Statistical Process Control

Statistical Process Control
Copyright 2006 John Wiley & Sons, Inc.

Lecture Outline Basics of Statistical Process Control Control Charts
Control Charts for Attributes Control Charts for Variables Control Chart Patterns SPC with Excel Process Capability Copyright 2006 John Wiley & Sons, Inc.

Basics of Statistical Process Control
Statistical Process Control (SPC) monitoring production process to detect and prevent poor quality Sample subset of items produced to use for inspection Control Charts process is within statistical control limits UCL LCL Statistical process control (SPC) is a methodology for monitoring a process to identify special causes of variation and signal the need to take corrective action when appropriate. When special causes are present, the process is deemed to be out of control. If the variation in the process is due to common causes alone, the process is said to be in statistical control. Histograms alone do not allow you to distinguish between common and special causes of variation. Control Chart Purpose: to monitor process output to see if it is random A time ordered plot representative sample statistics obtained from an on going process (e.g. sample means) Upper and lower control limits define the range of acceptable variation Copyright 2006 John Wiley & Sons, Inc.

Variability Random Non-Random common causes inherent in a process
can be eliminated only through improvements in the system Non-Random special causes due to identifiable factors can be modified through operator or management action Common causes of variation Random causes that we cannot identify Unavoidable e.g. slight differences in process variables like diameter, weight, service time, temperature Assignable causes of variation Causes can be identified and eliminated e.g. poor employee training, worn tool, machine needing repair Copyright 2006 John Wiley & Sons, Inc.

Control Chart 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 UCL LCL Sample number Mean Out of control Normal variation due to chance Abnormal variation due to assignable sources Copyright 2006 John Wiley & Sons, Inc.

Sources of Variation in Production Processes
Measurement Instruments Operators Methods Materials INPUTS PROCESS OUTPUTS Tools Human Inspection Performance Machines Environment Copyright 2006 John Wiley & Sons, Inc.

SPC in TQM SPC tool for identifying problems and make improvements
contributes to the TQM goal of continuous improvements The Control Process Define Measure Compare Evaluate Correct Monitor results A control chart is simply a run chart to which two horizontal lines, called control limits are added: the upper control limit (UCL) and lower control limit (LCL). The process for constructing and using a control chart includes preparation, data collection, determination of trial control limits, analysis and interpretation, estimation of process capability using the control chart data, and use a problem-solving tool. Copyright 2006 John Wiley & Sons, Inc.

Quality Measures Attribute Variable
a product characteristic that can be evaluated with a discrete response good – bad; yes - no Variable a product characteristic that is continuous and can be measured weight - length Copyright 2006 John Wiley & Sons, Inc.

Applying SPC to Service
Nature of defect is different in services Service defect is a failure to meet customer requirements Monitor times, customer satisfaction Service Organizations have lagged behind manufacturers in the use of statistical quality control Statistical measurements are required and it is more difficult to measure the quality of a service Services produce more intangible products Perceptions of quality are highly subjective A way to deal with service quality is to devise quantifiable measurements of the service element Check-in time at a hotel Number of complaints received per month at a restaurant Number of telephone rings before a call is answered Acceptable control limits can be developed and charted Copyright 2006 John Wiley & Sons, Inc.

Applying SPC to Service (cont.)
Hospitals timeliness and quickness of care, staff responses to requests, accuracy of lab tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance and checkouts Grocery stores waiting time to check out, frequency of out-of-stock items, quality of food items, cleanliness, customer complaints, checkout register errors Airlines flight delays, lost luggage and luggage handling, waiting time at ticket counters and check-in, agent and flight attendant courtesy, accurate flight information, passenger cabin cleanliness and maintenance Copyright 2006 John Wiley & Sons, Inc.

Applying SPC to Service (cont.)
Fast-food restaurants waiting time for service, customer complaints, cleanliness, food quality, order accuracy, employee courtesy Catalogue-order companies order accuracy, operator knowledge and courtesy, packaging, delivery time, phone order waiting time Insurance companies billing accuracy, timeliness of claims processing, agent availability and response time Copyright 2006 John Wiley & Sons, Inc.

