CHAPTER 8TN Process Capability and Statistical Quality Control Process Variation Process Capability Process Control Procedures Acceptance Sampling 2

Types of Statistical Quality Control

Basic Forms of Variation Assignable variation Common variation 3

Taguchi’s View of Variation Incremental Cost of Variability High Zero Lower Spec Target Upper Traditional View Incremental Cost of Variability High Zero Lower Spec Target Upper Taguchi’s View 30

Process Capability Process limits Tolerance limits How do the limits relate to one another? 28

Process Capability Index, Cpk Capability Index shows how well parts being produced fit into design limit specifications. As a production process produces items small shifts in equipment or systems can cause differences in production performance from differing samples. Shifts in Process Mean 29

Types of Statistical Sampling Attribute (Go or no-go information) Defectives refers to the acceptability of product across a range of characteristics. Defects refers to the number of defects per unit which may be higher than the number of defectives. Variable (Continuous) Usually measured by the mean and the standard deviation. 6

Statistical Process Control (SPC) Charts UCL Normal Behavior LCL 1 2 3 4 5 6 Samples over time UCL Possible problem, investigate LCL 1 2 3 4 5 6 Samples over time UCL Possible problem, investigate LCL 1 2 3 4 5 6 Samples over time 16

Control Limits are based on the Normal Curve x m z -3 -2 -1 1 2 3 Standard deviation units or “z” units. 14

Control Limits We establish the Upper Control Limits (UCL) and the Lower Control Limits (LCL) with plus or minus 3 standard deviations. Based on this we can expect 99.7% of our sample observations to fall within these limits. x LCL UCL 99.7% 15

STANDARD DEVIATION TABLE % Data Points # of Std Dev From Mean ------------------- --------------------------------- 68 95 95.5 99 99.7

Example of Constructing a p-Chart: Required Data 17

Statistical Process Control Formulas: Attribute Measurements (p-Chart) Given: Compute control limits: 18

Example of Constructing a p-chart: Step 1 1. Calculate the sample proportions, p (these are what can be plotted on the p-chart) for each sample. 19

Example of Constructing a p-chart: Steps 2&3 2. Calculate the average of the sample proportions. 3. Calculate the standard deviation of the sample proportion 20

Example of Constructing a p-chart: Step 4 4. Calculate the control limits. 21

Example of Constructing a p-Chart: Step 5 5. Plot the individual sample proportions, the average of the proportions, and the control limits 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Observation p 22

R Chart Type of variables control chart Shows sample ranges over time Interval or ratio scaled numerical data Shows sample ranges over time Difference between smallest & largest values in inspection sample Monitors variability in process Example: Weigh samples of coffee & compute ranges of samples; Plot

R Chart Control Limits From Table Sample Range at Time i # Samples

R Chart Example You’re manager of a 500-room hotel. You want to analyze the time it takes to deliver luggage to the room. For 7 days, you collect data on 5 deliveries per day. Is the process in control?

R Chart Hotel Data Sample Day Delivery Time Mean Range 1 7.30 4.20 6.10 3.45 5.55 2 4.60 8.70 7.60 4.43 7.62 3 5.98 2.92 6.20 4.20 5.10 4 7.20 5.10 5.19 6.80 4.21 5 4.00 4.50 5.50 1.89 4.46 6 10.10 8.10 6.50 5.06 6.94 7 6.77 5.08 5.90 6.90 9.30

R Chart Hotel Data Sample Day Delivery Time Mean Range 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 Largest Smallest 7.30 - 3.45 Sample Range =

R Chart Hotel Data Sample Day Delivery Time Mean Range 1 7.30 4.20 6.10 3.45 5.55 5.32 7.30 + 4.20 + 6.10 + 3.45 + 5.55 5 Sample Mean =

R Chart Hotel Data Sample Day Delivery Time Mean Range 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 2 4.60 8.70 7.60 4.43 7.62 6.59 4.27 3 5.98 2.92 6.20 4.20 5.10 4.88 3.28 4 7.20 5.10 5.19 6.80 4.21 5.70 2.99 5 4.00 4.50 5.50 1.89 4.46 4.07 3.61 6 10.10 8.10 6.50 5.06 6.94 7.34 5.04 7 6.77 5.08 5.90 6.90 9.30 6.79 4.22

