TM 620: Quality Management Session Seven – 9 November 2010 Control Charts, Part I Variables

Recall: What is Quality? Juran – Quality is fitness for use => we should be able to determine a set of measurable characteristics which define quality

Recall: What is Quality? Taguchi – Loss from quality is proportional to the amount of variability in the system Why? => if we reduce variation, we reduce loss from quality Less rework, reduction in wasted time, effort, and money

Quality Improvement The reduction of variability in processes and products Equivalent definition: The reduction of waste Waste is any activity for which the customer will not pay From Juran and Taguchi (and others); Define the three categories of activities

Recall: Cost of Quality Cost of Failure Cost of Control Total Cost Traditional View Quality Level Costs

Traditional Loss Function LSL USL x T “goalpost” specifications – as long as you kick the ball between the posts, it doesn’t matter where you hit. LSL T USL

Example (Sony, 1979) Comparing cost of two Sony television plants in Japan and San Diego. All units in San Diego fell within specifications. Japanese plant had units outside of specifications. Loss per unit (Japan) = $0.44 Loss per unit (San Diego) = $1.33 How can this be? Sullivan, “Reducing Variability: A New Approach to Quality,” Quality Progress, 17, no.7, 15-21, 1984.

Example LSL USL x T U.S. Plant (2 = 8.33) Japanese Plant (2 = 2.78)

Taguchi Loss Function T x x T The Taguchi loss function is a scientific approach to tolerance design. Taguchi suggests that no strict cut-off point divides good quality from poor quality. Rather, Taguchi assumes that losses can be approximated by a quadratic function so that larger derivations from the target correspond to increasingly larger losses. x T

Taguchi Loss Function T L(x) = k(x - T)2 L(x) k(x - T)2 x Nominal is best = quality deteriorates as the actual value moves away from the target on either side x T L(x) = k(x - T)2

Estimating Loss Function Suppose we desire to make pistons with diameter D = 10 cm. Too big and they create too much friction. Too little and the engine will have lower gas mileage. Suppose tolerances are set at D = 10 + .05 cm. Studies show that if D > 10.05, the engine will likely fail during the warranty period. Average cost of a warranty repair is $400.

Estimating Loss Function L(x) 400 10.05 10 400 = k(10.05 - 10.00)2 = k(.0025)

Estimating Loss Function L(x) 400 10.05 10 400 = k(10.05 - 10.00)2 = k(.0025) k = 160,000

Example 2 Suppose we have a 1 year warranty to a watch. Suppose also that the life of the watch is exponentially distributed with a mean of 1.5 years. The warranty costs to replace the watch if it fails within one year is $25. Estimate the loss function.

Example 2 L(x) f(x) 25 1 1.5 25 = k(1 - 1.5)2 k = 100

Example 2 L(x) f(x) 25 1 1.5 25 = k(1 - 1.5)2 k = 100

Single Sided Loss Functions Smaller is better L(x) = kx2 Larger is better L(x) = k(1/x2)

Example 2 L(x) f(x) 25 1

Example 2 L(x) f(x) 25 1 25 = k(1)2 k = 25

Expected Loss

Expected Loss

Expected Loss

Expected Loss

Expected Loss

Expected Loss Recall, X f(x) with finite mean  and variance 2. E[L(x)] = E[ k(x - T)2 ] = k E[ x2 - 2xT + T2 ] = k E[ x2 - 2xT + T2 - 2x + 2 + 2x - 2 ] = k E[ (x2 - 2x+ 2) - 2 + 2x - 2xT + T2 ] = k{ E[ (x - )2 ] + E[ - 2 + 2x - 2xT + T2 ] }

Expected Loss E[L(x)] = k{ E[ (x - )2 ] + E[ - 2 + 2x - 2xT + T2 ] } Recall, Expectation is a linear operator and E[ (x - )2 ] = 2 E[L(x)] = k{2 - E[ 2 ] + E[ 2x - E[ 2xT ] + E[ T2 ] }

Expected Loss Recall, E[ax +b] = aE[x] + b = a + b E[L(x)] = k{2 - 2 + 2 E[ x - 2T E[ x ] + T2 } =k {2 - 2 + 22 - 2T + T2 }

