J. Service Science & Management, 2010, 3: 106-109
doi:10.4236/jssm.2010.31013 Published Online March 2010 (http://www.SciRP.org/journal/jssm)
Multivariate Quality Loss Model and its
Coefficient Determination
Shuhai Fan, Xia Cao
Department of Industrial Engineering, Nanjing University of Technology, Nanjing, China.
Received August 28th, 2009; revised October 10th, 2009; accepted November 27th, 2009.
ABSTRACT
In the ea rly 1970s, based on single ind ex deducted from absolut e quality deviants, Genichi Taguchi proposed th e qual-
ity loss module. This module builds the foundation of his three-stage design theory, e.g. system design, parameter de-
sign and tolerance design. In actual production process, nevertheless, it is multiple quality indices that influence the
total quality. Consequently, the interaction of the quality indices should be imported into the module as a key factor.
Accordingly, based on several indices of relative quality deviation, introduce a 2-order multivariate quality module at
first. Next, extend the module to 3 or even higher orders. Then, improve the previous quality module by simplifying the
2-order module as a multivariate quality loss module. Finally, bring forward a significant solution to determinate all
the coefficients in th e multivariate quality loss module and describe its work flow as well.
Keywords: Quality Loss, Multiple Variables, Model Building
1. Introduction
In the early 1970s, Dr. Genichi Taguchi, the famous
quality management expert of Japan, carried on an inno-
vation research into the theories and methods of quality
management [1]. He established the famous “Taguchi
Three-stage Design Methods”, the system design, the
parameter design and the tolerance design. The core of
his theory was his quality loss model [2,3]. And the qual-
ity loss model was based on the absolute quality deviant,
which suited for the single quality index. However, in the
actual production process, many partial quality indices
often affected the total quality in a coactions mode. After
these interactions being considered, a multivariate quality
loss mode can be brought forward [4].
To this many partial quality indices, Chan and Ibrahim
studied a quality evaluation model using loss function for
multiple S-type quality characteristics [5]. To determi-
nate the coefficients of the multivariate quality loss mode,
AHP method or tolerance method can be used [6,7].
2. Model Building
The total quality is a synthetic quality, which is often a
coaction of some machining qualities and some assem-
bling qualities of every part [1]. And it can be regarded
as a function of every partial quality (including the ma-
chining qualities and the assembling qualities):
product1 2
( ,,... ...,)
hH
QfQQQQ
Assuming that f has continuous partial derivatives till
3-order in a neighborhood D of the origin, P0(0,0…0).
There are H partial quality indices altogether, q1, q2, …
qH. These partial quality indices are all continuous types;
their values are very small and stand for relative quality
deviants. When the values of q1, q2,…qH equal 0, it
stands for no quality deviant (Here we use the relative
quality deviants to get dimensionless. Thus a minor error
can be avoided in the derivation process of Taguchi’s
model.) [8]. Then the product quality can be expressed as
follows:
12
12
12
12
1
2
,1
( ,...,...,)(0,0...,0)(0,0...,0)
1
(0,0...,0)
2!
h
hh
H
hHQ h
h
H
QQh h
hh
fqq qqffq
f
qq R









while, R2 is the 2-order remainder term.
123
123
12 3
21
,, 1
1( ......)
3! hhh
H
QQQhHh hh
hh h
Rfqqqqqq 



(0 1)



A brief proof is as follows:
considering a single variant function ()Qt
12
( )(,...,...,)
hH
Qt fqtqtqtqt

, (1 1)t

Multivariate Quality Loss Model and Its Coefficient Determination 107
()Qt has continuous partial derivatives till 3-order for
(1 1)t .
Obeying the 2-order single variant Talylor’s formula
with Lagrange remainder term, we get:
23
11
( )(0)(0)(0)()
2! 3!
Qt QQtQtQ tt 
 
 ,(0 1)



Let t=1, we get
11
(1)(0)(0)(0)( )
2! 3!
QQQ QQ

   (1)
Follow the definition of ()Qt and the differential
method of composite function. We get
1
1
( )(,...,...,)
h
H
QhHh
h
Qtfqt qtqtq

 
12
12
12
1
,1
( )(,...,...,)
hh
H
QQhHh h
hh
Qtfqt qtqtqq

  
12 3
12 3
123
1
,, 1
( )(,...,...,)
hhh
H
QQQhHh hh
hhh
Qtfqt qtqtqqq

  
Then, when t =0,
1
(0) (0,...0,...,0)
h
H
Qh
h
Qf q



1
12
12
2
,1
(0) (0,...0,...,0)
hh
H
QQh h
hh
Qf qq

 

123
123
123
,, 1
(0) (0,...0,...,0)
hh h
H
QQQh hh
hhh
Qf qqq

 

while
12
(1)( ,...,...,)
hH
Qfqqqq,
(0) (0,...,0...,0)Qf
Substitute the results into (1)
The expansion equation can get proved.
To estimate the error term, we have:
f has continuous partial derivatives till 3-order; in the
neighborhood of (0,…,0,…0); the 3-order partial deriva-
tive has a bound M.
denote
22 22
12
1
...
H
H
h
h
qq qq 



, (0)


then
123
123
3
212
,, 1
(... )
3! 3!
H
hhh H
hhh
MM
Rqqqqqq



12
12
3
3
2
12
,1
=(...)
3! 3!
H
Hhh
hh
MM
qq qqq



 




12
12
3
22
,1
3
222
12
1()
3! 2
(...)
3!
H
hh
hh
H
Mqq
MHq qq












 
3
33
23
1
=3! 3!
H
h
h
MM
Hq H








2
2()Ro

 , (0)


