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Percentile-objective optimisation

Definition

Percentile-objective optimisation is a risk-aware stochastic optimisation formulation in which the decision θ is chosen to minimise a high percentile of the cost distribution — typically the 75th, 90th, or 95th — rather than the expected (mean) cost.

Formally, given a cost function C(θ; ω) that depends on decision θ and random outcome ω with distribution P:

  • Expected-value optimisation: θ* = arg min_θ E_ω[C(θ; ω)]
  • Percentile-objective optimisation: θ* = arg min_θ Q_α(C(θ; ω))

where Q_α is the α-th quantile (e.g. α = 0.75 for P75).

Related formulations: CVaR (Conditional Value-at-Risk), robust optimisation with uncertainty sets, worst-case optimisation. Percentile-objective is the simplest family — a single knob, same Monte Carlo evaluation loop, different aggregation.

Why not minimise the mean

Minimising E[cost] is the textbook answer and is correct on average, but it has blind spots:

  • Asymmetric cost functions. In inventory: a stockout costs a sale (high margin loss per unit); overstock costs holding + write-down (lower per-unit but predictable). The cost distribution is right-skewed — rare high-demand spikes cause catastrophic stockouts.
  • Mean is insensitive to tail shape. Two policies can have the same E[cost] but very different variance; the higher-variance one has more frequent bad outcomes — which the mean doesn't penalise.
  • Business survival favours robustness. One bad season can erase years of average-optimal performance. Risk-aware formulations penalise the bad-outcome tail explicitly.

Why P75 specifically

The choice of α = 0.75 is a balance:

  • Too low (α = 0.50, median) — essentially equivalent to mean in symmetric distributions; no tail protection.
  • Too high (α = 0.99, P99) — overfits to rare outlier realisations; becomes worst-case optimisation in disguise. Produces excessively conservative decisions.
  • α = 0.75 — enough tail weight to explicitly penalise upper-quartile bad outcomes, not so extreme that the optimiser chases rare tail samples.

Zalando is explicit about the bar: "This ensures our decisions protect against extreme, rare demand spikes."

Canonical instance (Zalando ZEOS)

The ZEOS replenishment recommender minimises the 75th percentile of the 12-week cumulative cost distribution across N Monte Carlo DES runs:

$$\theta^* = \arg\min_{\theta \in \Theta} Q_{0.75}!\left[ C_{\text{holding}}(\theta) + C_{\text{inbound}}(\theta) + C_{\text{outbound}}(\theta) + C_{\text{returns}}(\theta) + C_{\text{lost sales}}(\theta) \right]$$

Verbatim: "Instead of minimizing the average cost, we minimize the 75th percentile of the cost distribution. This ensures our decisions protect against extreme, rare demand spikes."

Empirical value (from the ablation study)

The paper's ablation isolates the percentile-objective contribution:

Forecast Objective GMV Uplift Availability Fill Rate
Probabilistic P75 22.11% 86.40% 91.14%
Probabilistic Mean 19.02% 81.27% 87.98%
Point P75 6.37% 77.76% 84.95%

Reading the delta (row 1 vs row 2): switching from mean to P75 on the same probabilistic forecast adds +3.09pp GMV uplift, +5.13pp availability, +3.16pp fill rate. This is the stability layer the Zalando post frames as "the final, critical layer". Probabilistic forecasting is the larger first-order lever (row 1 vs row 3: +15.74pp GMV), but the percentile objective supplies the tail-protection Monte Carlo guarantee.

Verbatim: "You need both. Switching from point forecasts to probabilistic ones provides the single largest gain. However, optimizing for the 75th percentile rather than the average provides that final, critical layer of stability, particularly in protecting the merchant against high-impact 'tail' events."

Prerequisites

  • Probabilistic cost estimate, not point estimate. If your evaluator only returns E[cost], there's no quantile to minimise. You need Monte Carlo realisations or an analytic quantile — which typically means Monte Carlo over a probabilistic forecast inside a DES.
  • Sufficient Monte Carlo samples to estimate the percentile stably. Estimating P75 reliably needs more samples than estimating the mean — rough rule: need enough samples that P75 is in the stable-density regime, not on the jagged sample-boundary edge. Zalando doesn't disclose their N but references "thousands of plausible futures per candidate policy".
  • Non-differentiable objective tolerance. The quantile of a Monte Carlo estimator is non-smooth in θ; the outer optimiser must be gradient-free.

Tradeoffs

  • Compute cost. P75 needs the full cost distribution, not just the mean. More samples per candidate θ. Offset by the fact that DES already computes full distributions — the percentile is a cheap aggregation on top.
  • Hyperparameter: which α? P75 is Zalando's choice. Other businesses may prefer P90, P95, or CVaR_α. The paper doesn't report sensitivity to α.
  • Conservatism cost. P75 is more conservative than mean — you pay some expected-value on average to gain tail protection. In Zalando's backtest this trade was uniformly positive, but in lower-variance environments (steady demand), the P75 penalty might exceed the tail-risk benefit.
  • Not the same as robust optimisation. P75 minimises a percentile of the sampled cost distribution; robust optimisation minimises the worst case over an uncertainty set. P75 is a middle ground between expected value and worst-case.

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