A three-layer supply chain integrated production-inventory model with idle cost and batch shipment policy

The paper describes an integrated/centralised supply chain model consisting of one supplier, one manufacturer and one retailer within a finite time horizon. The manufacturer produces, at a finite rate, in each lot. The lot production rate in a batch increases with a rate $\lambda$ in successive batch and the produced items are supplied to the retailer. The objective of the proposed model is to optimize the average total profit under the consideration of the proportional increase in the size of successive shipments within a batch production run and the production time of the supplier. The corresponding average profits of the supplier, the manufacturer and the retailer and the average total profit of integrated model are obtained. The results obtained in the numerical examples clearly establish that it is always beneficial in terms of profit when the size of the successive shipment is a variable. Therefore, size of the successive shipment should be variable in order to get more profit. A sensitivity analysis of the optimal solution with respect to changes of the parameter values is also carried out to strengthen the proposed model.