One Approach to Solving Vector Problems with Fractionally Linear Functions of the Criteria on the Combinatorial Set of Arrangements

Multiobjective Optimization on Permutations with Applications

“…Under a solution of Z(F, G) we mean an element or elements of one of the following sets [22][23][24]:…”

“…For instance, an element of the set ( 6) is called an ideal solution [4,10,[23][24][25], and it is the best for all the particular criteria, respectively, for MP as well. At the same time, Pareto optimality (see (7)) means that the value of any of the particular criteria can be increased only by reducing the value of at least one of the other particular criteria.…”

“…Since real-world systems are discrete-continuous by nature, they are modeled as partially integer programs. In particular, numerous problems of planning, management, design, and placement are modeled with the help of multiobjective problems whose solutions are of combinatorial nature, e.g., permutations, partial permutations, combinations, compositions, partitions, as well as their composition images [5,7,8,[22][23][24]. In this case, the search for an optimal solution is conducted on the correspondent combinatorial set or its proper subset.…”

“…The formulation of MCP on poly-partial permutations is formulated in [22,23], and an approach to its solution using polyhedral relaxation is offered from the group of cutting methods. In [24], the MCP formulation of linear-fractional optimization is presented, and a solution method is offered, which is based on explored properties of a feasible combinatorial domain. A connection between the linear combinatorial optimization problem (linear CP) and linear-fractional MCP is established for the case if the feasible domain is a set of combinations.…”

“…In [24], the MCP problem with a linear-fractional objective function on the set of permutations is solved by the coordinate method of localizing the value of a linear function for the solution of CP (the coordinate localization method, CLM). CLM is based on the construction of the CP graph and its decompositions into subgraphs and viewing a limited number of them.…”