Construction of control charts based on six sigma initiatives for the number of defectives with vary

International Research Journal of Engineering and Technology (IRJET) Volume: 03 Issue: 09 | Sep -2016

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Construction of control charts based on six sigma initiatives for the number of defectives with varying sample size Radhakrishnan. R1 and Balamurugan. P2 1Advisor,

Maharaja Group of Institutions, Perundurai, Tamilnadu

2Department

of Statistics, Government Arts College, Salem,

Tamilnadu – 636 007. pbalamuruganstat@gmail.com

Abstract – A control chart is a statistical device used for

realized that by measuring variations it will be possible

the study and control of repetitive process. W.A. Shewhart

to improve the working of the system. The procedure

(1931) of Bell Telephone Laboratories suggested control

was aimed at taking action to improve the overall

charts based on the 3 sigma limits. Now the companies in

performance. The companies, which are practicing Six

developed and developing countries started applying Six

Sigma, are expected to produce 3.4 or less number of

Sigma initiatives in their manufacturing process, which

defects per million opportunities. Radhakrishnan and

results in lesser number of defects. The companies

Sivakumaran (2008a, 2008b and 2008c) used the

practicing Six Sigma initiatives are expected to produce

concept of Six Sigma in the construction of sampling

3.4 or less number of defects per million opportunities, a

plans such as single, double and repetitive group

concept suggested by Motorola (1980). If the companies

sampling plans indexed through Six Sigma Quality Levels

practicing Six Sigma initiatives use the control limits

(SSQLs) with Poisson distribution as the base line

suggested by Shewhart (1931), then no point fall outside

distribution. Radhakrishnan (2009) suggested single

the control limits because of the improvement in the

sampling plan indexed through Six Sigma quality levels

quality of the process. In this paper an attempt is made to

(SSQLs) based on Intervened Random Effect Poisson

construct a control chart based on six sigma initiatives for

Distribution and Weighted Poisson Distribution as the

number of defectives with varying sample size specially

base line distributions. Radhakrishnan and Balamurugan

designed for the companies applying Six Sigma initiatives

(2010a and 2010b) constructed control charts based on

in their organization. Suitable Table – 4 is also constructed

six sigma initiatives for defects, mean, average fraction

and presented for the engineers to take quick decisions.

defectives, number of defectives, X bar using standard deviation, Exponentially Weighted Moving Average

Keywords: Control Chart, Process control, Six Sigma, Six

(EWMA), proportion defectives – number of defectives,

Sigma Quality Level.

Fraction

defectives,

Standard

deviation,

Standard

deviation with variable sample size, average number of nonconformities per multiple units, number of defects -

1. INTRODUCTION The concept of Six Sigma was introduced by Motorola (1980) by the engineer M.Harry who analyzed variations in outcomes of the company’s internal procedures and © 2016, IRJET

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average number of defects per unit and range. The control charts originated by W.A. Shewhart (1931) was based on 3 sigma control limits. If the same charts are used for the products of the companies which adopt six ISO 9001:2008 Certified Journal

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Volume: 03 Issue: 09 | Sep -2016

e-ISSN: 2395 -0056 p-ISSN: 2395-0072

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sigma initiatives in the process, then no point will fall outside the control limits because of the improvement in the quality. So a separate control chart is required to

3. CONDITIONS FOR APPLICATION

monitor the outcomes of the companies, which adopt six



sigma initiatives.

Number of units in a sample for different lots is further in advance.

In this paper an attempt is made to construct a



control chart based on six sigma initiatives for number of defectives with varying sample size. Suitable Table – 4 is also constructed and presented for the engineers to take

Human

involvement

should

be

less

in

the

manufacturing process 

The company adopts Six sigma quality initiatives in its processes

quick decisions.

4. CONSTRUCTION OF CONTROL CHART BASED ON 2. CONCEPTS AND TERMINOLOGIES

SIX SIGMA INITIATIVES FOR THE NUMBER OF

2.1 Upper specification limit (USL)

DEFECTIVES WITH VARYING SIZE SAMPLE

It is the greatest amount specified by the producer for

Fix the tolerance level (TL) and process capability

a process or product to have the acceptable performance.

(CP) to determine the process standard deviation (  6 ).

2.2 Lower specification limit (LSL)

Apply

It is the smallest amount specified by the producer for

the

value

of

 6

in

the

control

a process or product to have the acceptable performance.

limits c  4.831  6 

2.3 Tolerance level (TL)

on six sigma initiatives for the number of defectives with

It is the difference between USL and LSL,

varying sample size. The value of

TL = USL-LSL This is the ratio of tolerance level to six times

normal variate. For a specified TL and CP of the process, the value of  (termed as  6 ) is calculated from

standard deviation of the process.

TL USL  LSL  6 6

cp 

2.5 Subgroup size (k)

TL using a C program and presented in Table 4 6

for various combinations of TL and CP. The control limits

It is the number of units in a sample and ‘N’ is the number of samples.

based on six sigma initiatives for the number of defectives with varying size sample are

2.6 Quality Control Constants ( Q6 ) The constants

Q6 is obtained using

p(z  z6 )  1  1 , 1  3.4 x 106 and z is a standard

2.4 Process capability (Cp)

cp 

 , to get the control limits based  k

UCL 6

Q6 introduced in this paper to

determine the control limits based on six sigma initiatives for the number of defectives with varying

Central Line CL 6 LCL 6

   c  Q6   6   c

  k  

   c  Q6   6  

  k  

sample size.

