A new score function for optimizing transportation problems in complex Pythagorean fuzzy environment

TRANSPORTATION MANAGEMENT ANALYSIS USING VOGEL'S APPROXIMA-TION (VAM) AND METHOD OF DISTRIBUTION MODIFIED (MODI). This study aimed to analyze the problems of transportation, especially the problem transportation costs by using Vogel's Approximation (VAM) and modified distribution method (MODI). In this study VAM methods used to find the initial solution and then proceed with the search for the optimal solution by using MODI. The data obtained is data derived from PT.Pilar Inkasari Karya - East Jakarta on shipment per month for 3 years ie 2011 to 2013. The analysis showed that the optimization of the burden of transportation costs Rp 2.225.288.000 annually is obtained by using initial solution Vogel's Approximation method (VAM) and final solution to the Modified method Distribution (MODI). From these results it can be concluded that the method of Vogel's Approximation (VAM) and modified distribution method (MODI) can be used to solve the problem of transportation. Keywords : Vogel’s Approximation Method. Modified Distribution Method

Purpose: Transportation Problem is a Linear Programming application to physical distribution of goods and services from various origins to several destinations at a minimum cost.
 Methodology: In this study, five different methods were employed to solve transportation problems arising from unequal demand and supply of goods and variations. The methods considered were in terms of North West Corner Rule, Least Cost Method, Vogel’s Approximation Method, Row Minima Method and Column Minima Method were compared. Unbalanced transportation problems were resolved using Vogel’s Approximation Method (VAM) and Modified Distribution (MODI) methods.
 Findings: The methods compared produced different results with VAM generating the least transportation cost and better solution. With the MODI method, economic values were generated for the dual variables, uis and vjs associated with the source and demand points respectively.

Though, in the literature, many heuristic approaches were developed in getting an initial solution, VAM (Vogel’s approximation method) is considered to be a better efficient heuristic approach since it often provides an optimal or near optimal solution to the transportation problem. In general, transportation problems involved in supply-chain management fields are unbalanced (total supply > total demand or total supply < total demand) and large-scale problem size. Always, an unbalanced transportation problem is balanced before VAM procedure is applied. But, sometimes, using VAM with unbalanced feature can provide an improved VAM solution. To study this, a sensitivity analysis of VAM has been performed. Based on the sensitivity analysis of VAM, we can conclude that when we solve an unbalanced transportation problem using VAM procedure it is vital to solve the unbalanced transportation problem both ways with balancing and without balancing to get the initial costs of VAM and take the better one as the initial cost to the considered unbalanced transportation problem. Further, in solving large-scale transportation problems, an implementation of VAM is preferred due to time-consuming computations of VAM. In this paper, an attempt has been made to implement the coding of VAM successfully using C++ and compared to the existing coding of VAM from Nabendu Sen et al. [12] via many numerical examples. Based on the results of numerical examples, we can conclude that the correctness of the newly coded VAM is promising as compared with the previously coded one by Nabendu Sen et al. [12]. Keywords: Transportation problem, Vogel’s approximation method, Heuristic approach, Initial solution

Nowadays, getting substantial results for problems in operations research is crucial. The transportation problem can be solved very effectively with Vogel's Approximation Method (VAM) and discover a workable solution that is closer to the ideal solution. The basic principle of VAM is to allocate as many resources as possible to the cell with the smallest cost in the column or row with the greatest penalty. This cell will receive the maximum number of resources. A problem arises when the magnitudes of the least cost and the next-least cost are identical. Then, we devised a new method called the "New Vogel's Approximation Method (NVAM)" to provide a workable solution to the transportation problem that may then be used to determine the optimal solution. Transporting capacity to the required location is done by three distinct companies. The supply chain's participants seek to move their goods from warehouse to distribution centre, from plant to warehouse, and from distribution centre to retail store. In this study, the VAM method and the NVAM approach are used to determine the optimal transportation cost. These techniques are examined to provide potential supply chain management techniques.

“Mathematics is the queen of the sciences” – Carl Friedrich. In our daily life, planning is essential on various occasions, especially when resources are limited. Transportation problem include optimizing the allocation and distribution of goods from multiple sources to multiple destinations. It provides a structured mathematical structure for analyzing real-world logistics scenarios. Its importance in mathematics lies in its potential to convert complex logistical systems into solvable linear models, providing exact solutions through methods such as the Vogel’s Approximation Method (VAM), and the Modified Distribution Method (MODI). The aim of this paper is to examine improved or hybrid optimization techniques that can describe the limitations and provide more accurate, adaptable, and efficient solutions to transportation problems.

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

One of the popular operation research problems is transportation problem. Its solution is basically divided into two parts. Initially Initial Basic Feasible Solution (IBFS) is obtained then the result is used to calculate the optimal solution. The popular methods to find IBFS of transportation problem are North West Corner Method (NWCM), Least Cost Method (LCM), and Vogel's Approximation Method (VAM). In this paper, a novel approximation method is proposed to find out the IBFS of the transportation problem. There are five different examples used for which the IBFS are calculated using NWCR, LCM, VAM, and our proposed method. The results show that our proposed method provides the best result among them.

