Grey Repairable System Analysis

Grey Repairable System AnalysisRenkuan Guo*Department of Statistical Sciences, University of Cape Town, Cape Town, South AfricaCharles Ernie LoveFaculty of Business Administrations, Simon Fraser University, Burnaby, B. C. V5A 1S6, CanadaAbstract: In this paper, we systematically discuss the basic concepts of grey theory, particularly the grey differential equa-tion and its mathematical foundation, which is essentially unknown in the reliability engineering community. Accordingly,we propose a small-sample based approach to estimate repair improvement effects by partitioning system stopping timesinto intrinsic functioning times and repair improvement times. An industrial data set is used for ilustrative purposes in astepwise manner.Keywords: Grey theory, grey differential equation, intrinsic functioning times, repair improvement effects.1 Introductionhidden behind the collected data, and not the under-lying distribution. As long as a dynamic rule, even aDifferent from a mathematician, a statistician mustlocal and imprecise one is discovered, why should we beanalyze data collected from the real world and extractconcerned what distribution governs the data? In sum-rule(s) about reality from the data. Quite often, statis-mary, the adequateness of information depends uponticians luse existing sample distributions as a guidelinethe way we extract dynamics from the data. It may beto fit data and perform statistical hypothesis testing. Adistribution in mind is a priori knowledge in nature. Itby statistician's distributional thinking. Such a think-is often the exercise that for given data, according to re-ing logic has dominated statisticians for more than alated literature or his/ her own experience, a researchercentury, and has created today's large-sample statisti-estimates parameters according to a class of distribu-cal theory empire. A realistic data analyst should betion and an asymptotic variance-covariance matrix forconcerned with the final destination - the dynamicsestimators. As long as the statistical significance levelunderlying the data rather than the route of thinking.is satisfactory, most researchers will happily report andIn this paper, we are trying to engage in an effortpublish their findings. Therefore, a critical requirementto extract dynamics information underlying a discreteto facilitate statistical data analysis is that a sampledata sequence in terms of grey thinking, typically withsize be large enough. Such a prerequisite makes clas-a small sample size (as little as four). For this rea-sical statistics powerless whenever facing a small datason, we will use the next section to introduce the basicsample. It is obvious that statistical thinking - i.e.concepts of grey system theoryl1.drawing a conclusion from data, be corrected, but sucha correct intention is often haunted by small sample2 Grey processessize, which is often regarded as inadequate information.How much information is adequate is a matter to .2.1 Grey theory in generalbe debated. According to standard statistical thinking,a dynamic rule underlying collected data is discoveredA methodology to solve random uncertainty is prob-by distribution-fitting; therefore, large sample theoryability and statistics, i.e. to treat data in terms of sta-is inevitably employed. In this thinking pattern, onlytistical laws and prior (probability) laws. Since statis-information contained in a large enough sample is ad-tical laws are established via large samples, the moreequate. However, what is critical is the dynamic lawthe data the better will be the analysis. Therefore,straightforward statistical modeling is often dificult toManuscript received September 27, 2005; revised January 8,perform, and som中国煤化工°cost con-2006.siderations). If