A LINGUISTIC MODEL OF AGGREGATION IN VIRTUAL MANUFACTURING PROCESSES

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1 A LINGUISTIC MODEL OF AGGREGATION IN VIRTUAL MANUFACTURING PROCESSES Vasle CRĂCIUNEAN 1 ABSTRACT: The current objectves to ncrease the standards of qualty and effcency n manufacturng processes can be acheved only through the best combnaton of nputs, ndependent of spatal dstance between them. Electronc communcaton solves the problem of the dstance between resources. As a natural consequence, achevng these objectves s solved by vrtual producton processes. Due to the complexty of these producton processes they cannot be mastered wthout the help of accurate and complex models that reflect real-tme evoluton and breakdown of aggregate physcal and logcal actons focused on goals. Ths paper proposes a lngustc model for buldng vrtual producton systems by assemblng ntellgent elementary actons wth a hgh degree of ndependence. Thus from a lot of ndependent actons whch we call actons of aggregaton, we defne an aggregaton system of actons capable of generatng aggregate producton processes. In our model the aggregated producton process s a word n a language over the alphabet consstng of actons of aggregaton. KEY WORDS: acton of aggregaton, aggregaton system of actons, aggregated producton process, applcaton of aggregaton, applcaton of dsaggregaton. 1 INTRODUCTION Through the producton process we understand the transformaton of a gven set of resources (raw materals, sem-fnshed goods, energy, labor, equpment) nto fnshed products by aggregatng specfc actons accordng to ther manufacturng recpes. The techncal and technologcal progress has contrbuted to ncreased communcaton capacty and optmal use of the resources and actvtes regardless of ther dstrbuton n space leadng to the possblty of managng them vrtual. It s clear that technology s essental to manage vrtual actons but on the other hand the real need of collaboraton n vrtual organzatons creates the mpetus for the creaton of approprate technologes. Software and hardware technologes open opportuntes for developng new better ways of coordnaton and synchronzaton of the actons nvolved n producton processes. It s a fact that vrtual organzatons based on the archtecture of a dstrbuted manufacturng system, well organzed, can brng a sgnfcant ncrease n effcency and product qualty. Ths paper proposes a model for buldng vrtual producton systems by assemblng actons wth a hgh degree of ndependence. 1 Lucan Blaga Unversty of Sbu, Engneerng Faculty, Romana E-mal: cracunean@sln.ro 124 Constructon of an aggregate producton process from actons nvolves two dstnct actvtes: create new actons and creatng producton processes by aggregatng exstng actons. Therefore when we want to develop a producton process wll have to frst try to buld on exstng actons and just after that to buld new actons needed for our producton process requrement. Ths actons wll enlarge our base of actons. Our base of actons, on whch we buld producton process, together wth facltes for acton constructon and management s the actons nfrastructure. The actons nfrastructure conssts of three dstnct models: a standard acton model, an actons connecton model, and an acton deployment model. A standard acton model defnes what a vald acton s, and how to create a new acton n the actons nfrastructure. All reusable actons wll be bult, thus, accordng to the model of the standard acton. Each actons nfrastructure has a lbrary of reusable actons conformng to the standard acton model. The actons connecton model defnes a collecton of connectors and support facltes for actons aggregaton. Therefore, the connecton model determnes how to buld a producton process or a larger acton from exstng actons. The acton deployment model descrbes how the actons wll be mplemented n a work envronment. ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 12, ISSUE 3/2014

