Control system analysis of labour supply flows in production systems

• A review is made of labour supply chains using system dynamics.
• A review is made of supply chains using control system models.
• The model combines labour flow with a VMI inventory.
• The model is used to predict the stability of the combined VMI + labour supply chain.
• The results show that the parameters for stable operations of the inventory alone also apply to the combined chain model.

This paper examines the effects of the labour supply chain on the control of a two tier VMI inventory model based on the work of Towill and Disney for the case of an individual company.An extended labour plus the VMI model is used to produce an analytic expression for the combined system.A discrete z transform model of inventory is obtained using conventional algebra. This is then analysed to produce an algebraic expression and then the general stability solution is derived from the unit circle condition.The labour model alone is shown to be stable with the average employment duration and time to recognise a staff deficiency found to be the key parameters to achieve quick response to a labour shortage. Using MATLAB, the extended model incorporating the labour supply has been found to have a wider range of gains for stability, than without the labour component.The system is described by a linear control model of the production process and does not include production limits or other resource limitations. It also does not include any past history of sales demand and responses.Using the control measure integral of the time multiplied by the absolute error (ITAE), the combined labour and VMI model has been found to have different gains for optimum inventory than VMI alone, allowing a more flexible response to recruitment since labour costs are key cost parameters in the real world.This work extends the original control system model of Towill and Disney to include the effects of the Labour supply chain. This has not been published before.