Where to Use Control Charts
Process has a tendency to go out of control Process is particularly harmful and costly if it goes out of control Examples at the beginning of a process because it is a waste of time and money to begin production process with bad supplies before a costly or irreversible point, after which product is difficult to rework or correct before and after assembly or painting operations that might cover defects before the outgoing final product or service is delivered Copyright 2006 John Wiley & Sons, Inc.

SPC Errors Type I error Type II error
Concluding a process is not in control when it actually is. Type II error Concluding a process is in control when it is not. Copyright 2006 John Wiley & Sons, Inc.

Type I and Type II Errors
In control Out of control No Error Type I error (producers risk) Type II Error (consumers risk) No error Copyright 2006 John Wiley & Sons, Inc.

Control Charts Types of charts Attributes Variables
p-chart c-chart Variables range (R-chart) mean (x bar – chart) A graph that establishes control limits of a process Control limits upper and lower bands of a control chart Copyright 2006 John Wiley & Sons, Inc.

SPC Methods-Control Charts
Control Charts show sample data plotted on a graph with CL, UCL, and LCL Control chart for variables are used to monitor characteristics that can be measured, e.g. length, weight, diameter, time Control charts for attributes are used to monitor characteristics that have discrete values and can be counted, e.g. % defective, number of flaws in a shirt, number of broken eggs in a box Copyright 2006 John Wiley & Sons, Inc.

Process Control Chart Out of control Upper control limit Process
1 2 3 4 5 6 7 8 9 10 Sample number Upper control limit Process average Lower Out of control A control chart is simply a run chart to which two horizontal lines, called control limits are added: the upper control limit (UCL) and lower control limit (LCL). The process for constructing and using a control chart includes preparation, data collection, determination of trial control limits, analysis and interpretation, estimation of process capability using the control chart data, and use a problem-solving tool. A process is in control if no points are outside control limits; the number of points above and below the center line is about the same; the points seem to fall randomly above and below the center line; and most points (but not all) are near the center line, with only a few close to the control limits. Typical out-of-control conditions are represented by sudden shifts in the mean value, cycles, trends, hugging of the center line, hugging of the control limits, and instability. Copyright 2006 John Wiley & Sons, Inc.

Normal Distribution =0 1 2 3 -1 -2 -3 95% 99.74%

Traditional Statistical Tools
The Mean- measure of central tendency The Range- difference between largest/smallest observations in a set of data Standard Deviation measures the amount of data dispersion around mean Copyright 2006 John Wiley & Sons, Inc.

A Process Is in Control If …
… no sample points outside limits … most points near process average … about equal number of points above and below centerline … points appear randomly distributed Typical out-of-control conditions are represented by sudden shifts in the mean value, cycles, trends, hugging of the center line, hugging of the control limits, and instability. Copyright 2006 John Wiley & Sons, Inc.

Control Charts for Attributes
p-charts uses portion defective in a sample c-charts uses number of defects in an item Charts for attributes include p-, np-, c- and u-charts. The np-chart is an alternative to the p-chart, and controls the number nonconforming for attributes data. Charts for defects include the c-chart and u-chart. The c-chart is used for constant sample size and the u-chart is used for variable sample size. Copyright 2006 John Wiley & Sons, Inc.

