Markdown optimization at multiple stores

This article studies a markdown optimization problem commonly faced by many large retailers that involves joint decisions on inventory allocation and markdown pricing at multiple stores subject to various business rules. At the beginning of the markdown planning horizon, there is a certain amount of inventory of a product at a warehouse that needs to be allocated to many retail stores served by the warehouse over the planning horizon. In the same time, a markdown pricing scheme needs to be determined for each store over the planning horizon. A number of business rules for inventory allocation and markdown prices at the stores must be satisfied. The retailer does not have a complete knowledge about the probability distribution of the demand at a given store in a given time period. The retailer’s knowledge about the demand distributions improves over time as new information becomes available. Hence, the retailer employs a rolling horizon approach where the problem is re-solved at the beginning of each period by incorporating the latest demand information. It is shown that the problem involved at the beginning of each period is NP-hard even if the demand functions are deterministic and there is only a single store or a single time period. Thus, attention is focused on heuristic solution approaches. The stochastic demand is modeled using discrete demand scenarios based on the retailer’s latest knowledge about the demand distributions. This enables possible demand correlations to be modeled across different time periods. The problem involved at the beginning of each period is formulated as a mixed-integer program with demand scenarios and it is solved using a Lagrangian relaxation – based decomposition approach. The approach is implimented on a rolling horizon basis and it is compared with several commonly used benchmark approaches in practice. An extensive set of computational experiments is perfomed under various practical situations, and it is demonstrated that the proposed approach significantly outperforms the benchmark approaches. A number of managerial insights are derived about the impact of business rules and price sensitivity of individual stores on the total expected revenue and on the optimal inventory allocation and pricing decisions.