KABA ARALIKLI KATSAYILARA SAHIP ÇOK AMAÇLI DOĞRUSAL PROGRAMLAMA PROBLEMİNİN SIFIR TOPLAMLI OYUN İLE ÇÖZÜMÜ

Yıl 2024, , 97 - 113, 26.06.2024

Gizem Temelcan

https://doi.org/10.55071/ticaretfbd.1447939

Öz

Kaba sayılardan oluşan aralıklara sahip katsayılar içeren, çok amaçlı doğrusal programlama (MOLPRIC) problemi için bir çözüm önerisinde bulunulmuştur. Bu çalışmada ele alınan probleme uzlaşmacı çözümler kümesi önerilmiş olup çözüm algoritması iki aşamalı olarak düzenlenmiştir. İlk aşamada, MOLPRIC probleminin barındırdığı amaç fonksiyonlarının sayısı dikkate alınarak her bir tek amaçlı LPRIC kaba optimal çözümü bulunmuştur. İkinci aşamada ise MOLPRIC probleminin kaba optimal çözümünü bulmak üzere sıfır toplamlı oyundan yararlanılmıştır. Çok amaçlı problemlerin çözüm sürecinde amaç fonksiyonları arasındaki ödünleşim ağırlıklarının belirlenmesinde genellikle ağırlıklı toplam yöntemi kullanılmaktadır. Ancak amaç fonksiyonlarının sayısı arttığında bu geleneksel yöntem uygulamada zorluk çıkarabilmektedir. Dolayısıyla önerilen algoritmanın özgünlüğü, ikiden fazla amaç fonksiyonuna sahip MOLPRIC problemlerine kolay uygulanabilir olmasıdır. Bu motivasyonla, farklı amaç değerleri arasında sıfır toplamlı oyunun uygulanması, farklı uzlaşık çözümlerin bulunmasını sağlamaktadır.

Anahtar Kelimeler

Çok amaçlı doğrusal programlama problemi, Kaba sayılı aralık, Oyun teorisi, Uzlaşık çözüm

SOLUTION OF A MULTI-OBJECTIVE LINEAR PROGRAMMING PROBLEM HAVING ROUGH INTERVAL COEFFICIENTS USING ZERO-SUM GAME

Öz

In this paper, a set of compromise solutions is found for the multi-objective linear programming with rough interval coefficients (MOLPRIC) problem by proposing a two-phased algorithm. In the first phase, the MOLPRIC problem is separated into single-objective LPRIC problems considering the number of objective functions, and the rough optimal solution of each LPRIC problem is found. In the second phase, a zero-sum game is applied to find the rough optimal solution. Generally, the weighted sum method is used for determining the trade-off weights between the objective functions. However, it is quite inapplicable when the number of objective functions increases. Thus, the proposed algorithm has an advantage such that it provides an easy implementation for the MOLPRIC problems having more than two objective functions. With this motivation, applying a zero-sum game among the distinct objective values yields different compromise solutions.

Anahtar Kelimeler

Compromise solution, Game theory, Multi-objective linear programming problem, Rough interval coefficient

