## Abstract

We extend the classical linear assignment problem to the case where the cost of assigning agent j to task i is a multiplication of task i's cost parameter by a cost function of agent j. The cost function of agent j is a linear function of the amount of resource allocated to the agent. A solution for our assignment problem is defined by the assignment of agents to tasks and by a resource allocation to each agent. The quality of a solution is measured by two criteria. The first criterion is the total assignment cost and the second is the total weighted resource consumption. We address these criteria via four different problem variations. We prove that our assignment problem is NP-hard for three of the four variations, even if all the resource consumption weights are equal. However, and somewhat surprisingly, we find that the fourth variation is solvable in polynomial time. In addition, we find that our assignment problem is equivalent to a large set of important scheduling problems whose complexity has been an open question until now, for three of the four variations.

Original language | American English |
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Pages (from-to) | 1264-1278 |

Number of pages | 15 |

Journal | Discrete Applied Mathematics |

Volume | 159 |

Issue number | 12 |

DOIs | |

State | Published - 28 Jul 2011 |

## Keywords

- Assignment problem
- Bicriteria optimization
- Complexity
- Controllable processing times
- Resource allocation
- Scheduling

## All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics