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Statistical Process Control (SPC) Chapter 6 MGMT 326 Foundations of Operations Products & Processes Quality Assurance Introduction Managing Projects Managing Quality Product Design Statistical Process Control Strategy Process Design Just-in-Time & Lean Systems Capacity and Facilities Planning & Control Assuring Customer-Based Quality Product launch activities: Revise periodically Customer Requirements Ongoing activity Statistical Process Control: Measure & monitor quality Product Specifications Process Specifications Statistical Process Control (SPC) Basic SPC Concepts Types of Measures Variation Attributes Objectives Variables First steps SPC for Variables Mean charts Range charts and known , unknown Capable Processes = target = target Variation in a Transformation Process Inputs • Facilities • Equipment • Materials • Energy Transformation Process Outputs Goods & Services •Variation in inputs create variation in outputs • Variations in the transformation process create variation in outputs Variation in a Transformation Process Customer requirements are not met Inputs • Facilities • Equipment • Materials • Energy Transformation Process Outputs Goods & Services •Variation in inputs create variation in outputs • Variations in the transformation process create variation in outputs Variation All processes have variation. Common cause variation is random variation that is always present in a process. Assignable cause variation results from changes in the inputs or the process. The cause can and should be identified. Assignable cause variation shows that the process or the inputs have changed, at least temporarily. Objectives of Statistical Process Control (SPC) Find out how much common cause variation the process has Find out if there is assignable cause variation. A process is in control if it has no assignable cause variation Being in control means that the process is stable and behaving as it usually does. First Steps in Statistical Process Control (SPC) Measure characteristics of goods or services that are important to customers Make a control chart for each characteristic The chart is used to determine whether the process is in control Types of Measures (1) Variable Measures Continuous random variables Measure does not have to be a whole number. Examples: time, weight, miles per gallon, length, diameter Types of Measures (2) Attribute Measures Discrete random variables – finite number of possibilities Also called categorical variables The measure may depend on perception or judgment. Different types of control charts are used for variable and attribute measures Examples of Attribute Measures Good/bad evaluations Number of defects per unit Good or defective Correct or incorrect Number of scratches on a table Opinion surveys of quality Customer satisfaction surveys Teacher evaluations SPC for Variables The Normal Distribution = the population mean = the standard deviation for the population 99.74% of the area under the normal curve is between - 3 and + 3 SPC for Variables The Central Limit Theorem Samples are taken from a distribution with mean and standard deviation . k = the number of samples n = the number of units in each sample The sample means are normally distributed with mean and standard deviation x n when k is large. Control Limits for the Sample Mean x when and are known x is a variable, and samples of size n are taken from the population containing x. Given: = 10, = 1, n = 4 Then x 1 1 0.5 n 4 2 A 99.7% confidence interval for x is ( 3 x , 3 x ) 3 , 3 n n Control Limits for the Sample Mean x when and are known (2) The lower control limit for x is 1 LCL 3 x 3 10 3 n 4 10 1.5 8.5 Control Limits for the Sample Mean x when and are known (3) The upper control limit for x is 1 UCL 3 x 3 10 3 n 4 10 1.5 11.5 Control Limits for the Sample Mean x when and are unknown If the process is new or has been changed recently, we do not know and Example 6.1, page 180 Given: 25 samples, 4 units in each sample and are not given k = 25, n = 4 Control Limits for the Sample Mean x when and are unknown (2) 1. Compute the mean for each sample. For example, 15.85 16.02 15.83 15.93 x1 2. 4 15.91 Compute k x x m 1 k 25 m x m 1 25 m 15.91 16.00 ... 15.94 398.75 15.95 25 25 Control Limits for the Sample Mean x when and are unknown (3) For the ith sample, the sample range is Ri = (largest value in sample i ) - (smallest value in sample i ) 3. Compute Ri for every sample. For example, R1 = 16.02 – 15.83 = 0.19 Control Limits for the Sample Mean x when and are unknown (4) 4. Compute R , the average range k R R i 1 k i 0.19 0.27 ... 