I am a scientist, and I try to understand how the world works, particularly in my field of manufacturing and lean production. (I also try to teach others about this, e.g., through this blog.) Hence, in this post I will look at different equations used in lean. Somehow, there are not that many…
Takt Time
One of the most important equations in lean manufacturing is the takt time. This is most often seen as the customer takt, where it is the average time between the demand for one product or type of product.
\[ { Customer \ Takt \ Time = \frac{Customer \ Demand}{Work \ Time} }\]
However, this can also be a process or line takt, where it is the average time between the production of a product. In both cases, it is the long-term average, including all losses (the other value would be the cycle time, excluding all losses and assuming a perfect system).
\[ { System \ Takt \ Time = \frac{Good \ Parts \ Produced}{Work \ Time} }\]
The takt time is one of the starting points for many lean projects, as a common goal is to satisfy the customer demand, and to satisfy the demand of the customer, the following has to be roughly true.
\[ { Customer \ Takt \ Time \approx System \ Takt \ Time }\]
See also my post series on takt time.
OEE
Another common formula is the Overall Equipment Effectiveness, or OEE. It gives you a percentage of what a machine or system has achieved to what could have been possible theoretically with no losses. There are different ways you can calculate the OEE, all giving you the same result if done correctly. See also my post on good ways and bad ways to calculate the OEE. Sticking to my preferred ways, you could divide the cycle time by the takt time. Note to use the system takt, albeit the customer takt should be identical.
\[ { OEE= \frac{Cycle \ Time}{System \ Takt \ Time} }\]
You can also divide the number of good parts produced by the number of parts that should have been possible inder perfect circumstances.
\[ OEE = \frac{Good \ Units }{\frac{Total \ Time }{Cycle \ Time }}\]
And finally, you can divide the time that you would have needed for your production divided under ideal, perfect circumstances by the actual time needed.
\[ OEE = \frac{Good \ Units \cdot Cycle\ Time }{Total \ Time}\]
If done correctly, all these three equations should give you the same percentages. See also my post series on the OEE.
Lead Time (Little’s Law)
The lead time is the time needed for a part to pass though the system (or a segment of the system). The best way to calculate this is using Little’s law, one of the most useful equations for lean. You count your inventory in the system (or segment thereof), and multiply this by the average time between the departure of a part from the system (i.e., the system takt).
\[{ Inventory= \frac{Lead \ Time}{System \ Takt}} \]
Hence, to determine the lead time you calculate:
\[{ Lead \ Time = Inventory \cdot System \ Takt} \]
Overall, a great equation! See also my Eulogy for Little’s Law.
Kanban Calculation
A rather complex equation is the kanban calculation. And, when I say equation, it is more of an estimate, since there is a LOT of fuzziness in these numbers. In its pure basic form, the number of kanban is the lead time divided by the takt time, and adjusted for the number of parts each kanban represents. However, DO NOT USE THIS FORMULA, as this is a great simplification.
\[{ Number \ of \ Kanban = \frac{Lead \ Time}{Takt \ Time \cdot Number \ of \ Parts \ per \ Kanban }} \]
The problem is, that for the kanban formula, you cannot work with averages, but instead have to assume a worst (or at least pretty bad) case. After all, you don’t just want to have material available in average, but preferably ALL THE TIME. Hence, you have to divide the worst lead time you still want to cover by the fastest takt time that you expect within a lead time.
And this “worst case” assumption can blow up your kanban formula quite a bit, depending on what kind of situations you want to include, like breakdowns, lot sizes, large orders, peak demand, information transport time, waiting times at the kanban box, waiting times in queue for production (that one is particularly nasty!), and waiting time for sequence creation… and that is only if you have a production kanban. For a transport kanban, this again looks different and may include, in addition, the waiting time in the shipping queue and the waiting time for a truckload. Covering these equations in detail would be too much here for this short blog post, but luckily I have written an entire award-winning book, All About Pull Production.
Other Less Common Equations or Other Related Fields
The equations above are the ones I most commonly use for my work in lean manufacturing. However, there are of course more equations used in manufacturing that have a connection to lean. For example, quality-related aspects have their own metrics, as for example the defect rate and first pass yield, but these are mostly basic statistics. Much more advanced statistics are the Process Capability Index (CPk), calculating how far your median is away from your closest tolerance limit in terms of standard deviations (actually, in terms of three standard deviations for some reasons), but since most shop floors don’t really measure their standard deviations, this is not so common.
There are also equations related to inventory like the turnover rate, the days (or weeks?) of inventory, and the average inventory. There is also the Economic Order Quantity, an equation to determine the ideal order quantity. This equation is mathematically absolutely beautiful, but due to the flawed underlying assumptions (underestimating the cost of inventory) nearly useless.
Sometimes in lean, there is also the percentage of value-adding time, often calculated in connection with the value stream. Some people love to use this. However, I don’t find it that useful. But maybe it is because I don’t use it much, and if I would use it more maybe I would like it. Hence, if it helps you, feel free to use it! There are also fancier equations like the Kingman equation (or its alternatives), but these are more to understand the theory behind it and also not so much for practical use. The Power of Six is also an empirical relation between time and cost, or another empirical relation between the cost and the volume, but both are not used much, even though they can give some supporting arguments for lean improvements.
Of course, a lot of statistical tools are also used in lean, like pareto diagrams, scatter plots, histograms, or even statistical process control and more, but these are equations of statistic, but not lean manufacturing per se.
Summary
Come to think of it, these are not many equations. Lean is often more a collection of experiences and best practices, and much less hard theory. Creating a work standard or doing practical problem solving using the PDCA are such tried—and—true tools that do not have their own inherent math. Maybe this lack of hard numbers gives lean so much headache with cost accounting? Now, go out, use this handfull of equations to power your lean approach, and organize your industry!
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Thank you, Christoph. I write the lean equations as part of my work with eVSM so enjoyed this article in particular. Some of the more difficult ones I encountered were tied to interval calculations (EPEI) with mixed model production. A common question I get is about the sizing of inventory pockets in a supply network (cycle stock, safety stock for demand variance, safety stock for supply variance, policy stock). A future article on this would be great. Also something on Lean and environmental given our climate challenges.
Hi Dilesh, EPEI is indeed tricky. I think it is more of an algorithm than a calculation, and also more of an art than a science. I have to think more about the sizing of inventory pockets in a supply network, that is also a challenging topic! Thanks for commenting! Chris.