VYHCNMH and lok at* Corresponding author. E-mail address: rguo@stats.uct.ac.zathe real world froSxample by132International Journal of Automation and Computing 2 (2006) 131-144looking at system dynamics from the viewpoint of thelished based on discrete data, ie. a grey diferentialdegree of information availability, we will walk out fromequation.the shadow of large sample statistics.Researchers often believe that the time sequence ofIn modern control theory, system dynamics are clas-a single dimension can only provide very limited in-sified by the degree of information completeness, andformation. Some researchers believe there is a limi-accordingly diferent methodologies are developed fortation to treating data containing large relational sys-each. Table 1 offers a general picture for three dynamictems based on a“1-dimensional”point of view. How-systems classified using information adequacy criterion.ever，researchers of grey theory believe that a timesequence contains extremely rich information, whichTable 1 Three systems classifed via informationBlackGreyWhitepate in a dynamic system process. For example, letInformationindistinguishablebright and{Xo(t)} be a set taken from a variables time series. Inppearance:learthe real world, it is hard to assume a variable playingProcessnewreplacementold andits role alone, and rather some other p variables, saybetween new and old known{Zk(t),k: = 1,2,.,p}, also actively participate in aPropertychaotic and multi- levelpurespecific dynamic process. A common exercise in theignorant knowledgeMethodology negation sublimateconfirmation)ast was to select j principal control factors to buildAttitudeindulgence tolerancerigorousa mathematical model, while the exercise in grey the-Resultno solution no uniqueuniqueory is try to rebuild a dynamic process according tosolutioninformation contained in {Xo(t)}. .A critical feature of a grey system is informationNow let us state the two basic principles in greyincompleteness. The task of establishing a model un-theory. A grey concept is one whose connotationder the guidance of grey system theory is inevitably tois hazy and vague， but which has a clear bound-seek model built based on a small size data sample. Itsary and extension. Therefore, the basic feature of atarget is the establishment of a grey differential equa-grey system is information-incompleteness. Logically,tion, which emphasizes the exploration, utilization, andthe consequence of information incompleteness is non-processing dynamic information contained in the data.uniqueness. Basic principles in grey theory are there-fore derived as follows:1) Principle of incomplete informationGraycations of the principle of incomplete information arethe dialectical umity of“ittle’ and“Ilarge” (i.e. theamount of information), and the transformation be-tween“local" and“whole” information.2) Principle of process non-uniqueness - Because anobject under investigation in grey theory does not havecomplete information, criterion is often multiple foldand may often produce a multiple-to-multiple mappingfrom cause to effect. The representation of a processis therefore not umique. In detail, using grey theory tosolve real world problems often displays a form of non-unique solution, non-unique parameter to be identifi-Fig. 1 Three system dynamicsl2lable, non-unique model, non-unique decision method-It is commonly believed that only differential equa-ology, non-unique result, etc. .tions are suitable for continuous differentiable func-2.2 Interval grey numbers and operations .tions. The characteristic behavior of a grey system isin general represented by discrete time sequences. InGrey system dynamics are described using greyorder to build a diferential equation model, a modelernumbers, grey