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Crack the Code

Understanding safety stock and mastering its equations
  • Peter L. King
July/August 2011

Determining appropriate inventory levels is one of the most important and most challenging the the tasks faced by operations managers. If you carry too much inventory, you tie up money in working capital; if you don’t carry enough inventory, you face stockouts. Fortunately, the cycle stock portion of the inventory equation is straightforward. What keeps people up at night is safety stock.

 Safety stock simply is inventory that is carried to prevent stockouts. Stockouts stem from factors such as fluctuating customer demand, forecast inaccuracy, and variability in lead times for raw materials or manufacturing. Some operations managers use gut feelings or hunches to set safety stock levels, while others base them on a portion of cycle stock level— 10 or 20 percent, for example. While easy to execute, such techniques generally result in poor performance. A sound, mathematical approach to safety stock will not only justify the required inventory levels to business leaders, but also balance the conflicting goals of maximizing Customer Relations and minimizing inventory cost.

Safety stock determinations are not intended to eliminate all stockouts—just the majority of them. For example, when designing for a 95 percent service level, expect that 50 percent of the time, not all cycle stock will be depleted and safety stock will not be needed. For another 45 percent of cycles, the safety stock will suffice. But in approximately 5 percent of replenishment cycles, expect a stockout. (See Figure 1.)

While designing for a higher service level—say, 98 percent—would result in fewer stockouts, this requires significantly more safety stock. There must be a balance between inventory costs and Customer Relations. By using the methods and equations that follow, you can find safety stock levels to achieve your desired Customer Relations levels.

Variability in demand

Imagine the only variability you need to protect against is demand variability, and good historical data are available. The safety stock needed to give a certain level of protection simply is the standard deviation of demand variability multiplied by the Z-score—a statistical figure also known as standard score. For example, to satisfy demand with a 95 percent confidence level, according to statistical analysis, it’s necessary to carry extra inventory equal to 1.65 standard deviations of demand variability. This is equivalent to a Z-score of 1.65.

To further understand Z-score, imagine that no safety stock is carried. In this situation, the Z-score is zero. even so, there will be enough inventory to meet demand in 50 percent of cycles. If Z-score equals 1, the safety stock will protect against one standard deviation; there will be enough inventory 84 percent of the time. This percentage of cycles where safety stock prevents stockouts is called the cycle service level.

Figure 2 shows the relationship of desired cycle service levels to Z-score. As illustrated, the relationship is nonlinear: higher cycle service levels require disproportionally higher Z-scores and, thus, disproportionately higher safety stock levels. Typical goals fall between 90 and 98 percent, and—statistically speaking— a cycle service level of 100 percent is unattainable.

Rather than using a fixed Z-score for all products, set the Z-score independently for groups of products based on criteria such as strategic importance, profit margin, or dollar volume. Then, those stockkeeping units with greater value to the business will have more safety stock, and vice versa.

So far, it has been assumed that the demand periods equal total lead time, including any review period. When this is not the case, instead, calculate standard deviation based on periods equal to the lead time. For example, if the standard deviation of demand is calculated from weekly demand data and the total lead time including review period is three weeks, the standard deviation of demand is the weekly standard deviation times the square root of the time units, or √3.

Taking all this into consideration, the safety stock equation becomes:

The performance cycle includes the time needed to perform functions such as deciding what to order or produce, communicating orders to the supplier, manufacturing and processing, and delivery and storage, as well as any additional time required to return to the start of the next cycle.

Variability in lead time 

In the previous equation, safety stock is used to mitigate demand variability. However, when variability in lead time is the primary concern, the safety stock equation becomes: 

When both demand variability and lead time variability are present, statistical calculations can combine to give a lower total safety stock than the sum of the two individual calculations. In cases where demand and lead time variability are independent—that is, they are influenced by different factors—and both are normally distributed, the combined safety stock equation becomes:

In other words, the safety stock is Z-score times the square root of the sum of the squares of the individual variabilities.

But when demand and lead time variability are not independent of each other, this equation can’t be used. In these cases, safety stock is the sum of the two individual calculations:

Cycle service level and fill rate 
The previous equations are useful for predicting the safety stock needed to attain a certain cycle service level— a percentage of replenishment cycles. Sometimes, business leaders instead wish to control the amount of volume ordered that is available to satisfy customer demand—a quantity known as fill rate. Fill rate often is a better measure of inventory performance, as cycle service level merely indicates the frequency of stockouts without regard to their magnitudes. (See Figure 3.)

