Promotion Optimization for Multiple Items in Supermarkets

Promotions are a critical decision for supermarket managers who must decide the price promotions for a large number of items. Retailers often use promotions to boost the sales of the different items by leveraging the cross-item effects. We formulate the promotion optimization problem for multiple items as a non-linear integer program (IP). Our formulation includes several business rules as constraints. Our demand models can be estimated from data and capture the post-promotion dip effect as well as cross-item effects (substitution and complementarity). Since demand functions are typically nonlinear, the exact formulation is intractable. To address this issue, we propose a general class of IP approximations. For demand models with additive cross-item effects, we prove that it is sufficient to account for unilateral and pairwise contributions and derive parametric bounds on the performance of the approximation. We also show that the unconstrained problem can be solved efficiently via a linear program when items are substitutable and the price set has two values. For more general cases, we develop efficient rounding schemes to obtain an integer solution. We conclude by testing our method on realistic instances and convey the potential practical impact for retailers