Managing Inventory with Learning About Demands and Inter-Temporal Substitution Probability

Problem Definition: We consider a setting where firms sell the same perishable products in two phases during each period, an early-bird-discount phase and a regular-price phase. When the discounted product is stocked-out, an early-bird customer either purchases the regular-price product as the substitute (called inter-temporal substitution) or simply leaves.Academic/Practical Relevance: We study the inventory managers' optimal decisions on the total inventory level and the inventory level reserved for early-bird sales, where the demands for the early-bird-discount product and the regular-price product and the inter-temporal substitution probability are unknown and have to be learned from historical sales data.Methodology: We investigate four information scenarios based on whether or not lost sales are observable and substitution-induced demands can be separated out. We then build a Bayesian updating framework to learn both the demand distribution parameter and the substitution probability.Results: We find that the classic "stock more" result (that is, the Bayesian-optimal inventory level shall be kept higher than the myopic one) may or may not hold here, depending on the relative magnitude of the following four driving forces. One, the unobservability of lost sales requires "stock more". Two, the tangling of substitution demand and primary demand in the regular sales phase. To better observe the primary demand requires "stock more" for the discounted product, which reduces the substitution. Three, to better observe the substitution probability, one shall "stock less" the discounted product so that substitution happens more frequently. Four, compared to the regular-price product, the discounted product has a lower profit margin, dampening the firm's "stock more'' incentive.Managerial Implications: Our results shed light on the inventory manager's optimal inventory decisions with Bayesian learning, i.e., how much inventory allocated for early-bird-discount sales and how much for the regular-price sales when demand and substitution probability information is limited. Our results also complement the existing Bayesian inventory management literature by identifying new driving forces and deriving a general upper bound for the optimal solution