An Alternate Method for Finding Optimal Solution to Solid Transportation Problem under 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.

In this research, a new and efficient approach of finding an initial basic feasible solution to transportation problems is proposed. The proposed approach is named “Inverse Coefficient of Variation Method (ICVM)”, and the method is illustrated with seven numerical examples. Six existing methods; North West Corner Method (NWCM), Column Minimum Method (CMM), Least Cost Method (LCM), Row Minimum Method (RMM),Vogel’s Approximation Method (VAM), and Allocation Table Method (ATM) were compared with the proposed approach. It can be said conclusively that the proposed Inverse Coefficient of Variation Method (ICVM) provides an improved Initial Basic Feasible Solution to all the transportation problems used in the experiment. Further, the new method leads to the optimal solution to many of the problems considered.

There are two important approaches based on linear ranking functions for solving linear programming problems with cost coefficients as an auxiliary problem to obtain a fuzzy solution of fuzzy variable linear programming problem. The first approach uses the primal simplex method that assumes an initial primal feasible basic solution is at hand. The second approach is based on dual simplex method that begins with a basic dual feasible basic solution and proceeds by pivoting through a series of dual basic solutions until the associated complementary primal basic fuzzy solution is feasible. In this paper, we propose a new method called the primal-dual algorithm, which is similar to the dual simplex method and begins with dual feasibility and proceeds to obtain primal feasibility while maintaining complementary slackness. An important difference between the dual simplex method and the primal-dual method is that the primal-dual algorithm does not require a dual feasible solution to be basic. This algorithm is useful specially for solving minimum fuzzy cost flow problem in which finding an initial dual feasible solution turns out to be a trivial task.

This paper studied the balanced transportation problem. Many algorithms were developed to solve optimally this problem. The most commonly used method is the modified distribution method (MODI). The MODI algorithm starts by generating an initial basic feasible solution using the northwest corner method (NCM). In general, the cost of the initial basic feasible solution generated by NCM is too high and far away from the optimal cost. For this reason, we propose in this paper a new method to generate an initial basic feasible solution to the balanced transportation problem to solve this issue. This method, named Global Minimum Method (GMM), was tested and compared with the NCM, the minimum cost method (MCM), the Vogel's approximation method (VAM), and the optimal cost obtained by MODI. The computational results proved that the new heuristic is promising for small size instances and outperforms all other methods for big size problems.

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

In this paper we present a new algorithm for the optimal solution to bottleneck transportation problem with imprecise coefficients without changing to its crisp equivalent form which is very distinct from other existing methods. Here all the variables are assumed to be triangular fuzzy numbers. Using a parametric form of triangular fuzzy numbers such as left fuzziness index, right fuzziness index, modal value, we obtain all the efficient solutions to a bottleneck-cost fuzzy transportation problem. This method enables the decision maker to find next efficient solution from the current efficient solution which varies from other methods. A numerical example is solved to illustrate the proposed algorithm.

In dealing with real world transportation problems, several issues are frequently encountered as a result of uncontrollable factors. To deal this uncertainties, many authors have suggested transportation problems with intuitionistic fuzzy parameters. In this article, fully intuitionistic fuzzy transportation problem (InFTP) is considered in which the parameters are triangular intuitionistic fuzzy numbers. To solve this, initially intuitionistic fuzzy branch and bound technique are applied to get the initial basic feasible solution and then intuitionistic fuzzy modified distribution (InFMODI) method is applied to acquire the optimal solution of the fully InFTP. Also, a new ordering is developed here in which some properties of compensation, linearity, additive and partial order relations are satisfied. The optimal solution obtained from the proposed method satisfies the condition of optimality and feasibility. Finally, two numerical examples are provided to show the effectiveness of the proposed method.

Chapter 16 - Variants of the simplex method

A simplex algorithm for the Chebyshev solution of overdetermined systems of linear equations is described. In this algorithm, an initial basic feasible solution is available with no artificial variables needed. Also minimum storage is required and no conditions are imposed on the coefficient matrix. The algorithm is a simple and fast one. Numerical results are given.

