The graph theoretic formulation of the team formation problem based on the factor of competition

The object of the research is to increase the level of productivity of teamwork due to the effective selection of participants who demonstrate the highest level of productivity in cooperation. The presented research is aimed at the mathematical formalization of the problem of team formation based on the results of a series of competitions using graph-theoretic approaches. Each competition in this series involves teams with the same number of participants. The composition of the team necessarily changes for each subsequent competition. After the competitive series, the obtained information about the teams' composition and their results is evaluated for the success of the interaction of the participants, which can be used in the formation of successful teams. A graph-theoretic formalization of the team formation problem on a complete undirected weighted graph has been developed. The set of vertices of this graph corresponds to the set of potential participants. Each edge is weighted with a number that reflects the quality of the interaction between the two participants. A valid solution is to cover the graph with cliques, the size of which is determined by the number of team members. A mathematical model of a two-criterion problem with MAXSUM and MAXMIN criteria was built, where the first criterion evaluates the overall success of the created teams, the second criterion evaluates the «weakest link», allowing to choose the option that maximizes the minimum edge weights for each clique. A two-criterion objective function defines a Pareto set consisting of all Pareto optima in the set of admissible solutions. The algorithmic problem of finding the complete set of alternatives, which is a subset of the Pareto set of minimum power when the condition of equality of the objective functions for the complete set of alternatives and the Pareto set is fulfilled, is considered. The weight of the edges of the graph is calculated using the scores obtained during the series of competitions. In practice, the research results can be used as a basis for the development of team building techniques.