An Inventive Approach to Optimize Fuzzy Transportation Problem

The present paper wraps an innovative approach to optimize transportation problems through generalized trapezoidal numbers in a fuzzy environment. The main contribution here is to develop an innovative method to optimize the generalized fuzzy trapezoidal transportation problem and reduce the computational intricacy of the existing methods. Then again this method confers many improved results against classical North-West Corner and Least-Cost schemes in Fuzzy environment. An additional merit of the proposed scheme is that for several fuzzy transportation problems it furnishes the best possible way out directly. It is simple to understand and apply. The solution process is exemplified through two numerical examples and comparison with some standard existing methods.


Introduction
The challenge now a day has become how to get maximum output by minimum input. One of the most significant and effective tools of quantitative study to solve big business problems of the various organizations has been in the circulation of items referred to as transportation problems. In essence, the objective is to play down the cost of transportation within the prescribed time domain. The largest part of the literature is based on crisp transport problems, but in real circumstances the ambiguity in data requires the use of generalized fuzzy number and this is the main motivation behind this study. At the beginning of 1930, Tolstoi proposed a piece of writing on cargo-shipping (Schrijver, 2002). In this, he discussed the transportation based problem and expressed a variety of solution thoughts. The first-ever imperative contribution to the formulation along with the computational practice of the basic problem of transportation was made by Hitchcock (1941). Such a problem has been represented as standard LPP and solved by the simplex methodology. The modified distribution and stepping stone optimal testing methods (Samuel and Venkatachalapathy, 2013;Mathur et al., 2016, Mathur et al., 2018) express a dissimilar way to find out the approach of the simplex method. These tests followed the procedure to get initial basic feasible solution (IBFS) via Vogel approximation method (VAM), the North-west corner (N-W-C) rule or least-cost (L-C) techniques to test the optimality of results. On occasions, it is not possible to get particular data for finding cost of transportation and therefore indistinct data can't be signified by indiscriminate random variables, in such situations records can be characterized regarded as fuzzy numbers (FNs) under fuzzy logic (Zadeh, 1965;Bisht and Srivastava, 2017;Bisht et al., 2018;Barma et al, 2019;Srivastava and Bisht, 2019; A variety of transport problems are tackled within Zimmermann's fuzzy linear programming (Zimmermann, 1978). Later on, the model on fuzzy programming to unravel transport problems has been developed (Chanas et al., 1984;Chanas and Kuchta, 1996). With the passage of time, plenty of algorithms have been built to solve the fuzzy transportation problem (FTP). The important of these are the extension principle model to explain and solve FTPs (Liu and Kao, 2004), use of demands/supplies in the form of trapezoidal fuzzy numbers (TrFNs) (Gani and Razak, 2006), application of TrFNs for the examination of FTP (Dinagar and Palanivel, 2009) and description of new methods for solving FTP by imaging the ambiguity of the outlay of transportation along with order and delivery of the item (Kaur and Kumar, 2011;Kaur and Kumar, 2012).
In the present article, an innovative approach is defined to solve generalized trapezoidal FTPs. In the projected method, generalized TrFNs are brought into play to represent the transportation cost of the product and minimum of the demand supply algorithm is applied to find the initial basic feasible solution of the defuzzified transportation problem through classical ranking technique. The algorithm proposed here is easily applicable by the decision-makers in real-life problems of transportation.
The present document is put up as follows: the objective of generalized FNs is presented in Section 2, while Section 3 covers some fundamental definitions, arithmetic operations with ranking functions. Section 4 contains a description of the new algorithm. In Section 5, the new idea has been tested through two examples and the comparative work is done in Section 6. The final section 7 is focused on concluding remarks.

