Control theory applications to the production-inventory problem: a review

Mathematical models based on average or steady-state conditions are insufficient when dealing with dynamic situations faced by production–inventory systems in current business environments. Therefore, the use of mathematical tools based on control theory to handle time-varying phenomena has been reinvigorated in order to accommodate these new needs. Given the variety of research approaches in the field over several decades, there is a need to provide a review of this work. This review identifies some major research efforts for applying control theoretic methods to production–inventory systems. It is shown that in general, control theory is applied to reduce inventory variation, reduce demand amplification and optimize ordering rules. Some control theory tools applied are block diagram algebra, Mason's gain formula, Bode plots, Laplace transform, Z transform and optimal control. Basic approaches are classified within stochastic control theory and deterministic control theory. Two important issues are then identified within the deterministic models. First, separate efforts to integrate systems horizontally (i.e. supply chain) or vertically (i.e. hierarchical approach) are identified. Second, none of the reviewed models implemented a systematic way to calculate all the required model parameters. Some authors presented suggestions to optimize some parameters, but no reference was found that tried to obtain these parameter values from a real system.