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

Outline Statistical Process Control (SPC) Process Capability Control Charts for Variables Setting Mean Chart Limits (x-Charts) Setting Range Chart Limits (R-Charts) Process Capability Process Capability Ratio (Cp) Process Capability Index (Cpk )

Statistical Process Control (SPC) Variability is inherent in every process Natural or common causes Special or assignable causes SPC charts provide statistical signals when assignable causes are present SPC approach supports the detection and elimination of assignable causes of variation Points which might be emphasized include: - Statistical process control measures the performance of a process, it does not help to identify a particular specimen produced as being “good” or “bad,” in or out of tolerance. - Statistical process control requires the collection and analysis of data - therefore it is not helpful when total production consists of a small number of units - While statistical process control can not help identify a “good” or “bad” unit, it can enable one to decide whether or not to accept an entire production lot. If a sample of a production lot contains more than a specified number of defective items, statistical process control can give us a basis for rejecting the entire lot. The issue of rejecting a lot which was actually good can be raised here, but is probably better left to later.

Inspection Inspection is the activity that is done to ensure that an operation is producing the results expected (post production activity). Where to inspect At point of product design At point of product production (source) At point of product assembly At point of product dispatch to customer At point of product reception by customer

What to Inspect Variables of an entity Attributes of an entity Degree of deviation from a target (continuum scale) Lifespan of device Reliability or accuracy of device Attributes of an entity Classifies attributes into discrete classes such as good versus bad, or pass or fail Maximum weight of bag at airport Minimum height for exit seat

Data Used for Quality Judgments Variables Attributes Characteristics that can take any real value May be in whole or in fractional numbers Continuous random variables, e.g. weight, length, duration, etc. Defect-related characteristics Classify products as either good or bad or count defects Categorical or discrete random variables Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process.

How to Inspect: Methods Visual inspection Manual inspection (weigh, count ) Mechanical inspection (machine-based) Testing of device

Disadvantages of Inspection Most inspections are not done at the source Errors are discovered after its too late No link between error and the cause of errors Errors are often too costly to correct As products grow in number more staff are needed for inspection As products grow in number, more time is needed for inspection Most inspections involve the inspection of good parts as well as bad ones CHALLENGE: How could one achieve high product quality without having to inspect all goods produced?

Statistical Process Control A statistics-based approach for monitoring and inspecting results of a process, through the gathering, structuring, and analyses of product variables/attributes, as well as the taking of corrective action at the source, during the production process.

Class Example What can we learn from the results of the bodyguards?

Common Cause Variations Also called natural causes Affects virtually all production processes Generally this requires some change at the systemic level of an organization Managerial action is often necessary The objective is to discover avoidable common causes present in processes Eliminate (when possible) the root causes of the common variations, e.g. different arrival times of suppliers, different arrival times of customers, weight of products poured in box by machine

Assignable Variations Also called special causes of variation Generally this is caused by some change in the local activity or process Variations that can be traced to a specific reason at a localized activity The objective is to discover special causes that are present Eliminate the root causes of the special variations, e.g. machine wear, material quality, fatigued workers, misadjusted equipment Incorporate the good process control

SPC and Statistical Samples To conduct inspection using the SPC approach, one has to compute averages for several small samples instead of using data from individual items: Steps Each of these represents one sample of five boxes of cereal (a) Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight Frequency Weight # Figure S6.1

The solid line represents the distribution SPC Sampling To measure the process, we take samples of same size at different times. We plot the mean of each sample for each point in time The solid line represents the distribution (b) After enough samples are taken from a stable process, they form a pattern called a distribution Frequency Weight Figure S6.1

Attributes of Distributions (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 Variation Shape Frequency Figure S6.1

Identifying Presence of Common Sources of Variation (d) If only common causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable Prediction Weight Time Frequency Figure S6.1

Identifying Presence of Special Sources of Variation Prediction ? (e) If assignable causes are present, the process output is not stable over time and is not predicable Weight Time Frequency Figure S6.1

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 The mean of the sampling distribution (x) will be the same as the population mean m x = m This slide introduces the difference between “natural” and “assignable” causes. The next several slides expand the discussion and introduce some of the statistical issues. The standard deviation of the sampling distribution (sx) will equal the population standard deviation (s) divided by the square root of the sample size, n s n sx =