equations, and grey matrices, where greywith grey system theory extracts a relevant variable tonumbers are the elementary cell. In general, we call a .model a relational analysis; and further, based on thenumber whose ro中国煤化工hose accuinvestigation of the properties of the discrete functions,rate value is unknYHCNMH G'denoted asa dynamic model using a differential equation is estab-8R. Guo and C. E. Love/Grey Repairable System Analysis133Practically, interval grey numbers play very impor-wheretant roles. Ifa grey number 8∈[a',a"],a',a"∈ R,c = min{a'b',a'b" ,a4b',a4b"}then it is called an interval grey number. In otherwords, an interval grey number is one having both ac“= max{a'b' , a'b", a"b',a"b"}(8)lower bound a' and an upper bound a". For conve-iv)If*= +, andc'er > 0, then .nience, we denote an interval grey number 8([a' ,a"]),a',a"∈Ras8∈[a',a"]. In other words, an interval(a',a"]) + 8([b,b"])= 8([e',c"]) (9)grey number 8([a' ,a"]), a', a"∈R satisfies the prop-erty that inf{⑧([a',a"])} = a' and sup{8([a',a"])} =a". Ifa grey number 8([a' ,a"]) has the property thatc = min{a'/b',a'/b", a"/B',a"/b"}a'= a"=a,then grey number 8 becomes a whitec“= max{a'/b',a/b", a"/B',a"/b"} (10)number (a real number).Grey numbers are classified into characteristic andDefinition 1. Assume that A(8) is a set of greynon- characteristic types. A characteristic grey numbernumbers8. If for V8i, 8j∈A(8):is referred to as one that cannot find, or cannot findi)8i+8j∈A(8)temporally, a white number to represent it. For exam-ii)8;-8j∈A(8)ple, the total quantity of energy in the Universe is aili)8iX8j∈A(8) and .characteristic grey number. Contrary to a characteris-iv) 8;/8; ∈A(8) (the quotient operation musttic type, a non-characteristic grey number is referred toobey Lemma 2 iv), then A(8) is called a grey numberas one that can be represented by a white number nofieldmatter what prior information or other (statistical orTheorem 1. The set of all interval grey numberscognitive) approach is employed. If number a is a rep-([a',a"), a', a"∈R constitutes a grey number fieldresentative number of grey number 8, we call number aA(8).a corresponding whitening value of grey number 8, de-Definition 2. Assume that A(8) is a set of greynoted 8. Accordingly, the corresponding grey numbernumbers8. If for V8;, 8j, 8k∈A(8), we have8 is denoted as 8(a).i) 8i+8j=8j+ 8;Lemma 1. For Vk > 0 and a given interval greyii)(8;+8j)+8k=8j+ (8i+ 8k)number 8([a',a"]), a',a"∈R:ii) 30∈A(8) such that 8i+0= 8i-k8([a',a"])= 8([- ka", - ka])1)iv) For V8i∈A(8), 3- 8;∈A(8) such that8;+(-8i)=0Lemma 2. If * be an operator, then for any twov)(8;X8j)Xδk=8;x(8jX8k)interval grey numbers 8([a'，a"]) and 8([b', b"]):vi)31∈A(8)suchthat8ix1=1x8;=8;vi) (8i+8;)8k=8;X8k +8jX 8k .8(al ,a") * 8([b,b") = 8(c',c"])2)vii) 8ix(8j+8k)=8iX8j+8iX 8k,then A(8) is called a grey linear space.where the expressions of c' and c" depend upon the *Theorem 2. The set of all interval grey numbersoperator.8([a',a"]), a', a"∈R that have the same value char-i)If*= +, thenacter constitute a grey linear space A(8).(a',a"]) + (6,b")= 8([e',c"])3)2.3 Grey number whitening and the de-gree of greyc=a'+b'， c“=a“+ b“4)A grey number的may change its value about ani)If*=-, thenelementary value e, and therefore it is relatively easyto whitenize such a grey number 8. An elementary8(a',a"]) - 8([b,b"])=<([',c"]) (5)value can be regarded as a representative whiteningvalue and denoted as区= e. Correspondingly, a greynumber having an elementary value e can be denotedc =a'-b"， c"=a"-b'6)as 8(e) = e+ Ee; where Ee is known as a disturbingili) If*= x, thengrey element and中国煤化工terval greynumber⑧([a', a"whitening8(a',a"I)x 8(b6,b")= 8([C,c") (7)value 8(a',a"])IHCN MHGy0,1J.134International Journal of Automation and Computing 2 (2006) 131-144Definition 