Cutting-plane Algorithm For Solving Linear Semi-infinite Programming In Fuzzy Case
Volume 5, Issue 3, pp 212 - 218
Publication Date: October 15, 2012
Authors
Alireza Fakharzadeh
- Department of Mathematics, Shiraz University of Technology.
Somayeh Khosravi
- Department of Mathematics, Shiraz University of Technology.
Hamidreza Maleki
- Department of Mathematics, Shiraz University of Technology.
Abstract
This paper introduces a cutting-plane algorithm for solving semi-infinite linear programming problems in fuzzy case; the problem contains a crisp objective linear function and the infinite number of fuzzy linear constraints. In the first step; the designed algorithm solves a LP problem, which was created by the ranking function method based on a fuzzy sub-problem of the original one. In each iteration of the proposed algorithm, a cutting is created by adding a fuzzy constraint of the original problem to the fuzzy sub-problem. The convergence of the algorithm is proved and some numerical examples are given.
Keywords
- Semi-infinite linear programming
- Cutting-plane
- Fuzzy linear programming.
References
[1] R.E. Bellman, and L.A. Zadeh, "Decision making in a fuzzy environment", Management Science, 17(1970), 141-164.
[2] B. Betro, "An accelerated centeral cutting plane", Math. Program, 101(2004), 479-495.
[3] B. Betro, "Numerical treatment of Bayesian robustness problems", International Journal of Approximate Reasoning, 50(2009) , 279-288.
[4] M. A. Goberna and M. A. Lopez, "Linear Semi Infinite Programming", Alicant University, 1998.
[5] A. Ismael, F. Vaz and C. Eugnio Ferreira, "Air pollution control with semi-infinite programming", Applied Mathematical Modelling, 33(2009), 1957-1969.
[6] A. Ismael and F. Vaz, Edite M.G.P. Fernandes and M. Paula S.F. Gomes , "Robot trajectory planning with semi-infinite programming", European Journal of Operational Research, 153(2004), 607-617.
[7] K. Glashoff and S.A. Gustafson, "Linear Optimization and Approximation", Springer-Verlag, Berlin, 1983.
[8] Li, He., Huang , H. Guo and Lu. Hongwei, "Bivariate interval semi-infinite programming with an application to environmental decision-making analysis", European Journal of Operational Research, 211(2011), 452-465.
[9] T. Leon and E. Vercher "A purification algorithm for semi-infinite programming", European Journal of Operational Research, 57(1992), 412-420.
[10] H. R. Maleki, "Ranking function and their applications to fuzzy linear programming", Far East J. Math. Sci (EFMS), 4(2002), NO. 3, 283-301.
[11] P. Moulin, M. Anitescu, K.O. Kortanek and F.A. Potra, "The role of linear semi-infinite programming in signal-adapted QMF bank design", IEEE Trans. Signal Processing, 45(1997), 2160-2174.
[12] M. Roubnes, "Inequality constraints between fuzzy number and their use in mathematical programming Stochastic Versus Fuzzy Approaches To Multi objective Mathematical Programming Under Uncertainly", Kluwer Academic Publishers, (1991), 321-330.
[13] L. A. Zadeh, “Fuzzy sets”, Information and Control, 8 (1965), 338-353.