequency Weight Figure S6.1

© 2006 Prentice Hall, Inc.S6 – 13 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 Weight Central tendency Weight Variation Weight Shape Frequency Figure S6.1

© 2006 Prentice Hall, Inc.S6 – 14 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 Weight Time Frequency Prediction Figure S6.1

© 2006 Prentice Hall, Inc.S6 – 15 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 Weight Time Frequency Prediction?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Figure S6.1

© 2006 Prentice Hall, Inc.S6 – 16 Control Charts Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes

© 2006 Prentice Hall, Inc.S6 – 17 Types of Data  Characteristics that can take any real value  May be in whole or in fractional numbers  Continuous random variables VariablesAttributes  Defect-related characteristics  Classify products as either good or bad or count defects  Categorical or discrete random variables

© 2006 Prentice Hall, Inc.S6 – 18 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  x =   n x =x =x =x = 2.The standard deviation of the sampling distribution (  x ) will equal the population standard deviation (  ) divided by the square root of the sample size, n

© 2006 Prentice Hall, Inc.S6 – 19 Process Control Figure S6.2 Frequency (weight, length, speed, etc.) Size Lower control limit Upper control limit (a) In statistical control and capable of producing within control limits (b) In statistical control but not capable of producing within control limits (c) Out of control

© 2006 Prentice Hall, Inc.S6 – 20 Population and Sampling Distributions Three population distributions Beta Normal Uniform Distribution of sample means Standard deviation of the sample means =  x =  n Mean of sample means = x |||||||-3x-2x-1xx+1x+2x+3x-3x-2x-1xx+1x+2x+3x|||||||-3x-2x-1xx+1x+2x+3x-3x-2x-1xx+1x+2x+3x 99.73% of all x fall within ± 3  x 95.45% fall within ± 2  x Figure S6.3

© 2006 Prentice Hall, Inc.S6 – 21 Sampling Distribution x =  (mean) Sampling distribution of means Process distribution of means Figure S6.4

© 2006 Prentice Hall, Inc.S6 – 22 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

© 2006 Prentice Hall, Inc.S6 – 23 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

© 2006 Prentice Hall, Inc.S6 – 24 Setting Chart Limits For x-Charts when we know  Upper control limit (UCL) = x + z  x Lower control limit (LCL) = x - z  x wherex=mean of the sample means or a target value set for the process z=number of normal standard deviations  x =standard deviation of the sample means =  / n  =population standard deviation n=sample size

© 2006 Prentice Hall, Inc.S6 – 25 Setting Control Limits Hour 1 SampleWeight of NumberOat Flakes Mean16.1  =1 HourMeanHourMean n = 9 LCL x = x - z  x = (1/3) = 15 ozs For 99.73% control limits, z = 3 UCL x = x + z  x = (1/3) = 17 ozs

© 2006 Prentice Hall, Inc.S6 – = UCL 15 = LCL 16 = Mean Setting Control Limits Control Chart for sample of 9 boxes Sample number |||||||||||| Variation due to assignable causes Variation due to natural causes Out of control

© 2006 Prentice Hall, Inc.S6 – 27 Setting Chart Limits For x-Charts when we don’t know  Lower control limit (LCL) = x - A 2 R Upper control limit (UCL) = x + A 2 R whereR=average range of the samples A 2 =control chart factor found in Table S6.1 x=mean of the sample means

© 2006 Prentice Hall, Inc.S6 – 28 Control Chart Factors Table S6.1 Sample Size Mean Factor Upper Range Lower Range n A 2 D 4 D

© 2006 Prentice Hall, Inc.S6 – 29 Setting Control Limits Process average x = ounces Average range R =.25 Sample size n = 5

© 2006 Prentice Hall, Inc.S6 – 30 Setting Control Limits UCL x = x + A 2 R = (.577)(.25) = = ounces Process average x = ounces Average range R =.25 Sample size n = 5 From Table S6.1

© 2006 Prentice Hall, Inc.S6 – 31 Setting Control Limits UCL x = x + A 2 R = (.577)(.25) = = ounces LCL x = x - A 2 R = = ounces Process average x = ounces Average range R =.25 Sample size n = 5 UCL = Mean = LCL =

© 2006 Prentice Hall, Inc.S6 – 32 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

© 2006 Prentice Hall, Inc.S6 – 33 Setting Chart Limits For R-Charts Lower control limit (LCL R ) = D 3 R Upper control limit (UCL R ) = D 4 R where R=average range of the samples D 3 and D 4 =control chart factors from Table S6.1

© 2006 Prentice Hall, Inc.S6 – 34 Setting Control Limits UCL R = D 4 R = (2.115)(5.3) = 11.2 pounds LCL R = D 3 R = (0)(5.3) = 0 pounds Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 D 4 = 2.115, D 3 = 0 UCL = 11.2 Mean = 5.3 LCL = 0