It is the general form that is used to describe the total
quality index in the above formula. If the quality loss
form index is adopted, the model can be much more sim-
plified. Imply that when all the partial quality indices
equal 0, the total quality index takes the minimum, 0. It
is easy to see that the 0-order and 1-order derivative can
be gotten rid of. Then a new product quality loss model
can be built as follows:
212 1 2
112
2
product
11,
Hn
hhhhh
h
hhhh
L
Qwq wqq



(2)
Compared it with Taguchi’s model, we can see that (2)
is actually the extended multivariate form of Taguchi’s
quality loss model.
Here, Qproduct is a target total quality index and de-
scribed as the quality loss. All the n partial quality indi-
ces influence Qproduct in a coaction mode, and qh is the
relative quality deviant. Every partial quality deviant can
be gotten by statistic. Here we use the relative partial
quality deviant, and it can be expressed by the relative
quality value of the deviant from the target value and
severalfold tolerances. If the value of quality characteris-
tic lies beyond the tolerance, it should be multiplied by a
punishment factor.
where, 2
2
1
H
h
h
h
wq

, 1212
112
1,
n
hhh h
hhh
wqq

 are separately the
self-action item and the inter-action item of the partial
quality deviants. 2
h
w is the self-action influence weight
of the quality deviant 2h
q; 12
hh
w is the inter-action
weight of deviant 12 hh
qq .
Compared with our expansion equation, it can be got-
ten
22 2
11
(0,...,0,...0)(0,...,0,...0)
2! 2
hh
hQ Q
wf f
 

12 12
12 21
12 12
1(0,...,0,...0)
2!
1(0,...,0,...0) (0,...,0,...0)
2
=(0,...,0,...0) ()
hh
hh hh
hh
hhQQ
QQ QQ
QQ
wf
ff
f
hh
 

  

 
If necessary, we can also expand the product quality
model further more to U-order.

Multivariate Quality Loss Model and Its Coefficient Determination 108

12
12
12
()
product ...
2,,...1
10, 0..., 0...
!hh hu
u
u
UH
uQQQhhhU
uhhh
QfqqqR
u










()
U
U
Ro


When the quality model is expanded to 2-order, the
inter-actions of partial quality indices have been fully
considered. In fact, the impactions of items higher than
2-order are very little [10,11]. Thus for the simplification
of computation and the actual requirement, further higher
orders are never needed.
The above work suits the small-is-better and the nomi-
nal-is-best problems. The large-is-better problem can
also be reformed into a small-is-better problem.
3. Coefficient Determination
3.1 The Tolerance Limits Method
The quality loss model (2) can also be written as
2
product
11,


Hn
iiiij ij
iiij
Qwqwqq (3)
the coefficients wij can be determined using the following
method all these wij (i≤j) can constitute a upper triangular
matrix like Figure 1.
1) Determine the elements in Diagonal for an arbitrary
i (i=1,2,..., H), when all the qk=0, (k≠i) i
 is the toler-
ance limit of defective for self quality and Ai is the self
quality loss when defective happens.
Then we have
22
product
11,
()
Hn
iiiiijijiii
iiij
QAwqwqqw

 

So
2
/( )
ii i i
wA
2) Determine other elements for an arbitrary i,j
(i,j=1,2,...,H, i≤j), when all the qk=0, (k≠i, k≠j), i
ij

is the
tolerance limit of quality i for coaction quality i and j,
j
ij

Figure 1. Upper triangular matrix of coefficients then, we
can determine these coefficients
is the tolerance limit of quality j for coaction quality i
and j and Aij is the coaction quality loss when coaction
defective happens.
Then we have
02
11,
0, 20,20,0,
()( )
Hn
productijiiiijij
iiij
ijij
iiijjjijij ijij
QAwqwqq
ww w

 
 

Thus we have
00,2 0,20,0,
()()/
ijij
ijijiiijjjijij ij
wAww




where
002
/( )
ii ii
wA

, 002
/( )
j
jj
wA
This coefficient determination process can be de-
scribed in Figure 2.
Remark:
In some particular circumstances,
0, 00,0
,
ij
ij iijj


we have a simplified form
000 00
/
ijijijij
w AAA


 

3.2 Other Methods
To determinate the weight coefficients, the least square
method can also be adopt. However linear neural net-
works are more suitable [11–14]. The input weights of
the trained neural network are just the weight coefficients
of the model.
4. Conclusions
Based on the quality loss model of Taguchi, the total
quality model which is based on multivariate relative
quality deviants was studied. A 2-order product quality
model, which was based on several indices of relatively
quality deviation, was built. Then, the model was suc-
cessfully extended to 3 or even higher orders. To deduce
w11 w12w13w14 w15 w16w17
w22 w23w24w25 w26 w27
w33 w34w35w36 w37
w44 w45w46w47
w55 w56w57
w66 w67
w77
Figure 2. From the diagonal elements to other elements

Multivariate Quality Loss Model and Its Coefficient Determination 109
a simplified formula, a new simplified product quality
model (a multivariate quality loss model) was built and
the error of it was also analyzed. Finally, a useful
method of the coefficient determination was brought
forward to determinate all the coefficients in the mul-
tivariate quality loss model and the work flow of it was
also described.
However, in the complex product, the coupling factor,
structure layer and the number of parts may have a fur-
ther increase, and the quality loss model become more
and more complex. Then there are no enough data for the
determination of the coefficients and the calculation also
become more and more complex. To make full use of
historical data and similar data, a quick-response incre-
mental model of multivariate quality loss is needed. So it
should be in our further research.
5. Acknowledgment
This project was supported by National Natural Science
Foundation of China (70801036), Natural Science Foun-
dation of the Jiangsu Higher Education Institutions of
China (07KJB460045).
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