Solving mixed integer programming production planning problems with setups by shadow price information

In this study, we have solved production planning problems involving multiple products, multiple resources, multiple periods, setup times and setup costs. If a particular product is to be manufactured in a particular period, each required machine must be set up for that product exactly once during that period. This is a very generalized version of the capacitated lot sizing problem. Even though there has been plenty of research work on similar problems, models dealing with the degree of complexity presented in this study are rather scarce. Such production planning with setup decisions can be formulated as a mixed integer programming (MIP) problem. However, solving realistic MIP production planning problems is NP-hard; thus, obtaining optimal solutions is usually impossible. Heuristic methods are required to obtain good solutions efficiently. Algorithms developed to solve linear programming (LP) problems and advances in computer speed have made large-scale LP problems solvable in time for implementation. Solving an LP is relatively easier than solving an MIP for modern production planning problems. In this study, we propose a heuristic iterative algorithm between LP solution phases and setup decision computations for solving these difficult MIP production planning problems. By utilizing the shadow price information provided by the LP solution of the previous iteration, the setup decision computation converts an MIP problem into an LP problem, which can be efficiently solved in the current iteration. Extensive experiments show that the proposed heuristic algorithm performs well.