TY - JOUR

T1 - Complexity analysis of an assignment problem with controllable assignment costs and its applications in scheduling

AU - Yedidsion, Liron

AU - Shabtay, Dvir

AU - Kaspi, Moshe

N1 - Funding Information:
This research was supported by THE ISRAEL SCIENCE FOUNDATION (grant No. 633/08 ). Partial support by the Paul Ivanier Center for Robotics and Production Management, Ben-Gurion University of the Negev is also gratefully acknowledged.

PY - 2011/7/28

Y1 - 2011/7/28

N2 - 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.

AB - 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.

KW - Assignment problem

KW - Bicriteria optimization

KW - Complexity

KW - Controllable processing times

KW - Resource allocation

KW - Scheduling

UR - http://www.scopus.com/inward/record.url?scp=79957805825&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2011.04.001

DO - 10.1016/j.dam.2011.04.001

M3 - Article

AN - SCOPUS:79957805825

VL - 159

SP - 1264

EP - 1278

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 12

ER -