Joint inventory and pricing decisions with reference effects

This article considers a periodic review joint replenishment and pricing problem of a single item with reference effects. The demand is random and is contingent on the price history as well as the current price. Randomness is introduced with both an additive and a multiplicative random term. Price history is captured by a reference price, which is developed by consumers that are frequent buyers of a product or a service. The common reference price acts as a benchmark against which the consumers compare the price of a product. They perceive the difference between the price and the reference price as a loss or a gain and have different attitudes to these perceptions, such as loss aversion, loss neutrality, or loss seeking. A general way to handle the nonconvexity of the holding cost for nonlinear demand models is to make a transformation and use the inverse demand function. However, in reference price-dependent demand models, this brings the problem of a nonconvex action space. This problem is circumvented by defining an action space that preserves the convexity after a transformation. For the transformed problem, it is shown that a state-dependent order-up-to policy is optimal for concave demand models and concave transformed expected revenue functions that are not necessarily differentiable. It is shown that there are demand models with relative difference reference effects and loss-averse customers that satisfy the considered concavity assumptions. A computational study is performed to highlight the effects of joint inventory and pricing decisions under reference effects.