A Distributionally Robust Optimization Approach to Quick Response Models under Demand Uncertainty

Quick response is a widely adopted strategy to mitigate overproduction in the manufacturing industry, yet recent research reveals a counter-intuitive paradox: while it reduces waste from unsold finished goods, it may incentivize firms to procure more raw materials, potentially increasing total system waste. Additionally, existing models that guide quick response strategies rely on the assumption of a known demand distribution, whereas in practice, demand patterns are often ambiguous and historical data are scarce. To address these challenges, we develop a distributionally robust optimization (DRO) framework for the quick response model that builds robust policies even with limited data. We further integrate a novel waste-to-consumption ratio constraint into this framework, empowering firms to explicitly control the environmental impact of their operations. Our numerical experiments demonstrate that policies optimized for specific demand assumptions suffer severe performance degradation under distributional shifts, whereas our data-driven DRO approach consistently delivers superior robustness. Moreover, we find that the constrained quick response model resolves the central paradox: it can achieve higher profits with verifiably less total waste than a traditional, non-flexible alternative. These results resolve the quick response or not' debate by showing that the question is not \emph{whether} to use quick response, but \emph{how} to manage it. By incorporating socially responsible metrics as constraints, the quick response system delivers a win-win’ outcome for both profitability and the environment compared to traditional systems.

💡 Research Summary

The paper tackles a fundamental paradox in quick‑response (QR) manufacturing: while QR reduces waste from unsold finished goods, it can encourage firms to purchase more raw material, potentially increasing total system waste. Existing QR models assume a known demand distribution, an assumption that rarely holds in practice, especially for industries with short product lifecycles and limited historical data. To overcome these limitations, the authors develop a distributionally robust optimization (DRO) framework that delivers QR policies that remain effective under ambiguous demand information.

First, they prove that the expected profit function of the QR system is globally concave for any demand distribution. This structural property guarantees that the DRO problem admits a global optimum and can be solved efficiently. Two ambiguity sets are considered. The first is a moment‑based set defined by the known mean and mean‑absolute‑deviation (MAD). Within this set, the optimal QR policy takes a simple threshold form: produce an additional quantity only when observed demand exceeds a critical level that can be expressed analytically. This closed‑form solution offers immediate managerial insight and is easy to implement.

The second ambiguity set is data‑driven: a Wasserstein ball around the empirical distribution constructed from a limited sample of demand observations. By reformulating the worst‑case expectation over this ball as a second‑order cone constraint, the authors obtain a tractable SOCP that can be solved with standard convex‑optimization solvers. The SOCP formulation is asymptotically consistent, meaning that as the sample size grows the DRO solution converges to the true optimal policy.

A major methodological contribution is the incorporation of an explicit environmental performance metric, the waste‑to‑consumption (WTC) ratio, which measures deadstock per unit of satisfied demand. The WTC ratio is a fractional, expectation‑based constraint that is non‑convex and semi‑infinite. The authors show that it can be equivalently expressed as a single worst‑case expectation constraint. Consequently, for both the MAD and Wasserstein ambiguity sets the WTC‑constrained DRO problem remains tractable: it reduces to a linear program (LP) in the MAD case and to an SOCP in the Wasserstein case.

Numerical experiments compare three approaches: (i) policies optimized under a specific assumed demand distribution (e.g., uniform), (ii) the MAD‑based DRO policy, and (iii) the Wasserstein‑based DRO policy. The results demonstrate a “price of misspecification”: distribution‑specific policies suffer severe profit loss and increased waste when the true demand deviates even slightly from the assumed distribution. In contrast, both DRO policies exhibit robust performance across a wide range of demand shifts, including changes in mean, variance, tail heaviness, and mixture structures. Importantly, when the WTC ratio constraint is enforced, the DRO policies achieve higher expected profit than a traditional non‑flexible (single‑stage) system while guaranteeing a lower total waste level.

The study thus resolves the “quick response or not” debate. It shows that the decisive question is not whether to adopt QR, but how to manage it under demand ambiguity and environmental constraints. By integrating socially responsible metrics as hard constraints, the proposed DRO framework enables a win‑win outcome: firms can reap the financial benefits of QR while demonstrably reducing their environmental footprint. The paper provides both theoretical tools (global concavity, tractable reformulations) and practical guidelines (threshold policies, data‑driven SOCP) for managers and policymakers seeking sustainable, resilient manufacturing operations.