2 Based on ths realty and the belef that ths s the evoluton of producton processes, ths paper proposes a lngustc model for buldng complex producton processes startng from actons. Smplfyng thngs, n ths paper a producton process s a word n a language. All the actons of a producton process are desgned to operate n a parallel and dstrbuted envronment. In 3 we ntroduce the noton of an aggregaton system of actons (S) and the language L(S,f) assocated wth t. The aggregaton system of actons s a general framework that defnes a set of producton processes. In 4. we ntroduce the noton of aggregated producton process and functonal aggregated producton process. We beleve that the present paper provdes the necessary mechansms to buld statc or dynamc optmal producton processes by assemblng ndependent actons. 2 PRELIMINARY NOTIONS AND NOTATIONS Let X be a set. The famly of subsets of X s denoted by Ƥ(X). The cardnalty of X s denoted by X. The set of natural numbers, {0, 1, 2,....} s denoted by N. The empty set s denoted by Ø. An alphabet s a fnte nonempty set of abstract symbols. For an alphabet V we denote by V * the set of all strngs of symbols n V. The empty strng s denoted by λ. The set of nonempty strngs over V, that s V * -{λ}, s denoted by V +. Let Sub(w) denote the set of subwords of w. Each subset of V * s called a language over V. The length of a strng x V* (the number of symbol occurrences n X) s denoted by x. The number of occurrences of a gven symbol a V n x V * s denoted by x a. Let Symb(w) denote the set of symbol occurrences n w. 3 AGGREGATION SYSTEM OF ACTIONS In ths paper we consder that the atomc producton unt of the process s the acton. The actons and the resources on whch the actons are produced, are dstrbuted n a vrtual network of companes workng together. Actons are very nhomogeneous begnnng wth actons of desgn, procurement, logstcs, manufacturng, qualty control etc. There wll be software actons that wll store and dynamcally process all data on the evoluton of the producton process. Therefor we begn by defnng the noton of acton of aggregaton on wtch our model s based. Defnton 3.1. An acton of aggregaton a s a construct of the form a = (I,O,M) I={(r,c,e ) =1,2,...m ; r s a resource, c s the quantty of that resource, e s an event}; The resource r s called an nput resource and represents the necessary resource n the c quantty for the acton a, and e s called an nput event. O={(r o,c o,e o ) o=1,2,...n ; r o s a resource or a fnal product, c o s the quantty of the resource, e o s an event}; If r o s the resource then t s called the output resource and c o s the quantty of that resource, else r o s the fnal output product. The event e o s the output event. M s a set of data and assocated metadata such as duraton of acton, the mnmum allowed stock, costs, techncal data about the acton, software components, etc. Although vrtual producton processes are words n a language that are strngs of actons, the executon order s not strctly that from the word of aggregate producton process. Each component embeds n t a certan degree of ntellgence. It wll know how to check f t has provded the necessary context for startng, for example t wll know to check f t has the necessary resources for the smooth conduct. If ths condtons are not satsfed t wll wat untl these condtons are met. The wat wll last untl t gets a message about updatng the nput resources and then t wll restart the process wth context verfcaton. The acton wll know how to decrement stock, for example, wth consumed resources and ncrement the stock wth the resource created by t and also sendng the correspondng event. An acton of aggregaton also does a very mportant thng that s real-tme update of developments related to varous data such as