Parameterized integer programming models for multifactors representatives apportionment

This paper develops parameterized integer programming models for the problem of multifactors representatives apportionment. In the light of the idea of the optimal uniform approximation theory, we formulate such apportionment problem as three mathematical models. In all of the models, the total number of representatives is regarded as an adaptable parameter, the unfair degree of apportionment is evaluated using the norm-∞ of the vector, and the social characteristics of the different “units” are described by introducing the concept of the factor level matrix. In addition, we also attempt to describe some natural properties of each member in our models.By solving the proposed models, we can obtain an apportionment scheme which ensures that the deviation of the real representative number shared by each unit to the ideal one is as even as possible besides that the obtained apportionment scheme is optimal from mathematical point of view. For the solution of the models, so-called grid search algorithms are presented.A large amount of numerical experiments provide some helpful suggestion for the applications of the models.