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• K such that for any B e *}3 (E) , {0|£(<9)c B,B eq3(K)} e
(5)
Definition 2.5 (Liu [85]) The credibility distribution
Cr{de®\£(8)<x]
(6)
590
That is the credibility distribution Q>(x) is the accumulated credibility grade that the fuzzy variable £, takes a value less than or equal to a real-number x e R . Generally speaking, the credibility distribution G> is neither leftcontinuous nor right-continuous. Theorem 2.6 Let £ be a fuzzy variable on (0,
(7)
y>x
Definition 2.7 Let <& be the credibility distribution of the fuzzy variable ^ . Then function :M -»[0,+co) of a fuzzy variable £, is called a credibility density function such that
^(x)^^(y)dy,
Vx.
(8)
Example 2.8 A three-parameter half-sine fuzzy variable ^{rl,r^,ri') has a membership function sin [a- (* - r, )/2 (/-2 - r,)]
ifr, < x < r 2
/*(*) = 1
if r, < x
0
otherwise
(9)
Then credibility distribution of a three-parameter half-sine fuzzy variable ^(r{,r„r3) is 0,
*(*) =
sin[^-(x-rl)/2(r2-r,)]/2,
if x < r. ifr, <x
1/2,
ifr2<x
if
(10)
JC> J
and its credibility density function is $<x\\KC0S[x(x-^)l2(ri-ri)\l\_A(ri-ri)\ 0
'^ri^x
(11)
591 In reliability modeling, the multivariate fuzzy variable and its joint distribution are required. Definition 2.9 Let ( £ P £ 2 > ' " > 0 be a fuzzy vector defined on (0,
(12)
And the joint credibility density function 0 :[—oo,+oo]" —>[0,+oo] of fuzzy vector (£i,£ 2 >""'>0 is the function such that <3>(xl,x2,---,x„)= | J-—oo —ac
fy(xl,x2,---,xn)dxldx1---dx„
(13)
—oo
Holds for all (xl,x2,---,xn)e[-<x>,+<x>]", where «t>(x,,x2,--,xn) is the joint credibility distribution for fuzzy vector (£p£2>"">£n) • 3. Credibility Copula and Survival Copula In probability theory, copula is the concept for describing the dependent structure between two random variables and obtained more and more attention in financial risk analysis. Parallel to that in probability theory, we will define copula concept for credibility measure based fuzzy variables and will investigate the applicability in fuzzy reliability analysis. Definition 3.1 Let H(xt,x2) be the joint credibility distribution of the fuzzy variable pair (X,,X,) and the continuous marginal credibility distributions be ^ ( J C , ) and F2(x2) respectively. Then for any w, xw2 E [ 0 , 1 ] X [ 0 , 1 ] ,
C(u],u2) = H(Fi-l(u]),F2-[(u2))
(14)
where the inverse of credibility distribution F ( ) is defined by F'x (u) = inf {x e K: F(x) > u, Vu e [0,l]}
(15)
Now let us investigate the joint survival function of credibility fuzzy lifetimes (X,,Jf 2 ) , //(x,,x 2 ) , and their marginal credibility survival
592 distributions are defined Fl (x,) = 1 - Fi (x,) respectively. Then H(xl,x2)
= l-Fl(xl)-F2(x2)
by
Ft (*,) = 1 - Fl (JC, )
and
+ H(xl,x2)
=Fi(xl) + F2(x2)-l
+ C(Fl(xi),F1(x2))
=Fl(x1) + Ft(x1)-l
+
(16)
c(l-F1(x1),l-F2(x2))
Definition 3.2 Let C(W,,M 2 ) be the credibility copula for credibility fuzzy variable pair (Xt,X2) with continuous marginal credibility survival distributions ^ ( * , ) and F2{x2~) respectively. Then C(w,,w2) = w, +u2 - 1 + C ( 1 - M , , 1 - W 2 )
(17)
is called the survival copula (reliability copula). And the joint credibility survival function is H(xl,x2)
= C(Fl(xl),F2(x2))
(18)
However, we need to point out that the Axiom 5 stated in Section 2 is necessary to be replaced by an alternative one, which is (V,•) -based. The (V,•) -based credibility theory is called non-classical credibility theory by Liu (2006) and with the (V, •) -based credibility measure, we can define more important concepts like conditional credibility distribution, conditional expectation and others, which will help to facilitate a full structural development as that in probability theory and copula theory. Axiom 5*: let 0 - 0 , X@2 X . . . x 0 n with &k [k = 1,2,••-,«) being nonempty sets where Crk satisfy the first four axioms and furthermore satisfies
ifl((2Cr,{3})Al) ifminCr^}<0.5 Cr{(0p02,..,£„)}=
(19)
minCr^} \
for each {0x,02,...,9n)
if minCrA{^}>0.5 \
6 0 . In this case, we denote C r ^ C r ^ C ^ ' - ' - ' C r ^ .
4. Generalized Fuzzy Load-Strength Analysis of Complex System The fuzzy reliability of a complex system can be virtually treated as the interacting state between two factors: factor resulting from the fuzzy load or
593 functional demand of the system and the factor rooting within the strength or the intrinsic capacity of the system. If the complex system is a structure, say, a bridge, a roof, etc, the terms load and strength directly reflect the physical property of the structure. In general context, load (functional demand) and strength (intrinsic capacity) terms will represent virtual state variables of the complex system. In the concept of reliability, if the system fuzzy load is less than system fuzzy strength the system is safe and reliable otherwise, the system is unsafe and unreliable. Let Yx,Y2,---,Yn and Zx,Z2,---,Zm be two groups of control variables and the strength Xx = g(Zx,Z2,---,Zm)
and the load X 2 = r {YX,Y2,-• • ,Yn). Todinov
(2005) classifies the domain (Xx,X2)
into three areas: safe region, where
F = {(x,,x 2 ): Xx -X2 < 0, (x,,x 2 ) e [x, min ,x, raax ]x[x 2 min, x2max ]}
(19)
Failure surface Xx - X2 = 0, and safe region, where S = {(x,, x2): X, - X2 > 0, (x,, x2) e [x, min, x, max ] x [x2 min, x2 max ]}
(20)
The fuzzy reliability is thus R=
( 21 )
J"i, ^^(xi'^dxAi
where ^(x,,x 2 ) is the joint credibility density function of load-strength (XX,X2)
with a joint credibility distribution 0(x,,x 2 ) such that Hx»x2)
d2 = -^r®(xl,X2)
(22)
We notice that the partition of the domain of [\ min ,*,, miK ]x[x 2min ,x 2max ] can be converted to [ a , m i n , u l - m ] x [ u 2 m n , u 2 m ] , where M
l,min
M
2.rain
=
n
I ^l.min j '
M
-F(' \ ^2 V ^ . m i n ) '
l.max
M
=
2,max
M ( X\.man ) -F( \ ^2 ^ l . m a x )
—
^
and the safe region S = {(«,,w2): F~[ (ux)-F2x (i/2) > o} and therefore, the fuzzy reliability of the system can be presented by credibility survival (reliability) copula C(W 1 ,H 2 ) = M 1 + M 2 - 1 + C ( 1 - W , , 1 - W 2 ) , V ( « , , W 2 ) E S
(24)
594 5. Concluding Remarks In this paper, we briefly discussed the fuzzy characteristics of complex system reliability. We reviewed the concepts of credibility based fuzzy variable, the distribution and density function proposed by Liu (2004). Then we propose a credibility copula concept for the description of the fuzzy dependent structure, which is important in the investigation on complex system. Based on these credibility measure based fuzzy variable developments, we investigate fuzzy reliability for complex system in terms of the fuzzy load-strength concept and thus present a general theoretical foundation for fuzzy reliability research. References 1. 2. 3. 4. 5. 6.
A. Kaufmann, Introduction to the theory of Fuzzy Subsets Vol. I, Academic Press, New York (1975). B. D. Liu, Uncertainty Theory - An Introduction to Its Axiomatic Foundations, Springer (2004). B.D. Liu, A survey of credibility theory, Fuzzy Optimization and Decision Making 5(4), (2006). M. Todinov, Reliability and Risk Models: Setting Reliability Requirements, John Wiley & Sons Ltd, New York (2005). L. A. Zadeh, Fuzzy Sets, Information and Control 8, 338-353 (1965). L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1,3-28 (1978).
SMALL SAMPLE ASYMPTOTIC DISTRIBUTION OF COSTRELATED RELIABILITY RISK MEASURE RENKUAN GUO Department of Statistical Sciences, University of Cape Town Private Bag, Rondebosch 7700, Cape Town, South Africa
System reliability, as a quality index, is the capability to complete the specified functions accurately in mutually harmonious manner under the specified conditions within a specified time period. We notice that high costs are sometimes associated with the occurrence of tiny probability and therefore the reliability index alone would not fully characterize the consequence of system breakdown. Todinov (2005) proposed the cost of failure as a measure for system reliability risk and explored related models. However, the sparse data availability extracted from the system may haunt the modeling exercises. In this paper, we will merge cost of failure idea and the small sample asymptotic idea together for the investigation on the asymptotic distributions for the total cost of failures due to system failures and associated losses.
1
Introduction
In engineering context, system reliability, as a quality index, is the capability to complete the specified functions accurately in mutually harmonious manner under the specified conditions within a specified time period. Reliability is typically described by the probability of the system surviving (functioning) up to time t. Therefore, the classical reliability index just involves the system functioning capability not how much costs involved in the system operation and inevitably unrealistic in decision making on production scheduling and maintenance planning. On the contrary, the cost-related reliability risk measure is better and more realistic for understanding system availability and managing system reliability. Todinov (2005) proposed the cost of failure as a measure for system risk and accordingly investigated some related models. Therefore the study of the cost-related risk measure of a functioning system is essentially linking to the problem of finding the distribution function of the total loss, which is in nature a weighted sum of losses due to system failures. Furthermore, today's industrial systems are more and more complicated and fast-changed. Then the sparse data availability extracted from the system's operations may haunt the modeling exercises. Therefore, the so-called "small 595
596
sample asymptotics" or "saddle point approximation" is the proper mathematical solution to address the approximate distribution of total cost under small sample size. 2
Asymptotic Distribution for a Compound Risk Process
Denote the cost of loss due to the /' failure by Yi ( / = 1,2, • • • , « ) . Note that Yt is not necessarily just the repair cost but also the losses from production interruption and even the delay deliver of products etc due to the failure. Obviously, Yt > 0 is a random quantity and can be expressed by Yi=SlLl,i
= l,2,--,n
(1)
y&^L^, i— 1,2,---,n are independent bivariate random vectors with failure indicator (0 with probability p [ 1 with probability 1 - p and the loss I, > 0 due to the i failure having a common distribution FL(x) = Pr[L,<x]
(3)
or equivalently the loss L. > 0 has probability generating function PL (9). Therefore, the total loss due to the system failures at time / is a general compound process {C, : t > 0} N(l)
N(,)
C,=Z^ = S ^ , i=i
(4)
/=i
where {7V,,i > 0} is a regular counting process. We notice that the cost-related risk measure C, itself forms a compound process {C, - -<>0} where the summation index is a random number N, at time t > 0 . Then we have to find out the cumulant generating function of a compound process. Assume that the counting process {N, : ? > 0 } is a nonhomogeneous Poisson process with intensity or rate function A(f) > 0 and denote the corresponding probability generating function by PN (0) . Furthermore, it is assumed that loss Yt follows a common distribution function FY (•). {Nt :t > 0} and loss Yi are assumed to be independent. Then in terms of conditioning we have ( ' Pc, (9) = PHi (PL (0)) = exp - JA(u)du(\-PL V o
^ (0))
(5)
J
The required function is cumulant generating function via moment generating function. The relationship between a moment generating function and the probability generating function is
597 mc,(0) = Pc,{ee)
(6)
because Px (0) = E[0*] = E[e^9)x
] = mx (In 0)
(7)
Therefore KCi(0) = \nmc,(0) = \nPc,(ee) = (pY(ee)-\)\A{u)du
(8)
0
=(my(0)-l)JA(u)du 0
By imposing an additional assumption that Pr[i9f = l] = p, i.e., the failure probability does not vary. Then for loss Y: ='dtLl , its moment generating function can be simply expressed by mr(g) = E[eJ'L'] = p + (l-p)mL(q). With the above assumptions, we will have KCi(0) = (l-p)(mL(0)-l)JA(u)du
(9)
0
Then i
Kc,(0) =
mL(0)(\-p)JA(u)du
Kci(0) =
: mL(0)(l-p)JA(u)du
(10)
o
According to Field and Ronchetti (1991), the asymptotic distribution of C, can be expressed by fc,(x) = gc.{x)(l + 0(n-1))
(11)
where
(
gc W=
'
Y /2
[ ^ R j "PMM*)-**))
d2)
where the saddlepoint a0 is obtained by solving the following nonlinear equation for any given (x,t), x e R+, t ~> 0 K'Q(a0) = x i.e., solving the following equation
(13)
598 mL(a0) =
*
(14)
(l-p)JA(u)ctu 0
Once the asymptotic distribution of the cost-related risk measure of sample size n at time t > 0, gc (x) is obtained, we are often interested in finding the probability the accumulated loss exceeding the given risk threshold level x, at time t > 0 Pr[C,> C,]=)gCi(u)du
(15)
c,
3
Sample Asymptotic Distribution for Cost-Related Risk
Obviously the asymptotic distribution obtained in section 2 comes with quite strong assumptions which will limit the applicability of it. In this section, we will relax some of the assumptions but also tight some, for example, 9i=\
fO
with probability
p.
(16)
[1 with probability 1-/7, and the loss L, > 0 due to the / failure has a distribution Fj,(*) = Pr [!,<*]
(17)
and mean /Lt, = £[C,]. Therefore, the total loss due to the complete system failure is C{n) = tlYl=£tSlLl (=1
(18)
;'=I
Again because of the key role playing in small sample asymptotics is the cumulants of a distribution and the cumulant generating function, which is related to the moment generating function. We need another look at it. Given a random variable X and a probability distribution function Fx (x), if there exists an a > 0 such that mx(q) = E[eqX], V #| < a , then mx is called the moment generating function (abbreviated as mgf)of random variable X , or equivalently, of the distribution Fx (x). The logarithm of a mgfis called a cumulant generating function of random variable X . Kx(0) = \nmx(0) = Y^J
(19)
;=i Jm
where K.. , j = l,2,--- are called the cumulants. If Y = ^^XlXi weighted sum of m independent random variables, then ' =l
is a
599 m
KY(0) =
(20)
YjKXi(Ai0)
Therefore, for the random quantities Yj (/' = 1,2,•••,«) (defined in Eq. 1), the corresponding cumulant generating function K,(0) =
\nmr,(0)^^
(21)
' J'-
The average cumulant generating function of sum J ^ " = , ^ is then denoted by K„(0) =
(22)
±£K,(0) n (=1
We notice that from Eq. (1), Yt have mixed distributions respectively, with an atom at zero. As n—>-oo, the distribution of £ ) " = 1 ^ can be approximated by a continuous one, denoted by
=O
»»6[-J,J]'
(23)
'
and dK„(0)
lim sup
d0
dK(0) d0
=0
(24)
and lim sup
d2Kn(0) dd2
d2K(0) d02
=0
(25)
If the above-stated regularity conditions hold and let 0O e[—6,5] be a point satisfying K{0o) =
[dK(0)/d0]g_eo=x/n
(26)
Then, fa(x) = gn{x) + 0(n-2)
(27)
where 8» (x) = i pyrnK
ex
P (nK
(9o)-0ox)
(0O)
holds uniformly for x — J ^ / J < (3n where (3 > 0 is a fixed constant.
(28)
600
4
Asymptotic Distribution in Case of Failure and Loss Being Independent
The simplest case we are interested is that the /'* failure mechanism is independent of the loss structure due to the /"" failure. Then by noting that the independence between i9, and Lt leads to K,(e) = lnE[e^]
= In[p, + ( l - p , ) m u (0)]
(29)
and
£(*)=-2>l>+0-/>/K (*)] "
(30)
1=1
Then, under the regularity conditions, Eq. (27) and (28) can be stated as
M*) = g;(x) + o(n-2)
(31)
and g'„ (x) = * pxnK„(d0)
exp (nK„ (0O) - 0ox)
(32)
There are two cases worth to investigate: (1) The loss amounts are essentially known and fixed quantities, i.e., L, are constants. Then,
K(1(0) = - L 2>[>+(l-flK 1 ']
(33)
And accordingly,
W'\t£i^
(34)
and 1
' ȣ[>+(i-*)^] 2
(35)
(2) The loss random variables Z,, are gamma distributed with parameter (a,, /3,), r (a,, /3 f ). The gamma density function is then f,(x;a„{Jf)
= -^-x*'-le-»*,x>0
(36)
and T(ai)=\ua-'ie-"du
(37)
601 and accordingly, the mgfs for L, will be ( l — # / # ) "'
(38) n
/=i
L
J
Note that dm, (0) _q,f
0^
de ~p\
p,
d2mi{6) d92
-(«,+!)
at(a,+\) r
Pf
+1
ex^
)
(39)
1
P.
Therefore, .(«, + i)
Pi { ^)=-inttp,+(\-p,){\-01
PJ [}<)-"•
(40)
and
R-IC)
ifp-(i-A>;(^+(i-/'-)2"';(^)'".(^)+[(i-p->-(^2
(39)
[p,+(i-P,){i-eiP,r1 Then the asymptotic distribution for the total loss C(«)(with sample size n) is approximated by g"n (x). 5
An Asymptotic Distribution for Time-Dependent Failure Cost-Related Loss
The assumption on indicators i9; (in Eq. (2)) for developing the asymptotic distribution of cost-related cost is in some sense a stationary one, i.e., a stead limiting distribution assumption. More realistically, the individual default indicator 3i is time dependent, i.e., 3t= S.(t) . Accordingly, the individual default probability is time-dependent Pl=E[S,{t)]
= p,{t)
(40)
This is will create a wide applicability of cost-related risk-measure. As a matter of fact, for any given component, say, the /'* component (subsystem), if it is a repairable one, then the failure probability
A = £[$(/)] = />,(') In case of Gamma distributed loss Lt, we will have
(41)
602
K„(e,t) = ±£ln\pl(t) + (l-pl(t))(l-0/fi)-*]
(42)
and
{\-p,{t))a,(x^-{a^
K„(e) = - t 1
'
^
\U?L1
(43)
nkPl{t)+{\-Pl(t))i\-eip,r
and p
F: ,e)Jf
-W-*(OH(0Wl-p<('))*
»i(*)*»(0)+[il-p>(Oh(*)] ( 4 4 )
[/*«+o-A(0)o-«/»rI Then the asymptotic distribution for the total loss C («) (with sample size n) is given by g* (x) in terms of Eq. (31). 6
Conclusions
In this paper, we develop the small sample asymptotic distributions for the costrelated risk measure under different scenarios. With these approximate distributions, we can calculate the threshold probabilities and also the asymptotic confidence interval for given level 100(1— a)% (Kolassa and Tanner, 1999). We believe that these theoretical results establish the foundation for reliability analysis and decision-making. We will perform numerical analysis in future research. References 1. 2.
3.
4.
5.
J. Beran and D. Ocker, Small Sample Asymptotics for Credit Risk Portfolio, Journal of Computational & Graphical Statistics 14 (2), 339-351 (2000). C. A. Field and E. Ronchetti, Small Sample Asymptotics. Institute of Mathematical Statistics, USA, Lecture Notes - Monograph Series, 13 (1990). C. A. Field and E. Ronchetti, An Overview of Small Sample Asymptotics. Directions in Robust Statistics and Diagnostics (Part I). Editors, W. Stahel and S. Weisberg, Pringer-Verlag (1991). J. E. Kolassa and M. A. Tanner, Small Sample Confidence Regions in Exponential Families, Journal of the International Biometric Society 55 (4), 1291-1294(1999). M. Todinov, Reliability and Risk Models: Setting Reliability Requirements, John Wiley & Sons Ltd, New York (2005).
ESTIMATION OF FAILURE INTENSITY AND MAINTENANCE EFFECT UNDER TWO DIFFERENT ENVIRONMENTS
JONG-WOON KIM Railroad System and Safety Research Department, Korea Railroad Research Institute, 360-1, Woram-dong, Uiwang-si, Gyeonggi-do, 437-757, Korea WON-YOUNG YUN Department of Industrial Engineering, Pusan National University, Geumjeong-gu, Busan, 609-735, Korea JUN-SEO PARK, JAE-HOON KIM Railroad System and Safety Research Department, Korea Railroad Research Institute, 360-1, Woram-dong, Uiwang-si, Gyeonggi-do, 437-757, Korea The maintenance effect is a peculiar factor of repairable systems. Malik (1979) and Brown, Mahoney & Sivazlian (1983) proposed general approaches for the maintenance effects, where each maintenance reduces the age of the unit with respect to the rate of occurrences of failures. An important problem in failure data analysis has been that all parts of data have not always been collected under similar conditions. We consider an estimation problem of repairable systems under two different environments and Malik's proportional age reduction model. Failure intensities depending on environmental conditions and maintenance effect are estimated by the method of maximum likelihood. Simulation results are presented to illustrate the accuracy and the properties of the proposed estimation method. 1.
Introduction
Maintenance effect is a characteristic factor to repairable systems. Conventional statistical analysis for failure times of repairable systems takes into account one of the following two extreme assumptions, namely, the state of the system after maintenance is either as "good as new" (GAN, perfect maintenance model) or as "bad as old" (BAO, minimal maintenance model). Under GAN assumption, the failure process follows the renewal process and under BAO, the failure process follows the non-homogeneous Poisson process. It is well known in practice that maintenance may not yield a functioning item which is as "good as new". On the other hand, the minimal maintenance assumption seems to be too pessimistic 603
604
in realistic maintenance strategies. From this it is seen that the imperfect maintenance is of great significance in practice. Some authors assumed that maintenance restores the system operating state to somewhere between GAN and BAO. Malik (1979) proposed the approach modeling the improvement effect of maintenance, where maintenance reduces a part of the age, the operating time elapsed from the previous maintenance, while maintenance reduces the total system age in Brown, Mahoney & Sivazlian (1983). Malik's proportional age reduction model is considered in this article. Another important problem in failure data analysis has been that all parts of the data have not always been collected under similar conditions. For example, we often encounter the situation where a piece of equipment may have been used in different environments or may have a different age or modification status. Such different environments obviously affect the equipment's inherent reliability characteristics. Therefore, it may be useful to take account of the environmental factors in equipment reliability modeling. Several models have been used to account for the influence of the environments on a system. In this paper, the Weibull lifetime distribution is considered and it is assumed that the scale parameter of the Weibull distribution is changed by environmental conditions. This assumption is based on the property that the covariate just changes the scale parameter of the Weibull baseline hazard rate; X is changed into Aesn in the proportional hazards model. Figure 1 shows the problems of repairable systems with effective corrective maintenance under two different environments. In Figure 1, the scale parameter in the environment 1 is different to that by the environment 2. We estimate one common shape parameter and two different scale parameters of the Weibull distribution and the maintenance effect. The likelihood function is constructed and the genetic algorithm is used to find a set of value maximizing the likelihood functions. Notation p : the age reduction factor (0 < pt < l) mi : the number of the units operated under therthenvironment (i = 1,2) ntj : the number of failures of the/th units under the Mi environment tj . k : the &th failure time of the/th units under thetihenvironment t, ., : the termination time of the/th units under therthenvironment
605 —•*
*
A.1,1
^ , 1
—X 2,1,1
• • •
x {
A,l,2 •
UA,
'1,^,2
',»!,«,„
H
•••
'2,1,2
'.
?
'.l.*
• Environment 1
1^.*
H
1
2.1."i,i
2,1.*
*• Environment 2
I J ^ , 1
^ , 2
2."i.n:,^
^2^,«
Figure 1. Failure and maintenance process under two different environments
2. Literature Review Malik (1979) proposed an approach for modeling the improvement effect of maintenance, where maintenance is assumed to reduce the operating time elapsed from the previous maintenance in proportion to its age. On the other hand, in BMS's approach (1983), it is assumed that maintenance reduces system age. The two models can be expressed by the virtual age concept suggested by Kijima(1989). Higgins and Tsokos (1981) studied an estimation problem of failure rate under minimal repair model using the quasi-Bayes method. Tsokos and Rao (1994) considered an estimation problem for failure intensity under the Powerlaw process. Coetzee (1997) proposed a method for parameter estimation and cost models of non-homogeneous Poisson process under minimal repair. Park and Pickering (1997) studied an estimation problem to estimate parameters of failure process with failure data of multi-systems. Whitager and Samaniego (1989) estimated the lifetime distribution under Brown-Proshan imperfect repair model (1983). It is assumed that the data pairs (7},Z,-) are given, where T, is a failure time and Z, is a Bernoulli variable that records the mode of repair (perfect or imperfect). Lim (1998) studied an estimation problem using the EM algorithm when masked data (Z, is unknown) are given under Brown-Proshan imperfect repair model (1983). Lim and Lie (2000) extended Lim's work (1998) and considered first-order dependency between two consecutive repair modes. Shin, Lim and Lie (1996) proposed a method for estimating maintenance effect and intensity function in Malik's model. Jack (1997) estimated lifetime parameters and the degree of age rejuvenation when a machine is minimally
606
repaired on failures and imperfect preventive maintenance is also carried out. Pulcini (2000) used the Bayesian approach to estimate overhaul effect and intensity function under minimal corrective maintenance and effective preventive maintenance. Baxter, Kijima and Tortorella (1996), and Calabria and Pulcini (1999) dealt with some properties of the stochastic point process for the analysis of repairable units. Baker (2001) focused on fitting models to failure data. Martorell, Sanchez and Seradell (1999) proposed a age-dependent reliability dependent model considering effects of maintenance and working conditions which was applied to nuclear power plant. 3. Likelihood Function We consider the 2-parameter Weibull distribution whose probability density function and the survival function are given by: /(/)=A/?/'-'exp(-Af')
(1)
R{t) = exp(-Xtfi)
(2)
The method of maximum likelihood is used to estimate parameters. The likelihood function is of the form:
^>m$$&&%&
(3)
The likelihood function and log-likelihood function for the Weibull distribution are: 2
m.
(=i
j=\
L(^,A2,Ap)=nn^v",; nta-p'ri*-!))"" (4)
exp
-4*. -ptjr-ith
+
-p'ueJ *>t(t* -p**Y
«,(iia, +in/?)+\iip-iMtIJk
-ptij(kA
i=i y=i
(5)
•4r-PfJ-^k
-P*«H)Y+lth -P*J
607
4. Experimental Results Simulations are carried out to investigate some properties of the estimation. The number of simulation runs is set to be 100. Failure times are generated for some specified sets of parameters and estimations are performed in the generated failure data. Mean and standard deviation (Std), are calculated to show the performance of the estimation. The genetic algorithm is used in these experiments to find the set of values maximizing the log-likelihood function. Table 1 is the result from the experiments which are performed under four combinations of the number of units and that of failures per unit while the total number of failures is set to be equal. Table 1 shows that the estimation precisions for A,, A2 and p are more sensitive to the number of units than the number of failures per unit. Table 2 shows that the degrees of the difference between two scale parameters (A,, X2) do not highly influence the estimation precision. Table 1. Effect of the number of units vs the number of failures per unit
4 =3 2 5 12 30
Mean 3.13 3.16 3.11 3.!0
30 12 5 2
Std 0.90 0.77 0.66 0.48
KMean 7.30 7.43 7.56 7.14
7 Std 1.46 1.31 1.29 1.15
P-= 2 Mean 2.08 2.09 2.08 2.04
Std 0.24 0.19 0.19 0.17
P = 0.5 Mean Std 0.48 0.17 0.51 0.14 0.49 0.14 0.49 0.15
Table 2. Effect of the difference between two scale parameters
(«- a0,n
X "i
K
2 3 4 5
9 7 6 5
K Mean 1.98 3.10 4.26 5.33
P-= 2
K Std 0.36 0.60 1.06 1.26
Mean 8.39 7.44 6.59 5.64
Std 0.83 1.19 1.43 1.39
Mean 2.00 2.03 2.05 2.13
Std 0.21 0.19 0.21 0.23
= 5) P = 0.5 Mean Std 0.46 0.13 0.50 0.13 0.50 0.15 0.51 0.14
5. Conclusion In this study, the estimation problem is dealt with for repairable systems which are operated under two different environments. Malik's model is considered for the effect of corrective maintenance while there is no preventive maintenance. It is assumed that time to the first failure follows the Weibull distribution and only its scale parameter is changed by environmental conditions.
608
The likelihood function is given for the parameters, A,, A2, p and p, and the genetic algorithm is used in the experiments to find the sets of values maximizing the likelihood function. The simulation experiments show that the estimation precisions for A,, A2 and p are more sensitive to the number of units than the number of failures per unit and the degrees of the difference between two scale parameters do not highly influence the estimation precision. References 1. R. D. Baker, Data-based Modeling of the Failure Rate of Repairable Equipment. Lifetime Data Analysis 7, 65-83 (2001). 2. L. A. Baxter, M. Kijima and M. Tortorella, A Point Process Model for the Reliability of Maintained System Subject to General Repair, Commun. Statist. - Stochastic Models 12, 37-65 (1996). 3. J. F. Brown, J. F. Mahoney and B.D. Sivazlian, Hysteresis Repair in Discounted Replacement Problems, HE Transactions 15, 156-165 (1983). 4. M. Brown, and F. Proschan, Imperfect Repair, Journal of Applied Probability 20, 851-859 (1983). 5. R. Calabria and G. Pulcini, Discontinuous Point Process for the Analysis of Repairable Units, International Journal of Reliability, Quality and Safety Engineering 6, 361-382 (1999). 6. J.L. Coetzee, The Role of NHPP Models in the Practical Analysis of Maintenance Failure Data, Reliability Engineering & System Safety 56, 161168(1997). 7. J. J. Higgins and C. P. Tsokos, A Quasi-Bayes Estimate of the Failure Intensity of a Reliability-Growth Model, IEEE Transactions on Reliability 30,471-475(1981). 8. N. Jack, Analyzing Event Data from a Repairable Machine Subject to Imperfect Preventive Maintenance, Quality and Reliability Engineering International 13, 183-186 (1997). 9. M. Kijima, Some Results for Repairable Systems with General Repair, Journal of Applied Probability 26, 89-102 (1989). 10. T. J. Lim, Estimating System Reliability with Fully Masked Data under Brown-Proschan Imperfect Repair Model, Reliability Engineering & System Safety 59, 277-289 (1998). 11. T. J. Lim and C.H. Lie, Analysis of System Reliability with Dependent Repair Modes, IEEE Transactions on Reliability 49, 153-162 (2000).
609 12. M. A. K. Malik, Reliable Preventive Maintenance Scheduling, AIIE Transactions, 11, 221-228 (1979). 13. S. Martorell, A. Sanchez and V. Serradell, Age-dependent Reliability Model Considering Effects of Maintenance and Working Conditions, Reliability Engineering & System Safety 64, 19-31 (1999). 14. W. J. Park and E. H. Pickering, Statistical Analysis of a Power-Law Model for Repair Data, IEEE Transactions on Reliability 46, 27-30 (1997). 15. G. Pulcini, On the Overhaul Effect for Repairable Mechanical Units : a Bayes Approach, Reliability Engineering & System Safety 70, 85-94 (2000). 16. I. Shin, T. J. Lim and C. H. Lie, Estimating Parameters of Intensity Function and Maintenance Effect for Repairable Unit, Reliability Engineering & System Safety 54, 1-10 (1996). 17. C. P. Tsokos and A. N. V. Rao, Estimation of Failure Intensity for the Weibull Process, Reliability Engineering & System Safety 45, 271-275 (1994). 18. L. R. Whitager and F. J. Samaniego, Estimating the Reliability of Systems Subject to Imperfect Repair, Journal of the American Statistical Association 84, 301-309 (1989).
CHARACTERIZING A NEGATIVE BINOMIAL PROCESS FOR A GAMMA DISTRIBUTED FAILURE RATE*
W. H. KIM, S. E. AHN, C. S. PARK Department of Industrial Engineering, Hanyang University, Ansan 426-791,
Korea
Used as a mixing distribution for a random Poisson parameter, the gamma distribution leads to a negative binomial process. This appears to be a useful model for failure data, particularly for data from a number of repairable systems all of which follow a Poisson process but with different intensities. The hyper-parameters of the gamma distribution have different meanings according to the sources of randomness in the Poisson failure parameter. Two such sources are failure time and failure rate. Random failure time and random failure rate are interpreted in the resulting negative binomial average failure in terms of the number of failures and the intensity of a failure, respectively.
1. Introduction The important role in the area of reliability and life testing of the Poisson distribution and the associated exponential distribution of inter-failure time is well known. It does not appear that the negative binomial model has received much attention as an alternative to the Poisson model in the area of reliability. It has been used in analyzing failure data as count data but to the authors' knowledge the distribution of failure times or inter-failure times of a negative binomial distribution has not been developed for use in the area of reliability. A failure model assumes that failure intensity follow a Poisson distribution with a constant and estimated means. However, the assumption of a constant failure intensity has been questioned recently, since failure is generally uncertain and may even vary with time. Here we assume that failure numbers are described by a Poisson distribution with the random parameter following a gamma distribution. This implies that a negative binomial distribution is obtained by mixing the mean of the Poisson distribution with a gamma distribution (Agrawal, 1996). For example, consider the following situation. A manufacturer supplies quantities (e.g. one truckload) of some product to a central warehouse. The failure rates of such production constitute a renewal process, while the failure intensities for the product arrive at the warehouse according to a Poisson process.
This work has been supported by the research fund of Hanyang University (HY-2004-S), Korea.
610
611 It turns out that the distribution of the number of failures with a failure rate is negative binomial if and only if the failure rate has a gamma distribution (Engel, 1980). The main reasons for mixing the mean of the Poisson distribution with the gamma distribution are: (i) the mathematical tractability resulting from the fact that the gamma distribution is the natural conjugate to the Poisson distribution (Percy, 2002), and (ii) the considerable mathematical flexibility for fitting different distribution patterns. Many authors have found the gamma distribution to be sufficiently versatile for practical application. Burgin (1975) pointed out that the gamma distribution provides a sufficient reference to deal with most of the problems likely to occur in the application of the gamma distribution to failure control. Apostolakis and Mosleh (1979) studied the model for the evaluation of probabilities of rare events by combining the available information with a gamma distribution. Characterizations of the gamma and negative binomial distributions can be found in the literature. Gerber (1991) showed that using the generalized gamma distribution as a mixing distribution for an unknown Poisson parameter leads to a generalized negative binomial distribution. His generalized gamma and negative binomial distributions contain the gamma and negative binomial distributions as special cases. The purpose of this paper is to give an interpretation of the negative binomial failure by considering the sources of variability in the unknown Poisson parameter. Such variability comes from the unknown failure rate and the unknown failure time interval. In Section 2, the negative binomial failure as a mixed Poisson distribution is explained by a Bayesian model, where the mixing distribution is the prior distribution of the unknown Poisson parameter and the failure time is gamma distributed. In Sections 3 and 4, interpretations of the negative binomial failures are given when the sources of variability in the Poisson random failure rate are the unknown failure rate per unit failure time interval and the unknown failure time interval, respectively. 2. The Mixed Poisson distribution The Poisson distribution arises naturally in the study of data in the form of counts. For a homogeneous Poisson process {N(t),t ^ 0} with rate A, the number of events in any interval of length / is Poisson distributed with mean At. That is, for all s,t>0 Vr{N(t + s)-N(s) = y}=e-;l'^-,
v = 0,l,...
(1)
612
Note that the expected value of N(t) is X t which explains why X is called the rate of the process. We denote the distribution (1) as Poisson (y\At). We will further assume that the parameter X t is a random variable with prior distribution p(Xt) . We consider the special case when the random parameter At is gamma distributed with the shape and scale parameters of a (a positive integer) and /?>0 respectively, which is of the form pVt) = £-{kt)a-xe-Wt\ T(a)
(2)
and is denoted as gamma (At\ a,fi). Since the gamma distribution is a conjugate family for the Poisson likelihood (Percy, 2002), and Bayes' theorem states that the posterior distribution is related to the prior and likelihood distributions according to „,i , p(y\Xt)P(Xt) , p(Xt\y) = —— , p(>0
(3)
the posterior distribution of A t conditional on y is gamma (A t\ a + y, p +1). The unconditional marginal distribution p(y) can be obtained according to p(y)=jp(y,At)d(At),
(4)
where the integration is performed over the admissible range of X t. That is, by marginalizing the joint distribution of y and At, p(y,At) = p(y\At)p(At), with respect to A t, we can find the distribution p{y): p{y)=\p(y\At)p(At)d(At) =
Y"{Aty Pa o
y]
r
^t)a-\e-P(x,)d{Xt)
(«)
r(.a)yW + \r> J r(a + y) (a + y-1) a-\
P
n—i
fi + \
(5)
Equation (5) is known as the negative binomial distribution with parameters a and /?/(/? + l). We denote (5) as neg - bin (y \ a, /?). Note that the expected value of the random variable a + y in equation (5) is a times the mean
613 {P + \)lP of a geometric random variable, that is, E[a+y] = a(p + l)/p. For constant a , we have E [y] = a IP. Equation (5) shows that the negative binomial distribution is a mixture of the Poisson distribution where its parameter follows the gamma distribution, which can be expressed as follows: neg-bin{y\ a,p) = jPoisson(y\ Xt) gamma{Xt\a,P) d{Xt) .
(6)
More precisely, when the occurrences of some events (e.g. purchasing orders for a product or demands) follow Poisson (y\Xt) and the random mean rate X t follows a gamma failure time, the distribution of the number of such events is neg-bin (y\ a,P). Consider the random failure rate Xt in equation (1), which corresponds to the random number of failures which occurred during the time interval t when X is the failure rate per unit time interval. We have two cases allowing for the parameter Xt to be random; First, Xt is the random number of failures which occurred with a random failure rate per unit time interval during a fixed failure time interval t . Secondly, Xt is the random number of failures which occurred with a fixed failure rate per unit time interval during a random time interval t. By allowing one of the two variables X and t to be random and the other to be fixed, we have different interpretations of the unconditional marginal distribution in equation (5).
3. The Poisson Random Failure Rate with a Fixed Failure Time Interval When the Poisson random failure rate Xt is gamma{Xt\a,P) distributed and t is a fixed time interval, by the change of variable method we can see that the random failure rate per unit time interval X follows gamma (X\ a,pt). From equation (3) and the same derivation as in (5), the posterior distribution of X becomes gamma(X\a + y,pt + t) and the unconditional marginal distribution of y becomes
[ a-\
){pt+t)
ypt+t)
Note that equation (7) is equivalent to equation (5). Now we interpret the possible meanings of the parameters a and p. Because the prior mean failure rate Xt in the Poisson (y\Xt) likelihood of equation (1) is E (X t) - a I p, or equivalently E (X) = a I p t, the shape parameter a can be interpreted as the total failure number in the previous pt failure
614 time units. On the other hand, when we compare the form of the Poisson (y\ Xt) likelihood in equation (1) p(y\Xt)oz(Xt)y
e~Xt
oc^e-(')A
(8)
p{X)ozXa-xe-(p,)\
(9)
with the form of gamma {X\a,pi)
we can say that the prior distribution in equation (9) is equivalent to a total failure of a-\ in the previous pt failure time units. However, choosing equation (9) for X in equation (8) is still intended for use with alpt as the mean of the random X rather than (a -\)l pt. Note that if al pt converges to a constant, then al pt and (a-\)l pt are approximately the same for large Pt. In the posterior distribution of X given y which is described by gamma (X\ a + y,pt+t), the parameter (a + y) is referred to as the combined number of failures, whereas (Pt+t) is the combined total observed failure time units. The effect of gamma {X| a,pi) as a prior lead is to increase the observed number of failures y by a and increase the observed total failure time units t by pt. This gives a clear interpretation of the effect of the prior distribution in the analysis. The unconditional marginal distribution of y is given by equation (7) with mean and variance given by alp and a(P + \)lp2, respectively. It is further noted that the future failure y has a negative binomial distribution as in equation (7) and its expected value becomes alp. 4. The Poisson Random Failure Rate with a Fixed Failure Rate per Unit Failure Time Interval When the Poisson random failure rate Xt is gamma (Xt\ a,P) distributed and A is a fixed failure rate per unit time interval, by the change of variable method we can see that the random time interval t follows gamma(t\a,pX). By equation (3) and the same derivation as in equation (5), the posterior distribution of t given the failure datum y becomes gamma(t\a + y,px + X) and the unconditional marginal distribution of y becomes p(y) =
a + y-\^ a-\
PX
px + x) px+x
Note that equation (10) is equivalent to equation (5).
(10)
615 Consider a homogeneous Poisson process with a fixed unit time failure rate X and let N(t) be the total number of failures in the time interval [0, /) wherein t is a nonnegative random failure time. In the case of X = 1, Engel and Zijlstra (1980) showed that N{t) = y has a negative binomial distribution with parameters a and /?/(/?+ 1) if and only if t is gamma distributed with parameters a > 0 and p > 0 . This is equivalent to equation (6) when X = 1. For a fixed X, equation (6) becomes neg - bin (v | a, P X) = j Poisson (y | X t) gamma (t\a,pX)dt
.
