Abstract
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.
Highlights
Numerous practical problems are reduced to optimizing a nonlinear fractional functional in the form: F
The following tasks have been set: – to transform the original model of a fractional non linear optimization problem to the form typical for conventional problems of mathematical programming; – to devise a computational procedure to solve the problem of mathematical programming, derived in this case, in a single step
The method is based on the introduced special transformation of an original fractional-nonlinear structure of the optimized criterion to the form typical for standard problems of mathematical programming of arbitrary dimensionality
Summary
Numerous practical problems are reduced to optimizing a nonlinear fractional functional in the form: F It is clear that this uncertainty passes in transit into the problem’s objective function In this case, a conventional approach to solving the problem is optimization of the average [1–3]. The obvious disadvantage of the solution derived is the danger of obtaining a result that would grossly deviate from an optimum in some specific situations, which may occur by accident during operation of the analyzed object In this regard, a more appropriate approach is to modernize the criterion, which should be wisely chosen as the probability of obtaining a value for winning that is not below the assigned threshold. The use of more powerful methods of optimization of the first and se cond orders [6–8] is complicated due to the need to compute the gradient vector and the Hessian-matrix for functionals in the form (1) In this connection, it is a relevant task to construct a fast and accurate method of fractional-nonlinear optimization
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Eastern-European Journal of Enterprise Technologies
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
