Markdown optimization for an apparel retailer under cross-price and initial inventory effects

In this paper, we address the simultaneous determination of markdown prices and optimal initial inventory levels under the cross-price effects in a random demand setting for multi-product groups for an apparel retailer chain in Turkey. First, we formulate the problem as a Markov Decision Process that considers price-based substitution and complementary effects among products and maximizes the expected total profit over a finite horizon. Then, we find the approximate markdown policies of each product by using Approximate Dynamic Programming algorithm. We investigate how cross-price elasticity affects the markdown policies of each product by considering several relationships among them, such as the products are all substitute or all are complement or some are substitute and some are complement. In addition to this, we provide insights on how they affect the expected revenues when non-optimal and optimal initial inventory levels are considered. When cross-price effects are considered in case of non-optimal initial inventory levels, average revenue increases about 32% while it increases to 50% when optimal initial inventory levels are in case.