CONCEPT Cited by 1 source

Monte Carlo simulation under uncertainty

Definition

Monte Carlo simulation evaluates a decision or policy by drawing many random samples from the underlying uncertainty distributions and averaging the outcome across samples. In decision-under-uncertainty problems — inventory management, capacity planning, financial risk, scheduling — Monte Carlo replaces closed-form expectation calculation with sampled expectation, which is often the only tractable option when the cost function involves branching logic (stockouts), discontinuities (step-function storage costs), or non-linear-interactions with other probabilistic inputs.

Shape

For each candidate decision θ:

1. Sample — draw N realisations of the underlying random variables (demand, lead times, returns, prices…) from their forecast distributions.
2. Simulate — for each sample, run the forward simulator (inventory position evolves, orders are placed, demand is served or lost, holding costs accrue) to compute the realised cost.
3. Aggregate — average (or take percentile, CVaR, etc.) across the N samples to score θ.
4. Optimise — hand the score to an outer optimiser — typically gradient-free because Monte Carlo output is non-differentiable.

Why gradient-free optimiser

The Monte Carlo expectation is a non-differentiable, noisy function of θ — no clean gradient. Stochastic gradient methods can be made to work with reparameterisation tricks, but the operational simplicity of just sampling + evaluating fits naturally with black-box optimisers (CMA-ES, Bayesian optimisation, simulated annealing, Nelder-Mead over averaged batches).

Canonical instance (Zalando ZEOS)

The full ZEOS Replenishment Recommender cost objective:

$$Min\ Costs(\theta) = C_{storage}(\theta) + C_{lost\ sales}(\theta) + C_{overstock}(\theta) + C_{operations}(\theta) + C_{inbound}(\theta)$$

is evaluated via Monte Carlo over the 12-week probabilistic forecast (see concepts/probabilistic-demand-forecast) + probabilistic lead-time inputs. Then a black-box gradient-free optimiser searches θ-space for the minimum.

"All inputs are fed into a recommendation engine that leverages Monte Carlo simulations and black-box gradient-free optimisers for optimisation under uncertainty."

Sample count is not disclosed — a key cost lever noted in the source page's caveats.

Seen in

• sources/2025-06-29-zalando-building-a-dynamic-inventory-optimisation-system-a-deep-dive

Related

• concepts/probabilistic-demand-forecast — the source of the probability distributions Monte Carlo samples from.
• concepts/gradient-free-black-box-optimisation — the outer optimiser that consumes Monte Carlo cost estimates.
• systems/zeos-replenishment-recommender
• companies/zalando