Solving Integer and Mixed Integer Linear Problems with ABS Method

Solving mixed integer linear programming (MILP) problems is a difficult task due to the parallel use of both integer and non-integer values. One of the most widely used solution is to solve the problem in the real space and they apply additional iteration steps (so-called cutting-plane algorithms or Gomory’s cuts) to narrow down the solution to the optimal integer solution. The ABS class of algorithms is a generalized class of algorithms which, with appropriate selection of parameters, is suitable for the solution of both integer and non-integer linear problems. Here we provide for the first time a complete ABS-based algorithm for MILP problems by adaptation of the ABS approach to Gomory’s cuttingplane algorithm. We also provide a numerical example demonstrating the working principle of our algorithm.