Quest Journals Journal of Research in Applied Mathematics Volume 6 ~ Issue 3 (2020) pp: 28-32 ISSN(Online) : 2394-0743 ISSN (Print): 2394-0735 www.questjournals.org *Corresponding Author: A. P. Bhadane 28 | Page Research Paper APB’s method for the IBFS of a Transportation Problem and comparison with Vogel’s Approximation Method 1 A. P. Bhadane, 2 S. D. Manjarekar 1, 2 Department of Mathematics. L. V. H. Arts, Science and Commerce College, Nashik – 422003, Maharashtra (India) ABSTRACT: In this paper, we have given the new method as APB’s method for the IBFS of a transportation problem by number theoretic approach for finding out the initial basic feasible solution towards the transportation problems and compare it with Vogel’s Approximation method and have shown that the new APB’s method can become an alternative to Vogel’s Approximation method. Keywords: Transportation Problem, Congruence, Vogel’s Approximation AMS Subject Classification (2010): 90B06, 11A07, 90B99 Received 24November, 2020; Accepted 08 December, 2020 © The author(s) 2020. Published with open access at www.questjournals.org I. INTRODUCTION: The transportation problem [1] generally considered as a problems of multi – objective (like minimum cost and shortest path) combinatorial approach on the other hand as we know that the transportation problem were first proposed by Hitchcock in 1941. The standard transportation problems [4] mainly North – West Corner Method (NWCM), Least Cost Method (LCM) and Vogel‟s Approximation Method having important application in the area of physical distribution i.e. transportation of goods and services from several supply centers to several demand centers. As we saw towards [2] the VAM, its variants and some other methods, the flow of allocations is controlled by the DI [Distribution Indicator] or TOC [Total Opportunity Cost] tables. But these DI or TOC tables are formulated by the manipulation of cost entries only. None of them considers demand and/or supply entry to formulate the DI/ TOC table. To overcome this difficulty it is interesting to modify the given transportation problem as number theoretic approach using the congruence relation. We know that, the congruence relation ≡( ) is an equivalence relation [3] which tells us that   −  ↔ ≡( ) . The paper mainly consists of three parts. In first part algorithm for proposed method were given. In the second part, new APB‟s methods along with numerical example were explained. In the third part, we have compared the result with Vogel‟s Approximation method along with conclusion. II. ALGORITHM OF PROPOSED METHOD: The alternative method can be summarized into following steps applied for balanced transportation problem. Step I] Examine whether the transportation problem were balanced or not. If balanced, then go to next step. Step II]: Write the penalties over each rows by taking [     =1 ] (modulo m) and write the penalties over each column by taking [     =1 ] (modulo m) „respectively, where „m‟ is the val ue of supply and demand for the respective rows and columns. Step III] Select the row or column with the highest penalty and allocate as much as possible in the cell that has least cost in the selected rows or column and satisfies the given condition. If there is tie in the values of penalties, one can take any one of them where the minimum allocation can be made. Step IV] any row or column with zero supply or demand should not be used in computing future penalties.