Two echelon economic lot sizing problems with geometric shipment policy backorder price discount and optimal investment to reduce ordering cost

This article presents a single-vendor and a single-buyer joint economic lot size (JELS) production-distribution inventory model with the prime aim on; the effect of the investment on ordering cost reduction, back order price discount and reduction on lead time. The produced items are delivered to the buyer by adopting a geometric shipment policy. Two continuous review models are developed by assuming that the lead time demand follows a normal distribution and distribution-free. Two types of investments are incorporated to reduce the ordering cost. They are (i) logarithmic investment function and (ii) power investment function. The minimax distribution free approach is adopted in the distribution-free model to find the optimal values of the decision variables by minimizing the expected annual total cost of the system. Numerical examples are given to validate the proposed models. Sensitivity analysis is also performed to analyze the behavior of the key parameters on lot size, ordering cost, backorder price discount, the number of shipments from the vendor to the buyer in one production run and the expected annual total cost of the proposed models.