Brams-Taylor model of fair division for divisible and indivisible items

In this article, the fair division problem for two participants in the presence of both divisible and indivisible items is considered. Three interrelated modifications of the notion of fair division-profitably, uniformly and equitably fair divisions-were introduced. Computationally efficient algorithm for finding all of them was designed. The algorithm includes repetitive solutions of integer knapsack-type problems as its essential steps. The necessary and sufficient conditions of the existence of proportional and equitable division were found. The statements of the article are illustrated by various examples.