Dynamic Multi-product Procurement With Joint and Individual Setup Costs: Theory and Insights

Abstract

In practice, it is common, especially for online retailers, to bundle different products together during procurement to save transportation and handling costs. It is important to understand and theorize how to manage dynamic procurement by taking advantage of joint ordering in the presence of joint and individual setup costs. In this article, we characterize the structure of optimal policy for a periodic review multiproduct inventory system with multiple setup costs, including a joint setup cost and an individual setup cost for each product. By proposing the notion of $(\mathcal{K}\mathcal{K},\mathit{\eta })$ -quasi-convexity, we show that an optimal procurement policy for such a system follows the so-called ( $\sigma ,\omega ,\mathit{S}$ ) policy when demands increase stochastically over time: order up to $\mathit{S}$ for states in the region $\sigma$ , do not order for states in the region $\omega$ , and order certain quantities for states in neither $\sigma$ nor $\omega$ . To better understand the optimal policy, we provide the bounds for the optimal order-up-to levels and the boundary sets of the optimal policy. Under the convex single-period inventory costs, we also provide a lower bound deterministic system which can be asymptotically optimal as the coefficient of variations decreases to zero. Leveraging these operational insights, we propose five simple heuristic policies: the independent ( $\mathit{s},\mathit{S}$ ) policy, vector ( $\mathit{s},\mathit{S}$ ) policy, linear interpolation ( $\mathit{s},\mathit{S}$ ) policy, the deterministic approximation, and the weighted deterministic approximation policy. Extensive numerical experiments indicate that the last three heuristics perform well. In particular, the weighted deterministic approximation policy, whose average performance gap is < 1%, dominates the others in almost all our numerical experiments. Finally, we show that how our results can be extended to systems with more complex setup cost functions, such as time-varying, set-based, and quantity-dependent setup costs.