J. Service Scie nce & Management, 2009, 3: 204-208
doi:10.4236/jssm.2009.23024 Published Online September 2009 (www.SciRP.org/journal/jssm)
Copyright © 2009 SciRes JSSM
A Primary Robustness Optimization Strategy of
Multi-Item and Low-Volume Process*
Jianguo CHE
Department of Industry Engineering, Nankai University, Tianjin, China.
Email: cjg7705@yahoo.com.cn
Received February 8th, 2009; revised April 12th, 2009; accepted June 15th, 2009.
Multi-item and low-volume process is a production system with multi-input sour ce, interactio ns between inp ut va riab les,
and frequently changes of system state, etc. Strong interactions between input variables and time-varying of input vari-
ables cause poor robustness and large variation range of output quality, which produces high cost, heavy waste and
low efficiency of multi-item and low-volume process. Robustness optimization o f multi-item and low-volume process is a
new, important and need-to-deep research field with multi-item and low-volume production system prevails. It proposed
a strategy enhancing robustness of multi-item and low-volume process by Taguchi robust design. Firstly, build and
analyze a fitting output response model of multi-item and low-volume process after taking the ad justable variables (o r
time-varying variables) correspond ing to each item and interaction between inpu t variables into fitting output response
model of multi-item and low-volume process as input variables uniformly, and treating the parameter value of
time-varying variables corresponding to each item as level value of the adjustable variables (or signal factors) of proc-
ess. Secondly, present robustness evaluation index based on evidential theory, desirability function and dual response
surface etc. Finally, choose the proper experiment type and optimize the process. And then the robustness op timization
of multi-item and low-volume process can be reached.
Keywords: multi-item and low-volume process, robustness optimization, robust design, time-varying, interaction
1. Question Presenting
The mass production system cannot meet customer’s
demand on product’s quality, item, price and delivery
time for rigescent, low-efficient and laggard resource
allocation system. Multi-item and low-volume produc-
tion system becomes popular with the increasing cus-
tomer’s diversified, individual demand.
In multi-item and low-volume process (MILV proc-
ess), there exist followed questions as:
1) There are various production items, materials and
complicated process routes in MILV process, and the
process is affected by man, machine, material, method of
operation, measurement, environment and other influ-
ence factors (5M1E factors), which bring quality fluctua-
tion of process output, so multi-variation of MILV proc-
ess is a key problem.
2) Usually for MILV process, the parameter values of
input variablesor factors have to be renewed (such as
replacing material, adjusting the level values of process
variables, etc.) with one item being shifted, which
brought time-varying of MILV process (here we called
the adjustable influence factors time-varying variables).
Normally, even if MILV process is in control, due to
low volume of each item and frequent change of produc-
tion boundary and system state, the production process
can have been completed while the process didn’t get
stabilization, which leads to unstable output quality. So
time-varying is the second key problem of MILV proc-
3) The process is affected by the 5M1E factors, and
there exist interactions between the factors. When one
item has been shifted to another, the interaction will
change with this process state alters, which brings output
quality of next item unsteady and hard-to-control, and
increases source and range of variation. So interaction
between input variables of MILV process is third key
*This paper is financially funded by the Social Science Program o
Tianjin (TJJJ06-010), Doctoral Fund of Ministry of Education of China
(200800551012) and National Natural Science Foundation of China
From the view of system, MILV process is a typical
production system with multi-input source, multi-stage,
multi-response, time-varying, and strong interaction be-
tween input source variables of the process. Multi-varia-
tion, time-varying and strong interaction bring badly
robustness of the process, and increases variation of the
process and cost of poor quality of the firm.
Since the premise of monitoring the process with the
Shewhart control chart is steady-state stochastic process,
sufficient process capability, adequate and independent
identically distributed process data, and if MILV process
cannot meet these conditions, to monitor MILV process
with control chart will lead to heavy false alarms and
missing alarms, and greatly cut down the average run
length of in-control process. To some extent, MILV
process cannot reach indeed in-control state, and due to
quality control is a non-value added activity and pas-
sively adaptive quality policy. Simple quality control of
MILV process is not only difficult but less significant.
Robustness optimization is a systemic method to en-
hance output quality of MILV process, which can sys-
temically prevent and reduce out-of-control and output
quality fluctuation of process.
Robustness optimization of MILV process is more
complicated and significant than that of mass production
process. Now MILV process is mostly applied to vehicle,
machine and electronic industries and so on. Study on
robustness optimization of MILV process can not only
improve output efficiency and quality of process and
reduce waste and production cost of the firm, but have an
important theoretical significance and application value.
2. Literature Reviews
The principle of robustness optimization is early pre-
sented by the Japan scholar Dr. Genichi Taguchi, and it
is widely recognized and employed for its advantage
improving product quality [1].
