Mathematics and Education

Determining the Earliest Starting Times in Project Networks with Interval Activity Times Using Interval Max-Plus Algebra

2009-01-29T04:21:00.002+07:00

M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo

Prociding of The First International Seminar on Science and Technology (ISSTEC 2009). UII Yogyakarta. 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.

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ITERATIVE SYSTEM OF FUZZY NUMBER MAX-PLUS LINEAR EQUATIONS

2009-01-29T04:16:00.003+07:00

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

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4. Eigenvalues and Eigenvectors of Matrices over Fuzzy Number Max-Plus Algebra

2009-01-10T17:39:00.008+07:00

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).

3. Eigenvalues and Eigenvectors of Matrices over Interval Max-Plus Algebra

2009-01-10T17:29:00.003+07:00

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).

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2. EIGENVALUES AND EIGENVECTORS OF MATRICES OVER MAX-PLUS ALGEBRA

2009-01-03T08:06:00.006+07:00

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).

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EIGENVALUES AND EIGENVECTORS OF MATRICES OVER FUZZY NUMBER MAX-PLUS ALGEBRA ( Introduction )

2009-01-03T07:37:00.004+07:00

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.

EIGENVALUES AND EIGENVECTORS OF MATRICES OVER FUZZY NUMBER MAX-PLUS ALGEBRA (Abstract)

2009-01-03T00:12:00.004+07:00

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.

DEVELOPING HIGH SCHOOL MATHEMATICS’ CURRICULUM AND HAND-OUT .....

2009-01-03T00:04:00.003+07:00

DEVELOPING HIGH SCHOOL MATHEMATICS’ CURRICULUM AND HAND-OUT WITH INTEGRATING CONTRUCTIVISMS, CONTEXTUAL AND COLLABORATIVE APPROACH IN A LEARNING MODEL CALLED ‘MATEMATISASI BERJENJANG’

By: M. Andy Rudhito and Susento

In general, the research aims to develop High School Mathematics’ curriculum and handouts which integrating constructive, contextual and collaborative approach in a learning model called ‘Matematisasi Berjenjang’. Special purpose of the research in the year I (2007) is firstly to identify teachers’ problems and needs in the application of Curriculum 2006 particularly those related to constructive, contextual and cooperative approach. Secondly, it is to write a guidebook for developing and designing curriculum and handbooks (students’ guides and teachers’ manuals) in which the problems and needs as mentioned before to be addressed by integrating constructive, contextual and cooperative approach in a learning model, namely ‘Matematisasi Berjenjang’
This research belongs to development research. For year I this constitutes a design research consisted of two stages. First stage is an identification process of problems and needs. The resulted data are qualitative, i.e. descriptions of Syllabus and Learning Design, learning activities in classes, feedback from questionnaires and topics resulted from focus group discussion. The data are collected from taking sample of syllabus and learning design, recording class activities, completing questionnaires and focus group discussions. Second stage will be a design process based on results of the identification. The design includes a writing of curriculum developing handbook and students’ guide and teachers’ manuals.
Problems faced and related to the implementation of principals in Curriculum 2006 are that (i) teachers tend to explain step by step in solving problems and believe that mathematical competency can be achieved effectively through instrumental understanding development. (ii) students could not connect concepts to solve problems. (iii) teachers use learning methods which emphasize on explainations of general form at first and they give formal examples along with solving steps and then students imitate them. (iv) teachers are the only main source of learning and have not taken various media to facilitate of students’ knowledge construction process. (v) that teachers evaluate only from written formal mathematical problems solving and do not use other sources.
To deal with the problems above, (i) teachers need to give students opportunities for exploring and discussing problem solutions in their own ways. (ii) teachers need to develop students’ relational understanding in reaching mathematical competency. (iii) teachers need to start learning with contextual problem solving activities and in stages lead to formal mathematics. (iv) it is necessary to have alternative sources of learning that fit with constructivism, contextual and collaborative principles. (v) it is necessary also to have various ways of evaluation, for example written tests, project assignments and products and portfolios.
The guidebook for developing curriculum is composed to elaborate standard and basic competency handbook to be learning programmes covering main topics, sequences, and strategies. Mathematics learning strategies are applied to a learning model called ’Multistages Mathematics’.
Students’ guides play a role as students’ sources of learning so that they have materials and at once get motivations and guidances in the stage of mathematical understanding process facilitated by teachers. The structure of the guides follow a ’Matematisasi Berjenjang’ learning model. Teachers’ manuals are written as supplements to students’ guides in order for being teachers’ handbooks. They contain hints to be facilitators in mathematical learning process. These books also follow a ’Matematisasi Berjenjang’ learning model.

CONSTRUCTING MATHEMATICS LEARNING MODEL DESIGN THAT IS CONSTRUCTIVISTIC, CONTEXTUAL AND COLLABORATIVE FOR TRIGONOMETRY TOPIC

2009-01-02T23:54:00.006+07:00

This research aims to construct a mathematics learning model design that is constructivistic, contextual and collaborative for trigonometry topic in the Grade1 High Scholl which is tried out in the classroom. This research used classroom actions reseach method and did in the Pangudi Luhur Van Lith High School Muntilan Central Java.
The Early Scenario Learning Design which is constructed needs evaluated and repaired from management learning aspect in class, in order to become the Final Learning Scenario. The teacher still needs to pay attention to orientation activity eventhough the learners are ready to rush in to the exploration activity, control exploration and negotiation activity in order to did with efficient, and the negosiation activity should be still paid attention in order to could be done. The student seems ready to to learning activity with change method, can appear some problem solving strategy for the given problems, present the result of working bravely and discuss them, and did negotiation.

CONSTRUCTING LEARNING SIMULATION MODEL FOR QUADRATIC EQUATIONS TOPIC IN THE GRADE1 HIGH SCHOLL WITH “MATEMATISASI BERJENJANG” APPROACH

2009-01-02T23:39:00.006+07:00

This research aims to construct a learning simulation model for quadratic equations topic in the Grade1 High Scholl with “matematisasi berjenjang” approach. This simulation model contains the learning scenario and recording of learning activity implementation in class. This research used classroom actions reseach method and did in the Grade 1 Class Sang Timur High Scholl Yogyakarta.
The Early Scenario Learning Design which is constructed needs evaluated and repaired from management learning aspect in class, in order to become the Final Learning Scenario. The teacher should be given instruction briefly, manage cooperative class with calm and patient , give helping for student with appropriate according the student stage activity, conscious that time scheduling in the Early Scenario Learning Design is not strict. The student seems ready to to learning activity with change method, can appear some problem solving strategy for the given problems, present the result of working bravely and discuss them, the quickness and readiness student in activities stages which is happened are vary.

THE EXPLORATION WINGEOM TO SUPPORT SPACE GEOMETRY LEARNING AT HIGH SCHOOL

2009-01-02T23:27:00.004+07:00

This research aims at describing facility of Wingeom program which can be used to support space geometry learning at high school, developing procedures to use facilities of Wingeom for space geometry topics at high school, and arranging modul for space geometry learning at high school by using Wingeom.
The research found that Wingeom facilities that support space geometry learning in 3-dim submenu from Window menu at Wingeom window. The facilities is situated of this submenu is so complete to support space geometry, such that construct space units and their properties.The others facilities which is interested is animation facilities which is so easy to constructed and used. The facilities above could be used for almost topics in space geometry at high school. The topics which is could be aided such that: draw space units, draw diagonal of side and space, draw diagonal side, compute measure of space geometry, the animation for simetry and nets of space geometry units, draw cutting plane, draw point position, line and plane in space.