Statistical Process Control A. A. Elimam A. A. Elimam

Two Primary Topics in Statistical Quality Control n n Statistical process control (SPC) is a statistical method using control charts to check a production process - prevent poor quality. In TQM all workers are trained in SPC methods.

Two Primary Topics in Statistical Quality Control n n Acceptance Sampling involves inspecting a sample of product. If sample fails reject the entire product - identifies the products to throw away or rework. Contradicts the philosophy of TQM. Why ?

Inspection n n Traditional Role: at the beginning and end of the production process n n Relieves Operator from the responsibility of detecting defectives & quality problems n n It was the inspection's job n n In TQM, inspection is part of the process & it is the operator’s job n n Customers may require independent inspections

How Much to Inspect? n n Complete or 100 % Inspection. Viable for products that can cause safety problems Does not guarantee catching all defectives Too expensive for most cases n n Inspection by Sampling Sample size : representative A must in destructive testing (e.g... Tasting food)

Where To Inspect ? n n In TQM, inspection occurs throughout the production process n n IN TQM, the operator is the inspector n n Locate inspection where it has the most effect (e.g.... prior to costly or irreversible operation) n n Early detection avoids waste of more resources

Quality Testing n n Destructive Testing Product cannot be used after testing (e.g.. taste or breaking item) Sample testing Could be costly n n Non-Destructive Testing Product is usable after testing 100% or sampling

Quality Measures:Attributes Attribute is a qualitative measure Product characteristics such as color, taste, smell or surface texture Simple and can be evaluated with a discrete response (good/bad, yes/no) Large sample size (100’s)

Quality Measures:Variables A quantitative measure of a product characteristic such as weight, length, etc. Small sample size (2-20) Requires skilled workers

Variation & Process Control Charts n n Variation always exists n n Two Types of Variation Causal: can be attributed to a cause. If we know the cause we can eliminate it. Random: Cannot be explained by a cause. An act of nature - need to accept it. n n Process control charts are designed to detect causal variations

Control Charts: Definition & Types n n A control chart is a graph that builds the control limits of a process n n Control limits are the upper and lower bands of a control chart n n Types of Charts: Measurement by Variables: X-bar and R charts Measurement by Attributes: p and c

Process Control Chart & Control Criteria 1. No sample points outside control limits. 2. Most points near the process average. 3. Approximately equal No. of points above & below center. 4. Points appear to be randomly distributed around the center line. 5. No extreme jumps. 6. Cannot detect trend.

Basis of Control Charts n n Specification Control Charts Target Specification: Process Average Tolerances define the specified upper and lower control limits Used for new products (historical measurements are not available) n n Historical Data Control Charts Process Average, upper & lower control limits: based on historical measurements Often used in well established processes

Common Causes 425 Grams

Assignable Causes (a) Location Grams Average

Assignable Causes (b) Spread Grams Average

Assignable Causes (b) Spread Grams Average

Assignable Causes (c) Shape Grams Average

Effects of Assignable Causes on Process Control Assignable causes present

Effects of Assignable Causes on Process Control No assignable causes

Sample Means and the Process Distribution 425 Grams Mean Process distribution Distribution of sample means

The Normal Distribution -3  -2  -1  +1  +2  +3  Mean 68.26% 95.44% 99.97%  = Standard deviation

Control Charts UCL Nominal LCL Assignable causes likely Samples

Using Control Charts for Process Improvement  Measure the process  When problems are indicated, find the assignable cause  Eliminate problems, incorporate improvements  Repeat the cycle

Control Chart Examples Nominal UCL LCL Sample number (a) Variations

Control Chart Examples Nominal UCL LCL Sample number (b) Variations

Control Chart Examples Nominal UCL LCL Sample number (c) Variations

Control Chart Examples Nominal UCL LCL Sample number (d) Variations

Control Chart Examples Nominal UCL LCL Sample number (e) Variations

The Normal Distribution Measures of Variability: Most accurate measure  = Standard Deviation Approximate Measure - Simpler to compute R = Range Range is less accurate as the sample size gets larger Average  = Average R when n = 2

Control Limits and Errors LCL Process average UCL (a) Three-sigma limits Type I error: Probability of searching for a cause when none exists

Control Limits and Errors Type I error: Probability of searching for a cause when none exists UCL LCL Process average (b) Two-sigma limits

Type II error: Probability of concluding that nothing has changed Control Limits and Errors UCL Shift in process average LCL Process average (a) Three-sigma limits

Type II error: Probability of concluding that nothing has changed Control Limits and Errors UCL Shift in process average LCL Process average (b) Two-sigma limits

Control Charts for Variables Mandara Industries

Control Charts for Variables Sample Number1234RangeMean Special Metal Screw

Control Charts for Variables Sample Number1234RangeMean = Special Metal Screw

Control Charts for Variables Sample Number1234RangeMean = Special Metal Screw

Control Charts for Variables Sample Number1234RangeMean = ( )/4= )/4= Special Metal Screw

