Some Remarks on the History of Kanban | Alexei Zheglov

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“The Kanban method as we know it today has many other influencers and origins besides Ohno and TPS. Two such influencers were of course W. Edwards Deming and Eliyahu Goldratt. Demings 14 Points and the System of Profound Knowledge guide Kanban change agents worldwide. […] Thus the “watershed” of the Kanban method circa 2013 has many “tributaries” of which the TPS is only one. Those other sources should be studied by those how want to apply the Kanban method effectively as change agents.”

Michel Baudin‘s insight:

It takes nerve to write this sort of things.

Among the tools of TPS, the Kanban system is the only one that has been covered in the media from the beginning to the point of overexposure, because it combines a clever idea with objects you can see and touch.

What some software people did is borrow the names of both Lean and Kanban and apply them to theories with at best a tenuous relationship to the original.

That it worked for them as a marketing technique is to their credit, but I would not advise anyone wanting to learn about the Kanban system to read Deming, Goldratt, or Drucker, who is also referenced.

And TPS is not a “tributary” of the Kanban method. It is the Kanban method that is a tool of TPS, and useful only in the proper context, in conjunction with other tools in a well-thought out implementation.

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Supermarket sizing

Bosch’s Taojie Hua (涛杰 华) asked the following question:

How do you define a maximum limit for a supermarket?
Especially when the customer withdraws less than planned, and the lot can not be formed as a production signal, how can I react to that “deviation” by setting a proper max limit?

The response covers the following topics:

Supermarkets in Lean

First we have to clarify what we mean by a supermarket in a Lean manufacturing context. As the term has become popular, some plants have started using it for their warehouses, which is clearly excessive. Often, it is used for any kind of buffer on the shop floor, provided it is used to implement pull. I prefer to reserve the term for buffers from which users withdraw items in smaller quantities than are brought in. If pallets come in and go out, I don’t call it a supermarket, but, if 27-bin pallets come in and withdrawals take place 1 bin at a time, I do.

On a shop floor, supermarkets are found on the edges of manufacturing islands containing a group of cells or a production line and contain either incoming or outgoing materials.  A supermarket for incoming materials has more in common with the refrigerator in your kitchen than with the supermarket you buy groceries in. You need one when your plant Materials or Logistics organization is unable to deliver materials in a form that is suitable for direct use at a production work station.

Water spider at Solectron in  Mexico (2005)

Water spider at Solectron in Mexico (2005)

The supermarket is owned by Production, and more specifically by the first-line manager in charge of the cells or lines it serves. It is replenished  by  Materials or Logistics through periodic milk runs, but parts are withdrawn by experienced members of the production team — cell leaders or water spiders — and move from the supermarket to production on hand carts, gravity flow racks, or by hand. The parts arrive in the supermarkets in bins that are too large for the line side, and leave in kit trays, small bins, or single units.

You need a supermarket for outgoing materials when your production runs are multiples of the quantities needed downstream. This happens, for example, if you only know how to paint parts in batches of 50 with the same color, while assembly alternates colors one unit at a time. In outgoing supermarkets, materials are replenished by Production and withdrawn by Materials/Logistics.

Supermarket capacity

For incoming supermarkets, replenishment by milk runs is essential because it makes lead times predictable. I am assuming here that the upstream supply chain does not cause shortages. Making it work is no small feat, but this question is specifically on supermarkets. On the withdrawal side, you want to have the smoothest possible consumption rate for all items, so that you don’t have large ups and downs to contend with, which you achieve with  heijunka (平準化)  sequencing of production. Little’s Law then tell you that you have, for means:

\overline{Quantity\, on\, hand}\left ( Item \right )= \overline{Consumption\, rate}\left ( Item \right )\times \overline{Replenishment\, lead\, time}\left ( Item \right )

If you take the minimum quantity that Materials can deliver to the supermarket, on the average the Quantity on hand will be half of it. You know the Consumption Rate.  The Replenishment lead time is a multiple of the milk run pitch, plus the time needed for Materials to act on the pull signal, which depends on when the need is identified and how the signal is passed to Materials.

