the method used for solving an assignment problem is called

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Assignment problem

The problem of optimally assigning $ m $ individuals to $ m $ jobs. It can be formulated as a linear programming problem that is a special case of the transport problem :

maximize $ \sum _ {i,j } c _ {ij } x _ {ij } $

$$ \sum _ { j } x _ {ij } = a _ {i} , i = 1 \dots m $$

(origins or supply),

$$ \sum _ { i } x _ {ij } = b _ {j} , j = 1 \dots n $$

(destinations or demand), where $ x _ {ij } \geq 0 $ and $ \sum a _ {i} = \sum b _ {j} $, which is called the balance condition. The assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $.

If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem).

In the assignment problem, for such a solution $ x _ {ij } $ is either zero or one; $ x _ {ij } = 1 $ means that person $ i $ is assigned to job $ j $; the weight $ c _ {ij } $ is the utility of person $ i $ assigned to job $ j $.

The special structure of the transport problem and the assignment problem makes it possible to use algorithms that are more efficient than the simplex method . Some of these use the Hungarian method (see, e.g., [a5] , [a1] , Chapt. 7), which is based on the König–Egervary theorem (see König theorem ), the method of potentials (see [a1] , [a2] ), the out-of-kilter algorithm (see, e.g., [a3] ) or the transportation simplex method.

In turn, the transportation problem is a special case of the network optimization problem.

A totally different assignment problem is the pole assignment problem in control theory.

• This page was last edited on 5 April 2020, at 18:48.
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