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

This article investigates the optimal procurement strategy for a finite-horizon, periodic-review inventory system involving multiple products with both joint and individual setup costs. It introduces the concept of \((\mathcal{K}, \boldsymbol{\eta})\)-quasi-convexity to characterize the structure of the optimal policy, proving that it follows a \((\sigma, \omega, \mathbf{S})\) policy: ordering up to \(\mathbf{S}\) for inventory states in \(\sigma\), not ordering for states in \(\omega\), and ordering certain quantities otherwise. The study provides explicit bounds for the optimal order-up-to levels and reorder sets, and establishes a lower bound deterministic system that is asymptotically optimal as demand variability decreases. Motivated by these theoretical insights, five heuristics are proposed to approximate the optimal policy, with numerical experiments showing that the linear interpolation, deterministic approximation, and weighted deterministic approximation heuristics perform particularly well, often within 1% of optimality. Extensions to time-varying and more complex subadditive setup costs are also discussed.