3. Given V 8 ([a' , a"]) and the whiten-mapping. If both f and f-1 are continuous, then f ising of 8, taking a form 8([a',a"])=wa' +(1-w)a" forhomeomorphic.Vw∈[0,1] is called equal weight whitenization. Fur-Definition 8. For a grey number 8 with a typicalthermore, if w = 0.5, it is called equal weight averagewhitening weight function f,whitenization.Definition 4. Assume two interval grey numbers,2|a3 -a2( sa2 l(u)du Sas r(u)du0([a', a"]) and 8([b', b"]), and their equal weight whit-a3+ a2l2a3enizations 8(a',a"]) = wa'+(1-w)a" and 8(b6 ,b") =is called the degree of grey, of grey number 8.∞a'+(1 -∞)a", respectively. Ifw =四, we say that8([a',a"]) and 8([b' , b"]) have the same value, other-2.4 Grey differential equationwise, they do not have the same value.Theorem 3. The difference between two intervalThe core of grey system prediction theory is a greygrey numbers is not zero in general unless their valuesmodel composed of a large variety of grey differentialare the same, i.e. for 8([a',a"]), a',a"∈Requations. Diferential equations are powerful tools inmodern science because they reveal deeply the evolving8 ([a',a") - 8([a',a")dynamics of real world events. The common sense orf 0,the same value-takingslimitation of a differential equation is that it can onlyotherwisebe utilized to describe continuous differentiable objects.Mathematicians and statisticians often feel quite help-Similarly, the quotient resulting from two interval greyless when facing a discrete data sequence.numbers cannot in general be one.A natural question given rise to here is what is thenature of a differential equation, and under what con-(a',a") +8(a',a"])ditions does it exist?same value-takingsAssuming a diferential equation(o([兴"]),(12)c+βx=a(16)dhDefinition 5. A function fw is called a typicalwhitening weight function ifthen i is called a derivative of x, where x is calledlt“( l(x)， x∈[a1,a2)the background valuelx;一，and a and β are calledfw(x) =x∈[a2, a3]parameters. In other words, a first- order differential( r(x)， x∈(a;,a4]equation is composed of three components: derivativewhere l(a1)=0,l(a2)= 1, for Vx < y, andl(x) < l(y)( ), background value (x), and parameters (a and B).and r(a3)= 1,r(a4)=0, for Vx r(y).“In calculus, a common description of the existenceTheorem 4. Let X∈Y be a real-valued space andof the derivative of a function x(t) on T is if△t→0,Y C [0,1]. If mapping f : R→Y,x→f(x) satisfiesthe limitthe conditions of a typical whitening weight function,x(t+Ot)-x(t)_ dxthen f has the following properties:limOtdt(17)i) f(8)=σi) f(8)=Yexists.ii) GivenVA,BCX,ifAζ B, then f(A)C f(B)However, from the viewpoint of information, it isiv) IfA≠0, then f(A)≠0equivalent to say that the requirement for function x(t)v) f(A∪B)= f(A)∪f(B)to have a derivative is that the density of informationvi) f(A∩B)∈f(A)∩f(B).must be infinitely high. In other words, a function x(t)Definition 6. Let f be a typical whitening weighton T has infinitely large information density if Ot - →0function, and Y∈[0,1]. If functionx(t+Ot)-x(t)≠0.(18)f -1(y)= {x|f(x)=y, Vy∈Y}(14)and this is the condition required to maintain (17).exists, then it is called the inverse function of f.To further ex中国煤化工between aenaDefinition 7. Assume that X and Y are topolog- derivative and itsYHCNMHGeept cllelical spaces. Mapping f : R→Y, x H f(x) is a 1-1“equivalent projedR. Guo and C. E. Love/Grey Repairable System Analysis135Definition 9. Let A and B be two sets, and Pe be Thenan operation between A and B. For Va1,a2∈A, andd()=x(k;)-x(hi-1)，ki= 12+,.,mi (24)Vb∈B,ifa1Peb= a2Peb(19)is called the information increment at the ith level timethen b is called an equivalent projection with respectunitDefinition 14. Let X = (x(1i),x(2),-.,x(ni))to a1 and a2.Definition 10. Assume that Pe is an