Vector-Weighted Matchings

  • Dietmar Schweigert
Part of the Mathematics and Its Applications book series (MAIA, volume 329)


The theorem of efficiency shows that every efficient solution of a multiple objective linear program (MOLP) is the optimal solution of a parametric opti-mization problem. For the parametric optimization one uses weighted sums
$$ {\lambda _1}{c_1}({x_1},...,{x_n}) + ... + {\lambda _k}{c_k}({x_1},...,{c_n}) $$
for the k objective functions c i (x 1,…, x n), i = 1,…, k where the weights have the properties that and λ i > 0 and λ i = 1,…, k = 1. To get every efficient solution one has to organize effectively a search of these vectors (1, 0,…, 0) which de¬livers different efficient solutions. We decompose the poly tope given by the unit vectors (1,0,…,0),…,(0,…,0,1) into the poly topes which correspond to the efficient solutions. This decomposition can be performed by an algorithm using hyperplanes. Our methods will be examined for vector-weighted matchings but of course apply to every multiobjective linear program. We consider a simple bipartite graph G where the edges have weights w(e) = [w 1(e),…,[w n (e)] of vectors of positive reals. A matching M of G is called efficient if there is no other matching M’ with
$$ w(M) = [{w_1}(M),...,{w_n}(M)] < [{w_1}(M'),...,{w_n}(M')] = w(M') $$
where w i (M) is the sum of the ith components of the weights of all edges contained in the matching M. The problem to find the efficient matchings can be considered as a problem in multicriteria optimization. One can even present this problem as multiple objective linear program (MOLP). The solution of such a problem provides some unexpected difficulties.


Bipartite Graph Assignment Problem Linear Order Efficient Solution Preference Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Ch. R. Chegireddy, H. W. Hamacher, Algorithms for finding the fc-best perfect matchings. Discrete Appl. Math. 18 (1987) 155–165MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    H. W. Hamacher, M. Queyranne, K-best solutions to combinatorial optimization problems. Ann. Oper. Res. 4(1985/6) 12233–1143MathSciNetGoogle Scholar
  3. [3]
    H. Isermann, The enumeration of the set of all efficient solutions for a linear multiple objective program. Opl. Res. Q. 28 (1977) 711–725zbMATHCrossRefGoogle Scholar
  4. [4]
    D. Schweigert, Linear extensions and vector-valued spanning trees. Methods of operations research 60, Anton Hain Frankfurt 1990.Google Scholar
  5. [5]
    D. Schweigert, Lineatr extensions and efficient trees. Uni. Kaiserslautern, preprint 172 (1990)Google Scholar
  6. [6]
    R. E. Steuer, Multiple criteria optimization: Theory, computation and application. Wiley, New York, 1986zbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Dietmar Schweigert
    • 1
  1. 1.Fachbereich MathematikUniversität KaiserslauternKaiserslauternGermany

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