M. Zamirian - Islamic Azad University, Bojnourd Branch, Department of Mathematics, Bojnourd, Iran A. V. Kamyad - Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
In this paper a new approach for finding optimal path planning in a plane with stationary obstacles is discussed. At first, we consider a movable rigid object in a plane with stationary obstacles. The goal is to find the shortest path planning which brings rigid object from a given initial point to a given final point such that the length of path be minimized and distance between object and obstacles be maximized. By considering the length of path and the distance between rigid object with obstacles as objective functions, we obtain a multi-objective problem. Because of the imprecise nature of decision maker's judgment, these multiple objectives are viewed as fuzzy variables. Then we determine intervals for the value of objective functions such that these intervals for the distance between rigid object and obstacles are given by decision maker, and for the length of path is achieved by solving two optimal nonlinear problems (ONP). Now, we define a decreasing or increasing membership function for any objective functions on achieved intervals. Then the optimal policy is to find an optimal path which maximize all of membership functions, simultaneously. After a little calculation, we obtain an ONP. By solving this ONP, a (local) Pareto optimal solution for original goal is attained. Numerical example is also given.
M. Zamirian, A. V. Kamyad, Fuzzy Path Planning in a Plane with Stationary Obstacles, Journal of Mathematics and Computer Science, 2 (2011), no. 3, 417--424
Zamirian M., Kamyad A. V., Fuzzy Path Planning in a Plane with Stationary Obstacles. J Math Comput SCI-JM. (2011); 2(3):417--424
Zamirian, M., Kamyad, A. V.. "Fuzzy Path Planning in a Plane with Stationary Obstacles." Journal of Mathematics and Computer Science, 2, no. 3 (2011): 417--424