An alternate method for finding an optimal solution to Mixed Type Transportation Problem under a Fuzzy Environment

. In this paper, we develop a new algorithm for the initial fuzzy basic feasible solution to a transportation problem with imprecise parameters. All the decision variables are assumed to be triangular fuzzy numbers or trapezoidal fuzzy numbers or real numbers. By applying the efficient ranking function and arithmetic operations the solution is obtained without changing it into its crisp equivalent form. Using the parametric form of fuzzy numbers such as left fuzziness index, right fuzziness index, modal value and by using a proposed algorithm we obtained the initial basic fuzzy feasible solution to the problem. We further discuss the optimality by applying the modified distribution method. A numerical example is solved to show the effectiveness of the proposed algorithm.


Introduction
The linear programming model has been utilized in a wide range of fields such as inventory management, scheduling, production, transportation and distribution, finance, agriculture, etc. The class of transportation problems is one of the relevant and well-structured types of linear programming problems. Transportation Problem (TP) is a logistical problem for enterprises, particularly for transport and manufacturing companies. The transportation problem deals with delivering a homogeneous product from suppliers to targets. The purpose of the transportation problem is to decide the delivery strategy that reduces overall delivery cost that satisfying demand and supply points. The parameters of the transportation problem are the unit cost, which is the cost of shipping an item from a certain point of supply to a certain point of demand, the quantities available at the points of supply and the quantities needed at the points of demand.
In the actual situation, the parameters of a transportation problem may not be defined; it may be inexact due to some unexpected circumstances. Modeling of real-world mathematical optimization issues is practically impossible due to the existence of vague and incomplete information. Therefore, as the constraints and priorities are not clearly defined, the field of optimization faces the challenge of quantifying unknown data in a meaningful way. The definition of Fuzzy Sets could be used as an efficient decision-making technique to tackle this ambiguity. In 1965, Zadeh [10] introduced a fuzzy set as a sub-discipline of mathematics to capture the instability in real systems. In the field of optimization, the implementation of fuzzy set theory develops rapidly after the primary effort made by Bellman and Zadeh [2]. Fuzzy transportation problem (FTP) has received a great deal of researchers' attention as it is relevant to real-life problems with ambiguous phenomena. Chanas and Kuchta [3] suggested the theory of the optimal solution to the problem of transportation including fuzzy parameters and a methodology that determines this solution This imprecision may result from the lack of precise details or, possibly, from the flexibility of the company to plan its capabilities. Therefore, to solve transport problems effectively, conventional traditional methods should not be used. A Fuzzy Transportation Problem (FTP) is a problem where the demand, cost and supply for transport are fuzzy quantities. The use of FTP is therefore more appropriate for modelling and solving the problems of the real world. In a practical sense and a reallife problem, we meet uncertainty in some situations but not in each. The transportation problem where the parameters are either real numbers or fuzzy numbers is called a mixed type FTP.
To obtain the initial basic feasible solution for problem of fuzzy transportation containing both real numbers and fuzzy numbers, Nizam Uddin Ahmed et.al [7] used Robust's ranking feature and fuzzy zero-point method. Nirbhay Mathur [6] presented an innovative method for optimizing transportation problems through generalized trapezoidal numbers in a fuzzy environment. Senthil Kumar [8] used the PSK method to determine the solution to the Type 3 fuzzy transportation problem. Kumar [4] studied three different types in the Pythagorean fuzzy environment.
In this paper, we proposed a process for finding an optimal solution to a mixed type of fuzzy transport problem without being transferring it to a classical one. To develop the proposed method, we have used the proposed fuzzy ranking method and the fuzzy arithmetic operations introduced by Ming Ma et al. Also, we discuss the numerical example of a mixed type fuzzy transportation problem and its comparative study to illustrate the process developed in this chapter. The rest of the paper is organized as follows: In section 2 we define the fundamental concepts of fuzzy number, rank, and their arithmetic operations. Section 3 gives the formulation of a mixed type fuzzy transportation problem. In section 4, we present a method for the solution of a mixed type fuzzy transportation problem whose parameters are either fuzzy numbers or real numbers. In section 5, a numerical example is provided to show the efficiency of the presented method.

Preliminaries
Definition 2.1: Moreover, any arbitrary fuzzy number can also be written as , we define (I) Addition:

Mixed Type Fuzzy Transportation Problem
The fuzzy transportation problem can be expressed mathematically as follows:

Algorithm for Approximation Method (i)
Transform the mixed type fuzzy transportation problem into a fully fuzzy transportation problem by applying the concept of trivial Trapezoidal fuzzy number. (ii) Represent each fuzzy number in parametric form (iii) Look for the lowest cost for each row using the proposed ranking function and deduct it from all costs for the respective row. Do the same column-wise, and in each row and column, there will be at least one zero. (iv) Find the differences in each row and column between the smallest cost and the next smallest cost, and pick the row or column with the largest difference. Break the tie arbitrarily if there is a tie. (v) Choose a cell with zero cost in the chosen column (or) row and allocate the maximum feasible amount to that cell. Adjust the demand and supply. Repeat the steps (iii) to (v) unless all the criteria of demands and supplies are satisfied. (vi) Finally, we get the required optimal solution to the problem.

Numerical Example
Consider a mixed type fuzzy transportation problem discussed by Senthil Kumar [8]. Here the cost coefficient, supply, and demand are either fuzzy numbers are real numbers. Hence the fuzzy initial basic feasible solution in the form of fuzziness index and location index is given by The fuzzy optimal transportation cost is min (161 11r,176,191 11r) z = + − where 01 r . Therefore the fuzzy optimal transportation cost in terms of

Conclusion
We have suggested the approximation method in this paper to determine the optimal solution to the problem of mixed form fuzzy transport without converting to a crisp equivalent problem. To demonstrate the efficiency of the proposed process, a numerical illustration is given. From the example, we see that the optimal solution in the general form is given by (161,172,180191) whose crisp value is 176. In addition, compared to the existing methods, the fuzzy optimal solution determined by the proposed methods has minimized spreads which helps the decision-maker more.