Interval type-2 fuzzy linear programming: Uncertain constraints

This paper presents some definitions about of Interval Type-2 Fuzzy Constraints regarding Interval Type-2 Fuzzy Linear Programming models (IT2 FLP) such as the Interval Type-2 Fuzzy Feasible Region (IT2 FFR) and the concept of an Interval Type-2 Fuzzy Constraint (IT2 FC). In this proposal, a LP problem with uncertain right hand side parameters treated as Interval Type-2 Fuzzy sets is theoretically defined. As always, a real-valued solution should be found, so two optimization methods which deal with uncertain right hand side parameters are referred.

[1]  Cengiz Kahraman.  Fuzzy set applications in industrial engineering , 2007, Inf. Sci..

[2]  Arnold Neumaier.  On the Structure of Clouds , 2002 .

[3]  M. Melgarejo,et al.  A Fast Recursive Method to Compute the Generalized Centroid of an Interval Type-2 Fuzzy Set , 2007, NAFIPS 2007 - 2007 Annual Meeting of the North American Fuzzy Information Processing Society.

[4]  J. Wolfowitz,et al.  Introduction to the Theory of Statistics. , 1951 .

[5]  Pandian Vasant,et al.  Application of Fuzzy Linear Programming in Production Planning , 2003, Fuzzy Optim. Decis. Mak..

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

[7]  J. Mendel,et al.  α-Plane Representation for Type-2 Fuzzy Sets: Theory and Applications , 2009 .

[8]  C.A. Pena-Reyes,et al.  Implementing Interval Type-2 Fuzzy Processors [Developmental Tools] , 2007, IEEE Computational Intelligence Magazine.

[9]  Vladik Kreinovich,et al.  On the functional form of convex underestimators for twice continuously differentiable functions , 2007, Optim. Lett..

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

[11]  V. Kreinovich,et al.  Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables , 2002 .

[12]  Weldon A. Lodwick,et al.  Solving Large-Scale Fuzzy and Possibilistic Optimization Problems , 2004, IEEE Annual Meeting of the Fuzzy Information, 2004. Processing NAFIPS '04..

[13]  Jerry M. Mendel,et al.  $\alpha$-Plane Representation for Type-2 Fuzzy Sets: Theory and Applications , 2009, IEEE Transactions on Fuzzy Systems.

[14]  Jerry M. Mendel,et al.  Operations on type-2 fuzzy sets , 2001, Fuzzy Sets Syst..

[15]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[16]  J. C. F. Garcia,et al.  Linear programming with Interval Type-2 Fuzzy Right Hand Side parameters , 2008, NAFIPS 2008.

[17]  Jerry M. Mendel,et al.  Comments on "alpha -Plane Representation for Type-2 Fuzzy Sets: Theory and Applications" , 2010, IEEE Trans. Fuzzy Syst..

[18]  Franklin A. Graybill,et al.  Introduction to The theory , 1974 .

[19]  Vladik Kreinovich,et al.  Towards Optimal Techniques for Solving Global Optimization Problems: Symmetry-Based Approach , 2007 .

[20]  Vladik Kreinovich,et al.  Applications of Continuous Mathematics to Computer Science , 1997 .

[21]  M MendelJerry.  Comments on α-plane representation for type-2 fuzzy sets , 2010 .

[22]  J. Mendel.  Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions , 2001 .

[23]  George J. Klir,et al.  Fuzzy sets, uncertainty and information , 1988 .

[24]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[25]  Feilong Liu,et al.  An efficient centroid type-reduction strategy for general type-2 fuzzy logic system , 2008, Inf. Sci..

[26]  V. Kreinovich.  Computational Complexity and Feasibility of Data Processing and Interval Computations , 1997 .

[27]  Jerry M. Mendel,et al.  Type-2 fuzzy sets made simple , 2002, IEEE Trans. Fuzzy Syst..

[28]  Richard C. Waters,et al.  Tree Insertion Grammar: A Cubic-Time, Parsable Formalism that Lexicalizes Context-Free Grammar without Changing the Trees Produced , 1995, CL.

[29]  Jerry M. Mendel,et al.  Interval type-2 fuzzy logic systems , 2000, Ninth IEEE International Conference on Fuzzy Systems. FUZZ- IEEE 2000 (Cat. No.00CH37063).

[30]  V. Kreinovich,et al.  Fuzzy Set-Valued Random Variables , 2002 .

[31]  Vladik Kreinovich,et al.  Beyond Convex? Global Optimization is Feasible Only for Convex Objective Functions: A Theorem , 2005, J. Glob. Optim..

[32]  Jerry M. Mendel,et al.  Interval Type-2 Fuzzy Logic Systems Made Simple , 2006, IEEE Transactions on Fuzzy Systems.

[33]  Jerry M. Mendel,et al.  Type-2 fuzzy logic systems , 1999, IEEE Trans. Fuzzy Syst..

[34]  J. C. Figueroa García,et al.  Linear programming with Interval Type-2 Fuzzy Right Hand Side parameters , 2008, NAFIPS 2008 - 2008 Annual Meeting of the North American Fuzzy Information Processing Society.

[35]  Siegfried Gottwald,et al.  Fuzzy Sets and Fuzzy Logic , 1993 .

[36]  Aimo Törn,et al.  Models and Algorithms for Global Optimization , 2007 .

[37]  Arnold Neumaier.  Clouds, Fuzzy Sets, and Probability Intervals , 2004, Reliab. Comput..

[38]  J. Kacprzyk,et al.  Optimization Models Using Fuzzy Sets and Possibility Theory , 1987 .

[39]  George J. Klir,et al.  Fuzzy sets and fuzzy logic - theory and applications , 1995 .

[40]  Martin Fuchs,et al.  Clouds, p-boxes, Fuzzy Sets, and Other Uncertainty Representations in Higher Dimensions , 2009, Acta Cybern..

[41]  Ching-Lai Hwang,et al.  Fuzzy Mathematical Programming , 1992 .

[42]  Weldon A. Lodwick,et al.  Theoretical and semantic distinctions of fuzzy, possibilistic, and mixed fuzzy/possibilistic optimization , 2007, Fuzzy Sets Syst..

[43]  H. Zimmermann,et al.  Fuzzy reasoning for solving fuzzy mathematical programming problems , 1993 .