A Comparative Analysis of Various Optimization Methods for Solving Fully Fuzzy Transportation Problems

Öz

This study investigates the solution of a Trapezoidal Fuzzy Transportation Problem (FTP) using Basirzadeh’s Fuzzy Ranking Approach through various optimization methods. The FTP was drawn from Pandian and Natarajan’s 2010 article in Applied Mathematical Sciences. They used the Fuzzy Zero Point Method to solve the FTP. However, the fact that this method has eleven steps, each requiring fuzzy operations, makes it a compelling process. To point out this complexity and to illustrate how easy to get the result, Basirzadeh’s method has been adopted in this study. In this method, the fuzzy data is converted into crip data, and a transportation matrix is created. Then it can be solved by any transportation method and the results may remain crisp or they might be fuzzified. In this study, firstly, the FTP has been solved using the Fuzzy Zero Point Method. Not only the initial cost table and final allocation table have been shared with readers, but also all intermediate steps of the fully fuzzy solution have been demonstrated. Subsequently, some common optimization methods used in transportation problems such as Northwest Corner, Least Cost, Russel, and Vogel’s methods, have been applied to Basirzadeh’s method and the results have been compared. Russel’s Method seems to be the best among these methods because it yielded the lowest transportation cost.

Anahtar Kelimeler

• Fuzzy transportation problem
• ranking of fuzzy numbers
• fuzzy zero-point method
• northwest corner method
• least cost method
• Russel’s method

Tam Bulanık Ulaştırma Problemlerinin Çözümünde Çeşitli Optimizasyon Yöntemlerinin Karşılaştırmalı Analizi

Öz

Bu çalışma tümüyle yamuk bulanık sayılardan oluşan bir ulaştırma probleminin Basirzadeh’in bulanık sayıların sıralanmasına dayanan yöntemi benimsenerek, çeşitli optimizasyon yöntemleri ile çözülmesini konu almaktadır. Çözümlemelerin yapıldığı sayısal örnek, Pandian ve Natajaran’ın 2010’da Applied Mathematical Sicences dergsindeki makalelerinden alınmıştır. Pandian ve Natajaran, bulanık ulaştırma probleminin çözümünde bulanık sıfır noktası yöntemini kullanmışlardır. Fakat onbir adımdan oluşan ve her adımında bulanık sayılarla işlemler yapmayı gerektiren bu yöntemin uygulanması zorlu bir süreçtir. Bu zorluğu gözler önüne sermek ve sonucu elde etmenin ne kadar kolay olduğunu göstermek için, bu çalışmada Basirzadeh'in yöntemi benimsenmiştir. Bu yöntemde, bulanık veri keskin veriye dönüştürülmekte ve bir taşıma matrisi oluşturulmaktadır. Daha sonra herhangi bir optimizasyon yöntemi ile çözülebilmekte ve sonuçlar keskin kalabilmekte ya da bulandırılabilmektedir. Bu çalışmada, ilk olarak FTP, Bulanık Sıfır Noktası Yöntemi kullanılarak çözülmüştür. Sadece başlangıç maliyet tablosu ve son atama tablosu okuyucularla paylaşılmamış, aynı zamanda tam bulanık çözümün tüm ara adımları da gösterilmiştir. Ardından, aynı bulanık ulaştırma problemi bulanık sıralama yaklaşımı benimsenerek, Kuzey Batı Köşesi, En Düşük Maliyet, Russel ve Vogel’in optimizasyon yöntemleri ile çözülmüş ve sonuçları kıyaslanmıştır. Bu yöntemler arasında en iyi yöntemin, en düşük ulaşım maliyetini sağlayan Russel Metodu olduğu söylenebilir.

Anahtar Kelimeler

• Bulanık ulaştırma problemi
• bulanık sayıların sıralanması
• bulanık sıfır noktası yöntemi
• kuzey batı köşe yöntemi
• en düşük maliyet yöntemi
• Russel yöntemi
• Vogel yöntemi.

Kaynakça

1. Agrawal, A. & Singhal, N. (2024). An efficient computational approach for the basic feasible solution of fuzzy transportation problems, Int J Syst Assur Eng Manag, 15(7), 3337–3349.
2. Arockiasironmani, A. & Santhi, S. (2022). A new technique for solving fuzzy transportation problem using trapezoidal fuzzy numbers. Journal of Algebraic Statistics, 13(2), 2216-2222.
3. Balasubramanian, K. & Subramanian, S. (2018). Optimal solution of fuzzy transportation problems using ranking function, International Journal of Mechanical and Production Engineering Research and Development (IJMPERD), Vol. 8, Issue 4, 551-558.
4. Basirzadeh, H. (2011). An approach for solving fuzzy transportation problems. Applied Mathematical Sciences, 5(32), 1549-1566.
5. Chanas, S., Kolodziejczyk, W., & Machaj, A. (1984). A fuzzy approach to the transportation problem. Fuzzy Sets and Systems, 13(2), 211-221.
6. Chanas, S., Delgado, M., Verdegay, J. L., & Vila, M. A. (1993). Interval and fuzzy extensions of classical transportation problems. Transportation Planning and Technology, 17(2), 203– 218.
7. Chandran, S. & Kandaswamy, G. (2016). A fuzzy approach to transport optimization problem. Optimization and engineering, 17, 965-980.
8. Charnes, A. & Cooper, W. W. (1954). The stepping stone method of explaining linear programming calculations in transportation problems, Management Science, Vol. 1, No. 1, 49-69.

