Fuzzy set-valued and grey filtering statistical inferences on a system operating data Renkuan Guo Department of Statistical Sciences, University of Cape Town, Rondebosch, Cape Town, South Africa, and Ernie Love Faculty of Business Administrations, Simon Fraser University, Burnaby, Canada Abstract Purpose – Intends to address a fundamental problem in maintenance engineering: how should the shutdown of a production system be scheduled? In this regard, intends to investigate a way to predict the next system failure time based on the system historical performances. Design/methodology/approach – GMð1; 1Þ model from the grey system theory and the fuzzy set statistics methodologies are used. Findings – It was found out that the system next unexpected failure time can be predicted by grey system theory model as well as fuzzy set statistics methodology. Particularly, the grey modelling is more direct and less complicated in mathematical treatments. Research implications – Many maintenance models have developed but most of them are seeking optimality from the viewpoint of probabilistic theory. A new filtering theory based on grey system theory is introduced so that any actual system functioning (failure) time can be effectively partitioned into system characteristic functioning times and repair improvement (damage) times. Practical implications – In today’s highly competitive business world, the effectively address the production system’s next failure time can guarantee the quality of the product and safely secure the delivery of product in schedule under contract. The grey filters have effectively addressed the next system failure time which is a function of chronological time of the production system, the system behaviour of near future is clearly shown so that management could utilize this state information for production and maintenance planning. Originality/value – Provides a viewpoint on system failure-repair predictions. Keywords Systems theory, Preventive maintenance, Production planning Paper type Research paper 1. Introduction It is a well-known fact that management is concerning about the basic behaviour of his production system. Because if his field engineers could tell him what is the next stopping (failure) time and requires what type of maintenance or service, he can make a better production plan for his company and particularly avoid the unexpected system failure and sudden stopping, which would typically cause huge additional costs to production. Therefore, researchers ranging from reliability engineering to operations research and maintenance theory were all busy and keen to develop various optimal maintenance plans. Currently, the majority of maintenance planning is mathematical optimization based. The typical exercise there is assuming a set of system operating and maintenance The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at www.emeraldinsight.com/researchregister www.emeraldinsight.com/1355-2511.htm Fuzzy set-valued and grey filtering 267 Journal of Quality in Maintenance Engineering Vol. 11 No. 3, 2005 pp. 267-278 q Emerald Group Publishing Limited 1355-2511 DOI 10.1108/13552510510616478
Transcript
Fuzzy set-valued and greyfiltering statistical inferenceson a system operating data
Renkuan GuoDepartment of Statistical Sciences, University of Cape Town, Rondebosch,
Cape Town, South Africa, and
Ernie LoveFaculty of Business Administrations, Simon Fraser University,
Burnaby, Canada
Abstract
Purpose – Intends to address a fundamental problem in maintenance engineering: how should theshutdown of a production system be scheduled? In this regard, intends to investigate a way to predictthe next system failure time based on the system historical performances.
Design/methodology/approach – GMð1; 1Þ model from the grey system theory and the fuzzy setstatistics methodologies are used.
Findings – It was found out that the system next unexpected failure time can be predicted by greysystem theory model as well as fuzzy set statistics methodology. Particularly, the grey modelling ismore direct and less complicated in mathematical treatments.
Research implications – Many maintenance models have developed but most of them are seekingoptimality from the viewpoint of probabilistic theory. A new filtering theory based on grey systemtheory is introduced so that any actual system functioning (failure) time can be effectively partitionedinto system characteristic functioning times and repair improvement (damage) times.
Practical implications – In today’s highly competitive business world, the effectively address theproduction system’s next failure time can guarantee the quality of the product and safely secure thedelivery of product in schedule under contract. The grey filters have effectively addressed the nextsystem failure time which is a function of chronological time of the production system, the systembehaviour of near future is clearly shown so that management could utilize this state information forproduction and maintenance planning.
Originality/value – Provides a viewpoint on system failure-repair predictions.
Keywords Systems theory, Preventive maintenance, Production planning
Paper type Research paper
1. IntroductionIt is a well-known fact that management is concerning about the basic behaviour of hisproduction system. Because if his field engineers could tell him what is the nextstopping (failure) time and requires what type of maintenance or service, he can make abetter production plan for his company and particularly avoid the unexpected systemfailure and sudden stopping, which would typically cause huge additional costs toproduction. Therefore, researchers ranging from reliability engineering to operationsresearch and maintenance theory were all busy and keen to develop various optimalmaintenance plans.
