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An Algorithm for Projecting a Reference Direction onto the Nondominated Set of Given Points

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  • P. Korhonen
  • J. Karaivanova

Abstract

In this paper, we consider the problem of searching nondominated alternatives in a discrete multiple criteria problem. The search procedure is based on the use of a reference direction. A reference direction reflects the desire of the decision maker (DM) to specify a search direction. To find a set of given alternatives related somehow to the reference direction specified by the DM, the reference direction has to be projected onto the set of nondominated alternatives. Our purpose is to develop an efficient algorithm for making this projection. The projection of each given reference direction determines a nondominated ordered subset. The set is provided to a decision maker for evaluation. The decision maker will choose the most preferred alternative from this subset and continues the search from this alternative with a new reference direction. The search will end when no direction of of improvement is found. A critical point in the procedure is the efficiency of the projection operation. This efficiency of our algorithm is considered theoretically and numerically. The projection is made by parametrizing an achievement scalarizing function originally proposed by Wierzbicki (1980) to project any single point onto the nondominated set.

Suggested Citation

  • P. Korhonen & J. Karaivanova, 1998. "An Algorithm for Projecting a Reference Direction onto the Nondominated Set of Given Points," Working Papers ir98011, International Institute for Applied Systems Analysis.
    • Handle: RePEc:wop:iasawp:ir98011
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    References listed on IDEAS

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    1. Pekka Korhonen & Jyrki Wallenius & Stanley Zionts, 1984. "Solving the Discrete Multiple Criteria Problem using Convex Cones," Management Science, INFORMS, vol. 30(11), pages 1336-1345, November.
    2. Odile Marcotte & Richard M. Soland, 1986. "An Interactive Branch-and-Bound Algorithm for Multiple Criteria Optimization," Management Science, INFORMS, vol. 32(1), pages 61-75, January.
    3. Korhonen, Pekka, 1988. "A visual reference direction approach to solving discrete multiple criteria problems," European Journal of Operational Research, Elsevier, vol. 34(2), pages 152-159, March.
    4. Korhonen, Pekka J. & Laakso, Jukka, 1986. "A visual interactive method for solving the multiple criteria problem," European Journal of Operational Research, Elsevier, vol. 24(2), pages 277-287, February.
    5. Korhonen, Pekka J., 1986. "A hierarchical interactive method for ranking alternatives with multiple qualitative criteria," European Journal of Operational Research, Elsevier, vol. 24(2), pages 265-276, February.
    6. Stanley Zionts & Jyrki Wallenius, 1976. "An Interactive Programming Method for Solving the Multiple Criteria Problem," Management Science, INFORMS, vol. 22(6), pages 652-663, February.
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    Cited by:

    1. Jyrki Wallenius & James S. Dyer & Peter C. Fishburn & Ralph E. Steuer & Stanley Zionts & Kalyanmoy Deb, 2008. "Multiple Criteria Decision Making, Multiattribute Utility Theory: Recent Accomplishments and What Lies Ahead," Management Science, INFORMS, vol. 54(7), pages 1336-1349, July.
    2. Sinha, Ankur & Korhonen, Pekka & Wallenius, Jyrki & Deb, Kalyanmoy, 2014. "An interactive evolutionary multi-objective optimization algorithm with a limited number of decision maker calls," European Journal of Operational Research, Elsevier, vol. 233(3), pages 674-688.
    3. Pekka Korhonen & Margareta Soismaa & Aapo Siljamäki, 2002. "On the Use of Value Efficiency Analysis and Some Further Developments," Journal of Productivity Analysis, Springer, vol. 17(1), pages 49-64, January.

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