METAONTOLOGY DEDS: OPERATIONAL DYNAMIC SYSTEM ON CLASSES

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1 Volume2 Number3 September2011 pp METAONTOLOGY DEDS: OPERATIONAL DYNAMIC SYSTEM ON CLASSES VladimirV.Kulba 1,SergejNikolsky 2 1 InstituteofControlSciences,RussianAcademyofScience,Moscow,Russia 2 MoscowStateInstituteofElectronicsandMathematics,Moscow,Russia Corresponding author: Sergej Nikolsky Moscow State Institute of Electronics and Mathematics 3/12 Bolshoy Vuzovsky St., Moscow, Russia phone: nsn1946@yandex.ru Received: 21 April 2011 Abstract Accepted: 15 August 2011 In the present article the parametrical model of dynamic system on classes of objects in structural type COD(evt)[1] formally is under construction, allowing solving an automation problem on function of management with forecasting. The model is under construction on the basis of a coordination principle through a variation of factors of the expenses or security used in models of industrial systems. Keywords dynamic system, decision-making, cooridination principle. Introduction Thedecisionofaproblemofautomationofthe information technologies connected with function of management in organizational-technological systems, consists in consecutive synthesis of set of modelsofobjectofmanagement,setofproblemsofthe decision-making set on models of object of management and set of their decisions, corresponding to management. Researchofthedecisionofaproblemofautomation assumes use of the general form of representation of models of dynamics of object of management. Assuchformitisofferedtousedynamicsdefinitioninspaceofstatesoftheobjects,constructedin structuraltypecod(evt)onthebasisoftheontological approach to the analysis of process of modeling[1]. Articleisconstructedasfollows.InSec.1the sense of formal construction of dynamic system on classesisdefined.insec.2constructionofsuchsystemiscarriedoutonthebasisofacoordinationprinciple. In particular, through a variation of factors of the expenses used in models of economic dynamics. In Sec. 3 the industrial interpretation is discussed. Formal construction of dynamic system and classes In work the structural type has been constructed: COD(evt) =<Evt,A( ) >. CarrierCOD(evt)isthespaceofeventswith objectswhichrepresentssetofthreeofakind: Evt = { <ob,s,t > ob V (ob), s S (ob), t T }, where V (ob)thereisasetofelementaryconceptual objects. One elementary conceptual object ob is pair[1]: ob = ([ob],a(ob)), where[ob] thegeneralnameoftheobjectswhich are elements-copies of set-class of objects A(ob). Let sconsiderthatthestate s S(ob)ofobjectob iscapacity A(ob) ofaclassand(ob)oritssome function f( A(ob) ), e.g.: s = A(ob) or s = f( A(ob) ). (1) Thenthespaceofstatesofobjectobwillbe: S1(ob) = R 1. 28

2 Bydefinitioneventwithobjectobonanelement ofitsdynamicsisakindthree: evt(ob) =<ob,s,t >, wherethepair(s, t)iscalledasanelementofdynamics of object ob. In structural type COD(evt) whichsupportissetevt,itisconsideredtwobasic types dynamics of objects[1]: Elementary time dynamics or T-dynamics, defined as mapping (ob) : T S1(ob), where T theorderedsetofthemomentsoftime. Elementary operational dynamics or S dynamics, defined as operational correspondence: σ F :evt(ob 0 ) evt(ob1) evt(ob 2 ) connecting three of events: evt(ob 0 ), evt(ob 1 ), evt(ob 2 ). Onthebasisof(1)senseofT-dynamicsconsists thatcapacityofaclass A(ob)isdefinedasfunction oftime T.Thesenseof S-dynamicsisthatcapacity ofaclass A(ob)isdefinedbytheoperationdefined on elements of other classes. Thereby dynamics of conceptual elementary objects represent dynamics of classes corresponding to them. Let s choose as model oftime T = Z setofintegersandwewillreplace adesignationofanelementofdynamicsofclass A (ob), e.g. pair: (s, k) = ( A(ob), k), k Z (2) onsinglesymbol s(k). We will designate dynamics of