Control Charts for Attributes –P-Charts & C-Charts
Use P-Charts for quality characteristics that are discrete and involve yes/no or good/bad decisions Number of leaking caulking tubes in a box of 48 Number of broken eggs in a carton When observations can be placed into two categories. Good or bad Pass or fail Operate or don’t operate When the data consists of multiple samples of several observations each Use C-Charts for discrete defects when there can be more than one defect per unit Number of flaws or stains in a carpet sample cut from a production run Number of complaints per customer at a hotel Copyright 2006 John Wiley & Sons, Inc.

p-Chart UCL = p + zp LCL = p - zp
z = number of standard deviations from process average p = sample proportion defective; an estimate of process average p = standard deviation of sample proportion p = p(1 - p) n Copyright 2006 John Wiley & Sons, Inc.

p-Chart Example 20 samples of 100 pairs of jeans 1 6 .06 2 0 .00
NUMBER OF PROPORTION SAMPLE DEFECTIVES DEFECTIVE : : : 200 Copyright 2006 John Wiley & Sons, Inc.

p-Chart Example (cont.)
UCL = p + z = p(1 - p) n 0.10( ) 100 UCL = 0.190 LCL = 0.010 LCL = p - z = = 200 / 20(100) = 0.10 total defectives total sample observations p = Copyright 2006 John Wiley & Sons, Inc.

p-Chart Example (cont.)
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Proportion defective Sample number 2 4 6 8 10 12 14 16 18 20 UCL = 0.190 LCL = 0.010 p = 0.10 p-Chart Example (cont.) Copyright 2006 John Wiley & Sons, Inc.

P-Chart Example: A Production manager for a tire company has inspected the number of defective tires in five random samples with 20 tires in each sample. The table below shows the number of defective tires in each sample of 20 tires. Calculate the control limits. Sample Number of Defective Tires Number of Tires in each Sample Proportion Defective 1 3 20 .15 2 .10 .05 4 5 Total 9 100 .09 Solution: Copyright 2006 John Wiley & Sons, Inc.

c-Chart UCL = c + zc c = c LCL = c - zc where
c = number of defects per sample c = c Use only when the number of occurrences per unit of measure can be counted; non-occurrences cannot be counted. Scratches, chips, dents, or errors per item Cracks or faults per unit of distance Breaks or Tears per unit of area Bacteria or pollutants per unit of volume Calls, complaints, failures per unit of time Copyright 2006 John Wiley & Sons, Inc.

c-Chart (cont.) Number of defects in 15 sample rooms 15 c = = 12.67
: : 190 SAMPLE c = = 12.67 15 UCL = c + zc = = 23.35 LCL = c + zc = = 1.99 NUMBER OF DEFECTS Copyright 2006 John Wiley & Sons, Inc.

c-Chart (cont.) Copyright 2006 John Wiley & Sons, Inc. 3 6 9 12 15 18
21 24 Number of defects Sample number 2 4 8 10 14 16 UCL = 23.35 LCL = 1.99 c = 12.67 c-Chart (cont.) Copyright 2006 John Wiley & Sons, Inc.

C-Chart Example: The number of weekly customer complaints are monitored in a large hotel using a c-chart. Develop three sigma control limits using the data table below. Week Number of Complaints 1 3 2 4 5 6 7 8 9 10 Total 22 Solution: Copyright 2006 John Wiley & Sons, Inc.

x-bar Chart Mean chart ( x -Chart ) Range chart ( R-Chart )
uses average of a sample Range chart ( R-Chart ) uses amount of dispersion in a sample Control charts for variables data include: x- and R-charts;x - and s-charts; and individual and moving range charts. x - and s-charts are alternatives tox – and R-charts for larger sample sizes. The sample standard deviation provides a better indication of process variability than the range. Individuals charts are useful when every item can be inspected and when a long lead time exists for producing an item. Moving ranges are used to measure the variability in individuals charts. Copyright 2006 John Wiley & Sons, Inc.

Center line and control limit formulas
Constructing a X-bar Chart: A quality control inspector at the Cocoa Fizz soft drink company has taken three samples with four observations each of the volume of bottles filled. If the standard deviation of the bottling operation is .2 ounces, use the below data to develop control charts with limits of 3 standard deviations for the 16 oz. bottling operation. Center line and control limit formulas Time 1 Time 2 Time 3 Observation 1 15.8 16.1 16.0 Observation 2 15.9 Observation 3 Observation 4 Sample means (X-bar) 15.875 15.975 Sample ranges (R) 0.2 0.3 Copyright 2006 John Wiley & Sons, Inc.