9. Dantzig, G. B. (1951). Application of the simplex method to a transportation problem. Activity analysis and production and allocation, 209-213.
10. Ebrahimnejad, A. (2014). A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers. Applied Soft Computing, Volume 19, 171-176.
11. Ebrahimnejad, A. & Verdegay, J.L. (2018). A new approach for solving fully intuitionistic fuzzy transportation problems. Fuzzy Optimizaiton and Decision Making 17, 447–474.
12. Eljaoui, E. Melliani, S., & Chadli, L. S. (2018). Aumann fuzzy improper integral and its application to solve fuzzy integro-differential equations by laplace transform method. Advances in Fuzzy Systems, https://doi.org/10.1155/2018/9730502.
13. Fegade, M. & Muley, A. (2024). Solving fuzzy transportation problem using hexagonal number, IJCRT, Volume 12, Issue 3, 816-819.
14. Hitchcock, F. L. (1941). The distribution of a product from several sources to numerous localities. Journal of Mathematics and Physics. 20, 224-230.
15. Hunwisai, D. & Kumam, P. (2017). A method for solving a fuzzy transportation problem via Robust ranking technique and ATM. Cogent Mathematics, 4(1).
16. Kaufman, A. & Gupta, M. M. (1991). Introduction to fuzzy arithmetic theory and applications, Von Nostrand Reinhold Co.
17. Kaur, A. & Kumar, A. (2011). A new method for solving fuzzy transportation problems using ranking function, Applied Mathematical Modelling, Volume 35, Issue 12, 5652-5661.
18. Kaur, A. & Kumar, A. (2012). A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers, Applied Soft Computing, Volume 12, Issue 3, 1201-1213.
19. Khalaf, W. S. (2014). Solving fuzzy transportation problems using a new algorithm. Journal of Applied Sciences, 14(3), 252-258.
20. Khoshnavaa, A. & Mozaffari, M. R. (2015), Fully fuzzy transportation problem, Journal of New Researches in Mathematics, Vol.1, No.3, 40-54.
21. Kumar, A. & Kaur, A. (2011). Application of classical transportation methods for solving fuzzy transportation problems, Journal of Transportatıon Systems Engıneering And Informatıon Technology, Volume 11, Issue 5, 1201-1213.
22. Liu, S. T. & Kao, C. (2004). Solving fuzzy transportation problems based on extension principle, European Journal of Operational Research, 153, 661–674.
23. Maheswari, P. U. & Ganesan, K. (2018). Solving fully fuzzy transportation problem using pentagonal fuzzy numbers , J Phys.: Conf. Ser. 1000 012014.
24. Mathur, N., Srivastava, P. K., & Paul, A. (2016). Trapezoidal fuzzy model to optimize transportation problem, International Journal of Modeling, Simulation, and Scientific Computing, Vol. 7, No. 3, https://doi.org/10.1142/S1793962316500288.
25. Nishad, A. K. & Abhishekh. (2020). A new ranking approach for solving fully fuzzy transportation problem in ıntuitionistic fuzzy environment. Journal of Control, Automation and Electrical Systems, 31(4), 900–911.
26. Pandian, P. & Natarajan, G. (2010). A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Applied Mathematical Sciences, 4(2), 79-90.
27. Pratihar, J., Kumar, R., Edalatpanah, S. A., & Dey, A. (2021). Modified Vogel’s approximation method for transportation problem under uncertain environment, Complex & Intelligent Systems, 7, 29–40
28. Roy, H., Pathak, G., Kumar, R., & Malik, Z. A. (2020). A study of fuzzy transportation problem using zero-point method with ranking of trapezoidal fuzzy numbers, Bulletin Monumental, Volume: 21, Issue: 08, 24-30.
29. Samuel, A. E. (2012). Improved zero point method (IZPM) for the transportation problems, Applied Mathematical Sciences, Vol. 6, no. 109, 5421 – 5426.
30. Shore, H. H. (1970). The transportation problem and the vogel approximation method. Decision Sciences, 1(3-4), 441–457.
31. Srinivasan, R., Karthikeyan, N., Ranganathan, K., & Vijayan, D. V. (2021). Method for solving fully fuzzy transportation problem to transform the materials, Materials Today: Proceedings.
32. Tada, M. & Ishii, H. (1996). An integer fuzzy transportation problem, Pergamon Computers Math. Applic. Vol. 31, No. 9, 71-87.
33. Zadeh, L. A., (1996). Fuzzy logic = Computing with words, IEEE Transactions on Fuzzy Systems, 103 – 111.