operation ofbe a time series sequence with a time unit of infinitedensification. If 1i→0:absolute difference, i.d()=x(k;)-x(k;-1;)≠0，k;= 1,2..n (25)alPeb=|a-b|(20)thenX = (x(1i), x(2i),..,x(ni)) is called a time serieswhen a1Peb= a2Peb, i.e. |a1 - b|= |a2 - b|, operatorsequence with the connotation of a differential equationPe is called an arithmetically equivalent projection oror a grey differential sequence.a simple projection.lx,(d(3(k;)= lim(x(k;)-x(k;-1;)) (26)Definition 11. Assume that + Bx=a isa dif-ferential equation, x(t+ Ot) and x(t) are elements in theis therefore called a grey derivative of time series se-set of background values, and X = {x(t +△t), x(t)}.quence X = (x(1;), x(2).(n))) In general, d(k)i) Ifis denoted a grey derivative.Definition 15. EquationPex(t+△t)= Pex(t)(21)d(')(kx) + βx()(k:)=a(27)the derivative and background elements satisfy theequivalent projection relation;is called a grey-type differential equation.ii) If x is a background value andDefinition 16. If a grey-type differential equationsatisfies the following elementary conditions:x≠x(t + Ot), x(t+ Ot)∈X(22)i) Information density is infinitely highx≠x(t), x(t)∈ Xii) Background values are grey numbersii) The derivative and background value satisfy anAssuming that 8(t +△t) and 8(t) are the componentsequivalent projection relationofi, ifthen it is a grey differential equation.di8(t+ Ot)Pex= 8(t)Pex(23)2.5 GM(1,1) modelthen the background value and components of theGrey models play the core function in grey systemderivative satisfy an equivalent projection relation.Theorem 5. If a function x :T H R+, ie. atheory. For a full name, they should be called grey dif-ferential equation models. Commonly used grey mod-positive-valued function, then the elements in deriva-els are GM(1,1), GM(2,1), GM(1,h), GM(0,h), and thetive , and those in the set of background values withVerhulst model. The basic model is GM(1,1) (an abbre-dtrespect to the diferential equation n + βx = a satisfyviation for a first-order variable grey derivative). Wediscuss this type of model in this subsection.an equivalent projection relation.Definition 17. EquationTheorem 6. The elementary conditions whichconstituting a differential equation areax()(k) + βz(1)(k)=a，k=2,.,n (28)is called a one-variable first order grey differentialequation with respect to time series sequence X(1) =ili) The derivative and background value satisfy an(x(1)(1), x()2().x(1)()), whereequivalent projection relation.Definition 12. Let 1i and 1k denote an ith levelz(1)(k)= C[x(1)(k)+x(1)(k-1)]，k=2,-,n (29)time unit and a k:th level time unit respectively. If1; < 1k, then the ith level time unit is said to be denserand β is a developing coefficient, a grey input, and ax(0)than that of the kth level time unit.a grey derivative中国煤化工ation den-Definition 13. Let X = (.)().x(n;)) sity for a given:fHCN M H 2. modelisbe a time series sequence at the ith level time unit.called GM(1,1). FSal equation .136International Journal of Automation and Computing 2 (2006) 131-144dtdax(1)+βx(1) = a is called a whitened differential equa-estimated grey differential equation should be regardedas an equation with random cofficients (a,b)T rathertion, or the shadow equation of grey differential equa-than a deterministic one. Least-square theory providestion (28).The unknown parameter values (a, B) can be deter-quite detailed information, say the standard errors for(a,b)T and others. The estimated grey differentialmined in terms of the classical least-square approach.equation is still grey, but with additional informationWriting (28) asextracted using the least-square estimation approach.a + B(-z(1)(k))=x(0)(k)，k=2,3,.,n. (30)However, currently this is largely ignored by grey the-ory researchers.then a standard matrix form of