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

Abstract

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.

Keywords

• 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

Abstract

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.

Keywords

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

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