Taguchi parameter design better enables a product or
process to perform consistently as intend over a wide
range of operating conditions. The primary principle of
Taguchi parameter design with dynamic output charac-
teristics is to find an optimal level combination of control
factors (i.e. controllable influence factors of process)
which makes the output response of the production
process insensitive to variation of noise factors and sen-
sitive to variation of signal factors (i.e. variation of the
adjustable influence factors or time-varying input vari-
ables of the process). Systematic changing of the combi-
nation and levels of control factors, consistent with the
nonlinear relationships between those control factors and
output response and the linear relationships between the
signal factors and output response, leads to more robust
designs [1,2].
A graphical presentation of the concept is shown in the
Figure 1. The output response has a nonlinear relation-
ship with control factor C, and a linear relationship with
control factor
. The target value of the output response
Y (i.e. critical to quality) of the process when factor C is
located at level 1
and signal factor is given to a certain
value is labeled Y1. At this point, a small fluctuation
around 1
would cause a relatively large oscillation 1
of , which makes output quality of the process highly
unstable. When control factor C has a value of C2, the
output response value is shown as Y2. A control factor
fluctuation of the same magnitude (i.e. ΔC) produces a
less-pronounced oscillation of Y, shown as ΔY2. The goal
of Taguchi parameter design with robustness (also called
Taguchi robust design) is to find a level combination of
influence factors with the strongest anti-interference
ability through design of experiment and analysis of data,
which will minimize fluctuation of the output response
, and then to adjust value of the output response
the original target output by changing the level of
the linear influence factor
(also called signal factor or
time-varying input variables of the process) from 2
For the study of robustness optimization, the meas-
urement of process robustness is an important issue, and
signal to noise ratio (SN ratio) is early introduced by Dr.
Taguchi to define the robustness of production system.
Many criticisms are got to SN ratio for lack of mathe-
matical logic fundamental, and Reference [2] pointed out
that SN ratio was low efficient for losing more than sev-
enty percent data of process. In 1987, Leon, Shoemaker
and Kacker introduced Performance Measure Independ-
ent of Adjustment (PerMIA) by studying SN ratio, and
proved SN ratio was a PerMIA [3]. Reference [4] found
that most criticisms on Taguchi method were focused on
use of SN ratio by examining the viewpoints in the field
of robust design. Since SN ratio didn’t tell from control
factors’ influence on mean and variance, other people
also tried to build and analyze the models of criti-
cal-to-quality’s mean and variance respectively. And
many people presented many indexes measuring the ro-
bustness of process respectively [5], such as extension of
SN ratio, standard deviation or variance of output re-
sponse characteristics, vibration range of output response
characteristics or the ratio of vibration range to expecta-
tion of output response characteristics [6], rejection rate
Copyright © 2009 SciRes JSSM
Figure 1. Nonlinear effect of Taguchi robust design
or yield rate, Information Entropy of process quality [7],
At present methods of robustness optimization can be
cut into two catalogs [5]: one is traditional method of
robustness optimization based on experiential or semi-
experiential design, such as Taguchi robust design, de-
sign of experiment, response surface methodology etc.
Without getting process model, these methods find the
optimal parameter values of process through experiments
and explorations. Due to just considering single output
response characteristic problem with finite-level influ-
ence factors, optimization methods of this catalog are
difficult to fit second-order and above response surface
model, and these methods hardly get the global optimal
solution for the finite changing range and amount of pa-
rameters, they can be only adapted to the design with
mono-response, few variables and no constraint condi-
tion. Another catalog is what calls engineering robust-
ness or analytic robustness, to get the optimal parameter
values of process by computing engineering model with
optimization technology, including generalized linear
model, tolerance polyhedron, propagation of Error, state
space method, sensitivity analysis, stochastic model, and
hybrid robust design based on cost and quality model etc
[8]. These methods have a limited application scope due
to complicate solution process and requirement of exact
output quality model in advance.
Reference [5] pointed out, robustness optimization, the
combination of robust design and optimization design,
enables the robustness of process optimal solution by
adjusting the nominal value of process variables and
controlling the variation of variables. That is, robustness
optimization makes output response characteristics low
sensitive to the variations of process, and seeks the opti-
mal feasible solution of robust design in the meanwhile.
References [8,9] also divided the engineering model
into two catalogs: feasible robustness and sensitive ro-
bustness. He discussed seven robust design methods of
engineering model, and indicated that robust design is an
optimization problem, which is the kernel idea that the
fluctuation of design parameters leads to the variation of
the goals and constraint conditions, and the optimization
problem is addressed by the quantitative design with the
goal minimizing the variation.