start tme, the actual length, watng tme, costs, etc. We consder the set A of actons of aggregaton nvolved n a producton process, then we defne the precedence actons by two functons, namely: a bottom-up functon of precedence f: A Ƥ(A), defned as follows: f(a) = {b A O a ϵ I b }, ( ) a A and a top-down functon of precedence g:a Ƥ(A), defned as follows: g(b) = {a A O a ϵ I b }, ( ) b A. In the followng we defne some useful languages over the set A. The symbols from the A set, represent real elementary actons and therefore ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 12,ISSUE 3/

3 the semantc load of the elements of the vocabulary s well defned. Defnton 3.2. An aggregaton system of actons s a construct of the form A s a fnte set of actons called acton of aggregaton; f s a functon f: A Ƥ(A) named bottom-up functon of precedence; g s a functon g: A Ƥ(A) named top-down functon of precedence; L X s a language over A, ( ) X Ƥ(A). It s obvous that L X can be an empty language for some X Ƥ(A). Based on the functon f we defne the applcaton of aggregaton φ as follows: Defnton 3.3. An applcaton of aggregaton s an applcaton φ: Ƥ(A) Ƥ(A) defned as follows: φ X = x X f(x) X Ƥ A. In ths condtons φ n (X) can be defned as: φ 0 (X) = X ( ) X Ƥ(A); φ n (X) = φ(φ n-1 (X)). We observe that φ n (X) Ƥ(A) ( ) n N and ( ) X Ƥ(A), and therefore over the set φ n (X) we have the language L φ n (X) as defned n Defnton 3.2. The set {φ n (X); n 0, X Ƥ(A)} s fnte, because A s a fnte set. Smlarly, we defne the applcaton of dsaggregaton ψ: Defnton 3.4. An applcaton of dsaggregaton s an applcaton ψ: Ƥ(A) Ƥ(A) defned as follows: ψ X = x X g x X Ƥ A. Smlarly, ψ n (X) can be defned as: ψ 0 (X) = X ( ) X Ƥ(A); ψ n (X) = ψ (ψ n-1 (X)). We observe that ψ n (X) Ƥ(A) ( ) n N and ( ) X Ƥ(A), and therefore over the set ψ n (X) we have the language L ψ n (X) as defned n Defnton 3.2. The set {ψ n (X); n 0, X Ƥ(A)} s fnte, because A s a fnte set. In the followng we defne the language specfed by the aggregaton system S. We must pont out that one can defne two mportant languages specfed by the aggregaton system S, namely one startng from the bottom-up precedence functon f wtch we denote by L(S,f) and the other startng from the top-down precedence functon g wtch we denote by L(S,g). In ths paper we wll deal only wth the L(S,f) language, whch s why we use the aggregaton applcaton φ:ƥ(a) Ƥ(A) as defned above, by the bottom-up precedence functon f. Defnton 3.5. Let be an aggregaton system of actons. Then the language L(S,f) specfed by the aggregaton system of actons S s: L S, f = Lemma 3.1. Let X Ƥ(A) =0 be an aggregaton system of actons and the language then R(X) = =0 R(X) = R 1 (X) R 2 (X)(R 3 (X)) * R 4 (X) R 1 X = R 2 X = R 3 X = 1 1 = k= ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 12, ISSUE 3/2014

4 R 4 X = 2 2 = 1 k= 1 Proof. For X Ƥ(A) we consder the followng sequence: X, φ(x), φ 2 (X),, φ k (X), Because the set X A s fnte, t s obvous that ( ) and j such that φ (X) = φ j (X) and therefore L φ X = L φ j (X). Let 1 and 2 ( 1 < 2 ) be the smallest such values for and j. If we denote produces as output a resource or a fnal product and a specfc event. If we have a strng of actons of aggregaton α=a 1 a 2...a m a Symb(α) s of the form a = (I a,o a,m a ) then we denote and wth I α = I a a Symb (α) O α = O a a Symb (α) R 1 X = R 2 X = R 3 X = R 4 X = 1 1 = k= = 1 k= 1 then the Lemma 3.1 s proved. We have a prefx from 0 to 1 whch s not repeated, followed by R 3 whch can repeat from 0 to nfnte and then we have a remander (a part from a perod). Theorem 3.1. Let be an aggregaton system of actons. If all languages L X, X Ƥ(A) are of type {0,1,2,3} n the Chomsky herarchy, then the language L(S,f) s of type n the Chomsky herarchy. Proof. It follows mmedately from closng the languages from the Chomsky