Application of the Johnson Problem to Solve Applied Problems

This article discusses the application of Johnson’s algorithm to analyze the work of a volunteer help desk and the work of an online store if the initial data for the algorithm are random numbers from a given interval. For this purpose, a program has been developed for solving the Johnson problem for modeling work, i.e. simulating the processing of applications (orders, calls), written in the Python programming language using IDE – PyCharm 2021.1, Qt Designer and PyQt5, in which the processing time of applications is a random variable from a given interval. The analysis of the results obtained allows us to make recommendations on the number of applications that the service system can process if the processing time is random, but belongs to a given interval, as well as predict the waiting time for applications to arrive.

Detachments of Hypergraphs I: The Berge–Johnson Problem

A detachment of a hypergraph is formed by splitting each vertex into one or more subvertices, and sharing the incident edges arbitrarily among the subvertices. For a given edge-coloured hypergraph , we prove that there exists a detachment such that the degree of each vertex and the multiplicity of each edge in (and each colour class of ) are shared fairly among the subvertices in (and each colour class of , respectively).Let $(\lambda_1,\ldots,\lambda_m) K^{h_1,\ldots,\, h_m}_{p_1,\ldots,\, p_n}$ be a hypergraph with vertex partition {V1,. . .,Vn}, |Vi| = pi for 1 ≤ i ≤ n such that there are λi edges of size hi incident with every hi vertices, at most one vertex from each part for 1 ≤ i ≤ m (so no edge is incident with more than one vertex of a part). We use our detachment theorem to show that the obvious necessary conditions for $(\lambda_1\dots,\lambda_m) K^{h_1,\ldots,h_m}_{p_1,\ldots,p_n}$ to be expressed as the union 1 ∪ ··· ∪ k of k edge-disjoint factors, where for 1 ≤ i ≤ k, i is ri-regular, are also sufficient. Baranyai solved the case of h1 = ··· = hm, λ1 = ··· = λm = 1, p1 = ··· = pm, r1 = ··· = rk. Berge and Johnson (and later Brouwer and Tijdeman, respectively) considered (and solved, respectively) the case of hi = i, 1 ≤ i ≤ m, p1 = ··· = pm = λ1 = ··· = λm = r1 = ··· = rk = 1. We also extend our result to the case where each i is almost regular.