© 2006 Prentice Hall, Inc.S6 – 35 Mean and Range Charts (a) These sampling distributions result in the charts below (Sampling mean is shifting upward but range is consistent) R-chart (R-chart does not detect change in mean) UCLLCL Figure S6.5 x-chart (x-chart detects shift in central tendency) UCLLCL

© 2006 Prentice Hall, Inc.S6 – 36 Mean and Range Charts R-chart (R-chart detects increase in dispersion) UCLLCL Figure S6.5 (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) UCLLCL

© 2006 Prentice Hall, Inc.S6 – 37 Automated Control Charts

© 2006 Prentice Hall, Inc.S6 – 38 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)

© 2006 Prentice Hall, Inc.S6 – 39 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 UCL p = p + z  p ^ LCL p = p - z  p ^ wherep=mean fraction defective in the sample z=number of standard deviations  p =standard deviation of the sampling distribution n=sample size ^ p(1 - p) n p =p =p =p = ^

© 2006 Prentice Hall, Inc.S6 – 40 p-Chart for Data Entry SampleNumberFractionSampleNumberFraction Numberof ErrorsDefectiveNumberof ErrorsDefective Total = 80 (.04)(1 -.04) 100  p = =.02 ^ p = =.04 80(100)(20)

© 2006 Prentice Hall, Inc.S6 – – – – – – – – – – – – – Sample number Fraction defective |||||||||| p-Chart for Data Entry UCL p = p + z  p = (.02) =.10 ^ LCL p = p - z  p = (.02) = 0 ^ UCL p = 0.10 LCL p = 0.00 p = 0.04

© 2006 Prentice Hall, Inc.S6 – – – – – – – – – – – – – Sample number Fraction defective |||||||||| UCL p = p + z  p = (.02) =.10 ^ LCL p = p - z  p = (.02) = 0 ^ UCL p = 0.10 LCL p = 0.00 p = 0.04 p-Chart for Data Entry Possible assignable causes present

© 2006 Prentice Hall, Inc.S6 – 43 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 wherec=mean number defective in the sample UCL c = c + 3 c LCL c = c - 3 c

© 2006 Prentice Hall, Inc.S6 – 44 c-Chart for Cab Company c = 54 complaints/9 days = 6 complaints/day |1|1 |2|2 |3|3 |4|4 |5|5 |6|6 |7|7 |8|8 |9|9 Day Number defective – – – 8 8 – 6 6 – 4 – 2 – 0 0 – UCL c = c + 3 c = = LCL c = c - 3 c = = 0 UCL c = LCL c = 0 c = 6

© 2006 Prentice Hall, Inc.S6 – 45 Patterns in Control Charts Normal behavior. Process is “in control.” Upper control limit Target Lower control limit Figure S6.7

© 2006 Prentice Hall, Inc.S6 – 46 Upper control limit Target Lower control limit Patterns in Control Charts One plot out above (or below). Investigate for cause. Process is “out of control.” Figure S6.7

© 2006 Prentice Hall, Inc.S6 – 47 Upper control limit Target Lower control limit Patterns in Control Charts Trends in either direction, 5 plots. Investigate for cause of progressive change. Figure S6.7

© 2006 Prentice Hall, Inc.S6 – 48 Upper control limit Target Lower control limit Patterns in Control Charts Two plots very near lower (or upper) control. Investigate for cause. Figure S6.7

© 2006 Prentice Hall, Inc.S6 – 49 Upper control limit Target Lower control limit Patterns in Control Charts Run of 5 above (or below) central line. Investigate for cause. Figure S6.7

© 2006 Prentice Hall, Inc.S6 – 50 Upper control limit Target Lower control limit Patterns in Control Charts Erratic behavior. Investigate. Figure S6.7

© 2006 Prentice Hall, Inc.S6 – 51 Which Control Chart to Use  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 Variables Data

© 2006 Prentice Hall, Inc.S6 – 52 Which Control Chart to Use  Using the p-chart:  Observations are attributes that can be categorized in two states  We deal with fraction, proportion, or percent defectives  Have several samples, each with many observations Attribute Data

© 2006 Prentice Hall, Inc.S6 – 53 Which Control Chart to Use  Using a c-Chart:  Observations are attributes whose defects per unit of output can be counted  The number counted is often a small part of the possible occurrences  Defects such as number of blemishes on a desk, number of typos in a page of text, flaws in a bolt of cloth Attribute Data

© 2006 Prentice Hall, Inc.S6 – 54 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

© 2006 Prentice Hall, Inc.S6 – 55 Process Capability Ratio Cp =Cp =Cp =Cp = Upper Specification - Lower Specification 6   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