Currently, the majority of maintenance planning is mathematical optimization based.The typical exercise there is assuming a set of system operating and maintenance
The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at
conditions and converting these assumptions into mathematical models. Without anydoubts, certain assumptions imposed to production system might have some degree ofreflections to the functioning system as well as the system operators. However, wecannot see how these assumptions connect to the system reality in most of the cases inreliability engineering or related literature. Therefore the optimized state is at the bestan estimate of an imagined universal system under mathematically tractableassumptions. This kind estimate is clearly not the true reflection of a real system.
What a management term requires is the true statistical estimate extracted from itsown functioning production system because only estimate based on its own systemmake the team step forward on the solid ground. System state, particularly, the state ofcomplex system is often unobservable, thus requires statistical estimation from systemfunctioning and maintenance data. Nevertheless, another reality in reliability theory orstatistical theory is that the available theory and methodology cannot provide a way toevaluate the state of a functioning system convincingly although we do noticeresearchers in this field have turned their attention to estimate the system state fromfiled data.
The nice feature of statistical methods is that it does draw conclusion from data. Butstatistical inference requires reasonable large data set including Bayesian inferenceotherwise the estimate of the system state is not reliable and misleading. One of thereasons why the statistical method is so powerless is due to its object dealing withcontaining intrinsic character – random uncertainty. Probabilistic theory has ledscientific community into a deeper understanding of the real world surrounding us butthe methodology to extract the true state underlying the real world phenomenon mightbe short of deep sight of randomness. In other words, current existing statisticalmethodology is based on a correct theory but the methodology might be started from awrong kicking line.
In certain sense, the deterministic world outlook initiated the western science andengineering. Even today, quite a few branches of science and technology still usedeterministic model assumptions as the foundations. Although, philosophically,deterministic world outlook is no longer solid, people should be armed by stochasticworld outlook. However, the deterministic mathematics developed in the long scientifichistory should not be discarded. What we are trying to say that real world datacontaining randomness should be de-randomized and we can use the classicaldeterministic mathematical tools to deal the de-randomized data and dig out the truestate behind the data.
Gey system theory was first proposed and developed by Deng (1985). Since 1980s,grey system theory has been widely applied to control, medical, agriculture, militaryand engineering problems. In grey system theory, a dynamic model with a group ofdifferential equations called grey differential model is developed. To do this, Denginferred:
(1) A stochastic process whose amplitudes vary with time is referred to as a greyprocess.
(2) The grey modelling is based on the generating series rather than on the raw one.
(3) The grey derivative and grey differential equation are defined and proposed inorder to build a GM model.
(4) To build a GM model, only a few data (as few as four) are needed.
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Therefore, we develop a method based on Deng’s grey system GMð1; 1Þ filters forextracting the characteristic system functioning times and maintenance effects.However, are these extracted time sequences statistically representing what we believeand have practical values in operating and maintenance decision-making? We arerequired to examine them further. It is admitted that the random and vagueuncertainties does contain in the repair improvement (damage) estimates as well as theintrinsic system functioning times. To facilitate such reality, we then return back tostudy the system state as a random and fuzzy one. In the classical probability andstatistics, what is obtained from each statistical experimental trial (or run) is adeterministic point on the phase space (which is the set of possible observations –sample or elementary space). Such a statistical methodology is therefore is called apoint-wise statistics. However, in the great amount of the management and productionpractices, particularly, these involving human psychological measurements, what wefaced are no longer some isolated points. The obtained information from eachexperiment is usually a common subset or a fuzzy subset of the phase space. Classicalstatistics often ignored the whole interconnections between points therefore ignored anfact that the viewpoint of whole is an fundamental criterion for human being toperform partitions, selections, orderings and decision-makings. In set-valued statisticsthe outcome of each experiment is a subset of a certain discourse and therefore theclassical statistical methodologies no longer apply. As an extension to classicalstatistics the set-valued statistics will greatly expand the scope of applications. Ingeneral, the theory of random set and the falling shadow of random set is thefoundation of set-valued statistics. In certain sense, the current work is an extension tothe past covariate modelling for predicting a system behaviour by Love and Guo(1991), Guo and Love (2001, 2003, 2004).
This paper is organized as follows. The Section 2 introduces GMð1; 1Þ filteringtheory and Section 3 is used to report a set of grey GMð1; 1Þ filtering results, whichclearly shows that fuzzy set-valued statistics is necessary to explore. In Section 4, thecharacters and mathematical foundation of set-valued statistics – random set and thefalling shadow function is briefly reviewed. Chen and Chen fuzzy set-valued statisticalinference model is briefed in Section 5. As an illustration, a failure prediction exampleusing cement data is carried on in Section 6. Finally, we give a few concluding remarksin Section 6 and particularly point out future research directions.
