Control Chart for Variables

Outline Concept of variation The Control Chart Techniques State of Introduction Control Specifications Process Capability 2

variation The variation concept is a law of nature in that no two natural items in any category are the same.

Processes (กระบวนการ) variation Production process ในการผลิตชิ้นงานจำนวนมากค่าข้อมูลที่ได้จากผลิตภัณฑ์แต่ละค่าเช่น น้ำหนัก ความยืดหยุ่น ความแข็ง เป็นต้น จะมีรูปแบบการกระจายของค่าข้อมูลที่มีแน้วโน้มลู่เข้าสู่ค่าที่ควรจะเป็นค่าหนึ่ง () โดยธรรมชาตินั้นค่าข้อมูลที่ได้จากผลิตภัณฑ์แต่ละค่าอาจไม่เท่ากัน เป็นอิสระต่อกันและเบี่ยงเบนไปจากค่า  การเบี่ยงเบนนี้มีผลมาจาก สาเหตุสำคัญ 2 ชนิดคือ สาเหตุจากธรรมชาติ (Chance cause หรือCommon cause) และสาเหตุที่ระบุได้ หรือ สาเหตุที่เกิดจากความผิดผลาด (Assignable cause) Man, Machine, Method Processes (กระบวนการ) Products Hardware Service Raw Materials Variation of products Assignable cause ε Chance cause

variation The variation may be quite large and easily noticeable The variation may be very small. It may appear that items are identical; however, precision instruments will show difference The ability to measure variation is necessary before it can be controlled

variation There are three categories of variation in piece part production: Within-piece variation: e.g. Surface roughness Piece-to-piece variation: e.g. Difference in diameters among pieces produced at the same time Time-to-time variation (variation between groups/lots): e.g. Difference in product produced at different times of the day

variation Sources of Variation in production processes: Men + Materials + Methods + Machines + Environment Measurement Instruments Operators Methods Materials INPUTS PROCESS OUTPUTS Tools Human Inspection Performance Machines Environment

Variation causes Equipment: Material Sources of variation are: Toolwear Machine vibration Electrical fluctuations etc. Material Tensile strength Ductility Thickness Porosity etc. From your opinion; what are the causes of variation due to equipment? From your opinion; what are the causes of variation due to Materials?

What do you think are the causes of variation due to operators? Variation causes Sources of variation are: Environment Temperature Light Radiation Humidity etc. Operator Personal problem Physical problem etc. From your opinion; what are the causes of variation due to environment? What do you think are the causes of variation due to operators?

Variation cause There is also a reported variation which is due to the inspection activity. Variation due to inspection should account for one tenth of the four other sources of variation.

Variation causes Variation may be due to chance causes (random/common causes) or assignable causes. When only chance causes are present, then the process is said to be in a state of statistical control. The process is stable and predictable. (กระบวนการผลิตจะถือว่าอยู่ในสภาวะคงที่หรือมีเสถียรภาพ และสามารถทำนายได้ ก็ต่อเมื่อการเบี่ยงเบนหรือความแปรปรวนที่เกิดขึ้นในการผลิตมาจากสาเหตุธรรมชาติ (Chance cause) เท่านั้น หรืออาจกล่าวได้ว่าถ้ามีการเบี่ยงเบนของกระบวนการที่เกิดจากสาเหตุจากความผิดผลาด (Assignable cause) เราถือว่ากระบวนการยังไม่อยู่ในสภาวะคงที่ ในการควบคุมคุณภาพ (Control Chart) เราจะใช้แผนภูมิควบคุมเพื่อตรวจสอบสภาวะของกระบวนการผลิตว่าอยู่ภายใต้การความคุม และมีเสถียรภาพหรือไม่)

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

Control Charts Continuous Numerical Data Categorical or Discrete Numerical Data Control Charts Variables Attributes Charts Charts This slide simply introduces the various types of control charts. R X P C Chart Chart Chart Chart

Control Charts for Variables The control chart for variables is a means of visualizing the variations that occur in the central tendency and the mean of a set of observations. It shows whether or not a process is in a stable state.

Run Chart

Control Charts for Variables

Control Charts for Variables Example of a method of reporting inspection results

Variable Control Charts The objectives of the variable control charts are: For quality improvement To determine the process capability For decisions regarding product specifications For current decisions on the production process For current decisions on recently produced items

Control Chart Techniques Procedure for establishing a pair of control charts for the average Xbar and the range R: Select the quality characteristic (เลือกข้อมูลที่ได้จากการวัดที่ใช้เป็นตัวกำหนดคุณภาพของชิ้นงาน เช่น ความยาว ความกว้าง เส้นผ่าศูนย์กลาง ความแข็ง เป็นต้น) Choose the rational subgroup (เลือกกลุ่มตัวอย่างที่เหมาะสม เช่น เลือกความถี่การเก็บข้อมูล และจำนวนข้อมูลของกลุ่มตัวอย่าง ที่เหมาะสม) Collect the data Determine the trial center line and control limits (หาค่ากลางและค่าขอบเขตครั้งแรก) Establish the revised central line and control limits (ปรับค่า ค่ากลางและค่าขอบเขตที่ได้จากการคำนวนครั้งแรกให้เหมาะสม ซึ่งทำได้โดยการกำจัดข้อมูลที่เกิดจาก assignable cause) Achieve the objective

Rational Subgroup There are two schemes for selecting the subgroup samples: The instant-time method: Select subgroup samples from product or service produced at one instant of time or as close to that instant as possible The period-of-time method: Select from product or service produced over a period of time that is representative of all the products or services sampling 1 hr 2 hr 2 hr sampling 1 hr 2 hr 2 hr

Rational Subgroup The first scheme will have a minimum variation within a subgroup. The second scheme will have a minimum variation among subgroups. The first scheme is the most commonly used since it provides a particular time reference for determining assignable causes. The second scheme provides better overall results and will provide a more accurate picture of the quality.

