Aggregation of equations in integer programming

Background. The authors investigate the following generalization of the Diophantine equations aggregation: given$\sum_{j=1}^n a_{ij}x_j = a_i$, $i=1,...,m$ with integer coefficients, we need to find integer multipliers $f_1, f_2, ..., f_m$ such that all the vertices of the convex hull of integer non-negative solutions of the system are vertices of the convex hull of integer non-negative solutions of the equation $\sum_{i=1}^m f_i \sum_{j=1}^n a_{ij}x_j = \sum_{i=1}^m f_i a_i$. Materials and methods. The study included well-known methods of linear programming and geometry of numbers. Results. The existence of the generalized aggregating equation has been proved for any system of integer linear equations. A simple method to calculate the multipliers $f_1, f_2, ..., f_m$ has been developed for systems with non-negative coefficients. The obtainable lower bound has been received for the right-hand side of the generalized aggregating equation. The multidimensional knapsack problem has been reduced to the classical knapsack problem such that the right-hand side coefficient is less than at any of the existing aggregation methods. Conclusions. The new approach to aggregation broadens the area of its application and reduces coefficients of the aggregating equation.