(11)
The negative-binomial distribution in equation (11) can be interpreted by considering two independent Poisson processes with failure rates X and px . The Poisson likelihood in equation (11) represents the probability that exactly y independent failures, each of which has the exponential failure rate X, occur during a certain failure time interval [0,0- The distribution in equation (11) plays the role of a weighting factor for reflecting the effect of the length of / in determining the probability that exactly y independent failures with the exponential rate X occur during the failure time interval [0,0. This implies that the distribution gamma(t\a,pX) generates the failure time interval t which varies. We can now interpret the right-hand side of equation (11) as the probability that exactly y independent failures with the exponential rate X occur during the failure time interval [0,/) for all possible values of Xt wherein / is generated from gamma {t\ a,pX). Furthermore, the value of t is known as the sum of a independent and identically distributed exponential random variables with a mean of \/(PX). Therefore, the negative binomial distribution from equation (11) shows that exactly y independent failures with the rate X occur until the a th independent failure with the rate px occurs. The above negative binomial failure can be derived from the fact that a homogeneous Poisson process has the properties that the process from any point on is independent of all that has previously occurred, and also has the same distribution as the original process. In other words, the process has no memory, and hence exponential inter-arrival failure times are to be expected. The probability that exactly y failures occur in one Poisson process with the rate X until the a th failure occurs in another independent Poisson process with the rate px can be derived as follows. Let {Nx(t),t>0} and {N2(t),t>0} be two independent Poisson processes with respective failure rates X and p X . Also, let Ty denote the time epoch of the y th failure of the first process, and T% the time epoch of the a th failure of the second process. We seek P{Tly
616 Consider the special case y = a = 1. Since T\ , the time of the first failure of the JV] (0 process, and T,2, the time of the first failure of the N2 (t) process, are both exponentially distributed random variables with respective means MX and 1 ipX , it follows that P{r, 1
x
px+x
=—
•
(12)
This is the probability that one exponential random variable is smaller than another. Let us now consider the probability that two failures (i.e. y — 2) occur in the TV, (t) process before a single failure (i.e. a = 1) has occurred in the N2(t) process. To calculate P{T2
px + x)
' 1 U+i This reasoning shows that an occurrence of a failure is going to be a failure of the 7V,(0 process with probability X/(px + X) or a failure of the N2(t) process with probability PX/(PX + X), independent of all previous failures. In other words, the probability that the N^t) process reaches v before the N2(t) process reaches a is the product of two probabilities: (1) the probability of obtaining exactly a-\ failures from the N2(t) process in the first a + y-\ failures ( a -1 failures from Nx(t) process and y failures from N2(t) process), and (2) the probability of a failure on the a th failure from the N2(t) process. Thus, we have equation (10).
617 5. Conclusion The variances of the negative binomial distributions in equations (5), (7) and (10) are always greater than their corresponding means, in contrast to the Poisson distribution, whose variance is always equal to its mean. The negative binomial distribution is a two-parameter family that allows the mean and variance to be fitted separately, with variance at least as great as the mean. This is why the negative binomial can be used as a robust alternative to the Poisson distribution. In the limit as /?-><» with alp remaining constant, the underlying gamma distribution approaches a spike, and the negative binomial distribution approaches the Poisson distribution. The dominant factor in selecting a prior model for the random parameter Xt in the Poisson distribution of equation (1) is that the selected model represents the analyst's knowledge and experience concerning Xt. That is, the prior distribution should reflect the analyst's prior belief about Xt . The flexibility present in the gamma distribution of equation (2) through the choices of the parameters a and p allows the analyst to select the model that best expresses the current state of knowledge about Xt. It should be noted that a has the same meaning as the random variable y of our interest, i.e. the number of failures, and p plays a role in determining the time interval (see Section 3) or the failure rate (Section 4) in generating the values of the Poisson random parameter. Random failure time and random failure rate interpret the resulting negative binomial average failure in terms of the number of failures and the intensity of a failure, respectively.
References 1. 2. 3. 4. 5. 6.
Agrawal, N., and Smith, S. A., Naval Research Logistics, 43, 839 (1996). Apostolakis, G., and Mosleh, A., Nuclear Science and Engineering, 70, 135 (1979). Engel, J., & Zijlstra, M., Journal of Applied Probability, 17, 1138 (1980). Gerber, H. U., Insurance: mathematics and economics, 10(4), 303 (1991). Percy, D. F., European Journal of Operational Research, 139(1), 133 (2002). T. A. Burgin, Operational Research Quarterly, 26(3), 507 (1977).
ON NONPARAMETRIC TESTING EQUALITY OF RESIDUAL LIFE TIMES JAE-HAK LIM Department
of Accounting Hanbat National Taejeon 305-719, Korea
University,
DONG HO PARK Department of Information andStatistics Hallym Chuncheon 200-702, Korea
University,
A nonparametric procedure is proposed to test the exponentiality against the hypothesis that one life distribution has a greater residual life times than the other life distribution. Such a hypothesis rums out to be equivalent to the one that one failure rate is greater than the other and so the proposed test works as a competitor to more IFR tests by Kochar (1979, 1981) and Cheng (1985). Our test statistic utilizes the U-statistics theory to establish its asymptotic normality and consequently, a large sample nonparametric test is proposed. The power of the proposed test is investigated by calculating the Pitman asymptotic relative efficiencies against several alternative hypotheses. A numerical example is presented to exemplify the proposed test.
1. Introduction In reliability theory the concept of ageing plays a fundamental role of classifying the life distributions. The classes of increasing failure rate(IFR) and increasing failure rate average(IFRA) are based on the monotonicity pattern of failure rate of the distribution and the classes of decreasing mean residual life (DMRL) and new better than used in expectation(NBUE) are classified by the pattern of its mean residual life. The class of new better than used (NBU) is defined by utilizing the stochastic ordering of the residual life length. The dual classes of DFR, DFRA, IMRL, NWUE and NWU are defined by reversing the pattern of failure rate, mean residual life or residual life length. The border distributions for all of the above classes are in the class of exponential distribution. To test the null hypothesis that the distribution is exponential against the alternative that the distribution belongs to one of the above classes, many authors have proposed a number of nonparametric tests in the literature. Proschan and Pyke (1967), Barlow and Proschan (1969) and Ahmad (1975,2001) propose the IFR tests. For the NBU and NBUE alternatives, there 618
619 exist several nonparametric tests by Hollander and Proschan (1972), Koul (1977), Hollander and Proschan (1975) and Ahmad (2001), among many others. Hollander and Proschan (1975) and Aly (1990) also propose some classes of tests against the DMRL alternatives. Since Chikkagoudar and Schuster (1974) consider the problem of comparing two populations in terms of their failure rates, Kochar(1979, 1981) and Cheng (1985) develop several nonparametric procedures for testing the null hypothesis that two life distributions are equal against the alternative that one failure rate dominates the other. Such tests are proved to be useful to compare two used items with different underlying life distributions with regard to their degradation processes as they age. There exist other tests that compare two life distributions with respect to their NBU-ness or IFRA-ness by Hollander, Park and Proschan (1986) and Tiwari and Zalkikar (1991 ). Let X > 0 be a life length random variable with continuous survival function F(x) =P{X > x). Then the residual life time at age / , denoted by Xt, is a random variable with continuous survival function Fi(x) = F(x + t)/F(t), x,t>0. In this paper we propose a new nonparametric procedure to test the equality of two life distributions against the alternative that one life distribution has a greater residual life length than does the other life distribution. The following situation illustrates how our proposed test might be proved useful. Suppose that there are two groups of patients suffering from a certain type of cancer, one group being treated by a certain medical treatment and the other group being a placebo group. To check if the medical treatment is effective in treating that particular type of cancer, the medical authority wishes to test the hypothesis that the residual life length of the treatment group is stochastically greater than the residual life length of the placebo group after the treatment group receives the treatment for a certain length of time, while the placebo group is not treated at all for the same length of time. If such a hypothesis is accepted, then the treatment is verified to be effective and the medical authority draw a conclusion that the medical treatment may prolong the residual life of the patient with that particular type of cancer. To the best of our knowledge, no other tests have been proposed to test the stochastic ordering of two residual life lengths. However, it can be shown that the stochastic ordering of residual life lengths is equivalent to the stochastic ordering of failure rate. Thus, the efficiency of our proposed test can be evaluated by calculating the Pitman asymptotic relative efficiencies with respect to the more IFR tests suggested by Kochar (1979, 1981) and Cheng (1985).
620
In Section 2, we derive the test statistics for detecting greater residual life length property. Section 3 proposes a nonparametric test procedure by proving asymptotic normality of the proposed test statistic. The consistency of the test is also proved. Sections 4 illustrates a numerical example. 2. Test Statistic Let X and Y be nonnegative random lifetimes of two systems with continuous distribution functions F(t) and G(t) and let Xt and Yt be its corresponding residual life times (RLT) at age t, respectively. In this section we develop a test statistic for testing the null hypothesis H0 : F=G (the common distribution is unspecified) versus the alternative hypothesis SI
Ha : X, < Yt for all t > 0 , with strict inequality holding for some t, based on two complete random samples Xx ,...,Xm and Yx ,...,Yn taken from F and G , respectively. We assume that X_ = ( Xt,...,Xm ) and Y_ = ( 7 , , . . . , Yn) are independent. Ha states that the residual life time at age t is stochastically greater when the underlying distribution is G than when the underlying distribution is F . Note that under Ha, ~F{x + t)G^(t)
A(F,G)
oooo
= 2 \\{G{x + t)F{t)-F{x
+ t)G(t)}dG{x + t)dF{t)
00 oo
2
= \G (t)F(t)dF(t) 0
Q000
- 2 | \F{u)G
(t)dG(u)dF(t).
0/
Under H0 , A(F,G) = 0 and under Ha , A(F,G) > 0 . Thus A(F,G) can be used as a measure of departure from HQ and the larger value of A(F,G) indicates that Yt is stochastically greater than Xt. Let Fm{Fm) and Gn(Gn) denote the empirical (survival) functions formed by random samples X_ and Y_, respectively. A nonparametric test statistic for testing H0 versus Ha can be formed by substituting Fm and Gn in place of F and G in A ( F , G ) , respectively , as
621 Am,„=A(Fm,G„) CO
COCO
\G„2 (t)Fm (t)dFm(t) - 2 \ \I(u > t)Fm {u)G
n(t)dG„(u)dFm(t)
00
0
1 - j - j - d Z Z Z t / d V , >Xh)I(Yh m n h h J\ h - 21{Yh >Xh )I{Xh
>Xh)I{Xh>Xh)
>Yh ) I(Yh >Xh )]}
1
m
2n 2-zzzz^^,^2,y7l,^),
Where
hhh h I(a>b)=\ifa>b,=0ifa{Xx,X2J,,Y2) = I{Yx>X,)nY2>Xx)
-2I(YX>XX)I(X2>YX)I{Y2>
and I{X2>XX)
Jf,).
Note that Am n is a U-statistic utilizing the symmetric kernel for an estimable parameter (j){Xy ,X2,7, ,Y2 ) . Significantly large values of A m n may indicate that Y has a stochastically greater residual life at age t than X. Thus, our test is to reject H0 in favor of Ha if Am is too large. In the following section, we establish the asymptotic normality of Am „ and propose the two-sample residual life test. 3. Two-sample RLT Test The limiting distribution of Am n can be established by applying the Hoeffding's (1948) U-statistics theory. To evaluate the asymptotic variance of Am , we need the following conditional expectations. Direct calculations yield E{
°°
= G (*,)*TO"2(?(*,) J[G(A-,)-G(«)]dF(«), *i
£{<*(*,,*2,r,,y2) I *2} = fG2(u)dF(u)-2 0
\G(u)[G{u)-G(X2)]dF(u), 0
622
E{0(Xx,X2,Yx,Y2)
|F,}=
p(u)F{u)dF(u)-2F{Yx)p{u)dF(u), 0
0
E{{XX,X2JX,Y2)\Y2) Y2
Y2
_
= JG(u)F(u)dF(u)-2[ 0
p(u)F(u)dG(u) 0
oo
+ JF(u)F(Y2)dG(u). Thus, under H0 : F-G,ihc a] = Varij^
asymptotic null variance is obtained as
E{{XX ,X2 Jx ,Y2)
\Xk}]
*=i
+ Var[fjE{^(Xx,X2,Yx,Y2)
\ Yk}]
k=\
+ ^F\xx)}2
= Var{l-F(Xx) 2 105
2 105
+Var{^-F(YX)+^F3(YX)}2
4 105 '
The asymptotic distribution of Amn can be readily obtained by direct applications of Hoeffding's (1948) two-sample U-statistic theory (cf. Randies and Wolfe (1979)). The asymptotic null distribution of Amn is summarized as Theorem 3.1. Let X - lim ml N for 0 < /i <1. Then, the limiting distribution of N'2 {Amn - A(F,G)) is normal with mean 0 and a finite variance as N —>oo. Under H0, the limiting distribution of (105Ny2 Amnl2 is a standard normal distribution. Due to Theorem 3.1, the approximate a -level test is to reject H0 in favor oftfaif (l05N/lAm,„ .
>z ,
623
where za is the upper a - percentile point of the standard normal distribution. This is referred to as a two-sample RLT test. Assuming that F and G are continuous, Am is strictly greater than 0 under Ha and hence, the asymptotic normality of Am„ ensures the consistency of the two-sample RLT test against the class of (F, G) pairs for which Ha holds. The asymptotic unbiasedness of the two-sample RLT test can be proved by showing that Pr (
(105A0^„
for sufficiently large N,
>za)>a
whenever the alternative hypothesis holds true.
We first observe that pr((105A0
2
Am,„ > Z a ) = P r ( i V X ( A m n
-A(F,G)>
If A ( F , G ) = 0, the probability is equal to a. Under Ha, A(F,G) > 0 and thus, by the asymptotic normality of Am , the result follows for sufficiently large TV. 4. An Example In this section, we illustrate the use of two-sample RLT test based on the data set from Proschan(1963). Table 1 shows the life lengths (in hours) of the airconditioning systems of two different planes. Direct calculations yield Jm,n =—^TT.I.ZI.
and yl\05*NJmn
= -6.7209797£- 3
i\i2jlj2
12 = Vl05*45(-0.006721)/2 = -0.2310. Thus, the p-value
is obtained as P ( Z > -0.2310 ) = 0.5913, which strongly suggests to accept H0. The test suggests the equality of random residual life lengths of two underlying distributions, which agree with Hollander, Park and Proschan's
624
(1986) and Lim, Kim and Park's (2004) test results regarding the more NBUness. Table 1. Lifelengths of air-conditioning systems of planes 8045 and 7049
X (plane8045) 230 54 209 34 152 32 134 27 102 14 67 14 66
310 208 208 186 156 130 118
79 76 70 62 61 60 59
61 59
101 90
57
84
56 49 44
Y (plane7909)
44 29 26 25 24 23 20 14 10
References 1. L A . Ahmad, A nonparametric test for the monotonicity for a failure rate function, Comm. Statist. 4, 967-974 (1975). 2. I. A. Ahmad, Moment inequalities for aging families of distributions with hypotheses testing applications, Journal of Statistical Planning and Inference 92, 121-132 (2001). 3. E. A. A. Aly, Tests for monotonicity properties of the mean residual life function, Scand. J. Statistics 17, 189-200 (1990). 4. R. E. Barlow and F. Proschan, A note on test for monotone failure rate based on incomplete data, Ann. Math. Statist. 40, 595-600 (1969). 5. K. F. Cheng, Tests for the equality of failure rates, Biometrika 72, 211-215 (1985). 6. M. S. Chikkagoudar and J. S. Schuster, Comparison of failure rates using rank tests, J. Am. Statist. Assoc. 69, 411-413 (1974). 7. W. A. Hoeffding, A class of statistics with asymptotically normal distribution, Ann. Math. Statist. 19, 293-325 (1948). 8. M. Hollander, and F. Proschan, Testing whether new is better than used, Ann. Math. Statist. 43, 1136-1146 (1972). 9. M. Hollander, and F. Proschan, Tests for the mean residual life, Biometrika 62,585-593(1975). 10. M. Hollander, D. H. Park, and F. Proschan, Testing whether F is more NBU than is G, Microelectronics and Reliability 26, 39-44 (1986b).
625
11.
12. 13. 14. 15. 16. 17.
S. C. Kochar, Distribution-free comparison of two probability distributions with reference to their hazard rate, Biometrika 66, 437-441 (1979). S. C. Kochar, A new distribution-free test for the equality of two failure rates, Biometrika 68, 423-426 (1981). H. L. Koul, A test for new better than used. Comm, Statistist. A-Theory Method 6, 563-573 (1977). J. H. Lim, D. K. Kim and D. H. Park, Testing equality of NBU-ness at a specified age, Advances and Applications in Statistics 4, 97-108 (1967). F. Proschan and R. Pyke, Tests for monotone failure rate, Proc. Fifth Merk. Symp. Math. Statist. Prob. 3, 293-312 (2004). R. H. Randies and D. A. Wolfe, Introduction to the theory of nonparametric statistics, John Wiley, New York (1979). R. C. Tiwari, and J. N. Zalkikar, A new measure of IFRA-ness and its application to two sample test, Statistics 22, 3, 419-430 (1991).
P A R A M E T E R ESTIMATION OF T H E S H A P E P A R A M E T E R OF T H E G A M M A D I S T R I B U T I O N FREE FROM LOCATION A N D SCALE INFORMATION
H I D E K I N A G A T S U K A , HISASHI Y A M A M O T O Faculty of System Design, Tokyo Metropolitan University, 6-6 Asahigaoka, Hino-shi, Tokyo 191-0065, Japan E-mail: [email protected], [email protected] TOSHINARI KAMAKURA Department of Science and Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan E-mail: kamakura@indsys. chuo-u. ac.jp
The gamma distribution, having location (threshold), scale and shape parameters, is used as a model for distributions of life spans, reaction time, and for other types of non-symmetrical data. It has been said that the inference for the threeparameter gamma distribution is difficult because of nonregularity in maximum likelihood estimation although numerous papers have appeared over the years. On the other hand, the methodology for inference for the two-parameter gamma distribution have been established over the years. It is usual to avoid fitting the three-parameter gamma distribution and to fit the two-parameter gamma distribution to data in practice. In this article, we propose a new method of estimation of the shape parameter of the gamma distribution based on the data transformation free from location and scale parameters. The method is easily implemented with the aid of table or graph. A simulation study shows t h a t the proposed estimator performs better than the maximum likelihood estimator of the shape parameter of the two-parameter gamma distribution when the threshold is existent even though that is close to zero. Key Words: Gamma distribution; Threshold parameter; Location and scale parameter free; Maximum likelihood estimator; Order statistics; Method of moments.
1. Introduction The gamma distribution, having location (threshold), scale and shape parameters, is used as a model for distributions of life spans, reaction time, and for other types of non-symmetrical data. A random variable X has a gamma distribution if its probability density function (pdf) and cumulative 626
627
distribution function (cdf) are of forms: J(.;a,A7)
G{x\a,P,i)=
= ^
^
^
I g(z\<*,p,i)dz,
^
,
a>0,/J>0,x>7>(l)
a > 0,/3 > 0,x > 7.
(2)
This distribution includes x 2 distribution, Ehrang distribution and exponential distribution. It depends on three parameters a, (3 and 7. If 7 is equal to zero, the distribution is termed the two-parameter gamma distribution. The three-parameter gamma distribution has the problem of nonregular estimation when the three parameters are unknown. When a < 1, the likelihood function tends to infinity as 7 approaches the smallest observation. When a > 1, there exists a local maximum. However, if a is near to 1, the unstable results arise, even though it exceeds 1. Johnson et al.- recommends not to use the maximum likelihood estimators unless it is expected that maximum likelihood estimate of a is at least 2.5. It has been said that the inference for the three-parameter gamma distribution is difficult although numerous papers have appeared over the years. The fitting three-parameter gamma model to data has been avoided in many cases. On the other hand, two-parameter gamma distribution is used frequentry in practice. The methodology for inference for the two-parameter gamma distribution have been established over the years. The details of that, especially maximum likelihood estimation and the properties of maximum likelihood estimators, is well described in Bowman and Shenton ? . Johnson et aV mentioned that it is possible to assume 7 is zero for fitting the threeparameter gamma distribution in many cases. It is usual to avoid fitting the three-parameter gamma distribution and to fit the two-parameter gamma distribution to data in practice. In this article, we propose a new method of estimation of the shape parameter of the gamma distribution based on the data transformation free from location and scale parameters. The method do not require complicated calculation and iterative method and is easily implemented with the aid of table or graph. We focus on estimation of the shape parameter of the gamma distribution. The other parameters are easily estimated when the shape parameter is estimated. A simulation study shows that the proposed estimator performs better than the maximum likelihood estimator of the shape parameter of the two-parameter gamma distribution when the threshold is existent even though that is close to zero.
628
2. W - t r a n s f o r m a t i o n Let Xi,i = l , . . . , n (n > 3) be independently distributed with common distribution function (2) and X^, 1 < i < n, be the order statistic of order i. Now, we consider the following transformation:
w
=
X(j) - X,(1)
2,...,n-l.
^ x<, x (1)
(3)
We refer to this transformation as "W-transformation" (Nagatsuka and Kamakura' ). It is shown that the W-transformation statistics (3) are independent of the location 7 and scale /? as follows. For i — 1 , . . . , n, let Yjj) be ith order statistics of the random variables distributed identically with the standard distribution of (2) denoted by F ( x ) ( = G(x; a, 1,0)) then,
_ x(i) - x(1) _ 7 + pY(i) - ( 7 + jgy(1)) _ r w - r((i)
W{1) =
- ^ (n)
-
-X'I (i)
7 + PY(n) - (7 + /3F(1))
^(n) - Y{1
For the gamma distribution (2), we obtain the marginal distribution function of W(i) as follows:
= Pr(^)< W )=Prf^ W "^ (1) < w VA(n)-A(i) = p r ( y ( 1 ) < y w < « ; ( K ( n ) - K ( 1 ) ) + y(1)) /*oo
= / J0
n-2
/>oo
/
n(n - l)/(u)/(t/)
Ju
.fc=i-l
x {F{{l-w)u
+
wv)-F(u)}k + wv)}n-k-2
x {F(v) -F((l-w)u = /
n(n-
Jo Ju
n-2
E .fc=i-l
{ F ((1 - u/) F-\u)
dvdu
n-2 Jfc + w F~\v))
- «}' n-fe-2
dvdu,
(4)
where the function /(a;) is the pdf of standard distribution of (2), that is, f(x) = g(x;a, 1,0).
629
3. Estimation of a 3.1. The Mixture ofW(i)
Distribution
of the Marginal
Distribution
We consider the mixture of the marginal distributions of W^), i = 2 , . . . , n1 as follows: F{n\w)
= / / n{n-l)(v-u)n-3F((l-w)F-1{u) Jo Ju
+ wF-1{v))
n-2 Then, the mean of the mixture distribution (5) is given by
dvdu (5)
Mr. (a) /»00
/
h-F^(w)\
1+ n-2
dw
-„(„-!)/ / J* u> F l Jo Ju {v)-F-l{u)
/ JF-Hu)
F(t)dtdvdu. (6)
3.2. Estimation
of a
The mean of the mixture distribution (5) is also expressed as -i
n—1
5 > [ W ( i ) ] = i5[Wl.
(7)
whereW =£-£,?=! WW. From Eqs. (6) and (7), we obtain the following equation: /i„(a) = E[W}.
(8)
From Eq. (8), we present an estimator a, of a, which satisfies the following equation: Vn(a) = W.
(9)
It is readily seen that the estimator a are distributed independently of /3 and 7.
630
4. Simulation Study In order to compare numerically the maximum likelihood estimator of the shape parameter of the two-parameter gamma distribution (MLE) and the proposed estimator (Prop.) under the condition that 7 is near to zero, we performed a simulation study. Without loss of generality, we fix (3 = 1. We let a = 0.5, and 1.5 (we consider two very different shapes of the distribution) and 7 = 0.0(0.1)0.2(0.2)1.0. Performance of the estimators will depend on the number of the sample size. We let n = 10,25,50 and 100. For each configuration, 10000 simulations were carried out. The random number generator is from IMSL. The samples were discarded when the procedures failed to find parameter estimates. Table 1 shows the rate of successful runs for each estimator of a. For a = 0.5 and n = 25,50 and 100, the rates of successful runs for the proposed estimator are equal to or larger than the corresponding maximum likelihood estimator. Otherwise, that for the maximum likelihood estimator are equal to or larger than the corresponding proposed estimator. Table 2 shows the bias and root mean squared error (RMSE) for each estimator of a. The results are summarized as follows: • As the sample size increases, the performances of both estimators improve. • A s a becomes larger, the performances of both estimators change for the worse. • As 7 becomes larger, the performance of the maximum likelihood estimator changes for the worse. • the performance of the proposed estimator doesn't vary according to values of 7. • For 7 = 0, the maximum likelihood estimator has less bias and RMSE than the corresponding proposed estimator. • For 7 > 0.1, the proposed estimator has less bias than the corresponding maximum likelihood estimator. • For a = 0.5 and 7 > 0.2, the proposed estimator has less RMSE than the corresponding maximum likelihood estimator. • For a = 1.5 and 7 > 0.8, the proposed estimator has less RMSE than the corresponding maximum likelihood estimator. These results suggest that the proposed estimator is generally, but not uniformly, better than the maximum likelihood estimator when the threshold is existent.
631 Table 1. 7 0.0
0.2
0.4
n 10 25 50 100 10 25 50 100 10 25 50 100
a= =0.5 Prop. MLE 1.0000 0.9825 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9794 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9963 0.9850 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
The rate of successful runs (/? = 1.0) a—=1.5 MLE Prop. 1.0000 0.8746 1.0000 0.9849 1.0000 0.9995 1.0000 1.0000 1.0000 0.8679 1.0000 0.9826 1.0000 0.9988 1.0000 1.0000 0.9996 0.8707 1.0000 0.9846 1.0000 0.9988 1.0000 0.9999
7 0.6
0.8
1.0
a= =0.5 MLE Prop. 0.9845 0.9820 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000 0.9587 0.9826 1.0000 0.9992 1.0000 1.0000 1.0000 1.0000 0.9207 0.9842 1.0000 0.9966 0.9998 1.0000 1.0000 1.0000
a—1.5 MLE Prop. 0.9996 0.8730 1.0000 0.9828 0.9994 1.0000 1.0000 1.0000 0.9982 0.8737 0.9832 1.0000 0.9998 1.0000 1.0000 1.0000 0.8695 0.9968 0.9826 1.0000 0.9994 1.0000 1.0000 1.0000
Note: Based on 10000 random samples.
5. A n Example In order to illustrate the estimation procedure proposed in this paper, we fit the gamma distribution to the data (Davis?) using the proposed method. The data are hours to failure for 188 transmitting tubes. Greenwood and Durand ? attempted to fit an two-parameter gamma curve to the failure data and estimated a = 1.3538 and /3 = 23.79. The mean of the observed value of the W-transformation statistics (3) is 0.201631. Then the estimate of the shape parameter is given as the solution of Eq.(9). The estimate is also obtained easily from table or graph which show the values of /x„(o:). We calculate a = 1.31433. We also read a = 1.3 from Figure ?? (If greater accuracy is required, linear interpolation in table which shows the values of ^n(a) should be used). Then the maximum likelihood estimates of j3 and 7 of the gamma distribution (2) with a = 1.31433 are obtained as $ = 23.322 and 7 = 1.543. 6. Concluding remarks The new estimator of the shape parameter of the gamma distribution free from location and scale parameters was proposed. A simulation study showed that the proposed estimator performed better than the maximum likelihood estimator for two-parameter gamma distribution when the threshold is existent. The success rates of finding the parameter estimates were shown. It is seen that the failure rate for the proposed estimator is larger than the corresponding maximum likelihood estimator. As a future work, we will improve
632 Table 2. 7
n
0.0
10 25 50
0.1
100 10
25 50
0.2
100 10
25 50
0.4
100 10
25 50
0.6
100 10
25 50
0.8
100 10
25 50
1.0
100 10
25 50 100
Simulation Summary (/3 = 1.0)
a=0.5 MLE Prop. 1.035 1.291) 0.452 (2.077) 0.769 0.827) 0.106 ( 0.562) 0.696 0.721) 0.048 0.306) 0.666 0.678) 0.026 0.230) 1.035 1.291) 0.452 2.077) 0.769 0.827) 0.106 0.562) 0.696 0.721) 0.048 (0.306) 0.666 ' 0.678) 0.026 0.230) 1.908 2.454) 0.499 2.204) 1.368 1.476) 0.104 0.573) 1.236 [ 1.279) 0.044 0.302) 1.181 , 1.201) 0.027 0.230) 3.832 [ 4.919) 0.512 2.206) 2.723 [ 2.983) 0.105 0.616) 2.432 [ 2.531) 0.048 0.311) 2.311 ' 2.355) 0.024 0.224) 5.993 ( 7.519) 0.468 2.161) 4.307 [ 4.766) 0.102 0.550) 3.842 [ 4.028) 0.044 0.291) 3.630 ( 3.713) 0.031 0.230) 8.142 ( 9.890) 0.486 2.174) 6.176 ( 6.915) 0.106 ' 0.665) 5.464 ( 5.760) 0.043 ' 0.301) 5.108 ( 5.237) 0.027 0.226) 9.913 (11.612) 0.466 2.069) 8.226 [ 9.191) 0.098 0.503) 7.228 ( 7.645) 0.043 0.299) 6.805 ( 6.995) 0.027 ( 0.228)
a=1.5 MLE Prop. 1.005 ( 1.676) 1.210 ( 4.458) 0.564 0.782) 0.820 (3.012) 0.458 0.575) 0.415 1.743) 0.406 0.466) 0.187 0.870) 1.005 1.676) 1.210 4.458) 0.564 0.782) 0.820 (3.012) 0.458 0.575) 0.415 ( 1.743) 0.406 0.466) 0.187 0.870) 1.472 2.189) 1.238 ( 4.544) 0.944 1.149) 0.907 3.268) 0.803 0.897) 0.367 1.624) 0.747 0.795) 0.163 0.861) 2.359 3.100) 1.155 4.326) 1.683 1.883) 0.775 2.927) 1.512 1.605) 0.398 1.728) 1.429 1.474) 0.185 0.915) 3.376 4.242) 1.164 4.385) 2.512 2.750) 0.853 3.111) 2.261 2.365) 0.424 1.750) 2.164 2.213) 0.191 0.908) 4.474 5.479) 1.173 4.354) 3.361 ' 3.638) 0.936 3.315) 3.075 3.198) 0.395 1.680) 2.947 3.006) 0.208 0.948) 5.609 6.740) 1.182 4.360) 4.327 4.661) 0.858 3.120) 3.930 4.076) 0.418 1.767) 3.766 ( 3.836) 0.168 ' 0.871)
Note: Entries are the bias and (RMSE) for each estimator based on 10000 random samples.
this point. The proposed estimation procedure were illustrated with an example. We suggested to use table or graph which shows the values of j i n ( a ) - With the aid of table or graph, the estimates is easily obtained without complicated calculation and iterative method. The maximum likelihood estimator of a of two-parameter gamma distribution does not show the good performance when the threshold is existent. The almost estimators of a of three-parameter gamma distribution that have been proposed up to the present have some problems and require the simultaneous solution of nonlinear equations by iterative method. We recommend the use of the proposed estimator.
633 0.6
0.55 0.5
n=10 n=25 : n=50 :
\
n=200
!
!
I
i
]
:
!
0.45 0.4 0.35
S
0, 0.25 0.2 0.15 0.1 0.05
0
:
'•
X - ~ ~ — —
-
:
••"
""""
j^-::.''--- --''V7'V7
:
:
f//fy''r-' I
p-: :. I , , ,
0
,
1
2
3
4
I
5
6
:
I , , , i I , . , , I , i i i I ,
7
8
9
10
11
12
13
, I
14
15
a Figure 1. Graphs of p„(m) (n = 10, 25, 50, 100, 200)
References 1. Bowman, K. O. and Shenton, L. R., Properties of estimators for the gamma distribution, DefcA:er(1988). 2. Davis, D. J., An analysis of some failure data, J. Amer. Statist. Assoc. 47 113-150 (1952). 3. Greenwood, J. A. and Durand D., Aids for fitting the gamma distribution by maximum likelihood, Technometrics 2, 55-65 (1960). 4. Johnson, N. L. , Kotz, S. and Balakrishnan, N., Continuous Univariate Distributions, Vol.1, 2nd ed. , Wiley (1994). 5. Nagatsuka, H. and Kamakura, T., A new method of inference for Weibull shape parameter, J. Rliab. Eng. Assoc. Jap. 25, 583-597 (2003). 6. Nagatsuka, H. and Kamakura, T., Parameter estimation of the shape parameter of the Castillo-Hadi model, Commun. Statist. Theory Meth. 33, 15-27 (2004).
C O M P A R I S O N OF TWO I N F O R M A T I O N S T R U C T U R E S W I T H NOISE IN B A Y E S I A N DECISION ANALYSIS *
C. H. Q I A N A N D J. C H E N College of Management Science and Engineering, Nanjing University of Technology, 200 Zhongshan Road North, Nanjing, 210009, China E-mail: [email protected] T. NAKAGAWA Department
of Marketing and Information System, Aichi Institute of Technology, 1247 Yachigusa Yagusa, Toyota, Aichi,470-0392 Japan E-mail: [email protected]
In order to raise the efficiency of Bayesian decision analysis, the comparison analysis of added information structure with noise are considered by using the correlative coefficient of information structures. The entropy measures the indefiniteness of information system in information theory. It is possible to conduct the evaluation of information reliability, which through denning the decrease of object's indefiniteness by using of conditional entropy, and further to measure the volume of information content in the added information structures. An expression about information correlation and distance that matched to Bayesian decision analysis is derived by considering the standardization of information content. Such analysis demonstrates the correlative coefficient of information structures can be used to compare and appraise the information structures with noise.
1. Introduction Decision making has been widely studied by many authors. Refs. 1-3 present maximum expected utility theory forming a basis normative decision theory under uncertainty. A Bayesian decision maker (DM) has a priori probability distribution over the true state of nature. Upon receiving a signal that depends on the state realized by an information structure, the "This work is supported by NSFC(70471017) and Humanities & Social Sciences Research Foundation of MOE of China.
634
635
DM updates her priori belief.4 Blackwell 5~~6 compared different information structures, and many authors have done a series of significative work on this topic. 7 - 1 0 But there are several limitations in the application of information comparison. The well-known measure of uncertainty for a purely probabilistic system is the Shannon's entropy.11'12 Based on the entropy concept, several authors have proposed many different methods for modeling uncertainty. 13-16 Qiu 15 defined the concepts of sensitivity, value sensitivity and precision of information to compare different information. In this paper, we present an expression about information correlation and distance matched to Bayesian decision analysis by considering the standardization of information content in generalizing based on Refs. 14 -15. Such analysis demonstrates the correlative coefficient of information structures can be used to compare and appraise the information structures with noise.
2. Setup Consider decision maker under uncertainty. Let X = {xi, X2, • • • ,xn} be a finite set of states of nature, and let P(X) = {p{x\),p{x2), • • • ,p{xn)) be a priori probability distribution over X. It is assumed that P(X) assigns a positive probability to any state. That is, p(x) > 0 for every x G X. The set of actions available to the DM is denoted by A. The DM chooses an action from a € A and obtains a utility which depends on the realized state, x. This utility is denoted by u(a,x). We assume that the DM does not observe the true state of nature and takes an action that maximizes her expected utility. Then the expected utility, in the absence of information services, is given by V0 = Eu(a*,x)
= m&xy^ p(xi)u(a, Xi),
(1)
i
where a* is an optimal action. The DM uses information services in order to make a decision. We assume that an information structure is a pair (Y,P(Y\X)), where Y = {2/1.3/2, • • • , Vm} is the set of signals ( e.g., message, observation, or a family of mutually exclusive sets) and P(Y\X) is a collection of distribution on Y, one for each state. P(Y\X) is a stochastic matrix with n rows, the i-th row of P{Y\X) is the distribution over signals given the state x, and cell p(Vj\xi) of the i-th row is the conditional probability of receiving the signal yj 6 Y given that the state of nature is x*.
636
Before taking an action the DM from one of information services observes a signal y &Y which is correlated to the state of nature by P(Y\X). Upon receiving a signal yj, using Bayes rule, the DM updates her priori probability to the posteriori probability p(Xi|Wj,
~EiP(wWp(xi)' and then chooses actions so as to maximize her expected utility. Eu{a**\yj,x)
= max ^S2p(xi\yj)u{a,xi)1
( )
(3)
where a**\yj is an optimal action given the signal yj. The marginal probability p(yj) — J2iP(yj\xi)p(xi) i s occurrence probability for the signal yj, and we have P(Y) = P(X)P(Y\X).
(4)
Then, the expected utility is given by VY = Y^p(yj)Eu(a**\yj,x),
(5)
3
in the presence of the information service. Eu(a**\yj,x) > ^2iP(xi\yj)u(a*,Xi), we have ^2p(yj)Eu(a**\yj,x) 3
According to Eq.(3),
> ^p{yj)'Y^P{xi\yj)u(a*,xi) 3
= Eu(a*,x).
(6)
i
It shows that after accepting the signal from (Y,P(Y\X)), the expected utility will not reduce in average. In particular, if there exists Xi which satisfies p(xi\yj) = 1 for any yj, the mapping
(7)
given the state xt and the expected utility in the presence of the full information is given by Vx = ^2p(xi)u{a"*\xi,xi). i
For any P(Y\X),
it is evident that Vx > VY.
(8)
637
In general, there exists yj which satisfies p(xi\yj) < 1 for any a;;. It shows that the signal from (Y, P(Y\X)) is information with noise. P(Y\X) is the noise added on X, at the same time, we call Y is not better than X. Obviously, comparison of the qualities of information with noise is the same as of information structures. 3. Comparison of the Information Structures with Noise It is necessary to compare and choose the information before using. In general, the comparison of the information, indirectly by comparing the expected utility of the services, includes priori probability, the set of actions and the utility function. The utility gains V~x — Vo,Vy — Vo are called the value of full information and information with noise, respectively. But, we can not examine all the plans of action and the utility of each action in their states is difficult to evaluate in advance. These limit the comparison and appraising of the information structures. Furthermore, as to the provider of the information, the plans of action and their utility is indefinite which leads to no accordance of comparison. In this view, we should find another way to compare and appraise information. The well-known measure of uncertainty for a purely probabilistic system is the Shannon's entropy. It is possible to conduct the evaluation of information reliability formally by putting utility u(a,x) by — log(a;). Firstly, define the decrease of object's uncertainty by using of conditional entropy, then measure the volume of information content in the added information structures. The amount of information content is one of the criterions of information appraising. It will be more valuable after standardization. In this paper, an expression about information correlation and distance is derived by considering the standardization of information content, which can be used to compare and appraise the information structures. 3.1. Entropy
and Informational
Content
Let us represent the informational content of signals Y by the Shannon's entropy. 11 The uncertainty of states can be expressed by entropy
H{X) =-Y^pixjlogpixi),
(9)
where OlogO = 0. The entropy function attains its maximum where the uncertainty is maximal: at the distribution P(X). The uncertainty becomes
638
0 from H{X) when the DM obtains full information about the state realized. For the information structure with noise, upon receiving a signal yj from information structure (Y, P(Y\X)), the uncertainty of states becomes H
(X\yj) = ~ X]p(zi|2/j) logp(xi\yj),
(10)
i
the expected uncertainty is given by
H(X\Y) =-Y^piy^HWyj).
(11)
3
The conditional entropy H{X\Y) must be interpreted as the uncertainty about the state space X given the information provided by the structure (Y,P(Y\X)): It is evident that H(X\Y)
< H(X).
(12)
The equality holds iff (if and only if) X and Y are statistically independent, i.e., p{xi\yj) = p{xi) for any i,j. This equality denotes that the result of the observing of Y is helpless at reducing the uncertainty of X. Eq.(12) shows that after accepting the signal from (Y,P(Y\X)), the expected uncertainty of X will be partly eliminated in average. Obviously, H(X)—H(X\Y) is the decrease of uncertainty about the state space X given the information provided by the structure (Y,P(Y\X)). It must be interpreted as the volume of information content about state X in the added information structures (Y,P(Y\X)). In fact, we have 15 I(X, Y) = H{X) - H{X\Y)
= H(Y) - H{Y\X).
(13)
Definition 3.1. Mutual information content I(X,Y) defined by Eq.(13) is a measure of the statistical dependence between the two information X andy. For the full information, there exists x, which satisfies p(%i\yj) = 1 and H{X\yj) = 0 for any %, then H{X\Y) = 0, denoted X C Y. It shows that the information Y can eliminate the uncertainty of X completely. If H{X\Y) = H(Y\X) = 0, then call X = Y. Definition 3.2. Complemental information content D(X,Y) is a measure of the statistical distance between the two information X and Y, is defined by D{X,Y)
= H(X\Y)
+
H(Y\X).
(14)
639
It is evident that H(X) = I(X,Y)
+ H(X\Y),
(15)
and H(X,Y)
= I(X,Y)
+ D(X,Y),
(16)
where joint entropy H(X, Y) = - J^ij P(xi> Vj) ^°EP{xi-,Vj) denotes the uncertainty about state space 1 x 7 , and p(xi,yj) is the joint probability. 3.2. Information
Reliability
and
Comparison
Definition 3.3. If Y is an added information, we define reliability function of Y as px(Y) = I(X,Y)/H(X), which is a measure of the correlative coefficient between the two information X and Y. If Y and Z are different added information, and pxiY) > px{Z), then call Z is not better than Y. Let dx(Y)
= H(X\Y)/H(X),
we have the following properties:
(1) 0 < px(Y) < 1, 0 < dx(Y) < 1 and px(Y) + dx(Y) = 1. (2) px(Y) = 0 and dx{Y) = 1 iffX and Y are statistically independent. (3) px{Y) = 1 and dx(Y) = 0 iff X C Y. Blackwell compared different information structures. He defined two orders over the set of information structures: one in terms of decision problems and the other in purely probability terms. He showed the equivalence of these two orders. 5 - 6 Definition 3.4. (Blackwell) (Y,P(Y\X)) is more information than (Z,P(Z\X)) iffVy > Vz for all sets of terminal actions, all utility functions and all a priori probability. Theorem (Z,P(Z\X)) P(Z\X) =
3.1. (Blackwell) (Y,P(Y\X)) is more information than iff there exists a non-negative stochastic matrix B, such that P(Y\X)B.