Control Charts for Variables Sample Number1234RangeMean = ( )/4= )/4= Special Metal Screw

Control Charts for Variables Sample Number1234RangeMean R = x = Special Metal Screw

Control Charts for Variables Control Charts - Special Metal Screw R - Charts R = UCL R = D 4 R LCL R = D 3 R

Control Charts for Variables Control Charts - Special Metal Screw R - Charts R = D 4 = Control Chart Factors Control Chart Factors Factor for UCLFactor forFactor Size ofand LCL forLCL forUCL for Samplex-ChartsR-ChartsR-Charts (n)(A 2 )(D 3 )(D 4 )

Control Charts for Variables Control Charts - Special Metal Screw R - Charts R = D 4 = D 3 = 0 UCL R = (0.0020) = in. LCL R = 0 (0.0020) = 0 in. UCL R = D 4 R LCL R = D 3 R

Range (in.) Sample number UCL R = LCL R = 0 R = Range Chart - Special Metal Screw

Control Charts for Variables Control Charts - Special Metal Screw R = x = x - Charts UCL x = x + A 2 R LCL x = x - A 2 R Control Chart Factors Control Chart Factors Factor for UCLFactor forFactor Size ofand LCL forLCL forUCL for Samplex-ChartsR-ChartsR-Charts (n)(A 2 )(D 3 )(D 4 )

Control Charts for Variables Control Charts - Special Metal Screw R = A 2 = x = x - Charts UCL x = x + A 2 R LCL x = x - A 2 R UCL x = (0.0020) = in.

Control Charts for Variables Control Charts - Special Metal Screw R = A 2 = x = x - Charts UCL x = x + A 2 R LCL x = x - A 2 R UCL x = (0.0020) = in. LCL x = (0.0020) = in.

Average (in.) Sample number x = UCL x = LCL x = Average Chart - Special Metal Screw

Average (in.) x = UCL x = LCL x = Sample number   Measure the process   Find the assignable cause   Eliminate the problem   Repeat the cycle Average Chart - Special Metal Screw

Control Charts for Attributes MANDARA Bank UCL p = p + z  p LCL p = p - z  p  p = p (1 - p )/ n

MANDARA Bank UCL p = p + z  p LCL p = p - z  p  p = p (1 - p )/ n SampleWrongProportion NumberAccount NumberDefective Total147 p = n = 2500 Control Charts for Attributes

Control Charts for Attributes MANDARA Bank UCL p = p + z  p LCL p = p - z  p  p = ( )/2500 n = 2500 p =

Control Charts for Attributes MANDARA Bank UCL p = p + z  p LCL p = p - z  p  p = n = 2500 p =

Control Charts for Attributes MANDARA Bank UCL p = (0.0014) LCL p = (0.0014)  p = n = 2500 p =

Control Charts for Attributes MANDARA Bank UCL p = LCL p =  p = n = 2500 p =

Sample number UCL p LCL Proportion defective in sample p -Chart Wrong Account Numbers

Sample number UCL p LCL Proportion defective in sample p -Chart Wrong Account Numbers   Measure the process   Find the assignable cause   Eliminate the problem   Repeat the cycle

Process Capability Nominal value Hours Upper specification Lower specification Process distribution (a) Process is capable

Process Capability Nominal value Hours Upper specification Lower specification Process distribution (b) Process is not capable

Process Capability Lower specification Mean Upper specification Two sigma

Process Capability Lower specification Mean Upper specification Four sigma Two sigma

Process Capability Lower specification Mean Upper specification Six sigma Four sigma Two sigma

Upper specification = 120 hours Lower specification = 80 hours Average life = 90 hours s = 4.8 hours Process Capability Light-bulb Production C p = Upper specification - Lower specification 6s Process Capability Ratio

Process Capability Light-bulb Production Upper specification = 120 hours Lower specification = 80 hours Average life = 90 hours s = 4.8 hours C p = (4.8) Process Capability Ratio

Process Capability Light-bulb Production Upper specification = 120 hours Lower specification = 80 hours Average life = 90 hours s = 4.8 hours C p = 1.39 Process Capability Ratio

Process Capability Light-bulb Production Upper specification = 120 hours Lower specification = 80 hours Average life = 90 hours s = 4.8 hours C p = 1.39 C pk = Minimum of Upper specification - x 3s x - Lower specification 3sProcessCapabilityIndex,

Process Capability Light-bulb Production Upper specification = 120 hours Lower specification = 80 hours Average life = 90 hours s = 4.8 hours C p = 1.39 C pk = Minimum of (4.8) (4.8) ProcessCapabilityIndex,

Process Capability Light-bulb Production Upper specification = 120 hours Lower specification = 80 hours Average life = 90 hours s = 4.8 hours C p = 1.39 C pk = Minimum of [ 0.69, 2.08 ] ProcessCapabilityIndex

Process Capability Light-bulb Production Upper specification = 120 hours Lower specification = 80 hours Average life = 90 hours s = 4.8 hours C p = 1.39C pk = 0.69 ProcessCapabilityIndexProcessCapabilityRatio