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.

37 comments on “Safety Stocks: Beware of Formulas

  1. Comment in the Lean Six Sigma Worldwide discussion group on LinkedIn:

    Very well thought out – in the medical resupply side of things, we have typically relied on outdated formulas to calculate our min/max levels for most of our common stocked items. Rolling manufacturer back orders and raw materials can impact the simplest of items.

  2. Great topic Michel,

    Safety stock is not considered inventory waste in Lean because…..?

    I believe that when we understand the concept, it will make it easier to understand how to size.

  3. Comment in the Lean Six Sigma discussion group on LinkedIn:

    Surely safety stock is considered ‘waste’ in a lean system, it is simply something that we cannot do without at that time, but should aim to reduce. Given that the final lean principle is ‘perfection’ then we should be striving to achieve 1-piece flow and constantly look for ways to reduce safety stock. Just because an item is tagged as waste does not mean that we automatically get rid of it as the supporting system may not be able to cope without it. However, we should constantly revisit each area of waste to determine whether we can reduce it. Thoughts?

  4. Comment in the Lean Six Sigma discussion group on LinkedIn:

    I agree with Michel,
    Although the formula makes sense in theory you don’t usually have the data required to feed it:
    How do you know the average and standard deviation of the procurement lead time for each part purchased from each supplier?
    How do you know the average and standard deviation of your future customer demand?
    You don’t normally have enough reliable data to draw significant conclusions.

  5. Comment in the Lean Six Sigma discussion group on LinkedIn:

    I agree that most formulas make assumprtons about normality and averages that don’t apply. I introduce my students to the “standard” textbook formulas, but also what I usually add for forecast error and lead time variability. I also point out to them that when a product demand’s coefficient of variation is >.5 on a weekly basis, both forecasting and using kanbans become difficult. So I use my enhanced formulas, but only as a starting point for a simulation that takes all the demand and supply statistical patterns into account.

  6. Stock in and of itself is not waste, but unnecessary stock is waste.

    If you have variability in your supplies or your consumption, and you don’t have any safety stock you will have shortages that will stop production.

    If you can get suppliers to respond instantly to changes in your consumption rate, you don’t need safety stocks.

    If the variability in your consumption is self-inflicted, for example by bad planning and scheduling, then, by techniques like heijunka, you can reduce or eliminate the safety stock.

    If you need safety stock, then your challenge is to find just how much of it you need item by item. Keeping two months’ worth of everything on the shelf may seem simple, but it is likely to be overkill for many items, and it may not be enough for some. For example, if you use small quantities of an exotic alloy from a single worldwide supplier, it may be prudent to have two years’ worth of it.

  7. Comment in the APICS discussion group on LinkedIn:

    Safety Stock: beware of using it at all. The link is to an article that technically describes how safety stock calculations are often mathematically misrepresentative. Welcome to industry. Industry has been misusing mathematics for the better part of a century for all types of operational decisions – cost accounting being one of the biggest.

    Finding a better safety stock calculation will not fix the fact that using safety stock is like rearranging the deck chairs on the Titanic. The mechanism of safety stock itself is antiquated and does not really provide the safety that we think it does. Furthermore, it often exacerbates the bullwhip effect in supply chain. Finally, it is part of an antiquated formal planning method that is fundamentally inappropriate for today’s volatile and complex world.

    Carol Ptak and I wrote a white paper about safety stock versus a new alternative that is producing staggering results. You can access it here:

    More on the new total method itself can be accessed here:

    Comment in the Lean Six Sigma Worldwide discussion group on LinkedIn:

    Safety Stock formulas are misused but…using better ones will not dramatically improve performance. Safety stock is part of a larger formal planning system that is becoming more and more antiquated for the hyper volatile and complex circumstances today. Industry has been attempting to pptimize old and inappropriate rules for the last decade with little success. We have reached to point ot dimishing returns. Formal planning needs a fundamental overhaul, one that includes the elimination of the safety stock method.

  8. @Chad –
    If I understand you correctly, you are proposing to do away with safety stocks altogether. Let us look at the implications for specific items.

    You already have no safety stocks for model- and option- specific items you receive just-in-sequence from suppliers. You practice heijunka, you communicate your production sequence to the suppliers as you start executing it at the first station of final assembly, and the suppliers make and deliver the parts in that sequence at the station where they are used. It’s wonderful but there are not many customer-supplier relationships that are able to make it work.

    Now consider more generic items, like brackets, that you use on multiple products and for which you have an in-house Kanban loop between your incoming stores and one or more production lines. Let’s say you have milk runs every 30 minutes and have an item you use as an average of 1 unit every minute on a given line, with fluctuations so that, in 30 minutes, you use between 28 and 32. With Kanban rules, whenever you pick the first part in the bin, you pull the card and place it in the collection box for pick up by the next milk run and delivery on the following. If you issue the signal electronically, you may get the delivery one milk run earlier. If you use bins of 30, you may have only 30 units at the start of an interval between milk runs in which you will consume 32. On the other hand, if you use bins of 32, you will end up with a small amount of safety stock to protect you against the fluctuations. How do you propose to avoid this?

