Lead-Time Risk Estimator

where ρ = p_rare_delay. This captures both normal delivery and occasional severe delays in a single model.

Supplier reliability from supplier-reliability-aggregator then inflates the mixture variance:

Var[LT]_adjusted = Var[LT]_base × (1 + (1 − reliability))

— a robustness adjustment that penalises unreliable suppliers (Ch.14 §14.3).

Step-by-Step Execution

Step 1: Retrieve mixture distribution formula from knowledge base

search_knowledge_base(
    query="mixture distribution weighted sum components Gaussian mixture density probability",
    k=3
)

What to extract from the result:

• Mixture density formula (Ch.2 §2.3, eq. 2.17): p(x) = Σ ρᵢ × N(x | μᵢ, σᵢ²)
• Weights must sum to 1: (1 − ρ) + ρ = 1
• How to compute mean and variance of a mixture: E[X] = Σ ρᵢ × μᵢ Var[X] = Σ ρᵢ × (σᵢ² + μᵢ²) − E[X]²

Step 2: Retrieve quantile function definition

get_entity_by_number(number="2.4")

What to extract from the result:

• Quantile function definition (Ch.2, figure 2.4): quantile_X(α) = value x such that P(X ≤ x) = α
• Use α=0.95 for P95 and α=0.99 for P99 lead-time estimates
• Tail exceedance probability: P(LT > horizon) = 1 − CDF(horizon)

Step 3: Retrieve robustness / variance inflation concept

search_knowledge_base(
    query="robustness modeling errors sensitivity policy evaluation deviation uncertainty inflation",
    k=2
)

What to extract from the result:

• Ch.14 §14.3 — robustness analysis: evaluate policies under model deviations
• Apply supplier unreliability as a perturbation to the variance parameter: Var_adjusted = Var_base × (1 + (1 − reliability)) — less reliable supplier → wider distribution → higher P95/P99

Step 4: Fetch leadtime model for the purchase order

GET /api/v1/smart-query/procurement/purchase-orders/{purchase_order_id}

Extract leadtime_model_id.

Then fetch the model parameters:

GET /api/v1/smart-query/procurement/pipeline-summary
    ?supplier_id=<supplier_id>

From leadtime_models via the PO record extract: dist_family, p1, p2, p_rare_delay, rare_delay_add_days

Fallback: if no leadtime_model_id on the PO → fetch the global-scope leadtime model:

GET /api/v1/smart-query/procurement/pipeline-summary
    ?scope=global

Step 5: Fetch supplier reliability from supplier-reliability-aggregator

Requires posterior_mean_reliability from supplier-reliability-aggregator output for this supplier.

If supplier-reliability-aggregator was not run, use suppliers.reliability_score as a fallback.

Step 6: Build mixture distribution and compute risk metrics

Using the formula from Step 1 and quantile definition from Step 2:

# 1. Base distribution parameters
#    dist_family ∈ {normal, gamma, lognormal}
#    p1 = mean (or shape), p2 = std (or scale)

# 2. Shifted distribution for rare delay component
#    same dist_family but mean shifted by rare_delay_add_days

# 3. Mixture weights
w_base  = 1 − p_rare_delay
w_delay = p_rare_delay

# 4. Mixture mean (eq. 2.17 applied to means)
E[LT] = w_base × p1 + w_delay × (p1 + rare_delay_add_days)

# 5. Mixture variance (law of total variance)
Var_base    = p2²
Var_shifted = p2²
Var_mixture = w_base × (Var_base + p1²) + w_delay × (Var_shifted + (p1 + rare_delay_add_days)²) − E[LT]²

# 6. Reliability inflation (Ch.14 §14.3 robustness perturbation)
reliability = posterior_mean_reliability  (from `supplier-reliability-aggregator`)
Var_adjusted = Var_mixture × (1 + (1 − reliability))
σ_adjusted   = sqrt(Var_adjusted)

# 7. Quantiles (Ch.2 figure 2.4 — inverse CDF of adjusted mixture)
P95 = mixture_quantile(α=0.95, weights, dist_family, p1, p2, rare_delay, reliability)
P99 = mixture_quantile(α=0.99, ...)

# 8. Tail exceedance probability for planning horizon
P(LT > horizon) = 1 − mixture_CDF(horizon)

# 9. Risk classification
risk_class = ACCEPTABLE  if P(LT > horizon) < 0.05
             ELEVATED    if P(LT > horizon) < 0.20
             CRITICAL    if P(LT > horizon) ≥ 0.20

Step 7: Return result

Failure Modes

Example Walkthrough

User query: "We need SupplierX to deliver within 10 days for this critical order — what's the actual risk?"

<thinking>
I need to model the full lead-time distribution including rare delays,
inflated by SupplierX's actual reliability, then compute P(LT > 10).
</thinking>

Step 1 — search_knowledge_base("mixture distribution weighted sum components density probability")
→ Retrieved eq. 2.17: p(x) = Σ ρᵢ × N(x | μᵢ, σᵢ²)
→ E[X] = Σ ρᵢ × μᵢ,  Var[X] = Σ ρᵢ(σᵢ² + μᵢ²) − E[X]²

Step 2 — get_entity_by_number(number="2.4")
→ Retrieved: quantile_X(α) = x such that P(X ≤ x) = α
→ P(LT > horizon) = 1 − CDF(horizon)

Step 3 — search_knowledge_base("robustness modeling errors sensitivity variance inflation")
→ Retrieved: Ch.14 §14.3 — Var_adjusted = Var_base × (1 + deviation_factor)

Step 4 — GET /purchase-orders/po-uuid-abc → leadtime_model_id = LM-07
→ dist_family=normal, p1=7.0, p2=1.5, p_rare_delay=0.12, rare_delay_add_days=8

Step 5 — `supplier-reliability-aggregator` result:
→ posterior_mean_reliability = 0.765

Step 6 — Compute:
    E[LT]       = 0.88 × 7.0 + 0.12 × 15.0 = 7.96 days
    Var_mixture = 0.88×(2.25+49) + 0.12×(2.25+225) − 7.96²
                = 45.1 + 27.3 − 63.4 = 9.0
    Var_adjusted= 9.0 × (1 + 0.235) = 11.1   (reliability=0.765)
    σ_adjusted  = 3.33 days
    P95         = 7.96 + 1.645 × 3.33 ≈ 13.4 days
    P99         = 7.96 + 2.326 × 3.33 ≈ 15.7 days
    P(LT > 10)  = 1 − Φ((10 − 7.96) / 3.33) = 1 − Φ(0.61) ≈ 0.27

    risk_class = CRITICAL  (P(LT > 10) = 27% ≥ 20%)

Result:
  expected_lead_time_days = 7.96
  p95_lead_time           = 13.4 days
  p99_lead_time           = 15.7 days
  prob_exceed_horizon     = 0.27
  risk_class              = CRITICAL

There is a 27% chance SupplierX misses the 10-day deadline.
Recommend: either select an alternative supplier with lower P(LT > 10),
or place the order now with 6+ days buffer to absorb the tail risk.

Feeds Into

• multi-attribute-utility-scorer — uses risk_class as u_lt factor in utility scoring
• decision-confidence-estimator — uses risk_class in decision confidence assessment
• stochastic-dominance-filter — uses full distribution for stochastic dominance comparison across suppliers
• threshold-based-trigger-decision — uses expected_lead_time_days for pipeline coverage calculation