Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem

In this paper, we review some fuzzy linear programming methods and techniques from a practical point of view. In the rst part, the general history and the approach of fuzzy mathematical programming are introduced. Using a numerical example, some models of fuzzy linear programming are described. In the second part of the paper, fuzzy mathematical programming approaches are compared to stochastic programming ones. The advantages and disadvantages of fuzzy mathematical programming approaches are exemplied in the setting of an optimal portfolio selection problem. Finally, some newly developed ideas and techniques in fuzzy mathematical programming are briey reviewed. c 2000 Elsevier Science B.V. All rights reserved.

[1]  D. Dubois,et al.  Systems of linear fuzzy constraints , 1980 .

[2]  Hans Bandemer.  Modelling uncertain data , 1992 .

[3]  H. Ishibuchi,et al.  Local search algorithms for flow shop scheduling with fuzzy due-dates☆ , 1994 .

[4]  Richard Bellman,et al.  Decision-making in fuzzy environment , 2012 .

[5]  I. M. Stancu-Minasian,et al.  Stochastic Programming: with Multiple Objective Functions , 1985 .

[6]  S. Orlovsky.  Decision-making with a fuzzy preference relation , 1978 .

[7]  Jaroslav Ramík.  VAGUELY INTERRELATED COEFFICIENTS IN LP AS A BICRITERIAL OPTIMIZATION PROBLEM , 1991 .

[8]  Hisao Ishibuchi,et al.  Evidence theory of exponential possibility distributions , 1993, Int. J. Approx. Reason..

[9]  R. Słowiński.  A multicriteria fuzzy linear programming method for water supply system development planning , 1986 .

[10]  A. V. Yazenin,et al.  Fuzzy and stochastic programming , 1987 .

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

[12]  A. Celmins.  Least squares model fitting to fuzzy vector data , 1987 .

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

[14]  Jaroslav Ramík,et al.  Canonical fuzzy numbers of dimension two , 1993 .

[15]  A. Stuart,et al.  Portfolio Selection: Efficient Diversification of Investments , 1959 .

[16]  S. Kataoka.  A Stochastic Programming Model , 1963 .

[17]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

[18]  H. Tanaka,et al.  Fuzzy solution in fuzzy linear programming problems , 1984, IEEE Transactions on Systems, Man, and Cybernetics.

[19]  Francisco Herrera,et al.  Boolean programming problems with fuzzy constraints , 1993 .

[20]  J. Ramík,et al.  Inequality relation between fuzzy numbers and its use in fuzzy optimization , 1985 .

[21]  Constantin Virgil Negoita,et al.  The current interest in fuzzy optimization , 1981 .

[22]  Masatoshi Sakawa,et al.  Genetic algorithms with decomposition procedures for fuzzy multiobjective 0-1 programming problems with block angular structure , 1996, Proceedings of IEEE International Conference on Evolutionary Computation.

[23]  M. Inuiguchi,et al.  Possible and necessary optimality tests in possibilistic linear programming problems , 1994 .

[24]  Abraham Charnes,et al.  Programming with linear fractional functionals , 1962 .

[25]  Masahiro Inuiguchi,et al.  Robust Soft Optimization in Fuzzy Linear Programming Problems , 1996 .

[26]  Didier Dubois,et al.  A class of fuzzy measures based on triangular norms , 1982 .

[27]  Masahiro Inuiguchi,et al.  Minimax regret solution to linear programming problems with an interval objective function , 1995 .

[28]  S. Orlovsky.  On formalization of a general fuzzy mathematical problem , 1980 .

[29]  M. Sakawa,et al.  Fuzzy programming for multiobjective 0–1 programming problems through revised genetic algorithms , 1997 .

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

[31]  Masahiro Inuiguchi,et al.  An achievement rate approach to linear programming problems with an interval objective function , 1997 .

[32]  Ralph E. Steuer,et al.  Goal programming with linear fractional criteria , 1981 .

[33]  Jaroslav Ramík.  Fuzzy goals and fuzzy alternatives in goal programming problems , 2000, Fuzzy Sets Syst..

[34]  Hidetomo Ichihashi,et al.  Relative modalities and their use in possibilistic linear programming , 1990 .

[35]  辻村 泰寛,et al.  Solving Job-shop Scheduling Problem with Fuzzy Processing Time Using Genetic Algorithm , 1995 .

