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Abstract
Max-Plus algebra has been used to model and analyze network problems, such as railway networks, project scheduling, queuing systems, and production systems. In this network problem, Max-Plus Algebra is used for scheduling analysis. The assumption used in this analysis is a normal situation, there are no changes in the system that affect scheduling. One factor that is important and needs to be considered is the occurrence of delays in the system which affect scheduling. The objectives of this research are 1) to analyze the effect of system delays on scheduling, and 2) to determine interventions in the form of switching on the server to overcome delay propagation due to schedule delays at certain stages in the system. The simple production system in this research involves 3 servers. The arrangement of the three servers is a combination of parallel and series arrangements. The simulation in this research uses the Scilab. The switching carried out in this research is by speeding up the process on the server by 1 unit of time. The results of this research are 1) direct delay propagation occurs if the delay occurs on the server that contains the eigenvalues. If a delay occurs on a server that does not contain eigen values then there will be no delay propagation provided that the amount of delay on that server does not exceed (t – 1) where t is the time required for the entire process from that server to the next server. If the delay exceeds (t – 1) then delay propagation will occur. 2) Based on simulations using Scilab, delay propagation can be overcome by switching one or more servers according to the delay point. Delays on the server containing the eigenvalues can only be resolved by switching the server. Meanwhile, on servers that do not contain eigenvalues, switching can be done using a combination of these servers.
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References
F. (François) Baccelli, Synchronization and linearity: an algebra for discrete event systems. Wiley, 1992.
B. Heidergott, G. J. Olsder, and J. W. van der Woude, Max Plus at work: modeling and analysis of synchronized systems: a course on Max-Plus algebra and its applications. 2006.
D. Mustofani and A. Afif, “Pharmacy Service Queue Model Using Petrinet and Maxplus Algebra Journal of Mathematics and Mathematics Education,” 2018.
Y. Natalia, I. W. Sudarsana, and D. Lusiyanti, “ PASSENGER WAITING TIME MODELING ON INNER PALU CITY TRANSPORTATION LINES USING MAX-PLUS ALGEBRA,” 2019.
N. Nurwan and M. R. F. Payu, “MAX-PLUS ALGEBRA MODEL ON INAPORTNET SYSTEM SHIPS SERVICE SCHEME,” BAREKENG: Jurnal Ilmu Matematika dan Terapan , vol. 16, no. 1, pp. 147–156, Mar. 2022, doi: 10.30598/barekengvol16iss1pp147-156.
S. R. P. W. Pramesthi and F. Adibah, “SERVICE SCHEDULING OF ANOCYCLIC MULTICHANNEL QUEUE NETWORK SYSTEM WITH 5 SERVERS,” BAREKENG: Journal of Applied Mathematics, vol. 13, no. 1, pp. 039–046, Mar. 2019, doi: 10.30598/barekengvol13iss1pp039-046ar696.
R. Ragana Sakta and M. Rianti Helmi, “MAX-PLUS ALGEBRA AND ITS APPLICATION IN QUEUING SYSTEM,” Jurnal Matematika UNAND, vol. 11, no. 4, pp. 11-13. 271–283, 2022.
A. Permana, S. Siswanto, and P. Pangadi, “Eigen Problem Over Max-Plus Algebra on Determination of the T3 Brand Shuttlecock Production Schedule,” Numerical: Jurnal Matematika dan Pendidikan Matematika, pp . 23–30, Jun. 2020, doi: 10.25217/numerical.v4i1.702.
M. A. Rauf, L. Yahya, and A. Rezka Nuha, “HOUSING DEVELOPMENT PROJECT SCHEDULING MODEL USING PETRI NET AND MAX-PLUS ALGEBRA, ” 2021.
P. Majdzik, “A Feasible Schedule for Parallel Assembly Tasks in Flexible Manufacturing Systems,” International Journal of Applied Mathematics and Computer Science, vol. 32, no. 1, pp. 51–63, Mar. 2022, doi: 10.34768/amcs-2022-0005.
C. M. Rocco, E. Hernandez-Perdomo, and J. Mun, “Assessing manufacturing flow lines under uncertainties in processing time: An application based on max-plus equations, multicriteria decisions, and global sensitivity analysis,” Int J Prod Econ , vol. 234, Apr. 2021, doi: 10.1016/j.ijpe.2021.108070.
L. I. Setiawan, L. Simangunsong, and M. A. Rudhito, "Modelling and Analyzing Quadruped Robot Motion with Two Motors using Max-Plus Algebra," 2021.
P. E. M. Bhaghi, Z. A. K. W. Sabon, and M. A. Rudhito, “Modeling and analysis of hexapod robot motion with two motors using max-plus algebra,” in AIP Conference Proceedings, American Institute of Physics Inc., Dec. 2022. doi: 10.1063/5.0111009.
Z. Sya'diyah, "MAX PLUS ALGEBRA OF TIMED PETRI NET FOR MODELLING SINGLE SERVER QUEUING SYSTEMS," BAREKENG: Journal of Mathematical and Applied Sciences, vol. 17, no. 1, pp. 0155–0164, Apr. 2023, doi: 10.30598/barekengvol17iss1pp0155-0164.
Subiono, “Application of Max-Plus Algebra in Simple Production Systems and Its Simulation Using Matlab Subiono,” 2004.
Subiono, “Min-Max Plus Algebra and Its Application,” 2015.
M. A. Rudhito, “ALGEBRA MAX-PLUS AND ITS APPLICATIONS,” 2016.
V. Yan and I. Ilwaru, “POWER MATRIXES AND THEIR PERIODIC IN MAX-PLUS ALGEBRA The Power of Matrices and its Periodic in the Max-Plus Algebra,” 2014.
E. W. Rahayu, S. Siswanto, and S. B. Wiyono, “THE EIGEN AND EIGENMODE PROBLEM OF MATRIXES IN MIN-PLUS ALGEBRA,” BAREKENG: Journal of Applied Mathematics, vol. 15, no. 4, pp. 133-138. 659–666, Dec. 2021, doi: 10.30598/barekengvol15iss4pp659-666.
G. Ariyanti, “A NOTE ON THE SOLUTION OF THE CHARACTERISTIC EQUATION OVER THE SYMMETRIZED MAX-PLUS ALGEBRA,” BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 16, no. 4, pp. 1347–1354, Dec. 2022, doi: 10.30598/barekengvol16iss4pp1347-1354.
M. Hoekstra, “Control of Delay Propagation in Railway Networks Using Max-Plus Algebra,” 2020.
G. Vissers, “Max-Plus Extensions A Study of Train Delays,” 2022. [Online]. Available: http://repository.tudelft.nl/.