Technical

Paper BWB31903

Application of Process Capability Indices to

Measure Performance of Manufacturing Process

Wee Huai Chen

Faculty Applied Science and Technology

University Tun Hussien Onn Malaysia

(UTHM)

*Corresponding Author : [email protected]

Abstract

A process capability of a manufacturing process can be evaluate through

process capability indices with the help of several statistical tools and

software. The most common process capability indices that being implement in

manufacturing industry is Cp, Cpk, Cpm and Cpmk. Different process capability

indices have different interpretation for the process to monitor and measure

the performance of the process. A set of manufacturing dataset from automotive

industry is used as a case study in this paper. Results is obtained to measure

the performance of the boring operation using process capability indices.

Keywords:

Process Capability, Process capability indices, process performance, control

chart

1

Introduction

Process capability study is a

method of combining the statistical tools developed from the normal curve and

control charts with good engineering judgment to interpret and analyze the data

representing a process. The purpose of the process capability study is to

determine the variation spread and to find the effect of time on both the

average and the spread. (Nagesh,

1800) A process capability index is a numerical summary that compares

the behaviour of a product or process characteristics to engineering

specifications. These measures are often called capability or performance

indices. (Steiner,

Abraham, & Mackay, n.d.) Statistical process control

(SPC) has been widely used as a tool to assist the process capability, improve

quality and hence reducing cost in manufacturing process such as turning,

milling and grinding.

The original

process capability index is Cp, which a measure of a potential capability of a

process to meets the specification. Kane considered the Cpk index in order to

show the influence of a shift of the process mean on the ability of the process

to produce products within the tolerance values.(“Kane.pdf,”

n.d.) The Cpm index suggested

by Chan et al. involves the variation of production items with respect to the

target value and the tolerance limits that are present in the factory. (Technology,

2015) Process capability indices, as a process performance measure, have

become very popular in assessing the capability of manufacturing processes in practice

during the past decade. For example, Rado (1989) demonstrated how imprimis

technology, Inc. used the process capability indices for program planning and

growth to enhance product development. The Cp and Cpk indices have been used in

Japan and in the US automotive industry such as Ford Motor Company. (“Kane.pdf,”

n.d.)

1.1

Usage of Process Capability

Indices

Index

Estimation Equation

Usage

Cp

It estimates what the

process is capable of producing if the process mean were to be centered

between the specification limits. Assumes process output is approximately

normally distributed

Cpl

It estimates process

capability for specifications that consist of a lower limit only. Assumes

process output is approximately normally distributed.

Cpu

It estimates process

capability for specifications that consist of an upper limit only. Assumes

process output is approximately normally distributed.

Cpk

It estimates what the

process is capable of producing, considering that the process mean may not be

centered between the specifications limits.

Cpm

It estimates process

capability around a target T is always greater than zero. It assumes process output is

approximately normally distributed. Cpm is also known as the Taguchi

capability index

Cpmk

It estimates process

capability around a target T and accounts for an off-center process mean.

Assumes process output is approximately normally distributed.

Table 1: Usage of process capability indices

2

Methods

A set of manufacturing data is used for the analysis in this case study.

The data provided is about the bore diameter on the driver gear processed

by boring operation in an automotive industry. Statistical software, MINITAB is

used to construct the statistical process chart, control chart and process

capability chart to measure the performance of the process.

Sample no.

1

2

3

4

5

R

1

205.03

205.02

205.01

205.045

205.01

205.023

0.035

2

205.01

205.02

205.025

205.03

205.01

205.019

0.02

3

205.01

205.03

205.05

205.03

205.02

205.028

0.04

4

205.03

205.02

205.03

205.04

205.035

205.031

0.02

5

205.04

205.035

205.03

205.03

205.035

205.034

0.01

6

205.03

205.03

205.025

205.03

205.035

205.03

0.01

7

205.025

205.025

205.025

205.025

205.025

205.025

0

8

205.015

205.02

205.025

205.01

205.02

205.018

0.015

9

205.025

205.03

205.04

205.01

205.01

205.023

0.03

10

205.025

205.025

205.02

205.01

205.02

205.02

0.015

11

205.01

205.005

205.03

205.04

205.04

205.025

0.035

12

205.03

205.02

205.03

205.03

205.03

205.028

0.01

13

205.03

205.04

205.03

205.04

205.03

205.034

0.01

14

205.03

205.04

205.03

205.03

205.025

205.031

0.015

15

205.01

205.01

205.02

205.04

205.05

205.026

0.04

16

205.035

205.04

205.037

205.042

205.04

205.038

0.007

17

205.045

205.038

205.045

205.033

205.03

205.038

0.015

18

205.04

205.03

205.025

205.025

205.02

205.028

0.02

19

205.03

205.025

205.03

205.03

205.035

205.03

0.01

20

205.04

205.035

205.03

205.03

205.02

205.031

0.02

Table 2: Dataset of Bore Diameter

3

Results and Discussion

3.1 Process Capability Analysis

Before estimating a process

capability for any operation in manufacturing process, a few assumptions must

be hold. According to Kotz and Montgomery (2000), the assumptions is stated

below:

1.

The process must be in

state of statistical control.

2.

The quality characteristic

has a normal distribution.

3.

In the case of two sided specifications, the

process mean is centered between the lower and upper specification limits.

4.

Observations must be

random and independent of each other.