class A(ob) through: A(ob) = {s(k) k Z} In work[1] it is shown that operational dynamics σ F isdynamicsystemwhichisformallydefinedas interpretationint[1( ]aformalarrow1( )in space of events with objects: Int [1( )] = σ F. InthiscaseitissetEvt setofeventswithclasses.formalarrowisasymbolicaldesignofakind: 1( ) =< (s );dom[ cen ]cod >, wheresymbolsdom thearrowbeginning,cen the arrowcentre,cod the arrowend arecategorial variables, associeted with an internal arrowcorrespondence from 1( ). Hence, the problemofformalconstructionofdynamicsystem σ F on classes consists in construction of interpretation of aformalarrow1( )ineventspaceevt. By definition, interpretation Int[1( )] consists inachoiceofvaluesofvariablesdom,cenandcod. The decision of this decision-making problem of in Evttakestheformofassignmentsofvaluestothese variables: dom =evt(ob 0 ), cen =evt(ob 1 ), cod =evt(ob 2 ). Alsoitisdefinedatlevelofclasses: A(ob 0 ), A(ob 1 ), A(ob 2 ) (3) and elements of dynamics of these classes. We will name these classes initial, central and final. Coordination T-dynamics of classes Let sconsiderclassesa(ob 0 ),A(ob 2 ).Letdynamics O( )issetintheformoftheknownbalance equation[2]: s 3 (k + 1) = s 3 (k) + a(k) b(k), k Z,e.g.: s 3 (k) = O( ) or f( O( ) ). Let sdefineinterpretationint[1( )]inevtas following conformity: [ ] s 3 (k) ( ), dom s 0 (k) = a(k) (ob 0 ), cod s 2 (k) = b(k + 1) (ob 2 ). (4) To the given interpretation in Evt there corresponds the diagramme of time coordination shown in Fig. 1, which shows that interpretation(4) provides the coordination between elements of dynamics A(ob 0 )and A(ob 2 )duringtimemoments t = k, t = k + 1,atwhichtheconditionnotemptinessof aclass O( )issatisfiedatanymoment t = k. Fig. 1. Diagramme of time coordination. Really,forany k Zat s 3 (0) = 0performance ofacondition a(k) = b(k + 1)provides: s 3 (k) = a(k) > 0. Itmeansthat O( ) (ob 0 )andif A(ob 0 ) then O( ) also,on τ( ) = +1.Thus, coordination on time for dynamics A(ob 0 ) and A(ob 2 ),givenbyinterpretation(4)isdefinedby conditions Volume2 Number3 September

3 { s3 (k + 1) = s 3 (k) + a(k) b(k), k Z, a(k) = b(k + 1), b(k) = a(k 1), (5) which provide transition possibility dom cod, e.g. existenceofdynamicsystem σ F intheformofpair of co-ordinated dynamics: ( ) = ( A(ob 0 ), A(ob 2 )) satisfyingtoacondition(5).wenamethispairdynamics of transition. Coordination of dynamics of transition with T-dynamics of the central class on states From definition of an element of dynamics of aclass(2)followsthatforco-ordinatedontimedynamics coordination on states consists in a conformity establishment between the conditions entering into elements of dynamics, carried to some moment oftime t.let t = k + 1.Accordingto(5),atcoordinatedontimeInt[1( )]inevttheconformity defining value of an arrow takes place: [ ] s 3 (k + 1) ( ). (6) OntheotherhandInt[1( )]inevtincludesthe conformity defining value of the centre of an arrow: cen c(k + 1) (ob 1 ). Hence, simultaneously with(7) conformity takes place: [ ] c(k + 1) (ob 1 ). (7) To define coordination on states we will consider a following condition of balance of classes O ( ) and A(ob 1 )ondynamics ( ), c(κ + 1) = h(k + 1)s 3 (k + 1), (8) where h(k + 1) thecharacteristicofaclass A(ob 1 ) concerning a class O( ), corresponding to transition dynamics. Substitution in(9) expressions for s 3 (k + 1)from(6)leadstofollowingconditions { c(κ + 1) = h(k + 1)(s3 (k) + a(k) b(k)), (9) a(k) = b(k + 1), b(k) = a(k 1), defining coordination of elements of three T- dynamics A(ob 0 ), A(ob 1 )and A(ob 2 )concerning T-dynamicsofaclass O( ).Itiseasytosee, (10) are conditions of the coordination of dynamics oftransition ( )=( A(ob 0 ), A(ob 2 ))with dynamicsofthecentre A(ob 1 )onthestates,providingexistenceofdynamicsystem σ F asco-ordinated interpretationof