A hybrid approach of Hungarian method to find optimal solution for solving Fuzzy Transportation Problem using Hexagonal Fuzzy numbers

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INTRODUCTION
Transportation Problem is a branch of Linear Programming Problem which is used to find the optimal solution for transportation cost. Transportation Problem first introduced by Hitchcock (1941) and we can find brief introduction and solving methodology in [3,4,6,7,13,17,19]. [1, 2, 5, 10 -16 ] was introduced some new algorithm to derive the direct optimal solution for transportation problem and Fuzzy Transportation Problem using different fuzzy numbers.
Fuzzy theory was introduced by LA Zadeh in 1965 and proposed a mathematical model for dealing with uncertainty concepts and every problem will have several solutions. [9] given the basic definitions for fuzzy mathematical programming problems.

FUZZY TRANSPORTATION PROBLEM [FTP]:
The fuzzy transportation problems in which a decision maker is uncertain about the precise values of transportation cost, availability and demand. They are categorized into two types namely, Balanced Fuzzy Transportation Problem [BFTP] and Unbalance Fuzzy Transportation Problem [UFTP]. Unbalance fuzzy transportation problem is converted into balanced fuzzy transportation problem by introducing new dummy origin or dummy destination as per requirement.

DEFINITION: HEXAGONAL FUZZY NUMBERS [5]
The Fuzzy Number H is a Hexagonal Fuzzy is a hexagonal fuzzy number denoted ( , , , , , ; 1) and its membership function ( ) is given below: 62 Numerous types of numbers are employed and created using various techniques. While [1] presented an intuitionist fuzzy number, [8] employed trapezoidal fuzzy numbers and proposed a novel technique and used alpha-cut triangular numerals.

HYBRID APPROACH OF HUNGARIAN ALGORITHM FOR SOLVING HEXAGON FUZZY TRANSPORTATION PROBLEM [HFTP]:
Step 1: Check the given FTP is balanced or not balanced.
Step 2: If it is balanced move to Step 4, if it is not balanced move to Step 3.
Step 3: Convert the given UFTP into BFTP by adding dummy origin or dummy destination.
Step 4: Transform the HFTP into crisp transportation problem using Range technique [CP].
Step 5: Check whether the objective function is Minimum or Maximum, If its Maximum convert into Minimum with usual process.
Step 6: Check the Transportation matrix is square, if not make it square by adding dummy row or column with zero weightages.
Step 7: Apply the Hungarian Method.
Step 8: Get the optimum solution and identify the encircled elements.
Step 9: Consider the cell containing encircled zeros and allocate the supply and demand first in the encircled cell Step 10: Check the supply and demand are fully satisfied.
Step 11: If yes move to step 13 otherwise move to step 12.
Step 12: Use traditional method to find the optimal solution.
Step 13: Write down the optimum solution.

Methods
Optimal Solution Example 1 Method used in [5] 299.49 Proposed Method 265.5

CONCLUSION
In this article, the author introduced new hybrid approach of Hungarian method for finding optimal solution for FTP by using Hexagonal Fuzzy numbers. Numerical example problem is taken from the literature and solved the same problem using proposed method. A comparison is also made between proposed and literature method, it revels that the proposed method got the better optimal solution. Within short duration, one can get the optimum solution by using this proposed method.