Three stages algorithm for finding optimal solution of balanced triangular fuzzy transportation problems

In the literature, the fuzzy optimal solution of balanced triangular fuzzy transportation problem is negative fuzzy number. This is contrary to the constraints that must be non-negative. Therefore, the three stages algorithm is proposed to overcome this problem. The proposed algorithm consists of segregated method with segregating triangular fuzzy parameters into three crisp parameters. This method avoids the ranking technique. Next, total difference method is used to get the initial basic feasible solution (IBFS) value based on segregating triangular fuzzy parameters. While modified distribution algorithm is used to determine optimal solution based on IBFS value. In order to illustrate the proposed algorithm is given the numerical example and based on the result comparison, the proposed algorithm equality to the two existing algorithms and better than the one existing algorithm. The proposed algorithm can solve the fuzzy decision-making problems and can also be extended to an unbalanced fuzzy transportation problem.


INTRODUCTION
In real life, parameters of transportation problems (supply, demand, and transportation cost) sometimes are uncertain in values. The transportation cost, depends on fuel prices, congested routes and weather, while supply is caused by reduced quantities of raw materials, machine breakdowns and production failures. In addition, volatile market situations create uncertainty in demand. However, Zadeh [1] introduced transportation problem with the numbers of supply, demand, and transportation cost are represented by fuzzy number that is called Fuzzy Transportation Problem (FTP). The solving of FTP, many researchers changed the fuzzy parameters to crisp sets and solved by crisp transportation algorithm [2]- [12]. [13] used Yanger's ranking function to change L-R flat fuzzy parameters to crisp numbers and fuzzy classical transportation algorithm i.e. VAM (Vogel's Approximation Method), LCM (Least Cost Method) and NWC (North West Corner) to obtain L-R flat fuzzy optimal solution. [14] proposed the new ranking functionbased integral value to rank triangular fuzzy parameters and generalized fuzzy classical algorithm i.e. GVAM, GLCM and GNWC to determine a generalized triangular fuzzy optimal solution. [6] simplified the new ranking function that was proposed by [3] to rank trapezoidal fuzzy parameters and GLCM to obtain generalized trapezoidal fuzzy optimal solution. Chandran and Kandaswamy presented the score ranking method to change the triangular and trapezoidal fuzzy parameters and the modification of fuzzy LCM to find fuzzy optimal solution. [7] discussed the ranking method based mean of triangular fuzzy numbers and minimum demandsupply method to obtain crisp optimal solution. [9] proposed the classical ranking method and VAM-based harmonic mean method to obtain penalty value of each column and row. [15] introduced a score and accuracy function based on the Pythagorean fuzzy number and applied it on the fuzzy VAM to obtain fuzzy optimal solution. [16] presented the classical ranking function and the generalized minimum supply-demand to obtain triangular fuzzy optimal solution. [11] presented a segregated advancement scheme based minimum demandsupply method and stepping stone to obtain triangular fuzzy optimal solution. Furthermore, the direct approach of solving FTP such as the Zero point method to obtain fuzzy optimal solution [17]. Robust ranking to rank fuzzy number and Zero suffix method to obtain the fuzzy optimal solution Fuzzy dual matrix to obtain a fuzzy optimal solution [18], [19], Improved and revised Zero point [20]- [22]. The modification of Zero-point [23]. Particle Swarm Optimization algorithm (PSO) with fuzzy constraint and conjugate constraint [24]- [27].
The use of the ranking function in the solving of FTP has a very significant impact on the resulting fuzzy optimal solution. As the results of [13] which produce a negative fuzzy optimal solution, this is contrary to the constraints that must be non-negative in the FTP model. In addition, the ranking process takes a long time to compute so that it will affect the computational performance that is not good. Therefore, in this article, we use the segregated advancement scheme approach based on the total difference method and modified distribution method to produce a fuzzy optimal solution without ranking function.

