A bipolar approach in fuzzy multi-objective linear programming

Abstract The traditional frameworks for fuzzy linear optimization problems are inspired by the max–min model proposed by Zimmermann using the Bellman–Zadeh extension principle to aggregate all the fuzzy sets representing flexible (fuzzy) constraints and objective functions together. In this paper, we propose an alternative approach to model fuzzy multi-objective linear programming problems (FMOLPPs) from a perspective of bipolar view in preference modeling. Bipolarity allows us to distinguish between the negative and the positive preferences. Negative preferences denote what is unacceptable while positive preferences are less restrictive and express what is desirable. This framework facilitate a natural fusion of bipolarity in FMOLPPs. The flexible constraints in a fuzzy multi-objective linear programming problem (FMOLPP) are viewed as negative preferences for describing what is somewhat tolerable while the objective functions of the problem are viewed as positive preferences for depicting satisfaction to what is desirable. This approach enables us to handle fuzzy sets representing constraints and objective functions separately and combine them in distinct ways. After aggregating these fuzzy sets separately, coherence (or consistency) condition is used to define the fuzzy decision set.

[1]  Hans-Jürgen Zimmermann,et al.  Applications of fuzzy set theory to mathematical programming , 1985, Inf. Sci..

[2]  Suresh Chandra,et al.  On duality in linear programming under fuzzy environment , 2002, Fuzzy Sets Syst..

[3]  Gulcin Dinc Yalcin,et al.  Determining Weights in Multi-Objective Linear Programming under Fuzziness , 2011 .

[4]  Rudolf Felix,et al.  Efficient Decision Making with Interactions Between Goals , 2007, 2007 IEEE Symposium on Computational Intelligence in Multi-Criteria Decision-Making.

[5]  H. Zimmermann.  Fuzzy programming and linear programming with several objective functions , 1978 .

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

[7]  Brigitte Werners,et al.  Aggregation Models in Mathematical Programming , 1988 .

[8]  Kazuo Nakamura.  Some extensions of fuzzy linear programming , 1984 .

[9]  A. Tversky,et al.  Advances in prospect theory: Cumulative representation of uncertainty , 1992 .

[10]  B. Werners.  Interactive multiple objective programming subject to flexible constraints , 1987 .

[11]  Chang-Chun Tsai,et al.  Modeling and analysis of a manufacturing cell formation problem with fuzzy mixed-integer programming , 1997 .

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

[13]  Gürsel A. Süer,et al.  Effects of different fuzzy operators on fuzzy bi-objective cell loading problem in labor-intensive manufacturing cells , 2009, Comput. Ind. Eng..

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

[15]  Krassimir T. Atanassov,et al.  Intuitionistic fuzzy sets , 1986 .

[16]  M. O'Hagan,et al.  Aggregating Template Or Rule Antecedents In Real-time Expert Systems With Fuzzy Set Logic , 1988, Twenty-Second Asilomar Conference on Signals, Systems and Computers.

[17]  H. Zimmermann.  DESCRIPTION AND OPTIMIZATION OF FUZZY SYSTEMS , 1975 .

[18]  C. Duraisamy,et al.  Solving Fuzzy Multi-Objective Linear Programming Problems With Linear Membership Functions , 2011 .

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

[20]  Mohammadreza Safi,et al.  A NOTE ON THE ZIMMERMANN METHOD FOR SOLVING FUZZY LINEAR PROGRAMMING PROBLEMS , 2007 .

[21]  R. Felix.  Relationships between goals in multiple attribute decision making , 1994 .

[22]  R. Fullér.  On Obtaining OWA Operator Weights: A Sort Survey of Recent Developments , 2007, 2007 IEEE International Conference on Computational Cybernetics.

[23]  H. Rommelfanger.  Fuzzy linear programming and applications , 1996 .

[24]  Hans-Jürgen Zimmermann,et al.  Decision Making in Fuzzy Environment , 1985 .

[25]  H. Zimmermann,et al.  Latent connectives in human decision making , 1980 .

[26]  Aparna Mehra,et al.  Fuzzy Multiobjective Linear Programming: A Bipolar View , 2012, IPMU.

[27]  Masahiro Inuiguchi,et al.  A fuzzy satisficing method for large-scale multiobjective linear programming problems with block angular structure , 1996, Fuzzy Sets Syst..

[28]  M. Grabisch.  Fuzzy integral in multicriteria decision making , 1995 .

[29]  R. Fullér.  OWA Operators in Decision Making , 2003 .

[30]  R. Yager,et al.  PARAMETERIZED AND-UKE AND OR-LIKE OWA OPERATORS , 1994 .

[31]  Hideo Tanaka,et al.  On Fuzzy-Mathematical Programming , 1973 .

[32]  M. K. Luhandjula.  Compensatory operators in fuzzy linear programming with multiple objectives , 1982 .

[33]  Didier Dubois,et al.  Bipolar possibility theory in preference modeling: Representation, fusion and optimal solutions , 2006, Inf. Fusion.

[34]  C. R. Bector,et al.  Fuzzy Mathematical Programming and Fuzzy Matrix Games , 2005 .

[35]  H. Rommelfanger,et al.  Linear programming with fuzzy objectives , 1989 .

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

[37]  Didier Dubois,et al.  On the use of aggregation operations in information fusion processes , 2004, Fuzzy Sets Syst..

[38]  Didier Dubois,et al.  Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets , 2012, Fuzzy Sets Syst..

[39]  Amelia Bilbao-Terol,et al.  Pareto-optimal solutions in fuzzy multi-objective linear programming , 2009, Fuzzy Sets Syst..

[40]  José Manuel Cadenas Figueredo,et al.  Interactive decision-making in multiobjective fuzzy programming , 1994 .

[41]  Gaspar Mayor,et al.  Aggregation Operators , 2002 .

[42]  Didier Dubois,et al.  Possibility Theory, Probability Theory and Multiple-Valued Logics: A Clarification , 2001, Annals of Mathematics and Artificial Intelligence.

[43]  Miao-Ling Wang,et al.  A fuzzy multiobjective linear programming , 1997, Fuzzy Sets Syst..

[44]  S. Chanas.  The use of parametric programming in fuzzy linear programming , 1983 .