Objective of Generalized Fuzzy Number
To take a worthy judgment decision-maker must compare FNs before taking a decision. So comparison plays a pivotal role in the problems based on decision making. Proposal of the concept for comparison of normal FNs (Jain, 1976) and that of the perception of generalized FNs, where there is no restriction of membership function as a normal form (Chen, 1985) were two significant works in the field. Since then, so many efforts are made for comparison of generalized FNs. Application of generalized FNs in real-life problems has been discussed by various researchers (Kumar and Amarpreet, 2011;Samuel and Venkatachalapathy, 2013;Ebrahimnejad, 2014).
Normalization is also an important tool to compare the FNs. Process of normalization is adopted by many authors, but that results of normalization process consequences in the loss of data information which is obtainable in original data was remarkable (Kaufmann and Gupta, 1991). Later on, the reduction of the length and complexity of arithmetical operations was also vital (Hsieh, 1999).

Generalized Fuzzy Numbers
In The elementary concepts of fuzzy set, TrFN, generalized FN with several primary operations and classical ranking technique can be described as (Bisht and Srivastava, 2019a;Bisht and Srivastava, 2019b;Chhibber et al., 2019;Mathur and Srivastava, 2019;Nagar et al., 2019):

Classical Ranking Method
The classical ranking function has been applied to convert a generalized TrFN into a crisp number and also hence used it to compare generalized TrFN.
Now we have the following three possible cases:

Methodology
In the optimization of transport, problems are solved by means of the supposition that the transportation cost, availability, as well as demand, are mentioned in a specific way. But in certain cases, it is not possible to find crisp details of the elements which are members of the transportation problem. In such instances, the related measures of the problem may write together by means of fuzzy sets, which results in the FTP (Kumar and Amarpreet, 2011;Kumar and Ram, 2018).
In this section, an inventive method is incorporated to find the initial basic feasible solution (IBFS). The proposed procedure (algorithm) involves subsequent steps for solving the transport problem: Step I. Initially convert the given problem of transportation to a fuzzy LPP.
Step II. Set up the given problem into the fuzzy transportation matrix according to the objective to find the smallest transport cost.
Step III. Observe, if  Step V. Continue repetition of above step IV until the fuzzy transportation matrix in consideration shrink to the matrix of order 1 1 .
Step VI. Allot all ij x in the matching ) , ( j i element of the original fuzzy transport matrix.
Step VII. The IBFS and preliminary cost of fuzzy transportation are

Numerical Examples
This section puts forward two illustrations to demonstrate the utility of the projected approach. The illustrations are reviewed against some standard methods (Kaur and Kumar, 2012;Samuel and Venkatachalapathy, 2013). The recommended algorithm requires less computational time.

Comparative Study 6.1 Comparison Based on Allocation
To demonstrate the usefulness of the proposed scheme a healthy comparison has been exposed from Table 7 to 10. This comparison covers allocation by the proposed scheme and that made by some other standard methods (Kaur and Kumar, 2012;Samuel and Venkatachalapathy, 2013) respectively of Example 1 and 2 represented in Table 7 and 8.  (2012) Proposed approach Kaur and Kumar (2012) Proposed approach Kaur and Kumar (2012) Proposed

Comparison Based on Optimal Solution and Epochs
In addition, optimal solution and epochs involved getting optimal solution of Example 1 and Example 2 by generalized fuzzy N-W-C method, generalized fuzzy L-C method, generalized fuzzy Vogel approximation method (VAM), generalized fuzzy MODI and proposed generalized fuzzy min demand supply approach has been established in Table 9 and 10. Table 9. Comparison based on optimal solution and epochs (Example 1) There are several methods to solve generalized FTPs to obtain an IBFS but this method requires less calculation with respect to other methods. It is discussed in (Kaur and Kumar, 2012) that we cannot say which method gives optimal or near to optimal result in a few problems but in these cases, this method gives optimal solution.

Conclusion
In the present manuscript, a more efficient algorithm via generalized TrFNs is conferred to optimize transportation problem. In the proposed algorithm, the cost of transporting the items are represented by FNs based on the generalized trapezoidal system. The new algorithm consumes less time and it is an organized procedure. In view of the fact that the proposed process is faster in terms of calculation, it will be extremely supportive for assessment makers who are in the contract of logistic and supply chain. This study can be extended to FTPs constructed of higher generalized FNs.