Operations Management: Sustainability and Supply Chain Management PDF

Operations Management: Sustainability and Supply Chain Management Supplement 6 Statistical Process Control Learning Objectives (1 of 2) S6.1 Explain the purpose of a control chart S6.2 Explain the role of the central limit theorem in SPC S6.3 Build...

Operations Management: Sustainability and Supply Chain Management Supplement 6 Statistical Process Control Learning Objectives (1 of 2) S6.1 Explain the purpose of a control chart S6.2 Explain the role of the central limit theorem in SPC S6.3 Build -charts and R- charts S6.4 List the five steps involved in building control charts S6.5 Build p-charts and c-charts S6.6 Explain process capability and compute Cp and Cpk Statistical Process Control Application of statistical techniques to ensure that processes meet standards A process used to monitor standards by taking measurements and corrective action as a product or service is being produced The objective of a process control system is to provide a _________ __________when assignable causes of variation are present Statistical Process Control (SPC) Variability is inherent in every process – _________ or common causes – Special or ____________ causes Provides a statistical signal when assignable causes are present Quickens appropriate actions to eliminate assignable causes 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 “_____ ____________” Assignable Variations Also called special causes of variation – Generally this is some _________ 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 Samples (1 of 5) To measure the process, we take samples and analyze the sample statistics following these steps Samples (2 of 5) Figure S6.1 Natural and Assignable Variation (b) After enough samples are taken from a stable process, they form a pattern called a distribution Samples (3 of 5)  Figure S6.1 Natural and Assignable Variation (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 Samples (4 of 5) Figure S6.1 Natural and Assignable Variation (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 Samples (5 of 5) Figure S6.1 Natural and Assignable Variation (e) If assignable causes are present, the process output is not stable over time and is not predicable Sample Size Example Box 1 Box 2 Box 3 Box 4 Box 5 Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Control Charts Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes  Process Control Frequency Lower control limit Upper control limit Size (weight, length, speed, etc.) Figure S6.2 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 will be the same as the x= population mean μ 2) The standard deviation of the sampling distribution ( x ) will equal the population standard σ σx  deviation   divided by the n square root of the sample size, n Sampling Distribution Figure S6.4  Types of Quality Data ______________________ Product characteristic evaluated with a discrete choice Good/bad, yes/no ______________________ Product characteristic that can be measured Length, size, weight, height, time, velocity Control Charts for Variables Characteristics that can take any real value May be in whole or in fractional numbers Continuous random variables x -chart tracks changes in the ________ tendency R-chart indicates a gain or loss of dispersion These two charts must be used together Setting Chart Limits For x -Charts when we know  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  n σ  population (process) standard deviation n  sample size  Setting Control Limits You are working as quality control at Kellogg’s. The weight of boxes of Corn Flakes within a large production lot are sampled each hour. Nine (9) samples are taken and weighed each hour. Managers want to set control limits that include 99.73% of the sample mean. The population (or process) standard deviation (σ) is known to be 1 oz. How do you set your Upper and Lower control limits? Setting Control Limits (1 of 6) Randomly select and weigh nine (n 9) boxes each hour Average weight in 17 +13 +16 +18 +17 +16 +15 +17 +16 = = 16.1 ounces the first sample 9 Weight of Weight Of Weight of Weight Of Weight of Weight Of Sample Sample Sample Sample Sample Sample (Average of (Average of 9 (Average of 9 Hour 9 Boxes) Hour Boxes) Hour Boxes) 1 16.1 5 16.5 9 16.3 2 16.8 6 16.4 10 14.8 3 15.5 7 15.2 11 14.2 4 16.5 8 16.4 12 17.3 Setting Control Limits (2 of 6) x 16 ounces 12   n 9 Average mean   (Avg of 9 boxes)  z 3 of 12 samples  x  i 1   12  σ 1 ounce   Number of samples 12 Setting Control Limits (3 of 6) x 16 ounces n 9 z 3 σ 1 ounce Number of samples 12 Setting Control Limits (4 of 6) Setting Control Limits Super Cola bottles soft drinks labeled “net weight - 12 oz.” Indeed, an overall process average of 12 oz. has been found by taking 10 samples, in which each sample contained 5 bottles. You do not know the standard deviation (σ). How do you set your Upper and Lower control limits? Setting Chart Limits For x - Charts when we don't know  UCL x x  A2 R LCL x x  A2 R k R i where R i 1  average range of the samples k A2  control chart factor found in Table S6.1 x = mean of the sample means Ri  range for sample i k  total number of samples Control Chart Factors Table S6.1 Factors for Computing Control Chart Limits (3 sigma) Sample Size, Mean Factor, Upper Range, Lower Range, n A2 D4 D3 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 Setting Control Limits Range (Ri)= Sample Weight of lightest bottle Weight of