What Is An Unbalanced Assignment Problem? How It Is Solved With Hungarian Method


An assignment problem is a combination of optimization problems which occurs in the field of operations research. It is a special case and is the degenerate form of a transportation problem when each supply is 1 and each demand is 1. When a particular work is assigned to an agent or it involves assigning a number of tasks to an equal number of agents and these agents can be people, machines, vehicles and plants, etc. and the objective is to increase the total profit or minimize the total cost. The process can be applicable in different fields like assigning personnel to offices, machines to jobs, teachers to classes or salesmen to sales territories. Before we study more on this topic, we want to inform you that we have assignment services where you get assignments on more than 180+ subjects through top-notch experts at GotoAssignmentHelp platform. We provide assignments, essays, thesis help, and many more so join now on GotoAssignmentHelp platform to avail our services.

The assignment problem being a special case of transportation problem can be solved by transportation technique or simplex method (North-West corner rule, least cost method, Vogel’s approximation method, etc.). The most common and popular method to solve assignment problems is the Hungarian method which was developed and published by H.W. Kuhn. The Hungarian method was so named because the algorithm was based on the works of two Hungarian mathematicians: D. Konig and J. Egervary. Later the algorithm was reviewed and observed by James Munkres and was found to be strongly polynomial. The simplex method for linear programming was also followed to solve the assignment problem. The unbalanced assignment problem was solved by Kore’s new method that was proposed by him which was based on generating “ones” instead of “zeros” in the matrix. Many algorithms for solving cost minimization and profit maximizing assignment problems was investigated by Abdur Rashid. Sometimes, in assignment problems, the number of agents is not equal to the number of tasks and these problems are known as unbalanced assignment problems. Then the matrix will not be a square matrix and with these types of sums, the traditional method was to consider costs in such rows or columns as zero. After forming a balanced equation, the problem could be solved by the Hungarian method. If you find assignments and their numerical difficulties, the experts of GotoAssignmentHelp can help you solve them with the best tricks.

As the assignment problem is a special case of transportation problem it can be formulated as a linear programming problem where you have to take n tasks and solve them with n agents. Each agent has to complete only one task through this solving method. Here one should take the first set of constraints as an agent and the second set of constraints as a task. From many formulations, it is pointed out that the assignment problem is a 0-1 integer programming problem. You have to make a matrix of n x n type with ai = bj = 1. If you have problems with assignments, homework, thesis, case studies, avail our expert help from GotoAssignmentHelp.

Hungarian method

In Hungarian method you have to solve by the following steps:

Step 1: You have to determine the total opportunity cost matrix: the minimum element from each row of the given matrix should be subtracted from the elements of the respective rows and then continue the same with the column. The minimum element from each column of the matrix should be subtracted from the elements of the respective columns. Through this process, you obtain a total opportunity cost matrix.

Step 2: The minimum number of horizontal and vertical lines should be drawn to cover all the zeros obtained in the total opportunity cost matrix. If the order of the matrix is maintained then an optimal assignment can be made by the usual procedure so that the total opportunity cost involved is zero. And if the number of lines drawn is less than the order of the payoff matrix follows the next step.

Step 3: The uncovered element in the current total opportunity cost matrix should be the smallest. Then this smallest element should be subtracted from all the uncovered elements and be added at the intersection of the vertical and horizontal lines. Thus, you obtain a total opportunity cost matrix.

Step 4: repeat process 2 and 3 until you get total opportunity cost zero.

There are various numerical illustrations which you will find on our site, GotoAssignmentHelp for understanding the theory better so if you are having problems in assignment help, essays and homework you can reach us. We have a group of experts who are highly qualified and take urgent assignments so that you can relax. We also offer these assignments at affordable prices.

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