Finding algorithm of optimal subset structure based on the Pareto layers in the knapsack problem

An algorithm is developed for finding the structure of the optimal subset in the knapsack problem based on the proposed multicriteria optimization model. A two-criteria relation of preference between elements of the set of initial data is introduced. This set has been split into separate Pareto layers. The depth concept of the elements dominance of an individual Pareto layer is formulated. Based on it, conditions are determined under which the solution to the knapsack problem includes the first Pareto layers. They are defined on a given set of initial data. The structure of the optimal subset is presented, which includes individual Pareto layers. Pareto layers are built in the introduced preference space. This does not require algorithms for enumerating the elements of the initial set. Such algorithms are used when finding only some part of the optimal subset. This reduces the number of operations required to solve the considered combinatorial problem. The method for determining the found Pareto layers shows that the number of operations depends on the volume of the knapsack and the structure of the Pareto layers, into which the set of initial data in the entered two-criteria space is divided.