Lead Time Distribution Fitting

Lead times almost never follow a normal distribution. They are inherently right-skewed because a supplier can deliver early by a finite amount, but can be late by an infinite amount. Fitting a symmetric distribution ignores the long tail of extreme lateness, leading to chronic under-stocking. Lognormal and Gamma distributions are preferred because they naturally model this positive skew, accurately capturing the probability of rare but severe delays.

Selecting the correct distribution requires rigorous statistical testing to validate the model against empirical data:

• Kolmogorov-Smirnov Test: Measures the maximum distance between the empirical CDF and the theoretical CDF.
• Anderson-Darling Test: A refinement that gives more weight to the tails of the distribution, which is critical for safety stock where tail risk drives buffer size.
• Akaike Information Criterion (AIC): Balances goodness-of-fit with model complexity to prevent overfitting to historical noise.

Historical delivery data is often right-censored. If an order is still open, you only know the lead time is at least the time elapsed so far, not its final value. Ignoring open orders truncates the dataset and biases the distribution toward faster deliveries. Survival analysis techniques, such as the Kaplan-Meier estimator, must be used to incorporate these partial observations and produce an unbiased estimate of the true lead time distribution.

A single theoretical distribution often fails to capture the reality of a supplier's behavior. A supplier might have a fast mode for standard shipments and a slow mode for backordered items. A mixture model combines two or more distributions (e.g., two lognormals) to represent these distinct operational states. The resulting probability density function will have multiple peaks, providing a far more accurate fit than any single-distribution model.

Once a distribution family is chosen, its parameters must be estimated from historical data:

• Maximum Likelihood Estimation (MLE): Finds the parameters that make the observed data most probable. It is statistically efficient but sensitive to outliers.
• Method of Moments: Equates sample moments (mean, variance) to theoretical moments. It is simpler but often less accurate than MLE.
• Bayesian Estimation: Treats parameters as random variables and updates a prior belief with observed data to produce a full posterior distribution of parameters, naturally quantifying parameter uncertainty.

A supplier's lead time distribution is not static. It drifts over time due to seasonality, capacity changes, or raw material shortages. Fitting a single distribution to years of data averages out these distinct regimes, creating a model that is wrong for all of them. Change-point detection algorithms must segment the historical record into periods of statistical homogeneity, allowing for the fitting of separate, time-specific distributions that reflect the current operational reality.