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All Journal InPrime: Indonesian Journal Of Pure And Applied Mathematics Makara Journal of Technology
Magdalena, Ikha
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Chemical Engineering, Chemistry & Bioengineering Civil Engineering, Building, Construction & Architecture Computer Science & IT Electrical & Electronics Engineering Engineering Materials Science & Nanotechnology Mathematics Mechanical Engineering

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Development of A 2D Numerical Model for Pollutant Transport using FTCS Scheme and Numerical Filter Maitsa, Tias Ravena; Hafiyyan, Qalbi; Adityawan, Mohammad Bagus; Magdalena, Ikha; Kuntoro, Arno Adi; Kardhana, Hadi
Makara Journal of Technology Vol. 25, No. 3
Publisher : UI Scholars Hub

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Abstract

This study used the finite difference method to develop a numerical model for pollutant transport phenomenon simulation. Mathematically, the phenomenon is often described by the advection–diffusion differential equation, which is obtained from a combination of the continuity equation and Fick’s first law. The Forward Time Central Space (FTCS) scheme is one of the explicit finite difference methods and is used in this study to solve the model due to its simplicity in solving a differential equation. Yet, this method is currently unstable, which results in oscillations in the model. Thus, a numerical filter (Hansen) is added to the FTCS method to improve the stability of the model. The developed numerical model is applied to several 1D and 2D pollutant transport test cases. Simulation results are compared with those of existing analytical solutions to verify the developed model, and they show that the developed model can simulate the pollutant transport phenomenon well. Moreover, the numerical filter can increase the model stability.
Multi-Depot Vehicle Routing with Heterogeneous Vehicles using Nearest Neighbor Combined with Simulated Annealing Agustine, Debby; Naiborhu, Janson; Magdalena, Ikha
InPrime: Indonesian Journal of Pure and Applied Mathematics Vol. 7 No. 2 (2025)
Publisher : Department of Mathematics, Faculty of Sciences and Technology, UIN Syarif Hidayatullah

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.15408/ggqnkf48

Abstract

The Vehicle Routing Problem (VRP) is an essential component of contemporary logistics, which becomes more complex as the Multi-Depot Vehicle Routing Problem (MDVRP) and the Multi-Depot Vehicle Routing Problem with a Heterogeneous Fleet (MDVRPHF). The main objective of MDVRPHF is to meet all customer demands while minimizing total distribution costs by using vehicles with varying capacities. This paper proposes a metaheuristic framework that first uses the Nearest Neighbor (NN) algorithm to build initial routes and then employs the Simulated Annealing (SA) algorithm to optimize the arrangement of goods within each vehicle, ensuring capacity limits are met. Computational experiments using real-world inspired data, representing 20 items distributed from a Bandung depot to multiple customers with three heterogeneous vehicle types, showed that the proposed hybrid NN–SA method achieved an 18.4% reduction in total distribution cost compared to the NN method alone. These results indicate that this integrated approach offers a practical, computationally efficient solution to the complexities of MDVRPHF, establishing it as a useful tool for logistics planning. Keywords: Multi-Depot Vehicle Routing Problem; Heterogeneous Fleet; Nearest Neighbor; Simulated Annealing; Metaheuristics.   Abstrak Vehicle Routing Problem (VRP) merupakan bagian penting dari logistik kontemporer, yang kompleksitasnya meningkat menjadi Multi-Depot Vehicle Routing Problem (MDVRP) dan Multi-Depot Vehicle Routing Problem with a Heterogeneous Fleet (MDVRPHF). Untuk MDVRPHF, tujuan utamanya adalah memenuhi seluruh permintaan pelanggan sambil meminimalkan biaya distribusi total dengan memanfaatkan kendaraan berkapasitas berbeda. Makalah ini mengusulkan kerangka kerja metaheuristik yang pertama-tama menggunakan algoritma Nearest Neighbor (NN) untuk membentuk rute awal, kemudian algoritma Simulated Annealing (SA) digunakan untuk mengoptimalkan penataan barang di setiap kendaraan agar batas kapasitas terpenuhi. Eksperimen komputasi menggunakan data uji berbasis kondisi nyata yang merepresentasikan distribusi 20 item dari satu depot di Bandung ke beberapa pelanggan dengan tiga jenis kendaraan heterogen. Hasil penelitian menunjukkan bahwa metode hibrida NN–SA ini menghasilkan penurunan biaya distribusi total sebesar 18,4% dibandingkan metode NN murni, yang menunjukkan bahwa pendekatan terpadu ini memberikan solusi praktis dan efisien secara komputasi untuk kompleksitas MDVRPHF. Kata Kunci: Multi-Depot Vehicle Routing Problem; Armada Heterogen; Nearest Neighbor; Simulated Annealing; Metaheuristik. 2020MSC: 90B06, 90C59.
Co-Authors Adityawan, Mohammad Bagus Agustine, Debby Arno Adi Kuntoro Hadi Kardhana Hafiyyan, Qalbi Janson Naiborhu Maitsa, Tias Ravena