Single-Setup-Multiple-Deliveries for a Single Supplier- Single Buyer with Single Product and Backorder

This article investigates integrated production-inventory models with backorder. a single supplier and a single buyer are considered and shortage as backorder is allowed for the buyer. the proposed models determine optimal order quantity, optimal backorder quantity and optimal number of deliveries on the joint total cost for both buyer and supplier. two cases are discussed: single-setup-single-delivery (sssd) case and single-setup-multiple-deliveries (ssmd) case. two algorithms are applied for optimizing ssmd case: gradient search and particle swarm optimization (pso) algorithms. finally, numerical example and sensitivity analysis are provided to compare the total cost of the sssd and ssmd cases and effectiveness of the considered algorithms. findings show that the policy of frequent shipments in small lot sizes results in less total cost than single shipment policy.