Reference [10] considered that robust design mini-
mized the variation of noise factors and control factors,
to improve product quality, instead of removing source
of variation. He also divided robust design into two cata-
logs: one is to minimize the influence of noise factors’
variation on systemic performance, and another is to
minimize the influence of control factors’ variation on
systemic performance. And he indicated that Taguchi
design comes from the previous one, and the solution
process of two problems included three followed steps as:
1) to establish the output response model involved all the
primary control factors and noise factors by response
surface methodology. 2) to build the mean and variance
function according to the type of robust design problem
respectively. 3) to solve robust design problem with
compromise decision support method, etc.
Reference [11] considered that choosing proper vari-
ables to minimize the quality sensitivity to the uncertain
factors can get process robustness in robust design,
which advantage is to design the low-cost product ac-
cepting larger tolerance.
Response surface methodology, presented by Box and
Wilson, is the statistics technology modeling and ana-
lyzing multivariate problem based on design of experi-
ment. Earlier response surface methodology didn’t con-
sider noise factors, and in 1980 Myers and Montgomery
[12] introduced noise factors to build respectively mean
and variance fitting response models, and then robust
design based on response surface methodology is pre-
Reference [13] said that response surface methodology,
consisting of selection of parameters, local optimization
and global optimization, can be applied to robust pa-
rameter design. And the kennels of response surface
methodology are: 1) to take output response of process as
linear function of control factors, noise factors and their
interactions and build the function. 2) to select the proper
response surface design type, carry on the experiments
and get the experimental results of output response. 3) to
estimate results of parameter of linear model with least
square method. 4) to define the significant factors and
interactions with semi-normal distribution plot (or analy-
sis of variation, step regression, Cp statistics, etc), and
then get an estimation and explanation of the factors with
engineering analysis.
Copyright © 2009 SciRes JSSM
Now two methods of robust design based on response
surface methodology is brought as followed: one is to
establish mean and variance fitting models involved the
design variables (or control factors) respectively, which
is usually called dual response surface methodology.
Another is to establish response surface model involved
control factors and noise factors by experiments, and the
output response model of mean and variance is presented
based on Lucas’s propagation of error [14].
Dual response surface methodology is a robustness
optimization method building mean and variance fitting
response model respectively by design of experiments
and optimizing the model as a minimized constraints
problem. There are the advantages that strict mathemati-
cal logic fundamental, consideration of error distribu-
tions and interactions between influence factors, more
accurate solutions, higher optimization efficiency, higher
reliability and robustness and the disadvantages that ob-
taining some key parameters by experience can bring
repeated experiments and calculations when building the
response model for dual response surface methodology.
Again, to fit model will be highly complicated and diffi-
cult if interference variables or some High-dimensional
variables are involved in dual response surface model
The problems often arose in engineering optimization
are as followed: 1) random factors’ influence to quality
fluctuation. 2) The nonlinear implicit function relations
between design variables and response do not make for
optimization. 3) Continuous correction of design vari-
ables in optimization evidently increases experimental or
computational cost. For above problems, Reference [16]
introduced the six sigma robustness optimization method
by combining six sigma and dual response surface
methodology and illustrated with drawing and shaping
case of the ancon and tube-shape piece.
3. Robustness Optimization Strategy of
MILV Process
Robustness optimization of MILV process is a new, im-
portant and need-to-deep research field. We had a work
on quality improvement of MILV process with Taguchi
robust design, but due to a nonlinear function relation or
implicit function relation between input variables and
output response variable exist in MILV process, which
cannot meet the requirement of a linear function rela-
tionship for Taguchi robust design, and Taguchi robust
design cannot manage the strong interaction between
input variables and output response variable, and
time-varying variable problem, which only applies Ta-
guchi design in some specific MILV processes, and
greatly narrowed the applicable field of Taguchi robust
design. Furthermore, the robustness evaluation index of
Taguchi design SN ratio cannot tell the contribution of
high SN ratio from output response mean or variance, etc,
which leads to many criticisms, and due to there are dif-
ferent output target value and error requirement for each
different item of MILV process, and therefore there are
no comparability of SN ratio between different items, so
we have to find a revised Taguchi design method or
other methods to improve robustness of MILV process.
Generally, robustness design of MILV process need to
solve two primary problems followed: one is to build
fitting model of MILV process, another is to introduce
the robustness evaluation index of MILV process. And
we can realize the robustness optimization of MILV
process by design of experiment after solving two prob-
For the above problems, we believe that the interaction
and time-varying between/of input variables causes the
poor robustness of MILV process. Since the identical
modal of time-varying variables, we can take the fol-
lowed steps to obtain the robustness of MILV process as:
1) to treat time-varying variables and other input vari-
ables as input variables of MILV process uniformly, and
take the interaction between input variables into MILV
process model, and then establish MILV process fitting
output response model involved input variables, the in-
teractions between input variables and output response
variables. 2) to present robustness evaluation index of
MILV process. 3) according to the idea of Taguchi ro-
bust design with dynamic output characteristics, take
time-varying variables as input signal factors of MILV
process model, and take time-varying variable value
corresponding to each item and output response mean of
each item as the corresponding level of input signal fac-
tor and the corresponding parameter level of output re-
sponse variable of MILV process respectively, and then
design and optimize the robustness experiments. 4) the
fitting surface model of MILV process can be got by
solution and optimization of the model, and MILV proc-
ess robustness optimization with time-varying variables
and interactions between the input variables.