THE MODELING OF BALANCED TRIANGULAR FUZZY TRANSPORTATION PROBLEM
Based on the triangular fuzzy transportation problem with delivers and receivers. Given ̃= , , be the triangular fuzzy supply at the ℎ deliverer, ̃= , , be the triangular fuzzy demand at the ℎ receiver where = 1,2, … , ; = = 1,2, … , . Let ̃= ( , , ) be the per unit triangular fuzzy transportation cost from the ℎ deliverer to the ℎ receiver and ̃= ( , , ) be the number of triangular fuzzy approximation unit to assign from the ℎ deliverer to the ℎ receiver. The modeling of balanced triangular FTP can be formulated as follows, subject to ∑ =1̃=̃= 1,2, … , ,

THREE STAGES ALGORITHM FOR FINDING TRIANGULAR FUZZY OPTIMAL SOLUTION
In this section, the three stages algorithm is proposed to find fuzzy optimal solution of a balanced triangular fuzzy transportation problem. The proposed algorithm consists of a segregated scheme to partition triangular fuzzy numbers, Total Difference Method (TDM)1 [12] to find Initial Basic Feasible Solution (IBFS) and Modified Distribution method (MODI) to obtain optimal solution based IBFS.

A SEGREGATED SCHEME
The segregated method is based on the possibility of optimized output concept of balanced triangular fuzzy transportation problem when the triangular fuzzy parameters with the corresponding demand and supply be partitioned one by one. This scheme consists of pointwise segregation of each triangular fuzzy parameters such that the first element of each triangular fuzzy parameters will be defined first segregated transportation problem is denoted 1 . Similarly, the second and third elements of each triangular fuzzy parameters are denoted 3 and 3 respectively.

Modified Distribution Method
MODI to obtain the optimal solution of balanced triangular fuzzy transportation problem Algorithm 2 shown in Figure 2

Three Stages Algorithm
Three stages algorithm to obtain the optimal solution of balanced triangular fuzzy transportation problem shown Algorithm 3 shown in Figure 3.

NUMERICAL ILLUSTRATION
In this section , we illustrate the proposed method by using numerical example is adapted from Ebrahimnejad [28] in which a trading company wants to optimize the transportation cost of products. Example 1 A leading trading factory wants to obtain the fuzzy number of the commodity that should be distributed from each warehouse to each object such that the total fuzzy transportation cost is at a minimum. The factory has two production houses and three distribution centers. The parameters of transportation are fuzzy numbers that are represented by triangular fuzzy numbers because of disinformation real-life conditions. The summary of fuzzy parameters i.e. supply, demand and transportation cost (dollar) are shown in Table 1 with l as the least amount value, m as the most possible value and u as the greatest amount value. After implementing the MODI method shown in Algorithm 2(`)@ is used to obtain optimal solution based on the IBFS value of all three , = , , and resulted in minimal transportation cost for = 4535, = 7425 and = 12425. After combining the minimal transportation cost of all three , so that the fuzzy optimal solution of balanced triangular fuzzy optimal solution is (4525, 7425, 1245) where represents that the minimal transportation cost most likely will be $7,425 but certainly not less than $4,525. Meanwhile, if things are not in favor of decision-maker, it could be as high as $12,425.

CONCLUSION
In this article, the three stages algorithm consist of a segregated method, total difference method 1 and modified distribution method is proposed to optimize of balanced triangular fuzzy transportation problem. A segregated method scheme and TDM 1 are used to get IBFS value, meanwhile, the MODI method to obtain optimal solution based IBFS value. The numerical example is given to illustrate the justification of proposed algorithm. The Comparative study of results with the literature journal shows that the fuzzy optimal solution that is resulted by proposed algorithm equivalent to those by [13] and [11] and better than those by [2]. All the acquired allocations by proposed algorithm there are non-negative triangular fuzzy numbers whereas one allocation by [2] is negative and violates the basic rule of trading.