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A novel algorithm for solving optimal path planning problems based on parametrization method and fuzzy aggregation. (English) Zbl 1233.90253
Summary: In this Letter a new approach for solving optimal path planning problems for a single rigid and free moving object in a two and three dimensional space in the presence of stationary or moving obstacles is presented. In this approach the path planning problems have some incompatible objectives such as the length of path that must be minimized, the distance between the path and obstacles that must be maximized and etc., then a multi-objective dynamic optimization problem (MODOP) is achieved. Considering the imprecise nature of decision maker’s (DM) judgment, these multiple objectives are viewed as fuzzy variables. By determining intervals for the values of these fuzzy variables, flexible monotonic decreasing or increasing membership functions are determined as the degrees of satisfaction of these fuzzy variables on their intervals. Then, the optimal path planning policy is searched by maximizing the aggregated fuzzy decision values, resulting in a fuzzy multi-objective dynamic optimization problem (FMODOP). Using a suitable t-norm, the FMODOP is converted into a non-linear dynamic optimization problem (NLDOP). By using parametrization method and some calculations, the NLDOP is converted into the sequence of conventional non-linear programming problems (NLPP). It is proved that the solution of this sequence of the NLPPs tends to a Pareto optimal solution which, among other Pareto optimal solutions, has the best satisfaction of DM for the MODOP. Finally, the above procedure as a novel algorithm integrating parametrization method and fuzzy aggregation to solve the MODOP is proposed. Efficiency of our approach is confirmed by some numerical examples.

90C29Multi-objective programming; goal programming
90C30Nonlinear programming
93C10Nonlinear control systems
62C86Decision theory and fuzziness in statistics
91B06Decision theory
65K10Optimization techniques (numerical methods)
Full Text: DOI
[1] Borzabadi, A. H.; Kamyad, A. V.; Farahi, M. H.; Mehne, H. H.: Appl. math. Comput., Appl. math. Comput. 170, 1418 (2005)
[2] Latombe, J. C.: Robot motion planning, (1991) · Zbl 0817.93045
[3] Wang, Y.; Lane, D. M.; Falconer, G. J.: Robotica, Robotica 18, 123 (2000)
[4] Zamirian, M.; Farahi, M. H.; Nazemi, A. R.: Appl. math. Comput., Appl. math. Comput. 190, 1479 (2007)
[5] Balicki, J.: Int. J. Comput. sci. Netw. secur., Int. J. Comput. sci. Netw. secur. 6, No. 12, 1 (2006)
[6] Balicki, J.: Int. J. Comput. sci. Netw. secur., Int. J. Comput. sci. Netw. secur. 7, No. 11, 32 (2007)
[7] Miettinen, K.: Nonlinear multi-objective optimization, (1999) · Zbl 0949.90082
[8] Sakawa, M.: Fuzzy set and interactive multi-objective optimization, (1993) · Zbl 0842.90070
[9] Cohon, J. L.: Multi-objective programming and planning, (1985)
[10] Leberling, H.: Fuzzy set syst., Fuzzy set syst. 6, 105 (1981) · Zbl 0465.90081
[11] S. Bells, Flexible membership functions, http://www.louderthanabomb.com/Spark_Features.htm, 1999
[12] Zimmerman, H. J.: Inf. sci., Inf. sci. 36, 25 (1985)
[13] Zimmerman, H. J.: Fuzzy sets, decision making, and expert systems, (1987)
[14] Lootsma, F. A.: Fuzzy logic for planning and decision making, (1997) · Zbl 0904.90173
[15] Vasant, P. M.: Fuzzy optim. Decis. mak., Fuzzy optim. Decis. mak. 3, 229 (2003)
[16] Llir, G. J.; Yuan, B.: Fuzzy sets and fuzzy logics-theory and application, (1995) · Zbl 0915.03001
[17] Kamyad, A. V.; Mehne, H. H.: Int. J. Eng. sci., Int. J. Eng. sci. 14, 143 (2003)
[18] Teo, K. L.; Jenning, L. S.; Lee, H. W. J.; Rehbock, V.: J. austral. Math. soc. B, J. austral. Math. soc. B 40, 314 (1999)
[19] Rubio, J. E.: Control and optimization: the linear treatment of nonlinear problems, (1986)
[20] Tang, G. Y.: Syst. control lett., Syst. control lett. 54:55, 429 (2005)
[21] H.M. Jaddu, Numerical methods for solving optimal control problem using Chebyshev polynomials, PH.D thesis, School of Information Science, Japan Advanced Institute of Science and Technology, 1998 · Zbl 1038.93518
[22] Store, J.; Bulirsch, R.: Introduction to numerical analysis, (1992)