Cusum Charts

CuSum Chart AMT 2506 Shanmuga Raja .B (BVB0912004)

Module leader

: Mr. Sandeep .N M.S.Ramaiah School of Advanced Studies

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Introduction Cumulative sum, simply known as CUSUM chart is a sequential analysis technique. Implies the calculation of cumulative sum of deviation of each sample through which, • Change detection is monitored with high sensitivity.

• Efficient in detecting small shifts in the mean of process. • Stopping rule in case of out of control process. • Use all historic information unlike Shewart chart. • Can be used for variables as well as attribute data • Immediate action can be announced based on trend pattern. M.S.Ramaiah School of Advanced Studies

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Why CuSum? CuSum charts are applied because of two main reasons,  Sensitivity in small changes in mean. X-bar and Range or X-bar and S control chart with large subgroup increase sensitivity. But, Individual data chart doesn’t offer required sensitivity compared to CuSum.  Risk mitigation. Shewart charts depicts out of control process in +/-3 sigma levels with sufficient data. CuSum can propose control within 1-1.5 sigma with less data.

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Application CuSum charts are used as a management technique rather than shop floor chart because the change significance is shown through slopes and their interpretation require care. Used as a tool for, • Forecasting (actual v/s forecasted sales) • Employee engagement (recruitment, absenteeism) • Production levels (detection of process changes) • Plant breakdowns (maintenance performance) Widely used in continuous process like chemical processing, medical services to check the implementation level and bring them into control immediately. M.S.Ramaiah School of Advanced Studies

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Differences Individual chart showing the process within control based on runs.

Change in process control detected with same data using CUSUM.

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CuSum construction Lets have an example showing the downtime hours for 52 weeks tabulated as below Week

Sample

Week

Sample

Week

Sample

Week

Sample

Week

Sample

Week

Sample

1

24

11

24

21

27

31

21

41

27

51

20

2

27

12

27

22

22

32

19

42

20

52

19

3

22

13

24

23

24

33

20

43

21

4

35

14

23

24

27

34

28

44

25

5

27

15

25

25

28

35

20

45

19

6

23

16

27

26

23

36

23

46

22

7

27

17

29

27

26

37

25

47

24

8

28

18

28

28

31

38

21

48

19

9

25

19

24

29

29

39

18

49

25

10

20

20

29

30

25

40

21

50

28

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CuSum construction • Calculate the average downtime

• Select the target value, T Two things can be considered, which are selecting the mean as target or value near to mean whichever is appropriate. (T= 24.2) • Calculate the difference between individual and target x-T x 24 x - T -0.2

27 2.8

22 35 27 -2.2 10.8 2.8

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23 -1.2 7

CuSum construction • Calculate the CUSUM score (cumulative value from previous data),

x x-T ∑x - T

24 -0.2 -0.2

27 2.8 2.5

22 -2.2 0.3

35 10.8 11

27 2.8 13.8

23 -1.2 12.6

• Find the maximum and minimum CUSUM score to define the limits of scale. Min -0.2

Max 52.2

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CuSum construction • Determine scaling, unless a suitable convention is used the interpretation becomes difficult as shown below,

• The thumb rule used is to measure one observation in horizontal into 2 times the standard deviation in vertical axis. S= R / D2* 2s = 3.18 / 1.129 = 3.38 2s= 6.76 rounding off to 5 1x * Constant based on sample size.

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CuSum construction • Plot the CUSUM chart.

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CuSum Interpretation • Interpretation intent is on the slope and changes in the slope.

• The changes in slope can be considered as,

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CuSum Interpretation

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CuSum Control limits The control limits are different from what is used in Shewart charts. Control specification can be defined by, • V-Mask. • Decision interval. V-Mask: Most popular method in CUSUM to find, out of control plots. Week 29

In-control

Out-of-control M.S.Ramaiah School of Advanced Studies

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CuSum Control limits

F

• • • • • •

10 Samples

Draw horizontal datum line. Draw CB and DC with 5 standard deviation apart. Take a 10 observation distance from C to F. Draw vertical line AF with 10 standard deviation. Connect the line AB Repeat on other side M.S.Ramaiah School of Advanced Studies

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CuSum Control limits Decision Interval:

• Line AB is drawn from target to week 29. • Draw AC with 5 standard deviation. • At 10 observation from A draw line DE with 10 Standard error. • Join CE until it cuts the curve. If happens there is a change. M.S.Ramaiah School of Advanced Studies

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CuSum Limitations • CUSUM is relatively slower to respond in larger shifts. • Special patterns are hard to interpret and analyze. • Setting up a CUSUM chart is bit time consuming and laborious.

• The construction scheme changes for handling variable and attribute data. • For accurate results the CUSUM design should be carefully considered.

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References Steven Wachs (2010) “What is a CUSUM Chart and When Should I Use One?”, [online] available from [09 July 2013] Unknown

(2004) “Quality control methods”, [online] available from [10 July 2013]

Unknown (2005) “An introduction to cusum (cumulative sum) charts”, [online] available from [10 July 2013] John S. Oakland (2003) “Statistical process control”, 5th edn, Burlington MA: ButterworthHienemann. Wikipedia (2003) “Cusum”, [online] available from [10 July 2013]

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