Subgroup Size As the subgroup size increases, the control limits become closer to the central value, which make the control chart more sensitive to small variations in the process average As the subgroup size increases, the inspection cost per subgroup increases When destructive testing is used and the item is expensive, a small subgroup size is required

Subgroup Size From a statistical basis a distribution of subgroup averages are nearly normal for groups of 4 or more even when samples are taken from a non-normal distribution When a subgroup size of 10 or more is used, the s chart should be used instead of the R chart. The guide for the amount of sampling (or sample size) is show in the next slide ( subgroup size และ sample size ไม่ใช่สิ่งเดียวกัน subgroup size หมายถึงจำนวนตัวอย่างใน subgroup ส่วน sample size หมายถึงจำนวนของของชิ้นงานทั้งหมดที่ถูกสุ่มเก็บข้อมูล)

Subgroup Size Standard: ANSI/ASQ Z1.9-1993 Lot size Sample size 91-150 10 151-280 15 281-400 20 401-500 25 501-1,200 35 1,201-3,200 50 3,201-10,000 75 10,001-35,000 100 35,001-150,000 150 If a process produce 4000 pieces per day, then 75 total inspections (sample size) are suggested. Therefore, if a subgroup size is 4, the number of subgroup is 19 (75/4 ~ 19)

Determine the central line and control limit X-bar Chart A chart that tracks the changes in the means of the samples by plotting the means that were taken from a process. g X j i å = 1 Total number of samples Sample number The average of the means of the samples = g i X

Determine the central line and control limit R Chart A chart that tracks the change in the variability by plotting the range within each sample. The range is the difference between the lowest and highest values in that sample. g R i å = 1 Total number of samples Difference between the highest and lowest values in sample i Average of the measurement differences R for all samples = g Ri R

Determine the central line and control limit UCLx = x + 3x LCLx = x - 3x = UCLR = x + 3R LCLR = x - 3R = UCLx = x + A2R LCLx = x - A2R = UCLR = D4R LCLR = D3R

Note: All factors are based on the normal distribution. Source: E. L. Grant, Statistical Quality Control, 6th ed. pp. 57–59 (Copyright © 1998 by The McGraw-Hill Companies, Inc. 1988). Reprinted by permission of McGraw-Hill, Inc.

Krajewaksi, operation management 7th, 2004 Control Charts for Variables West Allis Industries The next series of slides presents Example 5.1. The series builds in steps to the conclusion of the Example showing the development of key equations along the way. The first example is of variables charting using x-bar and R-charts. Continuous process Produce 400 pieces per day Quality characteristic is the diameter of bolts Krajewaksi, operation management 7th, 2004 1

Krajewaksi, operation management 7th, 2004 Control Charts for Variables Subgroup Sample Number 1 2 3 4 R x 1 0.5014 0.5022 0.5009 0.5027 2 0.5021 0.5041 0.5024 0.5020 3 0.5018 0.5026 0.5035 0.5023 4 0.5008 0.5034 0.5024 0.5015 5 0.5041 0.5056 0.5034 0.5039 Special Metal Screw _ This slide and the next six show in a step-by-step fashion how the raw data is used to derive ranges and means for each observation. Example 5.1 Krajewaksi, operation management 7th, 2004 4

Range Chart - Special Metal Screw This slide shows the blank control chart for the R-chart with the mean and control limit values already plotted. This is a screen shot from the OM Explorer software. The process Range is comfortably in control. (The picky person might suggest there is not enough variability in the process, but with so few observations it is difficult to draw such conclusions.) Krajewaksi, operation management 7th, 2004 20

Krajewaksi, operation management 7th, 2004 Control Charts for Variables Table 5.1 Control Chart Factors Factor for UCL Factor for Factor Size of and LCL for LCL for UCL for Sample x-Charts R-Charts R-Charts (n) (A2) (D3) (D4) 2 1.880 0 3.267 3 1.023 0 2.575 4 0.729 0 2.282 5 0.577 0 2.115 6 0.483 0 2.004 7 0.419 0.076 1.924 8 0.373 0.136 1.864 9 0.337 0.184 1.816 10 0.308 0.223 1.777 Control Charts—Special Metal Screw X-Charts R = 0.0021 x = 0.5027 = UCLx = x + A2R LCLx = x - A2R = We return to the calculations, this time for the x-bar control limits. Example 5.1 Krajewaksi, operation management 7th, 2004 22

Krajewaksi, operation management 7th, 2004 Control Charts for Variables Control Charts—Special Metal Screw x-Charts R = 0.0021 A2 = 0.729 x = 0.5027 = UCLx = x + A2R LCLx = x - A2R = The A2 value is now shown with the other required values for the calculations. Example 5.1 Krajewaksi, operation management 7th, 2004 24

Control chart (Ex 5-2)