Solution and Control Chart (x-bar)
Center line (x-double bar): Control limits for±3σ limits: t Copyright 2006 John Wiley & Sons, Inc.

x-bar Chart Example OBSERVATIONS (SLIP- RING DIAMETER, CM)
SAMPLE k x R Example 15.4 Copyright 2006 John Wiley & Sons, Inc.

x- bar Chart Example (cont.)
UCL = x + A2R = (0.58)(0.115) = 5.08 LCL = x - A2R = (0.58)(0.115) = 4.94 = x = = = 5.01 cm x k 50.09 10 Retrieve Factor Value A2 Copyright 2006 John Wiley & Sons, Inc.

x- bar Chart Example (cont.)
UCL = 5.08 LCL = 4.94 Mean Sample number | 1 2 3 4 5 6 7 8 9 10 5.10 – 5.08 – 5.06 – 5.04 – 5.02 – 5.00 – 4.98 – 4.96 – 4.94 – 4.92 – x = 5.01 = x- bar Chart Example (cont.) Copyright 2006 John Wiley & Sons, Inc.

Control Chart for Range (R)
Center Line and Control Limit formulas: Factors for three sigma control limits Factor for x-Chart A2 D3 D4 2 1.88 0.00 3.27 3 1.02 2.57 4 0.73 2.28 5 0.58 2.11 6 0.48 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86 9 0.34 0.18 1.82 10 0.31 0.22 1.78 11 0.29 0.26 1.74 12 0.27 0.28 1.72 13 0.25 1.69 14 0.24 0.33 1.67 15 0.35 1.65 Factors for R-Chart Sample Size (n) Copyright 2006 John Wiley & Sons, Inc.

Second Method for the X-bar Chart Using R-bar and the A2 Factor (table 6-1)
Use this method when sigma for the process distribution is not know Control limits solution: Copyright 2006 John Wiley & Sons, Inc.

R- Chart UCL = D4R LCL = D3R R k R = where R = range of each sample
k = number of samples Copyright 2006 John Wiley & Sons, Inc.

R-Chart Example OBSERVATIONS (SLIP-RING DIAMETER, CM)
SAMPLE k x R Example 15.3 Copyright 2006 John Wiley & Sons, Inc.

R-Chart Example (cont.)
k R = = = 0.115 1.15 10 UCL = D4R = 2.11(0.115) = 0.243 LCL = D3R = 0(0.115) = 0 Retrieve Factor Values D3 and D4 Example 15.3 Copyright 2006 John Wiley & Sons, Inc.

R-Chart Example (cont.)
UCL = 0.243 LCL = 0 Range Sample number R = 0.115 | 1 2 3 4 5 6 7 8 9 10 0.28 – 0.24 – 0.20 – 0.16 – 0.12 – 0.08 – 0.04 – 0 – Copyright 2006 John Wiley & Sons, Inc.

Using x- bar and R-Charts Together
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 Copyright 2006 John Wiley & Sons, Inc.

Control Chart limits for ±3 sigma limits
Service at a bank: The Dollars Bank competes on customer service and is concerned about service time at their drive-by windows. They recently installed new system software which they hope will meet service specification limits of 5±2 minutes and have a Capability Index (Cpk) of at least 1.2. They want to also design a control chart for bank teller use. They have done some sampling recently (sample size of 4 customers) and determined that the process mean has shifted to 5.2 with a Sigma of 1.0 minutes. Control Chart limits for ±3 sigma limits Copyright 2006 John Wiley & Sons, Inc.

Control Chart Patterns
UCL LCL Sample observations consistently above the center line consistently below the Copyright 2006 John Wiley & Sons, Inc.

Control Chart Patterns (cont.)
LCL UCL Sample observations consistently increasing UCL LCL Sample observations consistently decreasing Copyright 2006 John Wiley & Sons, Inc.