    You may also have bulk items, like resin pellets for injection molding, that you manage on reorder point. The quantity remaining when you cross the reorder point is supposed to carry you until the next delivery. If you give yourself no safety stock whatsoever, then an accident on the freeway that delays the delivery truck for 30 minutes will stop your production. Same question: what do you propose to do?

    We could go on. If you have a plan for every part that involves safety stocks for none, what is it?

  9. Comment in the Lean Six Sigma Worldwide discussion group on LinkedIn:

    Strategic replenishment buffers in the DDMRP method are points that decouple processes. Where to place these buffers are extremely important as they compress lead times, absorb variablity and minimize working capital. Safety stock at its best can only absorb a certain amount of variability it does not decouple.

    Safety stock is a supplementary position to guard only against variation. Consider the dramatic increase in variation across every supply chain as complexity increases worldwide. Combine this fact with working capital reduction mandates and planners are now finding themselves in a huge conflict. If forecast error is high then the safety stock supplementary position can become quite an extraordinary commitment to statistically cover that error. Then begins the workarounds. And then the workarounds to the workarounds.

    Replenishment positions, however, are strategic and primary in nature. Replenishment positions are put into place with the idea of stopping or preventing the amplification and/or impact of variation instead of reacting against it when it happens (safety stock). It is the difference between the notions of a firewall versus a fire extinguisher. One isolates, prevents and protects while the other reacts only after a fire has occurred.

    These positions are a just a component of a completely overhauled formal planning process that recaptures the promise of technology and incorporates the demand driven approaches requried to compete in today’s supply chains.

    This is our latest paper on the headway that these new methods are making:

  10. @Chad – From your words, I still don’t see the difference. Safety stock is extra stock that is maintained to mitigate the risk of stockouts due to variations in supply and consumption.

    Where and how much of it you keep, and how you use it in your planning, does not strike me as changing what it is. By any other name, it is extra stuff you keep around in case you receive less or need more.

    Holding safety stock certainly decouples suppliers from users, in the sense that, for example, if Iran stops selling you oil, your strategic reserves — a safety stock — keep you from feeling an immediate effect.

    When you talk about preventing the amplification of variations, do you mean avoiding the bullwhip effect? To me that is what heijunka and a well thought out pull system do.

    Thanks for the link to your paper. My own publications can be found on Amazon but, rather than giving you homework, I would rather discuss the issues right here in this forum, as we might around drinks if we met at a conference.

  11. Michel – and particularly Chad, I think you are seeing the forest – instead of the trees.

    Michel is right, KISS is the way to go. Keep it simple stupid.

    In my experience in a large 24/7 aged manufacturing plant, with 16,000 SKU’s, and
    a marginal MRP system.

    The A,B,C and VMI practices take care of most of the “so called” stock outs, that may / or frankly in most cases, are just assumed stockouts – “over the weekend”.

    One also must consider EOQ’s, more than ever now. And for one to consider that safety stock (as Michel as mentioned several times) is NOT wasted material.

    It simply is material that will be used, but maybe, not this week, or until a months production input is need. Or gets damaged/lost.

    Unless it has a shelf life, it can generally be managed through production weekly counts, MRP updates, and/or better yet VMI programs.

    With VMI – the supplier manages the inventory and generally has MORE of an incentive to both mitigate stock outs, reducing inventory (financing) and simplify the supply chain. And opportunity costs mitigate any expense of a VMI program.

    Also if inventory is not “sold” to the business unit, until used, and safety stock, monitoring is done buy a vendor, including CI. Whats the worry? Why the formulas?

    Formulas and effciency gains, can be measured once the inventory levels are reduced overall, supply chain vendors are reduced and CI is in place.

    If I can get most inventory in a category, 24/7, an in hours of realising I may be running below ROP, and if this happens a few times a year, and typically planning causing most of the issues, what can purchasing possiblly do? We have done our job

    I remember the schedulers, productions comments that “we are always running out of pallets”, can’t ship without a pallet!

    After creating some data presentations showing pallets were at 98% on time delivery, (and the fact that production published at 75% on time delivery)..they shut up and did not mention it for sometime.(when I did heard purchasing had “stockout” again, production obviously did not mention my presentation/proof, to the plant manager or others too!

    We are just a good excuse for their poor planning and execution.Thats all.

    Again; VMI, EOQ and CI, will take care of most SKU’s and safety stocks, forget the formulas in a job shop or chase type, push type production environment.

    Lastly; if you are not instinctively monitoring your SKU’s, and part of communication , production meetings, or not visiting the plant floor, for storage and organizational reasons, you should definitely not rely on formuals, and actually do your job and know the “supply” and supply chain of any B,A items. C’s too.

    I see no need to overcomplicate a rather simple strategic process.

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  19. Comment on LinkedIn:

    Very interesting subject! When I first started training lean for a living, I used the diagram and formula you show in your post, simply because people kept asking me for a specific “recipe” to calculate the safety stock.

