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Ruang Vektor Eigen Suatu Matriks Atas Aljabar Max-Plus Interval Siswanto, Siswanto; Suparwanto, Ari; Rudhito, M. Andy
Jurnal Matematika dan Sains Vol 19 No 1 (2014)
Publisher : Institut Teknologi Bandung

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Abstract

Misalkan  himpunan bilangan real. Aljabar Max-Plus adalah himpunan Âmax = Â È {‑¥} dilengkapi dengan operasi maksimum Å dan plus Ä. Dapat dibentuk himpunan matriks berukuran n ´ n yang elemen-elemennya merupakan anggota himpunan Âmax, ditulis . Dibentuk himpunan I(Â)max yaitu himpunan yang anggotanya merupakan interval-interval tertutup dalam Âmax. Himpunan I(Â)max dilengkapi dengan operasi  dan  disebut aljabar Max-Plus interval. Selanjutnya, dapat pula dibentuk himpunan matriks berukuran n ´ n yang elemen-elemennya merupakan anggota himpunan I(Â)max, ditulis . Misalkan  dan , dengan , matriks interval A dikatakan tak tereduksi jika untuk setiap matriks  tak tereduksi. Jika tidak demikian matriks interval A dikatakan tereduksi. Dalam penelitian ini akan dibahas tentang ruang vektor eigen suatu matriks atas aljabar Max-Plus interval. Kata kunci : Ruang vektor eigen, Aljabar Max-Plus interval.   Eigenvector Space of a Matrix of Interval Max-Plus Algebra Abstract Let  be the set of real numbers. Max-Plus Algebra is the set Âmax = Â È {‑¥} equipped with the maximum operation Å and plus Ä. The set  is a set of n ´ n matrix with entries belonging to Âmax. Set I(Â)max i.e the set whose members are closed intervals in Âmax.The set I(Â)max equipped with the maximum operation  and plus  called interval Max-Plus algebra. Furthermore, we can also form the set of size n ´ n matrices whose elements are members of the set I(Â)max written . Suppose  and  where , the interval matrices A is irreducible if for any matrix  irreducible. Otherwise the interval matrix A is said reducible. In this research we will discuss eigenvector space of interval Max-Plus algebra matrix. Keywords : Eigenvector space, Interval Max-Plus algebra.
MATRIKS ATAS ALJABAR MAX-PLUS INTERVAL Rudhito, Marcellinus Andy; Wahyuni, Sri; Suparwanto, Ari; Susilo, Frans
Jurnal Natur Indonesia Vol 13, No 2 (2011)
Publisher : Lembaga Penelitian dan Pengabdian kepada Masyarakat Universitas Riau

Show Abstract | Download Original | Original Source | Check in Google Scholar | Full PDF (244.728 KB) | DOI: 10.31258/jnat.13.2.94-99

Abstract

This paper aims to discuss the matrix algebra over interval max-plus algebra (interval matrix) and a method tosimplify the computation of the operation of them. This matrix algebra is an extension of matrix algebra over max-plus algebra and can be used to discuss the matrix algebra over fuzzy number max-plus algebra via its alpha-cut.The finding shows that the set of all interval matrices together with the max-plus scalar multiplication operationand max-plus addition is a semimodule. The set of all square matrices over max-plus algebra together with aninterval of max-plus addition operation and max-plus multiplication operation is a semiring idempotent. As reasoningfor the interval matrix operations can be performed through the corresponding matrix interval, because thatsemimodule set of all interval matrices is isomorphic with semimodule the set of corresponding interval matrix,and the semiring set of all square interval matrices is isomorphic with semiring the set of the correspondingsquare interval matrix.
Semiring Pseudo-Ternary Maxrizal, Maxrizal; Suparwanto, Ari
Jurnal Matematika dan Sains Vol 19 No 2 (2014)
Publisher : Institut Teknologi Bandung

