Purpose -Lalwani et al. devised a controllable state-space model for a general APVIOBPCS production and inventory system. However, their procedure did not cater for production delays of other than one time unit. The authors have sought to devise a model that allows for any value of production delay. Design/methodology/approach -A discrete z transform model of APVIOBPCS inventory is obtained using conventional algebra and converted to a state-space model using a reachable control formulation. This is then analysed to produce an analytic expression for the eigenvalues and then the general stability solution is derived from the unit circle condition. Findings -This model allows a state-space model conversion from a discrete time input-output model using an exponential production delay with no loss of generality and is fully controllable and observable. Stability of these models can be obtained from the system eigenvalues and agrees with the authors' previously published stability boundaries using transform models.Research limitations/implications -The system is described by a linear control model of the production process and does not include production limits or other resource limitations. It does not include any past history of sales demand and responses. Practical implications -This work allows a model to be implemented in a spreadsheet of APVIOBPCS PIC that can be used for any production delay and can be modified to include different sales smoothing procedures. Originality/value -This present model is an extension and improvement of the model devised by Lalwani, in that it allows more accurate modelling of inventory production systems by permitting a more flexible selection of delay parameter values, closer to those of real systems.

Purpose – The purpose of this paper is to investigate a control engineering-based system model that allows for any value of production delay for a three-tier supply chain with information delays between tiers or systems with epos. Design/methodology/approach – A discrete z transform model of automatic pipeline, variable inventory and order based production control system three-tier supply chain is obtained using a state-space model using a reachable control formulation. This model provides a discrete time state-space model conversion using an exponential production delay with no loss of generality. Findings – This work allows a three-tier supply chain model to be computed via a spreadsheet using any production delay and can be modified to include different sales smoothing procedures. The model is fully controllable and observable. Stability of these models is obtained from the system eigenvalues and agrees with our previously published stability boundaries. Practical implications – The system is described by a linear control model of the production process and does not include production limits or other resource limitations, including history of sales demand and response. Originality/value – This present model is an extension of the model devised by White and Censlive, in that it allows accurate modelling of multi-tier inventory production systems by permitting flexible selection of delay parameter values for real systems.

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