Integrated single-vendor single-buyer inventory model with imperfect production and inspection by vendor

An integrated single-vendor, single-buyer inventory model is discussed in this research work with a situation where the production quality and post production inspection of items are imperfect. The imperfect production process produces few defective items at known probability distribution. An inspection of all items is conducted along with their production. The inspection process classifies items as perfect or defective. Perfect items are shipped to the buyer in lots and defective items are stored in the vendor's inventory to be dispose at a discounted rate at the end of each production cycle. As the inspection process is itself imperfect, some perfect items can be classified as defective, known as type I inspection error, with known probability distribution and defective items can be classified as perfect, known as type II inspection error, with known probability distribution, during the inspection process. Because of type I inspection error, some perfect items can be dispose at discounted rate as defective items, leading to loss of revenue. Similarly, due to type II inspection error, some defective items can be supply to the buyer and sale in market as perfect items. After use, defect will be detected in such items and get replace with a perfect one. All returned defective items are stored in the buyer's inventory and sent back to the vendor for disposal at end of each lot cycle. It leads to loss of goodwill and revenue. The minimum Expected Total Cost per production cycle is derive for integrated situation where both the vendor and the buyer work together to reduce minimum Expected Total Cost. There is a situation where the buyer does not cooperate with the vendor and places an order for items in a single lot. This is also discussed this paper and Minimum Expected Total Cost is compared with the integrated situation.