CRAMER’S RULE IN INTERVAL MIN-PLUS ALGEBRA
Abstract
A min-plus algebra is a set , where is the set of all real numbers, equipped with the minimum and addition operations. The system of linear equations in min-plus algebra can be solved using Cramer's rule. Interval min-plus algebra is an extension of min-plus algebra, with the elements in it being closed intervals. The set is denoted by equipped with two binary operations, namely minimum and addition . The matrix with notation is a matrix over interval min-plus algebra with size . Since the structure of min-plus algebra and interval min-plus algebra are analogous, the system of linear equations in interval min-plus algebra can be solved using Cramer's rule. Based on the research results, the sufficient conditions of Cramer's rule in interval min-plus algebra are for , and . The Cramer rule is .
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References
Y. I. Puspitasari and Y. Susanti, “Kongruensi Latis Distributif Terkecil pada Semiring dengan Additive Reduct Semilatis,” Pros. Semin. Nas. Aljabar, pp. 55–65, 2016.
H. Anton and C. Rorres, Elementary Linear Algebra. 2005.
Y. Kim and N. Pipattanajinda, “New method for finding the determinant of a matrix,” Chamchuri J. Math., vol. 9, no. January 2017, pp. 1–12, 2017.
G. P. T. Haneefa and Siswanto, “Petri Net Model and Max-Plus Algebra in Outpatient Care at Solo Peduli Clinic, Surakarta,” J. Phys. Conf. Ser., vol. 1776, no. 1, 2021, doi: 10.1088/1742-6596/1776/1/012047.
G. Ariyanti, A. Suparwanto, and B. Surodjo, “Karakterisasi Determinan Matriks atas Aljabar Maks Plus Tersimetri,” J. SILOGISME Kaji. Ilmu Mat. dan Pembelajarannya, vol. 3, no. 2, p. 48, 2019, doi: 10.24269/silogisme.v3i2.944.
Siswanto, V. Y. Kurniawan, Pangadi, and S. B. Wiyono, “Characteristic polynomial of matrices over interval max-plus algebra,” AIP Conf. Proc., vol. 2326, no. February, 2021, doi: 10.1063/5.0039779.
K. G. Farlow, “Max-Plus Algebra,” 2009.
S. A. Gyamerah, P. K. Boateng, and P. Harvim, “Max-plus Algebra and Application to Matrix Operations,” Br. J. Math. Comput. Sci., vol. 12, no. 3, pp. 1–14, 2016, doi: 10.9734/bjmcs/2016/21639.
G. J. Olsder and C. Roos, “Cramer and Cayley-Hamilton in the Max Algebra,” Linear Algebra Appl., vol. 101, no. C, pp. 87–108, 1988, doi: 10.1016/0024-3795(88)90145-0.
J. Plavka, “The weak robustness of interval matrices in max -plus algebra,” Discret. Appl. Math., vol. 173, pp. 92–101, 2014, doi: 10.1016/j.dam.2014.03.018.
Siswanto, A. Suparwanto, and M. A. Rudhito, “Strongly regular matrices and simple image set in interval max-plus algebra,” JP J. Algebr. Number Theory Appl., vol. 38, no. 1, pp. 63–78, 2016, doi: 10.17654/NT038010063.
Siswanto, Pangadi, and S. B. Wiyono, “Robust matrices in the interval max-plus algebra,” J. Phys. Conf. Ser., vol. 1265, no. 1, pp. 1–8, 2019, doi: 10.1088/1742-6596/1265/1/012029.
Siswanto, “PERMANEN DAN DOMINAN SUATU MATRIKS ATAS ALJABAR MAX-PLUS INTERVAL,” Pythagoras, vol. 7, no. 2, pp. 45–54, 2012.
Siswanto, “Aturan Cramer dalam Aljabar Maks-Plus Interval,” Mat. J., vol. 20, no. April, pp. 5–11, 2015.
Musthofa and D. Lestari, “Metode Perjanjian Password Berdasarkan Operasi Matriks atas Aljabar Min-Plus untuk Keamanan Pengiriman Informasi Rahasia (The Password Agreement Method Based on Matrix Operation over Min-Plus Algebra for Safety of Secret Information Sending),” J. Sains Dasar, vol. 3, no. 1, pp. 25–33, 2014.
A. W. Nowak, “The Tropical Eigenvalue-Vector Problem from Algebraic, Graphical, and Computational Perspectives,” p. 136, 2014, [Online]. Available: http://scarab.bates.edu/honorstheses/97.
Siswanto, A. Gusmizain, and S. Wibowo, “Determinant of a matrix over min-plus algebra,” J. Discret. Math. Sci. Cryptogr., vol. 24, pp. 1829–1835, 2021, doi: 10.1080/09720529.2021.1948663.
S. Siswanto and A. Gusmizain, “Determining the Inverse of a Matrix over Min-Plus Algebra,” JTAM (Jurnal Teor. dan Apl. Mat., vol. 8, no. 1, p. 244, 2024, doi: 10.31764/jtam.v8i1.17432.
S. Watanabe, “Min-Plus Algebra and Networks,” SpringerReference, vol. 47, pp. 41–54, 2014, doi: 10.1007/springerreference_21127.
Z. Nur, R. Putri, and V. Y. Kurniawan, “CRAMER ’ S RULE IN MIN -PLUS ALGEBRA,” vol. 18, no. 2, pp. 1147–1154, 2024.
A. R. Awallia, Siswanto, and V. Y. Kurniawan, “Interval min-plus algebraic structure and matrices over interval min-plus algebra,” J. Phys. Conf. Ser., vol. 1494, no. 1, 2020, doi: 10.1088/1742-6596/1494/1/012010.
M. A. Rudhito, “Matriks Atas Aljabar Max-Min Interval,” Pros. Semin. Nas. Sains dan Pendidik. Sains VIII, vol. 4, no. 1, pp. 2087–0922, 2013.
M. A. Rudhito and D. A. B. Prasetyo, “Sistem Persamaan Linear Min-Plus Bilangan Kabur dan Penerapannya pada Masalah Lintasan Terpendek dengan Waktu Tempuh Kabur.” Prosiding Seminar Nasional Sains dan Pendidikan Sains IX, pp. 826–834, 2014.
A. S. Septiany, S. Siswanto, and V. Y. Kurniawan, “Permanent and Dominant of Matrix over Interval Min-Plus Algebra,” BAREKENG J. Ilmu Mat. dan Terap., vol. 18, no. 3, pp. 2029–2034, 2024, doi: https://doi.org/10.30598/barekengvol18iss3pp2029-2034.
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