Input-Output Analysis on Pia Saronde Production Process Scheduling with Invariant Max-Plus Linear System
Sunarwin Ismail, Nurwan, Muhammad Rezky F. Payu, Lailany Yahya, Djihad Wungguli, Asriadi

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Negeri Gorontalo

Abstract

Max-plus algebra is one of the analysis methods of discrete event systems which have many applications on systems theory and graph theory. Max-plus algebra is a set of real number \(\textbf{R}\) combined with \(\varepsilon = -\infty\) equipped with operations max (\(\oplus\)) and plus (\(\otimes\)), can be denoted \((\textbf{R}_{\varepsilon},\oplus,\otimes)\) with \((\textbf{R}_{\varepsilon}=\textbf{R}\cup{\varepsilon})\) The production process of pia saronde is one of the problems that can be analyzed using max-plus algebra. The production process of this product is sequentially carried out by making skin dough, filling, baking, cooling, and packaging the pia. The max-plus algebra theory was used in this research to determine the optimal time in the production scheduling of pia saronde. Meanwhile, the max-plus invariant linear system (SLMI) is a max-plus algebraic theory with the Discrete Event System (DES) were used to solve production-related problems. SLMI analysis produces eigenvalues that represent the optimum production time. The results obtained the max-plus algebra model of \(x(k+1)=\bar{A} \otimes x(k)\), where \(\bar{A}=A\oplus B\otimes C \) and \(y=K\otimes x_0 \oplus H\otimes u \) for input-output SLMI analysis. From the matrix \(\bar{A}\) eigenvalue \(\lambda= 226\) and eigenvector \(v=[278\ 278\ 278 279\ 299\ 302\ 324\ 356\ 488]\) were obtained. Furthermore, the value of \(\lambda\) describes the pia production schedule at a time span of 226 minutes.

Keywords: Max-Plus Algebra, Input-Output Analysis, Invariant Max-Plus Linear System, Optimal Time

Topic: Mathematics