© 2006 Prentice Hall, Inc.S6 – 56 Process Capability Ratio Cp =Cp =Cp =Cp = Upper Specification - Lower Specification 6  Insurance claims process Process mean x = minutes Process standard deviation  =.516 minutes Design specification = 210 ± 3 minutes

© 2006 Prentice Hall, Inc.S6 – 57 Process Capability Ratio Cp =Cp =Cp =Cp = Upper Specification - Lower Specification 6  Insurance claims process Process mean x = minutes Process standard deviation  =.516 minutes Design specification = 210 ± 3 minutes = = (.516)

© 2006 Prentice Hall, Inc.S6 – 58 Process Capability Ratio Cp =Cp =Cp =Cp = Upper Specification - Lower Specification 6  Insurance claims process Process mean x = minutes Process standard deviation  =.516 minutes Design specification = 210 ± 3 minutes = = (.516) Process is capable

© 2006 Prentice Hall, Inc.S6 – 59 Process Capability Index  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 C pk = minimum of, Upper Specification - x Limit  Lower x -Specification Limit 

© 2006 Prentice Hall, Inc.S6 – 60 Process Capability Index New Cutting Machine New process mean x =.250 inches Process standard deviation  =.0005 inches Upper Specification Limit =.251 inches Lower Specification Limit =.249 inches

© 2006 Prentice Hall, Inc.S6 – 61 Process Capability Index New Cutting Machine New process mean x =.250 inches Process standard deviation  =.0005 inches Upper Specification Limit =.251 inches Lower Specification Limit =.249 inches C pk = minimum of, (.251) (3).0005

© 2006 Prentice Hall, Inc.S6 – 62 Process Capability Index New Cutting Machine New process mean x =.250 inches Process standard deviation  =.0005 inches Upper Specification Limit =.251 inches Lower Specification Limit =.249 inches C pk = = New machine is NOT capable C pk = minimum of, (.251) (3) (.249) (3).0005 Both calculations result in

© 2006 Prentice Hall, Inc.S6 – 63 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

© 2006 Prentice Hall, Inc.S6 – 64 Acceptance Sampling  Form of quality testing used for incoming materials or finished goods  Take samples at random from a lot (shipment) of items  Inspect each of the items in the sample  Decide whether to reject the whole lot based on the inspection results  Only screens lots; does not drive quality improvement efforts

© 2006 Prentice Hall, Inc.S6 – 65 Operating Characteristic Curve  Shows how well a sampling plan discriminates between good and bad lots (shipments)  Shows the relationship between the probability of accepting a lot and its quality level

© 2006 Prentice Hall, Inc.S6 – 66 Return whole shipment The “Perfect” OC Curve % Defective in Lot P(Accept Whole Shipment) – – – – 0 0 – ||||||||||| Cut-Off Keep whole shipment

© 2006 Prentice Hall, Inc.S6 – 67 AQL and LTPD  Acceptable Quality Level (AQL)  Poorest level of quality we are willing to accept  Lot Tolerance Percent Defective (LTPD)  Quality level we consider bad  Consumer (buyer) does not want to accept lots with more defects than LTPD

© 2006 Prentice Hall, Inc.S6 – 68 Producer’s and Consumer’s Risks  Producer's risk (  )  Probability of rejecting a good lot  Probability of rejecting a lot when the fraction defective is at or above the AQL  Consumer's risk (  )  Probability of accepting a bad lot  Probability of accepting a lot when fraction defective is below the LTPD

© 2006 Prentice Hall, Inc.S6 – 69 An OC Curve Probability of Acceptance Percent defective ||||||||| ||||||||| – – – – – – 0 0 –  = 0.05 producer’s risk for AQL  = 0.10 Consumer’s risk for LTPD LTPDAQL Bad lots Indifference zone Good lots Figure S6.9

© 2006 Prentice Hall, Inc.S6 – 70 OC Curves for Different Sampling Plans n = 50, c = 1 n = 100, c = 2

© 2006 Prentice Hall, Inc.S6 – 71 Average Outgoing Quality where P d = true percent defective of the lot P a = probability of accepting the lot N= number of items in the lot n= number of items in the sample AOQ = (P d )(P a )(N - n) N

© 2006 Prentice Hall, Inc.S6 – 72 Average Outgoing Quality 1.If a sampling plan replaces all defectives 2.If we know the incoming percent defective for the lot We can compute the average outgoing quality (AOQ) in percent defective The maximum AOQ is the highest percent defective or the lowest average quality and is called the average outgoing quality level (AOQL)

© 2006 Prentice Hall, Inc.S6 – 73 SPC and Process Variability (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) Lower specification limit Upper specification limit Process mean,  Figure S6.10