STATISTICAL PROCESS CONTROL: CONTROL CHARTS for VARIABLES Control charts are the tools that are used in SPC to indicate when special-cause variation is present in a process Control chart means of visualizing variations that occur in the central tendency and dispersion of a set of observations Variables control charts use actual measurement for charting To create a control chart, samples, arranged into subgroups, are taken during the process The centerline (CL) of this chart shows where the process average is centered, the tendency of the data The upper control limit (UCL) and lower control limit (LCL), calculated based on ± 3 sigma, describe the spread of the process Once the chart is constructed, it presents the user with a picture of what the process is currently capable of producing Since control charts show changes in the process measurements, they allow for early detection of process changes

CONTROL CHARTS for VARIABLES

CONTROL CHARTS for VARIABLES In fact, the variation concept is a law of nature in that no two natural items in any category are the same, the variation exists Even if variation small and appears same; precision instruments show differences Ability to measure variation necessary before it can be controlled Basically 3 categories of variation in piece part production: 1. Within piece – illustrated by the surface roughness of a piece wherein one portion of the surface is rougher than another portion 2. Piece to piece – light intensity of four consecutive light bulbs produced from a machine will be different (e.g. dimensions) 3. Time to time – illustrated by the difference in product produced at different times of the day (different outcomes e.g. tool wear) Source of variation (combination of 4M and etc.) Equipment (tool wear, machine vibration, workholding-device) Material (tensile strength, moisture content, ductility, porosity) Environment ( temperature, light, humidity, radiation) Operator (method, SOP followed, motivation level, training) Inspection (faulty inspection equipment, incorrect application)

CONTROL CHARTS for VARIABLES Objective of Variable Control Chart For quality improvement – indicate that there is a quality control program is missing the point To determine process capability – the true process capability can be achieved only after substantial quality improvement has been achieved For decisions in product specifications – once the true process capability is obtained, effective specifications can be determined Make decisions on recently produced items – release next process, customer or other disposition method, sorting, rework, reject

CONTROL CHARTS for VARIABLES The types of variables control charts include the following: 1. Average and range charts (X-Bar and R) - consists of two separate charts on the same sheet of chart paper - one graph tracks the sample mean X-Bar and the other tracks the sample range R - the sample size must be the same for all samples and usually consists of three to seven pieces - the mean and range for each sample are calculated, recorded, and charted - the charts is analyzed as it develops for indications of special- cause variation, and after about 25 samples - it is analyzed again to determine the location, spread, and shape of the distribution of measurements - it is most effective when the sample size is less than 10, range (R) chart is simple to construct

CONTROL CHARTS for VARIABLES 2. Average and standard deviation charts (X-Bar and S) - the X-Bar and S chart tracks X-Bar on one part of the chart and the sample standard deviation, S, on the other - the X-Bar and S chart does everything that the X-Bar and R chart does and is used in the same way - Although it is preferred to the R chart because the standard deviation is a better measure of variation than the range - X-Bar and S charts are not used as much as the other variables charts because the standard deviation is more difficult to calculate than the range - the X-Bar and S chart will be used more extensively in the future and may eventually become the standard control chart in use - when the sample size exceeds 10, the ranges does not truly represent the variation present in the process, the standard deviation S chart is used

CONTROL CHARTS for VARIABLES 3. Median and range charts ( and R) - exactly like the X-Bar and R chart, with one exception - the median is calculated and charted instead of the mean - the and R chart analysis also shows indications of special-cause variation and the location, spread, and shape of the distribution of measurements - some companies prefer the and R chart because the calculation of the median is easier and faster than the calculation of the mean - one reason that the X-Bar and R chart is usually preferred over the and R chart is that the X-Bar and R chart is more sensitive to variation analysis

CONTROL CHARTS for VARIABLES 4. Individual and moving range charts (X and MR) - this chart tracks consecutive individual measurement, X, on one chart and the piece-to-piece variability, MR, on the other - the X and MR chart is less reliable than the X-Bar and R chart because it is based on fewer data values - the X and MR chart is generally used when just a small number of data are available, such as for small or short production runs or for batch processes 5. Run charts - simplest measurement-type control chart - it track individual measurements against a target measurement - there is no measure of variation, and the run chart is very susceptible to over adjustment - the run chart is often misused as a control chart with samples of size one by periodically checking the measurement of just one piece

CONTROL CHARTS for VARIABLES Procedure to establish X-Bar (average) and R (Range) chart: 1 Select quality characteristics - measurable data i.e. numbers - 7 basic units – length, mass, time, temperature, substance, etc. - affecting performance and function of product - Pareto analysis – highest % rejects, high production costs, etc. - impossible to put X-Bar & R on all characteristics selective OR treat as attributes chart 2. Choose rational subgroup - rational subgroup which have variation within the group due only to chance causes - two ways selecting subgroup samples 1. Select subgroup samples at one instant of time or as close as possible 2. Select period of time product produced - lots must be homogeneous: same machine, same operator