[36]  M. K. Luhandjula.  Fuzzy optimization: an appraisal , 1989 .

[37]  Lotfi A. Zadeh,et al.  Fuzzy sets and systems , 1990 .

[38]  Hideo Tanaka,et al.  Fuzzy decisions in possibility programming problems , 1996, Soft Computing in Intelligent Systems and Information Processing. Proceedings of the 1996 Asian Fuzzy Systems Symposium.

[39]  Fred W. Glover,et al.  Tabu Search - Part I , 1989, INFORMS J. Comput..

[40]  Masahiro Inuiguchi,et al.  Possible and necessary efficiency in possibilistic multiobjective linear programming problems and possible efficiency test , 1996, Fuzzy Sets Syst..

[41]  M. K. Luhandjula.  Multiple objective programming problems with possibilistic coefficients , 1987 .

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

[43]  K. Asai,et al.  Fuzzy linear programming problems with fuzzy numbers , 1984 .

[44]  M. Gupta,et al.  FUZZY INFORMATION AND DECISION PROCESSES , 1981 .

[45]  R. Wets,et al.  Stochastic programming , 1989 .

[46]  M. K. Luhandjula.  Satisfying solutions for a possibilistic linear program , 1986, Inf. Sci..

[47]  Zbigniew Michalewicz.  A Hierarchy of Evolution Programs , 1996 .

[48]  Jaroslav Ramík,et al.  Fuzzy mathematical programming based on some new inequality relations , 1996, Fuzzy Sets Syst..

[49]  Masahiro Inuiguchi,et al.  A possibilistic linear program is equivalent to a stochastic linear program in a special case , 1995, Fuzzy Sets Syst..

[50]  Tetsuo Yokoyama,et al.  A Method for Fuzzy Multiple Objective Linear Fractional Programming Problems with Relationship Between Coefficients , 1994 .

[51]  Hans-Jürgen Zimmermann,et al.  Fuzzy mathematical programming , 1983, Comput. Oper. Res..

[52]  Masahiro Inuiguchi,et al.  The usefulness of possibilistic programming in production planning problems , 1994 .

[53]  Hiroaki Ishii,et al.  Two scheduling problems with fuzzy due-dates , 1992 .

[54]  M. Roubens,et al.  Comparison of methodologies for fuzzy and stochastic multi-objective programming , 1991 .

[55]  Zbigniew Michalewicz,et al.  Genetic Algorithms + Data Structures = Evolution Programs , 1996, Springer Berlin Heidelberg.

[56]  D. Dubois,et al.  FUZZY NUMBERS: AN OVERVIEW , 1993 .

[57]  S. Orlovsky.  Decision-making with a fuzzy preference relation , 1978 .

[58]  H. Ishibuchi,et al.  Identification of possibilistic linear systems by quadratic membership functions of fuzzy parameters , 1990 .

[59]  Shinji Kataoka.  Stochastic Programming -Maximum Probability Model- , 1967 .

[60]  Anders Hald,et al.  Statistical Theory with Engineering Applications , 1952 .

[61]  F. Herrera,et al.  EUROPEAN JOURNAL OF OPERATIONAL , 1992 .

[62]  Jacek Blazewicz,et al.  Scheduling under Resource Constraints , 1996 .

[63]  José L. Verdegay,et al.  Fuzzy Optimization: Recent Advances , 1994 .

[64]  C. Hwang,et al.  Fuzzy Mathematical Programming: Methods and Applications , 1995 .

[65]  Emile H. L. Aarts,et al.  Simulated Annealing: Theory and Applications , 1987, Mathematics and Its Applications.

[66]  George L. Nemhauser,et al.  Handbooks in operations research and management science , 1989 .

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

[68]  Roman Słowiński,et al.  Fuzzy Versus Stochastic Approaches to Multicriteria Linear Programming under Uncertainty , 1988 .

[69]  Masahiro Inuiguchi.  Stochastic Programming Problems versus Fuzzy Mathematical Programming Problems , 1992 .

[70]  Kazuo Nakamura,et al.  Canonical fuzzy numbers of dimension two and fuzzy utility difference for understanding preferential judgments , 1990, Inf. Sci..

[71]  Masahiro Inuiguchi,et al.  Membership function elicitation in possibilistic programming problems , 2000, Fuzzy Sets Syst..