herarchy at unon, catenaton, catenaton closure and from Lemma AGGREGATED PRODUCTION PROCESS We wll further look at an acton of aggregaton as a basc producton process whch allows as nput a set of resources n specfed quanttes and a specfed type of event and Defnton 4.1. An acton of aggregaton block of level k ntated by X s a construct of the form: Ɓ k (X) = (I k, O k, L k (X), B k ) L k (X)= I k = I a αε B k X O k = O a αε B k (X) B k s an object, named block-board, that can store nformaton of the form (resource, quantty, event). B k s used by the actons of aggregaton for communcaton. Defnton 4.2. Let be an aggregaton system of actons and X Ƥ(A). Then the language R X = ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 12,ISSUE 3/ =0 s called an aggregated producton network generated by X. Defnton 4.3. Let S be an aggregaton system of actons and R X = =0 be an aggregated producton network generated by X Ƥ(A). Then the aggregated producton network R(X) s a functonal aggregated producton network f for all,1,2 we have the acton of aggregaton block Ɓ k (X) = (I k, O k, L k (X), B k ) and I k+1 =O k. Lemma 4.1. Let

5 be an aggregaton system of actons and X Ƥ(A). If R X = =0 s a functonal aggregated producton network then: ) φ (ψ(x))=x ( )X Ƥ(A), ) ) ψ (φ (X))=X ( )X Ƥ(A), φ n (ψ n (X))=X ( )X Ƥ(A). v) ψ n (φ n (X))=X ( )X Ƥ(A). Proof. Ponts ) and ) are obvous from the defntons of φ and ψ. Statements ) and v) are proved by nducton. We prove only the asserton ) because v) has the same demonstraton. For n=1 we have the asserton ). We assume the statement s true for n,.e. φ n (ψ n (X))=X. We replace X wth ψ(x) and obtan φ n (ψ n (ψ(x)))= ψ(x). We apply φ on ths relatonshp and get φ n+1 (ψ n+1 (X)) = φ(ψ(x)) = X. Thus n each block, two control loops of the aggregated producton process close, one downward and one upward. Each block receves through breakdown a partcular path to follow by the ascendng block and reports back ts progress by aggregaton. Defnton 4.4. Let be an aggregaton system of actons, R(X) an aggregated producton network generated by X and α=α 1 α 2 α n, α L φ (X) a fnte word from R X. Therefore, A= (I, O, α, φ, ψ, B) s called an aggregated producton process : I = I α1 O = O α R X, n = n, n N s called an aggregated producton network generated by X and length n. We remark that R(X, n) contans all aggregated producton processes generated by X and length n, namely all aggregated producton processes of the form: A= (I, O, α, φ, ψ, B) : I = a X I a O = a φ k (X) α R(X,n), B s lke n the Defnton 4.2. In practce we are nterested n an aggregated producton processes that has a lot of nputs I and plenty of outputs O. Obvously, we need to search for our producton process n a subset of R(X, n), whch we denote R(X, Y, n) : X s a mnmal subset Z wth the property a Z I a n = mn O = I n a φ k (X) Y = {a φ n (X) O a O Ø} n R X, Y, n = L (φ k X ψ n k Y ) O a O a, n N α The φ and ψ applcatons are the restrctons of the φ and ψ applcatons from 3.3 and 3.4 defntons. B s an object, called process-board, that can store nformaton of the form (resource, quantty, event). B s used by the actons of aggregaton for communcaton. Defnton 4.5. Let S be an aggregaton system of actons and A=(I,O,α,φ,ψ,B) an aggregated producton process α=α 1 α 2 α n, α Lφ(X).Then the aggregated producton process A s a functonal aggregated producton process f ( ) =1,, n-1 we have I α+1 = O α. 128 Defnton 4.6. Let be an aggregaton system of actons. Then Defnton 4.7. The language R(X, Y, n) s called a network of acceptable aggregated producton process. As can be seen from the above there are a lot of equvalent acceptable producton processes n the sense that they make the same products. You should therefore choose the optmal aggregated producton process, problem whch wll be the subjects of future papers. Also, buldng generatve grammars for languages assocated wth an aggregaton system of actons wll be treated n future papers. 5 CONCLUSION The purpose of ths model s to generate aggregated producton processes by combnng ndependent actons. ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 12, ISSUE 3/2014

