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

TL;DR

The paper addresses demand uncertainty in quick-response production and the unintended environmental paradox where recourse flexibility can raise upstream waste. It develops a distributionally robust optimization framework with two ambiguity sets—mean-MAD and Wasserstein—and proves the profit function is concave, enabling tractable reformulations. A novel waste-to-consumption (WTC) ratio constraint is integrated, reformulated into a linear program under mean-MAD and into a second-order cone program under Wasserstein, delivering robust and environmentally aware policies. Empirical results show the DRO policies outperform benchmark quick-response schemes across distributional shifts and that the WTC constraint enables a true win-win, achieving higher profitability with verifiably lower total waste than traditional, non-flexible systems.

Abstract

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.

Figures (8)

• Figure 1: A visualization of the concave profit function $\Pi(x,q,y)$. Note that $d_y$ denotes the realized effective demand given by $d_y = (1-p)y$, where $y$ is a realization of the random demand $Y$.
• Figure 2: A comparison of optimal policies, expected profits, and WTC ratios for our DRO models against the benchmark by quick_res when the true demand distribution is uniform. Across all subfigures, the orange line represents quick_res model, the red line represents our DRO model with a MAD ambiguity set, and the green line represents our DRO model with a Wasserstein ambiguity set.
• Figure 3: A comparison of optimal policies, expected profits and WTC ratios for our DRO models against the benchmark by quick_res when the true demand distribution is a Lognormal distribution. Across all subfigures, the orange line represents quick_res model, the red line represents our DRO model with a MAD ambiguity set, and the green line represents our DRO model with a Wasserstein ambiguity set.
• Figure 4: A comparison of optimal policies, expected profits and WTC ratios for our DRO models against the benchmark by quick_res when the true demand distribution is a Beta distribution. Across all subfigures, the orange line represents quick_res model, the red line represents our DRO model with a MAD ambiguity set, and the green line represents our DRO model with a Wasserstein ambiguity set.
• Figure 5: A comparison of optimal policies, expected profits, and WTC ratios between the unconstrained quick response (QR) system and the no quick response (NQR) system. Across all subfigures, the red line denotes the QR system, and the yellow line represents the NQR system.

Theorems & Definitions (18)

• Proposition 1
• Proposition 2
• Theorem 1
• Theorem 2
• Lemma 1: Theorem 18 in kuhn2019wasserstein
• Proposition 3
• Theorem 3
• Proposition 4
• Proposition 5
• proof