Optimal Dynamic Promotion and the Knapsack Problem for Perishable Items (Extended Abstract)

Senior managers in retail industry make important decisions upon assortment planning, product pricing, and product promotion. While product assortment is a strategic decision taken over a long-term planning period (Kök et al., 2006), the latter two are both strategic and tactical: they can be used in dayto-day marketing decisions to dynamically adjust to demand variations. Within the food retail industry, the necessity, frequency, and complexity of pricing and promotion decisions are further magnified by perishability of food products. There is a strong need by retail managers for a “soft”marketing tool, whichwould dynamically allow them to improve sales and revenues, yet not altering product prices. For, dynamic pricing models may prescribe to change prices too often or in an “unsystematic” fashion, which contradicts discrete-time decision making, implementation costs, and retail brand image strategy. In addition, the price reduction must usually be done over all units of the product, thus losing possible profit from customers willing to pay the original, higher price. Furthermore, dynamic pricing results in a trade-off between markdowns and stockouts, since markdowns may damage producers, while stockouts may damage retailers. We design a revenue management model, in which demand is stimulated by moving a number of product units to a promotion space, rather than by price changes. Thus, we address the problem of filling a promotion location with limited space to maximize the expected revenue, which we have termed the Knapsack Problem for Perishable Items (KPPI). Examples of the promotion space include shelves close to the cash register, promotion kiosks, or a depot used for selling via the Internet. We solve the KPPI using a problem decomposition into single product unit subproblems. A natural mathematical setting for the KPPI subproblems is the restless bandit model (cf. Whittle, 1988; NiñoMora, 2002), a fundamental stochastic model for resolving a trade-off between exploration and exploitation in an optimal fashion. In our model the bandits (perishable items) are restless, because they can get sold regardless of being in the knapsack or not, the time horizon is finite, and we are to select more than one item for the knapsack, which is allowed to be filled partially, due to the heterogeneity of the items. Each product unit is assigned a promotion priority index, which captures the opportunity cost of promotion, as a function of its price, lifetime, expected demand, and expected promotion power. These indices are then used for each item as objective-function value coefficients in a (classic) knapsack problem, whose solution yields a well-performing heuristic for the KPPI. We thus mix up two models: the restless bandit problem and the knapsack problem.