the equation can be .produced in terms of least-square theory:.6 Lipschitz smooth sequenceA fundamental requirement is that the applicationX=y(31)[β]of a grey differential equation model requires a cer-tain degree of smoothness in a discrete data sequence.whereSmoothness in the mathematical sense is equivalent to- z()(2) ]「x(0)(2)infinite density in the sense of amount of information.1 -z(1)(3)x(0)(3)As we stated in subsection 2.4, one of the necessaryX=and y=(32)conditions to allw the existence of a grey differentiall x(0)(n)」equation is that information density is infinitely high.Translating the language of information into a geomet-which leads to an estimate for parameter (a,3):ric term is equivalent to saying that the data sequencemust be“smooth” enough.0=(xTx)-1xTy.(33)As with a grey differential equation for a discretedata sequence, it sounds controversial that we begin toBased on the estimated vector (a, b) for parameterinvestigate the smoothness of a discrete data sequence,(a, 3), and diferential equation theory, the predictedbecause differentiability everywhere is a basic featureequation (i.e. the response of the filtering function) isof a smooth continuous function. A data sequence isconstituted by discrete (single) points, and is not con-(1(k+1)= [r()(1)-号]e-0(k-1)+tinuous, and nowhere differentiable on the grounds ofwhich corresponds to the GM(1,1) differential equationclassical calculus. Therefore, it is not possible to usethe concept of a derivative and relevant mathematicaldX(1)+bX(1)=a.(35)results to study the smoothness of a data sequence. InItother words, it is necessary to change to a different an-Theorem 7. Let a positive-valued time series se-gle and use different mathematical concepts in order toquence x(0) = (x(0)(1), x()(),,x(.)(n)), and theinvestigate the character of a smooth continuous func-1-. AGO sequence X(1) = (x(1)(),()(),...<(1)(n))tion. The logic behind this new angle is that if a datasequence possses certain characteristics similar to thecounterpart of a continuous smooth function, then we()= 2x(0(i)(36)can treat the data sequence as a smooth one.i=1The first we can consider is the Lipschitz continu-and the average generating sequence of X(1), Z(1) =ity. An important idea is that if X(t) is not Lipschitz(x(1)(1), ()(2)....z(1)(n)), wherecontinuous on [a, 6)], then X(t) is not diferentiable on[a, b] and accordingly it is not smooth.z(")(k) = (x()(k) + (). .Definition 18. A function X(t) is called Lip-schitz smooth if it satisfies the Lipschitz condition:Then, the least-square parameter estimate (a, b) satis-|X(s)- X(t)| < [|s -t for some fixed positive numberfies the following equationL> 0, and for Vs,t∈[a, b].(a,b)T =(xTx)-1xTy.In a two dimensional space, X(t) is a curve, andhence a Lipschitz中国煤化工it smoothThe role of Theorem 7, indicates that the vector (a, b)Tcurve. AssumingYHCNMHGicesinis a least- square estimate of (a, β)T and therefore, thecontinuous functicPartition ofR. Guo and C. E. Love/Grey Repairable System Analysis137time interval [a, b].Definition 22. Assume that X is a LipschitzDefinition 19. (Partition) Let [a, b] be a finite in-smooth discrete data sequence and Y*= {Y1*,Y*,，-,terval, a sequence of numbers {to, t1,t2,.-,tn}, suchY*} is a . representative sequence given by Lipschitzthat a= to< t1 < t2,... < tn = b is called a partitionsmooth curve Y. If3ε ∈[0, 1] such that |d(Y*, X) -of [a, b].d(Y*,Z)|≌E， then the degree of smoothness ofDenote△= max{△tk}, where Otk = tk- tk-1.sequence X is said to be larger than 1/e. Sd =The role of a partition is actually to divide func-|d(Y*,X)- d(Y*, Z)|-1 is defined as the degree of Lip-tion X(t) into n segments, where the k:th segmentschitz smoothness of a sequence X.is denoted by [X(tk-1),X(tk)]. Let 境be an arbi-Proposition 2. If X(k)- X(k-1) = a is constanttrary interior point of interval [tk- 1,tk], i.e. Vt% ∈for all k (i.e. is straight-line distributed), thenSa=∞(tk- _1,tk). Then correspondingly X(k)≌X(t端) ∈and X is called a Lipschitz infinitely smooth sequence.