The meaning of the Blackwell theorem is both simple and intuitively clear. P(Z\X) = P(Y\X)B, i.e., the one information is represented as a transformation {garbling) of the other. That is, P(Y) = P(X)P{Y\X), i.e., P{Y\X) is the noise added on X, from P(Z) = P(X)P{Z\X), we have P(Z) = P{Y)B, i.e., B = P{Z\Y) and B is the noise added on Y. But this condition is so hard that to any A, u{a,x) and P(X) will satisfy the excepted utility of Vy higher than which of Vz- In practice,
640
we can only appraise the information structure partly. We can't judge the information by the Blackwell theorem if an information structure doesn't satisfy P{Z\X) = P(Y\X)B, but we can judge by px(Y). In fact, we have: Theorem 3.2. If(Y, P(Y\X)) px(Y)>px(Z).
is more information than (Z, P{Z\X)),
Proof. Let P{Z\Y) = B. p(zk\xi) = T,jP(yj\xi)p(zk\yj), Then,
H(X\Z) =
then
From P{Z\X) = P(Y\X)P(Z\Y), i.e., we have p(xi\zk) = T,jP(yj\zk)p{xi\yj).
-Y^p{zk)Y^P(xi\zk)^0E,p{xi\zk) k
i
= ~ ^Zp(zk) k
^ E P ^ I ^ ^ i
1
* ! ^ ) )
j
log[^p(2/j|zfc)p(ar»|j/j)]j
Let f(x) = -a; log a;, then f(x) is convex, i.e., / ( £ V AjZj) > £V A»/(xi) for any A» > 0 and ^ i A» = 1. Then we have H(X\Z)
-^p(zk)Y^p(yj\zk)Ylp(Xi\y^logp(Xi\y^
> k
j
i
= - E Ep(z^J')p(^) E ^ I ^ J ) 'ogPfel!/;) = H(X\Y)j
k
i
Thus px(Y)>px(Z).
•
The theorem shows that pxiX) c a n D e used to compare and appraise the information structures under the defined condition by Blackwell. The correlative coefficient muse be high between more information and the state space, but the opposite is not true. Under the defined condition, when B = P(Z\Y) is known, we have: 14 Definition 3.5. If Y and Z are different added information and P{Z\X) = P(Y\X)P(Z\Y), we define complementarity and redundancy of Y and Z, respectively, as d(Y,Z) = [H(Y\Z) + H{Z\Y)]/H{Y,Z) and p(Y,Z) = I(Y,Z)/H(Y,Z). The complementarity of information is a measure of the availability of the less information Z. When H{Y) > 0, H(Z) > 0 H(W) > 0, we have the following properties: (1) 0 < p(Y, Z) < 1, 0 < d{Y, Z) < 1 and p(Y, Z) + d(Y, Z) = 1.
641 (2) p(Y, Z) = 0 and d(Y, Z) = 1 iff X and Y are statistically independent. (3) p(Y, Z) = 1 and d{Y, Z) = 0 iff X = Y. (4) d{Y, Z) + d{Z, W) > d(Y, W) for any W.
References 1. J. von Neumann and 0 . Morgenstm, Theory of Games and Economic Behaviour, Princeton University Press, New Jersey (1944) 2. R. Keeney and H. Raiffa, Decisions with Multiple Objectives: Preferences and Value Trade-offs, John Wiley & Sons, New York (1976) 3. S. French, Decision Theory: an Introduction to the Mathematics of Rationality, Ellis Horood, Chichester, UK (1986) 4. H. Raiffa and R.O. Schlaifer, Applied Statistical Decision Theory, MIT Press, Massachusetts (1968) 5. D. Blackwell, Comparison of experiments. Proc. Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 93102 (1951) 6. D. Blackwell, Equivalent Comparison of Experiment. Annals of Mathematical Statistics, 24, 265-272 (1953) 7. B.K. Agnieszka, Sufficiency in Blackwell's theorem. Mathematical Social Sciences, 46, 21-25 (2003) 8. B. Eckwert and I. Zilcha, Economic implications of better information in a dynamic framework. Economic Theory, 24, 561-581 (2004) 9. I. Gilboa and E. Lehrer, The value of information - an axiomatic approach. Journal of Mathematical Economics, 20, 443-459 (1991) 10. O. Gossner, Comparison of information structures. Games and Economic Behavior, 30, 44-63 (2000) 11. C.E. Shannon, A mathematical theory of communication. Bell System Technical Journal, 27, 379-423 (1948) 12. C.E. Shannon, Communication theory of secrecy systems. Bell System Technical Journal, 28, 656-715 (1949) 13. A.D. Luca and S. Termini, A definition of non-probabilistic entropy in the setting of fuzzy sets theory. Information and Control, 20, 301-312 (1972) 14. C. Rajski, A Metric Space of Discrete Probability Distributions. Information and Control, 4, 371-377 (1961) 15. W.H. Qiu and M. Yan, Improvement of information-decision analysis. Control and Decision, 12, 353-356 (1997) (in Chinese) 16. T. Nakagawa and K. Yasui, Note on reliability of a system complexity considering entropy. Journal of Quality in Maintenance Engineering, 9, 83-91 (2003)
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PART VIII STOCHASTIC MODELS
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GREY RELIABILITY ANALYSIS UNDER Lj NORM RENKUAN GUO Department of Statistical Sciences, University of Cape Town Private Bag, Rondebosch 7700, Cape Town, South Africa YANGHONG CUI Department of Statistical Sciences, University of Cape Town Private Bag, Rondebosch 7700, Cape Town, South Africa
Grey theory initiated by Deng (1982) is a mathematical branch dealing with system dynamics having sparse data availability. Grey reliability analysis is thus advantageous because of the small sample size requirements, for example, the first-order one-variable grey differential equation only needs as little as four data points. However, the grey estimation of state dynamic law uses least-square approach, i.e., parameter estimation under L2 norm. Problems associated with L2 norm grey estimation are the model accuracy specifications. The L2 norm grey modeling campus often borrows the model fitting criteria from statistical linear model analysis, for example, using mean-sum-of squared errors as model fitting criterion and even using probability bound for it. These exercises are putting themselves in controversy. In numerical analysis and approximation theory, relative error is a standard approximation criterion although the L2 norm grey modeling campus also uses relative error as model accuracy measure. In this paper, we propose a L t norm based grey modeling and search the grey parameters in terms of simplex technique in linear programming. We will discuss briefly the grey reliability analysis under Li norm based grey state dynamics.
1. Introduction In reliability engineering modeling, or more specifically, in repairable system modeling, most of the reliability engineers are using the maximum likelihood theory for facilitating empirical analysis, which is in nature a large-sample based asymptotic theory for (asymptotic) confidence intervals and hypothesis testing. Nevertheless, we should be aware that small sample asymptotics developments in the statistical literature, for example, field and Ronchetti (1990, 1991), which facilitate approximations to the asymptotic distribution of interested quantity that models the true state of system. This approximation is still on the route of the probabilistic thinking - once the distribution of the system state is available then the information about the state is fully available. The real aim of modeling is actually to find the dynamic law of the system state. Therefore, the probabilistic thinking route is one of the possible choices. There are other choices of thinking logics, for example, fuzzy thinking, rough sets thinking, grey thinking or other thinking logics rooted in modern approximation 645
646
theory. Grey thinking is an approximation methodology aiming at directly revealing a system state dynamic relation (without the priori probabilistic distribution assumptions). However, the grey estimation of state dynamic law uses least-square approach, i.e., parameter estimation under L2-norm. Problems associated with L2 norm grey estimation may arise from the model accuracy specifications. The L2 norm grey modeling campus often borrows the model fitting criteria from statistical linear model analysis, for example, using mean-sum-of squared errors as model fitting criterion and even using probability bound for it. These exercises are putting themselves in controversy. In numerical analysis and approximation theory, relative error is a standard approximation criterion although the L2 norm grey modeling campus also uses relative error as model accuracy measure. In this paper, we propose a Lj norm based grey modeling and search the grey parameters in terms of simplex technique in linear programming. Finally, we will explore the L r norm based grey approximation of the individual repair improvements. 2. A Review of the L2-Normed GM(1,1) Model The most important grey differential equation model is the first order one variable (equal-spaced) grey differential equation model, abbreviated as GM(1,1) model, which is an efficient one with high predictive capacity. Given a (positive) discrete data sequence N [X<°>;f = [ | > (1)] ,jyo> (2)] , - , [ > » ( « ) ] ] , Equation^) is called the GM( 1,1) model with respect to observation sequence [^<0> 1 , (Liu et al, 2004) x{0) (k) + az{1)(Jfc) = fi, jfc = 2 , 3 , - , «
(1)
where the background value z^' (#) are generated by MEAN operator, z(l) (*) = MEAN(* 0) (*)) = 0.5(*(1) (k) + x{,) (k-1))
(2)
and jr (k) are generated by AGO operator, x(1)(/t) = A G O ( X ( 0 ) ) t = t ^ ( 0 , ( 0 ^
= 1 2
' '-'"
(3)
Using least-square approach (L2-nprm), the parameters a and ft can be estimated and denoted by a and J3 respectively. The filtering-predictive equation takes the discrete version of solution (with GM(1,1) least-square estimated parameter-values)
647
x{t)(k) = (x{l)(l)-fi/a)exp(-a(k-l))
+
fi/a
(4)
which is the solution to the differential equation with estimated parameters a
and J3 dx{,)/dt + ax{l)=J3 It [X^f
is
noticed
that
the
(5)
observational
data
sequence
is
= f p ° ) ( l ) J f a , [ ^ ( 2 ) ] * , . . . , [ ^ 0 » ( « ) J i 1 . The system dynamics is
assumed to be described by a first order linear (constant coefficient inhomogeneous) differential equation (5), and the theoretical system dynamics is described by f dx{i)/dt + ax{l) =/? (6)
\[d^ydt]it=x^{i) Then solution to (6) is
*<'>(0=f*W(l)-^exp(-a(,-l)) + £
(7)
a
which is called the whitenization solution to the grey differential equation GM(1,1) defined by Equation (1). Because the integrated level data is not extracted from system directly, rather created in accordance with some criteria, for example, AGO in Equation (3) and MEAN in Equation (2) proposed by Deng (1985), a natural question arises here what is the appropriate composition of jz' 1 ' (k), k = 2,3,••-,«} such that the integrated observation data sequence \x^'{k),k = 1,2, ••-,«} can be formed and lead to an optimal statistical estimation with respect to parameter a , j3, and the initial condition y. Deng (1985) argued that the only candidate at the level x (t) should be AGO based {z{l](k), k = 2 , 3 , - , / j ) , wherez(]](k) = 0.5x(x(1)(jfc) + x{>)(k-1)), while others, for example, Fan et al (2003) argued that the weight factor , denoted by co , between *'''(&) and r ' { k - l ) should not be predetermined rather than determined by the optimization procedure. In other words, the candidate, z{l](k) = cox{,\k) + (l-co)xi,}(k-\),
a,e[0,1]
should be considered and thus the L2-estimation estimation will be:
(8)
648
and the function x'1' (z) is given by (12): Then, it is required to solve the following nonlinear equation system for the parameters a,fi, and co :
Z^l[zo)(0-,o»(,)] =0 •
| ^ [ 2 ( , ) 0 ) - A 0 ] =o
do)
Mathematically, it is obvious that the optimal choice of a> is not necessary to choose as 0.5. Let us examine an example. Given a discrete data sequence X = {2.874, 3.278, 3.337, 3.390, 3.679}. We perform the parameter searching for two cases: Model (1): modeling with equal-spaced GM(1,1) model and Model (2): modeling based on Equ. 13-15 in terms of genetic algorithm. (In both models, we use x1'' (l) = 2.874 as initial values although this is debatable.) Table 1. The weight CO impact in GM( 1,1) models under L2-norm. Model (1) (2)
CC -0.03720 -0.04870
p 3.06536 2.91586
CO 0.5 (Deng) 0.048645
Squared error 0.006929752 0.002342187
It is obvious that in Model (2) a genetic algorithm-searched weight co = 0.0486454, This indicates that w is not necessarily 0.5 claimed by Deng [11-14] but it is a data-dependent parameter. Also the Model (2) gives much small squared error at xr -level, i.e., •/'' is minimized in the Model (2). However, if we think the computation convenience of GM(1,1) model which can carry on Excel, we would still appreciate the superiority of Model (1)GM(1,1) proposed by Deng [11].
649 3. An Investigation on L r Normed GM(1,1) Model Li-normed optimization was used in statistics with long history, particularly, in robust statistics. If we re-write Eq. (I) in the form x{0)(k) = /3 + a(-z{])(k)),k
=
2,3,-,n
en)
Then the L r normed formulation (L r Model I) will be
satk,(*)-[/'+«H,(*))]
(12)
We put a> as parameter searching list as Eq. 13 defined. The constraints for the L i-optimization problem (18) should be -2/(n + l)ia<2/(n 0
+ l)
(13)
where n is the sample size under consideration. Then let us write '< =x{0){k)-\/3
+ a(-z{l){k))\
ifx<0) ( * ) - [ > + a (-z ( , ) (*))' >0
+ a (-z(1) (*))]
if x(0) ( * ) - [ / ? + a (-z (l) (*))' <0
(14) e;
= XW (k)-\/3
The L] problem becomes n
mmY(el+e'k) s.t.
(15)
- 2 / ( « + l ) < a < 2 / ( « + l) 0<
min V j r ' f f c )
*M(ft)-(/J + ap>(ft))) x(0](k)
(16)
650
The term x (0, (Jfc)-(/3+a(-z (l, (*)]j) /x{0](k) is just the relative error contributed by /'* fitting. Therefore, the object function in L r Model I is a weighted sum of the (absolute-valued) relative errors. If we drop the weight term x(0) (k), then we have the L]-Model II formulation
=l5%)(e;"er)
(n>
which will gives the parameters (a,/3,u;) minimizing the sum of the (absolutevalued) relative errors. Li, Yamaguchi and Nagai (2005) argued that there are certain illogical treatments on the initial value, ;c'(0) , the derivative (or difference) approximation, atr'(t)/dt, t = k — 0,1,• • •,n , and the background value z — x^'(t) in classical grey theory proposed by Deng [11]. They proposed an approach in terms of cubic Hermite spline function for resolving these three problems. The Hermite interpolation function is #
( * ) = y,.Ft (t) + ytF ( / ) + h [y ' , G ( / ) + y \ G ( r ) ]
(18)
where (F 0 (,)=(W 2 )(1 + 20,/=|(0=< 2 (3-20
|G0(/HI-O2,
G,{ty-t\\-t)
(
}
with — , ^j_i <X<xi
(20)
X is called an interpolation point and ht is called an interpolation interval. The Hermite cubic spline function satisfies the following restrictions lim H\ (X) = lim //, (x)
Hl{xQ) = Hl{xn) = 0 Let yt = x(l) (/) and h,. = 1, and y] — dx^ (i)/dt, then following the results derived in Li et al (2005)
Y' = A~XB
(22)
651 where
r = [y*>y\>—>y,\
A=
b' =
2
1
0
•• 0
0
1/2 0
2 1/2
1/2 2
••0
0
•• 0
0
0
0
0
2
1/2
0
0
0 - • 1
(23)
2
[b0,b„...,bnf
with t>0 = , 3
3{yi-y0) U-M-JVI). i = l,2,-,»-l
(24)
*. = 3 ( 3 ' . - ^ . , ) Notice that >>,' (/ = 0,1,2, • • •, n) are the values of the derivative function of the cubic Hermite spline function Hit)
at interpolation points, i.e., y\ = H\ (/')
and the interpolation points of yt (/' = 0,1,2,• • •, n) are setting to y, = j r ' (/') = H3 (i), i.e., yt = r 1 ' (/) is just the background value used in L r optimization for parameter (a,(3). The parameters is dropped because the background value z is no longer obtained by weighing the adjacent 1-AGO values j r ' {k — l) and jr1' (k) proposed by Deng (1985). Then the Hermite cubic spline function based L]-formulation of GM(1,1) model is (abbreviated as 3HSPL L,-Model)
™pEK-|>+a(->'/)J
(25)
4. Evaluation of Individual Repair Improvement with Li-Normed M(l,l) Model Guo began his efforts in applying grey theory to repair efforts modeling since 2004. The motivation of grey theory modeling lies on two aspects. On the one hands, it facilitates a structure for sparse data modeling. On the other hand, grey modeling aims to provide individual estimated effect for each repair or PM (not
652
in statistical sense) directly. This is different from probabilistic or fuzzy modeling where the estimates are in average or cut-level sense. The grey estimate itself still contains intrinsic uncertainty. Although the individual estimated effect of repair is imprecise and not unique (as a matter of facts, it is a whitenization of the grey interval number) the repair effect does not describe the underlying mechanism (system changes under repair or PM) in statistical or probabilistic sense. In practices, the grey individual repair effect estimate provides reliability engineers or management the information for decisionmaking on production and maintenance planning. The key step in grey approximation to the individual repair effect is the partition of system functioning time into a system intrinsic functioning time and the repair effect (measured by time). This partition is carried on by GM(1,1) model. In other words, once the /,, -estimate of (a,/3) is obtained, we can use
^(k) = (^(0)-p/a)mv(-a(k)) iw(k) =
+
p/a
i{'){k)-x{'\k-l)
for obtaining the intrinsic functioning time sequence JC(0) = {xm(\),xi0)(2),-,xl0)(n)} which in nature the filtered values of x{0] under L, -norm. This is essentially the same as that under L2 -norm. For more details about L2 -normed treatments, see Guo (2005a-2005c), Guo and Love (2005). 5. Concluding Remarks In this paper, we reviewed L2-Normed GM(1,1) model and then we develop a set of Li-Normed GM(1,1) models. The grey reliability analysis under L r Norm is briefly discussed and we will carry data analysis in future research. Acknowledgment The author deeply thank Professors Nagai, Yamaguchi and Dr Li for kindly clarifying his understandings on their paper [12] in our communication e-mails. References 1. 2.
J. L. Deng. Control Problems of Grey Systems, Systems and Control Letters 1(6), March (1982) J. L. Deng, Grey Systems (Social • Economical), The Publishing House of Defense Industry, Beijing, (1985) (in Chinese)
653
3. 4. 5. 6.
7.
8.
9. 10.
11.
12.
13.
14. 15.
J. L. Deng, Grey Control System, Huazhong University of Technology Press, Wuhan City, China (1993), (In Chinese) J.L. Deng, The Foundation of Grey Theory, Huazhong University of Technology Press, Wuhan City, China (2002a) (in Chinese) J.L. Deng, Grey Prediction and Decision, Huazhong University of Technology Press, Wuhan City, China (2002b) (in Chinese) C.A. Field and Ronchetti, E., Small Sample Asymptotics, Institute of Mathematical Statistics, Lecture Notes-Monographs Series 13, USA, (1990) C.A. Field and Ronchetti, E., An Overview of Small Sample Asymptotics, in: W. Stahel, S. Weisberg (eds.), Directions in Robust Statistics and Diagnostics, Part I, Springer-Verlag, New York (1991) X.H. Fan, Q. M. Miao, and H.M. Wang, The Improved Grey Prediction GM(1,1) Model and Its Application, Journal of Armored Force Engineering Institute 17(2), 21-23 (2003) (in Chinese) R. Guo, Repairable System Modeling Via Grey Differential Equations, Journal of Grey System 8(1), 69-91 (2005a) R. Guo, A Repairable System Modeling by Combining Grey System Theory with Interval-Valued Fuzzy Set Theory, International Journal of Reliability, Quality and Safety Engineering 12(3), 241-266 (2005b) R. Guo, and C. E. Love, Fuzzy Set-Valued and Grey Filtering Statistical Inferences on a System Operation Data, International Journal of Quality in Maintenance Engineering - Advanced reliability modeling 11(3), 267-278 (2005c) G.D. Li, D. Yamaguchi, and M. Nagai, " New Methods and Accuracy Improvement of GM According to Laplace Transform", Journal of Grey System 8(1), 13-26(2005). S.F. Liu, Y. G. Dang, and Z. G. Fang, Grey System Theory and Applications, The Publishing House of Science, Beijing, China (in Chinese) (2004) S.F. Liu and Y. Lin, Grey Information, Springer, London (2006) K.L. Wen, Grey Systems: Modeling and Prediction, Yang's Scientific Research Institute, Taiwan (2004)
STATISTICAL ANALYSIS OF INTENSITY FOR A QUEUEING SYSTEM JAU-CHUAN KE Department of Statistics National Taichung Institute of Technology,
Taiwan
YUNN-KUANG CHU Department of Statistics National Taichung Institute of Technology,
Taiwan
Traffic intensity is an important measure for assessing performance of a queueing system. In this paper, we propose a consistent and asymptotically normal estimator of intensity for a queueing system with distribution-free interarrival and service times. Using this estimator and its associated estimated variance, a 100(l-a)% asymptotical confidence interval of intensity is obtained. A numerical simulation study is conducted to demonstrate performance of the proposed estimator and its associated estimated variance applied to interval estimations of intensity for a queueing system.
1. Introduction Traffic intensity (intensity) plays an important role in queueing models. It is defined as the ratio P=~, (1) M where 1/ X represents mean interarrival time and 1/ /u denotes mean service time, p can be interpreted as expected number of arrivals per mean service time in the limit, an important parameter, also called utilization factor that measures the average use of the service facility (see Gross and Harris [1]). The traffic intensity is a dimensionless quantity, but in teletraffic theory it is often quoted in 'erlangs', in honor of the pioneering effort of A.K. Erlang. This paper is aimed to propose statistical inference for the intensity p of a queueing system with distribution-free interarrival and service times. In section 2, we prove that the natural estimator p of p is strongly consistent and asymptotically normal. Based on the asymptotical normality of p , we can construct a 100(l-a)% confidence interval for p. A comprehensive simulation study is conducted in Section 3 to 654
655 demonstrate performance of p applied to interval estimation. Simulation results are shown by appropriate curves for illustrating performance of p in interval estimations. 2. Nonparametric Statistical Inference of Intensity Let X and Y represent the interarrival and service times of a queueing system, respectively. Then the intensity of the system is defined by P - ^ , Mx
(2)
where fix and fiy denote the mean interarrival time and mean service time of the system, respectively. The definition (2) is equivalent to definition (1). 2.1. Estimating Intensity Assume that X\, X2, ..., X„ is a random sample of Xand Yx, Y2, ..., Y„ is a random sample of Y. We use ( Xb Yt) to represent interarrival time and service time for the rth customer of a queueing system. Define X and Y to be the sample means ofXs and Ys, respectively. According to the Strong Law of Large Numbers (see Roussas [2], p. 196), we know that X and Y are strongly consistent estimator of fix and fiy, respectively. Thus a strongly consistent estimator of intensity p is given by p = YIX.
(3)
In practical queueing systems, the true distributions of X and Y are seldom known, so the exact distribution of p cannot be derived. But under the assumption that X and Y are independent, the asymptotical distribution of p can be developed as the following procedures. Firstly, according to the Central Limit Theorem (see Hogg & Craig [3]), we have ^(X-MX)^^N(0,
(4)
and ^i(Y-Mr)—^>N(0,(Tr),
(5)
where <JX and oy are variances of X and ^respectively. Next note that
J7i{p p)
^"^x(r
~//y}~Jly(-x~Mx)] Mx*
(6)
656
Therefore by the Slutsky's theorem (see Hogg & Craig [3]), we get •yfc(p-p)—°-+N(0,
(7)
where a2 = (/4°"y + MWX)/ f*x • Now, set &2=(X2S2+Y2S2X)/X4, where S2X =-£(X,
-X)2 and S2 = -i(Yi-Y)2
n 1=1
(8) .
n 1=1
Then
+ zal2&lJn)
Consequently, an approximate 100(1-a )% confidence interval of p is {p-zano14n,p + zall614n). 3.
(10)
Simulation Study
We simulate the confidence interval p + za/2<J/yln as follows. The levels of (A, ju) considered in the simulation process are set to (0.1, 1), (0.5, 1), and (0.9, 1) so that the intensity p is low (p=0.1), moderate (p=0.5), and high (/)=0.9), respectively. For each level of (A,//) , random samples of interarrival times ( xx,Xj,...,xn ) and service times ( yuy2,...,yn ) are drawn from X and Y, respectively. Then the estimate p and its associated variance a~ In are computed. The 90% confidence interval of p is given as yO + z 005
657 The simulation is replicated JV=1000 times and we record the fraction of times that the 90% confidence interval contains the true intensity p, which is called the coverage percentage. This process is repeated for ra=10, 20, ..., 1000. Since the number of confidence intervals containing the true intensity p follows a binomial distribution with A/=1000 and/?=0.9, the 99% confidence interval for the coverage percentage itself is 0.9+ 2.576^0.9(1 -0.9)/1000 = 0.9 + 0.0244, or (0.876,0.924). The simulation results for the performance of p in terms of confidence interval are presented from Figure 1 to Figure 3. Under each simulated queueing system, the coverage percentages of 90% confidence interval corresponding the three kind of intensity (low, moderate, and high) are plotted versus n from 10 to 1000. All three curves almost appear in the 99% confidence interval (0.876, 0.924) and fluctuate along the nominated 90% level provided that n reaches large enough (n > 50). On the other hand in Figure 4 (p=0.5), evaluating the chance of the three curves inside the 99% confidence interval, it appears that ordering these curves by their relative performance on coverage fraction produces E 4 /Hf /1 > M/E4/l > E4/M/l. coverage percentage 0.93 0.9: 0.92 0.91 0.9
• Hi"! IMS ;:".?5ti ; I'a < w. uiits i ;>Vi\'" ^ 3 ii ! ? '•'.• ? i '; 3i;S;'.iFi*1S! HH% •] "\t\ ff.jl :ihS | i : T~'- f- :H: '- P. xi%i Viisn/'f-i?/;
0.89 t
f?T * W ill' \ I W W 'I ti
:
' i
\! i
i'"j \'i
"i \\
!
ill
i* if
0.88 0.87|& 0.87 =0.1
0.86
=0.5 P =0.9
0.85
M/k JX system _ 0.84 0.83 0
100
200
300
400
500 n
600
700
800
900
1000
Figure 1. Coverage fraction of 90% confidence interval for different values of n and p .
658 coverage percentage 0.94 0 924 0.92
kmm m* II
.A U
0.9
fW.'
0..87D
p-#); —
0.86
-
p =0.1 -0.5
-. p
0.84
= o g
Ei/M/1 system 0.82
J 100
0
I 200
I 300
I 400
I 500 n
L 600
700
800
900
1000
Figure 2. Coverage fraction of 90% confidence interval for different values of n and p . coverage percentage 0.96
I
|
0.94
!
"W pi-.
.......
4
1 , :
0.9
|
!
0.9: /\ 0.92
•r
=.
i.
•
V\/-.| •
'•
'-
.........
— . — --
...
0.86
0.84
rh ;
i
i
;
i"
I
;
P
=o.p
p -0.5
!
:
0.82
r
........... p _Q g E.VH90/1.svstBrrL _ 4
4
"'"
i 0
100
200
300
400
500 n
700
800
900
1000
Figure 3. Coverage fraction of 90% confidence interval for different values of n and p
659 coverage percentage 0.94 i r-
0.9: 0.92 i
0.9
,1,
i f xii '(
i 0.87^
0.86
M/E / I system E /M/1 system
0.84
——
0.82 0
J 100
l 200
l 300
L 400
500 n
600
700
E4/HP°/1 system
J 800
l_ 900
1000
Figure 4. Coverage fraction of 90% confidence interval for different values of n ( p = 0.5 )
4.
Conclusions
This paper provides the interval estimations for the intensity p of a queueing system with distribution-free interarrival and service times. We show that the natural estimator p is strongly consistent and asymptotically normal with approximate variance a11 n . References D. Gross and C. M. Harris, Fundamentals of Queueing Theory 3"' Ed., NewYork, John Wiley, (1998). G. G. Roussas, A course in Mathematical Statistics. 2nd edition, Academic Press, (1997). R.V. Hogg and A.T. Craig, Introduction to Mathematical Statistics, Prentice-Hall, Inc., (1995).
APPLICATION FOR MARKET IMPACT OF STOCK PRICE USING A CUMULATIVE DAMAGE MODEL
SYOUJI NAKAMURA, MIWAKO ARAFUKA Department of Human Life and Information, Kinjo Gakuin University, 1723 Omori 2-chome, Moriyama-ku, Nagoya 463-8521, Japan E-mail: snakam@kinjo-u. ac.jp;arafuka@kinjo-u. ac.jp TOSHIO NAKAGAWA Faculty of Management and Information Science, Aichi Institute of Technology, 1247 Yachigusa, Yagusa-cho, Toyota 470-0392, Japan E-mail: [email protected] HITOSHI KONDO Faculty of Economics, Nanzan University, 18 Yamazato-cho, Showa-ku, Nagoya 466-8673, Japan E-mail: [email protected] This paper considers the problem of maximizing an expected liquidation profit of holdings, when the market impact of stock price is caused by the holdings sell-off. The cumulative damage model is applied to the fluctuations of stock price. We derive and analytically discusse an optimal sell-off interval of holdings to maximaize the expected liquidation profit of holdings.
1. I n t r o d u c t i o n When we have to sell off security holdings in a short term on the market, we need to consider a liquidation policy which maximizes the total amount of security holdings in consideration of their influence for the market price. We consider the following two stochastic models of liquidation policies for the security holdings: The security holdings 5 are sold off by one time or by deviding them into n blocks, S/n is sold off. Then, the market price decreases along with the amount of disposition according to an impact function. In addition, the market price also decreases from the supply-demand relation in the market, as the accumulation of selling orders increases gradually. But, the influence degree of the price becomes lowered if the security 660
661 holdings are broken down into small blocks, however, the dealing cost gradually increases. Conversely, if the security holding are sold off by dividing roughly, the market price greatly falls, however, the dealing cost decreases. That is, the disposition lot and market price of the security holdings have a trade-off relation. Another assumption is that the stock prices rise by selling off the stocks. In addition, the market impact to which stock prices drop sharply is assumed when it reaches the threshold price. In general, it is not easy to formulate the stochastic model of market impact because it depends on various factors. A consensus of a liquidation policy for security holdings has not been obtained yet, although various approaches in academic and business fields have been made. In this paper, we consider the market impact when the security holdings are sold off on the market, and derive analytically an optimal liquidation policy which maximizes its total amount. 2. Assumption of Model The market is composed as follows: The market price changes only according to this assumption. The following notations are used: So: Nominal value of security holdings at time 0. E(n): Real value of security holdings at time n. Co: Constant dealing cost per transaction. A: Parameter of market impact function. H'. Parameter of price restoration function.
Tablel: Relation between amount of contract and transaction fee. Amount of contract (yen) 1,000,000 2,000,000 5,000,000 10,000,000 20,000,000 30,000,000
Transaction fee (yen) 12,495 21,420 47,145 82,845 140,595 198,345
662
3. Model 1 It is assumed that the security holdings SQ at time 0 have to be sold off in a certain limit time. As liquidation methods, the security holdings S are sold off by one time or S/n sold off by dividing into n every time. Then, the market price decreases along with the amount of disposition according to an impact function. After a certain time has passed, the security price recovers to the price before the previous time. In addition, the market price also decreases from the supply-demand relation in the market, as the accumulation of selling orders increases gradually. But, the influence degree of the price becomes lowered if the security holdings are broken down into small blocks, however, the dealing cost gradually increases. Conversely, if the security holding are sold off by dividing roughly, the market price greatly falls, however, the dealing cost decreases (Table 1). When the security holdings So/n are sold off on the market, the amount of liquidation decreases exponentially, and is given by {So/n)(l — e~Xn/s°). Further, letting CQ be a dealing cost of transaction, we evaluate the present values of all costs by using a discount rate a(0 < a < oo). In addition, we consider the restoration function for an increasing market price rate and the decreasing market price rate by the market impact. The amount of liquidation of transaction at time 0 is ^(l_e-A„/5
0 )
_
C o
.
(1)
n This liquidation is restored exponentially, and is given by Q\ = (So/n)(l — T e-Xn/So }ev . Then, the amount of liquidation of transaction at time T is •{Q1(l-e-V«i)_Cd}e-°T Xn s
(2)
lT
the amount of liqui-
{Q2(l-e-A/«*)-co}e-Q2r.
(3)
Further, letting denote Q2 = (5o/n)(l - e~ l °)e> , dation at time T% is
Repeating the above procedures, we have generally Qj+1 = Qj(l - e-x^)e»T
(j = 0 , . . . ,n - 2),
Qo^^.
(4)
n Hence, the amount of n times liquidation of transaction is Ei(n) = £ 3=0
{Qj(l - e " ^ ) - c 0 } e~^T,
(5)
663
Thus, substituting (4) into (5), the expected total amount of liquidation per unit of security holdings is
Zi(n)=^p3.1. Numerical
Example
(n = l,2,...).
(6)
of Model 1
In Table 1, we compute the optimal number n* numerically when SQ = 108, A = 150,000,000, a = 0.01 and a transaction cost is c0 = S0 x 0.002625 + 119595.0. For example, when // = 0.006,n* = 35 blocks and E3(n") = 93,289.357.
°
Tune
Figure 1.
Market impact for Model 1
4. Model 2 The stock is sold off for T period, and its prices is assumed to rise gradually. However, as the market impact, the price drops sharply when it reaches the threshold price. The cumulative damage model[4] is applied to this model by replacing the change of stock prices with damage. When the clearance of the stock is continuously exercised, the stock price Z(0) = ZQ at time 0 is assumed to rise to Z(t) = at + zo(a > 0). When the stock price reaches a threshold K, it drops sharply and becomes
664
Table 2 Total amount of liquidation of security holdings So = 100,000,000, A = 150,000,000 and a = 0.01. n* 33 34 35 36 38 39
M 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012
£i (n*) (yen) 92,943,840 92,647,601 93,289,357 93,949,124 94,638,556 95,367,121
0 (Figure 2). In this case, a threshold K is a random variable with a distribution function K(x). The probability that the stock price drops sharply at time jT is denoted by Pr{jaT + ZQ > K} = K(jaT + ZQ), and the probability that the stock price drops sharply at time nT is n
j —1
J2K(JaT
+ zQ)l[K(iaT
j=i
+ zQ),
(7)
»=i
where K = 1 - K. Conversely, the probability that the stock prices does not drop sharply until time nT is
l[K(jaT + z0).
(8)
3=1
It is clearly shown that (7) + (8) = 1. Similarly, the mean time to the clearance of the stock is 3-1
n
Y^UT)K(jaT
n
+ z0) J} K(iaT + z0) + (nT) J J K(jaT + z0) j=0
j=\
n-l
j=l
j
l[K(iaT+z0) ]=0
(9)
.i=0
Then, the mean time until the stock price crash is »
3
l[K(iaT
+ zo)
(10)
3 =0 .t=0
The amout of stock is divided equally in n and is sold off by time nT(n = 1 , 2 , . . . ) . That is, the stock So/n at each jT(j = 1,2,... )are sold
665 off and stock price in jT is jaT + ZQ. In addition, when the stock price drops sharply, the amount of the clearance of the stocks is assumed to be 0. The expected liquidation of So at time nT is c
i
"•
P
J-1
+ z0) Y[ K(iaT + z0) Y,(iaT + zo)
C(n) = ^\Y/K(JaT n
_ 1
' 3= 1
U=l
i=l
n
+ 11 K(jaT + zo) ^{iaT
= — T^iJaT
+ z0)
+ zQ)T\R{iaT
+ z0)
(n = l , 2 , . . . ) .
(11)
Thus, the expected liquidation per unit of time for an infinite interval is, from (9) and (11), n
j
aT z
Y,u + o) n CW4
J =
'
n -i
R iaT
( +*>)
i ^
(n = l,2,...).
(12)
j=0 i=0
We seek optimal numbers n* that maximize C(n) and C{n). 4.1. Optimal policy for Model 2 We seek an optimal n* that maximizes C(n) in (11). If C(n) > C(n + 1) then -. n
*
n+1
- Y,(J*T + zo)Kj > — j J2UaT + **)Kh i,e., n
Y,{tiaT
+ z0)Kj - [(n + l)aT + z0}Kn+1}
> 0,
(13)
i=i
where Kj = Ili=oKjiiaT + zo) (j — 1,2,...). Letting L(n) denote the left-side of (13), L(n + 1) - L(n) = (n+ l){[(n + l)aT + z0]Kn+1 - [(n + 2)aT + zQ}Kn+2} =(n+l)Kn¥1{[(n+l)aT+z0}-[(n+2)aT+z0]Kl(n+2)aT+Zo]} =(n + l)Kn+1{K[{n
+ 2)aT + z0] - aT}.
(14)
666
It can be easily seen that K[(n + 2)aT + z0] is strictly increasing to 1 in n. Therefore, we have the following policy: Thus L(n) is also strictly increasing from L(l) or it increases after decreasing once for aT < 1. Therefore, then there exists a finite and unique n* which satisfies (13).
7
10
Figure 2.
4.2. Numerical
Example
13
16
19
22
25
28
31
34
Market impact for Model 2
of Model
2
In paticular, when K{x) = 1 - e~6x, Equation(13) is J2\UaT
+ z0)expl
-[{n+l)aT+z0}expl
\j(j + 1) -aT + zQ) -9
[n + l)(n + 2)
aT+{n+l)z0)
}}>_, (15)
For example, when S = 1, a = 5, z0 = 450, T = 1 and 6 = 4.0 x 10~ 7 in Table 3, the optimal sell-off interval is n* = 42, and the maiximam profit per unit time for an infinite is C(n*) = 13.27. Further, when 6 is large, the optimal sell-off interval is small, and the expected total amount of liquidation C(n*) is low. This reason is that if the probability of stock price drops sharply is large, the risk grows, and hence, the optimal sell-off interval is small.
667
Table 3: Total amount of liquidation of security holdings So = I,a = 5.0 and T = 1.
e 4 x 10-7 8 x 10 -7 12 x 10-7 16 x 10~7 20 x 10-7 24 x 10-7
n* C(n*) 42 13.27 27 19.26 18 27.64 14 34.82 11 43.64 10 47.75
5. Conclusions In analyzing the optimal liquidation behavior, we have made the trade-off between dealing cost and market impact, and considered the optimization problem: This paper has proposed the stochastic model in which the expected cost is quantitatively evaluated to liquidation. We have to formulate the market impact. The mechanism of the market impact is complex and the formulation is not easy. The market impact is different in a disposition of large security holding and a disposition of small of security holding. We have discussed analytically and numerically the optimal division frequency of security holdings which minimizes it.
References 1. Y.Hisata and Y.Yamai, Research Toward the Practical Application of Liquidity Risk Evaluation Methods, Monetary and Economic Studies, Vol.18, No.2, December (2000). 2. R.Oubil, Optimal Liquidation of Large security Holdings in Thin Markets, working paper, January (2002). 3. R.Jarrow and S.Turnbul, Derivative Securities. Thomson Learning Company, (1973). 4. T.Nakagawa, Maintenance Theory of Reliability. Springer, (2005). 5. D.R.Cox, Renewal Theory. John Wiley & Sons Inc, (1962). 6. D.Duffie and K.J.Singleton, Credit Risk. Princeton University Press, (2003).
SOME RESULTS O N A M A R K O V I A N D E T E R I O R A T I N G S Y S T E M W I T H MULTIPLE I M P E R F E C T R E P A I R
N. T A M U R A Department
of Electrical and Electronic Engineering, National Defense Academy, 1-10-20 Hashirimizu, Yokosuka, 239-8686, JAPAN E-mail: [email protected]
This paper considers a system which is inspected equally spaced points in time and whose deterioration follows a discrete time Markov chain with an absorbing state. After each inspection, one of the following actions can be taken: operation, imperfect repair m (1 < m < M) or replacement. When imperfect repair m is taken for the system which has been repaired n times in state i, it moves to state j with probability g^{n + 1). We study an optimal maintenance policy which minimizes the expected total discounted cost for unbounded horizon. It is shown that a generalized control limit policy is optimal under reasonable assumptions. We investigate structural properties of the optimal policy. Furthermore, numerical analysis is conducted to show that these properties could hold under weaker assumptions.
