Technical Author : [email protected] Abstract A process capability

Technical
Paper BWB31903

Application of Process Capability Indices to
Measure Performance of Manufacturing Process

We Will Write a Custom Essay Specifically
For You For Only $13.90/page!


order now

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

1.      
Grau, D. (2011). Testing capability indices for
manufacturing processes with asymmetric tolerance limits and measurement
errors, 73, 61–73. https://doi.org/10.1051/ijmqe/2011010

2.      
Kane.pdf. (n.d.).

3.      
Nagesh, P. (1800). THE
PROCESS CAPABILITY ANALYSIS – A TOOL FOR PROCESS PERFORMANCE MEASURES AND
METRICS – A CASE STUDY 2 . The basic capability indices commonly used in
manufacturing industries are Cp , Cpk Cpm and Cpmk ., 8(3), 399–416.

4.      
Steiner, S., Abraham,
B., & Mackay, J. (n.d.). Understanding Process Capability Indices.

5.      
Technology, Q. (2015).
A New Measure of Process Capability?: Cpm, (July 1988).

6.      
Kotlyar, Y. (2015). Application of Statistical Process
Capability Indices in Gear Manufacturing, (December 2014), 78–85.

7.       Douglas
C. Montgomery (2009). Introduction to
Statistical Quality Control, 6th ed., Elm

8.      
Burdick, R. K., C. M. Borror, and D. C. Montgomery
(2003). “A Review of Methods for

Measurement Systems
Capability Analysis,” Journal of Quality Technology, Vol. 35(4), pp.
342–354.

9.      
Montgomery, D.C. (2000). Introduction to
Statistical Quality Control, Fourth Edition, John Wiley and Sons, Inc.

10.   
Ali R?za Motorcu, Abdulkadir Gu¨ llu¨(2006),
Statistical process control in machining, a case study for machine tool
capability and process capability, Materials and Design Vol.27, 364–372.