Quality Control

Quality Control Quality Control is a testing procedure performed every hour (or every half hour, etc) in an ongoing process of production in order to see whether the process is running properly (“ is under control” or “not”). If the process is not under control, the process is halted to search for the trouble and remove it.

Statistical Process Control Statistical Process Control (SPC) Monitoring production process to detect and prevent poor quality Sample Subset of items produced to use for inspection Control Charts Process is within statistical control limits UCL LCL

Quality Measures Attribute Variable A product characteristic that can be evaluated with a discrete response good – bad; yes - no Variable A product characteristic that is continuous and can be measured weight - length

Process Control Chart Out of control Upper Control Limit (UCL) Process 1 2 3 4 5 6 7 8 9 10 Sample number Upper Control Limit (UCL) Process Average (CL) Lower (LCL) Out of control

A Process is in Control If … No sample points outside limits Most points near process average About equal number of points above and below centerline Points appear randomly distributed

Applying SPC to Service Hospitals Timeliness and quickness of care, staff responses to requests, accuracy of lab tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance and checkouts Grocery Stores Waiting time to check out, frequency of out-of-stock items, quality of food items, cleanliness, customer complaints, checkout register errors Airlines Flight delays, lost luggage and luggage handling, waiting time at ticket counters and check-in, agent and flight attendant courtesy, accurate flight information, passenger cabin cleanliness and maintenance

Applying SPC to Service Fast-Food Restaurants Waiting time for service, customer complaints, cleanliness, food quality, order accuracy, employee courtesy Catalogue-Order Companies Order accuracy, operator knowledge and courtesy, packaging, delivery time, phone order waiting time Insurance Companies Billing accuracy, timeliness of claims processing, agent availability and response time

Control Charts for Variables Mean (x-bar) Charts Tracks the Central Tendency (the average value observed) over time Range (R) Charts Tracks the Spread of the Distribution over time (estimates the observed variation)

Control Chart For Mean If α is not specified, then α be taken as 1%. (Z (1%) = 2.58)

Control Chart For Variance

Control Chart (For Std Deviation)

Control Chart For Range E(R*) = Mean of the Sample Ranges

Problem 1 Suppose a machine for filling cans with lubricating oil is set, so that it will generate fillings which form a normal population with Mean 1 gallon and Standard Deviation 0.03 gallon. Set up a Control Chart for controlling the mean (that is, find LCL and UCL), assuming that Sample Size is 6.

Problem 3 What Sample Size should we choose in Problem 1, if we want LCL and UCL somewhat closer together, say UCL – LCL = 0.5 Without changing the Significance Level?

Problem 5 How should we change the Sample Size in controlling the mean of a Normal Population, if we want the difference, UCL – LCL to decrease to half its original value?

Problem 7 Ten Samples of Size 2 were taken from a production lot of bolts. The values (length in mm) are as following: Assuming that population is normal with Mean 27.5 and Variance 0.024, set up a Control Chart for the Mean and graph Sample Means on the chart. Sample No 1 2 3 4 5 6 7 8 9 10 Length 27.4 27.5 27.3 27.9 27.6 27.8 27.7

Problem 9 Time Sample Values Graph the ranges of the given sample as Control Chart for Ranges, assuming that population is normal with Mean 5 and Standard Deviation 1.55. Time Sample Values Sample Range R 8:00 3 4 8 8:30 6 9:00 5 2 9:30 7 10:00 10:30 11:00 11:30 12:00 12:30

Problem 9 Graph the ranges of the given sample as Control Chart for Ranges, assuming that population is normal with Mean 5 and Standard Deviation 1.55. Time Sample Values Sample Range R 8:00 3 4 8 5 8:30 6 9:00 2 9:30 7 10:00 10:30 11:00 11:30 12:00 12:30

Problem 13 Find formulas for UCL, CL and LCL (corresponding to 3б Limits) in case of a Control Chart for the defectives, assuming that in a state of Statistical Control the fraction of defectives is p.

Problem 15 A so called C-Chart or Defects Per Unit Chart is used for the control of number of defects per unit (for instance, the number of defects per 10 meters of paper, the number of missing rivets in an airplane wing, etc). Set up formulas for UCL, CL and LCL corresponding µ ± 3б, assuming that X has a Poisson Distribution. Compute UCL, CL and LCL in a Control Process of number of imperfections in sheet glass, assume that this number is 2.5 per sheet on average, when process is under control.

Three Sigma Capability Mean output +/- 3 standard deviations falls within the design specification It means that 0.26% of output falls outside the design specification and is unacceptable. The result: a 3-sigma capable process produces 2600 defects for every million units produced

Six Sigma Capability Six sigma capability assumes the process is capable of producing output where the mean +/- 6 standard deviations fall within the design specifications The result: only 3.4 defects for every million produced Six sigma capability means smaller variation and therefore higher quality

Process Control Charts Control Charts show sample data plotted on a graph with Center Line (CL), Upper Control Limit (UCL), and Lower Control Limit (LCL).

Setting Control Limits

Problem 11 Since the presence of a point outside control limits for Mean indicates trouble (“the process is out of control”), how often would we be making the mistake of looking for nonexistent trouble, if we used: 1 – Sigma Limits 2 – Sigma Limits? Assume Normality.