In applied mathematics, the maximum generalized assignment problem is a problem in combinatorial optimization.
This problem is a generalization of the assignment problem in which both tasks and agents have a size.
Moreover, the size of each task might vary from one agent to the other.
This problem in its most general form is as follows: There are a number of agents and a number of tasks.
Any agent can be assigned to perform any task, incurring some cost and profit that may vary depending on the agent-task assignment.
Moreover, each agent has a budget and the sum of the costs of tasks assigned to it cannot exceed this budget.
It is required to find an assignment in which all agents do not exceed their budget and total profit of the assignment is maximized.
In the special case in which all the agents' budgets and all tasks' costs are equal to 1, this problem reduces to the assignment problem.
When the costs and profits of all tasks do not vary between different agents, this problem reduces to the multiple knapsack problem.
If there is a single agent, then, this problem reduces to the knapsack problem.
In the following, we have n kinds of items,
and m kinds of bins
A solution is an assignment from items to bins.
the total weight of assigned items is at most
The solution's profit is the sum of profits for each item-bin assignment.
The goal is to find a maximum profit feasible solution.
Mathematically the generalized assignment problem can be formulated as an integer program: The generalized assignment problem is NP-hard.
[1] However, there are linear-programming relaxations which give a
[2] For the problem variant in which not every item must be assigned to a bin, there is a family of algorithms for solving the GAP by using a combinatorial translation of any algorithm for the knapsack problem into an approximation algorithm for the GAP.
-approximation algorithm ALG for the knapsack problem, it is possible to construct a (
)-approximation for the generalized assignment problem in a greedy manner using a residual profit concept.
The algorithm constructs a schedule in iterations, where during iteration
a tentative selection of items to bin
The selection for bin
might change as items might be reselected in a later iteration for other bins.
The residual profit of an item
is not selected for any other bin or
is selected for bin
to indicate the tentative schedule during the algorithm.
The residual profit in iteration