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Abstract

Dalam makalah ini akan diperkenalkan definisi dan sifat-sifat semiring pseudo-ternary. Selanjutnya, akan diperkenalkan subsemiring pseudo-ternary dan ideal pada semiring pseudo-ternary. Lebih lanjut, ideal-ideal yang terbentuk pada semiring pseudo-ternary akan digunakan untuk membentuk semiring pseudo-ternary faktor.Pseudo-Ternary Semiring Abstract In this paper we introduce the notion of pseudo-ternary semiring. Furthermore, we will introduce pseudo-ternary subsemiring and ideals in pseudo-ternary semiring. Finally, ideals in pseudo-ternary semiring will be used for constructing pseudo-ternary factor semiring. Keywords:  Pseudo-ternary semiring, Factor pseudo-ternary semiring.
SYSTEMS OF FUZZY NUMBER MAX-PLUS LINEAR EQUATIONS Rudhito, M. Andy; Wahyuni, Sri; Suparwanto, Ari; Susilo, Frans
Journal of the Indonesian Mathematical Society Volume 17 Number 1 (April 2011)
Publisher : IndoMS

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22342/jims.17.1.10.17-28

Abstract

This paper discusses the solution of systems of fuzzy number max-plus linear equations through the greatest fuzzy number subsolution of the system. We show that if entries of each column of the coecient matrix are not equal to infinite, the system has the greatest fuzzy number subsolution. The greatest fuzzy number subsolution of the system could be determined by first finding the greatest interval subsolution of the alpha-cuts of the system and then modifying it if needed, such that each its components is a family of alpha-cut of a fuzzy number. Then, based on the Decomposition Theorem on Fuzzy Set, we can determine the membership function of the elements of greatest subsolution of the system. If the greatest subsolution satisfies the system then it is a solution of the system.DOI : http://dx.doi.org/10.22342/jims.17.1.10.17-28
ANALISIS LINTASAN KRITIS JARINGAN PROYEK DENGAN PENDEKATAN ALJABAR MAX-PLUS Rudhito, Andi; wahyuni, sri; suparwanto, Ari; Susilo, F
MATEMATIKA Vol 12, No 3 (2009): JURNAL MATEMATIKA
Publisher : MATEMATIKA

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Abstract

This paper proposes a method to critical path analysis in the project network using max-plus algebra approach.  The project network would be represented as a matrix over max-plus algebra. The dynamic of the project would be modeled and analyzed using max-plus algebra approach. The critical path analysis consists of determining earliest start time, latest completion time and float time. The finding show that the dynamic of the project is could be modeled in a systems of max-plus linear equations. The earliest start times of every node in the project are the solution of the system. The latest completion times of every node in the project are the solution of the modified system. The float time of every activity in the project could be detemined by modify and do some matrices operation over earliest start time vector and latest completion time vector. An example for modeling and computing a project using MATLAB example show that the result was appropriated with the critical path method (CPM).
Keteramatan Sistem Deskriptor Kontinu Musthofa, Muhammad Wakhid; Suparwanto, Ari
Jurnal Fourier Vol 1 No 1 (2012)
Publisher : Program Studi Matematika Fakultas Sains dan Teknologi UIN Sunan Kalijaga Yogyakarta

Show Abstract | Download Original | Original Source | Check in Google Scholar | Full PDF (953.392 KB) | DOI: 10.14421/fourier.2012.11.39-45

Abstract

In this paper the observability of continuous descriptor system of the form Ex(t)= Ax(t) Bu(t), x(0)=x0 will be studied, where  E,A, and B are constant matrices that may be singular and u(t) is piecewise continuous function which is differentiated (m-1) times, where m is the degree of nilpotency system. Two definitions about observability of descriptor systems  along with their characterizations given by Dai and Yip will be both discussed, then further the relationship and comparison between these characterizations will be presented.
Karakterisasi Determinan Matriks atas Aljabar Maks Plus Tersimetri Ariyanti, Gregoria; Suparwanto, Ari; Surodjo, Budi
JURNAL SILOGISME : Kajian Ilmu Matematika dan Pembelajarannya Vol 3, No 2 (2018): Desember 2018
Publisher : Universitas Muhammadiyah Ponorogo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.24269/silogisme.v3i2.944

