Motivated by the real-world problem of a logistics company, this paper proposes a novel distribution-free prescriptive analytics approach—termed kernelized empirical risk minimization (kernelized ERM)—to solve a complex two-stage capacity planning problem with multivariate demand and vector-valued capacity decisions and compares this approach both theoretically and numerically with an extension of the well-known sample average approximation (SAA) approach termed weighted SAA. Both approaches use integrated machine learning algorithms to prescribe capacities directly from historical demand and numerous features (covariates) without having to make assumptions about the underlying multivariate demand distribution. We provide extensive analytical insights into both approaches. Most important, we prove the universal approximation property for the kernelized ERM approach when using a universal (data-independent) kernel and show how out-of-sample guarantees can be derived for various kernels. We demonstrate the applicability of both approaches to a real-world planning problem and evaluate their performance relative to traditional parametric approaches that first estimate a multivariate demand distribution and then solve a stochastic optimization problem and a nonparametric approach (SAA). Our results suggest that the two prescriptive analytics approaches can result in substantial performance improvements of up to 58% compared with traditional approaches. Additional numerical analyses shed light on the behavior and performance drivers of the various approaches and demonstrate that in our case, the prescriptive approaches are much more robust to variations of exogenous cost parameters than traditional approaches. This paper was accepted by J. George Shanthikumar for the Special Issue on Data-Driven Prescriptive Analytics.

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