CONTROL CHARTS for VARIABLES - decision on size of sample empirical judgment & relates to costs choose n = 4 or 5 use R chart when n ≥ 10 use S chart - frequency of taking subgroup often enough to detect process changes - guideline of sample sizes/frequency using - say, 3000 parts/day, then 50 total inspections are suggested - if n = 4, suggested 19 subgroups would be a good starting point

CONTROL CHARTS for VARIABLES 3. Collect data - use form or standard check sheet - collect a minimum of 25 subgroups - does not matter Vertical or Horizontal (sample & subgroup)

CONTROL CHARTS for VARIABLES 4. Determine trial control limits - central line X-Bar and R-Bar - where A2, D4, D3 are factors – vary according to different n

CONTROL CHARTS for VARIABLES 5. Establish revised control limits - first plot preliminary data collected using control limits & center lines establish in step 4 - next step adopt standard values. If good control i.e. no out-of- control points - if there are points out-of-control discard from data Look at record – show an assignable cause – don’t use

CONTROL CHARTS for VARIABLES - limits for both charts become narrower after revised - revised limits used to report/plot future sub-group - for effective use – chart must be displayed and easily seen

CONTROL CHARTS for VARIABLES Final comments 1. Many analyst eliminate this revised step – but actually more representative of process 2. Formula mathematically same; 3. Initial estimate of process capability 6 True Cp is next 4. If use specification; nominal (target) value = . Range doesn’t change 5. Adjustments made to processes while taking data – not necessarily running defectives while collecting data 6. Process determines center line and the control limits, not design or manufacturing 7. When population values known easily obtained limits

CONTROL CHARTS for VARIABLES 6. Achieving objective Initiate control charts results in quality improvement - less variation in sub-group averages - reduction in variation of range Reduce frequency of inspection monitoring purpose – even once/month

CONTROL CHARTS for VARIABLES

CONTROL CHARTS SELECTION GUIDELINE

CHARTS AND TABLES FORMULA AND CONSTANT FOR CONTROL CHARTS

CHARTS AND TABLES FORMULA AND CONSTANT FOR CONTROL CHARTS

CONTROL CHARTS for VARIABLES STATE OF CONTROL When assignable causes eliminated and points plotted are new within CL process state of control Further improvement through changing basic process, system What are the characteristics of process in control? (natural pattern variation) - 34% within - 13% between - 2.5% of plotted points, - points located back & forth across center line random way - no points out of control - subgroup averages forms frequency dist which is normal distribution Control limits – establish at from center line Choice of is economic decision with respect to 2 types of error

CONTROL CHARTS for VARIABLES Process in control Individual parts will be more uniform – less variation and fewer rejects Cost of inspection will decrease Process capability easily attained Trouble can be anticipated before it occurs Percentage of parts fall between two values can be predicted with highest degree of accuracy, e.g. filling machines X-Bar & R charts can be used as statistical evidence for process control Predictable and stable process only chance causes present

CONTROL CHARTS for VARIABLES

CONTROL CHARTS for VARIABLES Process out of control A point falls outside control limits - assignable cause present - process producing subgroup average not from stable process - must be investigated, corrected - frequency distribution of

CONTROL CHARTS for VARIABLES Unnatural runs of variation even within limits

CONTROL CHARTS for VARIABLES Analysis for out-of-control 1. Freaks - is a single point that is beyond a control limit - it signifies that something changed dramatically in the process for a short time or that a mistake was made - out-of-control points marked with X’s Causes - sudden change material, mistake measurement, error in recording, omitted operation, damage in handling, etc.

CONTROL CHARTS for VARIABLES 2. Shifts - sets of seven or more consecutive points that are all on one side of the center line - something was introduced to the process that changed the whole process - usually temporary Causes - poorly trained operator, maintenance problem, change material, change method, new setup, change machine, etc.

CONTROL CHARTS for VARIABLES 3. Runs and Trends - points that are steadily climbing or steadily falling is called run - another pattern that is closely associated with a run is a trend - which 7 consecutive points are climbing or 7 points are steadily falling is classified as a run - trends are usually more gradual with more fluctuations and often indicate a slower process change Causes - material variability, loosening fixtures, operator fatigue, machine wear, better training & maintenance, etc.