6 A process as defned, wll run n a parallel and dstrbuted envronment. Startng the process mples parallel actvaton of all actons of aggregaton. Each acton wll get ts nput data from the block-board and wrte all ts output data also on the block-board. Of course, these operatons wll synchronze dynamcally. We note that we can get more functonal aggregated producton processes that do the same from the perspectve of the process, namely aggregated processes that have the same nputs and outputs. It s natural to ask the queston of choosng the best soluton for our process, optmal n terms of executon tme and cost. The problem of fndng the optmal aggregate can be put n two ways: statc determnaton of an optmum aggregated process or buldng an aggregated process that evolves over tme by changng hs actons accordng to ther effcency. 6 REFERENCES Mowshowtz A., Vrtual Organzaton: A vson of Management n the Informaton Age, The Informaton, 1994 Matthas Dehmer, Abbe Mowshowtz, and Frank Emmert-Streb - Advances n Network Complexty Wley-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, Wenhem, Germany Păun Gh., Mecansme generatve ale proceselor economce, Ed Tehncă, Bucureşt, 1988 Dassow J., Păun Gh., On the power of membrane computng, J. Unversal Computer Sc., 5, 2 (1999), 33-49, ( Calude Ch. S., Păun Gh., Computng wth Cells and Atoms an ntroducton to quantum, DNA and membrane computng, Taylor & Francs, 2001 Alfred V. Aho, Rav Seth, Jeffrey D. Ullman, Complers: Prncples, Technques, and Tools, Addson Wesley, Salomaa, A., Formal Languages, New York, Academc Press, Crăcunean, V., Proectarea Translatoarelor, Sbu, Edtura Unverstăț Lucan Blaga Sbu, 2014, ISBN Crăcunean, V., Stochastc grammars for ntellgent software systems modellng, Proceedngs of the 8 th WSEAS Internatonal Conference, Mathematcal Methods and Computatonal Technques n Electrcal Engneerng (MMACTEE), Bucharest, Romana 2006, pag , ISSN , ISBN Crăcunean, V., Stochastc Grammars n Smulaton Models WSEAS Transactons on Computers, Issue 2, Volume 6, February 2007, pag , ISSN Faban R., Crăcunean, V., Popa E. M., Intellgent system modelng wth total fuzzy grammars, Proceedngs of the 8 th WSEAS Internatonal Conference, Mathematcal Methods and Computatonal Technques n Electrcal Engneerng (MMACTEE), Bucharest, Romana 2006, pag , ISSN , ISBN Crăcunean, V., Faban R., Popa E. M., Package archtecture optmzaton n software applcaton desgn Proceedngs of the 8 th WSEAS Internatonal Conference, Mathematcal Methods and Computatonal Technques n Electrcal Engneerng (MMACTEE), Bucharest, Romana 2006, pag , ISSN , ISBN Crăcunean, V., Faban R., Popa E. M.,Total Fuzzy Grammar Drven Archtecture for Intellgent Software Sytems WSEAS Transactons on Computers, Issue 2, Volume 6, February 2007, pag , ISSN Crăcunean, V., Faban R., Popa E. M.,Algorthm for Optmal Decomposton of Software Applcatons WSEAS Transactons on Computers, Issue 2, Volume 6, February 2007, pag , ISSN Crăcunean, V., Aron C.E., Faban R., Hunyad I.D., Mult Level Recursve Specfcatons for Context Free Grammars - Proceedngs of the 11 th WSEAS Internatonal Conference on COMPUTERS, Agos Nkolaos, Crete Island, Greece, 2007, pag , ISSN , ISBN Crăcunean, V., Faban R., Hunyad I.D, Popa E. M.,Fx pont nternal herarchy specfcaton for context free grammars - Proceedngs of the 11 th WSEAS Internatonal Conference on COMPUTERS, Agos Nkolaos, Crete Island, Greece, 2007, pag , ISSN , ISBN ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 12,ISSUE 3/

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