Analysis of the Single-Vendor-Multi-Buyer Inventory Model for Imperfect Quality with Controllable Lead Time

The two-echelon single vendor-multi-buyer inventory model is an interesting topic and is suitable to many real conditions in the supply chain system. The existence of imperfect quality items and controllable lead time is one of the important assumptions in modern inventory analysis. This study analyzes the two-echelon single-vendor-multi-buyer inventory model with imperfect quality items presented as random variables, which follow the Binomial distribution. We use boundedness service level constraint to replace shortage cost term in the objective function. We also use the integrated scheme to formulate optimization problems. We prove a theorem about the nonconvexity properties of the objective function and then apply this result with the Karush-Kuhn-Tucker (KKT) conditions and the Lagrange multiplier method to obtain an optimum solution. Using numerical examples, we show that our model can help a vendor minimize the total cost of the inventory system and the number of lots in which a product is delivered from the vendor to all buyers under an uncertain lead time by reducing the number of imperfect items.