Comparative Study of Ranking Methods for Fuzzy Transportation

There are several methods that are used to solve the traditional transportation problems whose units of supply, demand quantities, and cost transportation are known exactly. These methods obtain basic solution, and develop it to the best solution through a series of consecutive calculations to obtain the optimal solution. The steps are more complex with fuzzy variables, so this paper presents the disadvantages of solutions of the traditional ways with existence of variables in the fuzzy form. This paper also presents a comparison between the results that emerged after using different conversion ranking formulas to convert from fuzzy form to crisp form on the same numerical example with a full fuzzy form. The problem has been then converted into a linear programming model, and the BIG-M method to be later used to find the optimal solution that represents the number of units transferred from processing or supply centers to a number of demand centers based on the known cost of transportation. Achieving the goal of the problem is by finding the lowest total transportation cost, while the comparison is based on that value. The results are presented in a comprehensive table that organizes data and results in a way that facilitates quick and accurate comparison. An amendment to one of the order formats was suggested, because it has different results compared to other formulas. One of the ranking equations is modified, because it has different results compared to other methods . .


Introduction
Transportation problem is classified as an important linear programming model which is solving means finding the optimal solution that represents the final optimum value of the total cost of transportation problems. Researchers [1] showed that the first transportation model was presented by Hitchcock. In 1965, the theory of fuzzy set was presented by [2]; whereas, the concepts of uncertainty and fuzzy set were developed by many researchers [3].
In general, the transportation model "classic model" represents the known data in the problem which is the cost of transportation of one unit from supply center to demand center. This model is solved by many different methods to find an optimal solution, such as lower cost LCM, north-west corner NWM, Vogel approximated method VAM, and stepping stone method SSM [4]. All these famous methods looking for an optimal distribution way to transport unites among cells of the model table with lowest total cost value.
Solving the model means finding the number of units that are transported from the number (i) of appropriate distribution supply centers to a number (j) of appropriate demand centers, so that the goal is to get the lowest cost of transferred units. These costs are organized in a table which is appropriate to the total number of distribution centers and the number of demand centers as described in Table-1 [5]. Table 1 is a number of units which transported from ( ) source to( ) demand. is a transportation cost for one unit from ( ) source to ( ) demand. S i is a number of unit which are available at ( ) source. D j is a number of unit which are demanded from ( ) destination.

Basic concepts
In this section, some definitions represent basic information of the proposed comparison [6,7]. Definition 1: A function ( ̌) be a ranking function, where H( ̌) is known by a set of fuzzy numbers into real numbers, such that is mapping each fuzzy number (triangular, trapezoidal or pentagon) into real numbers line. Definition 2: Let ̌ subset of universal set of real numbers R then it is said to be fuzzy set number if its membership function ̌( ) mapping domain element to closed interval [0,1] Membership function has the following properties.
1-It is represented by piecewise continues function or discrete points. 2-It holds a convex function property. 3-It is defined by many kinds of parameters as triangular, trapezoidal, pentagonal or octagonal [8].

4-If there exists
such that ̌( ) then ̌ said to be normal. The following Figure ) where are real numbers and its membership function ̌( )is written as follows [3]: Definition 4: A fuzzy numbers set X is said to be trapezoidal fuzzy number and expressed by ( ) where are real numbers and its membership function ̌( )is formed as follows: Other definitions such as Pentagonal, octagonal, etc. are defined similarly [9].

Mathematical Model and Environment of The fuzzy Transportation
Transportation problem and its available data include three main parts which follow the model of linear programing. The first part of transportation problems related to existence of the objective function that contains the total cost of transportation which depends on the number of units ( ) and costs that were assigned previously for each cell in the model of transportation problem. The objective function of linear programing is satisfied in terms of the first part that has the following form: ∑ ∑ (3) The second part is satisfied within the form of constraints to the sum of the required units that have been transported. Note that the number of these units cannot be more than number of available supply units [2]. ∑ Also, the number of units equipped not less than the number of units required from demand centers.
(5) In general, in transportation model, the number of available units in the supply sources is equal to the number of total demand [6].
∑ ∑ (6) The last requirement of the whole linear programing based on the meaning of non-negativity which is satisfied due to the numbers that are used real and positive units.
; for all i, j.
(7) The general mathematical formula for linear programming is represented by the following transportation model [10]. Minimize (Z): ∑ ∑ Subject to constraints: In many transportation problems the decision maker has no proven and uncertain information about the number of units that are available for transportation from supply centers and the number of requirements for all the following expressions ( ) ( ) ( ) ( ) The fact above is depending on the nature of the topic on which the problem was designed, and can represent these data with triple (triangular) points ( ) trapezoidal points ( ) pentagonal point or more [6,8].