Interpreting SPC Charts (a) In statistical control and capable of producing within control limits Frequency Lower Control Limit Upper Control Limit (b) In statistical control but not capable of producing within control limits This slide helps introduce different process outputs. It can also be used to illustrate natural and assignable variation. (c) Out of statistical control and incapable of producing within limits (weight, length, speed, etc.) Size Figure S6.2

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

Sampling Distribution Sampling distribution of means Process distribution of means x = m (mean) It may be useful to spend some time explicitly discussing the difference between the sampling distribution of the means and the mean of the process population. Figure S6.4

Steps In Creating Control Charts Take representative sample from output of a process over a long period of time, e.g. 10 units every hour for 24 hours. Compute means and ranges for the variables and calculate the control limits Draw control limits on the control chart Plot a chart for the means and another for the mean of ranges on the control chart Determine state of process (in or out of control) Investigate possible reasons for out of control events and take corrective action Continue sampling of process output and reset the control limits when necessary

In-Class Exercise : Control Charts 6/15 6/16 8AM 10AM 12AM 2PM 5 6 8 6.5 5.5 7 7.5 Calculate X bar and R’s for new data Calculate X double bar and R bar figures for new data Draw X bar chart Calculate LCL and UCL for X bar chart Draw lines for LCL and UCL and for X double bar in chart

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

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

Setting Control Limits 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 s = 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 n = 9 For 99.73% control limits, z = 3 UCLx = x + zsx = 16 + 3(1/3) = 17 ozs LCLx = x - zsx = 16 - 3(1/3) = 15 ozs

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

Setting Chart Limits 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 Table S6.1 x = mean of the sample means

Control Chart Factors 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 Table S6.1

Setting Control Limits Process average x = 16.01 ounces Average range R = .25 Sample size n = 5

Setting Control Limits Process average x = 16.01 ounces Average range R = .25 Sample size n = 5 UCLx = x + A2R = 16.01 + (.577)(.25) = 16.01 + .144 = 16.154 ounces From Table S6.1

Setting Control Limits Process average x = 16.01 ounces Average range R = .25 Sample size n = 5 UCL = 16.154 Mean = 16.01 LCL = 15.866 UCLx = x + A2R = 16.01 + (.577)(.25) = 16.01 + .144 = 16.154 ounces LCLx = x - A2R = 16.01 - .144 = 15.866 ounces

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

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 S6.1

Setting Control Limits Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 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

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 spread) UCL LCL Figure S6.5

Mean and Range Charts (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 mean) UCL LCL R-chart (R-chart detects increase in dispersion) UCL LCL Figure S6.5

Automated Control Charts

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)

Patterns in Control Charts 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. Normal behavior. Process is “in control.” Figure S6.7

Patterns in Control Charts 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.” Figure S6.7

Patterns in Control Charts 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. Trends in either direction, 5 plots. Investigate for cause of progressive change. Figure S6.7

Patterns in Control Charts 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. Run of 5 above (or below) central line. Investigate for cause. Figure S6.7

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

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

Process Capability Ratio Insurance claims process Process mean x = 210.0 minutes Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Cp = Upper Specification - Lower Specification 6s

Process Capability Ratio Insurance claims process Process mean x = 210.0 minutes Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Cp = Upper Specification - Lower Specification 6s = = 1.938 213 - 207 6(.516)

Process Capability Ratio Insurance claims process Process mean x = 210.0 minutes Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Cp = Upper Specification - Lower Specification 6s = = 1.938 213 - 207 6(.516) Process is capable

Process Capability Index Cpk = minimum of , Upper Specification - x Limit 3s 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

Process Capability Index New Cutting Machine New process mean x = .250 inches Process standard deviation s = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches

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

Process Capability Index New Cutting Machine New process mean x = .250 inches Process standard deviation s = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches Cpk = minimum of , (.251) - .250 (3).0005 .250 - (.249) Both calculations result in New machine is NOT capable Cpk = = 0.67 .001 .0015

Process Capability Comparison New Cutting Machine New process mean x = .250 inches Process standard deviation s = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches Upper Specification - Lower Specification 6s Cp = New machine is NOT capable .251 - .249 .0030 Cp = = 0.66

Interpreting Cpk Cpk = negative number Cpk = zero Cpk = between 0 and 1 Cpk = 1 Cpk > 1 Figure S6.8

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 Here again it is useful to stress that acceptance sampling relates to the aggregate, not the individual unit. You might also discuss the decision as to whether one should take only a single sample, or whether multiple samples are required.