1. Introduction In general, system deteriorates due to usage or age. Since deterioration is not evitable, most system can not remain in a good operating condition and eventually fail without maintenance action. To retain the desirable operating condition, it is essential to determine when and how to perform maintenance actions. To analyze this problem mathematically, various maintenance policies for stochastically failing systems have been widely investigated in the literature. The papers by Cho 1 and Wang 2 are excellent reviews of the area. When we can assume that the operating condition of a system is classified into a finite state, the deterioration process of the system could be described as a Markov process because of the tractability of the resulting mathematical problems. This model is called as Markovian deteriorating system. Derman 3 studied a discrete time Markovian deteriorating system where replacement is the only maintenance action possible, and established 668
669
sufficient conditions on the transition probabilities and the cost functions under which the optimal maintenance policy has a control limit rule. In real situations, however, system could be repaired and it might not be as good as new like replacement. That is, system after completion of repair might be younger and occasionally, it might be worse than before repair because of faulty procedure, e.g., wrong adjustments and bad parts. So, various stochastic models for systems with imperfect repair have been suggested and studied by Lam 4 , Kijima 5 , and Kijima and Nakagawa 6 ' 7 . Pham and Wang 8 provided a survey of recent studies on imperfect repair of two-state systems Focused on Markovian deteriorating system with imperfect repair, Douer and Yechiali9 proposed the idea of a general-degree of repair which is the action from any state to any better state at any time inspection and showed that, under reasonable assumptions, control limit policy holds. Chiang 10 studied a continuous time Markovian deteriorating system where it is costly to identify state through inspection and proposed an algorithm to find an optimal inspection policy. But control limit policy is not derived because of the complexity of the model. These studies don't assume multiple repair actions. In the case of medical treatment for persons who develop cancer, however, there are some options such as surgical treatment, drug treatment and radiation therapy. Then the effect and cost of treatment depend upon the option which is selected for patients and their conditions. In maintenance problem, we can interpret medical treatment as imperfect repair. Hopp and Wu 11 studied a Markovian deteriorating system with multiple maintenance actions. But this study doesn't introduce the idea of imperfect repair. Therefore it is important to pay attention to multiple imperfect repair. We propose a discrete time Markovian deteriorating system for which one can select replacement or one of M kinds of imperfect repair as maintenance action. It is shown that a generalized control limit policy holds under some reasonable assumptions and several structural properties of an optimal maintenance policy which minimizes a total expected discounted cost are provided. The outline of the paper is as follows. In the next section, the stochastic model is described in detail. In section 3, we derive a total expected discounted cost for unbounded horizon. Section 4 investigates several properties of an optimal maintenance policy. In section 5, numerical results are provided. Finally, some conclusions are drawn in section 6.
670
2. Model Description We propose the following model. The function of a system deteriorates with time, and the grade of deterioration is classified as one of N + 1 discrete states, 0, • • •, iV -f-1, in the order of increasing deterioration. State 0 represents the process before any deterioration takes place, that is, it is an initial new state of the system, whereas state N represents a failure state. The intermediates states 1, •••, N — 1 are deterioration states. The system is inspected at equally spaced points in time. Then the true state of the system is certainly identified. These states are assumed to constitute a discrete time Markov chain with an absorbing state. After each observation, we can select one of the following action. (1) action 1: we continue to operate the system until the next time. (2) action 2: we repair the system and operate it for one period. (3) action 3: we replace the system with a new one and operate it for one period. When the system in state i is operated, it moves to state j with probability pij at the next time. We assume that there are M kinds of imperfect repair. So, if repair m is performed for the system which has been repaired n — 1 times in state i, then it moves to state j with probability q^{n) after completion of repair m. We call q^{n) as repair probability. When replacement is selected for the system in state i, it becomes new, that is, the system moves to state 0 without fail. The time to replace or repair the system is assumed to be negligible. For these probabilities, we impose the following assumptions. In this paper, the term "increasing" means "nondecreasing"and "decreasing" means "nonincreasing", respectively. Assumption 1. For any h, YLj=hPij *s increasing in i. Assumption 2. For any h, m, n, Ylj=h ^Tji12)
JS
increasing in i.
Assumption 1 means that as the system deteriorates, it is more likely to make a transition to worse states. Assumption 2 implies that as the system deteriorates, it is less likely to be repaired to better states. Also, we assume that these probabilities have the next property. Assumption 3 . For any h, m, n, Ylj=h \Pij ~~ ^2i=o Qii(n)Pij) ing in i.
JS
increas-
Assumption 3 indicates that as the system deteriorates, the system which is operated until the next time is more likely to move to worse states
671 in comparison with the system which is operated until the next time after completion of imperfect repair. Furthermore, we assume that g£?(n) has the following property. Assumption 4. For any i, h, m, ^2j=hQij(n)
«* increasing in n.
The above property means that as the system undergoes imperfect repair, it is more likely to be repaired to worse states. The cost structure of this system is as follows: expected operation cost for the system in state i Ui repair cost when repair m is conducted for the system in state i ' i replacement cost for the system in state i These costs satisfy the next properties. Assumption 5. For any m, Ui, r™ and Ci are increasing in i. Assumption 6. For any m, Ui - Ci,
(1)
«< ~ ( r r + I>J?(")«i J ,
(2)
r? ~ Ci
(3)
are increasing in i. Assumption 5 means that as the system deteriorates, it is more costly to operate, repair or replace the system. Assumption 6 states that as the system deteriorates, in Eqs.(l) and (2), the merit of repair or replacement becomes bigger than that of operation, and in Eq.(3), the merit of replacement becomes bigger than that of replacement. 3. Mathematical Formulation In this section, we derive a total expected discounted cost for unbounded horizon. Let V(i,n) be the total expected discounted cost for unbounded horizon when the system which has been repaired n times stays in state i and an optimal maintenance policy is employed. We denote a discount factor by /? (0 < j3 < 1). Let Ha{i,n) be the total discounted cost when the system which has been repaired n times stays in state i and action o is selected. We denote by D(i,n) an optimal action for the system which has been repaired n times in state i. Then D(i,n) = 2„, indicates that an optimal action is to operate the system after completion of repair m (m = 1, • • • , M). We write the action that one of M kinds of repair is taken before operation as D(i,n) = 2.
672
By using the theory of Markovian decision process, Hi(i,n)
is given by
N
H1(i,n) = ui + pYlpijV(j,n).
(4)
3=0
Similarly, we get H2m(i,n)
and H3(i,n) N
as follows. (
N
Him{i,n) = »? + 5 > 3 > + 1) } Ul + ^PijV^n i=o
(
+ 1)
(5)
i=o
N
Hz(i,n)
= Ci + wo +
P^2p0jV(j,n).
Then V(i,n) is obtained by the following recursive equation. V(i,n) = min Hi(i,n) mm H2m (i, n), H3 (i, n)
(6)
(7)
Km<M
4. Structural Properties of an Optimal Maintenance Policy We derive several properties of an optimal maintenance policy which minimizes a total expected discounted cost for unbounded horizon. First, we can show that V(i,n) have the following property. Theorem 4.1. V(i, n) is increasing in i for any n, and increasing in n for any i. From this result, theorem 4.2 is derived. Theorem 4.2. There exist the states in and in for any n such that 1 for 0 < h < tn, 2 for in < h < in,
{
(8)
3 for 'in < h < N, where 0 < in < in < N + 1. Next, we examine properties of optimal maintenance policy under the following assumptions. Assumption 7. For any m, (i) r™ — rj™+1 is decreasing in i (ii) r™ — r™ +1 is increasing in i Assumption 8. For any h,, m, n, 0) Y^=h(lTj(n) n
00 12j=h(lij( )
~ qTj+l{n)) ~ Qij
n
( ))
is
decreasing in i.
is
increasing in i.
Then the optimal maintenance policy has the following structure. Theorem 4.3. For i and i' such that i < i1,
673
(i) if assumption 7-(i) and 8-(i) hold, then m>m' when D(i,n) = 2 m and D(i',n) — 2m<. (ii) if assumption 7-(ii) and 8-(ii) hold, then m < m' when D(i,n) = 2TO and D(i',n) = 2m>. Theorem 4.3 states that the optimal maintenance policy can be characterized by M + 2 regions at most. In the above theorem, assumption 7-(i) and 8-(i) state that as the system deteriorates, the merit of repair m becomes bigger in comparison with that of repair m + 1. Therefore, theorem 4.3-(i) is intuitively valid because it is appropriate to select smaller m with deterioration. Theorem 4.3-(ii) has the same interpretation as theorem 4.3-(i). As far, we consider that the system might be repaired to worse states than state immediately before repair. Next, we focus on the situation that the system is certainly repaired to better states before repair without fail. Corollary 1. Assume that J2)=h9ij(n) and i
*s increasing in m for any
i,h,n
N
Y^Cjin) = i> J2 « « ( " ) = ° f°r j=0
an
y »' m . n -
j=i+l
And, if (i) r™ — r™ +1 is decreasing in i, (») E}=h(9ij ( n ) -
when D(i,n) = 2 m and D(i',n) = 2m< where
Next, the following assumption is imposed to investigate the properties of an optimal maintenance policy with respect to the number of repairs. Assumption 9. For any h„ m, i, (i) ^j=h{
- q%+1(n)) is decreasing in n.
(") YJj=h(l?j(n)
~ 9ij+1(n))
is
increasing in n.
Then theorem 4.4 is obtained. T h e o r e m 4.4. For n and n' such that n
(i) if assumption 9-(i) holds, then m < m' when D(i,n) = 2 m and D(i,n') = 2m,, (ii) if assumption 9-(ii) holds, then m > ml when D(i,n) = 2 m and D{i,n')=2m,, where 0 < i < N.
674
Similar interpretation as theorem 4.3 holds for the above property. Furthermore, we can obtain an upper bound of the optimal number of repairs. Theorem 4.5. / / assumption 9-(i) holds and there exists n such that Efcotfo/W > c0 + u0 - nr0, or if assumption 9-(ii) holds and there exists n such that J2i=o 1oi( ) > Co + u0 - r0, then n is an upper bound of the optimal number of repairs. By using theorem 4.5, we can find an optimal maintenance policy numerically. 5. Numerical Results In this section, numerical analysis is carried out to show that our theoretical results could hold under weaker assumptions when gt™(n) = q£ for all i, j , m, n and M ~ 2. We consider a 5-state Markovian deteriorating system and define that P = (Pij), Qm = (gg), u = {Ui), r m = ( r f ) , c = (a). Parameters used for numerical analysis is as follows. f0.2 0.0 P = 0.0 0.0
0.2\ / 0 . 9 0.04 0.03 0.4 0.8 0.1 0.04 0.5 , Q 1 = 0.7 0.1 0.1 0.6 0.6 0.15 0.08 V0.5 0.2 0.09 Vo.o o.o o.o o.o 1.0/
Qs
0.2 0.1 0.0 0.0
/0.88 0.77 0.66 0.55 Vo.44
0.2 0.2 0.2 0.0
0.2 0.3 0.3 0.4
0.05 0.11 0.11 0.16 0.2
0.03 0.04 0.1 0.08 0.1
0.02 0.035 0.06 0.07 0.1
0.02 \ 0.045 0.07 0.14 0.16 J
0.02 0.035 0.06 0.07 0.1
/? = 0.99
0.01 \ 0.025 0.04 0.1 0.11 J
(9)
(10)
u = (45.0,55.0,70.0,90.0,115.0), c = (60.0,65.0,70.0,75.0,80.0),(11) r 1 = (13.5,32.5,33.5,43.5,53.5), r 2 = (7.0,18.0,28.5,40.0,51.0), (12) The optimal maintenance policy can be summarized by table 1, where D(i,n) — D(i) and V(i,n) = V(i) for all i, n.
Table 1. i V{i) D(i)
Example of optimal maintenance policy
0 735.71 1
1 2 765.50 784.29 1 22
3 803.38 2]
4 815.71 3
675 From table 1, we find t h a t theorems 4.1, 4.2 and 4.3 hold. Then Eqs.(9) and (10) don't satisfy assumption 3. Also E q s . ( l l ) and (12) don't satisfy assumption 6. These results indicate t h a t the properties obtained by us could hold under weaker assumptions. 6.
Conclusions
We proposed a Markovian deteriorating system where one of M kinds of repair or replacement can be selected as maintenance action and investigated structural properties of an optimal maintenance policy which minimizes an expected total discounted cost for unbounded horizon. It is found t h a t generalized control limit policy holds under reasonable assumptions. Also, we derive some monotonic properties of an optimal maintenance policy under intuitively valid assumptions. These properties are useful to find an optimal maintenance policy numerically. Future work is to obtain the same results under much weaker assumptions. References 1. Cho, D.I., A survey of maintenance models for multi-unit systems, European Journal of Operational Research, 5 1 , 1-23 (1991). 2. Wang, H., A survey of maintenance policies of deteriorating systems, European Journal of Operational Research, 139, 469-489 (2002). 3. Derman, C , On Optimal replacement rules when changes of states are Markovian, In: Mathematical Optimization Techniques (R. Bellman, Ed.), The RAND Corporation, 201-210 (1963). 4. Lam, Y., Geometric processes and replacement problem, Acta Mathematicae Applicatae Sinica, 4, 366-377 (1988). 5. Kijima, M., Some results for repairable systems with general repair, Journal of Applied Probability, 26, 89-102 (1989). 6. Kijima, M. and Nakagawa, T., A cumulative damage shock model with imperfect preventive maintenance, Naval Research Logistics, 38, 145-156 (1991). 7. Kijima, M. and Nakagawa, T., Replacement policies of a shock model with imperfect preventive maintenance, European Journal of Operational Research, 57, 100-110 (1992). 8. Pham, H. and Wang, H., Imperfect maintenance, European Journal of Operational Research, 94, .425-438 (1996). 9. Douer, N. and Yechiali, U., Optimal repair and replacement in Markovian systems, Communications in Statistics -Stochastic Models-, 10, 253-270 (1994). 10. Chiang, J.H. and Yuan, J., Optimal maintenance policy for a Markovian system under periodic inspection, Reliability Engineering and System Safety, 7 1 , 165-172 (2001). 11. Hopp, W.J. and Wu, S.C., Machine maintenance with multiple maintenance actions, HE Transactions, 22, 226-233 (1990).
T R A N S I E N T ANALYSIS OF I N T E R N E T - W O R M P R O P A G A T I O N B A S E D O N SIMPLE BIRTH A N D D E A T H PROCESSES *
K. T A T E I S H I , H . O K A M U R A A N D T . D O H I Department of Information Engineering, Graduate School of Engineering, Hiroshima University, 1~4~1 Kagamiyama, Higashi-Hiroshima 739-8527, JAPAN E-mail: {okamu, dohi}@rel.hiroshima-u.ac.jp
In this paper, we analyze the transient behavior of Internet worms. We show t h a t a stochastic SIS (Susceptible-Infected-Susceptible) model to describe the Internetworm propagation can be approximated by a simple birth and death process. Specifically, when there are a huge number of hosts, the propagation of Internet worms can be modeled by the simple birth and death processes. Deriving the probability generating function of the number of infected hosts, we formulate the probability mass function explicitly, and define some dependability measures for evaluating the transient behavior of Internet worms.
1. Introduction The Internet plays one of the most important roles among all the information technologies established during the last two decades. Because of growth of the Internet, it is possible to communicate easily with a number of users through the Internet. On the other hand, the Internet causes several social problems such as the Internet worm and the cyber-terrorism. In particular, the Internet-worm propagation is one of the most severe problems, because the damage caused by the Internet worms is spreading day by day, and their activities are becoming more and more malicious. Recently, some researchers attempt to predict the propagation process of Internet worm by mathematical modeling 1'2'3^5. Most of them are based on the continuous-time Markov chain (CTMC) to describe the Internetworm propagation. In their modeling, a number of-states are required to "This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Exploratory Research, Grant No. 15651076 (2003-2005).
676
677
represent the behavior of the Internet worms, because the number of states in the CTMC directly corresponds to the number of hosts located in the computer network under consideration. Therefore when the propagation processes of the Internet worms are modeled by the CTMC, it requires a huge number of states. That is, the CTMC modeling is not so tractable to evaluate the realistic Internet-worm propagation and has a limitation. In this paper, we consider a stochastic model to evaluate the probabilistic behavior of Internet-worm propagation based on a simple birth and death process. This model can be regarded as an approximate model with the well-known stochastic SIS (Susceptible-Infected-Susceptible) model which is used to represent the propagation of Internet worms as well as physiological viruses. One of the advantages of using the approximate model is to derive the probability generating function (PGF) of the number of infected hosts. In other words, the explicit probability mass function (PMF) of the number of infected hosts can be derived from the algebraic approaches based on the PGF. The PMF enables us to perform the transient analysis of propagation of Internet worms. More precisely, we define some dependability measures for the propagation of Internet worms based on the PMF, and investigate the sensitivity of these measures in numerical experiments. 2. Stochastic SIS Model The propagation of Internet worms is often modeled by the stochastic SIS model. The stochastic SIS model is an extensive model from the deterministic SIS model, where both of them are used for behavioral analysis in the epidemiology 6 . Consider two possible states for one host computer; a susceptible state (state S) and an infected state (state 7). The state S means that the corresponding host computer is eventually infected with the Internet worm. Once the host is infected, the state of the host becomes I. When the elimination of Internet worm is successful, i.e., the worm is removed, the state of the host becomes S again. Let {X(t);t > 0} denote a stochastic process representing the number of infected hosts at time t with PMF: pfc(t) = Pr{X(i) = k}.
(1)
Assuming that the total number of hosts is M, we have the transition probability of the number of infected hosts for the time period [t, t + At) as follows. Pr{X(t + At)=j\X(t)
= i}
678
' Pi(l - i/M)At + o(At), fiiAt + o{At), 1 - {pi{\ - i/M) + ip,}At + o(At), k o(At),
j =i+ 1 j =i-l j =i otherwise,
( ) [
'
where (3 and /x are the pairwise infection rate and the removal rate, respectively. Also, o(At) is the higher order function of At, i.e., limAt->o At = 0. By taking the limit of the above equation, At —> 0, we obtain the following differential-difference equations: jtPo(t) = 6lPl(t),
(3)
^Pfc(t) = €k-iPh-i(t)+5k+iPk+i{t)-(£k+fa)Pk(t),
(4)
A=l,...,Af-l, ^TPM(*)
= iM-\PM-i{t)
- (£ M + 6M)pM{t),
(5)
where ft = f3k{\ — k/M) and 8k = p,k. Prom Eqs. (3)-(5), the stochastic SIS model is represented by state-dependent parameters £& and 5k- Roughly speaking, the resulting stochastic process from the above differentialdifference equations is reduced to the birth-and-death processes. The stochastic SIS model, several kinds of birth-and-death processes are used in the epidemiology; for example, the SIR (Susceptible-Infected-Removal) model, the SIRS (Susceptible-Infected-Removal-Susceptible) model, the Predator-Prey model, etc. On the other hand, based on the stochastic SIS model, we can define some dependability measures such as hazard probability and extinction probability in order to evaluate and predict the propagation of Internet worms 4 . However, as mentioned before, since the stochastic SIS model consists of a huge number of states, the computation of these dependability measures have to be made approximately. 3. Approximation of Internet-Worm Propagation 3.1. Approximation
by Simple
Birth
and Death
Process
In this section, we propose an approximate model to represent the transient behavior of Internet-worm propagation based on the stochastic SIS model. First we prove that the stochastic SIS model can be approximated by the simple birth and death process in the case where the number of states is large. Suppose that the number of hosts is extremely larger than the number of infected hosts, that is, we make the following assumption:
679
• Assumption: the number of infected hosts is sufficiently smaller than the number of hosts in overall the Internet. Under this assumption, the fraction of susceptible hosts over the whole hosts is considerably close to one, i.e., (M — k)/M —> 1 as the number of infected hosts k extremely decreases. Therefore, in the formulation of £&, replacing (3(1 — k/M) by a constant value A yields the following differential equations: jtPo(t) = SlPl(t),
(6)
-jTPfc(0 = €k-iPk-i{t)
+ 5k+iPk+i{t) - (& + 5k)pk(t),
(7)
k = 1 , . . . , oo, where £& = Afc and 8k = fik. This stochastic process is classified into one of the birth-and-death processes, and particularly, the process with simple rates £& and 8k is called a simple birth and death process. From the above assumption, this model describes the situation where the number of infected hosts at the early stage of influences of Internet worms is quite small.
3.2. Transient Process
Analysis
based on Simple Birth and
Death
Consider a stochastic process {X(t);t > 0} representing the number of infected hosts at time t as the simple birth and death process. Then the PMF of the number of infected hosts and the initial number of infected hosts are denoted by pn(t) = Pr{X(t) = n\X(0) = N} and X(0) = N, respectively. The state-dependent parameters of the simple birth and death process are given by £fc = Afc and 6k = /xfc, where fc is the number of infected hosts, A and /J, are called the propagation rate and the removal rate of Internet worm, respectively. The PGF for the probability mass function Pk(t) is defined as oo
P(z,t) = E[zx^} = J2pk(t)zk.
(8)
k=0
From Eqs. (6) and (7), we have
^ ^ . = \M(1-Z)+XZ(Z-1)]2^1
an d
P(z,0) = ztf.
(9)
680
The above partial differential equation can be solved algebraically7. That is, the PGF of X(t) is given by
Next consider the PMF explicitly from the above PGF. For example, Kyriakidis 8 attempts to derive an explicit form of a simple birth and death process with catastrophe based on a moment approach from the PGF. Chao and Zheng9 and Zheng and Chao 10 discuss the transient analysis based on the inverse Laplace transform for somewhat different processes from a original simple birth and death process. This paper applies the inverse Z-transform into deriving the PMF. Based on the binominal theorem, Eq. (10) can be straightforwardly rewritten by
N
t=0 r
^-(A-M)* -
fj, ^
Ae-(A-/x)t
_ ^
. N—i |
ne-(x-n)t - \ ~ \e-(.x-i*)t - \ J •, N-i
(11) Me-(*-(.)i_A'
5
.
Since there is one term including z in Eq. (11), it is enough to take the inverse Z-transform for this term. Note that the PGF of negative binomial distribution is equal to this term. Therefore, we have the following inverse Z-transform:
Ae-^-»)'-A
r
(N + n - i - 1)! J Ae-< A -^' - A \ nl(N-i-l)\ ue-(A-")*-A
(12)
Finally the PMF of the number of infected hosts based on the simple birth
681 and death process can be derived as M-Y*
Pn{t) - }_,
N\{N + n - i - 1)1
ilnl(N
_ i _ ^N
J Ae-(A-">* - A»
_ {y J Ae -(A- M )« _ A
t=0
ne-(\-i*)t Ae
- x ~ \e-(\-n)t
-(A-^_Ar
^e-(\-u,)t 3.3. Dependability
_ AJ (13)
- Af '
Measures
We define two dependability measures to evaluate the transient behavior of Internet-worm propagation: extinction probability and hazard probability. The extinction probability, Pg;(i; N), is defined as the probability that none of hosts is infected with the Internet worm at an arbitrary time t provided that the number of infected hosts is N at the initial time t = 0. From the PGF or the PMF of the number of the infected hosts, the extinction probability is given by
«•<*">=(£££:;) -
<">
On the other hand, we define the hazard probability, Pn(t;N,H), as the probability that the number of infected hosts exceeds H at an arbitrary time t provided that the number of infected hosts is ./V at the initial time t = 0. Then we have P„(t; N, H) = Vx{X(t) > H\X(0) = N} oo
H-l
Pk(t) = i - £ ?*(*). k=H
(is)
fc=0
where pk (t) is the PMF of the number of infected hosts, as Eq. (13). Both of measures, PE(*; JV) and Pn(t;N,H), can estimate quantitatively the force of infection of Internet worms. 4. Numerical Example This section gives a numerical example to calculate the hazard probability. Figure 1 illustrates the behavior of hazard probability that the number of infected hosts exceeds 2N, where N is the initial number of infected hosts.
682
Figure 1.
Transient behavior of PH (t; N, 20).
In this case, we set N = 10, A = 0.5 and /J. = 0.3. As Fig. 1 shows, the probability, PH (t; 10, 20), gradually increases as the time t increases. Similarly, as the initial number of hosts increases, the probability also increases. In both increases, the hazard probability eventually converges to one. In this example, the model parameters are set as A > fi, and the force of propagation is stronger than the removal. Other numerical experiments in some similar situations cause the phenomenon where the hazrd probability converges to one. On the other hand, in the case of A < fi, we often observe that the propagation is eventually terminated with depending the initial number of infected hosts. Thus when we focus on the quantity of p = A//x, it can be regarded as a simple measure on the propation risk of Internet worms, and the termination of propagation requires this measure to be less than one.
5. Conclusion In this paper, we have derived the transient probability of the Internetworm propagation based on a simple birth and death process. The simple birth and death process has been derived as an approximate model for the stochastic SIS model under the assumption on the huge number of hosts.
683 This case can b e regarded as t h e propagation of Internet worms a t t h e early stage of their influences. In t h e approximate model based on t h e simple b i r t h and d e a t h process, b o t h P G F and P M F of t h e number of infected hosts can b e explicitly derived by an inverse Z-transform approach. Moreover, two dependability measures; probabilities of extinction and hazard, are denned t o estimate t h e quantitative force of infection of Internet worms. In future, we will analyze the transient behavior of existing Internet worms based on t h e proposed model, and examine t h e prediction ability of propagation of Internet worms with real d a t a .
References 1. J. O. Kephart and S. R. White, Measuring and modeling computer virus prevalence, Proceedings of the 1993 IEEE Computer Society Symposium on Research in Security & Privacy, pp. 2-15, 1993. 2. J. C. Wierman and D. J. Marchette, Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction, Computational Statistics & Data Analysis, 45, pp. 3-23, 2004. 3. H. Okamura, H. Kobayashi and T. Dohi, Dependence of computer virus prevalence on network structure -Stochastic modeling approach, Proceedings of Asian International Workshop on Advanced Reliability Modeling, pp. 379386, 2004. 4. H. Okamura, H. Kobayashi and T. Dohi, Markovian modeling and analysis of Internet worm propagation, Proceedings of 16th International Symposium on Software Reliability Engineering, pp. 149-158, 2005. 5. L. J. S. Allen and A. M. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Mathematical Biosciences, 163, pp. 1-33, 2000. 6. G. H. Weiss and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Mathematical Biosciences, 11, pp. 261-265, 1971. 7. L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Peason Education, Inc., New Jersey, 2003. 8. E. G. Kyriakidis, Transient solution for a simple birth-death catastrophe process, Probability in the Engineering and Informational Sciences, 18, pp. 233-236. 2004. 9. X. Chao and Y. Zheng, Transient analysis of immigration birth-death processes with total catastrophes, Probability in the Engineering and Informational Sciences, 17, pp. 83-106, 2003. 10. X. Zheng, X. Chao and X. Ji, Transient analysis of linear birth-death processes with immigration and emigration, Probability in the Engineering and Informational Sciences, 18, pp. 141-159. 2004.
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PART IX STATISTICAL QUALITY CONTROL
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IMPROVED INSPECTION SCHEDULE FOR A BATCH MODE PRODUCTION SYSTEM * JIH-AN CHEN* Department of Business Administration, Kao-Yuan University, NO.1821, Jhongshan Lujhu Township, Kaohsiung County 82151, Taiwan
Rd.,
YU-HUNG CHIEN Department of Statistics, National Taichung Institute of Technology, 129, Sec 3, San-min Rd., Taichung 404, Taiwan In this paper, we introduce and deal an inspection schedule for a lot-sizing production system. The system of production process has a general deterioration distribution with increasing failure rate and non-self-announcing failures. Rather than develop a nonMarkovian shock model, we focus on a quantile-based reliability model. This research will also provide a strategy of inspection based on the economic production quantity and examples of Weibull shock models will be given to illustrate this inspection schedule.
1. Introduction Rapidly changing markets and the explosion of product variety has increased automation and the need for sophisticated production system. The problem of the determination of economic production quantity (EPQ) for production processes has been well studied in literature. The role of the condition of production system in controlling quality and quantity is well known and production system must be maintained in conforming conditions through adequate maintenance programs. There are many studies, such as Rosenblatt and Lee (1986), on the effects of deteriorating processes on economic manufacturing quantity. The classical economic manufacturing quantity model assumes that the output of the production system is defect-free. When developing EPQ models, consideration of controlling the quality of the product has generally not been taken into account. Optimal inspection policies for products with imperfect qualities were also investigated considerably. For further review of the EPQ topics with imperfect production processes, the reader This work is supported in part by the National Science Council of Taiwan under grant NSC 932218-E-244-003. Corresponding author. Tel.: +886-7-6077074; E-mail address: [email protected].
687
688
can refer to Sheu (2004). Suppose that failure of the production system can be detected only by inspection of product. And, inspection can not determine either the level of deterioration or the remaining life of the production system. Such is the case in many protective items such as circuit breakers and protective relays, as well as in spare or standby. If the production system is found to be failed by performing inspection, it is replaced immediately with an identical item and the production process need not to be broken down. Many studies, such as Rahim (1994), have effort on the production system which is repairable. Being motivated by this, proposed that the length of sampling intervals should be chosen in such a way that the integrated hazard over each interval should be equal. Yang and Klutke (2000) were also deal inspection policies for maintaining deteriorating equipment with non-self-announcing failures. Their inspection policies utilize information from the inspection/repair history as well as the system lifetime distribution to schedule future inspections. If inspection, replacement and downtime cost are available, they also derived the expression for long-term expected cost. Our objective is to present inspection policies and propose some solution procedures for simultaneous determination of the inspection intervals and the number of inspection in a production run. 2. Inspection Schedule 2.1. The Quantile-based Reliability in a Production System The most widely used but weak inspection schedule is perhaps constant interinspection. It is to schedule inspections periodically with the advantage of being simple to implement and relatively easy to analyze. Such a schedule has constant inter-inspection ht=h, for all /', and inspection ages {h,2h, •••}. The periodic inspection schedule tends to over-inspect at less likely failure times and under-inspect at more likely failure times. Thus, it employs no information about the remaining life that is inherent in the sequence of previous inspection times. Yang and Klutke (2000) proposed an inspection schedule QBI( cc0) with quantile-based reliability. Let the sequence of inspection ages be {x„x2,---} and we denote t as the time to failure. And, t can be expressed as a random variable with a continuous, strictly increasing distribution function F{t). The quantile-based reliability a0 is fixed and known. Then, the inspection ages are JC, = sup{x>0: P{t>x}>a0} and
(1)
689 xi = sup {x > JCM : P{ t > x 11 > xM }>«„}, for i = 2,3, • • •.
(2)
Thus, x,. = F-\a[),
for i = l , 2 , - .
(3)
If the distribution of the time to failure is increasing failure rate, then F{x + hi)/F{x) is non-increasing in the time to failure x for all ht>0. Thus, hM < hi, for / = 1,2, • • • . For lot-sizing production system, the cycle time of operation system L can be expressed as £ = JA,=F- , (O 0 "),
(4)
where m is the known number of inspection in the cycle time of operation system. We denote such a schedule as QBI. 2.2. The Balance of Risk under Fixed Economic Manufacturing Quantity The problem to determine the economic production quantity and the production cycle time for lot-sizing production system has been well studied in literature. For a Markovian shock model, a uniform inspection scheme provides a constant integrated hazard rate over each interval when the number of inspection m is determined. And such a schedule has constant inter-inspection time h = L/m and inspection ages {h,2h,---,mh} . Being motivated by this, Rahim (1994) extended the idea to non-Markovian shock models and proposed that the interinspection times h, = x. - x,_, should be chosen in such a way that the integrated hazard over each interval should be equal, i.e.: ['r{t)dt JO
= --\Lr{t)dt, yyt
(5)
JO
and \^r{t)dt=\\(t)dt,
i = \,-,m,
(6)
where I r{t)dt represents the expected failure number over interval [a, b) and x0 = 0. It is noteworthy that keeping a constant integrated hazard over each sampling interval is equivalent to stating that the probability of a shift in an interval, given no shift until its start, is a constant for all intervals. The deterioration of production systems is inherent to most manufacturing industries. If the distribution of the time to failure is increasing failure rate, the inter-
690
inspection times \,h2,•••,hm are also non-increasing, as expected. According to the lot-sizing production cycle time and the number of inspection m, we have the inspection ages {x1,x2,---,x„}: i
x, = '£ihJ,i = l,-,m.
(7)
7-1
We denote such a risk balance schedule of as RB. 2.3. The Hybrid Inspection Schedule For most manufacturing industries, the quantile-based reliability is a specific aim for standard operation process. And, it is an unknown but important objective to determine the optimal number of inspection and the inspection ages during each production run. When the cycle time is given and under the known quantile-based reliability, we have the initial inspection ages. And,
i-l
i»l
Thus, we could determine the least number of inspection m from (8) and the alternative inspection ages {y„y2,•••,)>„} can be generated by Eq.(6). I'' r(t)dt=
[*r{t)dt = —- [Lr{t)dt, i = l , - , m ,
J >,-i
Jo
(9)
yfi Jo
where y0 = 0. We denote such a hybrid schedule as HYB. 3. Discussion 3.1. Equivalence between QBI and RB While the production cycle time and the number of inspection are predetermined, we can use equations (5) and (6) to obtain the inspection ages {x1,x1,---,xm}. At the first inspection age X,, the reliability i?(x,) can be expressed as P{t > xx} = R(x,) = exp {- £' r(t) di).
(10)
It implies that x, = R-\exV{-\\(t) dt}) = F-\aa),
(11)
691 where a0 is the quantile-based reliability. And, at the z'th inspection age is x =^hj
,i = 2,---,m , we have P{t > x,} = R(Xi) = exp{-£r(t) dt) = e x p { - £ £ r(t) dt)}.
(12)
From Eq.(6): P{t > x,} = exp {-/ • £ r(t) dt) = (exp {- £ r # df})' = a't.
(13)
It implies that x = F " ' ( a j ) . Because that the production cycle time has been predetermined, the reliability is also determined at the same time under fixed number of inspection. On the other hand, if the quantile-based reliability has been predetermined, we have also the same sequence of inspection ages with schedule RB under fixed number of inspection. 3.2. The Domination of Hybrid Inspection Schedule We may make better decision to a given scenario with no loss of generality. Firstly, fixed lot-sizing batch mode of production and constant production rate could be anticipated from economic manufacturing quantity or the characteristic of batch mode in most industrial. Thus, we have the known production cycle time for each production run. In a complete production run, the least number of inspection, an unknown but important objective, may also be determined. Under schedule QBI, we have the inspection ages x, = F~'(a'0) with reliability: R(x:) = exp{- j'rft) dt} . Under schedule HYB, we have the inspection ages fromEq.(8)toEq.(10): V'r(t)dt = — • \Lr(t)dt< — - V"r(t)dt=
R(v)
expf-fVodf}
Thus, y, <x, . And, -±£1 = h R (*,) exp{-[r(t)dt}
V'r{t)dt.
„ = exp{\ r(t)dt} .Because of J> '
y, < x, , the ratio —^- = exp{ ?r(t) dt) > e° = 1, i.e. a = R(y,) > R(x.) = aa x h R(x,) R( i) We could improve the reliability by adopt schedule HYB 4. Example To demonstrate the applications of our proposed procedure under nonMarkovian shock models, a few special cases of Weibull failure mechanisms are
692 considered in this section. Recall that the Weibull distribution function with scale parameter A and shape parameter ji is given by F(t) = l-exp{-(AtY} for t > 0. The corresponding failure rate is r(t) = (A/3) • (At)"''. For table 1, we use a0 = 0.990 and A = 0.05, p = 2.00. The mean time between failure (MTBF) is 17.7245. The first inspection age is 2.005 with reliability 0.990. Because of linearly increasing failure rate, the inter-inspections are decreasing. Table 1. Inspection schedule for time to failure distributed as Weibull(0.05,2.00)
i xj
i
2
3
4
5
6
7
2.005
2.836
3.473
4.010
4.483
4.911
5.305
K
2.005
0.831
0.637
0.537
0.473
0.428
0.394
R(*,)
0.990
0.980
0.970
0.961
0.951
0.941
0.932
y,
1.890
2.673
3.273
3.780
4.226
4.629
5.000
k
1.890
0.783
0.601
0.506
0.446
0.403
0.371
R(y,)
0.991
0.982
0.974
0.965
0.956
0.948
0.939
In table 1, suppose the production cycle time is 5 and the quantile-based reliability should be larger than 0.99. The number of inspection 7 can be determined from Eq.(8) and the inspection ages can be generated by Eq.(9). Under HYB schedule, the inter-inspections are decreasing similar to QBI schedule and the inspection ages advance against schedule QBI resulting in higher reliability. 5. Conclusion In this paper, we introduce, modify and compare some inspection policies for deteriorating lot-sizing production system subject to a non-Markovian random shock. These policies are simple to schedule and easy to implement, thus making it more realistic and reflective of situations of the known production condition. Weibull shock models with increasing failure rates are also considered as seen in the simulation example. Also, the modified inspection schedule is used to improve the reliability of the whole production cycle. References 1. 2. 3. 4. 5.
M.A.Rahim, HE Tran. 26, 2(1994) M.J. Rosenblatt and H.L. Lee. HE Tran. 18,48 (1986) S.H. Sheu and J.A. Chen, USS. 35, 69 (2004) S.H. Sheu, J.A. Chen and Y.H. Chien, AIWARM. 451 (2004) Y. Yang and G.A. Klutke, Prob. Eng. Inf. Sci. 14,445 (2000)
EVALUATION OF MULTI-PROCESS CAPABILITY BY A FUZZY INFERENCE METHOD T. W. CHEN Department of Mechanical Engineering, National Chung Hsing Taichung, Taiwan ROC
University
T. C. WANG Department of Mechanical Engineering, National Chung Hsing Taichung, Taiwan ROC Process capability indices C , Cpt , C ,„ and C
University
fitting for nominal-the-best type
quality characteristics, are an effective tool to assess process capability since these indices can reflect a centering process capability and process yield adequately. The index c introduced by Greenwich and Jahr-Schaffrath (1995) provides additional and individual information concerning the process accuracy and the process precision. Although c is useful to evaluate process capability for a single product in common situation, c
cannot
be applied to evaluate the multi-process capability. Referring to Vannman and Deleryd's (C;,, Q,)-plot, a fuzzy inference method is proposed in our study to evaluate the multiprocess capability based on values of a confidence box calculated from sample data. This method takes the advantages of fuzzy systems such that a grade instead of sharp evaluation result can be obtained. An illustrated example of ball-point pens demonstrates that the presented method is effective for assessment of multi-process capability.
1. Introduction Process capability indices (PCIs) are effective tools for the assessment of process capability indeed since the formulae of PCIs are easy to understand and straightforward to apply. Greenwith and Jahr-Schaffrath [1] introduced a new index CPI„ which provides an uncontaminated separation between information concerning process accuracy and process precision. It has been widely used to provide numerical measures on whether a production is capable of producing items within the specification limits preset by the designer. The index Cpp can be defined as:
c
+
--tefT feT693
694
where ^i is the process mean, a the process standard deviation, d the half the length of specification interval = (USL - LSL)/2, USL is the upper specification limit, LSL the lower specification limit, and T the target value. Now, let the inaccuracy index Cdr and imprecision index Cdp be defined as:
C =JL
* f
c
*=7
(2) (3)
Obviously, one can recognize that Cpp = (3Cdr)2 + (3Cdp)2 • Cpp (including Cdr and Cdp) provides additional information concerning the process accuracy and the process precision. Index Cpp detects process inaccuracy and process imprecision by using indices Cdr and Cdp. Thus, Cpp is a deter choice for engineers measuring process potentials and performance. Although Cpp is useful to evaluate process capability for a single product in a common situation, Cpp cannot be applied to evaluate process capability for that of multi-process. Thus, we extend the applicability of the contour plot for processes with multiple characteristics. In addition, we also apply the method developed by Chen et al. [2] who introduced a process capability plot, called the MCPCA control chart, which is an adjustment of Vannman and Deleryd's (Cdr, Cdp)- plot [3] where Cdr = (f.i - T)ld and Cdp = old. Referring to (Cdn Qp)-plot, we propose a fuzzy inference method to assess the process capabilities of multiprocess. The concept of fuzzy sets was first proposed by Zadeh [4]. Now, fuzzy theorems have been applied in many fields such as automatic control, manufacturing system and decision-making [5-8] in industry. In this paper, a fuzzy inference method is proposed such that process capability can be assessed. This fuzzy inference evaluation will consider Cdr and Cdp to formulate new indices as input and obtain a result value as output. In addition, illustrated example and evaluation procedure will be presented for ease of applications. 2. Process Capability Confidence Intervals The index Cpp can directly be used to assess process capability when 100% inspection is applied. Instead of 100% inspection, acceptance sampling is most likely to be useful when the testing is destructive, or the testing cost is extremely high, etc. Generally, only estimated capability index Cpp by using a sample can be obtained in practice. Each process has the same product specification and target value. Thus, the natural estimator of Cppl can be written as the following:
695
(Xi-TY
C
+
D2
-±r,i=\,2,...,k
D2
(4)
where D = d/3, X, and Sf are the sample mean and sample variance of process i with sample size ni. The probability density function of Cppi (see Chen [9]) is:
ff (*)
(nD2/a1)x
(nD2
^ - T ^ ) ^a J l^o
f
fK{^-x-y)fr](y)
J 0
(5)
where x > 0, 7; is distributed as chi-square with (1 + 2/) degrees of freedom and "-(A/2)(A/2)J
pn.)-
r-
Furthermore, the mean value and variance about C
' \~-ppi ppi) 1
Var (c •) v-'ppi I
is:
CD,
2a" nD4
nDA
Let /} and a be the unbiased estimators of p and a, then we have /} = X , a = c 4 x S and c4 = yJ2/(n — l) • r [ « / 2 j / r [ ( « - l ) / 2 ] . The factor c4 is a function of the sample size n and c4 approaches to unity when the sample size is large enough. Under normal assumption, (H-1)[(C 4 (T)/
M~ta/4,("-l)
XC
a
„
4 X~f='^ '4~n +ta/4,(n-l)
\{n-\)xcl X\-au(n-\)
xa2
XC
4
X
(«-l)xe 4 2 x(T 2 Xai^n-\)
a
~r=
(X,,X„)
(Y,,YU)
(6a)
(6b)
where ta/A ,„_n is the upper quartile of t distribution with (« - 1) degrees of freedom; X\_an{n-\) and x\i\{n-X) are the upper percentile of chi-square distribution with (« - 1) degrees of freedom. The joint confidence intervals of ju and a are used to formulate a confidence region and applied to reveal the process capability.
