A Novel Alternative Algorithm for Solving Integer Linear Programming Problems Having Three Variables

In this study, a novel alternative method based on parameterization for solving Integer Linear Programming (ILP) problems having three variables is developed. This method, which is better than the cutting plane and branch boundary method, can be applied to pure integer linear programming problems with m linear inequality constraints, a linear objective function with three variables. Both easy to understand and to apply, the method provides an effective tool for solving three variable integer linear programming problems. The method proposed here is not only easy to understand and apply, it is also highly reliable, and there are no computational difficulties faced by other methods used to solve the three-variable pure integer linear programming problem. Numerical examples are provided to demonstrate the ease, effectiveness and reliability of the proposed algorithm.

EFFICIENT ENUMERATION OF GRID POINTS IN A CONVEX POLYGON AND ITS APPLICATION TO INTEGER PROGRAMMING

This paper first presents an algorithm for enumerating all the integer-grid points in a given convex m-gon in O(K + m + log n) time where K is the number of such grid points and n is the dimension of the m-gon, i.e., the shorter length of the horizontal and vertical sides of an axis-parallel rectangle enclosing the m-gon. The paper next gives a simple algorithm which solves a two-variable integer programming problem with m constraints in O(m log m + log n) time where n is the dimension of a convex polygon corresponding to the feasible solution space. This improves the best known algorithm in complexity and simplicity. The paper finally presents algorithms for counting the number of grid points in a triangle or a simple polygon.