Example of R charts: 25

Example of R charts: From Exhibit TN7.7 25

R Chart Control Chart Solution UCL

`X Chart Type of variables control chart Shows sample means over time Monitors process average Example:

`X Chart Control Limits From Exhibit 7.7TN(n = 5) Sample Mean at Time i Sample Range at Time i # Samples

`X Chart Example You’re manager of a 500-room hotel. You want to analyze the time it takes to deliver luggage to the room. For 7 days, you collect data on 5 deliveries per day. Is the process in control? Alone Group Class

X Chart Hotel Data Sample Day Delivery Time Mean Range 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 2 4.60 8.70 7.60 4.43 7.62 6.59 4.27 3 5.98 2.92 6.20 4.20 5.10 4.88 3.28 4 7.20 5.10 5.19 6.80 4.21 5.70 2.99 5 4.00 4.50 5.50 1.89 4.46 4.07 3.61 6 10.10 8.10 6.50 5.06 6.94 7.34 5.04 7 6.77 5.08 5.90 6.90 9.30 6.79 4.22

Example of x-bar charts: Tabled Values From Exhibit TN7.7 25

Example of x-bar charts: Tabled Values From Exhibit TN7.7 25

`X Chart Control Chart Solution* UCL LCL

Process Capability

Process Capability Process Capability - TQM’s emphasis on “making it right the first time” has resulted in organizations emphasizing the ability of a production system to meet design specifications rather than evaluating the quality of outputs after the fact with acceptance sampling. Process Capability -

Process Capability Process limits – Tolerance limits -

Process Capability How do the limits relate to one another? You want: tolerance range > process range 1. Make bigger 2. Make smaller Two methods of accomplishing this:

Process Capability Measurement Cp index = Tolerance range / Process range What value(s) would you like for Cp? Larger Cp – The Cp index Assumes

Process Capability Depends On: Location of the process mean. Natural variability inherent in the process. Stability of the process. Product’s design requirements.

Natural Variation Versus Product Design Specifications

Process Capability Index Cp < 1: Cp > 1: As rule of thumb, many organizations desire a Cp index of at least 1.5. Six sigma quality (fewer than 3.4 defective parts per million) corresponds to a Cp index of 2.

LTL UTL

Process Capability Light-bulb Production UTL - LTL 6s CP = Upper specification = 120 hours Lower specification = 80 hours Average life = 90 hours s = 4.8 hours UTL - LTL 6s This slide presents the equation for the Process Capability Ratio, Cp. CP = Process Capability Ratio 25

Process Capability CP = Light-bulb Production Process Capability Ratio This slide substitutes in the values for the specification limits and the process standard deviation. Process Capability Ratio 26

Cpk Index = estimate of the process mean s = estimate of the standard deviation Together, these process capability Indices show how well parts being produced conform to design specifications.

Light-bulb Production

Another example of the use of process capability indices The design specifications for a machined slot is 0.5± .003 inches. Samples have been taken and the process mean is estimated to be .501. The process standard deviation is estimated to be .001. What can you say about the capability of this process to produce this dimension?

Basic Forms of Statistical Sampling for Quality Control Sampling to accept or reject the immediate lot of product at hand (Acceptance Sampling). Sampling to determine if the process is within acceptable limits (Statistical Process Control)

Acceptance Sampling Purposes Advantages Determine quality level Ensure quality is within predetermined level Advantages Economy Less handling damage Fewer inspectors Upgrading of the inspection job Applicability to destructive testing Entire lot rejection (motivation for improvement) 4

Acceptance Sampling Disadvantages Risks of accepting “bad” lots and rejecting “good” lots Added planning and documentation Sample provides less information than 100-percent inspection 5

Risk Acceptable Quality Level (AQL) Lot Tolerance Percent Defective (LTPD) 8