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assignment problem

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The {\em assignment problem} is a kind of optimization problem in which the goal is to reduce the total cost of a number of tasks by choosing a particular assignment of those tasks to a number of agents, or task executors. One example might be an algorithm for assigning pieces of work to individual processors of different capacities in a multiprocessor computer. A more real-world example might be Amazon deciding, given a finite number of airplanes, which distribution center should be used, with its fleet, to service a particular set of delivery requests in order to minimize the usage of fuel.

Mathematically, the assignment problem starts with a finite set of agents, $A$, and a finite set of tasks, $T$, where $\lvert A\rvert=\lvert T\rvert$. The cost of a particular agent executing a particular task is a cost, or \PMlinkescapetext{weight}, function $C:A\times T\to\Reals$. The goal is to find the bijective mapping $f:A\to T$ assigning agents to tasks such that the cost function
$$\sum_{a \in A}C(a, f(a))$$
is minimized.

The cost function can also be viewed as a square real-valued matrix $C$, where $C_{i,j}$ is the cost of having agent $i$ execute task $t$, so that the objective function can be written as
$$\sum_{a \in A}C_{a,f(a)}$$

The problem is ``linear'' because the cost function to be optimized as well as all the constraints contain only linear terms.

{\it This entry was adapted from the Wikipedia article \PMlinkexternal{Assignment problem}{} as of January 13, 2007.}