A bundling strategy for items with different quality based on functions involving the minimum of two random variables

A common strategy to increase sales and profit is to combine different types of products into bundles and sell at a discounted price. In this study, we consider the case where a wholesaler offers to sell two types of products through discount bundles. Each of the two types of products is purchased from a producer in lots that contain a percentage of imperfect quality items, which is a random variable having a known probability density function. Items received from the producer are inspected for imperfect quality using a 100% screening process. The perfect quality items are used to make the discount bundles, while the imperfect quality items are sold at a discounted price at the end of the screening period. Items of perfect quality of one type that are not bundled are kept in stock to be used in the next inventory cycle. A mathematical model is developed to determine the total profit function. A closed-form formula for the wholesaler’s optimal order quantity of each type of product is determined by maximizing the profit function. The optimal solution is given in terms of the expected values of functions involving the two random variables representing the percentages of perfect quality items. Numerical examples are provided to illustrate the model, and simulation is used to calculate the optimal solution.