Assignment problem with budget constraints

The focus area of this paper is on the assignment problem with budget constraints which is one of the application area of combinatorial optimization that operates on the domain of those optimization problemsin which the set of feasible solutions is discrete or can be reduced to discreteand in which the goal is to find the best solution. It is particularly concerned with solving the unconstrained assignment problems with Hungarian algorithm and the constrained assignment problem by cutting plane or outer linearization algorithm for solving the Lagrangian dual problem in which at each iterationthe function that approximates the dual function is optimized. The paper is divided in to two chapters. In the first chapterthe classical assignment problemthe problem of finding optimum (minimum or maximum) cost or profit assignment a set of workers or resources to jobs or activities to gather with its mathematical formulationsolution methods and special cases in assignment problems were considered. Under the second chapter we have considered generalization of the classical assignment problem concerning resource(or budget) constraints due to the variety of real life problems.