0.30 7.17 0.29 25 25 We will approximate 3 x by A2 R, where A2 is a number that depends on the sample size n. We get A2 from Table 6.1, page 182 Control Limits for the Sample Mean x when and are unknown (5) Factor for x-Chart 5. n = the number of units in each sample = 4. From Table 6.1, A2 = 0.73. The same A2 is used for every problem with n = 4. Sample Size (n) 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A2 1.88 1.02 0.73 0.58 0.48 0.42 0.37 0.34 0.31 0.29 0.27 0.25 0.24 0.22 Factors for R-Chart D3 0.00 0.00 0.00 0.00 0.00 0.08 0.14 0.18 0.22 0.26 0.28 0.31 0.33 0.35 D4 3.27 2.57 2.28 2.11 2.00 1.92 1.86 1.82 1.78 1.74 1.72 1.69 1.67 1.65 Control Limits for the Sample Mean x when and are unknown (6) 6. The formula for the lower control limit is LCL x A2 R 15.95 0.73(0.29) 15.74 7. The formula for the upper control limit is UCL x A2 R 15.95 0.73(0.29) 16.16 Control Chart for x The variation between LCL = 15.74 and UCL = 16.16 is the common cause variation. Common Cause and Special Cause Variation The range between the LCL and UCL, inclusive, is the common cause variation for the process. When x is in this range, the process is in control. When a process is in control, it is predictable. Output from the process may or may not meet customer requirements. When x is outside control limits, the process is out of control and has special cause variation. The cause of the variation should be identified and eliminated. Control Limits for R Factor for x-Chart 1. From the table, get D3 and D4 for n = 4. D3 = 0 D4 = 2.28 Sample Size (n) 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A2 1.88 1.02 0.73 0.58 0.48 0.42 0.37 0.34 0.31 0.29 0.27 0.25 0.24 0.22 Factors for R-Chart D3 0.00 0.00 0.00 0.00 0.00 0.08 0.14 0.18 0.22 0.26 0.28 0.31 0.33 0.35 D4 3.27 2.57 2.28 2.11 2.00 1.92 1.86 1.82 1.78 1.74 1.72 1.69 1.67 1.65 Control Limits for R (2) 2. The formula for the lower control limit is LCL D3 R 0(0.29) 0 2. The formula for the upper control limit is UCL D4 R 2.28(0.29) 0.66 fig_ex06_03 Statistical Process Control (SPC) Basic SPC Concepts Types of Measures Variation Attributes Objectives Variables First steps SPC for Variables Mean charts Range charts and known , unknown Capable Processes = target = target Capable Transformation Process Inputs • Facilities • Equipment • Materials • Energy Capable Transformation Process Outputs Goods & Services that meet specifications a specification that meets customer requirements + a capable process (meets specifications) = Satisfied customers and repeat business Review of Specification Limits The target for a process is the ideal value Example: if the amount of beverage in a bottle should be 16 ounces, the target is 16 ounces Specification limits are the acceptable range of values for a variable Example: the amount of beverage in a bottle must be at least 15.8 ounces and no more than 16.2 ounces. The allowable range is 15.8 – 16.2 ounces. Lower specification limit = 15.8 ounces or LSL = 15.8 ounces Upper specification limit = 16.2 ounces or USL = 16.2 ounces Control Limits vs. Specification Limits Control limits show the actual range of variation within a process What the process is doing Specification limits show the acceptable common cause variation that will meet customer requirements. Specification limits show what the process should do to meet customer requirements Process is Capable: Control Limits are within or on Specification Limits Upper specification limit UCL X LCL Lower specification limit Process is Not Capable: One or Both Control Limits are Outside Specification Limits UCL Upper specification limit X LCL Lower specification limit Capability and Conformance Quality A process is capable if It is in control and It consistently produces outputs that meet specifications. This means that both control limits for the mean must be within the specification limits A capable process produces outputs that have conformance quality (outputs that meet specifications). Process Capability Ratio C p Use C p to determine whether the process is capable when = target. USL LSL Cp 6 If C p 1, the process is capable, If C p 1 , the process is not capable. C p Example Given: Boffo Beverages produces 16-ounce bottles of soft drinks. The mean ounces of beverage in Boffo's bottle is 16. The allowable range is 15.8 – 16.2. The standard deviation is 0.06. Find C p and determine whether the process is capable. C p Example (2) Given: = 16, = 0.06, target = 16 LSL = 15.8, USL = 16.2 USL LSL 16.2 15.8 Cp 1.11 6 6(0.06) C p 1 The process is capable. Process Capability Index Cpk C pk USL LSL smaller , 3 3 If Cpk > 1, the process is capable. If Cpk < 1, the process is not capable. We must use Cpk when does not equal the target. Cpk Example Given: Boffo Beverages produces 16-ounce bottles of soft drinks. The mean ounces of beverage in Boffo's bottle is 15.9. The allowable range is 15.8 – 16.2. The standard deviation is 0.06. Find C pk and determine whether the process is capable. Cpk Example (2) Given: = 15.9, = 0.06, target = 16 LSL = 15.8, USL = 16.2 16.2 15.9 15.9 15.8 USL LSL C pk smaller , smaller , 3(0.06) 3 3 3(0.06) 0.3 0.1 smaller , smaller{1.67, 0.56} 0.56 0.18 0.18 Cpk < 1. Process is not capable. Statistical Process Control (SPC) Basic SPC Concepts Types of Measures Variation Attributes Objectives Variables First steps SPC for Variables Mean charts Range charts and known , unknown Capable Processes = target = target

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