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e-ISSN: 2395 -0056 p-ISSN: 2395-0072

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5. EXAMPLE

From the resulting Figure 1, it is clear that the process

The example provided by Acheson J. Duncan (1958)

is in control, since the entire subgroup numbers lie inside

is considered here. The following table gives the number

the control limits.

of missing rivets noted at aircraft final inspection:

5.2 Control limits based on six sigma initiatives

 c  21.98  2.62  k 8.4

for the number of defectives with varying

Weightage average (c) 

Square Yards Oilcloth Inspected

Inspection Lot No.

No. of defects per 100 square Yards 2.5 2.8 3.0 2.2 3.3 1.3

sample size Number of defects in Total Lot (c) 5 7 3 1.98 3.96 1.04 21.98

Number of units in a sample (k)

1 200 2.0 2 250 2.5 3 100 1.0 4 90 0.9 5 120 1.2 6 80 0.8 Total 8.4 Table 1: Missing rivets noted at aircraft

defectives with varying sample size The 3σ control limits suggested by Shewhart [22]

k

. The values are presented in Table 2.

Table 2: UCL and LCL values for 3 sigma control limits Inspection Lot number 1 2 3 4 5 6

k 2.0 2.5 1.0 0.9 1.2 0.8

© 2016, IRJET

3 c

k

3.43 3.07 4.86 5.12 4.43 5.43

UCL  c  3 c

k

LCL  c  3 c

6.05 5.69 7.48 7.74 7.05 8.05

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found from the Table-4 that the value of

 6 is 0.13. The

control limits for the number of defectives with varying sample size based on six sigma initiatives for a specified TL and

Q6 is c  4.831  6 

 . The UCL and LCL  k

values are presented in Table 3.

Table 3: UCL and LCL values for control chart based on six sigma Initiatives

5.1 Three Sigma Control limits for the number of

are c  3 c

For a given TL = 2 (USL-LSL =3.3-1.3) & CP = 2.5, it is

k

Inspection Lot number 1 2 3 4 5 6 From the

k

4.831

2.0 2.5 1.0 0.9 1.2 0.8 resulting

0.13 k

UCL 

LCL 

 4.831  c   0.13      k  

 4.831  c   0.13      k  

0.44 3.06 0.40 3.02 0.63 3.25 0.66 3.28 0.57 3.19 0.70 3.32 Figure 1, the lot number

2.18 2.22 1.99 1.96 2.05 1.92 5 goes

above the upper control limit and the lot number 6 goes

-

below the lower control limit. Therefore the process does not exhibit statistical control.

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Volume: 03 Issue: 09 | Sep -2016

e-ISSN: 2395 -0056 p-ISSN: 2395-0072

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Figure 1: Comparison of the process: 3 limits and control limits using six sigma initiatives 6. CONCLUSION

[3] Radhakrishnan, R., and Balamurugan, P., “Six sigma

In this paper, a procedure is given to construct a

based Control Chart for Fraction Defectives”, Journal of

control chart based on six sigma initiatives for the

Testing and Evaluation, Volume 39, Issue 4, Issue date

number of defectives with varying sample size with an

11th July, (accepted for publication), 2010a.

example. It is found that the process was in control even

[4]

when Six Sigma initiatives are adopted but it is very clear

“Construction of control charts based on six sigma

from the comparison that when the process is centered

Initiatives for the number of defects and average number

with reduced variation than the 3 sigma control limits,

of defects per unit”, Journal of Modern Applied Statistical

which indicate that the process is not in the level it was

Methods, (Accepted for publication), 2010b.

expected. So a correction in the process is very much

[5]

required to reduce the variations. The charts suggested

”Construction and Selection of Six sigma sampling plan

in this paper will be very useful for the companies

indexed through six sigma quality level”, International

practicing Six Sigma initiatives in their process. These

Journal of Statistics and Systems, 3(2), pp. 53-159,

charts will replace the existing Shewhart (1931) control

2008a.

charts in future when all the companies started

[6]

implementing Six Sigma Initiatives in their organization.

”Construction of Six Sigma Repetitive Group Sampling

Radhakrishnan,

Radhakrishnan,

Radhakrishnan,

Plan”,

International

R.,

R.,

R.,

and

and

Sivakumaran,

and

Journal

Balamurugan,

Sivakumaran, of

Mathematics

P.,

P.K.,

P.K., and

REFERENCES

Computation, 1(N08), pp. 75-83, 2008b.

[1] Acheson J. Duncan, “Quality Control and Industrial

[7]

Statistics”, Richard D. Irwin, Inc. Homewood, Illinois

”Construction and selection of conditional double

(sixth printing), pp. 280-282, 1958.

sampling plan indexed through six sigma quality levels”,

[2] Radhakrishnan, R., “ Construction of Six sigma based

Sri Lankan Journal of Applied Statistics, 9, pp. 74-83,

sampling plans”, a D.Sc. Thesis submitted to Bharathiar

2008c.

University, Coimbatore, India, 2009.

[8] Shewhart, W.A., 1931, “Economic Control of Quality of

Radhakrishnan,

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P.K.,

Manufactured Product”, New York: Van Nostrand, 1931. © 2016, IRJET

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