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Examples of expanding the problem, and clarify the notes on superclasses of problems |
Clean up the formal definition, remove links which were either 404, irrelevant, or way above the simple assignment problem. |
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An '''assignment problem''' is a type of problem in [[mathematics]] where the members of one set (referred to as ''agents'') must be assigned to the members of another set of equal size (referred to as ''tasks'') in such a way that each agent is assigned to exactly one task, each task has exactly one agent assigned to it, and the total ''cost'' of the assignment is minimized.
The cost of an assignment is determined from a ''cost function'', which gives the cost of assigning a particular agent to a particular task. The overall cost of the assignment is then equal to the sum of the costs for each agent (or the sum of the costs for each task, which is the same thing).
For example, suppose that a taxi firm has three taxis (the agents) available, and three customers (the tasks) wishing to be picked up
However, the assignment problem can be made rather more flexible than it at first appers. In the above example, suppose that there are four taxis available, but still only three customers. Then a fourth task can be invented, perhaps called "sitting still doing nothing", with a cost of 0 for the taxi assigned to it. The assignment problem can then be solved in the usual way and still give the best solution to the problem.
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Similar tricks can be played in order to allow more tasks than agents, tasks to which multiple agents must be assigned (for instance, a group of more customers than will fit in one taxi), or maximizing profit rather than minimizing cost.
The assignment problem is a special case of another optimization problem known as the [[transportation problem]], which
'''Formal mathematical definition'''
An '''assignment problem''' (or '''linear assignment''') gives two sets, ''A'' and ''T'', of equal size, together with a cost function ''C'' which maps pairs (''a'',''t'') (where ''a'' lies in ''A'' and ''t'' in ''T'') to [[real number]]s.
The problem then is to find the [[bijection]] ''P'' from ''A'' to ''T'' such that the sum of ''C''(''a'',''P''(''a'')) across all ''a'' in ''A'' is minimized.
The problem is "linear" because the cost function to be optimized as well as all the constraints can be expressed as linear equations.
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