Zones for Pattern Tests
UCL LCL Zone A Zone B Zone C Process average 3 sigma = x + A2R = 3 sigma = x - A2R 2 sigma = x (A2R) 2 3 2 sigma = x (A2R) 1 sigma = x (A2R) 1 1 sigma = x (A2R) x Sample number | 4 5 6 7 8 9 10 11 12 13 Copyright 2006 John Wiley & Sons, Inc.

Control Chart Patterns
8 consecutive points on one side of the center line 8 consecutive points up or down across zones 14 points alternating up or down 2 out of 3 consecutive points in zone A but still inside the control limits 4 out of 5 consecutive points in zone A or B Copyright 2006 John Wiley & Sons, Inc.

Performing a Pattern Test
B — B B U C B D A B D A B U C — U C A U C A U B A U A A D B SAMPLE x ABOVE/BELOW UP/DOWN ZONE Copyright 2006 John Wiley & Sons, Inc.

Sample Size Attribute charts require larger sample sizes
50 to 100 parts in a sample Variable charts require smaller samples 2 to 10 parts in a sample Copyright 2006 John Wiley & Sons, Inc.

SPC with Excel: Formulas

Process Capability Tolerances Process capability
design specifications reflecting product requirements Process capability range of natural variability in a process what we measure with control charts Product Specifications Preset product or service dimensions, tolerances e.g. bottle fill might be 16 oz. ±.2 oz. (15.8oz.-16.2oz.) Based on how product is to be used or what the customer expects Process Capability – Cp and Cpk Assessing capability involves evaluating process variability relative to preset product or service specifications Cp assumes that the process is centered in the specification range Cpk helps to address a possible lack of centering of the process Copyright 2006 John Wiley & Sons, Inc.

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 Copyright 2006 John Wiley & Sons, Inc.

Attribute Data 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 Copyright 2006 John Wiley & Sons, Inc.

Attribute Data Using a c-Chart: Observations are attributes whose defects per unit of output can be counted The number counted is 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 Copyright 2006 John Wiley & Sons, Inc.

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 Copyright 2006 John Wiley & Sons, Inc.

Design Specifications
Process Capability (b) Design specifications and natural variation the same; process is capable of meeting specifications most of the time. Design Specifications Process (a) Natural variation exceeds design specifications; process is not capable of meeting specifications all the time. Copyright 2006 John Wiley & Sons, Inc.

Process Capability (cont.)
(c) Design specifications greater than natural variation; process is capable of always conforming to specifications. Design Specifications Process (d) Specifications greater than natural variation, but process off center; capable but some output will not meet upper specification. Copyright 2006 John Wiley & Sons, Inc.

Process Capability Measures
Process Capability Ratio Cp = = tolerance range process range upper specification limit - lower specification limit 6 Copyright 2006 John Wiley & Sons, Inc.

Computing Cp Net weight specification = 9.0 oz  0.5 oz
Process mean = 8.80 oz Process standard deviation = 0.12 oz Cp = = = 1.39 upper specification limit - lower specification limit 6 6(0.12) Three possible ranges for Cp Cp = 1, as in Fig. (a), process variability just meets specifications Cp ≤ 1, as in Fig. (b), process not capable of producing within specifications Cp ≥ 1, as in Fig. (c), process exceeds minimal specifications One shortcoming, Cp assumes that the process is centered on the specification range Cp=Cpk when process is centered Copyright 2006 John Wiley & Sons, Inc.

Process Capability Ratio
Cp = Upper Specification - Lower Specification 6s 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 Copyright 2006 John Wiley & Sons, Inc.

Relationship between Process Variability and Specification Width

Process Capability Measures
Process Capability Index Cpk = minimum x - lower specification limit 3 = upper specification limit - x , Copyright 2006 John Wiley & Sons, Inc.

Computing Cpk Net weight specification = 9.0 oz  0.5 oz
Process mean = 8.80 oz Process standard deviation = 0.12 oz Cpk = minimum = minimum , = 0.83 x - lower specification limit 3 = upper specification limit - x , 3(0.12) Copyright 2006 John Wiley & Sons, Inc.