2. A review of GM(1,1) filteringA Grey system means that a system in which part of information is known and part ofinformation is unknown. In general, the analysis of system characteristics is based onthe statistical models relying on large sample size. However, a large sample set is notthat easy to be available; therefore, many systems are said in a status of poorinformation. The characteristic of the grey system is that we can utilize only a fewknown data by the way of accumulated generating operation (AGO) to establish aprediction model.
Let x ð0Þ ¼ ðx ð0Þð1Þ; x ð0Þð2Þ; . . . ; x ð0ÞðnÞÞ be a raw series and AGO be an accumulatedgenerating operator AGO,
x ð1Þ ¼ AGOx ð0Þ ¼ x ð0Þð1Þ;X2
i¼1
x ð0Þði Þ; . . . ;Xki¼1
x ð0Þði Þ; . . . ;Xni¼1
x ð0Þði Þ
!ð1Þ
Fuzzy set-valuedand grey filtering
269
then the following equation:
x ð0ÞðkÞ þ bz ð1ÞðkÞ ¼ a; k ¼ 1; 2; . . . ; n ð2Þ
is called a one-variable first order differential grey equation, where
z ð1ÞðkÞ ¼1
2x ð1ÞðkÞ þ x ð1Þðk2 1Þ� �
; k ¼ 1; 2; . . . ; n ð3Þ
and b is the developing coefficient, a is the grey input, and x (0) is a grey derivativewhich maximizes the information density for a given series to be modelled. This modelis called GMð1; 1Þ:
It is noticed that the unknown parameter values ða;bÞ can be determined in termsof the least square method. Writing equation (2) as
aþ b 2z ð1ÞðkÞ� �
¼ x ð0ÞðkÞ; k ¼ 2; 3; . . . ; n ð4Þ
Then a standard matrix form of the equation can be formed in terms of least-squaretheory:
Xa
b
" #¼ y ð5Þ
where
X ¼
1 2z ð1Þð2Þ
1 2z ð1Þð3Þ
..
. ...
1 2z ð1ÞðnÞ
26666664
37777775
and y ¼
x ð0Þð2Þ
x ð0Þð3Þ
..
.
x ð0ÞðnÞ
26666664
37777775
ð6Þ
which leads to the estimated parameter vector
a
b
" #¼ ðX TXÞ21X Ty ð7Þ
Based on the estimated vector ða; bÞ for parameter ða;bÞ and differential equationtheory, the predicted equation is
x̂ ð1Þðkþ 1Þ ¼ x ð0Þð1Þ2a
b
h ie2bðk21Þ þ
a
bð8Þ
3. GMð1; 1Þ filters for cement roller dataIn this section, we will continue to use the data set of operating data extracted from aCanadian cement plant for illustrative purpose. The plant’s roller mill data was usedrepeatedly in our past covariate analysis (Guo and Love, 2003).
We proposed an algorithm (Guo, 2004) for dealing with observation sequence inwhich the sojourn times are not equi-spaced. Then the current data sequence is
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augmented into a new sequence of 85 data points in which 31 data from the originalsequence with non-integer indices (except the first one) and the remaining 54integer-indexed with values created by linear interpolations. We summarize all 17GMð1; 1Þ groups and another two iterative computation results (by enhancing onemore data point each). The resulting estimates for ða;bÞ are combining into intervalsfor a and b, respectively because the grey nature of them, denoted as ^ða;bÞ
yX ð0Þðkþ 1Þ ¼ ð1 2 e2^bÞ X ð0Þð1Þ2^a
^b
� �e2^bk; k ¼ 1; 2; . . . ;m ð9Þ
Equation (9) is in nature a grey equation with grey parameters ð^a;^bÞ: However, itis a great advantage to represent the system characteristic age process in terms offuzzy random set processes. Since the coefficients in equation (9) is a sequence of greynumbers and thus extracting the information and investigate them in terms of fuzzyset-valued statistics methodology is definitely appropriate.
4. Fuzzy statistical model4.1 Fuzzy statistical experimentThe best way to understand the characters of set-valued statistics is to compare it tothe classical statistics. The Table I gives a systematic comparison between them(Table II).
