Jour of Adv Research in Dynamical & Control Systems, Vol. 12, No. 7, 2020 DOI: 10.5373/JARDCS/V12I7/20201983 ISSN 1943-023X 49 Received: 05 May 2020/Accepted: 10 June 2020 Two New Effective Methods to Find the Optimal Solution for the Assignment Problems H.A. Hussein, Department of Mathematics, College of Education for Pure Sciences, University of Babylon, Babil, Iraq. M.A.K. Shiker*, Department of Mathematics, College of Education for Pure Sciences, University of Babylon, Babil, Iraq. E-mail: mmttmmhh@yahoo.com Abstract--- Assignment problem (AP) is one of the main optimization problems, itis a private type of transportation problem (TP) in which every origin must have the ability to meet the request of any destination, i.e. any worker must be able to perform any job. The assignment problem is used to find one for one among a group of workers each of whom specializes for a specific job among a set of jobs, the main goal is to reduce gross cost (or reduce gross time) according to user requirements. This paper introduces two new methods (Al-Saeedi's 1st M. and Al-Saeedi's 2nd M.) to find a solution to the assignment problem. Moreover, some numerical examples were given to compare the results of the solution of the two new methods with the result of the solution of the Hungarian method. The two new methods are a systematic procedure, simple to apply and with minimal time and effort when using. The numerical experiment indicates that the two new methods are effective and promising. Keywords--- Linear Programming, Assignment Problems, Optimization Problems, Transportation Problems. I. Introduction Assignment problem (AP) is a special category of linear programming problems, it arises since the available resources like workers and others have different degrees of competence and ability to perform different activities. For this reason, the cost, profit, or time to perform various activities varies by the different assignees. We get the optimal allocation of the linear assignment problem from selecting m than the entries (precisely one element than each row and one element than each column) from the square cost matrix (  ) in which the primary goal is to allocate a numeral of resources to a similar numeral of activities on a one to one foundation so as to reduce the cost or maximize the profit. Consequently, the assignment problem is how to make assignments to improve the desired goal [1,2], it is known as the degenerate problem of the transportation problem (TP), because in a cost matrix (  ) of the assignment problem the gross number of allocated cells is , this problem refers to the cost allocated to the person to carry out the job assigned to him. Typical problems are assigning tasks or activities to assets or resources, i.e.  workers to  jobs, and so on [3]. The assignment problem is most often used in administrative discipline and the solution of manufacturing problems. In (1955), Kuhn developed the Hungarian method of the assignment problem, the reason for naming it with this name is because its basis liesby the effort of the Hungarian mathematician Egervàryin the year1931 [4]. Many researchers have advanced various techniques to solve generalized assignment problems, such as, S. Sudha and D.Vanisri [5] in 2015, A.R. Kumar and S. Deepain 2016 [6], H. D.Afroz and M. A.Hossenin 2017 [7] and M.Mismarin 2020 [8]. The AP is called balanced (Square Matrix) if the numeral of sources (workers) is equivalent to the numeral of tasks (jobs), i.e. , and it is called unbalanced (Non-Square Matrix) when the numeral of sources (workers) is not equivalent to the numeral of tasks ( jobs), i.e.  [9, 10]. We can convert the problem of unbalanced (Non-Square Matrix) assignment to a balanced (Square Matrix) assignment problem by entering the sufficient numeral of fake jobs (fake columns) or imaginary workers (fake rows) depending on the problem we face and that all costs for these new rows or new columns are zero where does not change of the objective function. The authors suggested many techniques to solve various optimization problems (see 11-21), but in this work, two new methods have been proposed (Al-Saeedi's 1st M. and Al-Saeedi's 2nd M.) to find a solution to the assignment problem as each of these two methods is easy to apply and guarantees access to the optimal solution to this problem with less time and effort when using. Moreover, many examples have been resolved and we have mentioned here some of them to compare the solution obtained by the two new methods with the solution obtained by the Hungarian method and these results were mostly equal.