In robustness optimization of MILV process, it is very
important and difficult to propose robustness optimiza-
tion index of MILV process. And then we will try to
propose the index based on evidential theory, desirability
function and dual response surface and other methods.
Furthermore, since Taguchi design with the inter-outer
array structure is difficult to solve the interactions be-
Copyright © 2009 SciRes JSSM
Copyright © 2009 SciRes JSSM
tween design variables in design of experiment, we have
to select other design type of experiments according to
the interactions. And in general, if we consider all the
interactions of input variables adequately, the best one is
full factorial design. But full factorial design will cause
high number and cost of experiments if there are many
input variables of MILV process, we have to choose few
variables for full factorial design. Since not all the in-
ter-actions between the input variables exist, and usually
we can know the variables between which the interac-
tions exists in advance, we can choose the proper design
such as Taguchi design, response surface methodology
etc, and then take the factors between which interaction
exists in the specific array to estimate the interactions, or
we can choose fractional factorial design with small ex-
periment number to optimize the MILV process. Fur-
thermore, if we cannot know the interactions between all
the factors, we can estimate the known interactions, and
analyze the goodness-of-fit of fitting model, and then do
the experiments until the satisfactory goodness-of-fit of
fitting model is obtained.
[1] R. Jeyapaul, P. Shahabudeen, and K. Krishnaiah, “Qual-
ity management research by considering multi-response
problems in the Taguchi method-a review,” International
Journal of Advanced Manufacturing Technology, No. 26,
pp. 1331-1337, 2004.
[2] G. E. P. Box, “Signal-to-noise ratios, performance criteria,
and transformations,” Technometrics, No. 30, pp. 1-40,
[3] R. V. León, A. C. Shoemaker, and R. N. Kacker, “Per-
formance measures independent of adjustment: An ex-
planation and extension of Taguchi’s signal-to-noise ra-
tios,” Technometrics, No. 29, pp. 253-285, 1987.
[4] V. N. Nair, “Taguchi’s parameter design: A panel discus-
sion,” Technometrics, No. 34, pp. 127-161, 1992.
[5] L. Z. Chen, “Robust design,” Machine Industry Press,
Beijing, 1999.
[6] D. B. Parkinson, “The application of a robust design
method to tolerancing,” Journal of Mechanical Design,
No. 122, pp. 149-154, 2000.
[7] J. C. Xu and Y. Z. Ma, “Measurement methods for robust
multivariate quality characteristics,” Systems Engineer-
ing-Theory & Practice, No. 19, pp. 45-48, 1999.
[8] A. Parkinson, “Robust mechanical design using engi-
neering models,” Journal of Mechanical Design, No. 117,
pp. 48-54, 1995.
[9] A. Parkinson, C. Sorensen, and N. Pourhassan, “A gen-
eral approach for robust optimal design,” Journal of Me-
chanical Design, No. 115, pp. 74-80, 1993.
[10] W. Chen, J. K. Allen, K. -L. Tsui, and F. Mistree, “Pro-
cedure for robust design minimizing variations caused by
noise factors and control factors,” Journal of Mechanical
Design, No. 118, pp. 478-485, 1996.
[11] A. D. Belegundu and S. H. Zhang, “Robustness of design
through minimum sensitivity. Journal of Mechanical de-
sign,” No. 114, pp. 213-217, 1992.
[12] R. H. Myers and D. C. Montgomery, “Response surface
methodology: Process and product optimization using de-
signed experiments,” Wiley Interscience Publication,
New York, 2002.
[13] A. C. Shoemaker, K. -L. Tsui, and C. F. J. Wu, “Eco-
nomical experimentation methods for robust parameter
design,” Technometrics, No. 33, pp. 415-427, 1991.
[14] J. M. Lucas, “How to achieve a robust process using re-
sponse surface methodology,” Journal of Quality Tech-
nology, No. 26, pp. 248-260, 1994.
[15] R. V. Geof and H. M. Raymond, “Combining Taguchi
and response surface philosophies: A dual response ap-
proach,” Journal of Qualify Technology, No. 22, pp. 38-
45, 1990.
[16] Y. Q. Li, Z. S. Cui, J. Chen, et al., “Six sigma robust
design methodology based on response surface model,”
Journal of Shanghai Jiaotong University, No. 40, pp.
201-205, 2006.