The Cereal Box Example We are the maker of this cereal. Consumer reports has just published an article that shows that we frequently have less than 15 ounces of cereal in a box. Let’s assume that the government says that we must be within ± 5 percent of the weight advertised on the box. Upper Tolerance Limit = (16) = 16.8 ounces Lower Tolerance Limit = 16 – .05(16) = 15.2 ounces We go out and buy 1,000 boxes of cereal and find that they weight an average of ounces with a standard deviation of .529 ounces. Copyright 2006 John Wiley & Sons, Inc.

Cereal Box Process Capability
Specification or Tolerance Limits Upper Spec = 16.8 oz Lower Spec = 15.2 oz Observed Weight Mean = oz Std Dev = .529 oz Copyright 2006 John Wiley & Sons, Inc.

What does a Cpk of mean? An index that shows how well the units being produced fit within the specification limits. This is a process that will produce a relatively high number of defects. Many companies look for a Cpk of 1.3 or better… 6-Sigma company wants 2.0! Copyright 2006 John Wiley & Sons, Inc.

Interpreting Cpk Cpk = negative number Cpk = zero
Cpk = between 0 and 1 Cpk = 1 Cpk > 1 Figure S6.8 Copyright 2006 John Wiley & Sons, Inc.

Process Capability A. Process variability matches specifications
Lower Specification Upper Specification A. Process variability matches specifications B. Process variability well within specifications C. Process variability exceeds specifications Copyright 2006 John Wiley & Sons, Inc.

±6 Sigma versus ± 3 Sigma Motorola coined “six-sigma” to describe their higher quality efforts back in 1980’s Six-sigma quality standard is now a benchmark in many industries Before design, marketing ensures customer product characteristics Operations ensures that product design characteristics can be met by controlling materials and processes to 6σ levels Other functions like finance and accounting use 6σ concepts to control all of their processes PPM Defective for ±3σ versus ±6σ quality Copyright 2006 John Wiley & Sons, Inc.

3 Sigma and 6 Sigma Quality
Process mean Lower specification Upper specification 1350 ppm 1.7 ppm +/- 3 Sigma +/- 6 Sigma Copyright 2006 John Wiley & Sons, Inc.

Six Sigma Quality A philosophy and set ofmethods companies use to eliminate defects in their products and processes Seeks to reduce variation in the processes that lead to product defects The name, “six sigma” refers to the variation that exists within plus or minus three standard deviations of the process outputs Copyright 2006 John Wiley & Sons, Inc.

How good is good enough? Motorola’s “Six Sigma”
6s minimum from process center to nearest spec Copyright 2006 John Wiley & Sons, Inc.

Motorola’s “Six Sigma”
Implies 2 ppB “bad” with no process shift With 1.5s shift in either direction from center (process will move), implies 3.4 ppm “bad”. Copyright 2006 John Wiley & Sons, Inc.

Chapter 5 Highlights SQC can be divided into three categories: traditional statistical tools (SQC), acceptance sampling, and statistical process control (SPC). SQC tools describe quality characteristics, acceptance sampling is used to decide whether to accept or reject an entire lot, SPC is used to monitor any process output to see if its characteristics are in Specs. Variation is caused from common (random), unidentifiable causes and also assignable causes that can be identified and corrected. Control charts are SPC tools used to plot process output characteristics for both variable and attribute data to show whether a sample falls within the normal range of variation: X-bar, R, P, and C-charts. Process capability is the ability of the process to meet or exceed preset specifications; measured by Cp and Cpk. Copyright 2006 John Wiley & Sons, Inc.

Determining Control Limits for x-bar and R-Charts
Appendix: Determining Control Limits for x-bar and R-Charts n A2 D3 D4 SAMPLE SIZE FACTOR FOR x-CHART FACTORS FOR R-CHART Factors Return Copyright 2006 John Wiley & Sons, Inc.

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