Expected Loss Recall, E[ax +b] = aE[x] + b = a + b E[L(x)] = k{2 - 2 + 2 E[ x - 2T E[ x ] + T2 } =k {2 - 2 + 22 - 2T + T2 } =k {2 + ( - T)2 }

Expected Loss Recall, E[ax +b] = aE[x] + b = a + b E[L(x)] = k{2 - 2 + 2 E[ x - 2T E[ x ] + T2 } =k {2 - 2 + 22 - 2T + T2 } =k {2 + ( - T)2 } = k { 2 + ( x - T)2 } = k (2 +D2 )

Example Since for our piston example, x = T, D2 = (x - T)2 = 0 L(x) = k2

Example (Piston Diam.)

Example (Sony) T x E[LUS(x)] = 0.16 * 8.33 = $1.33 LSL USL x T U.S. Plant (2 = 8.33) Japanese Plant (2 = 2.78) E[LUS(x)] = 0.16 * 8.33 = $1.33 E[LJ(x)] = 0.16 * 2.78 = $0.44

Tolerance (Pistons) Recall, 10 400 = k(10.05 - 10.00)2 = k(.0025) L(x) 400 10.05 10 400 = k(10.05 - 10.00)2 = k(.0025) k = 160,000

Tolerance L(x) Suppose repair for an engine which will fail during warranty can be made for only $200 400 200 10.05 10 LSL USL

Tolerance L(x) Suppose repair for an engine which will fail during warranty can be made for only $200 200 =160,000(tolerance)2 400 200 10.05 10 LSL USL

Tolerance L(x) Suppose repair for an engine which will fail during warranty can be made for only $200 200 = 160,000(tolerance)2 tolerance = (200/160,000)1/2 = .0354 400 200 10.05 10 LSL USL

Statistical Thinking All work occurs in a system of interconnected processes All process have variation Understanding variation and reducing variation are important keys to success

Variability A certain amount of variability is inescapable Therefore, no two products are identical The larger the variability, the greater the probability that the customer will perceive its existence

Sources of Variability Include: Differences in materials Differences in the performance and operation of the manufacturing equipment Differences in the way the operators perform their tasks Leave blank for students, make into a call-out

Variability and Statistics Variability is difference from the target Characteristics of quality must be measurable Therefore, Variability is described in statistical terms We will use statistical methods in our quality improvement activities Recall customer requirements vs. technical requirements Use management tools as first resort, save power tools for power situations

Breaking down workplace barriers Train people in the basics of the tools and methods Deal with those who look down on or fear numerical methods Encourage a data-driven culture of decision making Lose the jargon No “lying with statistics” Why do statistics sometimes fail in the workplace? Lack of knowledge about the tools General disdain for all things mathematical Cultural barriers in a company Statistical specialists have trouble communicating Statistics generally are poorly taught, emphasizing mathematical development rather than application People have a poor understanding of the scientific method Organizations lack patience in collecting data. All decisions have to be made “yesterday” Statistics are viewed as something to buttress an already-held opinion People fear using statistics Most people don’t understand random variation Statistical tools often are reactive and focus on effects rather than causes

Recall: Types of Errors Type I error Producers risk Probability that a good product will be rejected Type II error Consumers risk Probability that a nonconforming product will be available for sale Type III error Asking the wrong question

Data on Quality Characteristics Attribute data Discrete Often a count of some type Variable data Continuous Often a measurement, such as length, voltage, or viscosity We will use statistical methods for both types

Terms Specifications Target (or Nominal) Value Upper Specification Limit Lower Specification Limit Random Variation Non-random Variation Process stability Specs (manufacturing): desired measurements for the quality characteristics on the components and sub-assemblies that make up the product as well as the desired values for the quality characteristics of the final product Specs (service): typically in terms of the max amount of time to process an order or to provide a particular service Target: goal or desired value USL: largest allowable value for the given quality characteristic LSL: smallest allowable value for the given quality characteristic Random variation Centered around the mean Consistent amount of dispersion Nonrandom variation “Special Causes” Results from some event Dispersion and average of the process are changing Process that is not repeatable Process stability Random Variation Not nonrandom variation Process Charts

