How do you calculate the number of kanban cards?

A kanban loop is a reorder point counted in containers, and the flat safety factor in the textbook formula has no idea how variable demand is.

Julian R. Sterling · from Supply Chain Management

The number of kanban cards is demand during the replenishment lead time plus a safety allowance, divided by the container quantity, rounded up: N = (D × L + safety stock) / C. On an illustrative loop with 480 units a day, a 0.75-day lead time and 40-unit containers, the usual 10 per cent safety factor gives 10 cards, which protects only a 68.5 per cent cycle service level; sizing the safety stock for 95 per cent gives 13 cards, for 120 more units in the loop.

Worked in full in Supply Chain Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →

A kanban loop is a reorder point system in which the reorder point is counted in containers. Each card authorises one container. When a container is emptied, its card goes back to the supplying process and a full container must arrive before the cards still in the loop run out. So the cards must cover what is consumed while the replenishment is on its way, plus a margin for the days when consumption runs ahead. The textbook formula puts that margin in as a flat factor, α. That is where most loops are mis-sized.

The assumptions

One part, one kanban loop, illustrative figures
Input	Value
Average demand, units a day (D)	480
Standard deviation of daily demand	96
Waiting for card collection and in queue	0.40 day
Processing	0.25 day
Transport back to the point of use	0.10 day
Replenishment lead time (L)	0.75 day
Units per container (C)	40
Rule-of-thumb safety factor (α)	10%
Target cycle service level	95%
Unit cost	6.50 EUR

The lead time is the whole time a card spends away from the point of use, not the machine time. Here processing is only a third of it.

The calculation, step by step

Step 1: demand during the lead time

D × L = 480 × 0.75 = 360 units, or 9 containers of 40.

Step 2a: the textbook formula with a flat factor

N = D × L × (1 + α) / C = 360 × 1.10 / 40 = 9.90, rounded up to 10 cards.

Excel: =ROUNDUP(480*0.75*(1+10%)/40,0)

Step 2b: the safety allowance from a service level

Standard deviation of demand over the lead time = 96 × √0.75 = 96 × 0.866 = 83.1 units.

Safety stock = z × 83.1, with z = NORM.S.INV(95%) = 1.645, so 136.8 units.

N = (360 + 136.8) / 40 = 12.42, rounded up to 13 cards.

Excel: =ROUNDUP((480*0.75+NORM.S.INV(95%)*96*SQRT(0.75))/40,0)

Step 3: what the flat factor actually buys

Ten cards hold 400 units, 40 more than lead-time demand. z = 40 / 83.1 = 0.48, and NORM.S.DIST(40/83.1,TRUE) = 68.5%.

The result

Two ways of sizing the same loop
Method	Raw N	Cards	Units in the loop	Safety units	Cycle service level
Flat 10% factor	9.90	10	400	40	68.5%
Sized for 95%	12.42	13	520	160	97.3%

The flat factor leaves the loop short in roughly one replenishment cycle in three. A 10 per cent margin on 360 units is 36 units; the variability of demand over the lead time is 83.1 units, more than twice that. The factor is a percentage of the mean, and it has no way of knowing how variable the demand is.

Three extra cards put 120 more units into the loop, 780.00 EUR of stock, which costs 156.00 EUR a year at a 20 per cent holding rate. Rounding up to a whole card also lifts the service level above the target, to 97.3 per cent, because 13 cards hold 160 units of safety rather than the 136.8 needed.

What if the service level, the lead time or the container changes?

Cards needed by cycle service level, 0.75-day lead time, 40-unit containers
Service level	z	Safety stock	Raw N	Cards
90%	1.282	107	11.66	12
95%	1.645	137	12.42	13
99%	2.326	193	13.84	14

Cards and maximum units in the loop at 95%, by lead time and container size
Lead time	20 units	40 units	60 units	Max units, 40-unit
0.50 day	18	9	6	360
0.75 day	25	13	9	520
1.00 day	32	16	11	640

The lead time moves the loop far more than the service level does. Going from 95 to 99 per cent adds one card; cutting the card's waiting time so that the lead time falls from 0.75 to 0.50 day removes four and takes 160 units out of the loop. Smaller containers change the card count but barely the stock: at 0.75 day the loop holds 500, 520 or 540 units for 20, 40 or 60-unit containers. Container size is a handling decision; lead time is the inventory decision.

The common mistake

Using the processing time as the lead time. With 0.25 day the flat-factor formula gives 3.30, rounded to 4 cards instead of 10, 160 units, a third of a day of demand. Cards sit in a post box waiting for the next collection round, then in a queue at the supplying cell, and that time is consumed at the point of use just as surely as machine time. Measure the lead time as the card's round trip, and measure it more than once: if it varies, the safety allowance must cover that variance too, as in safety stock when lead time varies.

Takeaway

N = (D × L + z × σ_d × √L) / C, rounded up. It is the reorder point formula, counted in containers; see the reorder point with safety stock.
A flat α ignores variability: here 10 per cent buys a 68.5 per cent service level.
L is the card's full round trip. Shorten it before adding cards.

The service level table and the safety stock arithmetic behind the z values are in the free workbooks for this book.

Questions readers ask

What safety factor should you use in the kanban formula?

Derive it rather than pick it. The safety allowance should be z times the standard deviation of demand over the lead time. In the illustration that is 1.645 times 83.1, or 136.8 units, equivalent to a factor of 38 per cent of lead-time demand, not the customary 10 per cent, which buys only a 68.5 per cent service level.

Does a smaller kanban container reduce inventory?

Only slightly. At a 0.75-day lead time and a 95 per cent service level the loop needs 25 cards of 20 units, 13 of 40 or 9 of 60, holding at most 500, 520 or 540 units. Cutting the lead time to 0.50 day lowers the 40-unit loop from 520 to 360 units.

What lead time goes into the kanban formula?

The time a card spends away from the point of use: waiting to be collected, queueing, processing and transport. In the illustration that is 0.75 day, of which processing is 0.25. Using processing time alone with the same 10 per cent factor gives 4 cards instead of 10, against the 13 a loop sized for 95 per cent needs.

Read the whole case

This article is one calculation from Supply Chain Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.

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Also on this site

How do you calculate a reorder point with safety stock?Reorder point = demand over the lead time + safety stock: 11,040 + 7,547 = 18,587 units for part AF-7, compared with stock on hand plus stock on order.
How do you calculate safety stock when lead time varies?z times the root of L x demand variance plus d squared x lead-time variance: 7,547 units for AF-7 at 98 per cent, against 2,324 if lead time is treated as fixed.
How do you calculate safety stock from forecast error?Use the standard deviation of forecast errors, not of demand: for part AF-7 a 193.8 error cuts safety stock from 7,547 to 7,246 units, only 4.0 per cent.
Which ABC class should get the highest service level?Usually C, not A: for the same 99.82% unit fill rate, 99/95/90 per cent by class holds 1,458,856 euros of safety stock, the cost-optimal mix 13.5% less.
How do you convert a cycle service level into a fill rate?Fill rate is 1 minus sigma times G(z) over the order quantity. On part AF-7 a 98 per cent cycle service level is a 99.38 per cent fill rate.
Which costs more: a forecast 10 per cent low or 10 per cent high?On part AF-7 a forecast 10 per cent low costs 8,952.25 EUR a year; one 10 per cent high saves 2,168.93, because it lifts the board's 98 per cent to the optimum.

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