Measurement Process Capability – Trends and Approaches

MEASUREMENT SCIENCE REVIEW, Volume 1, Number 1, 2001

Measurement Process Capability – Trends and Approaches

Eva Kureková

Abstract

A measurement control system ensures that measuring equipment and measurement pro-
cesses are fit for their intended use and its important in achieving product quality objec-
tives. The following paper introduces theoretical foundations for determination of the mea-
suring process capability. Defines three capability indexes in short and recommended use
of the Cpm index.

Keywords

statistical process control, measurement process control, capability index, Taguchi function

1 Introduction bility. Both such efforts are determined by the tech-
nological development of the measuring instru-
Several years can be observed attempts to stan- ments. Due to the technical innovations (incorpo-
dardise evaluation of the processes (whether pro- rating electronics enabling predictive diagnostics,
duction or measurement) capability. Such attempts autocalibration, automatic correction of the out-
have not been successful yet. Many factors must put signal according to the status of the measuring
be taken into account – processes heterogeneous- instrument) variability of the measurement process
ness, changing requirements, wide variety of used is decreasing and the effort for its centering impro-
technological means. Therefore big production en- ve capability of the measuring instruments (see Fig.
terprises (General Motors, Volkswagen, Siemens, 1). Economical savings represent the direct impact
Bosch, etc.) have established their shop methodol- – possibility for better setting of the production
ogies for the evaluation of the processes capabili- process, decreasing of the number of nonconfor-
ty. Capability of measurement processes is searched ming products and resulting the increased produc-
similarly to that of production processes most of- tion efficiency.
ten.
2 Indexes of the measurement process
Process capability means ability of the process
to meet technological or other requirements, i.e. to capability
fulfill demands put on it.
Several types of the process capability indexes
Measurement process capability is determined exist. They differ one another by calculation meth-
by total variation caused by random reasons influ- od, by properties as well as by intended use. But
encing the process. Variation is caused by variabi- their design principle is approximately the same.
lity of the measured quantity values that are not The ratio of prescribed (required) accuracy and
connected to the measurement conditions and must really achieved process accuracy is always ob-
be excluded. Therefore it is recommended to per- served.
form preventive measurements. In those texts check
standards are measured by the measuring equip- According to philosophy of the quality control
ment being tested. approach, capability indexes of any process can
be divided to the capability indexes of the first and
Variability of the measured data (caused by the second generation.
measurement process) as well as the systematic
deviation from the required values is observed du- Design of the first generation capability index-
ring evaluation of the measurement process capa- es (Cp, Cpk) is based on classical philosophy of the
statistical process control. According to that phi-
Dr. Eva Kureková, Faculty of Mechanical Engineering, losophy all measurement results within required
STU, nám. Slobody 17, 812 31 Bratislava, Slovak Republic, tolerance interval are intended to be good. Mea-
e-mail: [email protected]

43

Theoretical Problems of Measurement. E. Kureková

Fig. 1 Influence of the measurement process variability and the effort for its centering

surements outside tolerance interval are considered CP = UTL−LTL (2)
to be bad. 6kσ

Second generation capability index (Cpm) is ris- where k = 3 to 10.
ing from new approach to the quality improvement Capability index Cp expresses only the po-
(Taguchi approach). It is not enough to know that
measurements are so called good (being within the tential process capability. It does not represent
tolerance interval) but important is knowledge on the position of the natural tolerance interval con-
how good they are. Such index enables to deter- sidering the position of the required tolerance
mine whether the values of the searched quality interval. Therefore it does not give a clear an-
index approach to the tolerance limits even when swer whether measured value of the searched
all measurement results fit within the tolerance. quality indicator fits within the tolerance inter-
val. Another disadvantage of the Cp index is the
2.1 Process capability index Cp fact that it does not represent the conformity of

Process capability index Cp is a simple relative the measurement process average mean X to a
number comparing value of the required process check standard nominal value X0.
variability (required tolerance interval) to a natu-
ral process variability (natural tolerance interval). 2.2 Process capability index Cpk
Capability index Cpk reacts to deviation of the
For measuring process given by expanded un-
certainty U, capability index Cp is calculated as measurement process average mean X from check
standard nominal value X0 (see Fig. 2). Index is
CP = 2U = U (1) calculated as
6σ 3σ

where s is a process standard Fig. 2. Design of the Cpk index

deviation.
Whenever requirements put

on measurement process are giv-
en by tolerance interval T, such
interval must be defined first.
Tolerance interval T is defined as
a difference between the upper
tolerance limit UTL and the low-
er tolerance limit LTL. That
means T = UTL – LTL. Capabil-
ity index Cp is calculated in this
case as

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MEASUREMENT SCIENCE REVIEW, Volume 1, Number 1, 2001

Cpk = U − ∆ −UKE (3) Observing equation (4) one can see that increasing
3σ process variability s2 causes increasing of the de-
nominator and therefore decreasing the Cpm index.
where UKE is the check standard uncertainty, Again retreating of the measurement process ave-

∆ = X0 − X . rage mean X from check standard nominal value
X0 increasing the denominator value and resulting
2.3 Process capability index Cpm decreases the Cpm index value.

