Abstract: During production, product quality is not always perfect, since it is directly affected by the reliability of the production process. To ensure the customer requirement of quality, quality screening is necessary. However, the inspector may make inspection errors during the screening process, which leads to two types of quality inspection risk, i.e., type-Ⅰ risk and type-Ⅱ risk. Due to quality inspection risk, shortages may sometimes occur. On the other hand, there is uncertainty outside demand, and it is fuzzy. In the meanwhile, the productivity is also changeable, which is also fuzzy. Based on these internal and external fuzzy demand and fuzzy productivity, as well as the quality inspection risks, a two-level production-inventory decision with fuzzy demand and fuzzy production rate is studied. In the production-inventory system, there is one manufacturer and one retailer. The manufacturer produces imperfect quality products and delivers the products to the retailer in small lots of equally sized shipments. Upon receipt of the products, the retailer will conduct a 100% inspection. As mentioned before, quality screening risk may occur for the retailer. By using JIT philosophy, the retailer has to determine the optimal order size and backordering quantity, and the manufacturer has to determine the optimal production batch and the optimal number of shipments between the manufacturer and the retailer per production cycle. The objective of this study is to minimize the total joint annual costs incurred by the manufacturer and the retailer. The signed distance method for fuzzy numbers is employed to transform the fuzzy total cost into the crisp cost. The total cost is proved to be a joint convex function of the size of the shipments and the backordering quantity of the retailer. Numerical examples show that with the increase of the expected defect rate, the optimal ordering quantity increases, the optimal backordering quantity decreases, and the optimal cost increases with a fast rapid speed. Type-Ⅰrisk makes total cost increase, but type-Ⅱ risk makes total cost decrease. The optimal ordering strategy is sensitive to type-Ⅰrisk, but not sensitive to type-Ⅱ risk. The contributions of this study are twofold. First, the inspection risk behavior is incorporated into the optimal decisions of the JIT production-inventory model. Second, the fuzzy demand and fuzzy production rate are considered to establish the model.

Key words: production-inventory, fuzzy demand, fuzzy productivity, quality screening risk