Badr Alghamdi
العربية
Lessons

Extreme value: pooling steadies a sum, a consolidation waits on the last

The box closes on the last supplier ready, so a maximum decides and not a sum. Type each promised day and spread, and on three suppliers the third is the first miss.

Your prediction before the experiment

Pooling makes the average spread smaller, so the day when the last supplier is ready also becomes steadier. Both traces are visible. Move the supplier-count control once to reveal the deciding pair of readings.
Pooling makes the average spread smaller, so the day when the last supplier is ready also becomes steadier. Both traces are visible. Move the supplier-count control once to reveal the deciding pair of readings.

What stays the same

Loading order
The order you typed
Closing rule
The last readiness day at the typed confidence
Contrast only
Pooled average spread never sets the capacity

Try a scenario

The maximum and the pooled contrast

Closing quantile and pooled spread by supplier prefixThe target holds 2 suppliers and the first miss is 3. At the full typed prefix, the last-ready closing day is 2.12 and the pooled average spread is 0.58.Last-ready closing day1.141.882.62closing targetlast readyPooled spread in days00.51pooled averagecurrent prefix123Suppliers in typed order
3 suppliers

This control follows the typed loading order. It never reorders the rows.

Typed suppliers and closing rule

%
days

Readiness movement

Lockstep is a typed stress case, not an estimated correlation.

Supplier rows
Supplier 1
days
day
Supplier 2
days
day
Supplier 3
days
day

Boundary

The readings describe a maximum and a pooled sum under the typed spreads and dependence case. They do not forecast a supplier, choose loading order, price freight, or plan a consolidation.

Here is the maths

The formula, in words

For independent readiness, multiply the probability that every supplier in the prefix is ready by a candidate day, then bisect until that product reaches the typed confidence. For lockstep readiness, apply one normal percentile to every row and take the largest day. Compute pooled average spread separately for contrast.

P(max⁡iXi≤x)=∏iΦ(x−μiσi)P\left(\max_i X_i\le x\right)=\prod_i\Phi\left(\frac{x-\mu_i}{\sigma_i}\right)

where XiX_i is a supplier's readiness day, xx the candidate day, Φ\Phi the normal cumulative distribution, μi\mu_i the promised day, and σi\sigma_i the spread.

A worked example

Three independent suppliers each have promised day 0 and spread 1. At 95% confidence, the last-readiness quantiles are about 1.64, 1.95 and 2.12 days for prefixes of 1, 2 and 3. A target of day 2 therefore holds 2 suppliers and first misses at 3. Over the same prefixes, pooled average spread falls from 1 to about 0.58 days.

What this does not model

You typed the normal assumptions. The calculation does not estimate dependence, predict a named supplier, optimize loading order, infer a freight cost, or let the pooled average spread determine the closing capacity.

What the sweep checks

Every supplier prefix from 1 through the typed list is checked in the order you typed. The first prefix whose maximum quantile is strictly above the target is the first miss, while equality remains inside.

Source basis

The mechanism uses public normal-distribution identities for maxima and pooling. Published inventory and freight-consolidation literature supplies application context, and no sold definition or procedure is reproduced.

Freight cost, a consolidation recommendation, and a supplier forecast remain outside this lesson and receive no answer.

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