A method for solving a fuzzy transportation problem via Robust ranking technique and ATM

In this research, we present a modified approach to “the least cost method” (LCM) of finding “the initial basic feasible solution” (IBFS) to the fuzzy transportation problem (FTP). The problem is in a foggy environment where the cost of transporting the product, the resources, and the demand in the destinations are fuzzy numbers. We solved the (FTP) by converting it to an equivalent crisp form using (a robust ranking technique). A numerical example is presented to demonstrate the suggested approach.

This paper presents problems in fuzzy transportation, which deals with fuzzy costs, fuzzy supplies, and demands of a fuzzy nature on any quantity transported. The paper deals with the minimization of the total fuzzy cost under the fuzzified decision-variables. The proposed method gives better optimum for fuzzy transportation problem (FTP) with unbalance and balance types. The method is intended to obtain a basic feasible solution (or “initial basic feasible solution”) (IBFS) of unbalanced fuzzy problems with triangular fuzzy number. A new and simple heuristic approach for obtaining optimum solution of triangular fuzzy unbalanced transportation problem is proposed that reduces the number of iterations in the optimization process. A given triangular fuzzy unbalanced TP is converted into a modified triangular unbalanced FTP by increasing the fuzzy-demand/fuzzy-supply of an origin and a destination, and the same is resolved by new method. Also an illustrative numerical example is discussed for proposed method solving a triangular FTP with m, n origins and destinations, respectively.

In this paper, we introduce a new method to solve Interval-Valued Transportation Problem (IVTP) to deal with those problems of transportation wherein the information available is imprecise. First, a newly proposed fuzzification method is used to convert the IVTP to octagonal fuzzy transportation problem and then with the help of ranking function proposed in this paper, the fuzzy transportation problem is converted into crisp transportation problem. Lastly, Initial Basic Feasible Solution (IBFS) of this problem is obtained using Vogel’s Approximation Method and the solution is improved using Modified Distribution (MODI) method. A numerical example with interval data is solved using the proposed algorithm to make comparison of the solution with some other methods. Also, a numerical example with parameters in the form of octagonal fuzzy numbers is illustrated to compare the effectiveness of the proposed ranking technique. The proposed fuzzification and ranking technique can be used in the other fields of decision making dealing with the data in the same form as considered in this paper.

Two methods have been used extensively for arriving at initial basic feasible solution (IBF). One of them is Northwest corner rule and the other on is Russell method (Hillier & Lieberman, 2005.) Both methods have drawbacks. The IBF obtained is either far from optimal solution or does not have adequate number of entries to initiate transportation simplex algorithm. The Northwest Corner rule gives an initial feasible solution that is far from optimal while the IBF solution obtained using Russell method doesn’t give enough number of entries to start the transportation simplex algorithm. Hence, there is a need for developing a method for arriving at initial basic feasible solution with adequate number of entries needed to initiate transportation simplex algorithm, which can then be used to get an optimal solution. A computer software has been developed based on the new proposed method for this purpose. The proposed new method has been validated through four simple but illustrative examples.

In literature, a variety of fuzzy transportation problems are discussed and solved by defuzzification of the fuzzy input values to the crisp values. This crisp-valued conversion, in the beginning, limits the features of fuzziness as it does not take into account the uncertainties occurring in the course of intermediate calculations. The present study is a successful effort to overcome this limitation by proposing a minimum demand supply-based methodology to obtain a fuzzy initial basic feasible solution (FIBFS) of the problem. A unique ordering of trapezoidal fuzzy numbers is proposed to make the demand and supply allocation in a fuzzy form. The proposed method has been applied to a special class of fuzzy transportation problems, having all the inputs viz. demand, supply, and cost as trapezoidal fuzzy numbers. Further, the incenter fuzzy ranking process is used to convert FIBFS in a crisp initial basic feasible solution (IBFS). The optimality of the IBFS thus obtained is tested through the stepping stone method. The results obtained by the proposed approach are compared with the fuzzy Hungarian-MODI approach and the presence of inherent uncertainties in FIBFS makes the former approach superior to the latter.

A new heuristic method of finding the initial basic feasible solution to solve the transportation problem

A Supply Selection Method for better Feasible Solution of balanced transportation problem

In the article, we focus on solving the fuzzy transportation problem (FTP) in which the cost is represented by a pentagonal fuzzy number but supply and demand are integers. First FTP is converted into a crisp valued TP by using the mid-range technique. Next, crisp valued TP is turned into a linear programming model and employs the Big-M procedure to solve. Next, the solution of the Big-M method proves that it directly gives the optimal solution obtained with the initial basic feasible solution (IBFS) and Fuzzy MODI approaches.

