Vendor‐managed inventory models for items with imperfect quality

In this paper, two models for supply chain under VMI settings are developed when the incoming lot of items contains a fraction of non‐conforming items. The first model is for a decentralised case where the vendor's profit is maximised. However, there is significant extra profit to be realised by optimising the performance of the entire supply chain. Therefore, another model for centralised supply chain under VMI configuration is introduced. This model seeks to optimise the system total expected profit. The additional profit realised by implementing the integrated solution is distributed between vendor and retailer. The distribution scheme is according to some weight such that no party obtains less profit than that generated by decentralised VMI model. Usually, this weight reflects the bargaining positions of the vendor and retailer. The optimal solutions are characterised by finding all Karush‐Kuhn‐Tucker points of the profit functions. Based on these Karush‐Kuhn‐Tucker points, efficient algorithms are devised to solve the models.