    However, I rapidly stopped doing that because when I tried out the formula on some historical data, using a very simple Excel model, I had a stockout by the 4th re-order. I tried with a couple more, which also failed. This reminds me of the “bat and a ball” puzzle that Dan Kahneman talks about in “Thinking, fast and slow”. The same mental bias appears over and over in business whenever we are given a mathematical formula or anything that has the imprimatur of a “trusted source”: we fail to expend even a moderate amount of work to test it.

    In the end I concluded that the reason for the failure of the recipe was not that the data was not normal (all 3 datasets passed the normality test), but that the formula uses standard deviations which apply only to the next occurrence. In other words, when we have stockouts, it is due to multiple “bad events” (abnormally high demand, long resupply, partial resupply, or a combination of these) occurring in a row.

    I suppose we could refine the formula to calculate the probability of such a string of bad events for a given period corresponding to a checkpoint before the stock theoretically reaches zero, and adapt our ordering decision according to the results. But this would miss the point: lean is not about single-loop, but about double-loop learning. In plain English, we need to learn how to reduce the safety stock, rather than how to calculate it correctly.

  20. Did you think that it is not correct to use standard deviation to calculate the Safety Stock?

    Because periods with demand below the average will increase SS according the formula!!!
    But we do not need SS at all if demand below the average. Stock = Lead Time Demand is enough for such periods without any SS.
    Only periods with demand bigger than the average should be considered for SS calculation.

    • I did not mean to imply any such thing, only that you should not blindly use formulas. Before you use a formula, you should know its underlying assumptions, and verify that they apply in your circumstances.

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  24. Hi Michel, I am a bit confused about the demand standard deviation per unit time. How to define unit time here? If the daily demand amount is group into weekly demand, what is the unit time here, a day or a week? The identical daily demand data series, if transformed into weekly demand or monthly demand data series, can produce different standard deviations. Then, in the situation, which standard deviation, weekly or monthly, should be taken as the right one required by the formula?

    • In principle, it doesn’t matter which unit you use. As long as you do it consistently, the end result should be the same. In practice, you should choose units that are matched to the clockspeed of your work. If you plan daily, the standard deviation of daily demand will be more meaningful to you than of yearly demand. In addition, the assumption that demand fluctuates around a steady average is unlikely to hold for a year.

      You seem motivated to use the formula. Remember the post is a warning against it!

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    • You may notice that I am explaining the formula, not advocating its use. The formula assumes that both consumption and replenishment lead times follow gaussian distributions. If they don’t, don’t use the formula. The post presents alternative methods of solving the problem.

  26. I have a doubt! When we talk about “L” (Lead Time), do we refer to the replenishment time from the time I place a purchase order until it reaches my warehouse, or the time between orders (Days of the year / Annual Demand / Optimum Q*?
    What happens when my ROP is greater than my Qmax?

    • A lead time is an amount of time between order issue and receipt of the goods, in whatever form the order is issued and however the goods are received. It’s a good idea to make a clear statement of the end points when you discuss a lead time, for example, “from the time I place the Kanban in the collection box to the time a full bin arrives.”

      If you use the reorder point system, and your reorder point is higher than the maximum amount of stock you can hold, you have a problem.

  27. Dear Michel, thank you for your kind reply.

    “If you use the reorder point system, and your reorder point is higher than the theoretical maximum amount calculated of stock” (I don´t have storage restriction)… so, What happens if I use the time between orders like lead time? That is, if my supplier takes 3 months to dispatch an order, and I place monthly orders, then only the first time the order will arrive in the third month, then, from there deliveries will begin arriving monthly. This in practice is very common. If this is so, to calculate the ROP I use the time between orders (1 month) to calculate dxL + SS and not the 3 months of Lead Time?

    • If you place monthly orders, you are not using the reorder-point system, and certainly should not use a formula that is for that system. Carmakers are often faced with the dilemma of ordering daily parts made overseas that have a 4-month lead time, and deal with the issue of working through a trading company that behaves like a local supplier and delivers daily. The trading company is essentially paid to take responsibility for the problem and deal with it, in the many forms it can take. Meanwhile, the carmaker can use the same methods to order parts from overseas and parts from the neighboring stamping shop and focus on the task of assembling them into cars.

  28. It looks like that formula has a sum of terms with different units:

    mu_L and sigma_L are in units of time
    mu_D and sigma_D are in units of amt / time

    This gives:

    mu_L * (sigma_D)^2 [=] (time) (amt / time)^2 [=] amt^2 / time

    (mu_D)^2 * (sigma_L)^2 [=] (amt / time)^2 time^2 [=] amt^2

    Terms with those units can’t rightly be summed, what am I missing here?

    • Yes, the formula is surprising because it doesn’t look homogeneous. I didn’t come up with it and don’t generally recommend it but its math is sound. I worked out the proof after seeing the formula — without proof — in several logistics books.
      You can check it out at Safety Stock: More About The Formula

      • That other post helped a lot, thanks for pointing me to it.

        So it turns out that the units do work out because demand-rate variance per time is precisely that, giving demand-rate standard-deviation per sqrt(time).

        But as you alluded to, it’s definitely counter-intuitive!

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