A Method to Convert Integer Variables of Mixed Integer Programming Problems into Continuous Variables

Abstract

A mixed integer programming problem, is an optimization problem that includes both integer decision variables and continuous ones. Many engineering and management problems are mixed integer programming problems, most of which are NP-hard and take a long time to approach the optimal solution. Therefore, this paper proposes a method to convert the integer variables of a mixed integer programming problem into continuous variables. Then a mixed integer programming problem becomes an equivalent nonlinear programming problem or linear programming problem. And then, well developed nonlinear programming or linear programming solvers can be employed to efficiently and effectively search for the solution of a mixed integer programming problem. In addition to describing processes for converting integer variables into continuous ones, this paper provides two examples to demonstrate the conversion process and the benefits coming from the conversion methodology in the process of approaching the optimal solutions. Once the mixed integer programming is converted to an equivalent one where the decision variables are continuous, a nonlinear programming problem solver such as a differential evolutionary algorithm, can be used to solve the new equivalent problem. We show the procedure by solving a practical mixed integer programming problem that arises in a typical mechanical design problem. The result shows our solution is better than the ones from other four published mixed integer programming solvers.KeywordsCombinatorial optimizationCombinatorial analysisEvolutionary computationsGlobal optimization