Formal Mathematical Definition
The formal definition of the assignment problem (or linear assignment problem) is
- Given two sets, A and T, of equal size, together with a weight function C : A × T → R. Find a bijection f : A → T such that the cost function:
-
- is minimized.
Usually the weight function is viewed as a square real-valued matrix C, so that the cost function is written down as:
The problem is "linear" because the cost function to be optimized as well as all the constraints contain only linear terms.
The problem can be expressed as a standard linear program with the objective function
subject to the constraints
The variable represents the assignment of agent to task, taking value 1 if the assignment is done and 0 otherwise. This formulation allows also fractional variable values, but there is always an optimal solution where the variables take integer values. This is because the constraint matrix is totally unimodular. The first constraint requires that every agent is assigned to exactly one task, and the second constraint requires that every task is assigned exactly one agent.
Read more about this topic: Assignment Problem
Famous quotes containing the words formal, mathematical and/or definition:
“This is no argument against teaching manners to the young. On the contrary, it is a fine old tradition that ought to be resurrected from its current mothballs and put to work...In fact, children are much more comfortable when they know the guide rules for handling the social amenities. It’s no more fun for a child to be introduced to a strange adult and have no idea what to say or do than it is for a grownup to go to a formal dinner and have no idea what fork to use.”
—Leontine Young (20th century)
“The most distinct and beautiful statement of any truth must take at last the mathematical form.”
—Henry David Thoreau (1817–1862)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)