Solving optimization problems with linear-fractional objective functions and additional constraints on arrangements

Abstract

The paper proposes an exact method to solve an optimization problem on arrangements with a linear-fractional objective function and additional linear constraints. The efficiency of the solution algorithm is analyzed by means of numerical experiments.

An approach to solving discrete vector optimization problems over a combinatorial set of permutations

“…To state the problem, we use the concept of a multiset A that is specified by its base S A ( ) and the multiplicity of its elements k a ( ) [9][10][11]. Let a multiset A a a a q = { } + + + = K , be given.…”

“…Definition 1 [9][10][11]. A set E A ( ) whose elements are n-samples of the form (1) from the multiset A is called an Euclidean combinatorial set if, for its arbitrary elements a a a a n ¢ = ¢ ¢ ¢ ( , , , ) 1 2 K and a a a a n ¢ ¢ = ¢¢ ¢¢ ¢¢ ( , , , ) 1 2 K , the conditions ( ) ( : ) a a j N a a n j j ¢ ¹ ¢ ¢ Û $ Î ¢ ¹ ¢¢ are satisfied.…”

“…According to [9][10][11], we call the set of inequalities of system (7), (8) , Î are different, some inequalities can be eliminated from the system of inequalities (7), (8). Taking into account the truth of condition (6) …”

“…As is shown in [9][10][11], the convex envelope of a set of permutations is a common permutable polyhedron whose set of vertices is equal to the set of permutations. This property of a permutable polyhedron allows one to reduce the solution of a problem defined on a discrete combinatorial set to the solution of the problem over a continuous feasible set.…”

Combinatorial optimization under uncertainty

“…The theory and methods of combinatorial optimization, as a part of discrete optimization [1][2][3][4][5][6][7][8][9], are being developed actively on the basis of immersing combinatory sets in a Euclidean space within the framework of Euclidean combinatorial optimization [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].…”