CONTROL CHARTS for VARIABLES 4. Cycles - pattern that repeat on a regular basis - concentration on the factors that change the process periodically Causes - operator fatigue, shift changes, periodic speed changes, power fluctuations, reliance on different suppliers, etc.

CONTROL CHARTS for VARIABLES 5. Grouping - this is another case in which one trouble classification may be embedded in another - grouping, or bunching, occurs when the points on a chart occur in clusters Causes - inconsistent materials & method, several upstream problems, shifting fixtures, differences in work quality, etc.

CONTROL CHARTS for VARIABLES 6. Instability - an erratic pattern that has large fluctuations on a control chart is classified as instability, or unstable mixture - more than one-third of the points lie outside the center ± band or more than 4% of the points fall in or beyond the outer bands - the chart has a steep, zigzag pattern Causes - frequent breakdowns and startups, overadjustment, poor sampling procedures, inconsistent materials, faulty gauge, etc.

CONTROL CHARTS for VARIABLES 7. Stable mixtures - a mixture pattern that has erratic ups and downs similar to the instability pattern but has very few points in the middle of the chart - usually indicates a mixing of two different stable distribution: one for the upper set of points and one for the lower set - five or more consecutive points outside the C zones or either side of the center line signal a stable mixture pattern Causes - different (suppliers, inspectors, operators, lines, gauges, etc.)

CONTROL CHARTS for VARIABLES 8. Stratification - the points hug the averages line on a control chart - this pattern is 14 or more consecutive points in the two C zones, within ± of the average value Causes - falsification of data, gauge scale too crude for the job, nonrandom, representative sampling, etc.

CONTROL CHARTS for VARIABLES SPECIFICATION LIMIT & PROCESS LIMIT Look at individual values and average values of x’s Individual x’s n = 84 - can be considered as population Average's n = 21 - sample taken and = same (in this case) Normally distributed individual x’s and average values having same mean, only the spread is different Relationship = population Standard Deviation of averages if n = 5 = 0.45 = population Standard Deviation of individual x’s SPREAD OF AVERAGES IS HALF OF SPREAD FOR INDIVIDUAL VALUES

CONTROL CHARTS for VARIABLES Assume Normal Distribution ‘Estimate’ population standard deviation

CONTROL CHARTS for VARIABLES Central Limit Theorem If the population from which samples are taken is NOT normal, the distribution of SAMPLE AVERAGES will tend toward normality provided that sample size, n, is at least 4 Tendency gets better as n And standardized normal for distribution of averages, Central Limit Theorem is one reason control chart works No need to worry about distribution of x’s is not normal, i.e. individual values

CONTROL CHARTS for VARIABLES Control Limits & Specification Control limit – limits for averages – established as a function of averages Specifications – allowable variation in size for individual values - estimate by design engineers Control limits, Process spread, Distribution of averages & distribution of individual are interdependent – determine by the process Control charts CANNOT determine process meets specification

CONTROL CHARTS for VARIABLES Process Capability & Tolerance When specification establish without knowing process capable of meeting it or not serious situations result Process capable or not – actually looking at process spread, which is called process capability ( ) Define specification limit as tolerance (T) i.e. T = USL – LSL 3 types of situation can result

CONTROL CHARTS for VARIABLES

CONTROL CHARTS for VARIABLES Process Capability Procedure (s – method) 1. Take subgroup size 4 for 20 subgroups 2. Calculate sample standard deviation, s, for each subgroup 3. Calculate average sample standard deviation 4. Calculate estimate population standard deviation 5. Calculate Process Capability = Procedure (R – method) 1. Same as no. 1 above 2. Calculate R for each subgroup 3. Calculate average Range, 4. Calculate 5. Calculate

CONTROL CHARTS for VARIABLES Process Capability ( ) and Tolerance USUALLY Cp = 1.33 (defect standard) MEASURE OF PROCESS PERFORMANCE Shortfall – measure not in terms of nominal or target value use Cpk

CONTROL CHARTS for VARIABLES

CONTROL CHARTS for VARIABLES Determine Cp = Cpk for a process with average 6.45, = 0.030 having USL = 6.50, LSL = 6.30 Process NOT capable since not centered. Cp > 1 doesn’t mean capable. Have to check Cpk

CONTROL CHARTS for VARIABLES Comment on Cp & Cpk 1. Cp does not change when process center (average) changes 2. Cp = Cpk when process is centered 3. Cpk ≤ Cp always this situation 4. Cpk = 1.00 de facto standard 5. Cpk ≤ 1.00 process producing rejects 6. Cp < 1.00 process not capable 7. Cpk = 0 process center is at one of specification limit (U or L) 8. Cpk < 0 i.e. – ve value, average outside of limits

CONTROL CHARTS for VARIABLES