heaviest bottle difference between the two A 11.50 11.72 B 11.97 12.00 C 11.55 12.05 D 12.00 12.20 E 11.95 12.00 F 10.55 10.75 G 12.50 12.75 H 11.00 11.25 I 10.60 11.00 J 11.70 12.10 Setting Control Limits Super Cola example Process average = 12 ounces labeled as “net weight 12 Average range =.25 ounces ounces” Sample size = 5 UCL = Mean = LCL = R – Chart Type of ___________ control chart Shows sample ranges over time – Difference between smallest and largest values in sample Monitors process variability Independent from process mean Setting Chart Limits For R-Charts Upper control limit (UCL R ) D4R Lower control limit (LCL R ) D3R where UCLR  upper control limit for the range LCLR  lower control limit for the range D4 and D3  values from Table S6.1  Setting Control Limits (R) Roy’s mail-ordering business want to measure the response time of its operators in taking customers orders over the phone. They recorded the time (in minutes) from five different samples of the ordering process with four customer order per sample How do you set your Upper and Lower range control chart limits? Setting Control Limits (R) Sample Observation (minutes) Sample range (Ri) 1 5, 3 , 6, 10 2 7, 5, 3, 5 3 1, 8, 3, 12 4 7, 6, 2, 1 5 3, 15, 6, 12 Control Chart Factors Table S6.1 Factors for Computing Control Chart Limits (3 sigma) Sample Size, Mean Factor, Upper Range, Lower Range, n A2 D4 D3 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 Setting Control Limits (6 of 6) Average range = 8 minutes Sample size = 4 D4 = UCL = D3 = Mean = LCL = Figure S6.5 Mean and Range Charts (1 of 2) (a) These sampling distributions result in the charts below Figure S6.5 Mean and Range Charts (2 of 2) (b) These sampling distributions result in the charts below  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 © 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Other Control Limits Table S6.2 Common z Values Desired Control Z-Value (Standard Deviation Limit (%) Required for Desired Level of Confidence) 90.0 1.65 95.0 1.96 95.45 2.00 99.0 2.58 99.73 3.00 Control Charts for Attributes For variables that are categorical – Defective/ nondefective, – good/ bad, – yes/no, – acceptable/unacceptable Measurement is typically counting defectives Charts may measure 1. ​Percent defective (p-chart) 2. ​Number of defects (c-chart) 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 σ p is estimated by UCL p p  zσ p p(1 p ) LCL p p  zσ p σˆ p  n where p  mean fraction (percent) defective in the samples z  number of standard deviations σ p  standard deviation of the sampling distribution n  number of observation in each sample p-Chart for Data Entry (1 of 3) Sample Number Of Fraction Sample Number Of Fraction Number Errors Defective Number Errors Defective n=100 n=100 1 6 11 6.06 2 5 12 1.01 3 0 13 8.08 4 1 14 7.07 5 4 15 5.05 6 2.02 16 4.04 7 5.05 17 11.11 8 3.03 18 3.03 9 3.03 19 0.00 10 2.02 20 4.04 80 p-Chart for Data Entry (2 of 3) UC L p =¿ LC L p =¿  P - Chart Example A Production manager for a Sample Number of Number of Proportion Defective Tires in each Defective tire company has inspected Tires Sample the number of defective tires in five random samples with 1 3 20 20 tires in each sample. The 2 2 20 table shows the number of 3 1 20 defective tires in each sample 4 2 20 of 20 tires. Calculate the 5 1 20 proportion defective for each Total 9 20 sample, the center line, and control limits to include 99.73% of the variations. Copyright © 2017, 2014, 2011 Pearson Education, Inc. All Rights Reserved chart Example n= z= = 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 c  mean number of defects per unit c  standard deviation of defects per unit Control limits (99.73%) c 3 c c-Chart for Cab Company c̄ = ¿ UCL𝑐=¯𝑐 +3 √ 𝑐¯ LCL𝑐=¯𝑐 −3 √ ¯𝑐  C -Chart Example Number of Week Complaints The number of weekly 1 3 customer complaints are 2 2 monitored in a large hotel 3 3 using a c-chart. Develop 4 1 three sigma control limits 5 3 using this data table. 6 3 What do you need to know? 7 2 n? z? c? 8 1 9 3 10 1 _ Total 22 Copyright © 2017, 2014, 2011 Pearson Education, Inc. All Rights Reserved  chart Example n= z = 3.00 ¿ 𝐶𝑜𝑚𝑝𝑙𝑎𝑖𝑛𝑡𝑠 =¿ ¿ 𝑆𝑎𝑚𝑝𝑙𝑒𝑠 Managerial Issues and Control Charts Three major management decisions: Select points in the processes that need SPC Determine the appropriate charting technique Set clear and specific SPC policies and procedures Figure S6.7 Patterns in Control Charts (1 of 6) Figure S6.7 Patterns in Control Charts (2 of 6) Figure S6.7 Patterns in Control Charts (3 of 6) Figure S6.7 Patterns in Control Charts (4 of 6) Figure S6.7 Patterns in Control Charts (5 of 6) Figure S6.7 Patterns in Control Charts (6 of 6) 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 _____________________ is a measure of the relationship between the natural variation of the process and the design specifications Cp and Cpk considers specification limits ________________________ Process Capability Ratio 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 Process Capability Index 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 Interpreting Cpk Figure S6.8 Meanings of Cpk Measures Learning Objectives When you complete this supplement you should be able to : 1. Explain the purpose of a control chart 2. Explain the role of the central limit theorem in SPC 3. Build -charts and R-charts 4. List the five steps involved in building control charts 5. Build p-charts and c-charts 6. Explain process capability and compute Cp and Cpk