Practical experiences showed that demand for 2.4 General remarks on capability indexes
a low percentage of check standard measurements
(or measurements performed directly on products If the process is centered, measurement process
or processes) falling outside the tolerance limit T is always capable when capability index value (Cp,
is not satisfactory. Few information are obtained Cpk, respectively Cpm) exceeds 1. Practical recom-
on spare measurements. Most interesting informa- mendation considers minimal admissible value
tion is about the quality of individual measurements 1,33. The reason is that certain variability is always
i.e. how far are measurements performed on check presented and process is never fully in statistically
standard from the nominal value X0 of the check controlled status. Therefore indexes value ³ 1,33
standard or from tolerance limits respectively. is recommended for established processes. Newly
adopted process should produce capability inde-
G. Taguchi and T. C. Hsiang designed a loss xes with values ³ 1,50, extremely precise measu-
function as a new approach to production process rements should have value ³ 1,66 [4], [6].
quality improvement in 1985 [5]. This dissipation
function can be adapted fully to the measurement Process standard deviation s is usually unk-
process. Its use is intended for decreasing the vari- nown in practical situations and must be estima-
ability around the target value of searched quality ted. Usually is estimated as
indicator, i.e. around the check standard nominal
value (see Fig. 3). 1∑s = n Xi − X )2 .
n −1
Process capability index Cpm defined by Taguchi (
is based on equation (1) and is defined as
i=1

Cpm = UTL−LTL = U (4) Its substitution to expressions (1), (2) and (3)
6kτ 3τ gives just estimation of individual indexes. It is a
random variable with probability distribution. The-
where tis standard deviation around the check refore calculation of the index interval estimation
standard nominal value X0. Taguchi advises to cal- is recommended. This interval contains the true
value of the index with probability of 1 - a. Obje-
culate this deviation as τ = σ 2+( X − X 0)2 . ctive proof of process capability (index gets value
exceeding 1,33) brings statistical test.

Fig. 3 Taguchi loss function 2.5 Example

Lets calculate capability indexes Cp, Cpk, Cpm
and compare them. Basic parameters of the mea-
surement process: check standard nominal value is
X0 = 20 mm, measurement process average mean
X = 20 mm; measurement process uncertainty U
= 0,02 mm, check standard uncertainty UKE can be
neglected, standard deviation s = 0,004 mm. Then
capability indexes

Cp = 0,02/(3×0,004) = 1,666

Cpk = (0,02-0)/(3×0,004) = 1,666

45

Theoretical Problems of Measurement. E. Kureková

Cpm = (0,02)/(3Ö(0,0042+02)) = 1,666; Manufacturing Engineers. McGraw Hill,
1997
While X0 = X , process is centered and all ca-
pability indexes are equal. However such status is [2] Janiga, I., Palenčár, R.: O jednom probléme s
very rare in technical practice. indexom spôsobilosti procesu. Proceedings of
the international conference Mechaniacl
After parameters change to X0 = 20 mm, X = Engineering 98, Bratislava, 1998
19,995 mm (other parameters remain the same),
capability indexes get following values: [3] Kane, V., E.: Proces Capability Indices.
Jounal of Quality Technology, vol. 18, No.1,
Cp = 0,02 / (3×0,004) = 1,666 (without change) 1986

Cpk = (0,02-(20-19,995)/(3×0,004)=1,25 [4] Palenčár, R., Kureková, E., Halaj, M.: Mea-
surement Proces Control. Proceedins of the
Cpm = (0,02) / (3Ö (0,0042+(20-19,995)2) = 1,04 international conference New trends in Auto-
mation of Energetic Processes’98, Zlín, 1998,
Process is not centered in this case while X0 ¹ X . pp. 360 - 365
Capability index Cp did not record falling-off the pro-
cess capability by remained variability. Indexes Cpk, [5] Taguchi, G.: Introduction to Quality Engineer-
Cpm are able to record infrigement of the required ing. Asian Productivity Organization, Tokyo,
value by untouched process variability. 1986

After another parameters change to X0 = 20 mm, [6] Terek, M., Hrnčiarová, Ľ.: Štatistické riadenie
kvality. Published by Ekonóm, Bratislava,
X = 19,99125 mm, s = 0,004 mm calculated ca- 1999, 293 pp.
pability indexes obtain following values:
[7] Hekelová, E.: Factors Improving the Quality
Cp = 0,02/(3×0,004)=1,666 (without change) of Services. In: Acta Mechanica Slovaca, Vol.
4, 2000, No. 2 (in Slovak)
Cpk = (0,02-(20-19,99125)/(3×0,003)=1,25
(without change) [8] Vdoleček, F.: Spolehlivost měřicí techniky v
regulaci. Proceedings of the international
Cpm= (0,02)/(3Ö(0,0032+(20-19,99125)2)=0,72 conference Řip 2000, Kouty nad Desnou, pp.
179 - 183, 2000
Capability index Cpk is not able to record chan-
[9] Wisweh, L., Sandau, M.: Quality Evaluation
ges of the measurement process average mean X of Measurement Instruments. Proceedings of
towards tolerance limit in the case of changed stan- the XVIth IMEKO World Congress, Vienna,
dard deviation. Only Cpm capability index registe- 2000
red such change.
[10] Palenčár, R., Halaj, M.: Metrological
3 Conclusions Assurance of the Quality Control Systems.
Published by Vydavateľstvo STU, 1998, 138
Presented paper defines in short three most ap- pp.
plicable methods for calculation of the measure-
ment process capability. Shows some deficiencies [11] Chudý, V. et all: Measuring of the Technical
of the Cp and Cpk capability indexes that are used Quantites. Published by Vydavateľstvo STU,
almost exclusively in nowadays practice. Capabil- 1999, 688 pp.
ity index Cpm represents best the real measurement
process capability. This paper was prepared under the support of
the VEGA grant agency, grants No. 1/7077/20 and
No. 1/8094/01.

References

[1] Bothe, D., R.: Measuring Process Capability:
Techniques and Calculations for Quality and

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