(X(tk-1), X()). Therefore, an interior point sequenceThe role of the straight-line distributed sequence isX = {X(1),X()，,X(n)} can be obtained. Sim-that it can be a criterion for specifying the order ofilarly, it should be noted that the lower point of theAGO.kth segment of X(t), i.e. [X(tk-1),X(tk)] is X(tk-1).2.7 Class ratio and smooth ratioAccordingly, a lower endpoint sequence can be denotedby Xi= {X(to), X(t1), X(2)，..,.X(tn-1)}. .The two ratios to be defined in this sub-section willDefinition 20. For a given Lipschitz smooth curvenot only serve the purpose of flling empty holes of end-Y(t) and an equal-spaced partition {to,t1,t2,...,tn}ing points technically in a discrete data sequence butwith t; = a + i(b - a)/n， the sequence Y*:also offer further insight into a sequence and therefore{Y',Y*，-,Yt}, where Y1* = Y(t;), i = 1,2,.n,criterion to justify the smoothness of a sequence.is called a representative sequence of the LipschitzDefinition 23. For sequence X = {X(1), X(2),smooth curve Y (t)., X(n)}, ratioProposition 1. If an arbitrary partition {to,t1,?(k)= X(k)/X(k-1)，k=2,3,..,n (39)t2,.,tn} of interval [a,b] and a Lipschtiz smoothcurve Y(t) and its corresponding representative se-is called the class-ratio for sequence X. Similarly, ratioquence Y* = {Y*,Y2*.+..,Y} are assumed, and a_X(k)curve X(t) satisfies the following two conditions:k=2,3,.,m (40)i) For any arbitrary interior sequence of X(t):Xj> X(i)= {X;(1), X(),，,X;(n)} and Xk= {Xx(1),Xe(2),... Xxe(n)}, distance d(Y*,Xj) =d(Y*,Xa)is called the smooth-ratio of sequence X .ii) d(Y*,X)= d(Y*,XI)It is obvious that these two ratios are related in thethen X(t) is also Lipschitz smooth.Now, based on Proposition 1 we can define a smoothfollowing way:data sequence.7(k+ 1)=0(k+1)(+()，k-=2,..n. (41)Definition 21. Let X = {X(1),X()，.,X(n)}p(k)be a discrete data sequenceZ = {Z(1), 2(2).，Z(n)},If sequence X contains empty holes X(1) and X(n),where Z(k;) = (X(k:- 1)+ X(k))/2, ie. the averagethen class-ratio generation will produce X(1) =(consecutive two values) generate a sequence of X, andX(2)/>(3) and X(n)= X(n- 1)n(n一1) respectively,Y*= {Yi*, Y*.-,Y} is the representative sequencewhile smooth-ratio generation will produce X(1) =of a given Lipschitz smooth curve Y. If X satisfies the|X(2)2|/(X(3)- X(2)) and X(n)= X(n-1)(1 +p(n-following two conditions: .1)) respectively.Definition 24. If a sequence X satisfiesi) X(k)< > X(i) for a sufciently large ki) p(k+ 1)/p(k)< 1,k= 2,3,..,n(42)i) do(Y*,X) > do(Y*, Z) where the distance isii) ρ(k;)∈[0,e],k= 2,3,.,n(43)defined byii)ε< 0.5(44)then X is a quasi-smooth sequence.d(U,V) = max{|U(k)- V(k)}Definition 25. Given a sequence X with emptyholes, if the new g中国煤化工isties quasi-then X is called a Lipschitz smooth discrete data se-smooth conditionsfYHCNM HGFon 25, thenquence.X9 is called a que-。-lence.138International Journal of Automation and Computing 2 (2006) 131-144Proposition 3. If sequence X is non-decreasingDeng pointed out in [4] and [5]. Furthermore, it shouldand satisfiesbe noted that current grey literature only permits the(i)~r(k) <2, for Vh= 2,3.