T-dynamicsinInt[1( )]inevt. As criterion of dynamics coordination we will consider concept of attainability. Let s say that T- dynamics ( )and A(ob 1 )areco-ordinated onstatesatthemomentoftime t = k + 1ifthesystem σ F isattainableatthemomentoftime t = k+1 e.g. if the condition: cod = b(k + 1) > 0 (10) issatisfied.itiseasytoseethatas a (k) = b(k + 1) owingtothefirstconditionin(10)sotheproblemaboutattainabilityofsystem σ F canbeformulated as the following problem of decision-making Z 0 < a 0, h 0 >withtwocriteria: max a(k)/a 0, 0 < a(k)a 0 = b(k + 1), maxh(k + 1)/h 0, 0 < h(k + 1) h 0, c(κ + 1) = h(k + 1)(s 3 (k) + a(k) b(k)), with set of admissible decisions: { < a(k), h(k + 1) > c(κ + 1) = = h(k + 1)(s 3 (k) + a(k) b(k))}. This problem is a problem on simultaneously reached maximum and it s any admissible decision there is a pareto-optimum. The best pareto-optimum decisionofthisproblemisthefreepoint(a 0, h 0 ). Inarticle[3]itwasshownthatalwaysitisnecessary to search for its optimum decision through afindingoftheoptimumdecisionofproblemz 1 < a 0, h 0 >: max[min {a(k)/a 0, h(k + 1)/h 0 }], c(κ + 1) = h(k + 1)(s 3 (k) + a(k) b(k)), with the same set of admissible decisions. Interrelationofproblems Z 0 < a 0, h 0 >and Z 1 < a 0, h 0 > consiststhatifthefreepoint(a 0, h 0 )istheadmissibledecisionitistheoptimumdecisionofboththese problems. Discussion Thedynamicsystem σ F asco-ordinatedinterpretation Int[1( )] in Evt is mathematical model of some industrial system σ, constructed in structural typecod(evt).thebasisofthismodelismadeby a condition of material balance(9) on the balance equation[2]: c(κ + 1) = h(k + 1)(s 3 (k) + a(k) b(k)) (11) which defines industrial system σ as dynamic whole [1] given in the structural form of set of objects(products) changing in time. The variables of Eq.(12) are following parametres of industrial systems σ: s 3 (k) quantityofproductionwhichisinprocess of manufacture in industrial systems σ on a intervaloftime τ( ) = +1; 30 Volume2 Number3 September2011

4 a(k) inputofindustrialsystems t = kwhichisequalto b(k + 1) plannedoutputofsystem t = k + 1; b(k) outputofindustrialsystems t = k; c(κ + 1) quantityofresourcesofindustrialsystems σ,usedinthecourseofmanufactureonan intervaloftime τ( ) = +1; h(k + 1) thefactorofsecurityinindustrialsystems σwhichcanbeconsideredbyresourcesof manufactureofaunitofproductionasfactorof expenses of resources(or local interpretation of factors of the expenses from Leonie s model) in the given system balance equation: s 3 (k + 1) = s 3 (k) + a(k) b(k), k Z. The balance equation entering in condition(12) is often used for management with forecasting[2]. Itiseasytoseethataccordingtothefirstcriterionthedecisionofproblem Z 1 < a 0, h 0 >isvalueof plannedoutput a(k)ofindustrialsystems σatthe momentoftime t = k + 1(inputofindustrialsystems t = k)dependingon predicted requirement for this production set in size b(k + 1). According to the second criterion received value a(k)shouldbeco-ordinatedwith h(k + 1) factor of security of industrial system σ. Hence,problem Z 1 < a 0, h 0 >representsfunction of co-ordinated management with forecasting, where parametre c(κ + 1) represents possibility of development of resourńes of industrial systems σ in time. Example In Introduction it was noted that automation of the information technologies connected with function of management combines three basic functions: modeling, decision making and set of decisions corresponding to management. The following example shows how this problem can be solved using structural type COD(evt) from metaontology DEDS[1]. Fromthelogicalpointofviewmodelingbasedon metaontology DEDS is the interpretation of dynamic system σ F definedthroughdescriptionofparameters oftheproblemofdecision-making Z 0 < a 0, h 0 >in the lexicon of specific manufacturing