The problems of choosing the cutting plane are increasingly important for economic applications, in particular the choice of the cutting plane in each iteration of the related ILP algorithm is very important in terms of convergence speediness. Several studies have been done on the matter, based on the study of the feasible cutting planes polynomial structures. Gomory [2] proved that the optimal cutting plane is the one that maximizes the number of feasible integer points the cut touches. A theorem introduced by Pick [3] allows calculating the area of each polygon whose vertices belong to a bi-dimensional lattice, as a function of the number of its internal and boundary lattice points. In 1957, John Edmund Reeve proposed a generalization of Pick’s theorem to the three dimensional case. Starting from results obtained by Caristi and Stoka [1] this paper considers two lattices with the non-convex cells represented in Figures 1 and 2. The probability is determined that a constant-length segment with random exponential distribution and random γ (2) distribution will intersect a side of the lattices.

cutting planes, lattice points, random sets, linear integer programming.