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mathematics and education skip to main skip to sidebar mathematics and education andy rudhito research site in mathatematics and mathematics education determining the earliest starting times in project networks with interval activity times using interval max plus algebra m andy rudhito sri wahyuni ari suparwanto f susilo procidi ng of the first international seminar on science and technology isstec 2009 uii yogyaka rta 24 25 januari 2009 abstract the activity times in a project network are seldom precisely known and then could be represented into the interval this paper aims to determine the earliest starting time for each node in the project networks with interval activity times using interval max plus algebra the finding shows that the project networks with interval activity can be represented as a matrix over interval max plus algebra the project networks dynamics can be represented as a iterative system of interval max plus linear equations the interval of earliest start time for each node in the project networks is a solutions vector of the system keywords max plus algebra earliest starting times project network interval full paper diposkan oleh m andy rudhito label earliest starting times interval max plus algebra project network iterative system of fuzzy number max plus linear equations iterative system of fuzzy number max plus linear equations prociding of international conference on mathematics and natural science 2008 fmipa itb bandung 28 30 oktober 2008 m andy rudhito sri wahyuni ari suparwanto f susilo abstract the activity times in a network are seldom precisely known and then could be represented into the fuzzy numbers with max plus algebra approach the network dynamic could be analyzed through its iterative system of fuzzy number max plus linear equations this paper aims to determine the existence and uniqueness of the solution of the iterative system of fuzzy number max plus linear equations the finding shows that if the square matrix of the systems is semidefinite then the solution exists the solution of the system could be determined the solution of the alpha cuts of the system firstly based on the decomposition theorem we can determine the membership function of the elements of solution vectors moreover the solution is unique if the square matrix of the systems is definite keywords max plus algebra system of linear equations fuzzy number full paper diposkan oleh m andy rudhito label fuzzy number max plus algebra system of linear equations 4 eigenvalues and eigenvectors of matrices over fuzzy number max plus algebra in this section we assume that readers have known some basic concepts of fuzzy set and fuzzy number further details can be found in zimmermann h j 1991 lee k h 2005 and susilo f 2006 diposkan oleh m andy rudhito label fuzzy mathematics max plus 3 eigenvalues and eigenvectors of matrices over interval max plus algebra in this section we will review some basic concepts of interval max plus algebra matrices over interval max plus algebra and the existence and uniqueness of interval max plus eigenvalue further details can be found in litvinov l g et al 2001 and rudhito a et al 2008a 2008b readmore click at t he pic ture bellow diposkan oleh m andy rudhito label interval mathematics max plus 2 eigenvalues and eigenvectors of matrices over max plus algebra in this section we will review some basic concepts of max plus algebra matrices over max plus algebra and its relations with graph theory and the existence and uniqueness of max plus eigenvalues further details can be found in baccelli et al 1992 and rudhito a 2003 readmore click at t he picture bellow diposkan oleh m andy rudhito label mathematics max plus eigenvalues and eigenvectors of matrices over fuzzy number max plus algebra introduction the max plus algebra can be used to model and analyze a networks like the project scheduling production system queueing networks etc bacelli et al 2001 rudhito a 2003 krivulin n k 2001 the networks modelling with max plus algebra approach is usually a max plus linear system equations and it can be written as a matrix equation the periodical properties of networks dynamics can be analyzed through the max plus eigenvalues and eigenvectors of matrices in its modelling recently the fuzzy networks modelling has been developed in this paper the fuzzy network refers to networks whose their activity times are fuzzy number the fuzzy scheduling can be read in chanas s zielinski p 2001 and soltoni a haji r 2007 the fuzzy queueing networks can read in lüthi j haring g 1997 and pardoa fuente 2007 when we follow the notions of modelling and analyzing of networks with max plus