Heuristic algorithms for the problem of integer balancing of a three-dimensional matrix with constraints of the second type

Abstract

The problem of integer balancing of a three-dimensional matrix with constraints of the second type is studied. The elements of the inner part (all three indices are greater than zero) of the three-dimensional matrix are summed in each direction and each section of the matrix; the total sum is also found. These sums are placed into the elements where one or more indices are equal to zero (according to the summing directions). The problem is to find an integer matrix of the same structure, which can be produced from the initial one by replacing the elements of the inner part with the largest previous or the smallest following integer. At the same time, variations of the sums of elements from those in the initial matrix should be less than 2 and an element with three zero indices should be produced with standard rounding-off rules. Heuristic algorithms for this problem are suggested: layering algorithm obtained as a generalization of a similar algorithm for the problem with constraints of the first type and a new matrix algorithm. The latter consists of three parts: search for the basic matrix, search for the maximum matrix, and matrix correction. Each of them is a cyclic change of the integer matrix using from one to three elements from the inner part. A modification of the matrix algorithm is suggested. The algorithm is directed to more uniform filling of the inner part of the integer matrix. Also, the complexity of all three algorithms is estimated. The comparative analysis of matrix algorithms based on the results of computing experiments is adduced.