Terms Nonconforming: failure to meet one or more of the specifications Nonconformity: a specific type of failure Defect: a nonconformity serious enough to significantly affect the safe or effective use of the produce or completion of the service

Nonconforming vs. Defective A nonconforming product is not necessarily unfit for use A nonconforming product is considered defective if if it has one or more defects Example: a detergent may have a concentration of an active ingredient that is below the lower spec limit but still performs acceptably

Classroom Exercise For a product or service in your job: Name a quality characteristic Give an example of a nonconformity that is not a defect Give an example of a defect

Types of Inspection Receiving In Process Final None One Hundred Percent Acceptance Sampling Difference between spot check and acceptance= in acceptance, inspectors that a statistically determined random sample and use a decision rule to determine acceptance or rejection of the lot based on the observed number of nonconforming items. The decision rule is called lot sentencing. Acceptance sampling was developed after SPC

Quality Design & Process Variation Lower Spec Limit Upper Spec Limit Acceptance Sampling 60 80 100 120 140 Statistical Process Control 60 140 Experimental Design 140 60

Variation and Control A process that is operating with only common causes of variation is said to be in statistical control. A process operating in the presence of special or assignable cause is said to be out of control.

Finding Trends and Special Causes Inspection does not tell you about a problem until it becomes a problem We need a mechanism to help us spot special causes when they occur We need mechanism to help us determine when we have a trend in the data

Statistical Process Control Originally developed by Walter Shewhart in 1924 at the Bell Telephone Laboratories Late 1920s, Harold Dodge and Harry Romig developed statistically based acceptance sampling Not recognized by industry until after World War II Shewhart was a contemporary of Deming and Juran Dodge and Romig also of Bell Labs statistically based acceptance sampling is an alternative to 100% inspection

Definition Statistical Process Control (SPC): “a methodology for monitoring a process to identify special causes of variation and signal the need to take corrective action when it is appropriate” (Evans and Lindsay)

The quality function is not responsible for quality!

Statistical Process Control Tools The magnificent seven The tool most often associated with Statistical Process Control is Control Charts We will spend the next few class on the different types of control charts

Common Causes Special Causes

Histograms do not take into account changes over time. Control charts can tell us when a process changes

Control Chart Applications Establish state of statistical control Monitor a process and signal when it goes out of control Determine process capability Note: Control charts will only detect the presence of assignable causes. Management, operator, and engineering action is necessary to eliminate the assignable cause.

Capability Versus Control Capable Not Capable In Control Out of Control IDEAL

Developing Control Charts Prepare Choose measurement Determine how to collect data, sample size, and frequency of sampling Set up an initial control chart Collect Data Record data Calculate appropriate statistics Plot statistics on chart

Types of Sampling Random Samples Systematic Samples Rational subgroups Each piece has an equal chance of being selected for inspection Systematic Samples According to time or sequence Rational subgroups A group of data that is logically homogeneous Computing variation between subgroups Rational Subgroups Subgroups or samples should be selected so that if assignable causes are present, the chance for differences between subgroups will be maximized while the chance for differences due to these assignable causes within a subgroup will be minimized.

Approaches to Rational Subgrouping Each sample consists of units that were produced at the same time (or as closely together as possible) Used when the primary purpose of the control chart is to detect process shifts Each sample consists of units of product that are representative of all units that have been produced since the last sample was taken Used when the purpose of the control chart is to make lot sentencing decisions

Next Steps Determine trial control limits Center line (process average) Compute UCL, LCL Analyze and interpret results Determine if in control Eliminate out-of-control points Re-compute control limits as necessary

Warning Limits on Control Charts Some suggest using two sets of limits on control charts: Action limits Set at 3-sigma When a point plots outside of this limit, a search for an assignable cause is made and any necessary corrective action is taken Warning limits Set at 2-sigma When one or more points fall in between the warning and action limits or very close to the warning limit, be suspicious

Typical Out-of-Control Patterns Point outside control limits Hugging the center line Hugging the control limits Instability Sudden shift in process average Cycles Trends

Shift in Process Average

Identifying Potential Shifts

Cycles

Trend

Western Electric Sensitizing Rules: One point plots outside the 3-sigma control limits Two of three consecutive points plot outside the 2-sigma warning limits Four of five consecutive points plot beyond the 1-sigma limits A run of eight consecutive points plot on one side of the center line Western Electric Handbook, 1956