4.2 Duality between two modelsAssume that P(U) is the collections of all the subsets of domain U and called the powerset of U. It is obvious to see that any set A , U will be an element of the power set, i.e.A [ PðU Þ: For ;u [ U ; define set Gu W ðB : B [ PðU Þ; u ] BÞ which is the superfilter of set-algebra PðU Þ: For any given u, Gu can be regarded as the subset of PðU Þ;i.e. Gu , PðU Þ: Thus a fuzzy statistical model on a discourse U can be converted into aclassical statistical model on the discourse PðU Þ:
For a given discourse U and its power set PðU Þ; define a set class CðGuÞ W {Gu :u [ U} and let B be a s-algebra containing CðGuÞ; i.e. CðGuÞ , B: Therefore,ðPðU Þ;BÞ is a measurable space. Actually, a measurable mapping from probabilityspace ðV;A;PÞ into measurable ðPðU Þ;BÞ is called random set on U, i.e.j : V!PðU Þ
j21ðBÞ ¼ {v [ jðvÞ [ B; ;B [ b} [ A ð10Þ
Intuitively, a random set j can be defined as a mapping from sample space V to asubset B of PðU Þ which satisfies that each pre-image of jðvÞ i.e. v [ V is a possibleexperiment outcome whose occurrence is associated with a probability measure.
4.3 Falling shadow functionTerm “falling shadow” is proposed by Wang (1985), basing on an image of a cluster ofcloud on the sky throwing shadow on the ground. Mathematically, assume that j is arandom set on discourse U. For ;u [ U ; 1jðuÞ ¼ P½v : jðvÞ ] u� is called the fallingshadow function. For a fixed u0 [ U ; 1jðu0Þ is called the falling shadow value of j atuo. Notice that 1jðuÞ ¼ P½v : jðvÞ [ Gu� ¼ PjðuÞ and therefore 1j is a real functiondefined on U. 1j is not only used to describe random set to a certain degree but also canexpressed as the membership function of the corresponding fuzzy subset on U :
Assume that ~A is a fuzzy concept which can be represented by a fuzzy subset of U :Gu W {B : B [ PðU Þ;B ] u} is the collections of all random sets which containu [ U and also represent the fuzzy concept ~A; ;B0 ¼ jðv0Þ [ Gu: The associatedprobability Po of jðv0Þ represents the degree of confidence for set B0 to describe fuzzyconcept ~A: Then the probability of Gu; 1jðuÞ; can be regarded as the membership offuzzy set ~A at u and denoted as 1 ~AðuÞ:
5. Fuzzy set-valued statistical inference on failure timeManager and floor engineer are mostly concerning a question: once the productionsystem starts, what is the next failure time? If they can figure out as early as possible,then their production and maintenance decision-making are ready. Grey system modelmay provide failure prediction. However, validation from other channels is alwaysnecessary. Chen and Chen (1984) developed a concrete model of fuzzy set-valuedstatistical inference summarized in Chen and Guo (1992). Assuming that X and Y are
Classical statistical model Fuzzy statistical model
Space Elementary space V, containing all therelevant factors and thus being anextremely high-dimensional Cartesianproduct space
A discourse U
Fixed An event A , V A fixed element u0 [ UVarying A variate v on V, once v is fixed, then all
the factors are fixed at their specific statelevel, respectively
A fuzzy concept ~a on U, a varying set A * isformed by constructing an uncertainpartition about a; each fixing of A * impliesa definite partition for ~a; which representsan approximation to the extension of ~a;
Condition A certain condition S A certain condition S
Table II.Comparisons betweenclassical and fuzzystatistical model
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two related discourses, the implication relation ~R from X to Y can be represented by thefalling shadow:
m ~Rðx; yÞ ¼1
n
Xni¼1
m ~Riðx; yÞ ð11Þ
in terms of n independent experimental sample of random sets,
{ ~R1; ~R2; . . . ; ~Rn}; ~Ri [ FðX £ Y Þ; i ¼ 1; 2; . . . ; n ð12Þ
For given ~A* [ FðXÞ; in terms of reasoning compositional rule,
~B* ¼ ~A*+ ~R ð13Þ
is obtained as the fuzzy inference conclusion.In order to continuously modify the original implication relation (no matter of prior
nature or sampling one) ~R by utilizing the data-based inference conclusionprogressively, define
~Rnþ1 ¼ ~A* £ ~B* ð14Þ
which has a membership function,
m ~Rnþ1ðx; yÞ ¼ m ~A* ðxÞ ^ m ~B* ð yÞ; ;ðx; yÞ [ ðX £ Y Þ ð15Þ
Then a fuzzy set-valued inference model with self-learning has a modified implicationrelation ~R* ; with a membership function,
m~R* ðx; yÞ W
vm ~Rðx; yÞ þ ð1 2 vÞm ~Rnþ1ðx; yÞ if correct
ðvm ~Rðx; yÞ2 ð1 2 vÞm ~Rnþ1ðx; yÞÞ _ 0 otherwise
8<: ð16Þ
However, sometimes the samples obtained in terms of statistical experimentation arenot subsets on X £ Y but the point ðx; yÞ on X £ Y : Particularly, when a probleminvolves many factors and thus the form of the random sample is multi-dimensionaldata ðx1; x2; . . . ; xm; yÞ: Therefore, it is necessary to extend the above model.
