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## Operations Management

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**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.**Statistical Process Control (SPC)**• 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 ) Outline**Statistical Process Control (SPC)**• Variability is inherent in every process • Natural or common causes • Special causes • SPC charts provide statistical signals when assignable causes are present • SPC approach support the detection and elimination of assignable causes of variation**Common Cause Variations**• Also called common causes • Affect virtually all production processes • Generally this required some change at the systemic level of an organization • Managerial action is often necessary • The objective is to discover when avoidable common causes are present • Eliminate the root causes of the common variations**Assignable Variations**• Also called special causes of variation • Generally this is 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 when special causes are present • Eliminate the root causes of the special variations • Incorporate the good causes**Each of these represents one sample of five boxes of cereal**# # # # # Frequency # # # # # # # # # # # # # # # # # # # # # Weight 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 Figure S6.1**The solid line represents the distribution**Frequency Weight 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 Figure S6.1**Central tendency**Variation Shape Frequency Weight Weight Weight 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 Figure S6.1**Prediction**Frequency Time Weight 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 Figure S6.1**?**? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Prediction Frequency Time Weight Identifying Presence of Special Sources of Variation (e) If assignable causes are present, the process output is not stable over time and is not predicable Figure S6.1**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**The mean of the sampling distribution (x) will be the same**as the population mean m x = m s n 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 sx = 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**(a) In statistical control and capable of producing within**control limits Frequency Upper Control Limit Lower Control Limit (b) In statistical control but not capable of producing within control limits (c) Out of statistical control and incapable of producing within limits Size (weight, length, speed, etc.) Interpreting SPC Charts Figure S6.2**Three population distributions**Mean of sample means = x Beta Standard deviation of the sample means Normal = sx = Uniform s n | | | | | | | -3sx -2sx -1sx x +1sx +2sx +3sx 95.45% fall within ± 2sx 99.73% of all x fall within ± 3sx Population and Sampling Distributions Distribution of sample means Figure S6.3**Sampling distribution of means**Process distribution of means x = m (mean) Sampling Distribution 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**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 Control Charts for Variables**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 Chart 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 UCLx = x + zsx= 16 + 3(1/3) = 17 ozs LCLx = x - zsx = 16 - 3(1/3) = 15 ozs Setting Control Limits For 99.73% control limits, z = 3**Variation due to assignable causes**Out of control 17 = UCL Variation due to natural causes 16 = Mean 15 = LCL Variation due to assignable causes | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 Out of control Sample number Setting Control Limits Control Chart for sample of 9 boxes**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 Setting Chart Limits**Sample Size Mean Factor Upper Range Lower Range**n A2D4D3 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 Control Chart Factors Table S6.1**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 UCLx = x + A2R = 16.01 + (.577)(.25) = 16.01 + .144 = 16.154 ounces UCL = 16.154 Mean = 16.01 LCLx = x - A2R = 16.01 - .144 = 15.866 ounces LCL = 15.866 Setting Control Limits**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**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 Chart Limits For R-Charts**Average range R = 5.3 pounds**Sample size n = 5 From Table S6.1 D4= 2.115, D3 = 0 UCLR = D4R = (2.115)(5.3) = 11.2 pounds UCL = 11.2 Mean = 5.3 LCLR = D3R = (0)(5.3) = 0 pounds LCL = 0 Setting Control Limits**(a)**These sampling distributions result in the charts below (Sampling mean is shifting upward but range is consistent) UCL (x-chart detects shift in central tendency) x-chart LCL UCL (R-chart does not detect change in mean) R-chart LCL Mean and Range Charts Figure S6.5**(b)**These sampling distributions result in the charts below (Sampling mean is constant but dispersion is increasing) UCL (x-chart does not detect the increase in dispersion) x-chart LCL UCL (R-chart detects increase in dispersion) R-chart LCL Mean and Range Charts Figure S6.5**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)**Upper control limit**Target Lower control limit Patterns in Control Charts Normal behavior. Process is “in control.” Figure S6.7**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**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**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**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**Upper Specification - Lower Specification**6s Cp = 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 mean x = 210.0 minutes**Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Upper Specification - Lower Specification 6s Cp = Process Capability Ratio Insurance claims process**Process mean x = 210.0 minutes**Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Upper Specification - Lower Specification 6s Cp = 213 - 207 6(.516) = = 1.938 Process Capability Ratio Insurance claims process**Process mean x = 210.0 minutes**Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Upper Specification - Lower Specification 6s Cp = 213 - 207 6(.516) = = 1.938 Process Capability Ratio Insurance claims process Process is capable**UpperSpecification - xLimit**3s Lowerx - Specification Limit 3s Cpk = minimum of , 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**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 (.251) - .250 (3).0005 Cpk = minimum of , 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 (.251) - .250 (3).0005 .250 - (.249) (3).0005 Cpk = minimum of , .001 .0015 Cpk = = 0.67 Process Capability Index New Cutting Machine Both calculations result in New machine is NOT capable**New process mean x = .250 inches**Process standard deviation s = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches Process Capability Comparison New Cutting Machine Upper Specification - Lower Specification 6s Cp = New machine is NOT capable .251 - .249 .0030 Cp = = 0.66**Cpk = negative number**Cpk = zero Cpk = between 0 and 1 Cpk = 1 Cpk > 1 Interpreting Cpk 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