Prescriptive Analytics for Flexible Capacity Management

Abstract

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.

Training Demand Prediction Models by Decision Error for Two-Stage Lot-Sizing Problems

“…Oroojlooyjadid et al applied a deep learning model to optimize the order quantities for the newsvendor problem based on features of the demand data [30]. Notz and Pibernik proposed a machine-learning approach termed kernelized empirical risk minimization to prescribe optimal decisions for capacity management problems [38]. The end-to-end framework seems attractive because it outputs decisions directly by accepting feature data as input.…”

A Deficiency of the Predict-Then-Optimize Framework: Decreased Decision Quality with Increased Data Size

“…Our work contributes to two primary streams of literature. First, it relates to contextual stochastic optimization problems, where decision makers observe previous samples of the uncertain parameter, along with features providing additional information about the uncertainty [3,5,[14][15][16][17][18]. Besides the widely-used PO framework, several advanced frameworks have emerged, which prescribe decisions directly from data.…”

“…Besides the widely-used PO framework, several advanced frameworks have emerged, which prescribe decisions directly from data. They include the smart predict-then-optimize framework [16,19], the weighted SAA (wSAA) framework [17,18], the empirical risk minimization framework [14,17], and the kernel optimization framework [5,14,17,20]. Theoretically, these works have demonstrated the asymptotic optimality of these frameworks, i.e., whether the data-driven solutions can converge to the fullinformation optimal solution as data size approaches infinity [9].…”

A Deficiency of the Weighted Sample Average Approximation (wSAA) Framework: Unveiling the Gap between Data-Driven Policies and Oracles

“…Our work contributes to two key areas of literature. Firstly, it relates to prescriptive analytics frameworks, in which decision-makers leverage contextual data about uncertainty to make informed decisions [1,2,[11][12][13][14]. Several advanced frameworks have been developed, including the smart predict-then-optimize framework [13,15], the wSAA framework [10,14], the empirical risk minimization framework [12,14], and the kernel optimization framework [2,12,14,16].…”