Motivated by the worldwide Covid-19 vaccine procurement, we study an inventory problem with an advance purchase contract which requires all ordering decisions to be committed at once. In reality, not only the demand is uncertain, but its distribution can also be ambiguous. Hence, we assume that only the mean and the variance are known and aim at minimizing the worst-case expected cost. We first show that our inventory model reduces to a robust conic optimization problem with a finite yet exponentially-sized uncertainty set. To gain tractability and err on the safe side, we propose two conservative approximations. Then to measure their approximation quality, we develop a progressive approximation based on a scenario reduction technique. All of the approximate models are expressed as standard polynomially-sized conic programs, which scale gracefully and allow us to incorporate additional distributional knowledge via a cone replacement. We quantify the benefit of committing to advance purchases, and we show that all approximations are close to being exact. Besides, we analytically derive the worst-case demand distribution and numerically use it to show that our robust policy is more resilient to the misspecification of the demand distribution than the state-of-the-art non-robust policies.
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