Kaynakça

• Akilbasha, A., Natarajan, G., & Pandian, P. (2017). Solving transportation problems with mixed constraints in rough environment. Int J Pure Appl Math, 113(9), 130–138.
• Ammar, E. S., & Brikaa, M. G. (2019). On solution of constraint matrix games under rough interval approach. Granular Computing, 4, 601–614. doi:10.1007/s41066-018-0123-4.
• Apolloni, B., Brega, A., Malchiodi, D., Palmas, G., & Zanaboni, A. M. (2006). Learning rule representations from data. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 36(5), 1010–1028.
• Arciszewski, T., & Ziarko, W. (1999). Adaptive expert system for preliminary design of wind bracings in steel skeleton structures. In Second Century of the Skyscraper (pp. 847–855). Springer.
• Atteya, T. E. M. (2016). Rough multiple objective programming. European Journal of Operational Research, 248(1), 204–210.
• Brikaa, M. G., Zheng, Z., & Ammar, E. S. (2021). Rough set approach to non-cooperative continuous differential games. Granular Computing, 6, 149–162. doi:10.1007/s41066-019-00179-1.
• Das, A., Bera, U. K., & Maiti, M. (2016). A profit maximizing solid transportation model under a rough interval approach. IEEE Transactions on Fuzzy Systems, 25(3), 485–498.
• Düntsch, I., & Gediga, G. (1998). Uncertainty measures of rough set prediction. Artificial Intelligence, 106(1), 109–137.
• El-Feky, S. F., & Abou-El-Enien, T. H. M. (2019). Hybrid algorithm for rough multi-level multi-objective decision making problems. Ingenierie Des Systemes d’Information, 24(1), 1–17. doi:10.18280/isi.240101
• Emam, O. E., Fathy, E., & Abohany, A. A. (2016). An interactive model for fully rough three level large scale integer linear programming problem. International Journal of Computer Applications, 155(12), 1–11.
• Fibak, J., Pawlak, Z., Słowiński, K., & Słowiński, R. (1986). Rough sets based decision algorithm for treatment of duodenal ulcer by HSV. Biological Sciences, 34, 227–249.
• Garg, H., & Rizk-Allah, R. M. (2021). A novel approach for solving rough multi-objective transportation problem: development and prospects. Computational and Applied Mathematics, 40(4), 149.
• Greco, S., Matarazzo, B., & Slowinski, R. (2001). Rough sets theory for multicriteria decision analysis. European Journal of Operational Research, 129(1), 1–47.
• Hamzehee, Ali, Yaghoobi, M. A., & Mashinchi, M. (2014). Linear programming with rough interval coefficients. Journal of Intelligent & Fuzzy Systems, 26(3), 1179–1189.
• Hamzehee, A., Yaghoobi, M. A., & Mashinchi, M. (2016). A class of multiple objective mathematical programming problems in a rough environment. Scientia Iranica, 23(1), 301–315.
• Khalifa, H. A. (2018a). Study on multi-objective nonlinear programming in optimization of the rough interval constraints. International Journal of Industrial Engineering & Production Research, 29(4), 407–413. doi:10.22068/ijiepr.29. 4. 407.
• Khalifa, H. A. (2018b). On solutions of linear fractional programming problems with rough-interval coefficients in the objective functions. Journal of Fuzzy Mathematics, 26(2), 415–422.
• Li, J., Mei, C., & Lv, Y. (2013). Incomplete decision contexts: approximate concept construction, rule acquisition and knowledge reduction. International Journal of Approximate Reasoning, 54(1), 149–165.
• Mitatha, S., Dejhan, K., Cheevasuvit, F., & Kasemsiri, W. (2003). Some experimental results of using rough sets for printed Thai characters recognition. International Journal of Computational Cognition, 1(4), 109–121.
• Munakata, T. (1997). Rough control: a perspective. In Rough Sets and Data Mining (77–88).
• Omran, M., Emam, O. E., & Mahmoud, A. S. (2016). On solving three level fractional programming problem with rough coefficient in constraints. Journal of Advances in Mathematics and Computer Science, 12(6), 1–13.
• Osman, M. S., Lashein, E. F., Youness, E. A., & Atteya, T. E. M. (2011). Mathematical programming in rough environment. Optimisation, 60(5), 603–611. doi:10.1080/02331930903536393.
• Pawlak, Zdzisław. (1982). Rough sets. International Journal of Computer & Information Sciences, 11(5), 341–356.
• Pawlak, Z., Słowiński, K., & Słowiński, R. (1986). Rough classification of patients after highly selective vagotomy for duodenal ulcer. International Journal of Man-Machine Studies, 24(5), 413–433.
• Rebolledo, M. (2006). Rough intervals—enhancing intervals for qualitative modeling of technical systems. Artificial Intelligence, 170(8-9), 667-685.
• Roy, S. K., Midya, S., & Yu, V. F. (2018). Multi-objective fixed-charge transportation problem with random rough variables. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 26(06), 971–996. doi:10.1142/S0218488518500435.
• Saad, O. M., Emam, O. E., & Sleem, M. M. (2014). On the solution of a rough interval bi-level multi-objective quadratic programming problem. International Journal of Engineering Innovation and Research, 3(6), 803–809.
• Singh, P., & Huang, Y. P. (2020). A four-way decision-making approach using interval-valued fuzzy sets, rough set and granular computing: a new approach in data classification and decision-making. Granular Computing, 5, 397–409. doi:10.1007/s41066-019-00165-7
• Sivri, M., Kocken, H. G., Albayrak, I., & Akin, S. (2019). Generating a set of compromise solutions of a multi objective linear programming problem through game theory. Operations Research and Decisions, 29(2), 77–88.
• Tao, Z., & Xu, J. (2012). A class of rough multiple objective programming and its application to solid transportation problem. Information Sciences, 188, 215–235.
• Tanackov, I., Badi, I., Stević, Ž., Pamučar, D., Zavadskas, E. K., & Bausys, R. (2022). A Novel Hybrid Interval Rough SWARA–Interval Rough ARAS Model for Evaluation Strategies of Cleaner Production. Sustainability, 14(7). doi:10.3390/su14074343.
• Temelcan, G., Albayrak, I., Kocken, H., & Sivri, M. (2020). Solving Fuzzy Multi-objective Linear Programming Problems Using Multi-player Zero-Sum Game. International Conference on Intelligent and Fuzzy Systems, 1483–1490. Springer.
• Temelcan, G. (2023). A solution algorithm for finding the best and the worst fuzzy compromise solutions of fuzzy rough linear programming problem with triangular fuzzy rough number coefficients. Granular Computing, 8(3), 479-489.
• Velázquez-Rodríguez, J. L., Villuendas-Rey, Y., Yáñez-Márquez, C., López-Yáñez, I., & Camacho-Nieto, O. (2020). Granulation in rough set theory: a novel perspective. International Journal of Approximate Reasoning, 124, 27–39.
• Xu, J., Li, B., & Wu, D. (2009). Rough data envelopment analysis and its application to supply chain performance evaluation. International Journal of Production Economics, 122(2), 628–638.
• Youness, E. A. (2006). Characterizing solutions of rough programming problems. European Journal of Operational Research, 168(3), 1019–1029. doi:10.1016/j.ejor.2004.05.019
• Zhao, J., Liang, J.-M., Dong, Z.-N., Tang, D.-Y., & Liu, Z. (2020). Accelerating information entropy-based feature selection using rough set theory with classified nested equivalence classes. Pattern Recognition, 107, 107517.