field. Consider the higher education industry as a field ofinterpretationthenthedynamicsystem σ F orindustrial systems σ is a institution of higher education.inthiscasetheparametersoftheproblemof decision-making Z 0 < a 0, h 0 >aredefinedasfollows: s 3 (k) numberofstudentsofinstitutionofhigher education t = k; a(k) numberofstudentsadmittedtotheinstitutionofhighereducation t = kwhichinthebestcaseisequalto b(k +1) demand or need of graduates of institution of highereducation t = k + 1; b(k+1) demandorneedofgraduatesofinstitutionofhighereducation t = k + 1; b(k) numberofstudentsgraduatingfrominstitutionofhighereducation t = k; c(κ + 1) classroomfoundationofinstitutionof highereducation σ,usedinthecourseofmanufactureonanintervaloftime τ( ) = [k, k + 1]; h(k + 1) factorofsecurityortheavailability σwhichcanbeconsideredtobetheresourcesof learning process needed for one student during an intervaloftime τ( ) = +1. Solution of decision-making problem Z 0 < a 0, h 0 >atthemomentoftime t = kisthevalue of a(k). The following table contains numerical solutionsof Z 0 < a 0, h 0 >forthecasewheninstitution of higher education σ produce specialists in 5 specialties S j, j = 1, 2, 3, 4, 5. Table 1 Numericalsolutionsof Z 0 < a 0, h 0 >. S j a 0 = b(k+1) s 3 (k) b(k) a(k) a(k) h 0 = h(k) h 0 = h N S S S S S Thevaluesofparameters s 3 (k), a(k), b(k)and b(k + 1)relatetoeachspecialty S j, j = 1, 2, 3, 4, 5. Solutions were obtained for two possible values of parameter h 0 : h 0 = h(k) = 8.1m 2 currentvalueofavailability t = k; h 0 = h N = 10.0m 2 normativevalueofavailability σ. The value of classroom foundation was chosen tobe: c(κ + 1) = c(κ) = 30214m 2. Theproblemofdecision-making Z 0 < a 0, h 0 > defines a(k)asafunctionofparameters(a 0, h 0 ), which defines the set of possible decisions of which mustbe chosenvalueof a(k) possiblesolution Volume2 Number3 September

5 atthemomentoftime t = k.thepossiblesolution corresponds to the management with forecastingifatthemomentoftime t = ktheparameter a 0 = b(k + T) plannedorforecasteddemand for specialists produced by institution of higher education σisgivenasafunctionoftimeparameter T = {0, 1, 2,..., K}. Thedecisionmakingproblem Z 0 < a 0, h 0 > mustbesolvedateachmomentoftime t = k.the tableshowsthatifatthemomentoftime t = k theplannedorforecasteddemandforspecialists S 1 isequalto175andthevalue h 0 = h(k)) = 8.1m 2 ischosentodeterminethevalueof a(k) a 0 b (k + 1)thenthedecisionwillbe a (k) = 168butif thevalueof h 0 = h N = 10.0m 2 thedecisionwillbe a (k) = 150. So it is possible to say that the solutions of decision-makingproblem Z 0 < a 0, h 0 >correspond to the management implementing the principle of trackingsystemwhensolution a (k)trackstheforecasteddemand b(k + 1). Conclusions The dynamic system σ F on classes in structural type COD(evt) is formally constructed. At system construction the coordination principle has been used.thereceivedresultscanbeusedforthesolution of a problem of automation of information technology of management in the organizationaltechnological systems, realised in the form of consecutive synthesis of model of system based on coordination principle the problems of decision-making set onmodelofsystemandsetoftheirdecisions,corresponding to management with forecasting based on coordination principle. References [1] Kulba V., Nikolsky S., Zaikine O., Ontological approach to modeling of discrete event dynamic system, Bulletin of the Polish Academy of Sciences. Technical Sciences, Modeling and optimization of manufacturing systems, 57, 3, , [2] Negoita K., Management applications of System Theory, Moscou, Ed. World, [3] Nikolsky S.N., Problem of managerial process automation on composite dynamics of classes [in Russian], Journal of Automation and Modern Technologies, 5, 18 23, Volume2 Number3 September2011