Optimizing Profit-to-Cost Ratios in Bakery Production Using the Hasan–Acharjee Fractional Programming Method

Fahliza Adisty, Riri Syafitri Lubis

Abstract


Production optimization under resource constraints can be effectively modeled using Linear Fractional Programming (LFP), where the objective function is defined as a profit-to-cost ratio. This study applies the Hasan–Acharjee method to optimize production planning in a household-scale bakery enterprise in Indonesia, considering four product types and three resource constraints (materials, labor, and equipment). The model was reformulated as a single linear program and solved using LINGO 21.0. Validation against the classical Charnes–Cooper transformation confirmed identical optimal solutions, demonstrating the robustness of the Hasan–Acharjee approach. Sensitivity and trade-off analyses further revealed how variations in costs and production capacity influence profitability. The results highlight both the theoretical relevance of the Hasan–Acharjee method in fractional programming and its practical applicability to small and medium-sized enterprises seeking efficient resource utilization under limited conditions.

Keywords


Charnes-Cooper; Hasan-Acharjee; Linear Fractional Programming; Optimization

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References


[1] B. Reynaldo, R. Widyati, and M. Irzal. “Pengembangan program pecahan linier dengan transformasi aljabar”. Jurnal Matematika Universitas Negeri Jakarta, vol. 5, no. 2, 2017. DOI: https://doi.org/10.21009/jmt.1.1.1.

[2] P. K. Swarup. Linear Programming: Methods and Applications. New Delhi: Sultan Chand & Sons, 2008.

[3] A. Charnes and W. W. Cooper. “Programming with linear fractional functionals”. Naval Research Logistics Quarterly, vol. 9, no. 3–4, pp. 181–186, 1962. DOI: https://doi.org/10.1002/nav.3800090303.

[4] S. Schaible. “Parameter-free convex equivalent and dual programs of fractional programming problems”. Zeitschrift für Operations Research, vol. 18, no. 4, pp. 187–196, 1974. DOI: https://doi.org/10.1007/BF01917205.

[5] G. R. Bitran and H. H. Yanasse. “Computational complexity of the capacitated lot size problem”. Management Science, vol. 30, no. 9, pp. 1123–1136, 1984. DOI: https://doi.org/10.1287/mnsc.30.9.1123.

[6] F. Hanum. “Program fraksional linear”. Jurnal Matematika dan Aplikasi Pemodelan, vol. 7, no. 1, pp. 21–32, 2008. DOI: https://doi.org/10.29244/jmap.7.1.21-32.

[7] G. B. Dantzig. Linear Programming and Extensions. Princeton, NJ: Princeton University Press, 1963.

[8] M. B. Hasan and S. Acharjee. “Solving LFP by converting it into a single LP”. International Journal of Research, vol. 8, no. 3, pp. 1–14, 2011. Available online.

[9] M. K. Zuhanda and E. S. M. Nababan. “Optimasi program linier pecahan interval”. Saintia Matematika, pp. 17–24, 2015. Available online.

[10] J. S. H. Kornbluth and R. E. Steuer. “Goal programming with linear fractional criteria”. European Journal of Operational Research, vol. 7, no. 2, pp. 204–207, 1981. DOI: https://doi.org/10.1016/0377-2217(81)90048-1.

[11] O. K. Gupta and A. Ravindran. “Branch and bound experiments in convex nonlinear integer programming”. Management Science, vol. 31, no. 12, pp. 1533–1546, 1985. DOI: https://doi.org/10.1287/mnsc.31.12.1533.

[12] M. W. Musthofa. “Penerapan algoritma Dinkelbach dan transformasi Charnes–Cooper pada pemrograman fraksional linear di UD Bintang Furniture”. Majalah Ilmiah Matematika dan Statistika, vol. 22, no. 3, 2022. DOI: https://doi.org/10.19184/mlm.v22i3.31615.

[13] N. V. Sahinidis. “BARON: A general purpose global optimization software package”. Journal of Global Optimization, vol. 8, no. 2, pp. 201–205, 1996. DOI: https://doi.org/10.1007/BF00138693.

[14] N. Nurul and E. Harahap. “Optimasi produksi T-shirt menggunakan metode simpleks”. Jurnal Matematika, vol. 20, no. 2, 2021. Available online.

[15] R. Y. Akbar and Mar’aini. “Optimasi produksi pada industri kecil dan menengah Karya Unisi dengan penerapan model linear programming”. Journal Inovasi Penelitian, vol. 2, no. 8, 2022. DOI: https://doi.org/10.47492/jip.v2i8.1255.

[16] R. S. Budianti, A. A. Nurahhman, H. Afriyadi, D. Ahmadi, and E. Harahap. “Penggunaan metode simpleks untuk memaksimalkan target sales pada penjualan paket internet”. Jurnal Riset dan Aplikasi Matematika, vol. 4, no. 2, pp. 108–114, 2020. DOI: https://doi.org/10.26740/jram.v4n2.p108-114.

[17] S. Basriati, E. Safitri, and R. Molina. “Optimalisasi keuntungan pengetaman kayu Berkah Mandiri dengan program pecahan linier menggunakan metode Hasan–Acharjee”. Jurnal Sains Matematika dan Statistika, vol. 6, no. 2, pp. 80–88, 2020. DOI: https://doi.org/10.24014/jsms.v6i2.10551.

[18] S. D. Endarwati, S. Khabibah, and Farikhin. “Program pecahan linier”. Jurnal Matematika, vol. 17, no. 1, pp. 19–23, 2014. Available online.




DOI: https://doi.org/10.18860/cauchy.v10i2.35738

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