Additional sensitizing rules: Six points in a row are steadily increasing or decreasing Fifteen points in a row with 1-sigma limits (both above and below the center line) Fourteen points in a row alternating up and down Eight points in a row in both sides of the center line with none within the 1-sigma limits An unusual or nonrandom pattern in the data One of more points near a warning or control limit

Classroom Exercise In small groups, choose two of the sensitizing rules. For each of your two rules, make up a (reasonable) situation where that rule would catch a problem and a (reasonable) situation where that rule might falsely identify a problem.

Final Steps Use as a problem-solving tool Compute process capability Continue to collect and plot data Take corrective action when necessary Compute process capability

Capability vs. Stability A process is capable if individual products consistently meet specification A process is stable only if common variation is present in the process

Process Capability Calculations PCR=(USL-LSL)/(6*sigma) where sigma=R-bar/d2

Process Capability The ability of the process to perform at the required level for the given quality characteristic Two of the methods for determining: Find the probability that the process will produce a part below the LSL plus the probability that the process will produce a part above the USL Process Capability Ratio PCR = CP = (USL – LSL) / 6σ

Process Capability Ratio Note There are many ways we can estimate the capability of our process If σ is unknown, we can replace it with one of the following estimates: The sample standard deviation S R-bar / d2

PCR and One Sided Specifications Upper specification only CPU = (USL – μ) / 3σ Lower specification only CPL = (μ – LSL) / 3σ We can use the same estimate for σ

PCR and an Off-Center Process CPK = min (CPU, CPL) Generally, if CP = CPK, then the process is centered at the midpoint of the specifications If CP ≠ CPK, then the process is off-center PCR=potential capability PCR(k) = actual capability

Commonly Used Control Charts Variables data x-bar and R-charts x-bar and s-charts Charts for individuals (x-charts) Attribute data For “defectives” (p-chart, np-chart) For “defects” (c-chart, u-chart)

Control Charts         x x  n We assume that the underlying distribution is normal with some mean  and some constant but unknown standard deviation . Let n x x   i n i  1

Distribution of x s s = n x  N (  ,  ) Recall that x is a function of random variables, so it also is a random variable with its own distribution. By the central limit theorem, we know that where, x  N (  ,  ) x s s = x x n

Control Charts x        x x  x

Control Charts  3  UCL & LCL Set at x  x LCL UCL & LCL Set at Problem: How do we estimate  &  ?  3  x

Control Charts m å x i x = i = 1 ® m m m å R R = i = 1 = f ( s ) ® s m

å Control Charts å x x = ® m m R R = = f ( s ) ® s m UCL = D R LCL = D i x = i = 1 ® m m m å R R = i = 1 = f ( s ) ® s m UCL = x + A R UCL = D R x 2 R 4 LCL = x - A R LCL = D R x 2 R 3

Example Suppose specialized o-rings are to be manufactured at .5 inches. Too big and they won’t provide the necessary seal. Too little and they won’t fit on the shaft. Twenty samples of 2 rings each are taken. Results follow.

X-Bar Control Charts X-bar charts can identify special causes of variation, but they are only useful if the process is stable (common cause variation).

Control Limits for Range UCL = D4R = 3.268*.002 = .0065 LCL = D3 R = 0

Special Variables Control Charts x-bar and s charts x-chart for individuals

X-bar and S charts Allows us to estimate the process standard deviation directly instead of indirectly through the use of the range R S chart limits: UCL = B6σ = B4*S-bar Center Line = c4σ = S-bar LCL = B5σ = B3*S-bar X-bar chart limits UCL = X-doublebar +A3S-bar Center line = X-doublebar LCL = X-doublebar -A3S-bar

X-chart for individuals UCL = x-bar + 3*(MR-bar/d2) Center line = x-bar LCL = x-bar - 3*(MR-bar/d2)

Next Class Homework Topic Preparation Ch. 11 (12) Disc. Questions 5, 6, 7 Ch. 11 (12) Problems 5 Topic Control Charts, Part II Preparation Chapter 12 (13)