This paper introduces a new approach to optimize the cost per unit of product for the Transportation Problem to achieve better outcomes. We present Basic Feasible Solution (BFS) approach compromised of five main steps: (1) Create a Matrix A = mod |Supply(si)-Demand(dj)| (2) Add the cost of each cell of cost matrix C with corresponding elements of Matrix A and Create Matrix B. (3) Mark number in Ascending order of each elements of Matrix B from 1 to mxn (4) If si ≠ dj, then ζ = |small(si, dj)|, else ζ = si or dj, Assign ζ in Matrix B to smallest number from 1 to mxn, and cut the rest of elements of row ζ or column ζ and subtract ζ from other than selected; and (5) Repeat step 4, until all the supply and demand become zero. This solution approach finds the basic feasible solution of TP with the same complexity for solving Vogel’s approximation method (VAM).KeywordsMount Order Method (MOM)Transportation Problem (TP)Basic Feasible Solution (BFS)Optimal Solution (OS)Stepping stone method

In this paper Vogel's Approximation Method and Least Cost Method for Transportation Problem were investigated to obtain the solution at a faster rate using Key-Value Pair approach. Computation experiments were carried out to investigate the effectiveness of Key-Value pair over 1900 problem instances for Vogel's Approximation Method and Least Cost Method each. It was seen that Key-Value pair greatly enhanced the rate of solving for both methods over traditional coding approach. Time comparison is shown for square balanced transportation problems.

The time minimization transportation problem focuses on reducing the total transportation time of goods from origins to destinations. In this research paper, we propose a novel method for achieving minimum transportation time that differs significantly from existing approaches. Our method introduces the use of the mean absolute deviation (MAD) to obtain an initial basic feasible solution. We then compare the results of our proposed method with established techniques, including the Vogel’s Approximation Method (VAM), Least Cost Method (LCM), and Modified Distribution Method (MODI). The findings demonstrate the effectiveness of our approach in minimizing transportation time, offering a competitive alternative to conventional methods.

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.

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.

<p>Maintenance of an aircraft requires reliable and responsible manpower. Manpower and manhour are needed to perform scheduled and unscheduled maintenance. C-Check maintenance requires a long time for the aircraft to operate again. Therefore, it is necessary to optimize the task card in the maintenance personnel group based on the required task card manhour. Simulation of task card distribution is needed to find the optimum C-Check work group. The method in this study uses Vogel's Approximation Method (VAM) as the main calculation and Northwest Corner (NWC) as the comparison, where VAM and NWC are included in operations research. This study aims to simulate the distribution of C-Check maintenance task cards by adjusting the manhour of the personnel group and the manhour task card required so that the maintenance personnel group can perform the task optimally. The analysis results obtained are manhour requirements in the task card group as demand, the ability of each maintenance personnel group as supply, and the multiplying factor between actual and maintenance program as transportation costs. The results of the calculation of manhour distribution using the Vogel's approximation method (VAM) calculation obtained a total manhours of 724.5 manhours and for the northwest corner (NWC) calculation obtained a total manhours of 902.8 manhours. Thus, the VAM calculation results in cheaper manhour costs compared to the NWC calculation.</p>

This research discusses to optimize the path and find out the distribution of minimum transportation costs using the VAM (Vogel’s Approximation Method) method, TOCM-SUM is then optimized through the stepping stone test. The VAM method is one of the methods of preparing a initial feasible solution by Determining the distribution table allocation to the cell that has the least cost. TOCM-SUM method to find a suitable initial solution of the transportation problem. The optimal stepping stone test aims to test the initial solution table by calculating the cost of empty cells that are passed by the stepping stone path.

The increase in annual turnover in a drug distribution company reflects positive growth but is often accompanied by a significant increase in operating costs that can reduce profit margins if not managed efficiently. This study aims to optimize distribution costs using a transportation method approach by comparing three initial solution methods, namely Vogel's Approximation Method (VAM), Russell Approximation Method (RAM), and Northwest Corner Method (NWC). The study was conducted at a drug distribution company with three central warehouses serving four distribution branches, based on distribution data for December 2024, which was arranged in the form of a transportation table. From the calculation results, the VAM method produced the lowest total distribution cost of IDR23,426,407. Evaluation of this solution was carried out using the Steppingstone method and the Modified Distribution Method (MODI) to determine whether the solution was optimal. The evaluation results show that all opportunity cost values ​​are positive, which means that no additional iterations or re-allocations are required so that the solution from the VAM method can be stated as an optimal solution. The novelty of this study lies in the integration of three initial solution methods with two optimal solution methods in one drug distribution case study, which has not been widely discussed in an integrated manner in previous studies. These findings provide strategic contributions to more precise and efficient logistics decision-making and support the sustainable operational growth of pharmaceutical distribution companies.