Let us consider an n-fold l-order (n, 1)-implication fuzzy inference. Assuming that Xand Y are two related real sets. A W { ~A1; ~A2; . . . ; ~Am} and B W { ~B1; ~B2; . . . ; ~Bm} aretwo fuzzy normal convex partitions on X and Y, respectively. In general A and B areprior type of fuzzy partitions based on features of a real problem. Assuming that nindependent random samples,
ðx1; y1Þðx2; y2Þ; . . . ; ðxn; ynÞ; xi [ X ; yi [ Y ; i ¼ 1; 2; . . . ; n ð17Þ
Denote S ¼ {ðxi; yiÞji ¼ 1; 2; . . . ; n}: For given i ¼ 1; 2; . . . ; n; observed data ðxi; yiÞcorresponds to fuzzy sets,
Therefore the two group of fuzzy sets { ~ai}ni¼1 and { ~bi}
n
i¼1 on fuzzy partitions A and Bof X and Y, respectively, where sample S are obtained. Fuzzy implication relation is~R ¼ ðrijÞm£k defined as a fuzzy relation from A and B; where rij [ ½0; 1� is interpretedas the degree of truth that if “x is ~Ai” then “y is ~Bj”. Let the ith row vector of ~R bedenoted as ~Ri ¼ ðri1; ri2; · · ·; rikÞ being regarded as the fuzzy subsets on B; i.e.
~Ri Wri1~B1
þri2~B2
þ · · · þrik~Bk
ð20Þ
Thus ~Ri is understood as the random set falling shadow estimate from the sample ofrandom sets of the fuzzy subset group,
If A is a common partition of X, i.e. all the ~Ai [ A are clear subsets, the above-statedmeaning of constructing ~Ri in terms of falling shadow is obvious. However, when ~A isa fuzzy subset, “xl [ ~Ai” cannot simply use “yes” or “no” to describe it, but it is usuallyto be described by the degree of xl [ ~Ai: The membership m ~Ai
ðxlÞ is just the quantityreflecting the degree of xl [ ~Ai: Therefore, a linear estimator is constructed for ~Ri
~̂Ri W
Xnl¼1
m ~AiðxlÞ ~bl
Xnl¼1
m ~AiðxlÞ
¼
Xnl¼1
alibi1
Xnl¼1
ali
;
Xnl¼1
alibi2
Xnl¼1
ali
; . . . ;
Xnl¼1
alibik
Xnl¼1
ali
0BBBB@
1CCCCA ð22Þ
As to computation details, for a given sample S, in terms of (6.65), two matrices can be,
An£m ¼
a11 a12 · · · a1m
a21 a22 · · · a2m
..
. ... . .
. ...
an1 an2 · · · anm
26666664
37777775
Bn£k ¼
b11 b12 · · · b1k
b21 b22 · · · b2k
..
. ... . .
. ...
bn1 bn2 · · · bnk
26666664
37777775
ð23Þ
and,
Wm£k ¼ An£mTBn£k ¼
w11 w12 · · · w1k
w21 w22 · · · w2k
..
. ... . .
. ...
wm1 wn2 · · · wmk
26666664
37777775
ð24Þ
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where
wij ¼Xnl¼1
ailblj; i ¼ 1; . . . ;m; j ¼ 1; . . . ; k ð25Þ
Let ai WXnl¼1
ali; then in terms of equation (22), the fuzzy implication relation,
~Rm£k ¼
w11=a1 w12=a1 · · · w1k=a1
w21=a2 w22=a2 · · · w2k=a2
..
. ... . .
. ...
wm1=am wm2=am · · · wmk=am
26666664
37777775
ð26Þ
For given x [ X ; it is obtained
~a* ¼ a*
1;a*
2; . . . ;a*
m
� �[ FðAÞ a
*
i ¼ m ~AiðxÞ; i ¼ 1; . . . ;m ð27Þ
Therefore in terms of reasoning compositional rule,
~a*+ ~R ¼ ~b* ¼ b*
1;b*
2; . . . ;b*
k
� �[ FðBÞ ð28Þ
Let y1; y2; . . . ; yk be the kernels of ~B1; ~B2; . . . ; ~Bk; respectively, then
y ¼Xkj¼1
b*
j yj
! Xkj¼1
b*
j