Assume you consume 1 unit every 25 seconds, the milk run pitch is 30 minutes, and Materials delivers in bins of 100 units. You consume 72 parts/pitch = 0.72 bins/pitch. If the milk runs are used to convey pull signals, as happens with the two-bin system or with hardcopy kanbans, replenishment may take up to 2 pitches. In this example, the 2-bin system would cause shortages, but a Kanban loop with two cards wouldn’t, because you pull the card when you withdraw the first unit from the bin and it is still 99% full. If, instead of using cards, you issue an electronic signal when you withdraw the first unit, Materials can act on it in the next milk run, meaning at most 1 pitch later. You still need room for two bins, because the current bin will still hold at least 28 parts when the replacement bin arrives.

In this example, the mismatch between the size of the delivered bins and the consumption rate forces you to hold enough excess material that you don’t need to worry about safety stocks. If it were instead perfectly matched, you could receive a bin of 72 parts like clockwork every 30 minutes, except that fluctuations in consumption occasionally would cause shortages, and you would need some safety stock to protect yourself against it.  Coming up with a sensible plan for any one item in your supermarket is not a major task, but you need such a plan for every item.

The speed with which signals circulate adjusts itself with fluctuations in consumption. The real question is whether your “customer withdrawing less than planned” should be treated as a fluctuation or a permanent drop. In the first case, there is no action required; in the second, you need to recalculate.  In any case, you need to periodically validate the parameters of your pull system to make sure they still reflect reality. In auto parts, it should be done at least quarterly.

Further reading

For details on pull systems, see Lean Logistics, Part IV, pp. 197-330. See also the two posts on Safety Stocks: Beware of Formulas and Safety Stocks: More about the formula.

A Lean Journey: Meet-up: Michel Baudin

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Interview on Tim McMahon’s A Lean Journey.
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Lean and Kanban: Poker Chips, Kanban and Buffets

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An approach to improving the experience of using a buffet that relies on capping the number of people with concurrent access. It is like the nightclub or museum management system in which a fixed number of visitors is allowed in, and the next one only allowed in when one leaves.

This is in the same spirit as the purely production control approach to Lean, in which you change production planning and scheduling but you don’t redesign the production line itself. For pointers on buffet design, see Waiting For Each Other.

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Hansgrohe uses Kanbans with RFID chips

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“Hansgrohe uses RFID-enabled kanban (signal) cards to track the flow of containers between its two production sites. The company now enjoys several benefits, including accelerated goods receipt and the certainty of having all the required components readily available for assembly.”

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The system has been used since 2008, and Hansgrohe provided the following pictures:

The cards look like regular kanbans.


The cards are read when placed in the mailbox on the left.

The full mailbox is read at once.

It should be noted that this system does not eliminate the recirculating cards, but simply replace bar codes with RFID chips as a means of integrating the Kanban system with the company’s ERP system, for the advantages of richer and faster data collection. It does not eliminate the manual handling of cards, at least internally to the plant.

The next step would be to eliminate the cards, attach the RFID tags to part bins, install readers on racks, and implement the replenishment logic electronically. But the readers would have to be substantially smaller than those shown in the pictures.



Safety Stocks: More about the formula

In a previous post on 2/12/2012, I warned against the blind use of formulas in setting safety stock levels. Since then, it has been the single most popular post in this blog, and commands as many page views today as when it first came out. Among the many comments, I noticed that several readers, when looking at the formula, were disturbed that three of the four parameters under the radical are squared and the other one isn’t, to the point that they assume it to be a mistake. I have even seen an attempt on Wikipedia to “correct that mistake.”

I was myself puzzled by it when I first saw the formula, but it’s no mistake.  The problem is that most references, including Wikipedia,  just provide the formula without any proof or even explanation. The authors just assume that the eyes of inventory managers would glaze over at the hint of any math. If you are willing to take my word that it is mathematically valid, you can skip the math. You don’t have to take my word for it, but then, to settle the discussion, there no alternative to digging into the math.

A side effect of working out the math behind a formula is that it makes you think harder about the assumptions behind it, and therefore its range of applicability, which we do after the proof. If you don’t need the proof, please skip to that section.

