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

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

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

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

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

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 )

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)

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.