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All Journal CAUCHY: Jurnal Matematika Murni dan Aplikasi Limits: Journal of Mathematics and Its Applications
Sutopo Sutopo
Universitas Gadjah Mada
Computer Science & IT Mathematics

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Submodul Maksimal Bertingkat Terdekomposisi pada Modul Bertingkat Mohammad Agus Kholilurrohman Kholilurrohman; Sutopo Sutopo; Ari Suparwanto
Limits: Journal of Mathematics and Its Applications Vol. 23 No. 2 (2026): Limits: Journal of Mathematics and Its Applications Volume 23 Nomor 2 Edisi Ju
Publisher : Pusat Publikasi Ilmiah LPPM Institut Teknologi Sepuluh Nopember

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.12962/limits.v23i2.5796

Abstract

Given  be a group with identity , and graded ring  type . By the graded ring can be formed graded module. A module  is called to be a graded module over graded ring  type  if there is family of subgroups of , say , for all  such that  and . The concepts of maximal submodule can also be defined in graded module. The graded proper submodule  of  is said to be a graded maximal if there is no graded other submodule that contains  properly except  itself. Not only usual module can be decomposed, but also graded module can be decomposed. This research discusses about graded modules with decomposed graded maximal submodules. By the decomposition, we obtain some properties that relate with graded maximal submodule, graded simple module, graded module homomorphism, graded torsion module, and graded direct summand.
Determining Logistics Route Capacity Using Maximal Solutions of Two-Sided Max-Min Linear System with Python Implementation Zakia Nur Ramadhani Putri; Ari Suparwanto; Sutopo Sutopo
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 2 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i2.44534

Abstract

Max-min algebra, equipped with maximum and minimum operations, provides a natural framework for modeling systems with bottleneck constraints, such as logistics and transportation networks. An algorithm for solving two-sided linear systems of the form A x = B x in max-min algebra has been proposed, for which the maximal solution can be constructed explicitly for the 1 n case under three conditionsunconstrained, with a given upper bound, and with a given lower boundand generalized to the m n case, terminating after at most m iterations and producing a unique maximal solution with complexity O(m2n). However, no implementation has been made available, which limits its use for larger, practically sized problems: the algorithm may require up to m computational cycles, so the computational burden grows with the matrix sizeparticularly the number of rowsand manual calculation becomes more prone to error. This paper addresses this gap by providing a Python implementation of the algorithm, along with a numerical application to a logistics distribution network. The Python implementation makes the algorithm accessible to practitioners, and the logistics example demonstrates how the model can be used to determine optimal road capacities. This work provides a practical computational tool for solving capacity planning problems in logistics and transportation.
Co-Authors Ari Suparwanto Mohammad Agus Kholilurrohman Kholilurrohman Zakia Nur Ramadhani Putri