696 Referring to Vannman and Deleryd's (Cdr, Q ; )-plot, we propose a new method to assess the process capabilities of multi-process. Since Cdr = {/u - T)ld and Cdp = a/d, thus the confidence region in the plot of Cdr -Cdp should be changed as: Upper-right coordinate: ((Xu - T) I d, Yu I d) = (X ru , Yru)
(7a)
Bottom-right coordinate: ((Xu - T) I d, Yu I d) = (X ru , Yr,)
(7b)
Upper -left coordinate: ((X, - T) I d, Yu I d) = (X rl , Yru)
(7c)
Bottom -left coordinate: ((X, - T) I d, Y, I d) = (X rl , Yrl)
(7d)
where Xu, X,, Yu and Yt are described in equation (6). And the maximum estimated index (C ) could be calculated as: v
pp /max
(CPP )max = max[ (3Xm f + (3Yru ) 2 , (3Xrl f + (3Kra ) 2 ]
(8)
To reduce the influence of sampling errors, we shall use values of mentioned confidence box described in equation (7) to afford a more reliable assessment. In the plot of Cdr -Cdp, there are two process capability values for PI and P2 as seen in Fig. 1, one can recognize that the process capability is adequate if the confidence box is inside of the line Cpp = 1 (process P2) and process capability is inadequate if the confidence box is outside of the line Cpp = 1 (process PI). The larger distance of this confidence box to the coordinate original point (0, 0), the worse process capability. Let the nearest distance from the coordinate original point to each confidence box be Rmin in Fig. 1 and the most far distance from the coordinate original point to each confidence box be Rmax, then Rminar\d Rmax can be defined referring to equation (7) as: Rmm = min (Jxl+Y?u, Rmax = max{]x2m
fx2ru+Y*, jx2+Yr2u,
^X^+Y2,)
+ Y2U,]x2ru + Y2 ,]x2rl + Y2U ,^X2+
(9) Y2,)
(10)
In our study, Rmi„ and Rmax are used to represent the process capability for each model and a reliable assessment is achieved by using the magnitudes of Rminand
3. Fuzzy Inference Method for Process Capability In this section, a fuzzy inference method is proposed to assess the process capability for multi-process. As stated in section 3, Rmin and Rmax are used to
697 assess the process capability. The larger values of Rminand Rmm for each model, the worse process capability. Let [ Rmmi, Rm3xi ] and [ RmmJ, RmaxJ ] be the nearest and the most far distances in the plot of C& -Cdp for the process / and j , respectively. Consider [Rminj, ^ m a x ,] and [Rminj-, ^max_,-] as set of two lines in same axis, then the comparison of two processes can be represented by statistical method. In order to distinguish the equal grade of capability (for process i and j) in different intersection, a method to incorporate the fuzzy inference with a process capability index is then proposed. An approximating rule-based reasoning approach is used for quantitative analysis. In our study, the process i is said to be superior to process^' when the value of inference result is positive. The larger of result value, the more capable of process i than that of process j . Oppositely, the negative value of result implies that the process / is inferior to process j . In other words, the process i is said to be completely better-quality than that of process j when the value of inference result is equal to 1; the process i is said to be completely same-quality to that of process j when the value of inference result is equal to 0 and the process / is said to be completely worse-quality than that of process j when the value of inference result is equal to -1. The result value of inference within {0, 1} or {-1, 0} is used to represent the different grade of capability. The h processes are tested two at a time, there are hC2 = h(h - l)/2 possible paired comparisons. Let the indices 8 and y be defined as R X_
— R
mini
max/
/ii\
max(RmiSKJ,Rmmj) R
—R
max/
/ =
: maX
min /
—•
., _.
(12)
(-Rmax/'^maxy)
Then the fuzzy inference systems are composed of two inputs and one output. Generally, the procedure of fuzzy analysis consists of four steps: definition of input/output fuzzy variables, fuzzy rules, fuzzy inference and defuzzification [6]. (1) Definition of input/output fuzzy variables. In our study, we adopt the triangular and trapezoid types as MFs for the sake of simplicity and easy to describe the asymmetric property. The triangular MF is specified by three parameters {a, b, c) which determine the three corners of triangle. Furthermore, the trapezoid MF is specified by four parameters {a,b,c,d}.
698 The universe of input variables is defined in {-1, 1} as shown in Fig. 2. Membership functions of input 8 are defined as N4 (negative), N3, N2, Nl, ZE(zero) and PO (positive), respectively. Also, input y are defined as NE (negative), ZE, P\ (positive), P2, P3 and P4, respectively. In addition, the output variables are composed of seven triangular MFs for representing Z3 (inferior), L2, LI, EQ, 51 (superior), 52 and 53 as shown in Fig. 3. (2) Fuzzy rules. Fuzzy rules are important to successful inference result [7]. A rule base represents the experience and knowledge of experts. The fuzzy rules are similar to the intuitional thinking of a human. A fuzzy inference system, composed of two inputs and one output, could employ this kind of fuzzy rule as If xl is An and x2 is Ai2 thenyis Bt (for/= 1 tow), In this study, the fuzzy inference system is applied to assess the process capability by using the confidence interval values of Rmin and Rmax (defined in equation 7). Thirty-three if-then rules are employed in our study. They are: Rule \:if{8
is PO) and(y
is PA) then (result is L3).
Rule 2: if(5
is PO) and(y
is P3) then (result is 13).
Rule 33: if {8 is N4) and(y
is NE) then (result is 53).
The tabulated fuzzy rules are listed in Table 1. The fuzzy rules indicated in Table \,if(8 is PO) and(y is ZE) or (y is NE) in addition to / / ( 8 is ZE) and (y is NE), are never happened since the definition of two input variables always exists y>8. Table 1. The tabulated fuzzy rules.
8 PO ZE N\ N2 Nl NA
PA
Pi
P2
P\
Li 12 12 L\ L\ EQ
Li L2 12 L\ EQ S\
LI 12 L\
Li LI
EQ
EQ S\ S\
Si 52 52
ZE
NE
EQ 51 52 52 53
Si 53 53 53
\
(3) Fuzzy inference. Fuzzy inference is an inference procedure to derive conclusion based on a set of if-then rules. In this paper, the Mamdani inference
699
>
US
\ "f
j! -0.6
N..-1 °
/'
P1
-0.4
K
V
" • . „ \
-0.2
Hi
0
0.2
0.4
0.6
0.8
Fig. 1 Multi-process capability analysis plot.
input membership functions
-0.4
-0.2
NE 0.8 0.6 0.4 0.2
0 0.1 Input 8
TO
PI
/
YYVVY A A A A
IvVVv^\
-0.1 0
0.2
0,4
Fig. 2. Membership functions of input variables. output membership functions
Fig. 3. Membership functions of output variable.
700
method [8] that employs the maximum-minimum product composition to operate fuzzy if-then rules is adopted. Let the rule be: if xx = Ax and x2 = A2 then y = B, then the result of inference can obtain a fuzzy set with MF of B- as jUB;(y) = max{min[//4. (*,),MA;2(*2)>HR,(*i>X2>>0j}>
(13)
where VR, (xi,x2,y)
= min[,«4i ,MAi2,MB, (>')] •
(4) Defuzzification. The fuzzy sets of B- are obtained by step (3), then the defuzzification is used to find a crisp value y e Y which represents the fuzzy sets. The frequently used defuzzification methods have: weight, area and height method in [6]. The weight defuzzification method is used in our study, and then we have \yB{y)dy y* = —r )B{y)dy
•
(14)
Y
Result of fuzzy inference, performed by a Matlab Logic Fuzzy Toolbox [10], is then used to represent the process capability for each model. 4. Procedure of Fuzzy Evaluation and Illustrated Example Below, an example is taken to state the proposed procedure in detail. To illustrate how the procedure may be applied to collected data, the following case on four manufacturing processes of ball-point pens is presented. In this case, four models of ball-point pens that are named as PEN1, PEN2, PEN3 and PEN4 models. The fitness of the cap to the body is considered as an important characteristic to avoid losing the cap of the ball-point pen or hard to pull the cap out; that is, the fitness cannot be too tight or too loosely. The critical measurement is how much strength is required to pull out the cap from the body of the ballpoint pen and the strength unit is measured in kilogram. The specification limits are set to 2.5±1.0, that is, the upper/lower specification limits are set to USL = 3.5, LSL = 1.5, and the target value is set to T= 2.5. Table 2 displays the concise information of these four models. The fuzzy evaluation procedure is stated as follows:
701 Step 1 :Determine the sample size n = 60 for all manufacturing processes, then the values of mean and standard deviation are calculated as indicated in Table 2. Also, the significant level is given 0.05. Step 2: Compute the values of c4 and a . Also, calculate four coordinates values of each confidence box described in equation (7). Step 3: Compute the nearest and the most far distances (Rmi„ and Rmax) in equations (9) and (10) for each process. Table 2: Process capability value for four processes.
Process
x,
s,
PEN1
2.52
0.202
0.2011
0.1720
0.2661
0.6373
PEN2
2.51
0.250
0.2489
0.2168
0.3252
0.9519
D
/Y
D
min /
A
PEN3 2.44 0.300 0.2987 0.2502
0.4054
1.4788 *
PEN4 2.59 0.197 0.1962 0.1662
0.2887
0.7499
max/
max /
Step 4: Compute the indices 8 and y for two-process pairs (there are six pairs) and thus to obtain the fuzzy evaluation results through the proposed fuzzy inference system. The above calculations are performed through developed Matlab program. As indicated in Table 3, one can recognize the model of PEN1 is the best one among these four processes since all values of inference result are positive for competing pairs (PEN1 to PEN/, for _/=2, 3 and 4). Furthermore, model of PEN3 is the worst process. Table 3: Fuzzy inference results. Pairs (;' toy)
1 ^min/ '
max/ J
PEN 1 to 2
[0.1720,0.2661]
PEN 1 to 3 PEN 1 to 4
I
min / ' ^ m a x / J
8
Y
Result
[0.2168,0.3252]
-0.4710 0.1515
+0.50
[0.1720,0.2661]
[0.2502, 0.4054]
-0.5756 0.0392
+0.87
[0.1720,0.2661]
[0.1662,0.2887]
-0.4040 0.3461
+0.16
PEN 2 to 3
[0.2168,0.3252]
[0.2502,0.4054]
-0.4651 0.1860
+0.42
PEN 2 to 4
[0.2168,0.3252]
[0.1662,0.2887]
-0.2208 0.4889
-0.35
PEN 3 to 4
[0.2502,0.4054]
[0.1662,0.2887]
-0.0948 0.5900
-0.65
5. Conclusion Process capability indices of Cpm and CPI, are proved that it can reflect the centering process capability and process yield adequately and is used to afford a
702
numerical measure on whether a production is capable of producing items within the specification limits. The index Cpp provides individual information concerning the process accuracy and process precision. Referring to Vannman and Deleryd's (Cdr, C
A NEW PROCESS IMPROVEMENT CAPABILITY INDEX OF CONSIDERING COST TO SELECT SUPPLIER
K. S. CHEN Department of Industrial Engineering
& Management,
National Chin-Yi Institute of Technology, Taichung, Taiwan, R.O.C. E-mail •'
[email protected]
S. L. YANG Institute of Production System Engineering and
Management,
National Chin-Yi Institute of Technology, Teaching, Taiwan, R.O.C.
J. M. HUANG, C. Y. HSIEH Department of Mechanical Engineering, National Chin-Yi Institute of Technology, Taichung, Taiwan, R.O.C. A lot of businesses bring process capability index into be measurement quality tool. Process capability index Cpm is now in widespread uses in industries because of process capability index Cpm amply react to loss of the process and yield of the process. In general, manufactures select suppliers just only use the index Cpm to evaluate the suppliers' process capability. However, coming the supply chain competitive times, manufacturers have to realize how to select suppliers and subcontractors to be key point work of the supply management. Therefore, for original suppliers, we matched up the process capability index Cpm and process improvement capability index CPIM to evaluated, measured the suppliers' process capability and reduced manufactures' improvement cost. When suppliers' process capability not enough, we can further make use of process improvement capability index CPIM to measure suppliers' process improvement capability, decrease internal company's process of improvement cost effectively, and improve product's quality and production to get up to the goal of business operation forever.
1.
Introduction
Quality is by no means a new concept in modern business James R. Evans and William M. Lindsay (2002). Now, a lot of businesses bring in process capability index to be measurement quality tool (Mats Deleryd, 1999). Measure the 703
704
process quality of products need to compare the process mean fl and the process standard deviation (7 at the same time. Due to the process of products result from each product hasn't been the same specifications. In fact, process capability index definition is based on above view. Process capability index is a function value for process distribution parameter value, and specifications limit. Kane (1986) proposed that process capability indices Cp and Cpk, however, process capability indices Cp and Cpk can't react to the process loss because that indices Cp and Cpk is based on yield definition. Therefore, Chan et al. (1988) propound the index Cpm which is an index can be responds the process loss sufficiently. Pearn et al. (1992) provided that take index Cpm measure process loss also can really react to the process capability and the process loss. Besides, Govaerts (1994) pointed out the relationship between index Cpm and the process yield is Yield%^20(3 Cpm)-1 when index Cpm value largely enough. It is clear that the index Cpm not only reaction a process loss but also reaction a process yield. Define to of the index Cpm as follows (1): c
USL-LSL
=
d
_
1
.
(1)
According to Phadke (1989) & Pearn & Chen (1997) pointed out that the process loss includes the following two factors: (A) one factor is the process specification not on target (precise of the process not enough) (B) the other one factor is the process variance extremely very big. These two factors will make the index Cpm to be changed. When these two factors loss small will have high index value Cpm. Conversely, when these two factors loss upper and then will have low index value Cpm (at this time mean that process capability is not enough). Obviously, index of Cpm is an excellence index to measure the process capability. However, Chen et al. (2005) pointed out the enterprises (manufacturers) need to realize that how to select and contract their suppliers and manufacturers and apply supply chain management to be an important task in the current competitive global business environment. The competitiveness of an enterprise can be increased by using strategic alliances to integrate supply chain parties and activities. Although manufacturers and subcontractors themselves could take index Cpm measured product's quality, and evaluated the process capability. However, when manufacturers want choose one of suppliers to continued collaboration to take the index of Cpm selected new suppliers and evaluated subcontractors, the index Cpm has no reaction for manufacturers who want to have a completely result of measured standard for process improvement capability of the primal suppliers. Hence, this paper will basis for the index Cpm model to consider the improvement cost and focus on this problem to provided a
705
process improvement capability index, so that the index Cpm can judge and evaluate the process capability of original suppliers with process improvement capability index CPiM. When the process capability Cpm for suppliers not enough, we will think over to improve precise or accuracy of the process capability first. And then we could use process improvement capability index CPiM to measure whether suppliers' process improvement capability superiority or not. Manufacturers by operating these two indices, process capability index Cpm and process improvement capability index CPIM to find out real potential, capable and continued collaboration's suppliers; enhance quality of the process, get more profit with suppliers, and advance the competition for whole supply chain. 2.
Concept of Cost to Set up a Process Improvement Capability Index Model
Above section I, process capability insufficiently (Cpm not arrive standard of quality's needed) includes the following two factors: one factor is the process precise not enough, the other factor is the process accuracy deficiency. Usually, low improvement cost originate from precise not enough, high improvement cost come from accuracy shortfall. The index Cpm can be measured the process capability but couldn't to be evaluate the process improvement capability. Assuming two factories, A. and B. factories produce the same products simultaneously. Where Tis target value, d- (USL-LSL)/2 and T± d is product's upper and lower specification limits. Hypothesizing CI is a cost unit to improve the process precise and C2 also is a cost unit to improve the process accuracy. Supposing supplier A. that the process average of product move did distance from target (The process mean fJ.) and the process standard deviation a is d/3, the index value Cpm will equal to 0.9. According to Chen et al. (2001) pointed that the process capability is not enough need to be improvement when process capability index value Cpm is smaller than 1. Based on definition of six sigma for American Motorola Corporation allows that the process shift dIA distance from not on target (six sigma tolerate the process shifts under 1.5 a ) . So that, the process of supplier A. shifts distance in the six sigma tolerances range. Thus, accuracy shortfall is a really reason for the process capability not enough. After analysis by quality team, maybe scarcity of materials in market is a main effect factor due to the origin short of materials led materials often needed to be changed, led to the process variance big due to difference bench come from difference materials sources and a lot of cost needed to input in improvement quality of product. For convenient to solve this problem, maybe increase purchase of cost, check product's quality from materials, or monitor and control product's quality. On the other hand, supplier B. shifts d/3 distance from target, standard deviation is did, at this time the index value Cpm as same as
706
equal to A. supplier. (Cpm = 0.9) It is very clear that the process precise over dIA from tolerance range, so that the process capability not enough is a really reason for the process precise. After quality team analysis, maybe deflective machine parameters led precise insufficiency. As general, solving this problem needed to input a quite low cost to improve product's quality, such as adjusts the machine parameters back. Comparing with A. and B. suppliers, supplier A. spends decuple cost to improve accuracy of the process then B. supplier. The result is that supplier B. increases a little cost to improve precise of the process supplier A. higher then improvement cost. According to above view of improvement cost, defined process improvement capability index as following (2): c
-
* 3VC,a 2 +C 2 /? 2
(2)
Based on above, process improvement capability index Cp]M focus divergent improvements cost on suppliers. For advantageous application of business, we simplify process improvement capability index CPIM (make r =C2/ C|) as below (3):
C
l 2
3-yja +r/32
0)
To carry on, we were individually count out process improvement capability value are 0.53 and 0.31 from A. and B. suppliers'. Although A. and B. suppliers' process capability all not enough (both of the index value Cpm are equal to 0.9). Thus, B. supplier's process improvement capability is higher then A. supplier's process improvement capability. Obviously, B. supplier will be a top priority to input a shade of improvement cost in for manufactures also according to the index CPIM to select suppliers and promote suppliers' process capability of products.
3.
Estimation of CPlM
On the assumption that characteristic of the process X is normally distributed with mean fx and variance a 2 . Let Xh_X„ be a random sample from normal distribution. Has the same estimation ways with the index Cpm. In this paper, we took the sample mean * = £>,/» to estimate at//and the sample standard deviation $=[£" (*,-]o2/(w-i)~|"2 to estimate atcr. The nature estimation of CPIM as following (4):
707
C„u =
. 1 , 3ja2+r/32
(4)
With a-S I d and ft = (X -T)l d . And then we derive the expected value, variance in (4) as following (6):
x
y £f y=o
( ^ / a » 2 ) > T ( ( H - l ) / 2 + y) y! r(«/2+y)
Var(CPIM)-CPIM2(l+^r)(^-) a
(nB2/a22)J j\ n-\
^
4.
j 2
n 2
2 7
^ «
1
n
2
2
(»/J 2 /a 2 2y r((#i-l)/2 + y)
yAi-^V}. «
j 2
2-
n + 2j-2
' " ' " f t —f. 2
v
n^77r
2F/(
1
y+
1
2' 2' (6>
J
Conclusion
Quality is by no means a new concept in modern business. It's a principal competition of company to find out many methods to reduce operation cost effectively, raise high quality product, and increase production. Manufacturers have to put more improvement cost to assist their suppliers control and supervise suppliers' products quality or purchase new machines. For original suppliers, we matched up the process capability index Cpm and process improvement capability index CPiM to evaluated, measured the suppliers' process capability and reduced manufactures' improvement cost. Lately, When the process capability Cpm for suppliers not enough, we will think over to improve precise or accuracy of the process capability first. And then we could use process improvement capability index CPIM to measure whether suppliers' process improvement capability superiority or not. Manufacturers by operating
708
these two indices, process capability index Cpm and process improvement capability index CPiM to find out real potential, capable and continued collaboration's suppliers; enhance quality of the process, get more profit with suppliers, and advance the competition for whole supply chain. References 1. 2.
3.
4. 5. 6. 7. 8.
9. 10.
L.K. Chan, S.W. Cheng and F.A. Spiring, A New Measure of Process Capability: Cpm, Journal ofQuality Technology 20, 162-175 (1988). K.S. Chen, M. L. Huang, and R. K. Li, Process capability analysis for an entire product, International Journal of Production Research 39(17), 4077-4087(2001). K.S. Chen, K.L. Chen and R.K. Li, Contract manufacturer selection by using the process incapability index CPP , International Journal of Advance & Manufacturer Technology 26, 686-692 (2005). M. Deleryd, A pragmatic view on process capability studies, International Journal of Production Economics 58, 319-330 (1999). B. Govaerts, Private Communication to Kotz, S (1994). R. E. James, M. L. William, The Management and Control of Quality, Thomson, 1-2 (2002). V.E. Kane, Process capability indices, Journal of Quality Technology, 18, 41-52(1986). W. L. Pearn, S. Kotz, N. L. Johnson, Distributional and inferential properties of process capability indices, Journal of Quality Technology 24, 216-231(1992). W.L. Pearn, K.S. Chen, Process Improvement Capability Analysis Based on Expected Process Loss, JCSA 35(2), 151-160 (1997). M.S. Phadke, Quality Engineering Using Robust Design, AT&T Bell Laboratories, New Jersey, (1989).
JOINT OPTIMIZATION OF PROCESS MEAN AND TOLERANCE LIMITS FOR MULTI-CLASS SCREENING
SUNG HOON HONG Department of Industrial & Information Systems Engineering, Chonbuk National University, Chonju, Chonbuk 561-756, Korea IK JUN CHOI Department of Industrial & Information Systems Engineering, Chonbuk National University, Chonju, Chonbuk 561-756, Korea MIN KOO LEE Department of Information and Statistics, Chungnam National University, 220 Gung-dong, Yuseong-gu, Daejeon 621-759, Korea HYUCK MOO KWON Department of Systems and Engineering, Pukyong National University, Yongdang-dong, San 100, Nam-gu, Pusan 608-709, Korea Most models reported in the literature treat the determination of process mean and tolerance limits as two separate research fields. In this paper, the problem of jointly determining the optimum process mean and tolerance limits for each market is considered in situations where there are several markets with different price/cost structures. A profit model is constructed which involves selling price, production cost, penalty cost, and inspection cost. A Taguchi's quadratic loss function is utilized for developing the economic model for determining the optimum process mean and tolerance limits. A numerical example is given.
1.
Introduction
Consider the problem of selecting the optimum mean value for a continuous production process. All items are inspected to determine whether its quality characteristic satisfies a predetermined lower specification limit. Conforming items are sold at a regular price, whereas all others are reprocessed or sold at a 709
710 discount. Typical quality characteristics under consideration are weights, volume, number and concentration. Items produced by a production process may deviate from the process mean because of variations in materials, labor and operation conditions. The process mean may be adjusted to a higher value in order to reduce the proportion of the nonconforming items. Using a higher process mean, however, may result in a higher production cost. Consequently, the decision of selecting a process mean should be based on the tradeoff among production cost, payoff of conforming items, and the costs incurred due to nonconforming items. This problem has been studied by several researchers. Hunter and Karma solved the problem of selecting the optimum process mean when the nonconforming items are sold at a reduced price [11]. Bisgaard et al. extended Hunter and Kartha's model to a situation where the nonconforming items are sold at a price proportional to the amount of ingredient used [4]. Golhar considered the problem of selecting the optimum process mean in a canning process [8]. Boucher and Jafari, and Al-Sultan discussed situations in which the items are subjected to lot-by-lot acceptance sampling rather than complete inspections [1] [5]. Arcelus and Rahim developed a model for jointly selecting optimum target means for both a variables and an attributes quality characteristics [1], and Chen and Chung considered an economic model for determining the most profitable target value and the optimum measuring precision level for a production process [6]. Tang and Lo [16] and Lee and Jang [13] determined the optimum process mean and the tolerance limits when a surrogate variable is used in inspection. Hong and Elsayed studied the effect of the measurement error on the determination of the optimum process mean for a filling process [9]. In all the previous work, the inspected items are classified into two quality grades only; conforming items are accepted and nonconforming items are rejected. But, it is common practice to grade the outgoing items on the basis of the quality and then sell them in different markets. This practice has been used for chemical materials and primary materials such as lumber, wheat, cotton, and butter (England and Leenders) [7]. Different grades of a product may be sold at different selling prices under different names or marketed in different chain stores or areas under the same brand name (Tang) [15]. Economic inspection procedures under similar situations are considered by several authors; Tang [15], Bai and Hong [3], Kim et al. [12], Lee and Jang [13], and Hong et al. [10]. In this paper, economic models are developed for jointly determining the optimum process mean and the tolerance limits for each market in situations where there are several markets with different price/cost structures. A Taguchi's quadratic
711 loss function is utilized for developing the expected profit function model. The loss caused by imperfect quality may include loss of goodwill, warranty, replacement cost, and handling cost. Classical concept in the field of optimum target value determination assumes that this loss is a constant when an item does not conform to product specifications and is zero otherwise. However, Taguchi [15] argued that this cost concept was incorrect. Instead, he suggested that a quadratic function of the deviation from the product target value could better measure the true loss. This function has received widespread attention and been used by several researchers. This paper is organized as follows: An economic models is formulated and the procedure for obtaining the optimum solution is derived in Section 2. An example is used to illustrate the solution procedure, and sensitivity analyses are performed in Section 3. 2. The Model Let 7 be a performance variable representing the quality characteristic of interest and r denote the target value of 7 . We assume that 7 is a "larger is better" variable and normally distributed with unknown mean value u and known variance <7 . Suppose that a product can be sold to several different markets. When products are sold to market / , the selling price is A,, and the item with y < r causes a loss of C, (y, r) = a,- (y - T) which is a quadratic function. a ( is a positive constant, and y is the observed value of 7 . This function was strongly advocated by Taguchi (1984) and has received widespread attention. If y > x , Ci (y, r ) = 0 . Now consider the case where At>Aj and Ci (y, T) < Cj (y, r) , or A, = A} and C, (y, r ) < C} {y, r) . It is easy to verify that market j is dominated by market i and thus a product should be sold to market / rather than market j . Therefore only the markets which are not dominated need to be considered. Assume that there are m markets which are not dominated. It is not profitable to ship the low quality products to an ordinary market because of the penalty cost C,- (y, r ) . Therefore, market m is considered as an alternative with one of following modes; sell the products at a discount, scrap the products, etc. Without loss of generality, it is assumed that A, > A} and C, (y, r) > Cj (y, r) for all i<j. The condition Cj (y, T) > Cj (y, r) is equivalent to the condition a, > ay.. Since a smaller index market / requires higher quality products than a lager index market j , an appropriate inspection procedure is as follows. 1.
Take measurement y for each incoming item.
712
2.
Let Sj , i = 1,2,..., m , be real numbers such that Sl>S2>...>Sm = -oo and 80 = oo . If St < y < 5t_x , i = 1,2,...,m, ship the item to market / . Note that if Sj = 8-_j, the item will not be shipped to market / .
The item is shipped to market i whenever Sj < y < St_i, i = 1,2,..., m . Therefore, the expected revenue for market i is
A^~lg(y)dy,
(i)
where giy) is the probability density function of Y which is a normal density function with mean jJ. and variance <J . The production cost per item is c0+cty which is proportional to the quantity y . c 0 and C\ are positive constants (Bisgaard et al. (1984)). The expected production cost per item thus becomes Ko(c 0 +cly)g(y)dy = C0 +cxfly . The expected penalty cost for market / caused by imperfect quality is $" Cj(y,T)g(y)dy.
(2)
Therefore, the expected profit per item is given by m
EP=-sy-c0-cxfiy+Y,
*
£'"ki(y)g(y)dy.
(3)
where sy is the inspection cost of Y and kt (y) = At - Ct (y, r) . The optimum values of (/J ,8],S2,---,8m) can be obtained by maximizing equation (3). We first determine the optimum tolerance limits St =8j (jUy) , i = l,2,...,m , for given fj.y and then determine /dy maximizing the expected profit. For a given value of jU , the expected profit is maximized by choosing the values of 8\,S2,---,Sm that maximize the fourth term in equation (3). An upper bound of the fourth term for given jU is J[ {maxkj(y)}g(y)dy. This value is clearly attained by shipping the item to market i whenever y e.lx•, , i = l,2,...,m , where It is the set of real numbers y satisfying the inequalities kj(y)> kj(y) for all j ^ i , simultaneously. Since k\ (y) — k: (y) for j >\ is a nondecreasing function of y , it is clear that 7| is given by the interval / , =[Si ,oo) where Sj , i = 1,2,...,m — 1 , are the smallest real numbers satisfying the inequalities kj (y) > k (y) for all / > /, simultaneously. Similarly, if / , is not empty, it is of the form I2= [S2 ,8^ ) . If I2 is empty, we let S2 = Sx . By the
713 same reasoning, if I{, i = 3,4,..., m — 1, is not empty, it is of the form / — \8t ,8',-_, ) . If Ij is empty, we let 8X = S t_x . Since kt (y) is a function of the parameters ( A { , at, T) , it is clear that the optimum values of (8^ ,S2 ,...,Sm ) also depend on the same parameters and do not depend on the value of fJ. . Inserting the optimum values of (8X ,82 ,...,8m ) into equation (3), we obtain EP=-sy
- c 0 -cxny
+ ^Jmaxki(y)}g(y)dy.
(4)
Setting the partial derivative of equation (4) with respect to fl following equation is obtained ^Jmaxki(y)}(y-My)g(y)dy
- cxa2
= 0.
to zero, the
(5)
It is difficult to find closed form expressions for the solution of equation (5). Numerical studies over a wide range of parameter values of (r, At, al, cr Cj) indicate that equation (5) has unique solution and it represents a maximum point. Search algorithms such as Newton-Rapson or bisection method can be used for finding the value of jUy . 3. A Numerical Example Consider a packing plant of cement industry. The plant consists of two processes; a filling process and an inspection process. Each cement bag processed by the filling machine is moved to the loading and dispatching phases on a conveyor belt. Inspection is performed by continuous weighting feeders (CWFs). A CWF measures the mA (milli ampere) X of the load cell of the cement bag, which is positively correlated with the weight Y of the cement bag. From theoretical considerations and past experience, it is known that the variance of Y, ay =(1.25%) , and that X for given Y = y is normally distributed with mean 4.0 + O.O&y and variance (0.05mA)2 . That is, X and Y are jointly normally distributed with, unknown means (JUX,JU ) , known variances ax =(0.112m/i) , ay =(l.25kg) , and correlation coefficient p = 0.894. The weight marked on each bag is 40kg, and it is the target value r. The cement bag can be sold to foreign, domestic, or discount markets. The low quality product is scraped because of the penalty cost. The selling price in the foreign market is higher than that of the domestic market. The cost caused by imperfect quality in the foreign market is also higher than that of the
714 domestic market, because of differences in the costs of identifying and handling a nonconforming item, labor cost, transportation cost, etc. We will consider the foreign market as market 1, the domestic market as market 2, the discount as market 3, and the scrap as market 4. The selling prices and the estimated penalty cost coefficients in dollars are as follows: Foreign market Domestic
market
Discount
Scrap Price (A,) Penalty cost coefficient ( a . )
40 10.5
39 6.5
24 0.75
0 0
The production cost in dollars is 6.0 + 0.6y which is proportional to the quantity y , and the inspection costs are sx =$0.2 and sy =$1.3 . Using these values, we obtain (/j.*,£,*,S2*,S$) = (41.74, 39.50, 38.38, 34.34). Hence the cement bags are sold to the foreign market if y > 39.50, to the domestic market if 38.38 < y < 39.50 , or to the discount market if 34.34 < y < 38.38 . If >- < 3 4 . 3 4 , they are scraped. In this case the expected profit per item is $7,333. The optimum values of /J and their expected profits are given in Table 1 for selected values of <Jy for 0.50 (0.25) 2.50. It shows that /u can be set significantly lower as ay decreases. These results agree with our intuition that // can be set r if
s;
s2*
V
Expected
°y 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50
40.56 40.95 41.35 41.74 42.13 42.51 42.88 43.24 43.60
39.5 39.5 39.5 39.5 39.5 39.5 39.5 39.5 39.5
38.38 38.38 38.38 38.38 38.38 38.38 38.38 38.38 38.38
34.34 34.34 34.34 34.34 34.34 34.34 34.34 34.34 34.34
8.229 7.937 7.636 7.333 7.034 6.738 6.447 6.158 5.873
Profit
4. Conclusions We have considered the problem of jointly determining the optimum process mean and tolerance limits for each market in situations where there are several markets with different price/cost structures. A Taguchi's quadratic loss function is utilized for developing the profit model. It is difficult to find closed form
715 expressions for the optimum process mean as well as to show analytically that the solution is optimum. Numerical analyses over a wide range of parameter values, however, indicate that the expected profit functions are indeed unimodal for the process mean. A numerical search such as Muller's method is used to find the optimum process mean. Extensive sensitivity analyses show that the optimum process mean and tolerance limits are very insensitive to the changes of cost parameters. Numerical results also show that the optimum process mean tends to increase and the expected profit tends to decrease as the process variation <J increases. Numerical studies are performed by using FORTRAN and IMSL (International Mathematical and Statistical Libraries) subroutines on a Pentium II PC. In most cases the results can be obtained within a few minutes. A possible area of further investigation would be the extension of the model to the cases where the parameters (cr , c , A , , A 2 ) are unknown. References 1.
2.
3. 4. 5.
6.
7. 8. 9.
K. S. Al-Sultan, An Algorithm for the Determination of the Optimal Target Values for Two Machines in Series with Quality Sampling Plans, International Journal of Production Research 12, (1994). F. J. Arcelus and M. A. Rahim, Simultaneous Economic Selection of a Variables and an Attribute Target Mean, Journal of Quality Technology 26, 125-133 (1994). D. S. Bai, and S. H. Hong, Economic Design of Sampling Plans with MultiDecision Alternatives, Naval Research Logistics 37, 905-918 (1990). S. Bisgaard, W. G. Hunter and L. Pallesen, Economic Selection of Quality of Manufactured Product, Technometrics 26, 9-18 (1984). T. O. Boucher and M. A. Jafari, The Optimum Target Value for Single Filling Operations with Quality Sampling Plans, Journal of Quality Technology 23, 44-47 (1991). S. L. Chen and K. J. Chung, Selection of the Optimal Precision Level and Target Value for a Production Process: the Lower-Specification-Limit Case, HE Transactions 28, 979-985 (1996). W. B. England and M. R. Leenders, Purchasing and Materials Management (5th Edition, Homewood, IL: Richard D. Irwin), (1975). D. Y., Golhar, Determination of the Best Mean Contents for a Canning Problem, Journal of Quality Technology 19, 82-84 (1987). S. H. Hong and E. A. Elsayed, Setting Optimum Mean for Processes with Normally Distributed Measurement Error, Journal of Quality Technology 31,338-344(1999).
716 10. S. H. Hong, E. A. Elsayed and M. K. Lee, Optimum Mean Value and Screening Limits for Production Processes with Multi-Class Screening, International Journal of Production Research 37, 155-163 (1999). 11. W. G. Hunter and C. P. Kartha, Determining the Most Profitable Target Value for a Production Process, Journal of Quality Technology 9, 176-181 (1977). 12. C. T. Kim, K. Tang and M. Peters, Design of a Two-Stage Procedure for Three-Class Screening, European Journal of Operational Research 79, 431-442(1994). 13. M. K. Lee and J. S. Jang, The Optimum Target Values for a Production Process with Three-class Screening, International Journal of Production Economics 49, 91-99 (1997). 14. G. Taguchi, Quality Evaluation for Quality Assurance, Romulus, MI: American Supplier Institute, (1984). 15. K. Tang, Design of Product Grading Procedures, Decision Sciences 21, 434-445 (1990). 16. K. Tang and J. Lo, Determination of the Process Mean When Inspection is Based on a Correlated Variable, HE Transactions 25, 66-72 (1993).
SEPARATE RESPONSE SURFACE MODELING FOR MULTIPLE RESPONSE OPTIMIZATION YOUNG JIN KIM Department of Systems Management and Engineering, Pukyong National Busan 608-739, Republic of Korea
University
In spite of the growing importance of multiple response optimization, there have been few research efforts in this area. This article proposes a versatile optimization model by employing the concept of quality loss function and response surface modeling. Taguchi (1986) proposes the use of quality loss function to measure a societal cost incurred by the customer on a monetary scale. Mostly applied to a single quality characteristic problem, however, the quality loss function may also be extended to multiple response systems by combining performances of individual quality characteristics into a single objective function. The overall performance of a multiple response problem is then evaluated by obtaining mean and variance responses for each quality characteristic, and covariance responses among quality characteristics.
1. Introduction Most manufacturing industries have been faced with the problem involving simultaneous optimization of several quality characteristics that could be considered as the basis for the product selection by customers. One of the major concerns in manufacturing design is to find settings of design parameters that result in a satisfactory combination of quality characteristics. This generally requires a trade-off or balancing between quality characteristics. Response surface methodology (RSM) has widely been used in design optimization problems. Especially, Myers and Carter (1973) proposed a dual response surface technique, which models process mean and variance as separate surfaces, to achieve a robust design for single quality characteristic. Several approaches using RSM have been proposed to optimize multiple response systems and include desirability function (Derringer and Suich 1980, 1994), utility function (Myers, Khuri, and Carter 1989), and quality loss function (Pignatiello 1993, Ames et al. 1997, and Vining 1998). Taguchi (1986) perceived that product quality is closely related to the manufacturing imperfection and quality loss is always incurred to the society if the quality characteristic(s) of a product deviates from the nominal target value(s). Quality loss function (QLF) is widely accepted since it provides a 717
718 monetary indication of product quality by relating the deviation of a product characteristic from its target value to the monetary loss. As discussed in Pignatiello (1993) and Ames et al. (1997), QLF can be extended to multiple response systems by combining performances of multiple quality characteristics into a single objective function. We will use the concept of QLF for multiple response optimization. Pignatiello (1993) proposed two different strategies for multiple response optimization: a direct strategy and a partitioned strategy. A direct strategy models and estimates QLF itself as a response of interest. In a partitioned strategy, the sum of variance and sum of squared bias in QLF are modeled separately. He proposed a two-step procedure in which the variance portion is first minimized and the squared bias term is then minimized with given minimum variance. He did not provide the reasoning for the preemptive optimization of the variance portion. Moreover, rather unrealistic assumption has been made by strictly partitioning design variables into three nonoverlapping components: design variables that affect process dispersion, that affect process mean, and that affect neither the mean nor the dispersion. Ames et al. (1997) proposed a multiple response optimization scheme that minimizes weighted sum of individual QLF's. Vining (1998) pointed out that their model ignores the correlation structure among the responses and proposed a squared error loss approach based on the distance function of Khuri and Conlon (1981). Even if addressed the issue of correlation among responses, this approach may have difficulty in separately analyzing process mean and variance-covariance structure since they are embodied in estimated functions of individual responses. On the other hand, individual responses can not be analyzed separately within the framework proposed by Pignatiello (1993) since all the responses are mixed up in the estimated functions. This problem may be overcome by employing a dual response surface approach since the process mean and variance for individual responses are modeled and estimated separately. This paper provides a multiple response optimization scheme by employing the concept of QLF and extends a dual response surface approach to multiple response systems. The proposed model is aimed at minimizing QLF for multiple responses. To do this, we need to know estimated functions for process means, variances, and covariances. Within the proposed framework, the mean and variance for individual responses are modeled separately through a dual response surface approach. Furthermore, covariances between individual
719 responses are also estimated using the data obtained through design of experiment. 2. Quality Loss Function for Multiple Responses Most manufacturing product has more than one quality characteristics and its quality is typically evaluated by the customer on the basis of multiple quality characteristics. Let rxl vector of quality characteristics denoted by y = [yi>y2>'">yr]' • Assume that y follows an /"-dimensional multivariate normal distribution with mean vector M = [juuju2,---,^r]' and variancecovariance matrix S and all yi,s(i = \,2,---,r) are functions of the same set of p input variables x = {xx,x2,---,xp) . Furthermore, the target values for individual quality characteristics are known and denoted by a vector T = [ r 1 , r 2 , - - , r r ] ' . With this, a quadratic loss function for multiple responses can be defined by (Pignatiello 1993) Z(y(x)) = ( y ( x ) - T ) ' K ( y ( x ) - T ) .