3.2 Construction of and R chart

Before interpreting the process capability, we must ensure that the

process is in the state of statistical control. Hence, a control chart is

constructed to monitor the both the mean value of the quality characteristic

and its variability. Since the sample size of this data is small which is

smaller than 10, X-bar and R chart is plotted instead of X-bar and s chart

which is more suitable for larger sample size. R chart is used to determine

whether the process variation is in control before the X-bar chart being

interpret. If the R-chart is not in control, then the control limits on the

X-bar chart are consider as meaningless and not accurate.

Control limits for chart:

UCL = = 205.049 +

(0.577) (0.029) = 205.06560

LCL = = 205.049

– (0.577) (0.029) = 205.03219

Control limits for R chart:

UCL = = 2.114 x0.029

=0.06124

LCL = = 0.00 x 0.029 =0.0000

Figure 1: and R chart for case study data

Based on the figure above, all

points lie within the upper and lower control limits and there is no outlier

detected in both and R chart. This indicates that the boring

operation is in statistical control.

3.3 Estimation of Process

Capability Indices

3.3.1 Process Capability Index, Cp

= = 1.34

Cr = (1 / Cp) x 100 = (1/ 1.34) x

100 = 74.62 %

This indicate that the boring

operation uses 74.62 % of specification band.

3.3.2 Process Capability Index,

Cpk

= min () = min () = min (0.026,

2.66)

Therefore, Cpk

= 0.026

3.3.3 Process Capability Index,

Cpm

Cpm = = = 0.33

3.3.4 Process Capability Index,

Cpmk

Cpmk = = = 0.0063

Based on the results calculated above, Cp value is

1.34 which is slightly larger than 1.33, this means that the boring operation

is a fairly capable process. But the Cpk value is smaller than the Cp value,

this indicates that the process is off-centered. Although the process can be

considered as under statistical control stable over time, yet there has been 4,

64,626.00 rejected products out of 1 million products due to the shift of the

process mean towards upper specification limit as shown in figure 1. In order to reduce the

scrap, it is necessary to shift the process mean as close as possible to the target

value (i.e., 205.00 mm).

3.4 Construction of and R chart after adjusting the process mean

Control limits for chart:

UCL = = 205.001 +

(0.577) (0.033) =205.020

LCL = = 205.001

– (0.577) (0.033) =204.982

Control limits for R chart:

UCL = = 2.114 x0.033

=0.070

LCL = = 0.00 x 0.033 =0.0000

Figure 2: and R chart for case study data after

adjusting the process mean

The figure above depicts that all

the data are plotted within the upper and lower specification limits and there

is no outlier present in chart as well as R chart. This shows that the

boring operation is in statistical control and the process is stable over time.

3.3 Estimation of Process

Capability Indices

3.3.1 Process Capability Index, Cp

= = 1.182

Cr = (1 / Cp) x 100 = (1/ 1.182) x

100 = 84.60 %

This indicate that the boring

operation uses 84.60 % of specification band.

3.3.2 Process Capability Index,

Cpk

= min () = min () = min

(1.158,1.205)

Therefore, Cpk

= 1.158

3.3.3 Process Capability Index,

Cpm

Cpm = = = 1.18

3.3.4 Process Capability Index,

Cpmk

Cpmk = = = 1.155

Figure 3: Process Capability Analysis after Adjusting

the Process Mean

The upper specification limit (USL) and lower

specification limit (LSL) is used to define customer requirements and to

evaluate whether the process produces items that meet the requirements. The

histogram bar is compared to the lines to assess whether the measurement are

within the specification limits. Based on the figure above, the histogram bar

is within the upper and lower specification limits which is 205.05mm and 204.95mm

respectively. Therefore, we can conclude that the process is under statistical

control and capable of meeting the specification limits.

According to the results we obtained from the

calculation, we can see that the value for process capability Cp is 1.182, this

indicate that the boring process is fairly capable process since it larger than

1.00 but yet smaller than 1.33. The process will continue to produce conforming

products as long as the process is under statistical control. On the other

hand, the Cpk value is 1.158 which is smaller than the Cp value, 1.182. This

means that the process is slightly off-centered but still within the

specification limits. In addition, figure above shows that rejections as

few as 149 gears as scrap and 255 gears as rework out of 1 million gears. By

identifying and solve the causes of variation, the scrap and rework of gears

can be reduce as well as achieve a better Cp and Cpk value which is 1.33.

4 Conclusion

Process capability indices Cp Cpk, Cpm and Cpmk is used to examined a

case study of boring operation in automotive industry. Before interpreting the

process capability, several assumptions were tested by using statistical tools.

With the help of those statistical tools and statistical software, results are

obtained as above. Process capability for the original process condition shows

that there will be a lot of scrap produced in the process as the process mean

approximately equal to one of the specification limit. An action is taken to

reduce the scrap produced which is to shift the process mean as close as

possible to the target value. After adjusting the process mean, the process

seems to be under statistical control and capable of meeting the

specification limits although the value of Cp and Cpk are not satisfying

enough. The causes of variation should be examined in order to improve the

current process.

ACKNOLEDGEMENT

This acknowledgement is dedicated to my lecturer for Statistical for

Quality Improvement subject, Dr Shuhaida binti Ismail. I am thankful for her

patient and effort in teaching our class with full dedication. Not to mention

that Dr Shuhaida has provided us with those interesting topics for the

technical writing. I also would like to express my gratitude toward individuals

(reference) for making useful discussion related to the research topics.

References

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2.

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4.

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