Math prerequisites

As math goes, it is not complicated. It only requires a basic understanding of expected value, variance, and standard deviation, as taught in an introductory course on probability.

In this context, those who have forgotten these concepts can think of them as follows:

  • The expected value E(X) of a random variable X can be viewed, in the broadest sense, as the average of the values it can take, weighted by the probability of each value. It is linear, meaning that, for any two random variables X and Y that have expected values,

E[X+Y] = E[X]+E[Y]

and, for any number a,

E[a\times X]= a \times E[X]

  • Its variance is the expected value of the square of the deviation of individual values of X from its expected value E(X):

Var(X) = E[X-E(X)]^{2}= E[X^{2} -E(X)^{2}]

Variances are additive, but only for uncorrelated variables X and Y that have variances. If

E[[X-E(X)] \times [Y-E(Y)]]= 0


Var(X+Y) = Var(X)+Var(Y)

  • Its standard deviation is

\sigma = \sqrt{Var[X]}

Proof of the Safety Stock Formula

Fasten your seat belts. Here we go:

As stated in the previous post, the formula is:


  • S is the safety stock you need.
  • C  is a coefficient set to guarantee that the probability of a stockout is small enough.
  • The other factor, under the radical sign, is the corresponding standard deviation.
  • μL and σL are the mean and standard deviations of the time between deliveries.
  • μD and σD are the mean and standard deviation rates for the demand.

  \sqrt{\mu{_{L}^{}}\times\sigma_{D}^{2}+\mu_{D}^{2} \times \sigma_{L}^{2}} is the standard deviation of the item quantity consumed between deliveries, considering that the time between deliveries varies.

μD and  \sigma_{D}^{2} are the mean and variance of the demand per unit time, so that the demand for a period of length T has a mean of \mu_{D} \times T , a variance of  \sigma_{D}^{2} \times T, and therefore a standard deviation of \sigma_{D} \times \sqrt{T}. See below a discussion of the implications of this assumption.

Note that the assumptions are only that these means and variances exist. At this stage, we don’t have to assume more, and particularly not that times between deliveries and demand follow a particular distribution.

If D(T) is the demand during an interval of duration T, since:

we have:

If we now allow T to vary, around mean μL with, standard deviation σL , we have:

and therefore:


That’s how the variance ends up linear in one parameter and quadratic in the other three!


 \sigma\left [ D \right ]= \sqrt{\mu _{L}\times\sigma_{D}^{2} + \mu _{D}^{2} \times \sigma_{L}^{2} }


Note that all of the above argument only requires the means and standard deviations to exist. There is no assumption to this point that the demand or the lead time follow a normal distribution. However, the calculation of the multiplier C used to calculate an upper bound for the demand in a period, is based on the assumption that the demand between deliveries is normally distributed.


The assumption that the variance of demand in a period of length T is \sigma_{D}^{2} \times T implies that it is additive, because if  T = T_{1} + T_{2}, then \sigma_{D}^{2} \times T = \sigma_{D}^{2} \times T_{1} + \sigma_{D}^{2} \times T_{2}.

But this is only true if the demands in periods T_{1} and T_{2} are uncorrelated. For a hot dog stand working during lunch time, this is reasonable: the demands in the intervals between 12:20 and 12:30, and between 12:30 and 12:40 are from different passers by, who make their lunch choices independently.

On the other hand, in a factory, if you make a product in white on day shift and in black on swing shift every day,  then the shift demand for white parts will not meet the assumptions. Within a day, it won’t be proportional to the length of the interval you are considering, and the variances won’t add up. Between days, the assumptions may apply.

More generally, the time periods you are considering must be long with respect to the detailed scheduling decisions you make. If you cycle through your products in a repeating sequence, you have an “Every-Part-Every” interval (EPEI), meaning, for example, that, if your EPEI is 1 week, you have one production run of every product every week.