(1)
where AT is an r x r vector of positive loss coefficients and represent the losses incurred when y deviates from T. Note that if K is a diagonal matrix, it implies that there are no interactions between any responses and the quality loss function becomes just a sum of r single response quality loss functions, i.e.,
i(y(x)) = X*/0'l-(x)-r/)2. (=i
th
where k, is the z' diagonal element of K. And non-diagonal elements of K are related to the incremental losses only when pairs of quality characteristics are simultaneously off the target. It can easily be shown that the expected quality loss is given by E[L(y (x))] = (M - T)'K(M - T) + trace(KS),
(2)
or equivalently (Kapur and Cho 1996),
£Wy«)] = X 4 t t -r,)2 + ^ ] + Z I X h + ( ^ -^(Mj -TJ)], i=\
(3)
i=2 j=l
where ktj and atj are the (i,j) element of A" and S, respectively, and cr,2 is the t diagonal element of Z. If r = 2 , the expected quality loss is given by
720
E[L(y(x))] = kx[(Ml -T02+V?\+
k2\jt2
-r2)2+a22\ (4)
QLF is widely accepted since it provides a monetary indication of product quality by considering the deviation of a quality characteristic from its target value. For multiple response optimization, minimizing quality loss is a simple and persuasive approach to finding the best settings of the design variables. Another advantage of QLF approach is that it does not require much information on preferences while those kinds of information are indispensable in constructing desirability functions and utility functions. One of the basic features of the desirability and utility functions is the subjectivity on the part of the user in evaluating. Moreover, improperly assessed desirability functions or utility functions can lead to inaccurate results. 3. Model Development When the true response function is unknown, it could be approximated over a limited experimental region by a polynomial representation through response surface methodology. Consider the situation in which multiple responses of quality characteristics y = [yu v2,---,yr]' depend on the same set of variables x = (x],x2,---,xp). It has been very common to estimate each response as a quadratic response of the form. p
p
p
Aw^o+X/^+ZZ/v^ • (=1
(5)
1=1 _/=/
Kim and Lin (1998) pointed out that such a model works well when the variance of the response is relatively small and stable, but when the variance is not a constant, classical response surface methodology could be misleading. For the robust design of single quality characteristic, Vining and Myers (1990) proposed a dual response approach, which models process mean and variance as separate responses, to tackle such a problem. The concept of a dual response approach can be extended to the multiple quality characteristics. It is obvious that the biases, variances, and covariances of quality characteristics should be reduced to decrease quality loss. To estimate process means and variances for individual responses, a typical dual response approach can be appropriately employed. Let yukj- be the fh experimental value of kth quality characteristic at the wth design point. The sample mean and variance for each
721 quality characteristic can be used to get the estimated functions for process mean and variance. The sample mean and variance for tih quality characteristic at the M"1 design point are given by i J".
yUk=—Y.
m
i and
4 = — T X (y>*i - *«* >2 •
yUkj
7=1
(6)
7=1
where m is the number of replications at each design point. Equation (5) can be used to get the estimated functions for process mean and variance. Let jik (x) and ak (x) represent the fitted functions for the mean and the standard deviation of the kth quality characteristic, respectively. Assuming a second-order polynomial model for the response functions, the fitted functions on the basis of the sample mean and variance yield to p
p
X
p
h (*) = Po + X P' > Z Z PiJXiXJ P
°k W = n+X
+
P
r x
P
+
< < H S YVX'XJ •
(7)
Similarly, we can get the estimated functions of mean and standard deviation for every response. Thus, the biases and variances can be taken care of by employing a typical dual response surface approach. Now, we only need to know the estimated functions for covariances as well as means and variances to develop our QLF minimization model. To do this the concept of a dual response surface approach will also be extended to covariances just like mean and variance. It is well known that a sample covariance is an unbiased estimator of a covariance. Thus a sample covariance can be used to find estimated functions of covariance. The sample covariance between kth and f" quality characteristics at the wth design point can be expressed as 1 m uki = —7y(y U kj~y U k)(y u ij-y u i),
s
for k*i.
(8)
7=1
Assuming a second-order polynomial model for the response function again, an estimated function for covariance between A* and f" quality characteristics
722 P
P
a x
<*«(*) = «0 + X ' i 1=1
+
P
S 2 ,=i
a
Vx'xJ
(9)
j=t
Similarly, estimated functions for covariance between any pair of responses can be obtained. Introducing estimated functions of means, variances, and covariances in the expected quality loss, the optimization model can simply be described as E[L(y(x))] = ] T *, [(A to - r, ) 2 + af (x)] Minimize
'"
+
,
x +
ZZ^h( ) (AW-^)(^W-r7)] 1=2 7=1
where x e fi and Q represent the feasible region. Major advantages of the proposed model over the previous ones come from its modeling versatility. The model by Pignatiello (1993) uses an estimated function for expected quality loss itself. So, individual responses cannot be analyzed within the framework and one can get only an overall picture of the multiple response systems. Later, Vining (1998) modeled individual responses separately and discussed the variance-covariance structure of multiple response systems. However, the results may turn out to be misleading if the variances or covariances are not stable. In this regard, separate modeling of process mean, variance, and covariance seems to be attractive and makes it possible examining multiple response systems more rigorously and extensively. Furthermore, individual responses can be analyzed separately with the proposed model. For example, suppose keeping the mean of the first quality characteristic on its target should be pre-emphasized over the other criteria. Then, we just need to add the constraint Mi(x) = Ti t 0 t n e proposed model. However, other approaches cannot be applied to this kind of situations. Separate modeling process means, variances, and covariances may provide us with a great deal of opportunities to explore individual responses in multiple response systems, so the proposed model can be considered as a versatile approach compared with existing models. 4. Conclusions We have proposed a multiple response optimization model which minimizes QLF by extending the concept of a dual response surface approach to the multiple response systems. Individual process means, variances, and
723
covariances are modeled separately and estimated as in a dual response surface approach. Introducing estimated functions into the QLF, the optimization model is to minimize QLF. Separate modeling for each process parameter provides us with modeling versatility. As a concluding remark, we would like to point out that the weights representing relative importance of individual responses could be introduced to the proposed model. Let W be the importance weight matrix of individual responses. Then, the QLF can be written as
£(y (*)) = ( y « - T) W (
y
(x) - T).
Taking expectations on both sides yield to £[£(yO))] = (M - T ) ' K ^ ( M - T) + trace(KJfE). Introducing estimated functions, one can get the optimization model which considers relative importance of responses. How to get the weight matrix W is beyond the scope of this paper and needs to be examined further. References 1.
A.E. Ames, N. Mattucci, S. MacDonald, G. Szonyi and D.M. Hawkins, Journal of Quality Technology, 29, 339 (1997). 2. N. Artiles-Leon, Quality Engineering, 9, 213 (1996). 3. R.E. Chapman, Quality Engineering, 8, 31 (1995). 4. E. Del Castillo and D.C. Montgomery, Journal of Quality Technology, 25, 199(1993). 5. G. Derringer and R. Suich, Journal of Quality Technology, 12, 214 (1980). 6. K.C. Kapur and B.R. Cho, HE Transactions, 28, 237 (1996). 7. A.I. Khuri and M. Conlon, Technometrics, 23, 363 (1981). 8. K.J. Kim and D.K.J. Lin, Journal of Quality Technology, 30, 1 (1998). 9. D.K.J. Lin and W. Tu, Journal of Quality Technology, 27, 34 (1995). 10. J.J. Pignatiello, HE Transactions, 23, 5 (1993). 11. G.G. Vining, Journal of Quality Technology, 30, 309 (1998). 12. G.G. Vining and R.H. Myers, Journal of Quality Technology, 22, 38 (1990).
A GLOBAL CRITERION APPROACH TO MULTIPLE RESPONSE OPTIMIZATION YOUNG JIN KIM Department of Systems Management and Engineering, Pukyong National Busan 608-739, Republic of Korea
University
BYUNG RAE CHO Department of Industrial Engineering, Clemson Clemson, SC 29634, United States
University
Most manufacturing industries have been faced with the problem involving simultaneous optimization of several quality characteristics that may be considered the basis for the product selection by customers. Compared with the case of single quality characteristic, however, the design optimization of multiple quality characteristics has received little attention. This paper employs the concepts of the dual response surface technique and the MSE criterion to the optimization of multiple quality characteristics. The proposed model is aimed at simultaneously minimizing the MSE of individual quality characteristics. However, the optimal solution to one quality characteristic may result in poor performances in other quality characteristics. Thus, a tradeoff among quality characteristics is required and the design optimization of multiple quality characteristics may be viewed as a multiple objective programming problem. A global criterion approach (Lai and Hwang 1994) is employed to set up an optimization model for multiple quality characteristics.
1. Introduction Most manufacturing industries have been faced with the problem involving simultaneous optimization of several quality characteristics that could be considered as the basis for the product selection by customers. The design optimization with multiple quality characteristics is to find settings of design variables that result in a satisfactory combination of quality characteristics. This generally requires a tradeoff or balancing among quality characteristics. The response surface methodology (RSM), which is designed to find optimal settings of design variables to optimize the response (or a set of responses), has widely been used in design optimization problems. Especially, there have been many progresses in design optimization associated with single quality characteristic. Recently, a dual response surface technique, developed by Myers and Carter (1973), has received a lot of attention. Unlike the classical 724
725
RSM, the dual response surface approach models the process mean and variance as separate responses. Vining and Myers (1990) first applied the dual response approach to achieve robust design, and minimized variance subject to the constraint that process mean equals to the target value. Later, Lin and Tu (1995) pointed out that their approach may rule out better conditions by forcing process mean to a fixed value, and proposed a procedure based on the minimization of mean squared error (MSE). Considering that the quality of a product is closely related to both the bias and variance of the process (Taguchi 1986), the MSE approach is quite appealing since it incorporates the variance as well as bias. Besides, there have been several efforts to further develop the dual response surface approach including Castillo and Montgomery (1993), Copeland and Nelson (1996), Kim and Lin (1998), and Kim and Cho (1998). Compared with the case of single quality characteristic, the design optimization of multiple quality characteristics has received little attention. Furthermore, there are so many opportunities to extend the concept developed in the design optimization of single quality characteristic, such as a dual response surface approach, to the optimization of multiple quality characteristics. This paper extends the concepts of the dual response surface technique and the MSE criterion to the optimization of multiple quality characteristics. The proposed model is aimed at simultaneously minimizing the MSE of individual quality characteristics. However, the optimal solution to one quality characteristic may result in poor performances in other quality characteristics. Thus, a tradeoff or balancing among quality characteristics is required and the design optimization of multiple quality characteristics can be viewed as a multiple objective programming problem. A global criterion approach (Lai and Hwang 1994) is employed to set up an optimization model for multiple quality characteristics. The proposed model first identifies the individual optimal solutions, which minimize the MSE of each quality characteristic, and then provides the compromise solution that balances the generalized distances (Khuri and Cornell 1987) from the individual optimal solutions. The MSE's are incommensurable due to the differences in relative magnitudes of individual quality characteristics. A generalized distance removes the incommensurability. 2. A Dual Response Surface Approach and MSE Criterion The RSM is designed to find the optimal settings for a set of design variables that optimize the response. Typically, such a problem is focused on the mean value of the response, so the classical RSM works well when the variance of the response is stable. As pointed out in Lin and Tu (1995) and Kim and Lin (1998),
726
the classical RSM does not work well if the variance is not stable. Vining and Myers (1990) first made use of the dual response surface approach, developed by Myers and Carter (1973), to resolve this problem. The dual response surface approach models the process mean and standard deviation as separate responses as follows: p
p
p
i=\
i=\
j=i
p
p
p
y x +
<7« = n+Yu ' i HH 1=1
Y
'Jx'xj
(l
)
1=1 j=i
where //(x) and o-(x) represent the fitted functions for the mean and the standard deviation, respectively. Vining and Myers (1990) proposed the following model to achieve the robust design of nominal-the-best quality characteristic. Minimize
(2)
subject to x e Q . Considering that the quality of a product is closely related to the variance as well as the bias of the process (Taguchi 1986), the MSE approach is quite appealing since it incorporates both of them. Furthermore, the MSE criterion can easily be applied to the optimization of multiple quality characteristics as shown in the next section.
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3. Model Development 3.1. Multiple Objective Programming Consider the situation in which the quality of a product is determined by n quality characteristics, Yl,Y2,---,Yn , that depend on the same set of input variables x = {xx,x2,---,xp). To estimate process mean and variance for each quality characteristic, a typical dual response surface approach can be employed. Let yukJ be the jth experimental value of kth quality characteristic at the uth design point. The sample mean and variance for each quality characteristic can be used to get the estimated functions for process mean and variance. The sample mean and variance for kth quality characteristic at the uth design point are given by i Jn
I
yUk =—Y,yUkj mr-i
_m
and s
lk = — { Y i y u k j -y»k)2 • OT-1 —
where m is the number of replications at each design point. Letting juk (x) and ak (x) denote the estimated functions of mean and standard deviation of kth quality characteristic, respectively, juk (x) and ak (x) can be found as in equation (1). Thus, one can get the estimated functions of mean and standard deviation and the optimization model based on MSE criterion can be constructed as in equation (2) for each quality characteristic. However, the optimal solution for one quality characteristic may not be the optimal to the other quality characteristics. Thus, the optimization of multiple quality characteristics can be written as a multiple objective programming problem Minimize MSE, (x) = (/}, (x) - r, )2 + a? (x)
Minimize MSE„ (x) = (//„(x) - r„ )2 + a2n (x) subject to x e Q , where MSE,(x) and r,- represent the MSE and target value of the quality characteristic Y{, respectively. The simplest way to solve this problem is to minimize the sum of the objective functions. However, keeping that the relative magnitudes of MSE's are different from each other in mind, this naive approach may yield to the solution close to the point which minimizes the MSE of the quality
728
characteristic with largest magnitude. For example, consider a situation where two quality characteristics jointly determine the quality of a product. Suppose the MSE of one quality characteristic ranges from 1 to 2 and that of the other from 100 to 200 within the feasible region. If the optimal solution for each quality characteristic is different from the other, the optimal solution that minimizes the sum of MSE's may be close to the point which minimizes the MSE of the quality characteristic with larger magnitude. Consequently, minimizing the sum of MSE's is not convincing due to the incommensurability among MSE's. 3.2. Proposed Optimization Model In the context of multiple objective programming, a global criterion approach provides a compromise solution, which balances the performances of individual objective functions, while removing the incommensurability among objective functions. We shall apply the global criterion approach to the optimization of multiple quality characteristics. One can get the optimal solution for each quality characteristic by solving optimization model as in equation (2). Let MSE* and MSE/ denote the MSE's of the quality characteristic Yt and K(for j*i) at the point that minimizes MSE,(x), respectively, then a payoff matrix can be developed as in Table 1. Further, define MSE," as the maximum of MSE/ for all j&i . The generalized distance of MSE,-(x), denoted by of, (x), can be written as |MSE,W-MSE-| [MSET-MSE*] As seen in equation (3), the generalized distance function removes the differences in relative magnitudes of individual objective functions by properly normalizing. A design optimization of multiple quality characteristics requires a simultaneous optimization. If a minimum operator is employed for incorporating multiple objectives, a design optimization problem can be written as Minimize 9 subjectto
dt{\)<6,
i = 1,2,---, n , and x e Q .
The above formulation, which is actually a min-max problem, balances the generalized distances of individual MSE's. The proposed model not only
729 balances the performance of each quality characteristic, but also removes the differences in relative magnitude of individual MSE's. Table 1. A payoff matrix.
Setting
MSE,(x)
MSE 2 (x)
MSE,(x)
x (,)
MSE*
MSE,2
MSE;
MSE"
x (2)
MSEJ
MSE j
MSE^
MSEj
x (;)
MSE)
MSE 2
MSE*
MSE,"
x(">
MSE),
MSE2,
MSE'„
MSE*„
..
MSE„(x)
X '' indicates the point at which MSE,(x) is minimized.
4. Conclusions This paper proposes an optimization model for multiple quality characteristics by employing the concept of a dual response surface approach and the MSE criterion. The process mean and variance for each quality characteristic are modeled separately. It is shown that the design optimization of multiple quality characteristics can be modeled as a multiple objective programming problem, in which individual objective functions are to minimize the MSE's of individual quality characteristics. Individual objectives are normalized by defining generalized distance. Thus, the proposed model balances the generalized distances of quality characteristics based on a max-min formulation. As a concluding remark, we would like to point out that the weights representing relative importance of each quality characteristic could be introduced to the proposed model. Let wi be the pre-specified constant which represents the relative importance of the quality characteristic Yt. Then, the optimization model can be written as
730
Minimize 9 subject to Wjdj (x)<0, y
;i
i = 1,2, • • •, n
w, =1 and x e Q .
References 1. A.E. Ames, N. Mattucci, S. MacDonald, G. Szonyi and D.M. Hawkins, Journal of Quality Technology, 29, 339 (1997). 2. N. Artiles-Leon, Quality Engineering, 9, 213 (1996). 3. R.E. Chapman, Quality Engineering, 8, 31 (1995). 4. E. Del Castillo and D.C. Montgomery, Journal of Quality Technology, 25, 199(1993). 5. G. Derringer and R. Suich, Journal of Quality Technology, 12, 214 (1980). 6. A.I. Khuri, and M. Conlon, (1981). Technometrics, 23, 363 - 375. 7. A.I. Khuri and J.A. Cornell, Response Surfaces: Designs and Analyses. Marcel Dekker, Inc., New York, NY, (1987). 8. K.J. Kim and D.KJ. Lin, Journal of Quality Technology, 30, 1 (1998). 9. Y.J. Lai and C.L. Hwang, Fuzzy Multiple Objective Decision Making: Methods and Applications. Springer-Verlag, New York, NY, (1994). 10. D.KJ. Lin and W. Tu, Journal of Quality Technology, 27, 34 (1995). 11. J.J. Pignatiello, HE Transactions, 23, 5 (1993). 12. G.G. Vining, Journal of Quality Technology, 30, 309 (1998). 13. G.G. Vining and R.H. Myers, Journal of Quality Technology, 22, 38(1990).
DETERMINATION OF MEAN VALUE FOR A PRODUCTION PROCESS WITH MULTIPLE PRODUCTS MIN K 0 0 LEE Department of Information and Statistics, Chungnam National University, 220 Gung-dong, Yuseong-gu, Daejeon 621-759, Korea SUNG HOON HONG Department of Industrial & Information Systems Engineering, Chonbuk National University, Chonju, Chonbuk 561-756, Korea HYUCK MOO KWON Department of Systems and Engineering, Pukyong National University, Yongdang-dong, San 100, Nam-gu, Pusan 608-709, Korea We consider the problem of determining the optimum target value of the process mean for a production process where multiple products are processed. Every outgoing item is inspected, and each item failing to meet the specification limits is scrapped. Assuming that the quality characteristics of the products are normally distributed with known variances and a common process mean, the common process mean is obtained by maximizing the expected profit which includes selling prices, costs of production and inspection, and losses due to the scraps. A method of finding the optimum common process mean is presented and an illustrative example from electronic device production process is given.
1. Introduction The problem of selecting the most profitable mean value is considered for a continuous process. All products are inspected to determine whether their quality characteristic satisfies predetermined lower limit. Conforming products are sold at a regular price, whereas all others are reprocessed or sold at a discounted price. Typical quality characteristics applicable for such strategy are weights, volume, number and concentration. Products produced by a production process may deviate from the process mean because of variations in materials, labor and operational conditions. The process mean may be adjusted to a higher value in order to reduce the proportion of the nonconforming items. Using a higher process mean, however, may result in a higher production cost. Consequently, the decision of selecting a process mean should be based on the 731
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tradeoff among production cost, payoff of conforming items, and the costs incurred due to nonconforming items. Techniques to determine the optimum process mean have been discussed and developed for more than 40 years. Bettes (1962) solved the problem of choosing the optimum values for the process mean and upper specification limit. Hunter and Kartha (1977) considered the problem of selecting the optimum process mean when underfilled items are sold at a reduced price. Bisgaard et al. (1984) extended Hunter and Kartha's model to a situation where underfilled items are sold at a price proportional to the amount of ingredient used. Golhar (1987) considered the problem of selecting the optimum process mean in a canning process; cans filled above a lower specification limit are sold at a fixed price and the underfilled cans are emptied and refilled at a reprocessing cost. Boucher and Jafari (1991), and Al-Sultan (1994) discussed situations in which the items are subjected to lot-by-lot acceptance sampling rather than complete inspections. Tang and Lo (1993), Lee and Jang (1997), and Lee et al. (2001) determined the optimum process mean and the screening limits when a surrogate variable is used in inspection. Elsayed and Chen (1993) determined optimum levels of process parameters for products with multiple characteristics, and Arcelus and Rahim (1994) developed a model for simultaneously selecting optimum target means for both variable and attribute quality characteristics. Chen and Chung (1996) considered an economic model for determining the most profitable target value and the optimum measuring precision level for a production process. Hong and Elsayed (1999) studied the effects of measurement errors on process target, Pfeifer (1999) showed the use of an electronic spreadsheet program as a solution method, and Hong et al. (1999) presented the problem of jointly determining the optimum process mean and screening limits in situations where there are several markets with different price/cost structures. Most recently, Rahim and Shaibu (2000) and Lee et al. (2004) applied the Taguchi loss function to determine the optimum process target and variance. Kim et al. (2000) proposed a model for determining the optimal process target with the consideration of variance reduction and process capability. Teeravaraprug and Cho (2002) designed the optimum process target levels for multiple quality characteristics, and Rahim et al. (2002) considered the problem of selecting the most economical target mean and variance for a continuous production process. Finally, Duffuaa and Siddiqui (2003) considered process targeting with multi-class screening and measurement error.
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Although the quality engineering literature related to this issue contains a vast collection of work, some questions still remain unanswered. In all the previous studies, they considered the case where only one product type is produced. In some situations, however, multiple products may be produced through a common production process. For example, many types of electronic devices are processed in some metal plating process. The plating thickness of electronic device depends on the volume of the electronic device. In this situation, the previous studies cannot be applied directly. Assuming that the quality characteristics of multiple products are normally distributed with known variances and a common process mean, the optimum target value of the common process mean is found by maximizing the expected profit function which involves selling price, production, inspection, scrap costs. The proposed model is demonstrated with an illustrative numerical example and sensitivity analyses are performed. 2. The Model Consider a production process where multiple products are produced continuously through a common process. Suppose that the quality characteristics (X{, X2,..., Xn ) of products are independently and normally distributed with a common process mean JU and known variances (cr, , <J2,..., an ) . Every item is inspected prior to the shipment to determine whether they are conforming to the lower and upper specification limits L- and £/, or not. If L-, < X-, < Uj, then the item is sold at a fixed selling price per item of the i
product Ai for / = 1, 2, • • •, n . Otherwise, the
item is scrapped at a scrap cost S . The production cost per item is assumed to be equal for different product types, i.e., B + Cxt, where B and C represent the fixed per item and the unit cost of a quantity of material. Let Tt be the indicator variable, which takes 1 when an i
product type is produced and 0
otherwise. It should be noted that £[7^] = a. for i = \,2,---,n th
denotes the production proportion of the / The profit function P(x{ ,x2i...,x„;//)
, where at n
product type and thus V
= P can then be written as
a
=\.
734
\A, -B-Cxt -C,), (-B-Cxi -C,-S), P= (-B - Ox, -Cj- S),
forT, = 1,Lj < x ,
0,
(1)
for ^ = 0,
where C and C ; arethe unit cost of a quantity of material and the inspection cost per item, respectively. Taking the expectatioin on both sides of equation (1) yields
£(/>) = £
a,. JM-B-Cxi-CMixJdXi-a,
-a, ^(B +
j(B + Cxi +
CI+S)fi(xi)dxi
Cxi+C1+S)fi(xi)dxi
(2)
Ui
where /.[•) represents the probability density function of Xt . After some algebra, equation (2) can be rewritten as
E{P) = Yjai{Ai+S)U Ui-M
-
— B-C/u-C,-S,
(3)
1=1
where ^(-) and O(-) are the standard normal density function and the standard normal distribution function, respectively. The optimum common process mean jU can then be obtained by maximizing E(P) , of which solution procedures are discussed in the next section. 3. The Optimum Solution The optimum solution to equation (3) can be obtained by taking the first and second derivatives with respect to (J., which are given by dE(P) _
and
^ajjAi+S)
t
-
\Li-M)
I
°i
•c,
J
(4)
735
dzE{P) 2
dM
=-z
a,(Ai+S)
r
(U,-M)*
ut-^
i=\
(5) respectively. It is reasonable to set the process mean greater than the lower specification limits and smaller than the upper specification limits, i.e., JU > L, and // < £/, for all / , for which equation (5) is negative. Therefore, the expected profit function is concave when jU > Li and fi < Ut for all i, and thus the global maximum is guaranteed at the value of // satisfying dE(P) I d/U = 0 . Consequently, the optimum common process mean JU is the value of jJ, such that
aM+S)
I ai J
(=! jU>Lj
-
and // < t/(. for all /
•-c,
(6)
Computation search algorithms, such as bisection search and golden section search, may be employed to obtain the optimum process mean JU from equation (6). In most cases, the optimum solution has been obtained within a few seconds using a simple computer program using FORTRAN and IMSL (International Mathematical and Statistical Libraries) subroutines on a 586PC. An Illustrative Example: Consider a production process where three types of electronic devices are passed through a common copper plating process. The plating thickness of electronic devices depends on the volume of the electronic devices. From the past data, it is known that the plating thicknesses {Xs ,X2,Xi)of three electronic devices are normally distributed with known variances ( <x,2 =(\.\/jm)2,a\ ={\.2fjm)2,al =(l.2jum)2 ) and a common process mean JU . The production proportions of three electronic devices are GC] — 0.4 a n d a 2 = a3 = 0.3 . Suppose that the cost components and lower specification limits of (X], X2, X3) are ^ , = $30.5, ^ 2 = $32.5, ^ 3 = $ 3 4 . 5 , = $6.0 , C = $1.0 , C 7 =$0.8 , 5 1 = $2.5 , L, = 13.25//m , L2=\2>.5fjm 3= 14.0/urn , £/,= 18.O/0w , L 2 =18.25//m and Z, 3 =18.75/^K The optimum common process mean jU is obtained from equation (6) employing the bisection search algorithm, which yields //* = 15.759//W with the expected profit E(P) = $6.8337 .
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The optimum common process mean fj,
for the above example are given
in Table 1 for selected values of the production proportion ax, CC2, and a3 ranging from 0.1 to 0.9. The computational results agree with our intuition that * the optimum common process mean jU decreases as CCl and CC2 increase, whereas the process mean is to be set at a larger value as GC3 increases.
Table 1. fX for selected values of Of], a2
a2
a,
«3
(a, =a 3 )
a
(«2 = 3 )
and a3 .
*
(a, =a2)
0.1
15.902
0.1
15.829
0.1
15.634
0.2
15.857
0.2
15.813
0.2
15.703
0.3
15.808
0.3
15.797
0.3
15.770
0.4
15.759
0.4
15.782
0.4
15.835
0.5
15.707
0.5
15.766
0.5
15.898
0.6
15.652
0.6
15.751
0.6
15.958
0.7
15.594
0.7
15.735
0.7
16.018
0.8
15.534
0.8
15.720
0.8
16.076
0.9
15.471
0.9
15.706
0.9
16.135
4. Conclusion We suggested a model for determining the optimum common process mean for a production process where multiple products are processed. An economic model is constructed which involves selling prices, costs of production and inspection, and losses due to the scraps. We assumed the quality characteristics of the products are independently and normally distributed with known variances and a common process mean. The optimum common process mean is obtained by maximizing expected profit function. The solution is shown to be unique and optimum under the reasonable assumption that the proportion of
737
defective items is less than 50%. However, closed form expression for the optimum common process mean is not obtained, and a numerical search algorithm such as bisection method is used. In this study, we considered the case where inspection is performed directly the quality characteristics of interest. In some situations, it is impossible, or not economical, to inspect the quality characteristics of interest It will be of interest to consider the case where inspection is performed the surrogate variables correlated with the quality characteristics of interest. References 1.
2.
3.
4. 5.
6.
7.
8.
9. 10.
K.. S. Al-Sultan, An Algorithm for the Determination of the Optimal Target Values for Two Machines in Series with Quality Sampling Plans, International Journal of Production Research 12, (1994). F. J. Arcelus and M. A. Rahim, Simultaneous Economic Selection of a Variables and an Attribute Target Mean, Journal of Quality Technology 26, 125-133(1994). D. C. BETTES, Finding an Optimum Target Value in Relation to a Fixed Lower Limit and an Arbitrary Upper Limit, Applied Statistics 11, 202-210 (1962). S. Bisgaard, W. G. Hunter and L. Pallesen, Economic Selection of Quality of Manufactured Product, Technometrics 26, 9-18 (1984). T. O. Boucher and M. A. Jafari, The Optimum Target Value for Single Filling Operations with Quality Sampling Plans, Journal of Quality Technology 23, 44-47 (1991). S. L. Chen and K. J. Chung, Selection of the Optimal Precision Level and Target Value for a Production Process: the Lower-Specification-Limit Case, HE Transactions 28, 979-985 (1996). S. DUFFUAA and A. W. SIDDIQUI, Process Targeting with Multi-Class Screening and Measurement Error, International Journal of Production Research 41, 1373-1391 (2003). E. A. ELSAYED and A. CHEN, Optimal Levels of Process Parameters for Products with Multiple Characteristics, International Journal of Production Research 31, 1117-1132(1993). D. Y. Golhar, Determination of the Best Mean Contents for a Canning Problem, Journal of Quality Technology 19, 82-84 (1987). S. H. Hong and E. A. Elsayed, Setting Optimum Mean for Processes with Normally Distributed Measurement Error, Journal of Quality Technology 31,338-344(1999).
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11. S. H. Hong, E. A. Elsayed and M K. Lee, Optimum Mean Value and Screening Limits for Production Processes with Multi-Class Screening, International Journal of Production Research 37, 155-163 (1999). 12. W. G. Hunter and C. P. Kartha, Determining the Most Profitable Target Value for a Production Process, Journal of Quality Technology 9, 176-181 (1977). 13. Y. J. KIM, B. R. CHO and M. D. PHILLIPS, Determination of the Optimal Process Mean with the Consideration of variance Reduction and Process Capability, Quality Engineering 13, 251-260 (2000). 14. M. K. Lee and J. S. Jang, The Optimum Target Values for a Production Process with Three-class Screening, International Journal of Production Economics 49, 91-99 (1997). 15. M. K. LEE, S. H. HONG and E. A. ELSAYED, The Optimum Target Value under Single and Two-Stage Screenings, Journal of Quality Technology 33, 506-514 (2001). 16. M. K. LEE, S. B. KIM, H. M. KWON and S. H. HONG, Economic Selection of Mean Value for a Filling Process Under Quadratic Quality Loss, International Journal of Reliability, Quality and Safety Engineering 11,506-514(2004). 17. P. E. PFEIFER, A General Piecewise Linear Canning Problem Model, Journal of Quality Technology 31, 326-337 (1999). 18. M. A. RAHIM, J. BHADURY and K. S. AL-SULTAN, Joint Economic Selection of Target Mean and Variance, Engineering Optimization 34, 1-14 (2002). 19. M. A. RAHIM and A. B. SHAIBU, Economic Selection of Optimal Target Values, Process Control and Quality 11, 369-381 (2000). 20. K. TANG, and J. LO, Determination of the Process Mean when Inspection Is Based on a Correlated Variable, HE Transactions 25, 66-72 (1993). 21. J. TEERAVARAPRUG, and B. R. CHO, Designing the Optimal Process Target Levels for Multiple Quality Characteristics. International Journal of Production Research 40, 37-54 (2002).
SOME ADVANCED CONTROL CHARTS FOR MONITORING WEIBULL-DISTRIBUTED TIME BETWEEN EVENTS J.Y. LIU, M. XIE, T.N.GOH Department of Industrial and Systems Engineering, National University of Singapore 10 Kent Ridge Crescent, 119260, Singapore Time-between-events (TBE) data are available in industries such as manufacturing, maintenance, and even in service. Recently, control charts have been shown to be useful for the time-between-events to detect changes in the statistical distribution, especially the mean change. A common assumption for control chart design is the time between occurrences of events is exponentially-distributed. However, this is valid only when the events occurrence rate is constant. In this paper, a version of exponentially weighted moving average (EWMA) chart is developed for monitoring Weibull-distributed TBE data. The Average Run length (ARL) and Average Time to Signal (ATS) properties are examined, and an example is given for illustration.
1. Introduction Control charts have been shown to be effective for process monitoring not only in manufacturing but also in maintenance and service where time-between-event (TBE) is an important factor to show the capability and stableness of the process (Ye et al. 2003; Wang and Tsung, 2005; and Zhou et al. 2006). Here, the word events can stand for nonconforming items in a manufacturing process, failures in a maintenance process, accidents in a traffic system, diseases in health care, etc. The events occurrence rate of such a process can be monitored by a control chart which plots the time between events occurs, namely, time-between- events (TBE) chart. TBE charts are especially suitable when the events rarely occur and therefore it is quite difficult to form rational subgroups as the traditional Shewhart control charts require. There have been increasing interests on TBE charts recently. Some researchers suggested employing a control chart based on probability control limits like the Cumulative Count of Conforming (CCC) chart (e.g. Calvin, 1983; Goh,1987; and Bourke,1991), the Cumulative Quantity Control (CQC) chart (Chan et al. ,2000) or their extensions (Xie et al. 2002 and Schwertman,2005). Others proposed to apply the Cumulative SUM (CUSUM) and the Exponentially Weighted Moving Average (EWMA) methods for TBE data directly, as shown by Gan (1998) and Lucas (1985). Moreover, Shewhart 739
740
control charts can also be used to monitor TBE data after a proper transformation, see Radaelli (1998) and Jones and Champ (2002). However, most of the current studies on TBE charts are based on the assumption that the occurrence of events can be modeled by a homogeneous Poisson process, thus the time between two successive events follows exponential distribution. However, the assumption is true only when the events occurrence rate is constant, and as a result may limit the scope of TBE chart. A possible extension is to use Weibull distribution to simulate various TBE situations (including exponential) with non-constant events occurrence rate by varying its scale and shape parameters. This is especially useful in reliability monitoring, where events occurrence rate is rarely constant due to the aging property. Control charts for Weibull data have been studies by many scholars. Earlier studies focus on Shewhart control charts (e.g. Nelson, 1979; Ramalhoto and Morais, 1999). Xie et al. (2002) developed another control chart, named /-chart, for monitoring exponential and Weibull distributed time between failures based on probability control limits. Moreover, the CUSUM and EWMA charts can also be applied for monitoring Weibull-distributed data. Chang and Bai (2001) proposed a heuristic method of constructing X, CUSUM, and EWMA chart for skewed populations with weighted standard deviations. Hawkins and Olwell (1998) studied the optimal design of CUSUM for Weibull data with fixed shape parameter. Borror et al. (2003) investigated the robustness of TBE CUSUM for Weibull-distributed and Lognormal-distributed TBE data. However, few methods have been proposed using EWMA chart. Zhang and Chen (2004) developed a lower-sided and upper-sided EWMA chart for detecting mean changes of censored Weibull lifetimes with fixed censoring rate and shape parameter. In this study we considered the EWMA for complete Weibull data with known parameters. The rest of this paper is organized as follows: Section 2 briefly introduces two existing TBE charts for monitoring Weibull data, and then Section 3 describes the construction of the Weibull EWMA chart with an illustrative example. Section 4 discusses the Average Run Length (ARL) and Average Time to Signal (ATS) properties of the Weibull EWMA chart. Finally, some conclusions are given in Section 5. 2. Control Charts for Weibull-distributed TBE Data Let A',, X2, denote a sequence of time between events data, which are independent Weibull random variables with probability density function:
741 /
/(*)=
N'T"'
Afi. \rj
eKP> ,x>0,/3>0,5>0
(1)
where ft is the scale parameter and rj is the shape parameter. 2.1. The Cumulative Quantity Control (CQC) Chart The CQC chart, also referred as /-chart in Xie et al. 2002, monitors the time between two successive events based on probability control limits. If the TBE is Weibull-distributed and the acceptable false alarm probability is a, the upper control limit (UCL), central line (CL) and lower control limit (LCL) of the CQC can be calculated as: i/-i
l/>7
LCL = P In
2-a
,CL = p[\n(2)f\UCL = 0
a
(2)
so that the probability of a point is located above UCL or below LCL is equal to all. 2.2. The Weibull CUSUM Chart The basic statistics for upper and lower-sided CUSUM chart are S;
=max{0,S;_l+(Xl-k)}
5:=min{0,5 / l 1 +(^, -k)}
(3)
where k is the reference value and can be determined by the following formula when X follows Weibull distribution. k=
pr
(4)
/?, and /?o are the out-of-control and in-control scale parameter, respectively. An out-of-control alarm will arise if the control statistics go beyond the control limits, i.e., S~ < -h or 5,+ > h. Hawkins and Olwell (1998) provides the detailed calculation methods of ARL for Weibull CUSUM chart. 3. The EWMA Chart for Monitoring Weibull-distributed TBE Another alternative is to use EWMA chart for monitoring the Weibulldistributed TBE. The statistic of two-sided Weibull EWMA chart is
Z,=AX,+{\-X)Z,_
(5)
742
where ?, is the smoothing constant that satisfies 0 < k <1. Usually the starting value is set to be the process target, i.e. Z0=/u0, where /J0 the mean of Weibull data. With this definition, it can be deducted that £(Z,) = //„, Var{Z,)=a^-^j\-{\-Xf]
(6)
Therefore, the UCL, CL and LCL can be calculated by UCL = n0+Lv JVar(Z,);CL = na;LCL = //„ - LjVar{Z,)
(7)
where the Lv and LL are the design parameters which influence the width of the control limits. An out-of-control alarm will arise when X exceeds either UCL or LCL. Since the time between events is always positive, the LCL will be set to be zero if the calculated LCL is less than zero. For large value oft, the variance of Z, will approximate constant, and the upper and lower asymptotic control limits hu and hL are given by
An out-of-control signal will arise when Zt< /zL or Zt > hv. Here is an illustrative example of the Weibull EWMA chart. Table 1 shows a set of time between failures data for monitoring the reliability of a process. The first 20 observations are simulated following Weibull distribution with shape parameter r/=2 and scale parameter/?= 10 hours. The next 20 observations were generated following Weibull distribution with r\=2 and /?=5 hours. A twosided Weibull EWMA chart is designed so that the in-control ARL=370 (A=0.10, LV=LL=2.70). Fig. 1 shows the Weibull EWMA chart for the data in Table 1. An out-of-control alarm is raised from the 33rd point, which indicates that the mean time to failure may have decreased. Therefore, engineers need to check the process and try to find out the reasons for it so as to further improve the reliability of the process. 4. Properties of Weibull EWMA chart The ARL properties of an EWMA scheme can be approximated using Markov Chain approach similar to that described by Brook and Evans (1972). The continuous state Markov chain can be evaluated by discretizing the infinite-state transition probability matrix.
743 Table 1 .Time between failures (TBF) data for Weibull EWMA chart
EWMA
UCL
LCL
Failure No.
TBF (hours)
EWMA
UCL
LCL
8.37
6.31
685
5.33
21
1.13
6.69
7.82
4.35
2
3.25
6.01
7.11
5.06
22
6.05
6.63
7.82
4.35
3
4.43
5 85
7.28
4.89
23
3.53
6.32
7.82
4.35
4
6.62
5.93
7.40
4.77
24
4.25
6.11
7.83
4.35
Failure No.
TBF (hours)
1
5
4.48
5.78
7.49
4.68
25
1.70
5.67
7.83
4.35
6
6.44
5.85
7.56
4.61
26
2.61
5.36
7.83
4.35
7
10.36
6.30
7.62
4.55
27
4.43
5.27
7.83
4.34
8
11.13
6.78
7.66
4.51
28
5.99
5.34
7.83
4.34
9
10.37
7.14
7.69
4.48
29
2.22
5.03
7.83
4.34
10
7.92
7.22
7.72
4.45
30
3.50
4.88
7.83
4.34
11
5.65
7.06
7.74
4.43
31
3.48
4.74
7.83
4.34 4.34
12
10.83
7.44
7.76
4.41
32
2.41
4.50
7.83
13
4.20
7.11
7.77
4.40
33
1.43
4.20
7.83
4.34
14
7.52
7.16
7.78
4.39
34
2.75
4.05
7.83
4.34
15
9.97
7.44
7.79
4.38
35
4.59
4.11
7.83
4.34
16
5.94
7.29
7.80
4.37
36
4.75
4.17
7.83
4.34
17
7.77
7.34
7.81
4.37
37
1.29
3.88
7.83
4.34
18
9,00
7.50
7.81
4.36
38
1.47
3.64
7.83
4.34
19
6.04
7.36
7.81
4.36
39
4.72
3.75
7.83
4.34
20
6.90
7.31
7.82
4.35
40
1.68
3.54
7.83
4.34
J 0 -i
15 0 -i
0
1
?
r—
10
,
1?
— - ,
20
,
25
r-
30
,
35
.
,
40
Failure No
Figure 1. The two-sided EWMA chart for monitoring Weibull distributed time between failures
744
Consider a two-sided Weibull EWMA chart with design parameters A, hv and hL, the interval between the lower and upper control limit (hL, hv) is divided into m subintervals of width w. w can be expressed as: W =
h -h ?LL^L m
(9)
The EWMA control statistics Z, is said to be in transient sate (j) at time (t) if hL+jw
j = K + 0 + 0.5)w>y = °X •••m-X
(10)
The control statistics Zt is regarded as in the absorbing state m if the point goes outside the control limits, i.e. Zt> hv or Zt
_p\K+pv-{\-*)m, \ A
P„=\X.
c^+(y+l>v-(l-^K]
c x
'
A
< t < ^ } + ^ , >^±*i},/.„,1„.„_l
/>,„;= 0,7= 0,1,.../H-l P
i=0X..m-l }j=0,l,...,m-l
(11)
=1
r mm
Let Q be the matrix of transition probabilities obtained by deleting the last row and column of P. The vector of ARLs 6 can be calculated with 9 = {I-Q)'\
(12)
where 1 is an m*\ vector of Is and I is a m*m identity matrix. The elements in the vector 0 are the ARL's when the EWMA chart starts in various states. The first element in the vector 6 gives the average run length for the Weibull EWMA chart starting from zero. Let the rth element be the ARL given that the EWMA chart starts from, r can be is achieved by Z
o-hl.
;i3)
745 where [C] stands for the largest integer not greater than C. Hence, the Average Time to Signal (ATS), which presents the average time until an alarm is observed, can be obtained by
ATS^B^X,
=E(R)E(x)=ARL-j3-r
(14)
where R is the number of points ploted until an alarm signals. ARL and ATS are useful measurements to judge the efficiency of a TBE chart for detecting changes in statistical distributions. Table 2 lists some ARL and ATS values for the Weibull EWMA chart in the illustrative example (>.=0.10, LV=LL=2.70). The shape parameter is fixed at 2.0, and scale parameter varies from 2 tolO. The in-control scale /?=10 hours, and in-control ARL=370. Fig. 2 presents the ARL curve for the Weibull EWMA chart, from which we can see that the Weibull EWMA chart is very sensitive to the scale parameter shifts. The ATS value implies that the average time to an out-of-control alarm will be around 46 hours for detecting the scale parameter's change from 10 hours to 5 hours. Table 2. Some ARL and ATS values for the Weibull EWMA 2
5
6
8
8.5
9
10
ARL
5.34
10.38
15.16
64.94
120.67
249.19
370.84
ATS
9.47
45.99
80.63
460.40
908.99
1987.51
3286.48
Scale/?