Abstract

Aljabar maks plus merupakan suatu struktur aljabar ( , Å, Ä) yang tidak mempunyai elemen invers terhadap operasi Å. Dengan kata lain, jika aÎ  maka tidak ada bÎ  sehingga aÅb = bÅa = e, kecuali jika a = e. Oleh karena itu, dikembangkan suatu struktur yang lebih luas yang disebut aljabar maks plus tersimetri, dinotasikan dengan ( , Å, Ä) dengan  = / di mana  suatu relasi ekuivalensi. Dengan adanya struktur ini, maka elemen di dalam aljabar maks plus tersimetri akan mempunyai elemen negatif terhadap operasi Å. Akibatnya, determinan matriks atas aljabar maks plus tersimetri dapat didefinisikan. Dalam tulisan ini akan dikembangkan karakterisasi determinan matriks atas aljabar maks plus tersimetri, khususnya di dalam hubungannya dengan adjoint. Hasil utama yang diperoleh yaitu untuk suatu  dengan   aljabar maks plus tersimetri, elemen nol e dan elemen identitas e berlaku sifat , di mana notasi   menyatakan relasi setimbang.
PEMODELAN ALJABAR MAX-PLUS DAN EVALUASI KINERJA JARINGAN ANTRIAN FORK-JOIN TAKSIKLIK DENGAN KAPASITAS PENYANGGA TAKHINGGA ANDY RUDHITO, M.; WAHYUNI, SRI; SUPARWANTO, ARI; SUSILO, F.
Jurnal Matematika Vol 1 No 1 (2010)
Publisher : Mathematics Department, Faculty of Mathematics and Natural Sciences, Udayana University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.24843/JMAT.2010.v01.i01.p10

Abstract

This paper aims to model and determine the service cycle completion time of noncyclic fork-joinqueueing networks with infinite buffer capacity, using max-plus algebra. The finding show that thedynamics of the noncyclic fork-join queuing networks with infinite buffer capacity can be modeledinto a matrix equation over max-plus algebra. We can also show that the service cycle completion timeof queuing networks is a max-plus eigenvalues of the matrix in the equation.slklsklslsllsllllllllllllllllllllll
SIFAT PERIODIK JARINGAN ANTRIAN SERI TERTUTUP DENGAN PENDEKATAN ALJABAR MAX-PLUS Rudhito, M Andy; Wahyuni, Sri; Suparwanto, Ari; Susilo, Frans
MATEMATIKA Vol 14, No 2 (2011): JURNAL MATEMATIKA
Publisher : MATEMATIKA

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Abstract

  Abstract. This article discussed about the properties of  closed periodic queuing network series susing max-plus algebra. The result showed that the properties of  closed periodic dinamic queuing network series can be determined by using the concept of eigen values ??and eigen vectors of max-plus matrix in the network model. Through the max-plus eigen vector fundamental, can be determined faster early time departure of customers of departure to the next customer periodically, with a large period of max-plus eigenvalue
THE BEST SOLUTION FOR INEQUALITIES OF A O CROSS X LOWER THAN X FROM B O DOT X USING HIGH MATRIX RESIDUATION OF IDEMPOTENT SEMIRING Susilowati, Eka; Suparwanto, Ari
Jurnal Sains Dasar Vol 5, No 1 (2016): Jurnal Sains Dasar
Publisher : Yogyakarta State University

Show Abstract | Download Original | Original Source | Check in Google Scholar | Full PDF (1062.184 KB) | DOI: 10.21831/jsd.v5i1.12663

Abstract

Abstract A complete idempotent semiring has a structure which is called a complete lattice. Because of the same structure as the complete lattice then inequality of the complete idempotent semiring can be solved a solution by using residuation theory. One of the inequality which is explained is  where matrices A,X,B with entries in the complete idempotent semiring S. Furthermore, introduced dual product , i.e. binary operation endowed in a complete idempotent semirings S and not included in the standard definition of complete idempotent semirings. A solution of inequality  can be solved by using residuation theory. Because of the guarantee that for each isotone mapping in complete lattice always has a fixed point, then is also exist in a complete idempotent semirings. This of the characteristics is used in order to obtain the greatest solution of inequality . Keywords: complete lattice, complete idempotent semiring, dual Kleene Star, dual product, residuation theory