Annals of Operations Research

, Volume 147, Issue 1, pp 287–316 | Cite as

Multi-criteria assignment problem with incompatibility and capacity constraints

  • Bernard RoyEmail author
  • Roman Słowiński


The considered assignment problem generalizes its classical counterpart by the existence of some incompatibility constraints limiting the assignment of tasks to processing units within groups of mutually exclusive tasks. The groups are defined for each processing unit and the constraints allow at most one task from each group to be assigned to the corresponding processing unit. The processing units can normally process a certain number of tasks without any cost; this capacity can be extended, however, at some extra marginal cost that is non-decreasing with the number of additional tasks. Each task has to be assigned to exactly one processing unit and has some preference for the assignment; it is expressed for each pair ‘task-processing unit’ by a dissatisfaction degree. The quality of feasible assignments is evaluated by three criteria: g 1-the maximum dissatisfaction of tasks, g 2-the total dissatisfaction of tasks, g 3-the total cost of processing units. If there is no feasible assignment, tasks and processing units creating a blocking configuration are identified and all actions of unblocking are proposed. Formal properties of blocking configurations and unblocking actions are proven, and an interactive procedure for exploring the set of non-dominated assignments is described together with illustrative examples processed by special software.


Assignment problem Multi-criteria combinatorial optimization Incompatibility constraints Blocking configuration Actions of unblocking Non-dominated assignments Interactive exploration 


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  1. Ahuja, R., T. Magnanti, and J.B. Orlin. (1993). Network Flows. Prentice-Hall, Englewood Cliffs.Google Scholar
  2. Figueira, J. (2002). “On the Integer Bi-Criteria Network Flow Problem: A Branch-and-Bound Approach.” Cahier du LAMSADE no. 191, University of Paris Dauphine, Paris.Google Scholar
  3. Ford, L.R. and D.R. Fulkerson. (1962). Flows in Networks. Princeton University Press, Princeton, NJ.Google Scholar
  4. Roy, B. (1996). Multicriteria Methodology for Decision Aiding. Kluwer Academic Publishers, Dordrecht.Google Scholar
  5. Słowiński, R. (2001). MASCOME: M ulti-criteria A ssignment S ubject to C ons-traints O f M utual E xclusion. Software system developed in the Laboratory of Intelligent Decision Support Systems, Poznań University of Technology, Poznan.Google Scholar
  6. Sedeno-Noda, A. and C. Gonzalez-Martin. (2001). “An Algorithm for the Biobjective Integer Minimum Cost Flow Problem.” Computers & Operations Research, 28(2), 139–156.CrossRefGoogle Scholar
  7. Ulungu, E.L. and J. Teghem. (1994). “Multi-Objective Combinatorial Optimization Problems: A Survey.” Journal of Multiple-Criteria Decision Analysis, 3(2), 83–104.CrossRefGoogle Scholar
  8. Ulungu, E.L. and J. Teghem. (1995). “The Two Phase Method: An Efficient Procedure to Solve Bi-Objective Combinatorial Optimization Problems.” Foundations of Computing and Decision Sciences, 20(2), 149–165.Google Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.LAMSADE, University of Paris DauphineParisFrance
  2. 2.Institute of Computing SciencePoznań University of TechnologyPoznańPoland
  3. 3.Institute for Systems ResearchPolish Academy of SciencesWarsawPoland

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