11
12
13
14
15
18
20
ARL
Scale/?
89.19
35.14
19.89
13.55
10.24
5.98
4.74
ATS
869.48
373.76
229.12
168.14
136.14
95.40
84.07
scale parameter
Figure 2. The ARL curve of the Weibull EWMA chart.
746 5. Conclusions Control charts for Weibull-distributed TBE can be very useful for monitoring reliability processes. In this paper, an EWMA scheme for the monitoring of Weibull-distributed TBE data is proposed, and the ARL and ATS properties are investigated. The results show that Weibull EWMA chart is sensitive to the process shifts and can be very effective for reliability monitoring. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
T.W. Calvin, IEEE Trans. Comport. Hybr. 6, 323 (1983). T.N. Goh, Qual. Assur. 13, 18(1987). P.D. Bourke, J. Qual. Technol. 23,225(1991). L. Y. Chan, M. Xie and T. N. Goh, Int. J. Prod. Res. 38, 397(2000). F.F. Gan, J. Qual. Technol. 30, 55(1998). J.M. Lucas, Technometrics 27, 129(1985). G. Radaelli, Total Qual. Mgnt. 9, 33 (1998). L.A. Jones and C.W.Champ, Qual. Reliab. Engng. Int. 18, 479 (2002). P.R. Nelson, IEEE Trans. Reliab. 28, 283(1979). M. F. Ramalhoto and M. Morais, J. Appl. Stat. 26,129 (1999). M. Xie, T. N. Goh and P. Ranjan, Reliab. Eng. Syst. Safe. 77, 143(2002). Y.S. Chang and D.S. Bai, Qua!. Reliab. Engng. Int.M, 397(2001). D.M. Hawkins and D.H. Olwell, Cumulative Sum Charts and Charting for Quality Improvement, New York: Springer, (1998). C. M. Borror, J.B. Keats and D.C. Montgomery, Int. J. Prod. Res. 41, 3435(2003). L.Y. Zhang and G.M. Chen, J. Qual. Technol^, 321(2004). D. Brook and D.A. Evans, Biometrika 59, 539(1972). N.C. Schwertman, Qual. Reliab. Engng. Int. 21,743(2005). N.Ye, S.Vilbert and Q.Chen, IEEE Trans. Reliab. 52, 75(2003). S. Zhou, D. Sun and J. Shi, IEEE Trans. Autom. Sci. Engng. 3, 60 (2006). K.Wang and F. Tsung, Qual. Reliab. Engng. Int. 21,677(2005).
THE RELATIONSHIP BETWEEN PCIS AND 6 a
L. Y. OUYANG, C. H. HSU Graduate Institute of Management Science, Tamkang Taipei, Taiwan, R.O.C. E-mail [email protected]
University,
Six Sigma has already become an efficient improvement technique adopted by a great number of enterprises. Numbers of Sigma has become a tool of measuring process capability in some enterprises. However, many enterprises still use Process Capability Indices (PCIs) to measure the process capability. The paper will research the relationship between PCIs and number of Sigma. In bilateral specifications, the paper will research the relationship between the PCIs which are Cp, Cpk, Cpm and Cpmk and number of Sigma. In unilateral specifications, the paper will research the relationship between the PCIs which are Cpu and Cpl and number of Sigma. If supplier and buyer use different tools to measure the process capability, the relationship can decrease the communicate noise.
1.
Introduction
In the 1980s and early 1990s, Motorola was one of many U.S. and European corporations whose lunch (along with all other meals and snacks) was being eaten by Japanese competitors. Motorola's top leaders conceded that the quality of its products was awful. They were, to quote one Motorola Six Sigma veteran, "In a word of hurt." Like many companies at the time, Motorola didn't have one "quality" program, it had several. But in 1987, a new approach came out of Motorola's Communications Sector—at the time headed by George Fisher, later top exec at Kodak. The innovative improvement concept was called "Six Sigma" (Pande et al., 2000). Six Sigma is named after the process that has six standard deviations on each side of the specification window. Such a process produces 3.4 defects per one million opportunities in the long term (Wyper and Harrison, 2000). Based on Tong et al.(2004), six sigma has been initiated using statistical tools and techniques in business, transactional, and manufacturing process. It has been proven to be successful in reducing costs, improving cycle times, eliminating defects, raising customer satisfaction, and significantly increasing profitability. In the first 5 years of Six Sigma implementation, Motorola achieved saving of SUS 2.2 billion. Other companies followed, e.g. 747
748
GE, ABB, Bombardier and Allied Signal (Wyper and Harrison, 2000). Apparently, Six Sigma has already become an efficient improvement technique adopted by a great number of enterprises. Numbers of Sigma has become a tool of measuring process capability in some enterprises. However, many enterprises still use Process Capability Indices (PCIs) to measure the process capability. If supplier and buyer use different tools to measure the process capability, the communicate noise will be maybe happened. So the relationship between number of Sigma and PCIs is an important topic. Because PCIs can differentiate two parts which are unilateral specifications and bilateral specifications, the paper will research the relationship between the two parts of PCIs and number of Sigma. In bilateral specifications, the paper will research the relationship between the PCIs which are Cp, CPk, Cpm and Cpmk and number of Sigma. In unilateral specifications, the paper will research the relationship between the PCIs which are Cpu and Cpi and number of Sigma. Based on Linderman et al.(2003), Motorola set this goal so that process variability is ± 6crfrom the mean. They further assumed that the process was subject to disturbances that could cause the process mean to shift by as much as 1.5
The Relationship Between ka and PCI when the Process Mean shifts 1.5cr
Because the specifications are different in different product, manager of process can't evaluate process performance from //and a right away. For the above reason, Juran (1974) combined process parameters with product specifications to bring up the idea of Process Capability Lndices(PCIs). Following the above idea, Kane(1986) proposed that Cp, Cpk evaluated process capability of bilateral specifications. And then Chan et al.(1988) used the idea of loss function to improve Cpk and to propose Cpm. Pearn et al. (1992) proposed another index called Cpmk which can reflect the degree that process mean shifts the process centering. The section defines Cp, Cpk, Cpm, and Cpmk to compute the corresponding value in 6a, 5a, Aa and 3owhen the process mean shifts 1.5a. The indices Cp, Cpk, Cpm, and Cpmk have been defined as fallows:
749
c =*o
3a
d-\fi-f\ \r i
c
pk
3a
=
d
' C pm
d
-\jU-T\
3^1^-T)1
pmk
The section will research the corresponding values of Cp, Cpk, Cpm, and Cpmk when the process mean shifts 1.5 a (i. e., I \i-T I =1.5
_d_ 3
d-\ju-T\ 3a
ka-l.5a 3a
c> 'pk = pm
k 3J325
2
3^a +(ju-Tf
2
k-1.5
ka 3-sla +(l.5a)2
2
d-\ju-T\ pmk
k 3
=
3-Ja +{M-T)
2
kcr-1.5a 2
=
3ja +(\.5a)
2
fc-1.5 3J325
The corresponding values which are Cp, Cpk, Cpm and Cpmk can be computed as follows table 1 in 6a, 5a, 4a and 3a when the process mean shifts 1.5 a .
750 Table 1: the corresponding value of PCIs of bilateral specifications when the process mean shifts 1.5
yield
cP
Cpk
^pm
^prnk
3(7
0.9331928
1.00
0.50
0.55
0.28
4(7
0.9937903
1.33
0.83
0.74
0.46
5a
0.9997673
1.67
1.17
0.92
0.65
6a
0.9999966
2.00
1.50
1.11
0.83
Because Six Sigma was allowed that the process mean can shift 1.5cr off the target based on Linderman et al.(2003), the process mean shifts right or left 1.5cr off the target in smaller-the-better larger-the-better process. If we will find the minimum value of Cpu of arriving 6 a standard, the process mean must shift right 1.5cr off the target. Because the value of Cpu is more and more large when the process mean shifts left. For the same cause, if we will find the minimum value of CP| of arriving 6 cr standard, the process mean must shift left 1.5 a off the target. When that the process quality level arrived ka (i. e., d = ka) and the process mean shifts 1.5cr, two unilateral specifications PCIs which are Cpu and Cpi can be showed as fallows: _USL-n pu
=
3cr
=
la
H-LSL "'
(k-l.5)cr
=
3cr
k-\.5 3
(k-l.5)cr 3cr
=
£-1.5 3
The corresponding values which are Cpu and Cpi can be computed as follows table 2 in 6
yield
L-pu
Cpi
la
0.9331928
0.50
0.50
4(7
0.9937903
0.83
0.83
5(7
0.9997673
1.17
1.17
6a
0.9999966
1.50
1.50
751 3. Conclusions Because Six Sigma has already become an efficient improvement technique adopted by a great number of enterprises, numbers of Sigma has become a tool of measuring process capability in some enterprises. No papers research the relationship between PCIs and numbers of Sigma perfectly yet. So the paper researched the relationship between PCIs and numbers of Sigma. In the second section, the relationship between PCIs and numbers of Sigma was built, and the corresponding values which are Cp, Cpk, Cpm, Cpmk, Cpu and Cpi were computed in 6a, 5a, 4a and 3
6. 7.
8.
9.
R. A. Boyles, Process capability with asymmetric tolerances, Communications in Statistics-Simulation and Computation 23, 615-643 (1994). L. K. Chan, S. W. Cheng and F. A. Spiring, A new measure of process apability: Cpm, Journal of Quality Technology 20(3), 162-175 (1988). J. M. Juran, Juran's quality control handbook, 3 rd Edition, McGraw-Hill, New York (1974). V. E. Kane, Process capability indices, Journal of Quality Technology 18, 41-52(1986). K. Linderman, R. G, Schroeder, S. Zaheer and A. S. Choo, Six Sigma: a goal-theoretic perspective, Journal of Operations Management 21, 193-203 (2003). P. S. Pande, R. P. Neuman and R. R. Gavanagh, The Six Sigma Way, McGraw-Hill, New York (2000). W. L. Pearn, S. Kotz and N. L. Johnson, Distributional and inferential properties of process capability indices, Journal of Quality Technology 24, 216-231 (1992). J. P. C. Tong, F. Tsung and B. P. C. Yen, A DMAIC approach to printed circuit board quality improvement, International Journal of Advance Manufacturing Technology 23, 523-531 (2004). B. Wyper and A. Harrison, Deployment of Six Sigma methodology in human resource function: a case study, Total Quality Management 11, NOS4&5, 720-727 (2000).
C O N F I D E N C E INTERVALS FOR M E A S U R E S OF VARIABILITY IN B A L A N C E D M I X E D G A U G E R&R STUDY
D . J. P A R K Division of Mathematical Sciences College of Natural Science Pukyong National University 599-1 Daeyeon 3-Dong Nam-Gu, Pusan, 608-737, South Korea E-mail: [email protected]
We consider a design for a balanced mixed model with covariates considered as fixed effects and one factor as random effects. In this paper, some methods for constructing confidence intervals on measures of variability in repeatability and reproducibility are provided. The objective of this study is to compare the confidence intervals on repeatability and reproducibility to apply for R&R study. A numerical example is provided.
1. Introduction In many measurement studies, a gauge is used to obtain replicate measurements on units by several different operators, setups, or time periods. One of the common experiments employed in industry is to properly monitor manufacturing process, e.g., repeatability and reproducibility. The variability is inherent in the measurement system, which we generally think of as the precision of the gauge. Burdick et al.(2005) define that repeatability represents the gauge variability when it is used to measure the same unit(with the same operator or setup or in the same time period). Reproducibility is referred to as the variability arising from different operators, setups, or time periods. Borror et al.(1997) considered the capability of a gauge as designing a statistical experiment in which several manufactured parts are measured under controlled conditions. The gauge capability is expressed as variance components, one variance components for the repeatability of the gauge and the other variance component for the reproducibility of the gauge. These variance components are analyzed using 752
753
ANOVA method that leads to a direct and convenient method of estimating the variance components in an experimental design. 2. The Balanced Mixed Model A traditional gauge study utilizes a random two-factor design with a random sample of operators and of parts. That is, o operators are randomly chosen to measure p randomly selected parts from manufacturing process. Each measurement is made n times by each operator. The measurements, Yijk, are modeled by Yljk = lx + P, + Oj + (PO)ij + Eijk i = l,.-,p;j
(1)
= l,...,o;fc= l,...,n
where n is a constant as overall mean, Pi, Oj, (PO)ij, Eijk are jointly independent normal random variables with means of zero and variances aP, a o> a%0> a n d a%> respectively. Adamec and Burdick(2003) extended results of the typical two-factor repeatability and reproducibility (R&R) study to a situation with three random factors. Unlike this type of traditional model, the measurements of parts, Yij, can be linearly related to a covariate (or concomitant variable), Xij. I ope r a t o r s ^ operating machines) are randomly sampled and J measurements are made for each operator(or operating machine). Table 1.
Breaking Strength Data.
(Y = strength in pounds and X = diameter in 1 0 - 3 inches) Machine
1
Machine
2
Machine
3
Y
X
Y
X
Y
X
36
20
40
22
35
21
41
25
48
28
37
23
39
24
39
22
42
26
42
25
45
30
34
21
49
32
44
28
32
15
Montgomery(1984) published a data set in Table 1 of this experiment. We here assume that three different operating machines were randomly chosen and they measured a monofilament fiber strength for a textile company. The process engineer is interested in determining the measurement of variability in the breaking strength of the fiber measured by the three machines
754
randomly chosen and the measurement of variability among three machines. However, the strength of a fiber is related to its diameter, with thicker fibers being generally stronger than thinner ones. A random sample of five fiber specimens is selected from each machine. The fiber strength(Yy) and corresponding diameter(Xij) for each specimen are shown in Table 1. Then the replication of measurements, Yjj, is modeled by Y{j = n + pXij +Oi + E^ i = l,...,I;j
(2)
= 1,..., J
where \i is a constant, /3 is linear regression coefficient indicating the dependency of measurements Yij, X^ is a covariate, Oj and E^ are jointly independent normal random variables with means of zero and variances O"Q and a\, respectively. This model has both fixed effects for a covariate and random effects for operating machines so that it is a mixed model with two error terms. One possible partitioning for ANOVA for model (2) is shown in Table 2. Table 2.
ANOVA for model(2)
sv
DF
SS
Mean
1
IJY2
Covariate after Mean
1
MQ\&xxa
Operating Machines adjusted for Covariate after Mean
I- 1
dyya
Error
IJ -- 7 - 1
•5yyw
Total
IJ
^i^-jYij
i
Jxxw)
\ P^Oxxw
HQ\&xxa
~T~ ^>xxw)
P^Jxxw
The notation in ANOVA Table 2 is as follows: $w = Sxyw/Sxxw, \^xya Y..j
~r ^xyw)/\^xxa i ^xya
=
i &xxw))
J*ji\Xi.
^xxa
A )(li
J^iy-^-i.
A...J > ^yya
— Y..), £>xxw = 2-li2-Jj\Xij
2
Y,iT,j(Yij-Yi.) , Sxyw = Y,iY,j(Xij-Xi.)(Yij-Yi.). of model (2), RA/{d2E + Ja20) ~ X?_2> RW/
ti\V
=
^yyw
P\y&xxwi
PA
=
—
Pc = J2-'i\ii.
— A j J , Jyyw
=
Under the assumptions where R ~ Xjj-i-i A =
^xya/^xxai
and liA
anu
n\y
are independent. Reproducibility in model (2) is the variation attributed to using different operating machines to measure the specimens. Repeatability represents the
755
variation within specimens on the same operating machine. The variance component that measures reproducibility is <JQ and the variance component that describes repeatability is a\: reproducibility
®O
(3) a
repeatability
=
a
E
W
Parameters to make the estimates are shown in Table 3. Table 3.
Parameters of interest in model (2)
Parameter
Definition
IP = o"o
Measurement of variability attributed to using different operating machines
1M = &%
Measurement of variability within specimens on the same operating machine
3. Confidence Intervals on Measurement of Variability Using the distributional properties in model (2) the parameters of interest in Table 3 are written as functions of expected mean squares. E(SA) =
(5)
= a
E = ®w
where S^ = RW/(U - I - I) and S\ = RA/(I - 2). Modified large sample method that Burdick and Graybill(1992) discussed can be employed for constructing confidence intervals on individual variance components. Another method for constructing confidence intervals on the parameters in Table 3 could be SAS(Statistical Analysis System) procedure MIXED employing Restricted Maximum Likelihood Estimation(REML)which are generally preferred to the ANOVA estimators. The concept of generalized inference that Tsui and Weerahandi(1989) introduced for testing hypotheses can also be used to construct confidence intervals on the parameters.
756
3.1. Modified Large Sample
Method
3.1.1. Confidence Interval on ~JM Using equation (5) measurement of repeatability 7 M is written as 1M
= 0W.
(6)
Since modified sum of squares Rw/°% is a chi-square random variable with IJ — I — 1 degrees of freedom. The exact 100(1 — a)% confidence interval on repeatability 7 M is c2
c2
F
(7)
'F
where ir(a/2:<^1,d/2) is the 100(1 — a/2) percentile F—value with df\ and dfo degrees of freedom. 3.1.2. Confidence Interval on 7 P Using equation (5) measurement of reproducibility jp is written as IP =
j
•
(8)
An approximate 100(1 — a)% confidence intervals on 7 P is constructed using the method of Ting et al.(1990) 1 SA ~ Sw - (GxSA + H2Sw J
+ Gi2SASw)?,
&A ~~ $w + (HiSA + G2SW + where d
= 1-
1/F(Q/2:/_2J0O),
Hi2SASw)i
G 2 = 1 - l/Fia/2.u_j_hoo),
l / - f ( l - a / 2 : / - 2 , o o ) ~ 1, # 2 = l/F(l-a/2:IJ-I-l,oo)
~ 1> Gi2
GlFf-HR/Ft, Fx = ^ ( a / 2 : / _ 2 , / j _ / _ i ) , # 1 2 = and F 2 = i r ( 1 _ Q / 2 :/-2,/j-/-i)3.2. Restricted
Maximum
(9)
Likelihood Estimation
#i =
= [(-Fl - I ) 2 ~
[(1-F2)2-H*F$-Gl]/F2,
Method
Measurements of repeatability 7 M and reproducibility 7 P can be obtained using the PROC MIXED of SAS that reports asymptotic standard errors for restricted maximum likelihood estimators. The likelihood-based statistics to estimate variance components is Wald statistics Z which is computed as the parameter estimate divided by its asymptotic standard error. The asymptotic standard errors are computed from the inverse of the second
757 derivative matrix of the likelihood with respect to each of the variance parameters. Wald statistics Z is valid for large samples but unreliable for small data sets. 3.2.1. Confidence Interval on 7 M An approximate 100(1 - a)% confidence interval on measurement of repeatability 7 M is thus obtained using asymptotic standard error by REML method: v\ 5% 2 -X(i/i,l-<*/2)
v\ &%
T
2 X(Vua/2)
(10)
where v\ is degrees of freedom defined as v-i = 2 x Z\, Z\ is Wald statistics defined as Z\ = d\/S.E.(a2E), and b\ is the REML estimator. 3.2.2. Confidence Interval on "/p Similarly, an approximate 100(1 — a)% confidence interval on measurement of reproducibility •yp is obtained using asymptotic standard errors for REML estimators: vi <3Q 2 X(j/ 2 ,l- a /2)
'
VI <5O 2 X(i/2,a/2)
(11)
where v2 is degrees of freedom as v2 = 2 x Z\, Z2 is Wald statistics defined as Z2 = &O/S.E.(&Q), and &Q is the REML estimator. 3.3. Generalized
Confidence
Intervals
Application of the generalized inference concept requires a generalized pivotal quantity(GPQ) which extends the standard notion of a pivotal quantity. 3.3.1. Confidence Interval on ^M Since Ry/jo-\ ~ X / J _ J _ I > @W quantity as follows:
m
equation (5) can be written as a pivotal
6w _(IJ-I-
l)s2w
= —w*—
(12)
where s ^ is an observed value of S^ and W* = (IJ — I—l)Syy/o~%. Define R\ as the solution for 6\y- The distribution of R\ is completely determined
758 by W* using Monte Carlo methods. An approximate 100(1—a)% confidence interval on repeatability is defined as (13)
[-^la^'-Rli-o/a]
where i?i o / 2 and Ril_a/2 are the percentile a/2 and 1 - a/2 of the distribution of R\, respectively. 3.3.2. Confidence Interval on 7p Since RA/{&% + JO-Q) ~ Xi-2> ®o in equation (5) can be written as a generalized pivotal quantity as follows: (T 7 _ T _ 1 U 2
{I ~ 2 ) 42 U*
e0-)
(IJ-I-l)sw W*
(14)
where s\ is an observed value of S\ and U* = (I - 2)S2A/{o-2E + J
are
where i?2Q/2 d R2l_a/2 bution of R2, respectively. Table 4. Parameter 7F = ° o
Method MLS GEN
=o\
the percentile a/2 and 1 — a / 2 of the distri-
Two-sided 90% Confidence Intervals using d a t a in Table 1
REML
1M
(15)
Lower Limit
Upper Limit
Interval Length
0
52.58
52.58
0.22
3338.76
3338.54
0
52.71
52.71
MLS
1.42
6.12
4.70
REML
1.43
6.24
4.81
GEN
1.43
6.19
4.76
MLS, REML, and GEN in Table 4 represent modified large sample method, restricted maximum likelihood estimation method, and generalized confidence intervals, respectively. Because of limited space, we did not show simulation results for three methods. The generalized confidence intervals were calculated based on 10,000 simulated values of R\ and Z?2- Since Wald
759 statistics Z is valid for large sample, it is not surprising for Table 4 results t h a t the R E M L method generates wider interval length for reproducibility parameter ^p t h a n MLS and G E N methods.
4.
Conclusion
W h e n practitioners meet an experimental design where the measurements of parts, Yij, are linearly related t o a covariate, Xij, and the measurements are made for operating methods(or operating machines) randomly sampled in manufacturing process, they can use three methods provided in this paper to construct confidence intervals on repeatability and reproducibility. In order to measure variability of repeatability three methods generate desirable confidence interval lengths. However, MLS and G E N methods are recommended t o measure the variability of reproducibility when sample sizes are moderately small. REML method is useful only for large samples.
References 1. E. Adamec and R. K. Burdick, Confidence Intervals for a Discrimination Ratio in a Gauge R&R Study with Three Random Factors, Quality Engineering 15(3), pp.383-389 (2003). 2. C. M. Borr, D. C. Montgomery, and G. C. Runger, Confidence Intervals for Variance Components from Gauge Capability Studies Quality and Reliability Engineering International 13, pp.361-369 (1997). 3. R. K. Burdick, C. M. Borror, and D. C. Montgomery, Design and Analysis of Gauge R&cR Studies, ASA-SIAM Series on Statistics and Applied Probability (2005). 4. R. K. Burdick and F. A. Graybill, Confidence Intervals on Variance Components, Marcel Dekker, Inc. (1992). 5. D. C. Montgomery, Design & Analysis of Experiments 2ed. John Wiley & Sons, Inc. (1984). 6. N. Ting, R. K. Burdick, F. A. Graybill, S. Jeyaratnam, and T.-F.C. Lu., Confidence Intervals on Linear Combinations of Variance Components, Journal of Statistical Computation and Simulation, 35, pp. 135-143 (1990) 7. K. Tsui and S. Weerahandi, Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters, Journal of the American Statistical Association, 84, pp. 602-607 (1989)
T H E R A P E U T I C DECISION M A K I N G FOR U N C E R T A I N T Y AVERSE PATIENT *
T . S A T O W A N D H. K A W A I Tottori University Faculty of Engineering, 4-101 Koyama-Minami, Tottori 680-8552, Japan E-mail: {zwsatow, kawai}@sse.tottori-u.ac.jp
A therapeutic strategy is planned by a doctor who takes charge of medical treatment. The strategy is based on a criterion which maximizes an expected utility of patient. In general, the axiom of Savage's expected utility is used to measure the patient's utility. However, Savage's utility can not be used when several probability measures are nominated as a candidate for the derivation of the utility. For measuring a patient's subjective value, health-related QOL scale is considered as one of most important issue. As a recent topic, Q-TWiST(Quality-adjusted Time Without Symptoms and Toxicity method) is a powerful tool to measure the variable subjective value of patient. In order to derive Q-TWiST, the survival probability of patient has to be estimated from censored or non-censored clinical data. Q-TWiST takes different values when there are two or more recommended estimation methods. Which values should be adopted by the viewpoint of patient? For the purpose of solving the above-mentioned problems, Q-TWiST which is composed by several probability measures is proposed. The sensitivity analysis concerning the utility value which depends on the state of the patient is made.
1. Introduction 1.1. Quality
of Life
An extension of survival time has been considered as a criterion for highquality medical treatment, because it is easy to do an objective assessment. Misunderstanding of which life-prolongment is the profit of patient was caused by an overemphasis of life-sustaining treatment. Therefore, the situation which increases the patient's load is caused by intervention which does not consider patient's well-being. Such an overemphasis on the objective "This work is supported by Grant-in-Aid for Young Scientists (B) 17710137.
760
761 assessment can satisfy only with the unilateral side of wide-ranging health care. As a standard concept of the world, the goal of health care is not only helping patient's long life but also enhancing Quality Of Life(QOL). In the current constitution of the World Health Organization(WHO), the following principles are basic to the happiness, harmonious relations and security of all peoples: Health is a state of complete physical, mental and social well-being and not merely the absence of disease or infirmity1. As we can face with informed consent, patient's independence and dignity begin to be being esteemed. This would be an end of paternalism in the relation between doctor and patient. How should we evaluate patient's profit, i.e. patient's well-being? One of answers to the question is QOL which attracts attention in recent years. A lot of health services and outcome researches were carried out in many countries in 1980's. As a result, some health-related QOL measures were proposed, and the application to a medical practice site started, too. WHO defines QOL and tries to spread the concept. WHOQOL is "An individual's perception of their position in life in the context of the culture and value systems in which they live and in relation to their goals, expectations, standards and concerns. It is a broad ranging concept affected in a complex way by the person's physical health, psychological state, personal beliefs, social relationships and their relationship to salient features of their environment" 2 . In addition, MOS 36-Item Short-Form Health Survey(SF-36) can be also given as the typical health-related QOL 3 . These QOL measure only the "quality" of a patient's prognosis. Quality-Adjusted Life Years(QALYs) was proposed as one method of implementing the trade-off between "quality" and "quantity". As an extended model of QALYs, Quality-adjusted Time Without Symptoms and Toxicity method(Q-TWiST) was proposed 4 . Q-TWiST is used when we want to decide a treatment method.
1.2. Quality-adjusted Toxicity
Time Without Symptoms
and
Q-TWIST is a statistics technique proposed as a model by whom the side reaction by the therapeutic intervention etc. can be considered for the cancerous patient 5 ' 6 - 7 . Q-TWiST consists of 3 health states; TOX(toxicity), TWiST(time without symptoms or toxicity) and REL(relapse). TOX is time duration with conscious side reaction by treatment. TWiST is time duration without conscious side reaction and symptom. REL, sometimes calls PROG (progression) is time duration from admission of relapse to the
762
death. Utility coefficients are given to each health state. Q-TWiST is defined that the sum of the products of the proportion of survived patient of each health multiplied by the utility value of each health state. In a survival time analysis, Kaplan-Meier or Cutler-Ederer methods are used in comparing between treatment groups. The Kaplan-Meier method is recommended when the number of cases is less than or equal to 50. On the other hand, the Cutler-Ederer method is adopted when there are a lot of numbers of cases.
1.3. The aim of work The threshold value is about 50 as previously stated. Which method should be adopted when the number of cases is about 50? In other words, you can not judge which estimation to have to use. As another instance, some different case studies for the same disease are reported under the same estimation method. By the difference of estimation method or of case report, survival curves take different sharp. As a result, Q-TWiST takes a different value. Therefore, the possibility that different treatment method is recommended is caused. It causes confusion to the decision making of therapeutic strategy. In general, the behavior of decision maker follows a rule which maximizes his(her) expected utility. This is the concept of Savage's 8 expected utility theory. A probability measure underlies the behavior of decision maker, and a unique probability measure is assigned to future events. On the other hand, even if any unique probability measures are assigned, it is known that behaviors not expressible exist. As a famous instance is known as Ellsberg paradox 9 . In order to solve the Ellsberg paradox, Gliboa and Schmeidler 10,11 ' 12 developed Savage's axiom. In their axiom, the ambiguity can be expressed by two or more probability measures. Under these measures, the decision maker behaves for maximizing an expected utility which is calculated by using the most pessimistic probability. It is called uncertainty averse(or ambiguity averse) since the most pessimistic probability is used. In this research, it thinks about decision making for planning a treatment strategy when the patient is uncertainty averse. The conception and definition of uncertainty averse is introduced in section 2. In section 3, the formulation of our Q-TWiST is proposed. The sensitivity analysis of Q-TWiST is in section 4. Finally, a summary and problem in the future are described in section 5.
763
2. Uncertainty Averse Let S be a state space, and T be an algebra which consists of events. A function ip : T —» [0,1] is called probability capacity when the following conditions are satisfied. V # ) = o , ^ ( s ) = i,
(i)
(\M, B € .F)J4 C B => V(^) < V W
(2)
It says that the probability capacity is convex if the following inequality is satisfied. (\/A,Bef),
iP(AuB)
+ i>(ADB)
> rp{A)+^{B).
(3)
Further, it is called probability charge when the equal sign is always satisfied. Let define the core of ip. The core of probability capacity tp is Q{iP) = { p | p e probability charge, (yA e T), ip(A) < p(A) < ip'{A)} , (4) where rl>'(A) = l-,l>(Ac).
(5)
c
The notation A is a complementary set of A. The ip' : T —» [0,1] is called conjugate. The core contains all probability measures that are compatible with the information conveyed by the probability capacity ip and the conjugate ip' 13 . Now, we interpret the state space 5 as a state space which expresses the condition of patient. Let X be a clinical treatment set, and / and g are functions to translate S to X. We call each function an act. A set H consists of close and convex probability measures on (S,T). A notation >indicates the order of preference. If you prefer B more than A, then B x A. For / and g, if
/ x g <=^> min j / u(f(s))p{ds)\p
e H I > min I / u{g(s))p(ds)\p G H I , (6)
then acts / and g is called uncertainty averse. The function u is a utility function. Such preference is called maxmin expected utility (MMEU). It was axiomatized by Gilboa and Schmeidler11. On the other hand, Schmeidler12
764
axiomatizes the order of preference for the probability capacity. For u X —» R, probability charge ip, f > g <==> m i nJ / « ( / ( s ) ) # t e ) j > min i^J u( 5 ( s ))V(d*)| .
(7)
This is called Choquet expected utility(CEU). Especially, if ip is convex, then J u(f(s))i>(ds) = min Uu(f(s))p(ds)\p
£ H\ .
(8)
3. Q-TWiST 3.1. Q-TWiST
Derivation
Q-TWiST is calculated by the mean duration of 3 clinical health states and their utility coefficients. Usually, the utility coefficient for TWiST is assumed to be 1. The utility coefficient for death is 0. Utility coefficients for TOX and REL depend on the individual. If utility coefficients for TOX and REL are /^TOX and ^REL respectively, then Q-TWiST can be calculated by the following equation. QTWiST = fiToxL[TOX]
+ L[TWiST] + »RELL[REL}.
(9)
In general, coefficients /XTOX and //REL are weights between 0 and 1. L[TOX], L[TWiST], and L[REL] in Eq.(9) are the mean duration with being each state, respectively. The outline of this model is as follows. The patient can select one method from n kinds of treatment methods. There are m probability measures to estimate the survival probability. In this subsection, we use a treatment method k (= 1, 2, • • • ,n) and probability measure i (= 1,2, • • • , m). Usual Q-TWiST is defined on a unique probability measure. Let T^t be the length of the TOX period, T$i be the time to relapse, and T ^ be the time to end point. In general, the end point is interpreted as the death of patient. However, it should be noted that it is not all. A function Sji(t) is the survival estimator for S^^t) = P{T^ > t}. By using the estimator Sj^t) for each treatment method, Q-TWiST which is derived in Eq.(9) is rewritten as follows.
QTWiSTi(fc) = /iTOxTOXi(A;) + TWiST^A;) + where
^RELREL^/C),
(10)
765
TOMk)
= f " SiMdt, Jo TWiST^fc) = p {Sl/t) - SlM} RELi(fc) =
MREL p
{SIM
(11) dt,
(12)
~ S£i(t)} dt.
(13)
The upper bound rfc denotes the time to the end of evaluation period. As an estimation method of rjt, a median value at the follow-up study period is used. However, there is criticism to use of the median follow-up time. Shuster14 was pointing out that the definition of the median follow-up time is different in abstracts at ASCO and AACR. Figure 1 shows the survival probability for each health state. Q-TWiST is obtained by multiplying the area of each state by its utility value.
60
120
ISO
240
300
360
420
480
Days
Figure 1.
3.2. Uncertainty Averse
Partitioned survival probability
Q-TWiST
In the previous subsection, we used a unique probability measure i to derive Q-TWiST. Now it is assumed that some appropriate measures appear as a candidate of Q-TWiST. For arbitrary time v t (> 0), a minimum element of the survival probability vector is defined as follows. SM£(0 = Min {S^(t),
S* 2 (0, ••• , S £ m ( 0 } .
(14)
The Q-TWIST value when the uncertainty averse patient selects treatment method k is QTWiST(fc) =
MTOXTOX(A;)
+ TWiST(A;) + ^RELREL(A;),
(15)
766
where
TOX(fc)= / Jo
SMf(t)dt,
.
TWiST(fc) = J " J5M2fc(i) - SM?(t)} dt, REL(k)=
|^M 3 f c (i)-5M 2 f e (t)}df.
(16)
(17) (18)
Therefore, the best treatment for the uncertainty averse patient, k*, is k* = {k; Max QTWiST(fc), k = 1,2,-- • ,«} . 3.3. Sensitivity
(19)
Analysis
A sensitivity analysis for Q-TWiST is useful to select a treatment method. It assumes that the difference between QTWiST(A) and QTWiST(B) is G(A,B), i.e. G{A, B) = QTWiST(>4) - QTWiST(B).
(20)
If G(A, B) takes a positive value then the treatment A is preferred (avoided) than the treatment B, i.e. G(A,B) > (<)0 <=> A >- (~<)B. The relation G(A, B) — 0 is rewritten asiollows. _ REL(g) - KEL(A) MTOX- TOx(A)-TOX(B)MREL
TWiST(i4) - TWiST(ff) TOX(y4)-TOX(S) '
(
'
Eq.(21) can be considered as a threshold line which decides the recommended treatment method. If a plot (//REL,/^TOX) is in the left hand side of Eq.(21) then the treatment A should be selected. On the other hand, if a plot (/iREL,MTOX) is in the right hand side of Eq.(21) then the treatment B should be selected. (Refer to Figure 2.) 4. Conclusion We proposed the basic theoretical formulation of Q-TWIST for the uncertainty averse patient. The sensitivity analysis for Q-TWiST is introduced. Usually, the utility value /i-rox and ^REL are estimated from QOL information. However, the case, where enough QOL data cannot be collected happens frequently. It is necessary to devise the estimation method of appropriate weight when QOL information is not enough.
767
_ 8KI.;»; - HKl.l/tt ! / CIVMC - TOX.,,:, -.T
/
A » B /
/ * -< B
Figure 2. Sensitivity analysis with a threshold line
References 1. World Health Organization, BASIC TEXTS Forty-Fourth Edition, WORLD HEALTH ORGANIZATION, (2004). 2. World Health Organization, WHOQOL Measuring Quality of Life, WORLD HEALTH ORGANIZATION, (1997). 3. http://www.sf-36.org/ 4. R.D. Gelber and A. Goldhirsch, A New Endpoint for the Assessment of Adjuvant Therapy in Postmenopausal Women with Operable Breast Cancer, Journal of Clinical Oncology, 4, 1772-1779, (1986). 5. P. P. Glasziou, B. F. Cole, R. D. Gelber, et at, Quality Adjusted Survival Analysis with Repeated Quality of Life Measures, Statistics in Medicine, 17, 1215-1229, (1998). 6. A. Goldhirsch, R. Gelber, R. Simes, et al., Costs and Benefits of Adjuvant Therapy in Breast Cancer: A Quality-Adjusted Survival Analysis, Journal of Clinical Oncology, 7, 36-44, (1989). 7. M.A. Nooija, J.C. J.M. de Haesb, L.V.A.M. Beexc, et al. for the EORTC Breast Cancer Group, Continuing Chemotherapy or not after the Induction Treatment in Advanced Breast Cancer Patients: Clinical Outcomes and Oncologists ' Preferences, European Journal of Cancer, 39, 612-621, (2003). 8. J.L. Savage, The Foundations of Statistics, John Wiley, New York, (1954). 9. D. Ellsberg, Risk, Ambiguity, and the Savage Axioms, Quarterly Journal of Economics, 75, 643-669, (1961). 10. I. Gilboa, Expected Utility with Purely Subjective Non-Additive Probabilities, Journal of Mathematical Economics, 16, 65-88, (1987). 11. I. Gilboa and D. Schmeidler, Maxmin Expected Utility with Non-Unique Prior, Journal of Mathematical Economics, 18, 141-153, (1989). 12. D. Schmeidler, Subjective Probability and Expected Utility without Additivity, Econometrica, 57, 571-587, (1989). 13. K.C. LO, Correlated Equilibrium under Uncertainty, Mathematical Social Sciences, 44, 183-209, (2002). 14. J.J. Shuster, Median Follow-Up in Clinical Trials, Journal of Clinical Oncology, 9, 191-192, (1989).
THE AUTOCORRELATED GWMA CONTROL CHART
SHEY-HUEI SHEU Department of Industrial Management, National Taiwan University of Science and Tchnology, 43, Keelung Road, Section 4, Taipei 106, Taiwan
SHIN-LI LU Department of Industrial Management, National Taiwan University of Science and Technology, 43, Keelung Road, Section 4, Taipei 106, Taiwan
Traditionally, using a control chart to monitor a process assumes that process observations are normally and independently distributed. In fact, for many processes, products are either connected or autocorrelated and, consequently, obtained observations are autocorrelative rather than independent. In this scenario, applying an independence assumption instead of the autocorrelation for process monitoring is unsuitable. This study examines a generally weighted moving average (GWMA) with a time-varying control chart for monitoring the mean of a process based on autocorrelated observations from a first-order autoregressive process (AR( 1)) with random error.
