Coordinated Fuzzy Autocatalytic Set of Fuzzy Graph Type- 3 of an Incineration Process 1 SUMARNI ABU BAKAR, 2 TAHIR AHMAD & 3 SABARIAH BAHARUN 1,2 Theoretical & Computational Modeling for Complex Systems (TCM), Ibnu Sina Institute for Fundamental Science Studies & Department of Mathematics, Faculty of Science UTM, Skudai, Johor, MALAYSIA. 3 Malaysia-Japan International Institute of Technology (MJIIT), Universiti Teknologi Malaysia, International Campus, Jalan Semarak, Kuala Lumpur, MALAYSIA 1 sumarni@tmsk.uitm.edu.my , 2 tahir@ibnusina.utm.my , 3 drsabariah@ic.utm.m y Abstract: - Fuzzy Autocatalytic Set of Fuzzy Graph Type-3 (FACS) has provided a graphical model of an incineration process. Six important variables identified in the incineration process are represented as nodes and the catalytic relationships are represented by fuzzy edges in the graph. In this paper a non-coordinated FACS of the model is transformed into coordinated FACS. Preliminary analysis of the new graph revealed some relations to left Perron vector of transition matrix for fuzzy graph of FACS. Key-Words: - Fuzzy Autocatalytic Set, Fuzzy Graph, Incineration Process, Directed Laplacian for FACS. 1 Introduction The emergence of fuzzy graph to autocatalytic sets has instigated a new concept named Fuzzy Autocatalytic Set (FACS) [1, 2]. It is defined as a subgraph where each of whose nodes has at least one incoming link with membership value ( ) ( ] E e e i i ∈ ∀ ∈ , 1 , 0 µ . A clinical waste incineration process in Malacca [1, 2] is modelled using this concept where it incorporates fuzzy graph of type-3, ( ) E V G F , 3 , where both the set of vertex, V and edge, E are crisp, but the edges have fuzzy heads and tails. As for incineration process, six variables that play vital roles in the process are waste (v 1 ), fuel (v 2 ), oxygen (v 3 ), carbon dioxide (v 4 ), carbon monoxide(v 5 ) and other gases including water (v 6 ). The vertices of the graph correspond to the variables and a directed link from vertex i to vertex j indicates that variable i catalyzes the production of variable j. When 3 F G is specifically considered in the construction of FACS, the description of its fuzzy head, fuzzy tail and fuzzy edges connectivity of the edges are given in [1, 2]. Several new results of FACS which linked to Perron-Frobenius (PF) Theorem have been discussed in previous study [1, 2]. These include formation of new definition of transition matrix for fuzzy graph of FACS [3] and Directed Laplacian matrix of FACS [4]. Some properties of the matrices were then established which comprised of stochasticity and its relevancy to PF Theorem [5, 6]. The dynamics model of the graph using left Perron vector of transition matrix for fuzzy graph of FACS which resulted on the evolution of variables on a longer time scale had been investigated [7]. Furthermore the least importance variables correspond to the least value of the element in the left Perron vector was established. In the following section, transformation of non- coordinated graph of FACS to coordinated graph is presented using Directed Laplacian matrix of FACS and adapted technique of one dimensional optimization [8]. 2 Directed Laplacian Matrix of FACS The definition of Directed Laplacian matrix for fuzzy graph of FACS is given as below. Definition 1 [4] Suppose P * transition matrix for fuzzy graph of FACS with P* u,v a fuzzy value of moving from u to v. We let 1 denote the all 1s vector then P * 1 = 1. If we assume the graph is strongly connected and aperiodic, then, from PF Theorem, there exist a unique (row) vector, ϕ for which ϕ ϕ = * P with () 0 > v ϕ for all v and () ∑ = 1 v ϕ which is called the Perron vector of P * . Let Φ the diagonal matrix with ( ) () v v v ϕ Φ = , , then directed Laplacian of FACS is Mathematical Models and Methods in Modern Science ISBN: 978-1-61804-106-7 42