Transportation problem (TP) in operations research is a widely used optimization technique to study the problems concerned with transporting goods from production places to sale points. The TP may have one or more objectives such as minimization of transportation cost, minimization of distance with respect to time, and so on. There is a systematic method to solve such problems. For this, we find the Initial Basic Feasible Solution (IBFS) to the given problem. North West corner method, least cost method, Vogel’s approximation method are the standard methods one uses to find the IBFS. In recent years, there are several other methods are proposed to solve such problems. In this paper, we propose a new technique named as Direct Sum Method (DSM) and its effectiveness is compared with the standard methods. The result shows that it is easy to compute and near to the optimum solution of the problem.

This research presents transportation modeling approach to solve transportation problems that flow of items from a certain number of suppliers with various production capacities to a certain points of destination. The aim of the paper is to achieve minimum cost of transportation flow for an initial basic feasible solution (IBFS) which consequently will be closer to the optimal solution of the model. Classical transportation methods for distribution are discussed in details and their IBFS are calculated and compared to the results of newly introduced methodology. The results show that the new method has an advantage over the classical transportation methods through achieving a competitive cost for the IBFS.

The transportation problems (TPs) are a fundamental case-study topic in operations research, particularly in the field of linear programming (LP). The TPs are solved in full resolution by using two types of methods: initial basic feasible solution (IBFS) and optimal methods. In this paper, we suggest a novel IBFS method for enhanced reduction in the transportation cost associated with the TPs. The new method searches for the range in columns of the transportation table only, and selects the maximum range to carry out allocations, and is therefore referred to as the maximum range column method (MRCM). The performance of the proposed MRCM has been compared against three traditional methods: North-West-Corner (NWCM), Least cost (LCM) and Vogel’s approximation (VAM) on a comprehensive database of 140 transportation problems from the literature. The optimal solutions of the 140 problems obtained by using the TORA software with the modified distribution (MODI) method have been taken as reference from a previous benchmark study. The IBFSs obtained by the proposed method against NWCM, LCM and VAM are mostly optimal, and in some cases closer to the optimal solutions as compared to the other methods. Exhaustive performance has been discussed based on absolute and relative error distributions, and percentage optimality and nonoptimality for the benchmark problems. It is demonstrated that the proposed MRCM is a far better IBFS method for efficiently solving the TPs as compared to the other discussed methods, and can be promoted in place of the traditional methods based on its performance.

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.

It is well known that the G. B. Dantzig’s simplex method is not a good algorithm due to the reason that it is not a polynomial algorithm for use to handle computational complexity; however, it is the popular basic, most effective and widely used way to solve linear programming problems so far. Herewith, various improvements on the simplex procedure have filled many volumes of works. There are two key works that deserve our attention: how to get an initial feasible basis in the form of a unit matrix; how to get the optimal solution as quickly as possible. The basic thought of the simplex method is aimed at the standard form of a linear programming problem:max { }, , , ,C X A X b X b m nT m n = ≥ ≤ (1)Improving the optimal value of the object function via a transform from a basic feasible solution to another basic feasible solution start off at the point where a basic feasible solution is given in its feasible region; and then one obtains the optimal solution or arrives at a judgment that no optimal solution exists. This thought is realized by using a basis transformation in reality. The basis transformation is an operation of iteration by choosing a pivot via determining an entrance variable and an extraction variable according to a given rule. In practical uses, we find that theAlso, assume that B is a basis, and the iteration coefficients A and b are A j( )ai , b = (bi ). By letting XN = 0 , we obtain the corresponding basic solution (Editorial Group of Textbook of Operations Research. 1990, J. L. Xue. 1992, G. H. Wei, J. L. Fu, etc. 1987, Y. L. Niu, 1994)X b b b B b= ⎡⎣ ⎤⎦ = ⎡ ⎣⎢⎤ ⎦⎥1 21 0 00 , , , , , ,…the value of the object function Z C B bB T 1 , andthe criterion σ σ σ[ ,σ , , ]1 2 n T .

In this paper to solve the initial basic solution for transportation problem and Minimizing transportation time and getting optimal solutions are always considered as important factors while solving transportation problem. This paper shows a new approach for finding initial basic solution for transportation problem which reduces cost of transportation more than any transportation method such as Northwest Corner Method(NCM), Least Cost Method(LCM), northwest, Vogel’s approximation, Total Opportunity Cost Matrix-Minimal Total (TOCM-MT) and so on. This method has been illustrated by taking an Numerical example.