On Solving Optimization Problems with Ordered Average Criteria and Constraints

The problem of aggregating multiple numerical attributes to form overall measure is of considerable importance in many disciplines. The ordered weighted averaging (OWA) aggregation, introduced by Yager, uses the weights assigned to the ordered values rather than to the specific attributes. This allows one to model various aggregation preferences, preserving simultaneously the impartiality (neutrality) with respect to the individual attributes. However, importance weighted averaging is a central task in multiattribute decision problems of many kinds. It can be achieved with theWeighted OWA (WOWA) aggregation though the importanceweightsmake the WOWA concept much more complicated than the original OWA. In this paper we analyze solution procedures for optimization problems with the ordered average objective functions or constraints. We show that the WOWA aggregation with monotonic preferential weights can be reformulated in a way allowing to introduce linear programming optimization models, similar to the optimization models we developed earlier for the OWA aggregation. Computational efficiency of the proposed models is demonstrated.

[1]  Michel Grabisch,et al.  Fuzzy aggregation of numerical preferences , 1999 .

[2]  Ronald R. Yager,et al.  Constrained OWA aggregation , 1996, Fuzzy Sets Syst..

[3]  W. Ogryczak,et al.  LP solvable models for portfolio optimization: a classification and computational comparison , 2003 .

[4]  Michael Clarke,et al.  Symbolic and Quantitative Approaches to Reasoning and Uncertainty , 1991, Lecture Notes in Computer Science.

[5]  Ronald R. Yager,et al.  Including importances in OWA aggregations using fuzzy systems modeling , 1998, IEEE Trans. Fuzzy Syst..

[6]  J. Kacprzyk,et al.  The Ordered Weighted Averaging Operators: Theory and Applications , 1997 .

[7]  W. Ogryczak.  An Implementation of Variable Upper Bounds via SUB Methodology , 1992 .

[8]  Wlodzimierz Ogryczak,et al.  On solving linear programs with the ordered weighted averaging objective , 2003, Eur. J. Oper. Res..

[9]  Marina L. Gavrilova,et al.  Computational Science and Its Applications - ICCSA 2007, International Conference, Kuala Lumpur, Malaysia, August 26-29, 2007. Proceedings, Part I , 2007, ICCSA.

[10]  Janusz Kacprzyk,et al.  The Ordered Weighted Averaging Operators , 1997 .

[11]  R. Yager.  Quantifier guided aggregation using OWA operators , 1996, Int. J. Intell. Syst..

[12]  Wlodzimierz Ogryczak,et al.  On Decision Support Under Risk by the WOWA Optimization , 2007, ECSQARU.

[13]  Wlodzimierz Ogryczak,et al.  Multiple criteria linear programming model for portfolio selection , 2000, Ann. Oper. Res..

[14]  OgryczakWLodzimierz,et al.  Dual Stochastic Dominance and Related Mean-Risk Models , 2002 .

[15]  Ernesto Damiani,et al.  A WOWA-based Aggregation Technique on Trust Values Connected to Metadata , 2005, STM.

[16]  Maria Grazia Speranza,et al.  Conditional value at risk and related linear programming models for portfolio optimization , 2007, Ann. Oper. Res..

[17]  Dyneley Hussey,et al.  Annals of Opera , 1943 .

[18]  V. Torra.  The weighted OWA operator , 1997, International Journal of Intelligent Systems.

[19]  Xinwang Liu,et al.  Some properties of the weighted OWA operator , 2006, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[20]  Wlodzimierz Ogryczak,et al.  Dual Stochastic Dominance and Related Mean-Risk Models , 2002, SIAM J. Optim..

[21]  David F. Nettleton,et al.  Processing and representation of meta-data for sleep apnea diagnosis with an artificial intelligence approach , 2001, Int. J. Medical Informatics.

[22]  Ronald R. Yager,et al.  On ordered weighted averaging aggregation operators in multicriteria decisionmaking , 1988, IEEE Trans. Syst. Man Cybern..

[23]  R. Słowiński.  Fuzzy sets in decision analysis, operations research and statistics , 1999 .

[24]  Vicenç Torra,et al.  The WOWA operator and the interpolation function W*: Chen and Otto's interpolation method revisited , 2000, Fuzzy Sets Syst..

[25]  Włodzimierz Ogryczak,et al.  On goal programming formulations of the reference point method , 2001, J. Oper. Res. Soc..

[26]  Marc Roubens,et al.  Fuzzy Preference Modelling and Multicriteria Decision Support , 1994, Theory and Decision Library.

[27]  Aïda Valls,et al.  Using classification as an aggregation tool in MCDM , 2000, Fuzzy Sets Syst..

[28]  Włodzimierz Ogryczak,et al.  Multiple criteria optimization and decisions under risk , 2002 .

[29]  W. Ogryczak,et al.  On equitable approaches to resource allocation problems: the conditional minimax solutions , 2002 .

[30]  H. Larsen,et al.  Importance weighted OWA aggregation of multicriteria queries , 1999, 18th International Conference of the North American Fuzzy Information Processing Society - NAFIPS (Cat. No.99TH8397).

[31]  Michael M. Kostreva,et al.  Relocation problems arising in conservation biology , 1999 .

[32]  F. Glover,et al.  The simplex SON algorithm for LP/embedded network problems , 1981 .

[33]  Ronald R. Yager,et al.  On ordered weighted averaging aggregation operators in multicriteria decision-making , 1988 .

[34]  L. Schrage.  Implicit representation of variable upper bounds in linear programming , 1975 .

[35]  Wlodzimierz Ogryczak,et al.  WOWA Enhancement of the Preference Modeling in the Reference Point Method , 2008, MDAI.

[36]  Ronald R. Yager,et al.  Essentials of fuzzy modeling and control , 1994 .

[37]  Wlodzimierz Ogryczak,et al.  On Optimization of the Importance Weighted OWA Aggregation of Multiple Criteria , 2007, ICCSA.