In a warehouse, product-specific items don’t need replenishment lead times below the EPEI. If you are using an item once a week, you don’t need it delivered twice a day. You may instead receive it once a week, every other week, every three weeks, etc. And the weekly consumption will fluctuate with the size of the production run and with quality losses. Therefore, it is reasonable to assume that its variance will be \sigma_{D}^{2} \times T where T is a multiple of the EPEI, and it can be confirmed through historical data.

You can have replenishment lead times that are less than the EPEI for materials used in multiple products. For example, you could have daily deliveries of a resin used to make hundreds of different injection-molded parts with an EPEI of one week. In this case, the model may be applied to shorter lead times, subject of course to validation from historical data.

How natural disasters test Lean supply chains

Via Scoop.itlean manufacturing

The floods in Thailand are the latest. Before, there was the Fukushima earthquake and, going back further in time, the Aisin Seiki fire of 1997 in Japan and the Mississippi flood of 1993…   Each time, the press has faulted Lean for making the economic disruptions caused by theses events worse. The actual record is that the vigilance inherent in Lean Logistics and the strength of customer-supplier relationships in a Lean Supply Chain are in fact key to a rapid recovery.

In 1993, Toyota logisticians in Chicago reserved all the trucking available in the area a few days before the flood cut off the rail lines to California, thereby allowing the NUMMI plant to keep working during the flood.

In 1997, when the Aisin Seiki fire deprived Toyota in Japan of its single source of proportioning valves, other suppliers came to the rescue in what the Wall Street Journal a few months later called the business equivalent of an Amish barn raising.

You can, and should protect production against routine fluctuations. That is what tools like Kanbans are countermeasures for. But there is no way you can afford to protect your business against all possible, rare catastrophic events. What you can and must do instead is be vigilant and prepared to respond quickly and creatively to whatever nature or society might throw at you.
Via the Bangkok Post

Safety Stocks: Beware of Formulas

A formula you find in a book or learn in school is always tempting. It is a “standard.” If you follow it, others are less likely to challenge your results. These results, however, may be worthless unless you take a few precautions. Following are a few guidelines:

  1. Don’t use a formula you know nothing about. Its validity depends on assumptions that may or may not be satisfied. You don’t need to know how to prove the formula, but you need to know its range of applicability.
  2. Examine your data. Don’t just assume they meet the requirements. Examine their summary stats, check for the presence of outliers, generate histograms, scatter plots, time series, etc.
  3. Don’t make up missing data. If you are missing the data you need to estimate a parameter, find what you can infer about the situation from other parameters, by other methods. Do not plug in arbitrary values.
  4. Make your Excel formulas less prone to error by using named ranges rather than cell coordinates. If a formula is even slightly complicated, referring to variables by names like “mean” or “sigma” makes formulas easier to proof-read than with names like “AJ” or “AK.”

The safety stock formula for the reorder point method

Safety stock is a case in point. The literature gives you a formula that is supposed to allow you to set up reorder point loops with just the minimum amount of safety needed to prevent shortages under certain conditions of variability in both your consumption rate and your replenishment lead time. It is a beautiful application of 19th century mathematics but I have never seen it successfully used in manufacturing.

Let us look more closely at what it is so you can judge whether you would want to rely on it. Figure 1 shows you a model of the stock over time when you use the Reorder Point method and both consumption and replenishment lead time vary according to a normal distribution. The amount in stock when the reorder point is crossed should be just sufficient to cover your needs until the replenishment arrives. But since both replenishment lead time and demand vary, you need some safety stock to protect against shortages.

Figure 1. The reorder point inventory model

If your demand is the sum of small quantities from a large number of agents, such as sugar purchases by retail customers in a supermarket, then the demand model makes sense. In  a manufacturing context, there are many situations in which it doesn’t. If you produce in batches, then the demand for a component item will be lumpy: it will be either the quantity required for a batch or nothing. If you use heijunka, it will be so close to constant that you don’t need to worry about its variations.

What about replenishment lead times? If in-plant transportation is by forklifts dispatched like taxis, replenishment lead times cannot be  consistent. On the other hand, if it takes the form of periodic milk runs, then replenishment lead times are fixed at the milk run period or small multiples of it. With external suppliers, the replenishment lead times are much longer, and cannot be controlled as tightly as within the plant, and a safety stock is usually needed.