algebra approach we can use the analyzing of periodical properties of the dynamic can be do through eigenvalues and eigenvectors of matrices over max plus fuzzy number in its modelling for this reasons this paper will discuss eigenvalues and eigenvectors of matrices over max plus fuzzy number before we proceed the essential considerations we will reviewed the notions of eigenvalues and eigenvectors of matrices over max plus algebra and eigenvalues and eigenvectors of matrices over interval max plus algebra outline article next sections posting 2 eigenvalues and eigenvectors of matrices over max plus algebra 3 eigenvalues and eigenvectors of matrices over interval max plus algebra 4 eigenvalues and eigenvectors of matrices over fuzzy number max plus algebra 5 conclusion references bacelli f et al 2001 synchronization and linearity new york john wiley sons chanas s zielinski p 2001 critical path analysis in the network with fuzzy activity times fuzzy sets and systems elsevier science b v lee k h 2005 first course on fuzzy theory and applications spinger verlag berlin heidelberg litvinov g l sobolevskii a n 2001 idempotent interval anaysis and optimization problems reliab comput 7 353 377 2001 arxiv math sc 010180 lüthi j haring g 1997 fuzzy queueing network models of computing systems proceedings of the 13th uk performance engineering workshop ilkley uk edinburgh university press july 1997 pardoa marıa jose fuente david de la 2007 optimizing a priority discipline queueing model using fuzzy set theory computers and mathematics with applications 54 2007 267 281 rudhito andy 2003 sistem linear max plus waktu invariant tesis program pascasarjana universitas gadjah mada yogyakarta rudhito andy wahyuni sri suparwanto ari dan susilo f 2008a aljabar max plus interval prosiding seminar nasional matematika s3 ugm yogyakarta 31 mei 2008 2008b matriks atas aljabar max plus interval prosiding seminar nasional matematika s3 ugm yogyakarta 31 mei 2008 soltoni a haji r 2007 a project scheduling method based on fuzzy theory journal of industrial and systems engineering vol 1 no 1 pp 70 80 spring susilo f 2006 set and logika fuzzy serta aplikasinya edisi kedua graha ilmu yogyakarta zimmermann h j 1991 fuzzy set theory and its applications kluwer academic publishers usa diposkan oleh m andy rudhito label fuzzy mathematics max plus eigenvalues and eigenvectors of matrices over fuzzy number max plus algebra abstract presented at the 3rd international conference on mathematics and statistics icoms 3 institut pertanian bogor indonesia 5 6 august 2008 the activity times in a network is seldom precisely known and then could be represented into the fuzzy numbers with max plus algebra approach the periodical properties of the network dynamic could be analyzed through the eigenvalues and eigenvectors of matrices over max plus algebra in the its modelling this paper aims to determine the eigenvalues and eigenvectors of matrices overfuzzy number max plus algebra the result of this paper can be used to analyze the periodical properties of the network dynamic with its activity times which is represented using the fuzzy numbers this paper is a theoretical investigation based on literature and computation using matlab program the maximum and addition operations of the fuzzy number is defined through its alpha cuts which are the closed intervals the eigenvalues and eigenvectors of matrices overmax plus algebra is extended into eigenvalues and eigenvectors of matrices overfuzzy number max plus algebra through eigenvalues and eigenvectors of matrices over interval max plus algebra the finding shows that eigenvalues and eigenvectors of matrices over fuzzy number max plus algebra could be determined the eigenvalues and eigenvector of every its alpha cuts matrices firstly based on the decomposition theorem we can determine the membership function of the eigenvalues and membership functions of the elements of eigenvectors corresponding to the eigenvalues moreover the eigenvalue is unique if the matrices is irreducible diposkan oleh m andy rudhito label fuzzy mathematics max plus older posts home subscribe to posts atom labels curriculum 1 earliest starting times 1 fuzzy 3 fuzzy number 1 interval 2 learning material 1 learning model 2 math education 4 mathematics 5 max plus 5 max plus algebra 2 media 1 project network 1 system of linear equations 1 max plus articles guest book view shoutbox shoutmix chat widget visitor mathematics feedjit live traffic feed feedjit live blog stats about me m andy rudhito dosen program studi pendidikan matematika universitas sanata dharma yogyakarta email arudhito yahoo co id dan rudhito usd ac id view my complete profile design by finalsense