--,ngrey differential equation modeling of a positive real-(i) p(k:+ 1)/p(k)< 1, for Vk=2,3,..,nvalued discrete data sequence. However, Guo et al6]then for any ε∈[0,1] and Vk: = 2,3,..,n, p(k) < 1have proposed two classes of transformation in termsimplies 7(k+1)∈[0,1 + e].of a class ratio as criterion, such that a transformeddata sequence can be grey diferential equation mod-2.8Quasi-exponential formof greyeledprocessDefinition 26. Let X = {X(1),X()...,X(n)}B Grey intrinsic functioning timebe a given sequenceRepairable system modelling is an important re-i) If X(k)= Oexp(Bk), Vθ,β≠0fork= 1,2,，..,n,search field in reliability engineering. Based on stan-then X = {X(1),X().,X(n)} is called a homoge-dard statistical theory, system operating- failure-repairneous sequence.recurrences form a stochastic process and thus re-ii) If X(k) = θexp(Bk)+T，Vθ，β，τ≠0, forpairable system modelling is in nature a stochasticVh = 1,2,.,n, then X = {X(1),X(2)，..,X(n)} isprocess modelling, as demonstrated in [7 ~ 17]. How-called a non-homogeneous sequence.ever, a logical thinking is to estimate the individualTheorem 8. SequenceX = {X(1), X()，.,.X(n)}repair effect for a“perfect" modelling purpose. Guo etis homogeneous if and only if r(k) = constant holdsal[18~21] started to explore this issue in terms of fuzzyfor Vk:= 1,2，.,n.set theory and gradually evolved into grey approachTheorem 9. Assume that X = {(1)...(n)}for the individual repair effect evaluation problem asis a given sequenceexplored in following subsections.i)Iffor Vk= 1,2,.,n, γ(k)∈[0,1]，then X hasa3.1 Interpolation-least square fitting met-negative exponential law (β < 0).ii) Iffor V以k= 1,2,..n, 7(k)∈[1,], then X hashoda positive exponential law (β > 0).If we have system functioning and failure times (orii) If for Vk= 1,2..,n, /(k)∈(3,7],δ=T-β,then X has a grey exponential law with an absolutetem stopping times, and if we denote system stop-degree of grey δ.ping times as {Tr,T,...,TL}, it is immediately no-iv) Ifδ < 0.5, X is called a sequence having a quasi-ticeable that we have a situation in which there is no .exponential law.direct or original sequence {Xi",X2',，.,X} read-Theorem 41. Assume that X(0) is a nonnegative .ily available for analysis. If we first apply 1-AGO toquasi-smooth sequence; then X(1)= 1- AGOX(0) has{Tr, T2,., TL} to obtain {t,t,.,tz}, wherea quasi-exponential law.It is worth mentioning that the age stochastic{t,t2,.t}= AGO{T,I,.,TL} (47)process:it is obvious thatN(t)A(t) = ao expt+σW+22log(1+Yk)t;=>T， j= 1,2,.,L.(48)(-)仁1=1(45Now the“original" observation sequence will beproposed by Guo and Lovel3) (which can be regardedas a grey process) in terms of the least-square- fttedx(0)(s;)=ti，i= 1,2,., L.(49)GM(1,1) model also demonstrates an exponential formIt can also be noted that {x(0)(s1), x0(2),.. x(O()}of the process. Actually,is not equidistant spaced, becausesi+1-si≠8j+1-8j，立()(k)= (4 - v()())exp(-u(k:- 2)) (46)for Vi≠j,i,j= 1,2,，., L. Therefore, in terms of thefollowing steps, we will create an equal-gap“original"whereφ=a/(1 + 0.5b) and v=a/(1 + 0.5a).Also, if the class ratio r(k) of a discrete data se-sequence (i.e. equi-spaced):1) Divide {hv T. and obtain a newquence X is of size n, then the admissible interval forsubscript sequenc中国煤化工r(k) is [exp(-2/(n + 1),exp(2/(n+ 1))] for a quasi-exponential law fitting discrete data sequence X asMYHCNMH{s1, S2,..,MH-9}. (50)R. Guo and C. E. Love/Grey Repairable System Analysis139It is obvious that the values in the sequence {81, 82,sequences into 17 sub-data sequences. GM(1,1) mod-...,sL} are mostly non-integers. Therefore it is re-eling was performed for each sub-sequence, and the 17quired that a mixed real-valued indexed sequence {81,GM(1,1) groups computational results were summa-i2,s2,i3...,iL,8L} and a corresponding sequencerized in Table 3. For each group, the starting timex{!)= {(0(1)x(O)(i2), x()(s2), x(3),, x()(in),listed in Table 3 is just the value of“x(0)(1)”.x(0)(sL)} be created. It is necessary to point out thatTable 2 Cement Roller data recordl22,23]the symbol is does not necessarily represent a singleTBWCinteger and there should be integers between Si-1 and108009:33....; 1200141422) Determine the integer(s) is such thats;-1