Global Journal of Science Frontier Research, F: Mathematics and Decision Science, Volume 22 Issue 4

(X. Q. Hu, 2006) (M. J. Liu, 2013) (H. Z. Zhang, 2009) (L. W. Huang, 2007) (Y. Wang, 2005). In (J. L. Du, 2010), an enhanced Hungarian algorithm, the "add zero row approach," was presented to handle the incomplete assignment problem based on a study of the standard Hungarian algorithm. The Hungarian method was first introduced by (T. M. Chang, 2004) and, it was used to solve a common assignment problem, such as a marriage assignment. (Ma., 2014) suggested a new method, the "difference method," for solving the non-standard assignment problem: "tasks more than the number of people." This method is more straightforward than the standard algorithm because it does not require using a new matrix to replace the original coefficient matrix at the beginning and instead solves the problems directly on the old coefficient matrix. (Qiu., 2013) suggested an enhanced Hungarian algorithm for studying multiple maintenance scheduling problems in hostile environments. A quick order reduction optimization approach based on the classical Hungarian algorithm was proposed (J. X. Ren, 2014) to increase the efficiency of the distribution of cloud computing activities. The order of the matrix is quickly lowered and, the computing efficiency is increased by deleting the matrix elements that are determined. Reference (R, 2014) used the Hungarian technique to investigate the dynamic power allocation of weapon-targets by changing it into an assignment issue. Furthermore, the traditional Hungarian algorithm has been used to tackle business and technical challenges in a variety of disciplines (P. Hahn, 1998) (Kuhn., 2012) (E. M. Loiola, 2007) (T. Tassa, 2008) (S. Promparmote, 2006) (M. H. Paul, 2013). According to many authors, the unbalanced assignment problem has many solutions, all of which assume that all jobs are finished. Kumar (Kumar, 2006) came up with a fresh approach to address the problem of uneven assignments. The decision-maker can allocate several tasks to a single component using his methodology. The Lexi Search Approach, developed by Haragopal and Yadaiah (V. Yadaiah, 2016), is a more effective technique for dealing with imbalanced assignment problems that yield the same outcomes as Kumar (Kumar, 2006). In the same year, Kumar's (Kumar, 2006), Haragopal's and Yadaiah's (V. Yadaiah, 2016) methods were surpassed by an approach provided by Betts and Vasko (Vasko, 2016). II. M athematical F ormulation Consider the processing cost of the jth job on the ith component be Cij, where i = 1, 2..., m; j = 1, 2..., n. The challenge is to create an ideal work assignment method that ensures that every task is finished while keeping the overall cost of doing so as low as possible. Mathematical model of an unbalanced assignment problem can be expressed as, Minimize: = ∑ =1 ∑ =1 Subject to constraints ∑ ≥ 1, = 1,2, . . . . . . . . , =1 (1) ∑ ≥ 1, = 1,2, . . . . . . . , =1 (2) = 0 1 © 2022 Global Journals 1 Year 2022 30 Global Journal of Science Frontier Research Volume XXII Issue ersion I V IV ( F ) An Improved Hungarian Algorithm for a Special Case of Unbalanced Assignment Problems N otes