Method of Solving Problem of Conditional Optimization on Combinatorial Set of Arrangements

Abstract

The paper considered a formulated optimization problem on a combinatorial set of arrangements and suggested a method for its solution taking into account satisfaction of conditions imposed on gains of restrictions and objective function. The method consists of three steps. The first step constructs normalization and compliance matrices which provide transformation of arrangements set elements to a necessary form for the objective function and the given restrictions. The second step consists in finding the first support solution taking into account the arrangements set property. It is worth noting that to find the first support solution it is sufficient to calculate gains of restrictions. If the feasible solution satisfies presented inequalities, the initial data are fixed to be the verification conditions for the next improved solution. The value of objective function is determined by calculating objective function gains without the need to calculate the entire previous function. The third step provides finding the optimal solution at direct improvement of the obtained support solution. This step formulated sufficient and necessary conditions to search for the optimal solution, considered numerical examples of searching for externa of functions on the arrangements set and also presented numerical experiment for the case of |Ak3| with growing number of sample units of the arrangements set (k). It is also worth noting that the number of steps of searching for the optimal solution does not significantly increase at a sharply increased number of elements of arrangements set. Analyzing the indicator of percentage ratio of the considered points number when searching for the optimal solution and the number of elements of arrangements set it should be noted its considerable reduction that indicates to efficiency of the proposed method. So this method application allows us to find the function extremum on the set of arrangements over the finite number of steps.