Construction of the fractional-nonlinear optimization method - Raskin, Lev; Sira, Oksana (Eastern-European Journal of Enterprise Technologies, 2019)
A method for solving the fractional nonlinear optimization problem has been proposed. It is shown that numerous inventory management tasks, on the rational allocation of limited resources, on finding the optimal paths in a graph, on the rational organization of transportation, on control over dynamical systems, as well as other tasks, are reduced exactly to such a problem in cases when the source data of a problem are described in terms of a probability theory or fuzzy math. We have analyzed known methods for solving the fractional nonlinear optimization problems. The most efficient among them is based on the iterative procedure that sequentially improves the original solution to a problem. In this case, every step involves solving the problem of mathematical programming. The method converges if the region of permissible solutions is compact. The obvious disadvantage of the method is the uncontrolled rate of convergence. The current paper has proposed a method to solve the problem, whose concept echoes the known method of fractional-linear optimization. The proposed technique transforms an original problem with a fractional-rational criterion to the typical problem of mathematical programming. The main advantage of the method, as well its difference from known ones, is the fact that the method is implemented using a single-step procedure for obtaining a solution. In this case, the dimensionality of a problem is not a limiting factor. The requirements to a mathematical model of the problem, which narrow the region of possible applications of the devised procedure, imply: 1) the components of the objective function must be separable functions; 2) the indicators for the power of all nonlinear terms of component functions should be the same. Another important advantage of the method is the possibility of using it to solve the problem on unconditional and conditional optimization. The examples have been considered.