1. Introduction Statistical Process Control (SPC) is a significant quality control issue in which data analysis is employed to determine whether the process is under control. One primary aim of SPC is to detect instantly a process shift and adopt the necessary corrective actions to improve process quality. Control charts are widely applied tools for monitoring processes. How to best select and apply a control chart, therefore, is a critical task. Alwan and Reoberts [1] applied an appropriate time series model to illustrate autocorrelated processes and used control charts to monitor forecast errors of the process. Montgomery and Mastrangelo [2] proposed a method which fitted a proper time series model for original data and then constructed the exponentially weighted moving average (EWMA) control chart for the residuals. Thombs and Padgett [3] studied Shewhart charts when process observations can be modeled as an AR(1) process with random error. Wardell, Moskowitz and Plante [4] considered four charts in the presence of data correlation and 768
769 compared the performance of the four charts when the process can be described as an autoregressive moving average process of order 1 and 1 (ARMA (1,1)).. Lu and Reynolds [5] considered the performance of EWMA charts of the residuals and observations. Studies have demonstrated that the EWMA chart of observations is more efficient than the Shewhart observational chart in detecting small shifts of autocorrelated processes. Sheu and Griffith and Sheu and Lin [6] proposed and applied an expanded EWMA control chart, called the generally weighted moving average (GWMA), to enhance the detection ability of control charts. They indicated that the GWMA control chart performs substantially better than both the Shewhart and EWMA control charts for monitoring small process mean shifts. This work focuses on the GWMA control chart of autocorrelated observations to monitor the process mean. The GWMA control chart performance is compared with that of the EWMA control chart. Numerical simulation is used to assess the ARL properties of various process mean shifts and adjusted parameters with different levels of autocorrelation. 2. The AR (1) Process with Random Error The AR(1) process with random error is presented. It is assumed that Xt can be written as Xt=nt+s, ,t = 1,2,3 (1) where nt is random process mean at time t and £t 's are independent and have a common normal distribution with mean 0 and variance a2 . Supposing that fxt be specialized as anAR(l) process with process mean £ 0 ,thus, /*/=(!-)£o + M-\ +a, ,t = 1,2,3 (2) where a, is the white noise of the AR(1) process at time t and 0 is the AR(1) parameter satisfying \<j>\ < 1. We may assume that a, 's are independent normal random variables with mean 0 and variance a2, independent of the s, 's. If we suppose that the starting value ju0 follows a normal distribution with mean £0 and variance cr2/(l-02), then ju, will be a normal distribution with mean £ 0 and variance alM = — a —, \-tt>2
for all t > 1
hence, the mean and variance of process observations X, 's respectively are:
Var(Xl) = a2x=*l
+
a2£=^-I
+
<j2
770
One can conveniently define y/ = —j- =
*"
to be the proportion of the
process variance that is due to the AR(1) process. The covariance between Xt and Xl+l is tjxr^, the autocorrelation coefficient between Xt and Xt+l is P = 4>¥ •
The AR(1) process observed with random error is equivalent to an ARMA( 1,1) process (see Box, Jenkins, and Reinsel [7] ). The ARMA(1,1) process can be represented as *,=(l-0£o+#rr-i+*,-<*,-! (3) where b, is the random white noise of the ARMA( 1,1) process at time t and follows independent normal random variables with mean 0 and variance a\, £0 is the mean of the observation, 9 is the MA parameter, and ^ is the same AR parameter as in Eq. (2). Most applications, it is interpretable based on the AR(1) with random error; however, for some purposes, the ARMA(1,1) model can be convenient for acquiring original observations and estimating parameters. Hence, as long as parameters satisfy Q<9<<j><\ and
771 = (P0 -P,)Xj
+
{Px -P2)Xj_l+-
+ {Pj_l-PJ)Xl +PjMo
(4)
The expected value of Eq. (4) is E(Yj) =E[(P0-Pl)Xj+(P]-P2)Xj_i+-
+ (PJ_l-PJ)Xl+PJM0]
=#,(5)
The variance of Eq. (4) is Var(Yj) = f(P0 -j>) 2 + (/> -P2)2 + - + {PH -Pj)2]-a2
(6)
Supposing L denotes the width of the control limits, the central limit (CL ), upper control limit (UCL ) and lower control limit (LCL ) of the GWMA control chart can be written as UCL^ft+LjVariYj) CL = ^ LCL^/Jo-LjVariYj)
(7)
When statistics Y • falls outside the range of control limits, it indicates that the process is out of control and some actions should be taken adequately. 4. Constructing Autocorrelated GWMA Control Charts Generally, process data are assumed to be independent in conventional control charts. This work, investigates GWMA control charts for monitoring the mean of autocorrelated observations based on an AR(1) process with random error. Notably, it has the property of a time series AR(1) and independent data. Therefore, we propose that the data from an AR(1) process with random error are close to practical autocorrelated processes. According to Eqs. (1) and (4), the autocorrelated GWMA control statistic, Yj can be written as Yj =tp(M
= j-i +1)(//,. + e.) + P(M >j)Z0
(8)
The expected value of Eq. (8) will be E(Yj) = fp(M
= j-i + l)E(fi, + ei) + P(M > j)E(40) =
(9)
1=1
The variance of Eq. (8) will be Var(Yj) = Var(tP(M
= j-i +1)(//,. + *.) + P(M > j)^)
;=i
= Qj (*l +
+ \)P{M =j-k
+l ^ a l
(10)
772 J
-
where Qj = J^[P(M = i)] .The time-varying control limits of the (=1
autocorrelated GWMA control chart will be UCL = £0+LjVar(YJ)
LCL = 40-L^Var(Xj)
(11)
Particularly, we choose Pj = qJ where design parameter q is constant (0
02)
(=1
The variance of Eq. (12) will be Var(Zj)=Qjal+2
P2 M-
^ L2 _ [ g Z g - - « ^ 1Y JH + T ^ ^ < 13 ) \-q 1-& l-^" \+q
where p = l - g . W h e n 7 increases to 00, Var(Zj) will increase to a limit value, VariZu)=-P-£4-o>+ol) \ + q \-
(14)
The fixed-width control limits of the EWMA control chart of autocorrelated observations will be UCL = <^0+Laz CL = {0
(16)
LUL = gQ-L(jz Hence, the EWMA control chart becomes a special case of the GWMA control chart of autocorrelated observations when a = 1. 5. Performance Measurement and Comparison Average Run Length simply is the average number of points plotted before an out-of-control signal is observed. Generally, control chart performance is measured by the ARL. An ARL should be sufficiently large to reduce false alarms when a process is under control and adequately small to detect shifts
773
rapidly when a process is out of control. Numerical analyses and computer simulations in this work are used to estimate ARL of a control chart, (see Roberts [8] and Crowder [9]). The following steps are recommended: Stepl. Give parameters
774 Table 1 .ARLs of GWMA charts with time-varying control limits when (// = 0 . 9
+ -= 0.4 5
* = 0.8
a = 0.5
q==0.5 a = 0.9 a = 0.7
a = 1.0
a = 0.5
a = 0.7
q=0.5 a = 0.9
a = 1.0
L=2.950
L=2.934
L=2.928
L=2.926
L=2.754
L=2.744
L=2.747
L=2.750
0.00
370.65
369.82
370.32
370.30
370.46
370.58
370.55
370.87
0.25
217.11
227.56
239.87
243.42
279.09
287.53
292.91
294.11
0.50
89.51
97.32
106.60
110.31
156.58
167.11
174.08
175.92
0.75
42.80
46.24
50.63
52.48
87.30
94.87
99.45
101.30
1.00
23.37
24.64
26.90
28.11
51.71
56.20
59.67
61.00
1.25
14.44
14.68
15.66
16.15
32.44
34.93
37.18
37.92
1.50
9.54
9.46
9.89
10.19
21.33
22.65
23.99
24.45
2.00
4.96
4.80
4.83
4.88
9.54
9.91
10.38
10.60
3.00
1.93
1.88
1.85
1.85
2.12
2.11
2.14
2.16
q=i0.9
a = 0.5
or = 0.7
a = 0.9
a = 1.0
a = 0.5
q=0.9 a = 0.7 a = 0.9
L=2.778
L=2.659
L=2.616
L=2.613
L=2.439
L=2.353
0.00
370.55
370.27
370.25
370.36
369.83
370.28
370.57
370.03
0.25
115.50
117.92
136.51
147.65
207.97
222.29
243.59
253.84
0.50
44.48
42.75
46.90
50.52
96.80
102.90
118.60
127.45
0.75
23.49
21.88
22.79
23.89
52.92
54.71
62.29
67.08
1.00
14.43
13.14
13.39
13.81
32.27
32.52
36.56
39.31
1.25
9.61
8.73
8.75
8.93
20.87
20.61
22.63
24.13
8
L=2.365
a = 1.0 L=2.384
1.50
6.84
6.15
6.11
6.21
13.82
13.45
14.61
15.50
2.00
3.87
3.50
3.43
3.46
6.18
5.84
6.21
6.55
3.00
1.68
1.56
1.53
1.53
1.51
1.44
1.44
1.48
6. Conclusions In sophisticated industries with automatic process development, quality characteristics in many applications no longer meet a standard independent assumption.Lu and Reynolds [5] argued that the EWMA control chart of autocorrelated observations is more efficient than Shewhart control chart in detecting small process mean shifts. This work considered a GWMA control chart for monitoring the process mean in which the observations can be modeled as an AR(1) process with random error. This study utilized a simulation to acquire the ARL of the autocorrelated GWMA control chart. The autocorrelated GWMA control chart proved superior to the autocorrelated EWMA control chart for detecting small process mean shifts at low levels of autocorrelation. However, for high levels of autocorrelation, the autocorrelated GWMA control chart also
775
performs well at the shifted size within 3a, especially in small process mean shifts. The GWMA control chart of autocorrelated observations reduces the type II errors when the process mean is out of control and parameter a < 1. Hence, when the process is autocorrelated, the GWMA control chart is superior for detecting small process mean shifts. References 1. L. C. Alwan and H. V. Roberts, Time-Series Modeling for Statistical Process Control, Journal of Business and Economic Statistics, 6, 87-95. (1998). 2. D. C. Montgomery and C. M. Mastrangelo, Some Statistical Process Control Methods for Autocorrelation Data, Journal of Quality Technology 23,179-193(1991) 3. C. S. Padgett, L. A. Thombs and W. J. Padgett, On the a -risk for Shewhart Control Charts, Communications in Statistics-Simulation and Computation 21,1125-1147.(1992). 4. D. G Wardell, H. Moskowitz and R. D. Plante, Control Charts in the Presence of Data Correlation, Management Science 38, 1084-1105. (1992). 5. C. W. Lu and M. R. Reynolds Jr., EWMA Control Charts for Monitoring the Mean of Autocorrelated Processes, Journal of Quality Technology 31, 166-188 (1999a). 6. S. H. Sheu and T. C. Lin, The Generally Weighted Moving Average Control Chart for Detecting Small Shifts in the Process Mean, Quality Engineering 16,209-231.(2003). 7. G E. P. Box, G. M. Jenkins and G. C. Reinsel, Time Series Analysis: Forecasting and Control, 3rd, Prentice-Hall, Englewood Cliffs, New Jersey, (1994) 8. S. W. Roberts, Control Chart Tests Based on Geometric Moving Average, Technometrics 42, 97-102. (1959). 9. S.V. Crowder, Design of Exponentially Weighted Moving Average Schemes, Journal of Quality Technology, 21, 155-161. (1989).
AN EXTENDED EXPONENTIALLT WEIGHTED MOVING AVERAGE CONTROL CHART FOR MONITORING POISSON OBSERVATIONS SHEY-HUEI SHEU Department of Industrial Management, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei 106, Taiwan TSE-CHIEH LIN, SHIH-HUNG TAI Department of Industrial Management, Lunghwa University of Science and Technology, 300 Wan-Shou Road, Section I,Kueishan, 33306 Taoyuan, Taiwan
The c chart is often used to monitor the number of non-conforming products. When small shifts in the nonconformities of process mean result from assignable causes, classical c charts are relatively inefficient in detecting small shifts in the nonconformities of process mean. Therefore, the Poisson exponentially weighted moving average (Poisson EWMA) control scheme is another superior alternative to c charts. In this paper, we extended the Poisson EWMA control chart. This generalized chart, is called herein the Poisson generally weighted moving average (Poisson GWMA) control chart. This study presents the Poisson GWMA control chart for monitoring the Poisson counts. Simulation is used herein to evaluate the average run length (ARL) properties of the c chart, the Poisson EWMA control chart and the Poisson GWMA control chart. The Poisson GWMA control chart is superior to the Poisson EWMA control chart as measured by ARL. An example is also given to illustrate this study.
1. Introduction Attribute data are based on counts, or the number of times a particular event is observed. The events may be the number of nonconforming items, the number of defects, or any other distinct occurrences that are operationally defined, Including for example, defects per printed circuit board, the number of blemishes per tire and the number of accidents per month in a plant. Such counts are often well fitted by a Poisson distribution. In many manufacturing and business processes, it is often important to monitor various counts that can be modeled using Poisson random variables (see, e.g., Refs, 1-3). The control limits and performance measures for the c
776
777 chart is based on the Poisson distribution. The c chart is perhaps the simplest type of control chart for monitoring the Poisson counts. When small shifts in the nonconformities of process mean result from assignable causes, classical c charts are relatively inefficient in detecting small shifts in the nonconformities of process mean. Therefore, the Poisson-EWMA control scheme is another superior alternative to c charts. Gan (4) introduced a modified version of the EWMA control chart, which monitors the mean of the Poisson distribution. In Gan's method, no information is lost because the exact EWMA statistic is maintained and plotted. Borror, Champ and Rigdon (5) evaluated the average run lengths (ARLs) of the Poisson EWMA chart for various in-control means, using the Markov chain approach and simulation. The Poisson EWMA control chart has smaller ARL than the c-chart. Sheu and Lin (6) introduced the generally weighted moving average (GWMA) control charts to enhance the detection ability of control charts. The GWMA control chart is more sensitive in detecting small shifts in the process mean, and can spot small shifts in the initial process, due to its added adjustment parameter a . In this study, we propose the GWMA chart to monitor the mean of a Poisson distribution. It is called the Poisson GWMA control chart. Simulation is performed to evaluate the ARL properties of the Poisson GWMA control chart. 2. The Poisson GWMA Control Chart For a sequence of independent samples, let q. represent the probability of the occurrence of event A at the j-th sample and 1 — q . represent the probability of the occurrence of event B at the j-th sample. Events A and B are assumed to be complementary and mutually exclusive. Let M count the number of samples until the first occurrence of event A since the previous occurrence of event A. L e t P ; = P(M > j) . That is, Pj is the probability that the event A does not occur in the first j samples. We_assume throughout that the domain of Pj is {0,1,2, •••} and that 1 = P o > P i > ••• . We use the notation {Pj} as an abbreviation for ^sequence of probabilities. The sequence {Pj } is supposed to be known. Let Pj = P(M = j) = Py_, - Pj = PH (1 -
Zi.)
(Note: assumes probabilities > 0 for all j , which may not be the case if probability distribution changes over time for unstable processes.).
778
Hence, event A occurs with probability q • = 1 - _ J J Pj-l occurs with probability i _ q. = E' '
at the ;'-th sample. Event B
at they'-th sample.
Pj-\
Therefore, 00
m=\
= P(M = \) + P{M = !) + ••• + P(M =j) + P(M>
= {P0-Pl) =1
+ (Pl-P2)
+ - + (Pj-l-Pj)
j)
(!)
+ Pj
If (Po - P i ) > (Pi -Pi) > •••> (Pj-i -Pj), then the current sample is weighted most heavily, and the preceding samples are assigned lower weightings; the earliest sample is weighted least. P(M = j) can be regarded as the weight in the weighted moving average. The weight of the current sample is P(M = 1) and the previous observation has a weight of P(M = 2) . Accordingly, P(M > j) is weighted c 0 , where C0 is the target mean. Hence, all of the samples are considered, but the current sample is weighted most heavily. The weights decrease with the age of the samples. Let Y. denote the generally weighted moving average in the plotted test statistics at time j {j = 1,2,3, •••) and C represents the observation at the time j . C •, j — 1,2,3, • • • are assumed to be independent identically distributed Poisson random variables with mean c, where c is the mean number of defects per inspection unit. When the process is in control, C = C 0 , where C0 is the target value of mean number of defects per inspection unit. The Y0 represents the starting value that is set by the practitioner. Setting Y0 = C0 is convenient. For the Poisson GWMA control chart the sample statistic is a weighted average of the current observation C . and all previous observations, with the current observation most heavily weighted. Consequently, Y, can be configured as, Y = P ( M = 1)C, + i W = 2 ) C , . _ , +-+P(M=j)C] = (P0 -Pi)Cj +(/>, -Pi)CH +-+{PJA -Pj)Cx
+P(M>j)c0 +Pjc0
(2)
779
LetPj =qJ .where design parameter q is constant (0 < q < 1) , (j = 0,1,2,3, • • •) and a is adjustment parameter determined by the practitioner. Thus, the expected value of Eq. (2) can then be calculated as, E(YJ)=E[(q0" -/)Cj
+ ( / -qT)CH
+-+{q(Hf
~qfY\ + / c 0 ]
p )
The variance is, Var(Yj) = [(qoa -q[°f
+ ( / -q2")2 + ... + {q(Hf
-qf )2}Var{C)
= [(<7°" - / ) 2 + ( ? r -q2")2+... + (qU-,r ~qf)2]c0
(4)
where, Q} = (9°° - qx° f + (ql" - q2" )2 + • • • + (q^" - qJ" )2 . Let L denotes the multiplier to determine the width of the control limits, the control limits can be written as,
CL =c 0
(5)
LCL = c0-L^JQjCQ If Cj , j — 1,2,3, • • • are independent identically distributed Poisson random variables, then the Poisson GWMA statistic given in Eq. (2) will be nonnegative and the Poisson GWMA chart will not signal at a LCL that is less than or equal to zero. Thus, if the computed value of LCL is less than zero, then we set LCL=0. The Poisson GWMA statistics are then plotted on a control chart with the UCL and LCL, and if any point exceeds the control limits, then the Poisson GWMA chart shows an out-of-control signal. 3. Comparison of control charts The ARL is a useful measure for the design of control charts. The ARL is defined as the average number of points plotted before an out-of-control signal is given. When the process is under control, the ARL should be sufficiently large to avoid false alarms.
780 When the process is out of control, the ARL should be sufficiently small to rapidly detect shifts. However, the ARL can be obtained through numerical analysis and computer simulation (see, e.g., Refs, 7-10). The process is said to be in control when C = c0 and out of control when the mean shifts to some other value, say, C = C, . The early detection of a shift to C, > C0 is desirable and so adjustments can be made to bring the process mean back in control. The early detection of a shift to C, < C0 is desirable to allow any beneficial changes to be made. A downward shift in C may be caused by errors in the inspection of the units, rather than by an actual change in the process. For example, a new inspector may not have been trained properly to inspect the process output. In such a case, quick detection is desirable. Simulations under the following conditions were conducted to evaluate the ability of the Poisson GWMA chart to detect shifts. First, the in-control ARL is maintained at approximately 1234, and second, the in-control ARL is maintained at approximately 1384. First, the in-control ARL is maintained at approximately 1234. The top of Table 1 compares the ARLs of the Shewhart c chart, of Gan (4) Ceil EWMA control charts (CEWMA), of Borror, Champ and Rigdon (5) Poisson EWMA control charts (PEWMA), and of Poisson GWMA control charts (PGWMA) proposed herein. The lower and upper control limits (LCL, UCL) of the Shewhart c, CEWMA (A =0.27), PEWMA (X =0.27) control charts are given by (5, 35), (13,27) and (25.864, 14.136), and the in-control ARLs are 1218.6, 1234.5 and 1233.4, respectively. The in-control process mean c 0 is 20. For a fair comparison with CEWMA and PEWMA chart, in-control ARL of Poisson GWMA control chart with design parameter q=0.73 and different adjustment parameters a {a =1.00, a =0.90, a =0.80, a =0.75, and a =0.50) is maintained at approximately 1234 by changing the width of the control limitsL (Z=3.320,1=3.331,1=3.344,1=3.352, andi=3.398). Table 1 presents the ARLs for various values of c when the in-control process mean is ° =20. The Poisson GWMA control charts are superior to the Shewhart c chart based on ARL. For most of process mean shifts, the Poisson GWMA control charts with q=0.73 yield lower ARLs than the Ceil EWMA control charts and the Poisson EWMA control charts.
781 Table 1. Comparison of the ARLs of the Poisson GWMA Control Charts and Other Control Charts Shewart Gan's Borror et al's c-chart CEWMA PEWMA .parameters °
X=\
X =0.27
A =0.27
34.2 131.6 553.7 2426.2 4876.2 1218.6 266.1 76.0 27.5 12.2 6.4
7.9 18.9 138.5 5566.2 89688.2 1234.5 78.8 18.1 8.4 5.4 4.0
3.4 4.8 8.5 24.2 197.2 1233.4 85.2 17.7 7.8 4.8 3.5
\
10 12 14 16 18 20* 22 24 26 28 30
Poisson GWMA 9=0.73 a =1.00 a =0.90 a =0.80 a =0.75 #=0.50 £=3.320 £=3.331 1=3.344 £=3.352 £=3.398 2.8 2.9 3.0 3.7 3.1 4.4 4.3 4.5 5.8 4.7 8.0 8.0 8.1 10.5 8.3 22.1 21.2 21.1 25.4 23.8 198.9 176.4 157.6 149.7 138.4 1233.4 1233.3 1233.5 1233.5 1233.3 83.9 76.9 71.0 68.4 64.4 17.0 16.3 15.8 15.7 17.5 7.1 7.1 7.1 7.2 8.3 4.2 4.2 4.3 4.3 5.0 2.9 2.9 3.0 3.4 3.0
* The in-control process mean is 20 (Cg = 20) . The actual process mean is c.
4. Conclusions This paper presents a way to extended the exponentially weighted moving average control chart to the case of Poisson data for fraction nonconforming. The GWMA control charts proposed for monitoring the Poisson counts are superior to Shewhart c charts in terms of ARL considerations. For most shifts in the process mean, Poisson GWMA control charts have shorter ARLs than modified EWMA control charts and Poisson EWMA control charts. Due to its superiority in detecting shifts, the Poisson GWMA control chart for fraction non-conforming should be given high priority by quality control practitioners. References 1. A. J. Duncan, Quality Control and Industrial Statistics, 5th ed., Irwin, Homewood Illinois, (1986). 2. D. C. Montgomery, Introduction to Statistical Quality Control, 4th ed., John Wiley & Sons, New York, (2001). 3. H. Gitlow, A. Oppenheim and R. Oppenheim, Quality Management: Tools and
782
4.
5. 6. 7. 8.
9. 10. 11.
Methods for Improvement, 2nd ed., Irwin, Burr Ridge, Illinois, (1995). F. F. Gan, Monitoring Poisson Observations Using Modified Exponentially Weighted Moving Average Control Charts, Communications in Statistics-Simulation and Computation 19, 103-124(1990). C. M. Borror, C. W. Champ and S. E. Rigdon, Poisson EWMA Control Charts, J. Qual. Technol. 30, 352-361 (1998). S. H. Sheu and T. C. Lin, The Generally Weighted Moving Average Control Chart for Detecting Small Shifts in the Process Mean, Qual. Eng., 16(2), (2003). S. W. Roberts, Control Chart Tests Based on Geometric Moving Averages, Technometrics 1, 239-250 (1959). S. V. Crowder, A Simple Method for Studying Run Length Distributions of Exponentially Weighted Moving Average Control Charts, Technometrics 29, 401-407(1987). P. B. Robinson and T. Y. Ho, Average Run Lengths of Geometric Moving Average Charts by Numerical Methods, Technometrics 20, 85-93 (1978). C. H. White and J. B. Keats, ARLs and Higher-Order Run-Length Moments for the Poisson CUSUM, J. Qual. Technol. 28, 363-369 (1996). IMSL., User's Manual Statistical Library, Version 1.1, IMSL Inc., Houston, TX, (1989).
BUILD UP THE PRODUCT SATISFACTION PERFORMANCE MEASUREMENT BY USING 6-SIGMA METHODOLOGY AND S/N RATIO APPROACH - AN EXAMPLE OF PDA C. J. TAO Department of Industrial Engineering & Management, National Chin-Yi Institute of Technology Taichung, Taiwan, R.O.C S. C. CHEN 12
Department of Industrial Engineering & Management, National Chin-Yi Institute of Technology Taichung, Taiwan, R.O.C H. C. TSUNG
Department of electric machinery, National Chin-Yi Institute of Taichung, Taiwan, R.O.C
Technology
In the original, there is only personal information management function build in the personal digital assistant (PDA). But under the circumstance that by the importance, PDA is less than mobile phone and by function it is not as good as Notebook PC, it's product features are being overlooked, if we could clarify the factors that will affect the satisfaction and importance in the promotion course that would be helpful for the spreading of product and the market development. In order to know the satisfactory and respective on the product feature, brand name, service conveniences of PDA as to clarify the factor that will affect the purchasing desire of PDA so that might help for the IA promotion in the future. Therefore, this article will take the example of PDA and utilize the DMAIC methodology of 6-Sigma to build up the measurement and improvement model to promote the customer overall satisfaction performance. At first, we define the questions and measurement model and then using the questionnaire to measure the performance of customer satisfaction and importance), and then construct the satisfaction performance metrics and overall satisfaction performance control chart thus be used as the analyzing tool to find out the key improvement items and review items, and then, using the cause effect diagram to make analysis and find out the corrective action, then implementing these corrective action in the critical review and improvement items. In the final, focus on the critical and improvement items re-do the questionnaire survey and build up the overall satisfaction performance control chart to control and sustain the execution result of the relevant corrective actions. Through the comprehensive measurement and improvement model of this article, the enterprise could quickly and effectively measure, analyze, improve and control their service quality and then under the reasonable cost condition to effectively promote overall customer satisfaction
783
784 performance to create high value-added quality competitiveness and effectively enhance the profit gaining capability of enterprise. This is where the abstract should be placed. It should consist of one paragraph giving a concise summary of the material in the article below. Replace the title, authors, and addresses with your own title, authors, and addresses. You may have as many authors and addresses as you like. It is preferable not to use footnotes in the abstract or the title; the acknowledgments of funding bodies etc. are to be placed in a separate section at the end of the text.
1. Introduction After the coming of so called Post PC Era of personal computer industry which emphasizing the compact and easy-to-use product has become the new cosset product in the market and caused every company are thinking over of transfer and involve into the R&D of IA (Information Appliance). The IA product is not only implementing the concept of 4C: Computer, Communication, Consumer and Contents into the Internet networking, but also possess the convenience of portability. In view of the relevant research on new product in the past, some topics are on the innovation proliferation and few are focusing on researching the consumer evaluation to the product feature attributes. Therefore this article will take the example of PDA to applying the DMAIC methodology that described in 6-Sigma deviation of Michael (2002) to promote the overall customer satisfaction performance. At first, with reference to the theses of Dabholker et al. (1966) and Parasuraman et al. (1985, 1991), designing the relevant questionnaire on the PDA. There are total 5 quality items: Function, Specification, System, Service Convenience and Brand Name; by that we deployed out 30 quality feature items and use these as the questions. And then use these 30 questions to carry out the performance measurement of satisfaction and importance performance, and then build up the performance matrix and overall satisfaction performance control chart as the tool of analysis, so that we can find out the key items for review and improvement. And then deploy these items by using the cause-effect diagram to analyze and find out the corrective action for improvement. At the last, focus on the key item of review and improvement to re-design the questionnaire and redo the survey measurement and then set up the overall satisfaction performance control chart to control and maintain the executing effectiveness of the relevant corrective actions. By the comprehensive measurement and improvement model of this thesis, the enterprise of the PDA could focus on the quality items and under the consideration of time and cost, to do an effective and quick measurement and in accordance with the overall customer satisfaction performance value to find out the quality items that need review cost and improve the satisfaction. Then under the reasonable cost condition to promote the overall customer satisfaction
785 effectively and create high value added quality competitiveness and the capability of gaining profit. 2. The Definition of Service Quality Measurement Model The definition of service quality by the view point of Dabholker et al. (1966) and Parasuraman, A. et al. (1985, 1991), In the following, we will in reference to the concept of Hung et al. (2003), Lin et al. (1005) and Chen et al. (2005, 2006), focus on the expectation importance of customer for every quality item and the perception satisfactory level after usage to define an index, the relation formula can be shown as below respectively: m i nn
jUj -
Pl =
Ps =
R jus - min
R
(1)
(2)
Where, Ps represents Satisfactory index value, Pt represents Importance index value, us is for average value of satisfactory, and fij is for average value of importance, min = 1 represents the minimum value of k scale table, R = k - 1 represents the full range of k scale table. These two index value both are set between (0, 1). Because the index value of Pi and P s can not reflect the variation level of questionnaire so that in this article we utilize the S/N concept of Taguchi (1989) to modify the index value and by comparing them with P, and P s to for finding the variation level of index value to interpret the meaning of questionnaire contents and find out the more properly corrective actions, the relation formula as shown as:
SNi = A . s
(3)
PI
SNS= A .
(4)
In the following we will modify the Performance Evaluation Matrix proposed by Hung et al. (2003) ,Lin et al. (1005) and Chen et al. (2005, 2006) and use it as the evaluation tool.. This performance evaluation matrix can express both the importance and satisfactory of every quality service item Due to the above coordinates can not be accurately and objectively judge how many items need to be improved or reviewed, therefore, in the following we will use the difference of the coordinate value of quality importance minus the
786 coordinate value of satisfaction as the overall satisfactory index value, the relation formula is as shown in the following: SN,.S = SN, - SNS
(5)
SNi represents the importance performance index, SNS represents the satisfaction performance index and SNi_s represents the overall satisfactory performance index, their values are within the range of -1 to +1. When the overall satisfactory performance value SNi_s is positive, means need to improve the satisfaction thus need to increase the resource to promote the satisfaction to proper block; similarly, when the overall satisfactory performance value SNi.s is negative, means need to review if need to decrease the resource investment to lower the cost. 3. The Definition, Measurement and Analysis of PDA's Quality Satisfaction In accordance to the above mentioned evaluation model, this article will investigate the importance and satisfaction of the 5 quality compositions, and these 5 quality compositions are: "function", "specification", "system", "brand name" and "service convenience". Then we deploying these 5 quality compositions into 30 question features and use them as the question items of the questionnaire and focus on the Taiwan area survey the customer in the information industry market. In total, we released 100 copies of questionnaire after retrieval and arrangement, deducting the general public that did not purchase the PDA commodity and ineffective 37 copies that incomplete and with same answer for all questions, we totally collected 63 copies effective questionnaire. The effective retrieval rate is 63%. In the following, we will carry out the confidence analysis on the questionnaire. The most used internal confidence level is Cronbach's alpha coefficient. In according to the viewpoint of Gary (1992) if the confidence coefficient is greater than 0.90, it means this scale has an excellent confidence level; and a scale with the confidence coefficient greater than 0.80 is acceptable. But there is no consistent standard among the scholars on the confidence level. The scholars such like as DeVellis (1991) and Nunnally (1978) are considering the minimum acceptable coefficient of confidence level is 0.70. After the analyzing of statistic software the average Cronbach's a value of the importance of the 5 quality compositions of PDA is over 0.7487, that is, for the question item of this questionnaire to the person for answering the importance of that question there is over 74.87% confidence level that his answer is consistent, the Cronbach's a value of the other compositions
787
are all over 0.7134, In summary of the above analysis, as to the importance and satisfaction of this questionnaire the confidence level is acceptable. On the analysis respect of importance and satisfaction of product and service quality of PDA, at first we make statistics on every quality question to calculate the importance value P! and satisfaction value P s , then in accordance with the above formula to get the importance performance value SN b satisfactory performance value SNi.s, and at last calculate the quartile Ql, Q2, Q3 and standard deviation value of the Pi and SNr value of importance and the P s and SNS value of satisfaction, as well as their 3 coordinates, by using these coordinates to draw out the box-and-whisker plot and fill the data into table 1. Table 1. Quality satisfactory and importance performance table SNS
SN,.S
27.105 7.75
14.5936 9.286872
-1.53687
8.75038 14.30955 11.61502
10.2178 3.080622
5. Touch panel
18.9682 17.39017 24.0002
26. Service convenience 27. Brand name reputation 28. Payment flexibility
12.3044
13.57105 16.90483
-1.26665
10.84859 5.564523
29. Passion of serviceman
22.74527
30. Professional Knowledge
24.9998 13.97972
11.84981 11.22332 11.67372
13.7765 -1.49202
18.03053
15.24178
2.788745
22.55131
17.83259
Measured question
sitionS~~-^
Function
Service
1. Easy to control 2. Stock News 3. Entertainment 4. Image processing
1" quartile Q, 2,Ki quartile Q2 3'd quartile Q3 Standard Deviation
SN,
12.33785
12.5114
12.3852
-6.05624 -6.77332 10.8955
9.68628 8.085816
Due to the evaluation matrix is established by the quality performance value P, and P s , and P[ and P s value is the performance values that established by the average and target value which only considering the shift and not considering the deviation, thus causes the coordinates of every quality feature are too cumulative and can not accurately find out the quality items that need to be reviewed and improved. Therefore, this article will applying the SN ratio concept of Taguchi (1989) to add the parameter of standard deviation to set up
788
the evaluation index value of SNi and SNS and use these indices value to set up the performance matrix, and using the 3 coordinates of the quartile values Ql, Q2, Q3 of importance SN[ and satisfactory value SNS to draw the box of boxand-whisker plot,; and then using the minimum and maximum coordinate as the whisker of the box and use it to measure the variation degree deviation of importance and satisfaction and judging the bad or good of the overall satisfactory level by the block that box-and-whisker plot locates, as shown in figure 1: 70
•
1
• %
»
B
&
• ' 7 * * *>*
<£ S it
,*«
M
•%
***«
J
/ /
0 [
5
10
15
2D
»
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Importmce SNi Figure 1. The evaluation matrix of quality performance SNT and SNS before improvement
From figure 5, we can see the box-and-whisker plot locates in improvement zone it means that most of the quality feature with the importance lower than it's expecting satisfactory level, that is, most of the quality feature can not meet customer's expected satisfaction. Since the evaluation matrix of ion SNi and SNS do not easy to define the upper and lower control limit, and thus hard to find out the quality items that need be improved and reviewed. Therefore, this thesis will adopt the concept of s control chart to define the control chart of overall satisfaction. At first, using the above formula and the relative distance between coordinate on the quality performance evaluation matrix to the target line, to transfer the 2 dimensional coordinate of performance value SNT and SNS of the performance matrix into one dimensional coordinate of quality satisfactory performance value SNi-s- If the following, we adopting the above formula to set up the central line, upper and lower control limit. We define the central line T value is 0 (zero) and the upper and lower control limits are plus and minus one standard deviation value. Due to 1 standard deviation value is 6.56815 so that the upper control limit UCL will be 6.56815 and the lower control limit LCL is -6.56815. In the end, draw the overall satisfactory
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performance value SNi.s of every quality item on the overall satisfactory performance control chart, as shown in figure 2:
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Figure 2. The overall satisfactory performance control chart before improvement
From the figure 2, we can see in the performance control chart of SN[.S, the area above the upper limit is the improvement zone and there are 9 items locate in this area they are: 1st item - easy to control, 3rd item - video entertainment, 5th item - touch panel, 12th item - expandable function, 15th item - document processing, 24th item - down rate, 25th item - time to repair, 29th item - service passion and 30th item - professional knowledge; Therefore, we need to increase the resource investment to promote the satisfactory level. The area below the lower control limit is the review zone, there are 2 items locate in this area, and they are: 21 st item - brand name and 22nd item - market share. 4. The Improvement and Control of Product and Service Quality Features of PDA At first, focus on the 9 items, then proceeding the 4P analysis on process, man and facility to get 9 critical cause they are: when design the PDA to take customize and customer layer variation into consideration, enhance the video entertainment technique of PDA, enhance the touch control technique and quality, design the survey questionnaire to find out the actual need of customer, establish the relation with Microsoft to set up the co-platform with OFFICE, enhance the quality of battery power supply system, increase the repair station, enhance the training of service altitude and considering the importance of customer's need to the quality features, simplify the product design. In the end, for the action limitation we propose 2 counter plan, they are: set up the cost control mechanism should considering the design budget and investment benefit
790 and set up the schedule control for the above mentioned task and related personnel and in reference with competitor's product for review, revise in periodically. Besides, according to the above analysis there are 2 items that need to be reviewed for considering decreasing the resource investment to lower the cost, After discussion and analysis we concluded there are 2 main factors: We find out 3 corrective actions in accordingly: 1. Due to the customer is not respecting the brand name so we need to consider to decrease the promoting advertisement cost and promoting the product bundled with the other relevant products. 2. Increase the product value-added to the customer to expand the market growth rate. 3. Enhance the market survey to design the PDA product that really meet customer's requirement. In the end, we propose 2 counter action plans: 1. Review and approve the confidence level of the market survey questionnaire and analyzing the practice process. 2. Set up control mechanism for advertisement promotion; strictly control the investment benefit of advertisement. For the respect of control, at first focus on specific customer layer carry out the planned critical corrective action, counter plan and execution steps, then focus on the deviated question item re-design the questionnaire and in accordance with the coordinate position of importance and satisfaction on the performance matrix (SNi, SNS) calculate the overall satisfactory performance value SNi_s of every service quality item and set up the overall satisfactory performance control chart, accordingly. Use the control chart as a toll for controlling the service quality. And follow up the above formula we can calculate the SNi_s value of every service quality item and set up a control chart with control limit been set between -20 to +20. By using this overall satisfactory performance control chart, the managers can measure and detect if the above deviated service quality item is out of the control limits. If they are not within the control limit, then we need to submit the related corrective action strategy and counter plan. For the real example of this thesis, currently, the deviated product quality function all within the control limit, as shown in figure 3. It means we have achieved the expecting improvement target therefore need to standardization the process to set up the quality plan and standard operation procedure of after improvement, and expand the education and training for related process personnel.
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Figure 3. Overall performance control chart after improvement
5. Conclusion By using the DMAIC evaluation methodology of 6-sigma provided by this thesis the industrial enterprise can focus on the related industry's product quality and service quality, For the example case of this thesis, from the process of evaluation analysis we found 2 quality items the relevant actions they are: consider to reduce the brand name promotion advertisement expense and promote the product by combining with the other related products and enhance the market survey to design the PDA product that meet the customer requirement to promote the value-added that the product brought to the customer to enlarge the market growth rate and quality. Besides, there are 9 service items that need improvement they are: easy to control, video entertainment, touch panel, expansion of function, document processing, down time rate, time to repair, service passion, and professional knowledge, respectively. Then we found out 9 relevant corrective actions for improvement, they are: while designing the PDA consider the customized and differentiation of customer level, enhance the video entertainment technique of PDA, enhance the function and quality of the touch panel, design a questionnaire to enhance the market survey to understand the real need of customers, build up the coplatform with Microsoft Office, increase the quality of the battery's power supply system, increase the repair station quantity, enhance the training of service altitude and considering the importance of customer requirement to the quality features and simplify the product design, respectively. The study of this thesis found that due tot eh main function of personal digital assistant (PDA) can not meet the customer's basic main need; therefore, the overall satisfactory level is lower than its importance.
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AUTHOR INDEX
Ahn, S.E., 569, 610 Akiba, T., 107 Arafuka, M., 660
Ding, J.Y., 48 Dohi, T., 219, 243, 251, 395, 427, 443,451,577,676
Bae, CO., 142 Bae, S.J., 267 Bertolini, M, 116 Bevilacqua, M., 116
Esaulova, V., 559
Cha, J.H., 3 Cha, S.W., 486 Chakraborty, T., 369 Chang, CM., 310 Chang, W.L., 318 Chaturvedi, S.K., 11 Chen, C.H., 329 Chen, J, 634 Chen, J.A., 337, 345, 687 Chen, J.W., 80 Chen, K.S., 703 Chen, M.C., 195,203 Chen, S.C, 783 Chen, T.W., 693 Cheng, C.Y., 195,203 Chien, Y.H.,337 345, 687 Cho, B.R., 724 Cho, G.S., 24, 32 Choi, C.H., 353 Choi, I.J., 709 Choi, S.K.., 211 Choi, S.S., 549 Chu, Y. K.., 654 Chukova, S., 361 Chung, I.H., 185 Chung, S.W., 494 Chung, Y.J., 64 Cui, L., 97 Cui, Y. H., 645
Giri, B.C., 369 Goh, T.N., 739 Gotoh, H., 577 Guo, R., 585, 595, 645
Finkelstein, M., 559 Fujiwara, T., 395, 435
Ha, Y.J., 227 Ham.J.K., 518 Han, S.W., 553 Hayakawa, Y., 361 Hiroshima, T., 124 Hong, J.W., 301 Hong, S.H., 709, 731 Horikoshi, Y., 132 Hsieh, C.Y., 703 Hsu, C.H., 747 Huang, J.M., 703 Hwang, D.H., 518 Hwang, H.S., 24, 32 Imaizumi, M., 40 Ishii, T., 395 Ito, K., 235 Iwamoto, K., 243,251 Jin, L., 132 Johnston, M.R., 361 Jung, J.S., 486 Jung, M., 541 Kaio,N., 219, 243 793
794 Kamakura, T., 626 Kambayashi, Y., 124, 176 Kang, C.W., 267 Kang, K.M., 385 Kang, T.H., 494 Karim, M.R., 377 Kawai, H., 760 Ke, J.B.,48, 80 Ke, J.C., 654 Kim, CM., 502, 510, 541 Kim, Dohoon, 56 Kim, Do Hoon, 259 Kim Dong Hyeok, 525 Kim, D.K., 403 Kim, H.M., 569 Kim, HoGyun, 142, 185 Kim, Hu Gon, 64 Kim, H.S., 411 Kim, Jae Hoon, 603 Kim, Jae Hwan, 150 Kim, J.R., 56 Kim, J.S., 150 Kim, Jin Woo, 486 Kim, Jong Woon, 603 Kim, M, 477 Kim, M.S., 486, 518 Kim, S.J., 267 Kim, S.M., 518 Kim, W.H., 569, 610 Kim, Y.B., 301 Kim, Y.J., 717, 724 Kimura, M., 40 Kimura Mitsuhiro, 419 Kondo, H., 660 Kwon, H.M., 709, 731 Lee, B.J., 267 Lee, E.Y.,211 Lee, G.L., 158 Lee, H.J., 486 Lee, H.K., 533 Lee, H.W., 275
Lee, J.K., 486 Lee, M.K., 709, 731 Lee, O.S., 525 Lee, S.W., 533 Lie,C.H., 301 Lim, H.S., 477 Lim, J.H., 259, 618 Lim, K.E.,211 Lin, C.S., 72 Lin, T.C., 776 Liu, J. Y., 739 Lu, S.L., 768 Maruo, T., 251 Maeji, S., 293 Mishra, R., 11 Mizutani, S., 285 Moon, J.S., 486 Mukuda, M., 166 Nagatsuka, H., 107, 626 Nakagawa, T., 235, 285, 293, 634, 660 Nakamura, S., 660 Naruse, K., 293 Nishimaki, T., 285 Oh,G.T., 518 Okamura, H.,251,427, 577, 676 Osaki, S., 219 Ouyang, L.Y., 747 Paik, C.H., 64 Park, C. S., 569, 610 Park, D.H., 403,411,618 Park, D.J., 752 Park, J.H., 301 Park, J.S., 603 Park, J.W., 518 Park, J.Y., 435 Park, N.I., 275
795 Park, S.J., 477 Qian, C.H., 634
Wang, K.H., 48, 80 Wang, M., 310 Wang, T.C., 693 Woo, C.S., 549
Rinsaka, K., 443, 451 Xie, M., 739 Satow, T., 760 Seo,J.H., 510, 541 Sheu, S.H., 72, 768, 776 Shibata, K.,451 Shin, G.H., 301 Shin, J.C., 486 Shingyochi, K., 176 Suzuki, K., 132, 377 Tai, S.H., 776 Tamura, N., 668 Tamura, Y., 459 Tao, C.J., 783 Tateishi, K., 676 Teramoto, K., 235 Tokuno, K., 467 Tsujimura, Y., 124, 166, 176 Tsung, H.C., 783
Yamachi, H., 124 Yamada, S., 411,459, 467 Yamamoto, H., 107, 124, 176, 626 Yanagi, S., 89 Yang, S.L., 703 Yang, X., 97 Yasui, K., 40 Yeh, R.H., 318 Yeo, I.K., 403 Yuge, T., 89 Yum, B.J., 477 Yun, W.Y., 185, 227, 353, 385, 494, 603 Zhao, X., 97
Advanced Reliability Modeling II Reliability Testing and Improvement lity Modi (AIWARM) is the the i hers , from no
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improvement". The contributions in bility and maintenance engineering,
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