On Some Optimization Problems on Permutations

Abstract

Numerous studies consider combinatorial optimization problems and their solution methods, since a large number of practical problems are described by means of combinatorial optimization models. Among these problems, the most prominent ones are function optimization problems on combinatorial configurations. Many of the studies mentioned above propose approaches and describe methods to solve combinatorial optimization problems for linear and fractionally linear functions on combinatorial sets such as permutations and arrangements. This work describes new approaches and methods to solve some maximization problems for linear, fractionally linear and quadratic functions on permutation set. Algorithms for solving these problems are given. For linear function, we provide a considerably easy method to find the permutation on which the function attains its maximum value. We describe the general algorithm to find fractionally linear function maximum. We consider the cases in which variables of numerator and denominator do not intersect, and when the numerator function and the denominator function have one common variable. We also describe the case in which the numerator function and the denominator function in total have incomplete variable list, i.e. when the total quantity of variables is less than the quantity of numbers in the permutation set. As for solving the problem of finding quadratic function maximum on permutation set, it turned out that there is no universal algorithm to solve this problem for any quadratic function. In this paper, we describe a method of finding maximum on permutation set for quadratic function consisting of two items. We provide examples of solving the considered optimization problems. Keywords: function, permutation, set, transposition, coefficients.