Let us assume that all the conditions shown in Figure 1 are met. Then there is a formula for calculating safety stock that you can find on Wikipedia or in David Simchi-Levy’s Designing and Managing the Supply Chain (pp. 53-54).  Remember that it is only valid for the Reorder Point method and that it is based on standard deviations of demand and lead time that are not accessible for future operations and rarely easy to estimate on past operations. The formula is as follows:


  • S is the safety stock you need.
  • C  is a coefficient set to guarantee that the probability of a stockout is small enough. You can think of it a number of standard deviations above the mean item demand needed to protect you against shortages. In terms of Excel built-in functions, C is given by:

C = NORMSINV(Service level)

Service levelC
90.0% 1.28
95.0% 1.64
99.0% 2.33
99.9% 3.09
  • The other factor, under the radical sign, is the corresponding standard deviation.
  • μL and σL are the mean and standard deviations of the lead time.
  • μD and σD are the mean and standard deviation of the demand per unit time, so that the demand for a period of length T has a mean of μD xT and a standard deviation of σDx √T

Case study: Misapplication of the safety stock formula

This formula is occasionally discussed in Manufacturing or Supply Chain Management discussion groups, but I have only ever seen one attempt to use it,  and it was a failure. It was for the supply of components to a factory, and 14 monthly values were available for demand, but only an average for lead times.

The first problem was the distribution of the demand, for which 14 monthly values were available. This is too few for a histogram, but you could plot their cumulative distribution and compare it with that of a normal distribution with the same mean and standard deviation, as in Figure 2. You can tell visually that the actual distribution is much more concentrated in the center than the normal model, which is anything but an obvious fit.  Ignoring such objections, the analyst proceeded to generate a spreadsheet.

Figure 2. Actual versus normal cumulative distribution

The second problem is that he entered the formula incorrectly, which was not easy to see, because of the way it was written in Excel.  The formula in the spreadsheet was as follows:


then, looking at the spreadsheet columns, you found that they were used as follows:

  • AJ  for Standard Deviation of Daily Demand, and
  • AL for Average Replenishment time.

And therefore the first term under the square root sign was σDL2 instead of μLxσD2.

The third problem was that the formula requires estimates of standard deviations for both consumption and replenishment lead times, but no data was available on the latter. To make the formula produce numbers, the standard deviations of replenishment lead times was arbitrarily assumed to be 20% of the average.

For all of these reasons, the calculated safety stock values made no sense, but nobody noticed. They caused no shortage, and the “scientific” formula proved that they were the minimum prudent level to maintain.

Sizing safety stocks in practice

There is no universal formula to determine an optimal size of safety stocks. What can often be done is to simulate the operation of a particular replenishment loop under specified rules. For a simulation of a Kanban loop using Excel, see Lean Logistics, pp. 208-213.

No calculation or simulation, however, is a substitute for keeping an eye on what actually happens on the shop floor during production. One approach is to separate the safety stock physically from the regular, operational stock and monitor how often you have to dig into it. If, say, six months go by without you ever needing it, you are probably keeping too much and you cut it in half. With a Kanban loop, you tentatively remove a card from circulation. If no shortage results, then the card was unnecessary. If a shortage occurs, you return the card and look for an opportunity to improve the process so that the card can be removed.

The Original Kanbans

Via Scoop.itlean manufacturing

The kanban has met many adventureson its way to becoming a popular tool for the limitation of tasks, projects and works in process. As superhero origin stories go, kanban has an interesting one. As long ago as 8th century Japan, guidelines were set down for the forms and functions of kanban as corporate logos and shop signs. Just as the study of the use and evolution of forms of kanban as an improvement tool is illustrative as to the development of management various industries from manufacturing to software development, an examination of kanban as Japanese shop signs is instructive of the historical and cultural changes that took place.
Read more: Lean Manufacturing Blog, Kaizen Articles and Advice | Gemba Panta Rei

How Manufacturing Software Should Adapt to Support Lean Principles

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This article takes a critical look at the debate between lean manufacturing and MRP software advocates, and how to find middle ground.