Transportation plays a major role in the industrial and corporate world and so as uncertainty. Several ways have been given in the literature to deal with the imprecision originated from various uncontrollable factors and conditions. In this paper, we consider transportation problems with uncertainty in transportation costs and propose an alternate algorithm to find its initial basic feasible solution. The grade value for the zero costs is defined and is used to find the initial solution. Numerical examples with transportation costs represented by different kinds of fuzzy numbers have been added to illustrate the proposed methodology. A comparison of the obtained results with those obtained by the existing ones indicates that the proposed algorithm yields better results than the existing methods of finding the initial basic feasible solution like fuzzy north-west method, fuzzy least cost method and the fuzzy Vogel’s approximation method.

This paper leads to an Algorithm/technique to solve optimal solution occurring in Transportation problems. In Transportation Problems by presenting a new algorithm to choose the Absolute differences of boundary cost cells. This New Algorithm lesser time than the existing Transportation method to get the optimal solution using initial basic feasible solution. Proposed technique/algorithm is better choice to get optimal solution without finding initial basic feasible solution and hence the proposed algorithm is useful to get optimal solution Transportation problems. Key words:- Transportation Problems, Absolute differences of boundary cost cells, Modified Distribution (M.O.D.I) Method, Initial Basic feasible solution, Optimum solution. DOI : 10.7176/MTM/9-1-07

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 consist 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 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 velue. 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 then the one existing algorithm. The proposed algorithm can solve in the fuzzy decision-making problems and can also be extended to an unbalanced fuzzy transportation problem.

Transportation problem has been one of the most important applications of Linear programming. Transportation problems have become vastly applied in industrial organizations with multiple manufacturing units, warehouses and distribution centers. In this stugy, the methods of finding initial basic feasible solution of balanced transportation problem are studied and compared to find among them the best in terms of efficiency. The initial basic feasible solution tableaus of all the methods are constructed using data collected from Katsina State Transport Authority. The costing of the allocated cells associated with the initial basic feasible solutions of the five methods are computed and compared with that of optimal solution which was found to be N1,098,000:00. It was observed that Vogel’s approximation method, Least-Cost method and Column minimum method yielded better starting solutions. The North-West Corner method and Row minimum methods though simple to compute yielded starting solutions far from the optimal solution. In addition, Vogel’s approximation method is more difficult and requires more iteration. The best transportation network for Katsina State Transport Authority was obtained.Keywords: Comparison, Transportation algorithms, Initial Basic Feasible Solution, Optimal solution, Linear programming solver

Total opportunity cost matrix – Minimal total: A new approach to determine initial basic feasible solution of a transportation problem

Nowadays, algorithms and computer programs, which are going to speed up, short time to run and less memory to occupy have special importance. Toward these ends, researchers have always regarded suitable strategies and algorithms with the least computations. Since linear programming (LP) has been introduced, interest in it spreads rapidly among scientists. To solve an LP, the simplex method has been developed and since then many researchers have contributed to the extension and progression of LP and obviously simplex method. A vast literature has been grown out of this original method in mathematical theory, new algorithms, and applied nature. Solving an LP via simplex method needs an initial basic feasible solution (IBFS), but in many situations such a solution is not readily available so artificial variables will be resorted. These artificial variables must be dropped to zero, if possible. There are two main methods that can be used to eliminate the artificial variables: two-phase method and Big-M method. Data envelopment analysis (DEA) applies individual LP for evaluating performance of decision making units, consequently, to solve these LPs an IBFS must be on hand. The main contribution of this paper is to introduce a closed form of IBFS for conventional DEA models, which helps us not to deal with artificial variables directly. We apply the proposed form to a real-data set to illustrate the applicability of the new approach. The results of this study indicate that using the closed form of IBFS can reduce at least 50 % of the whole computations.

In this paper, we discuss a fuzzy transportation problem(FTP) involving trapezoidal fuzzy numbers. We planned another algorithm to find an initial basic fuzzy feasible solution which will be very nearer to the optimal solution. The optimal solution of the given fuzzy transportation problems is verified by using the fuzzy version of Modified Distribution method without transforming into its correspondent classical form. A few numerical problems are solved to illustrate the proposed algorithm and the solution is compared with the solutions obtained by using an existing algorithms.