In this paper, we are interested in the controller design for constrained production systems subject to uncertainties on the demand and the production delays. The case study focuses on the inventory regulation problem in production systems which must respond to the customer demands of finite products. Such systems are characterized by the presence of delays due to production processes, the saturation of the input command and the constraints due to the finite capacities of stocks. In our study, we assume that (i) the customer demands are considered to be unknown but bounded by a given value, (ii) both the control input and the inventory output are subject to assigned constraints, and (iii) the production delay is defined with an uncertainty interval. Our model includes two factors that commonly have an impact on the supply chain performances and cause the bullwhip effect: the variability of the customer demand and the uncertainty on the lead time. The proposed approach is based on a saturated predictor-feedback structure, in which the constraints and the physical limitations of the production system are taken into account. The concepts used in this approach are the BIBO-stability and the D-invariance properties. We examine then the bullwhip effect phenomenon, which is an important observation in supply chain management. In order to study the robustness of the control system state feedback, the proposed approach gives necessary and sufficient conditions on the controller parameters, for which the system requirements will be completely met, and permits to ensure the bounds of solutions for the control parameters. The accuracy of the proposed methodology is illustrated through simulation result which demonstrates that the bullwhip effect can be reduced, but not completely eliminated, by using a saturated command and a predictor-feedback structure.