Clustering Vertex-Weighted Graphs by Spectral Methods

Spectral techniques are often used to partition the set of vertices of a graph, or to form clusters. They are based on the Laplacian matrix. These techniques allow easily to integrate weights on the edges. In this work, we introduce a p-Laplacian, or a generalized Laplacian matrix with potential, which also allows us to take into account weights on the vertices. These vertex weights are independent of the edge weights. In this way, we can cluster with the importance of vertices, assigning more weight to some vertices than to others, not considering only the number of vertices. We also provide some bounds, similar to those of Chegeer, for the value of the minimal cut cost with weights at the vertices, as a function of the first non-zero eigenvalue of the p-Laplacian (an analog of the Fiedler eigenvalue).

Editorial for the Special Issue on “Fault Trees and Attack Trees: Extensions, Solution Methods, and Applications”

Fault Trees are well-known models for the reliability analysis of systems, used to compute several kinds of qualitative and quantitative measures, such as minimal cut-sets, system failure probability, sensitivity (importance) indices, etc [...]

A graphical method for breaking logical loops based on multi-tree structure

Logical loops or circular logics, interpreted as circular supporting relations among systems, remain a longstanding challenge in the probabilistic safety assessment (PSA). Logical loops are commonly found in complex industrial systems. Due to the existence of the logical loops, the minimal cut sets cannot be directly obtained. In order to solve this problem, the logical loops should be broken properly. This paper proposes a graphical method based on multi-tree structure. By constructing the simplified multi-tree, logical loops both in linearly and non-linearly interrelated systems are solved. To illustrate this method, examples of linearly interrelated systems and non-linearly interrelated systems are given in this paper. As a supplement, this method is applied to the well-known complex logical loops in the nuclear power plant. It shows that this method is highly intuitive and efficient by means of graphs.

Speeding up the core algorithm for the dual calculation of minimal cut sets in large metabolic networks

Background The concept of minimal cut sets (MCS) has become an important mathematical framework for analyzing and (re)designing metabolic networks. However, the calculation of MCS in genome-scale metabolic models is a complex computational problem. The development of duality-based algorithms in the last years allowed the enumeration of thousands of MCS in genome-scale networks by solving mixed-integer linear problems (MILP). A recent advancement in this field was the introduction of the MCS2 approach. In contrast to the Farkas-lemma-based dual system used in earlier studies, the MCS2 approach employs a more condensed representation of the dual system based on the nullspace of the stoichiometric matrix, which, due to its reduced dimension, holds promise to further enhance MCS computations. Results In this work, we introduce several new variants and modifications of duality-based MCS algorithms and benchmark their effects on the overall performance. As one major result, we generalize the original MCS2 approach (which was limited to blocking the operation of certain target reactions) to the most general case of MCS computations with arbitrary target and desired regions. Building upon these developments, we introduce a new MILP variant which allows maximal flexibility in the formulation of MCS problems and fully leverages the reduced size of the nullspace-based dual system. With a comprehensive set of benchmarks, we show that the MILP with the nullspace-based dual system outperforms the MILP with the Farkas-lemma-based dual system speeding up MCS computation with an averaged factor of approximately 2.5. We furthermore present several simplifications in the formulation of constraints, mainly related to binary variables, which further enhance the performance of MCS-related MILP. However, the benchmarks also reveal that some highly condensed formulations of constraints, especially on reversible reactions, may lead to worse behavior when compared to variants with a larger number of (more explicit) constraints and involved variables. Conclusions Our results further enhance the algorithmic toolbox for MCS calculations and are of general importance for theoretical developments as well as for practical applications of the MCS framework.

Analisis Penurunan Capaian Target Produksi Bijih Nikel Menggunakan Metode Fault Tree Analysis di PT Ifishdeco Provinsi Sulawesi Tenggara

Abstrak Proses produksi berarti menghasilkan suatu produk yang bernilai guna. Dalam suatu perusahaan pertambangan, produksi merupakan hal yang sangat penting, sehingga diperlukan perencanaan yang matang. Perusahaan menargetkan jumlah bahan galian yang akan diproduksi baik dalam jangka panjang, menengah maupun jangka pendek. Operasi produksi PT. Ifishdeco menggunakan bantuan alat gali muat excavator Komatsu PC 300 dimana terjadi penurunan produksi bijih nikel sebesar 19,94%. Target produksi sebesar 149.934 mt, namun capaian produksi hanya sebesar 116.603mt. Oleh karena itu, penelitian ini dilakukan untuk mengetahui faktor utama dan nilai probabilitas penyebab turunnya capaian produksi. Salah satu metode yang dapat digunakan untuk menganalisis faktor utama penyebab penurunan capaian produksi adalah menggunakan Fault Tree Analysis (FTA). Persamaan logika top down dari Fault Tree Analysis disubstitusi ke dalam aljabar Boolean hingga diperoleh minimal cut set. Minimal cut set merupakan persamaan akhir yang merincikan top down. Top event penurunan capai target produksi memiliki empat top down, yaitu faktor pengisian, kesiapan fisik alat, efektivitas kerja, dan waktu edar alat gali muat. Berdasarkan nilai probabilitas basic event tertinggi yang bernilai 1, maka faktor utama penyebab penurunan capaian produksi adalah dari faktor umur pakai alat tua, penjadwalan perawatan tidak teratur, kualitas alat buruk, dan proses selective mining. Abstract Analysis of Decrease in Nickel Ore Production Targets Using the Fault Tree Analysis Method in PT Ifishdeco, Southeast Sulawesi Province. Production process meant produce product with beneficial value. At the mining company, production was very important, so it needed to be well-planned. Company was making target amount of digging material to be produced in long term, middle term, and short term. Production operation at PT Ifishdeco utilized digger loader equipment namely excavator Komatsu PC 300. There was decreasing in nickel ore production of 19,94%. Production target was 149,934 mt, but the company was only able to meet 116,603 mt. Objective of this study was to know main factor and probability value which caused the low production performance. One of methods to be used to analyze the main factor was Fault Tree Analysis (FTA). Top down logical equations of Fault Tree Analysis was substituted into Boolean algebra to get minimal cut set. Minimal cut set was a formula of the top down and used to calculate probability. Top event of the decreasing of the production target had four top downs which were filling factor, mechanical availability, effectivity of use, and cycle time of digger loader. Based on the highest probability of basic event which value was equal to 1, the main factor caused the decreasing of productivity were lifetime of equipment, unscheduled of maintenance, low quality of equipment, and selective mining. Kata Kunci: Excavator, Work Effectivity, Boolean, Cut Set, Probability