[72]  Fred Glover,et al.  Tabu Search - Part II , 1989, INFORMS J. Comput..

[73]  Hidetomo Ichihashi,et al.  Fuzzy Programming: A Survey of Recent Developments , 1990 .

[74]  S. Orlovsky.  Mathematical Programming Problems with Fuzzy Parameters , 1984 .

[75]  Masahiro Inuiguchi,et al.  Extensions of Fuzzy Relations Considering Interaction Between Possibility Distributions and an Application to Fuzzy Linear Program , 1990 .

[76]  Masahiro Inuiguchi,et al.  Solution Concepts for Fuzzy Multiple Objective Programming Problems , 1990 .

[77]  Didier Dubois,et al.  Fuzzy constraints in job-shop scheduling , 1995, J. Intell. Manuf..

[78]  R A Aliev,et al.  Applications of Fuzzy Sets , 2001 .

[79]  Hidetomo Ichihashi,et al.  Possibilistic Linear Programming with Measurable Multiattribute Value Functions , 1989, INFORMS J. Comput..

[80]  Hidetomo Ichihashi,et al.  Fuzzy and semi-infinite mathematical programming , 1992, Inf. Sci..

[81]  Masahiro Inuiguchi,et al.  Modality Constrained Programming Problems Introduced Gödel Implication and Various Fuzzy Mathematical Programming Problems , 1991 .

[82]  Masatoshi Sakawa,et al.  Fuzzy Sets and Interactive Multiobjective Optimization , 1993 .

[83]  D. Dubois,et al.  The mean value of a fuzzy number , 1987 .

[84]  Jaroslav Ramík.  Fuzzy Preferences in Linear Programming , 1991 .

[85]  Masahiro Inuiguchi,et al.  Portfolio Selection under Independent Possibilistic Informaiton , 1996, Proceedings of IEEE 5th International Fuzzy Systems.

[86]  J. Ramík,et al.  A single- and a multi-valued order on fuzzy numbers and its use in linear programming with fuzzy coefficients , 1993 .

[87]  Masaaki Ida.  Optimality on Possibilistic Linear Programming with Normal Posiblity Distribution Coefficient , 1995 .

[88]  Ivo G. Rosenberg,et al.  Joint canonical fuzzy numbers , 1993 .

[89]  R. Weiner.  Lecture Notes in Economics and Mathematical Systems , 1985 .

[90]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[91]  Didier Dubois,et al.  Ranking fuzzy numbers in the setting of possibility theory , 1983, Inf. Sci..

[92]  Masahiro Inuiguchi,et al.  Portfolio selection under independent possibilistic information , 2000, Fuzzy Sets Syst..

[93]  G. Klir.  Where do we stand on measures of uncertainty, ambiguity, fuzziness, and the like? , 1987 .

[94]  Francisco Herrera,et al.  Fuzzy boolean programming problems with fuzzy costs: A general study , 1996, Fuzzy Sets Syst..

[95]  Peter Kall,et al.  Stochastic Programming , 1995 .

[96]  Jaroslav Ramík.  Some problems of linear programming with fuzzy coefficients , 1993 .

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

[98]  J. Ramík.  Extension principle in fuzzy optimization , 1986 .

[99]  Hideo Tanaka,et al.  Genetic algorithms and neighborhood search algorithms for fuzzy flowshop scheduling problems , 1994 .

[100]  Mitsuo Gen,et al.  Solving Job-shop Scheduling Problem with Fuzzy Processing Time Using Genetic Algorithm , 1995 .

[101]  Hiroaki Ishii,et al.  One machine scheduling problem with fuzzy duedates , 1994 .

[102]  A. M. Geoffrion.  Stochastic Programming with Aspiration or Fractile Criteria , 1967 .

[103]  Hidetomo Ichihashi,et al.  Modality constrained programming problems: A unified approach to fuzzy mathematical programming problems in the setting of possibility theory , 1993, Inf. Sci..

[104]  S. Orlovsky,et al.  Two Approaches to Multiobjective Programming Problems with Fuzzy Parameters , 1984 .

[105]  Hidetomo Ichihashi,et al.  Relationships between modality constrained programming problems and various fuzzy mathematical programming problems , 1992 .

[106]  R. Słowiński,et al.  Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty , 1990, Theory and Decision Library.