General Form Of Assignment Problem In Franklin
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The assignment problem in the general form can be stated as follows: “Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to assign each facility to one and only one job in such a way that the measure of effectiveness is optimised (Maximised or Minimised).”
For example, suppose an accounts officer has 4 subordinates and 4 tasks. The subordinates differ in efficiency and take different time to perform each task. If one task is to be assigned to one person in such a way that the total person hours are minimised, the problem is called an assignment problem.
Explanation: The Hungarian method is a well-known algorithm for solving assignment problems, which involves finding the optimal assignment of n workers to n jobs. However, the Hungarian method can also be used to solve other optimization problems, such as transportation problems and linear programming problems.
The assignment problem in the general form can be stated as follows: “Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to assign each facility to one and only one job in such a way that the measure of effectiveness is optimised (Maximised or Minimised).”
The assignment problem consists of finding, in a weighted bipartite graph, a matching of a given size, in which the sum of weights of the edges is minimum. If the numbers of agents and tasks are equal, then the problem is called balanced assignment. Otherwise, it is called unbalanced assignment.
For example, suppose an accounts officer has 4 subordinates and 4 tasks. The subordinates differ in efficiency and take different time to perform each task. If one task is to be assigned to one person in such a way that the total person hours are minimised, the problem is called an assignment problem.
(i) We first reduce the row by subtracting every element of the row with the smallest element of that row. (ii) Next, we do the column subtraction by subtracting every element of the column with the smallest element of that column. (Note that the elements should be greater than zero in the cells).
The Generalized Assignment Problem has shown to be NP-hard and therefore efficient algorithms are needed, especially for large problems.
Although an assignment problem can be formulated as a linear programming problem, it is solved by a special method known as Hungarian Method because of its special structure.
