Multidimensional assignment problem: Difference between revisions

Revision as of 19:08, 8 February 2022

Revision as of 22:08, 8 February 2022

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{{Short description|Generalization of linear assignment problem from two to multiple dimensions}}

{{Draft topics|computing}}

{{AfC topic|stem}}

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The '''assignment problem''' is a fundamental [[combinatorial optimization]] problem which was introduced by Pierskalla<ref>{{cite article |last=Pierskalla |first=William P. |title=Letter to the Editor—The Multidimensional Assignment Problem | journal=Operations Research 16(2) |publisher=INFORMS |date=1968 |page=422-431 |url=https://pubsonline.informs.org/doi/abs/10.1287/opre.16.2.422}}</ref>. This problem can be seen as a generalization of the linear [[assignment problem]]. In words, the problem can be described as follows:

: An instance of the problem has a number of ''agents'' (i.e., ''cardinality'' parameter) and a number of ''job characteristics'' (i.e., ''dimensionality'' parameter) such as task, machine, time interval, etc. For example, an agent can be assigned to perform task X, on machine Y, during time interval Z. Any agent can be assigned to perform a job with any combination of unique job characteristics at some ''cost''. These costs may vary based on the assignment of agent to a combination of job characteristics - specific task, machine, time interval, etc. The problem is to minimize the ''total cost'' of assigning the agents so that the assignment of agents to each job characteristic is an [[injective function]], or [[one-to-one function]] from agents to a given job characteristic.

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Alternatively, describing the problem using graph theory:

:The multidimensional assignment problem consists of finding, in a [[weighted graph|weighted]] [[multipartite graph]], a [[Matching (graph theory)|matching]] of a given size, in which the sum of weights of the edges is minimum.

==Formal definition==

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The problem is generally [[NP-hard]]. In other words, there is no known [[algorithm]] for solving this problem in polynomial time, and so a long computational time may be needed for solving problem instances of even moderate size (based on dimensionality and cardinality parameters).

== References ==