FUZZY DYNAMIC LINEAR PROGRAMMING IN ENERGY SUPPLY PLANNING
Abstract
Linear programming is an important field of optimisation. Many practical problems can be expressed as linear programming problems and be solved with a simplex method. When all data in a linear program are determined and quantities are known in advance, the simplex algorithm, i.e. the simplex method, is explicit. However, in special cases the coefficients in the linear programming problem can be a) fuzzy numbers or b) functions of time with specific requests. In this manner, we have either fuzzy linear programming in the first situation or continuous dynamic linear programming in the second. The synthesis of both methods is a fuzzy dynamic linear programming problem, which is explored in this article and represents a new method in the theory of linear programming problems. Following the theory, we have some different procedures for solving an energy supply planning problem, Fabijan, Predin, [1], Usenik, [2]. One rational possibility is to define the mathematical model as a problem of fuzzy dynamic linear programming and solve it with the new simplex procedure. In this article, the simplex method for this possibility is proposed. At the end of the article, a numerical example is shown.
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References
D. Fabijan, A. Predin: Optimal energy capacities and technologies for permanent and reliable electricity supply, considering risk control, JET Journal of Energy Technology, Volume 2 (2009), p.p. 69–84.
J. Usenik: Fuzzy approach to optimize energy capacities for permanent and reliable electricity supply, considering risk control, JET Journal of Energy Technology, Volume 3 (2010), p.p. 13–26.
V. Rupnik: Zvezno dinamično linearno programiranje, Založba Obzorja Maribor, 1978. V. Rupnik: Stability conditions on continuous dynamic linear programming, Journal of Mathematical Analysis and Applications, Volume 119, Issues 1–2, October–November 1986, p.p. 171–181.
J. Usenik: Generalizirano zvezno variabilno dinamično linearno programiranje, Ph.D. thesis, Univerza v Ljubljani, 1983.
J. Usenik: Generalized Continuous Variable Dynamic Linear Programming (G-c/b/A- CDLP), Some selected papers, University of Ljubljana, 1997.
H.-J. Zimmermann: Fuzzy Set Theory and its Applications, Kluwer Academic Publishers (2001), Fourth edition.
S.H. Nasseri, E. Ardil: Simplex method for Fuzzy Variable Linear Programming Problems, World Academy of Science, Engineering and Technology, Issue 8 (2005), pp. 198–202.
A. Ebrahimnejad, S.H. Nasseri, Hosseinzadeh F. Lotfi,: Bounded linear programs with trapezoidal fuzzy numbers, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 18, No 2 (2010), p.p. 269–286.
T. Liang: Interactive multi-objective transportation planning decisions using fuzzy linear programming, Asia-pacific Journal of Operational Research, Vol. 25, No. 1 (2008), pp. 11– 31.
M. Jimenez, M. Arenas, A. Bilbao, M.R. Rodriguez: Linear programming with fuzzy parameters: An interactive method resolution, European Journal of Operational Research, Volume 177, Issue 3, 16 March 2007, pp. 1599–1609
G.B. Dantzig: Linear Programming and Extensions, Princeton University Press, (Princeton, New Jersey, 1998), eleventh printing.
E.H. Ruspini, P.P. Bonissone, W. Pedrycs: (editors-in-chief): Handbook of Fuzzy Computation, Institute of Physics Publishing Ltd, (Dirac House, Temple Back, Bristol, 1998).
T.J. Ross: Fuzzy Logic with Engineering Applications, second edition, John Wiley & Sons Ltd., (The Atrium, Southern Gate, Chichester, West Sussex, England, 2007).
S. Chen, C. Hwang, P. Hwang: Fuzzy Multiple Attribute Decision Making, Springer-Verlag, (Berlin, Heidelberg, New York, 1992).
J. Usenik: Generalised continuous variable dynamic linear programming in energy systems, Journal of Energy Technology, Vol. 3, No. 4 (2010), pp. 19–31.
J. Usenik: Continuous dynamic linear programming approach in the transportation problem, ISEP 2011, Ljubljana.
J. Usenik: Fuzzy approach in the transportation problem, ISEP 2011, Ljubljana.
Institute of Statics and Dynamics of Structures (TU Dresden), 2005,
http://www.uncertainty-in-engineering.net/uncertainty_models/fuzziness/, 5 November 2011.