Shortcoming of the Existing Methods
There are several methods of solution apply algorithms similar to those used in traditional problems, and develop it to include fuzzy data after definition of some operations and properties. Meanwhile, some of shortcoming points arise while applying the algorithms.
1-The algorithms of the famous methods to obtain the basic solution for traditional transportation problem are incompetent when it used to solve a model that contains fuzzy triangular, trapezoidal or pentagonal data [8]. Additionally, some of these problems its data consist of two sets of membership and non-membership, and this resulted in increases the complexity of arithmetical operations [5].
2-The algorithms of the developed methods for solving the fuzzy data need to have many additional calculations in order to obtain the basic solution, and then develop it to reach the optimal solution [11].
3-Some researchers used the general model of linear programming to solve the fuzzy model by dividing it into problems equal to the number of variables in a single cell. This procedure doubles the number of iterations that used in the algorithm of solution [12]. 4-While applying some original algorithms to solve a fuzzy transportation problem because of using subtraction operations, some negative numbers appear in the occupied cells that represent the number of transferred units according to transportation problem model. The negative signal is not realistic and not correspond to the nature of used data [8].

Ranking Functions ( ):
In order to avoid the shortcoming that were presented by solving the transportation model which includes data in the form of fuzzy numbers, the ranking function is used for the purpose of converting the data of the problem from fuzzy number to crisp number (R). Thus, ranking function shortens the procedures to reach to the optimal solution. The problem is first converted into a linear programming problem, and then is solved by using a software program (TORA) that characterized by precision and the lowest number of procedures.
To study the results and compare the elements of optimal solution in every format of ranking formulas, the following numerical example in the Table-2 shows a full fuzzy formula data of transportation problem with parameters designed as trapezoidal form. The value of the objective function that obtained from using ranking formula, should be between the objective function of first parameters as the lower limit in Table-3 and objective function of the fourth parameters as the upper limit . The value of the objective function (the upper value) (16*22 +1*10 +8*12+22*11) =700. Therefore, the value of the objective function Z with any ranking formula must be . The following various ranking formulas are applied on the same numerical example to convert the data from fuzzy to crisp form.

The first formula of ranking function:
Let ( ̌ ) be a fuzzy number then ( ̃) represents the Ropust ranking technique for trapezoidal numbers [3,13].
For example (5, 6, 8, 10) = ∫ ) ( ) The ranking formula is applied on all data of the problem. Then, the results appeared in crisp form, and placed on a similar Table-5. The optimal solution is The value of objective function is: ∑ ∑ =7.25*0.25 +9.25 *6. 0 +19 *13.75+29.25*0.5 +19.75*9.5 =520.81. It is obvious from the results of the total cost Z by using the TORA Program is lower than the cost produced by using the Least Cost Method and then Stepping Stone Method.

The second formula of ranking function:
This formula is applied on the original problem. [12] ( ̌) ( ̌) ( ) In another form of the same formula: By applying the same trapezoidal fuzzy example: Likewise, all data in table 5 is converted by using the current ranking formula. The results are then converted into a linear programming model and by using TORA Program to obtain the optimal solution as shown in the following Table-7. The value of objective function: ∑ ∑ 5.3. The third formula of ranking function: ) By applying the same steps as in the second model of the ranking function, the following results are obtained in T 8 For example ( ̌) ( ) ( ) By applying the same steps as in the previous models of the ranking function. The results of optimal solutions are bolded in Table-9. The value of objective function: ∑ ∑ 5.5. The fifth formula of ranking function [15]: where: ( ̌) ( ) and ( ̌) Similarly, the data is converted by using the fifth ranking formula in the Table-10 When the current ranking formula is applied ranking function 13, the results are quite different from the results obtained from using other formulas in this paper. The reason for that is the incompatibility with the transport model data. 5.6. The sixth formula of ranking function: For example: ( ̌) = *(7/18) =13.27 By applying the same steps as in the previous models of the ranking function, the crisp results are placed in Table-11 The optimal solutions is bolded in same table. The value of the objective function is: ] Note that the current result of the total transportation cost Z=78.05 is quite different from the other results of the previous formulas, and it is out of the limits. The reason for that difference is finding the center of the trapezoidal shape that has been segmented in to triangles and then finding the center of the resulting triangles as shown in the Figure- Therefore, the ranking formula can be adjusted by removing the weight ratio (7/18) of the trapezoidal variables.
The adjusted form of the formula is: ( ̌) (15) The obtained results of the adjusted formula is Z=516.31 by applying the data of Table-12.  Table-13.

6-Results
The aim of this study is to compare between various ranking formulas to obtain the optimal solution in order find the minimum value of total cost of transportation.The data and results that placed in the table for comparison and analysis, columns A-D are trapezoidal fuzzy numbers for numerical example, columns E-K represent the results of applying ranking formulas, column L represents result of LCM and column M represents result of SSM as shown in Table-