Interval min-plus algebraic structure and matrices over interval min-plus algebra

Awallia, A. R.; Siswanto, Siswanto; Kurniawan, Vika Yugi

doi:10.1088/1742-6596/1494/1/012010

Cited by 3 publications

(2 citation statements)

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“…Element identity (neutral element) for minimum (⨁′) operations is 𝜀 ′ = +∞ and element identity for plus (⊗ ′ ) operations is 𝑒 = 0 (Tam, 2010). According to (Awallia et al, 2020) if 𝑎, 𝑏, 𝑐, 𝑥, 𝑦 ∈ ℝ 𝜀 ′ then min-plus algebraic properties apply.…”

Section: Min-plus Algebramentioning

confidence: 99%

Determining the Inverse of a Matrix over Min-Plus Algebra

Siswanto,

Gusmizain

2024

JTAM

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Linear algebra over the semiring R_ε with ⊗ (plus) and ⨁ (maximum) operations which is known as max-plus algebra. One of the isomorphic with this algebra is a min-plus algebra. Min-plus algebra that is the set R_(ε^' )=R∪{ε'}, with ⊗^' (plus) and ⨁' (minimum) operations. Given a matrix whose components are elements of R_(ε^' ) is called min-plus algebra matrices. Any matrix can be connected by an inverse. In conventional algebra, a square matrix is said an invertible matrix if the det⁡〖(A)〗≠0. In contrast to max-plus algebra, a matrix is said to have inverse condition if it meets certain conditions. Some concepts from the max-plus algebra can be transformed to the min-plus, because of their structural similarity. This means that the inverse matrix concept in max-plus can be constructed into a min-plus version. Thus, this study will explain the inverse of a matrix over the min-plus algebra, property of multiplying two invertible matrices, and connection between invertible matrix and linear mapping used the literature study method, with literature sources such as books, journals, articles, and theses. The data analysis technique used in this research is qualitative data analysis technique. Then, this article has a principal result that is matrix A∈R_(ε^')^(n×n) has a right inverse if and only if there are permutations of σ and the value of λ_i<ε', i∈{1,2,3,…,n} such that A=P_σ ⊗^' D(λ_i ) which is the inverse of matrices. Furthermore, if B is the correct inverse that satisfies A⊗^' B=E then B⊗^' A=E and B is uniquely determined by A.

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Section: Min-plus Algebramentioning

confidence: 99%

Determining the Inverse of a Matrix over Min-Plus Algebra

Siswanto,

Gusmizain

2024

JTAM

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“…It is assumed that the time when raw materials enter the system together with the time when the product leaves the system, or can be written + 1 = ( ) so that obtained = [ ⊕ ⊗ ].. Then the production process schedule can be formed by determining the periodic time of the production process from the Eigen values of matrix and the starting time of each processor starts working is determined from the eigenvector matrix [8]. Using the algorithm described by Subiono [11] and Awallia [1] and using the help of Scilab software, the results as shown in Figure 2. These results require more iterations compared to previous studies, because the amount of data is greater and the type of matrix may be reducible so that the existence and singularity of Eigen values need to be checked first.…”

Section: Figure 1 T3 Shuttlecock Production Flowchartmentioning

confidence: 99%

Eigen Problem Over Max-Plus Algebra on Determination of the T3 Brand Shuttlecock Production Schedule

Permana

Siswanto

Pangadi

2020

Numerical: J.Mat and Pend.Mat

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The production process is included in the Discrete Event System (DES). The DES independent variable generally depends on the event, so an event is influenced by the previous event. Max-plus algebra can be applied in the DES problem to change the system of nonlinear equations obtained into linear equations. Max-plus algebra is a set of real numbers combined with equipped with operations max and plus ⊗ or can be denoted with . An effective and efficient production process needs to pay attention scheduling steps well. The purpose of this research is to determine the Shuttlecock T3 production schedule using eigenvalue and eigenvector in max-plus algebra. The research method in this research is study of literature and observation. Literature study is carried out by studying references about max-plus algebra, especially material related to scheduling problems, while observation are carried out in the process of taking data of the Shuttlecock T3 production process in Surakarta. The linear equation system that is formed based on the results of the observation is then presented in the form and . The periodic time and initial system production time are determined from the eigenvalue and eigenvector matrix where . The results of the research showed that the production system run periodically every 249 minutes, then the best time for each processing unit to start working can be determined, as well as the Shuttlecock T3 production schedule according to the working hours more effective and efficient can be determined too.

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