Statistical Process Control Chapter 4 1

Customer Requirements Product launch activities: Revise periodically Product Specifications Process Specifications Statistical Process Control: Measure & monitor quality Meets Specifications? Ongoing Activities Fix process or inputs No Yes Conformance Quality

Transformation Process Inputs Facilities Equipment Materials Energy Outputs Goods & Services Transformation Process Variation in inputs create variation in outputs Variations in the transformation process create variation in outputs

Specifications and Conformance Quality Product specification: quality characteristics that a product or service must have to meet customer requirements A product which meets its specification has conformance quality. Capable process: a process which consistently produces products that have conformance quality. 2

Capable Transformation Process Inputs Facilities Equipment Materials Energy Outputs Goods & Services that meet specifications Capable Transformation Process

Capability and Conformance Quality (2) If the process is capable and the product specification is based on current customer requirements, outputs will meet customer expectations.

Objectives of Statistical Process Control (SPC) To produce a product or service with conformance quality and to do so consistently. Understand current performance Detect evidence of process change 3

Types of Variation Common cause (random) variation: systematic variation in a process. Results from usual variations in inputs, output rates, and procedures. Special cause (non-random) variation: a short-term source of variation in a process. Results from changes or abnormal variations in inputs, outputs, or procedures. 4

Statistical Process Control (SPC) SPC distinguishes between common cause and special cause variation. A process is in control if there is no special cause variation (predictable) A process will be capable if it is in control and consistently produces output that meets specifications. A capable process has conformance quality. 5

Characteristics of Goods & Services Attributes: Evaluated with discrete choices Good/bad, yes/no (p charts) Count of defects (c charts) 6

Characteristics of Goods & Services (2) Variables: Can be measured and take on continuous values Length, diameter, weight, height, time, velocity, temperature, pressure Types of charts X and R (sample size < 10) X and S (sample size > 10) 7

Control Chart Format Upper Control Limit (UCL) Process Mean Measure Lower Control Limit (LCL) Sample Measure 9

Hypothesis Test H0: The process mean (or range) has not changed. (null hypothesis) H1: The process mean (or range) has changed. (alternative hypothesis). If the process has only random variations and remains within the control limits, we accept H0. The process is in control.

Process In Control Intuitive Explanation No sample points outside limits Most points near process average About equal number of points above and below the centerline Points appear randomly distributed Quality control analysts use more precise tests.

Terminology We take periodic random samples n = sample size = number of observations in each sample 11

X and R Charts for Variables X = Sample mean Measure of central tendency Central Limit Theorem: X is normally distributed. R = Sample range Measure of variation R has a gamma distribution 12

Slip-ring diameter (cm) Data for Examples 4.3 and 4.4 Slip-ring diameter (cm) Sample 1 2 3 4 5 X R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 … … … … … … … … 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15 Note: n = number in each sample = 5 15

Calculate X and R for Each Sample 5 = 4.98 R = range = maximum - minimum = 5.02 - 4.94 = 0.08 16

Calculate X and R R = 0.08 + 0.12 + 0.08 + … + 0.10 = 0.115 17

The Normal Distribution 95% 99.74% -3s -2s -1s m=0 1s 2s 3s 13

Control Limits for X 99.7% confidence interval for X: (X - 3s, X + 3s). For small sample sizes (n < 10), this may be approximated as (X - A2R, X + A2R). A2 is a factor which depends on n and is obtained from a table. 18

3s Control Chart Factors Sample size x-chart R-chart n A2 D3 D4 2 1.88 0 3.27 3 1.02 0 2.57 4 0.73 0 2.28 5 0.58 0 2.11 6 0.48 0 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86 19

Control Limits for X and R For X: LCL = X - A2R = 5.01 - 0.58 (0.115) = 4.94 UCL = X + A2R = 5.01 + 0.58 (0.115) = 5.08 For R: LCL = D3R = 0 (0.115) = 0 UCL = D4R = 2.11 (0.115) = 0.243 20

SPC Procedures Decide what characteristics of the product or service should be measured. Establish consistent measurement procedures. Collect data and set trial control limits, based on confidence intervals. Look for out-of-control points Take remedial action to eliminate the causes of out- of-control points. 8

SPC Procedures (2) Compute the revised control limits, using only the "in-control" points on the original chart. Use the control charts to monitor and control quality. Take remedial action when non-random patterns occur. If process conditions change permanently, collect data under the new process conditions and construct new control charts.

Process Capability A process must be in control before you can decide whether or not it is capable. Control charts measure the range of natural variability in a process Specification limits are set to meet customer requirements. Process cannot meet specifications if natural variability exceeds tolerances (i.e., one or both control limits are outside specification limits) 21

Process Meets Customer Requirements Upper specification limit UCL X LCL Lower specification limit 9

Process Does Not Meet Customer Requirements UCL Upper specification limit X Lower specification limit LCL 9

3-Sigma Quality Use 3-s control limits for x. Corresponds to 3 defects per thousand If a product has 250 parts and each has 3-s quality, P[at least 1 bad part] = 0.528 23

6-Sigma Quality Use 6-s control limits for x. Control limits are (X- 2A2R, X + 2A2R). Corresponds to 3.4 defects per million If a product has 250 parts and 6-s control limits for each are within specifications, P[at least 1 bad part] <0.001 23