Solving combinatorial optimization problems with fuzzy weights

In this paper a combinatorial optimization problem with fuzzy weights is discussed. In order to choose a solution the concept of a necessary soft optimality is adopted. It is shown that the fuzzy problem can be reduced to solving a family of interval problems with the maximal regret criterion. Two general methods of solving the interval problems are presented. The first one is based on a branch and bound technique and the second method is based on a mixed integer programming formulation. Both techniques are general and can be applied if the underlying interval problem is NP-hard.

[1]  Shinkoh Okada,et al.  Fuzzy shortest path problems incorporating interactivity among paths , 2004, Fuzzy Sets Syst..

[2]  Stefan Chanas,et al.  Maximum flow in a network with fuzzy arc capacities , 1982 .

[3]  C. Klein.  Fuzzy shortest paths , 1991 .

[4]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[5]  Arie Tzvieli.  Possibility theory: An approach to computerized processing of uncertainty , 1990, J. Am. Soc. Inf. Sci..

[6]  Masahiro Inuiguchi.  On Possibilistic/Fuzzy Optimization , 2007, IFSA.

[7]  Adam Kasperski,et al.  The robust shortest path problem in series-parallel multidigraphs with interval data , 2006, Oper. Res. Lett..

[8]  Adam Kasperski,et al.  An approximation algorithm for interval data minmax regret combinatorial optimization problems , 2006, Inf. Process. Lett..

[9]  Hande Yaman,et al.  The robust spanning tree problem with interval data , 2001, Oper. Res. Lett..

[10]  Dorota Kuchta,et al.  Discrete fuzzy optimization , 1999 .

[11]  Pascal Van Hentenryck,et al.  A Constraint Satisfaction Approach to the Robust Spanning Tree Problem with Interval Data , 2002, UAI.

[12]  Roberto Montemanni,et al.  A branch and bound algorithm for the robust shortest path problem with interval data , 2004, Oper. Res. Lett..

[13]  Daniel Vanderpooten,et al.  Complexity of the Min-Max (Regret) Versions of Cut Problems , 2005, ISAAC.

[14]  Igor Averbakh,et al.  Interval data minmax regret network optimization problems , 2004, Discret. Appl. Math..

[15]  Timothy Soper,et al.  A shortest path problem on a network with fuzzy arc lengths , 2000, Fuzzy Sets Syst..

[16]  Masahiro Inuiguchi,et al.  Robust optimization under softness in a fuzzy linear programming problem , 1998, Int. J. Approx. Reason..

[17]  Roberto Montemanni,et al.  A branch and bound algorithm for the robust spanning tree problem with interval data , 2002, Eur. J. Oper. Res..

[18]  Daniel Vanderpooten,et al.  Complexity of the min-max and min-max regret assignment problems , 2005, Oper. Res. Lett..

[19]  S. Chanas,et al.  Real-valued flows in a network with fuzzy arc capacities , 1984 .

[20]  M. R. Rao,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[21]  Adam Kasperski,et al.  On combinatorial optimization problems on matroids with uncertain weights , 2007, Eur. J. Oper. Res..

[22]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[23]  Olga Kosheleva,et al.  IEEE International Conference on Fuzzy Systems , 1996 .

[24]  Eduardo Conde,et al.  An improved algorithm for selecting p items with uncertain returns according to the minmax-regret criterion , 2004, Math. Program..

[25]  A Gerodimos,et al.  Robust Discrete Optimization and its Applications , 1996, J. Oper. Res. Soc..