Kanban Sizing

390. Kanban Sizing

The classical Toyota Production System formula for sizing kanban (replenishment-authorization cards) between two production stages:

𝐾 = ⌈ \frac{𝐷 \cdot (𝑇_{𝑝} + 𝑇_{𝑤}) \cdot (1 + 𝛼)}{𝐶} ⌉

where:

$𝐾$ : number of kanban cards needed
$𝐷$ : demand rate (units per period)
$𝑇_{𝑝}$ : processing lead time (production time for one container)
$𝑇_{𝑤}$ : waiting lead time (queue, transport)
$𝛼$ : safety factor (typically 0.05–0.30)
$𝐶$ : container size (units per card)

390.1. Logic

Each kanban authorizes one container of parts. Total inventory in the loop = $𝐾 \cdot 𝐶$ units.

Required inventory must cover:

Demand during the (processing + waiting) lead time: $𝐷 (𝑇_{𝑝} + 𝑇_{𝑤})$
Plus safety: $(1 + 𝛼)$ multiplier

Round up so $𝐾$ is an integer.

390.2. Example

Assembly line uses gear-boxes at $𝐷 = 100$ /hour. Supplier has $𝑇_{𝑝} = 0.5$ hr processing, $𝑇_{𝑤} = 1$ hr waiting. Container holds $𝐶 = 10$ gear-boxes. Safety factor $𝛼 = 0.2$ .

𝐾 = ⌈ \frac{100 \cdot 1.5 \cdot 1.2}{10} ⌉ = ⌈ 18 ⌉ = 18 cards

Total inventory loop: $18 \cdot 10 = 180$ gear-boxes.

390.3. Two-card kanban (Toyota)

Toyota’s classic system uses two card types:

Production kanban (P-kanban): authorizes the producer to make a container
Withdrawal kanban (W-kanban): authorizes downstream to pull from producer’s output buffer

P-kanban + W-kanban coordinate the pull. Modern simplified versions use just one card type.

390.4. Compared to CONWIP

	Kanban	CONWIP
Granularity	Per-part-type, per-stage	Global cap on all WIP
Setup	More cards to manage	Simpler
Mix flexibility	Lower (each card is specific)	Higher (any part can fill cap)
Best for	Repetitive, stable mix	Variable mix, fewer-part-types

390.5. When kanban sizing breaks down

The formula assumes:

Stationary demand — varying demand requires resizing $𝐾$ periodically
Reliable supplier — if upstream stations break down, fixed $𝐾$ won’t compensate
Small mix — separate kanbans per part-type only practical for moderate part counts
Short lead times — long lead times require many cards, defeating leanness

In practice, $𝐾$ is recalibrated regularly (monthly / quarterly) based on observed performance.

390.6. Lean reduction strategies

The formula highlights levers:

Reduce $𝑇_{𝑝} + 𝑇_{𝑤}$ (faster production, less queueing) → fewer cards needed
Smaller $𝐶$ → more cards needed but smaller batches, more flexibility
Lower $𝛼$ (reduce variability, more reliability) → fewer cards needed

Each improvement reduces inventory while maintaining service.

390.7. See also

CONWIP — alternative discipline
Takt Time
Heijunka
Factory Physics