Cycle service level reflectsvolume of stockouts the frequency of stockouts

FIgure 3: Cycle service level and fill rate

As illustrated in Figure 3, when supply and demand are relatively stable—that is, when standard deviations of demand and lead time are low—fill rate will tend to be higher than cycle service level. While stockouts still occur, the magnitude of stockouts tends to be small. Conversely, when demand or lead time variability is high, fill rate will be lower than cycle service level, and the volume of stockouts will be high.

To better understand these equations, consider the following example of a warehouse that holds large rolls of plastic film. The film gets sold to processors who cut it into shorter, narrower rolls for food packaging, pallet wrapping, or as a dielectric material in industrial capacitors.

The following are the relevant data for one unit type: 
Weekly demand = 50 rolls 
Standard deviation of weekly demand = 10 rolls 
Standard deviation of lead time = 0 
Production cycle = weekly 
Production capacity = 496 rolls/week.

The lead time is very stable and predictable. Process reliability is high enough that the manufacturing lead time never exceeds seven days, and the lead time of transport from the manufacturing facility to the warehouse never exceeds one day. Because this gives a standard deviation of lead time of zero, the safety stock requirements can be calculated from the original equation:

The desired cycle level is 95 percent; that is, the business can tolerate stockouts of this product on no more than 5 percent of the replenishment cycles, or slightly more than two per year. Using the chart in Figure 2, the Z-score is found 1.65. Performance cycle, which affects replenishment of the warehouse inventory, is the sum of the seven-day manufacturing time and the one day needed to arrive at the warehouse, for a total of eight days. T1, the time increment used to calculate σD, is seven days. Thus:

If lead time were variable, more safety stock would be required to meet performance goals, and the safety stock equation becomes:

In this example, lead time varies with a standard deviation of half a day, or approximately 0.07 weeks. Thus:

These results demonstrate that demand variability is the dominant influence on safety stock requirements: Its effect is almost 10 times that of lead time variability. With the recognition of what factors dominate an equation, it becomes easier to focus improvement efforts. In this case, if a reduction in safety stock is desired, it is far more productive to reduce demand variability than lead time variability. Conversely, if a high level of Customer Relations is not required, safety stock can be lowered to a more appropriate level.

Once safety stock levels have been established, inventory levels should be monitored on an ongoing basis to determine if the inventory profile is as expected. Is the safety stock being consumed in about half of the cycles? Are service level targets being realized? If not, before any adjustments are made, perform a root cause analysis to see if any special causes are responsible for the deviations from expected results.

Safety stock alternatives
The previous calculations may result in safety stock recommendations higher than what business leaders feel they can carry. The good news is there are alternatives to mitigate variability other than safety stock.

One is implementing an order-expediting process for preventing stockouts when safety stock is insufficient to cover all random variation. This practice is especially appropriate for products that cost more to produce (and thus cost more to carry in inventory).

In one example involving an expensive but relatively lightweight product, total supply chain costs were reduced significantly. The company carried small amounts of safety stock in overseas warehouses, and it relied on air freight to cover peaks in demand. The cost of shipping a small percentage of total demand via air was minimal compared to the cost of carrying large amounts of safety stock of the valuable material on an ongoing basis.

Another alternative to carrying safety stock is to consider a make-to-order (MTo) or finish-to-order (FTo) production environment. If lead times allow, MTo eliminates the need for most safety stock. Meanwhile, FTo allows for less differentiation in safety stock than finished-product inventory, which lowers demand variability and reduces safety stock requirements. FTo and MTo also are well suited for situations where customers are willing to accept longer lead times for highly sporadic purchases.

In the end, safety stock can be an effective way to mitigate demand uncertainty and lead time variability while still providing high service levels to customers. But before proceeding with a plan, first understand how to determine appropriate levels of safety stock—and what degree of protection they provide.

To read more about cycle service level and fill rate, check out this informational document from Peter L. King, CSCP.  

Peter L. King, CSCP, is founder and president of consulting firm Lean Dynamics LLC and author of Lean for the Process Industries: Dealing with Complexity. Previously, he was a principal consultant in the lean technology division of DuPont. he may be contacted at

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  1. Chaitanya November 30, 2012, 10:32 AM


    Very good article and gives insight into safety stock calculation. But it will be helpful how Target days supply effects the SS calculation in addition to Z score, lead time and Std deviation.



  2. Varun November 19, 2013, 07:46 AM
    Target days supply should have no effect on SS calcualtions. IT is merely a goal set for how many days of inventory you want to carry. To get to that target involves next steps involving reduction in supplier lead time and MOQ definitions. Quite an extensive topic but the goal is to reduce SS at all times

  3. sean July 21, 2015, 10:13 AM
    I'm a big fan of this article and pass it on frequently to help explain why we use the equations shown above. I had a great question raised recently: "Does your equation take into consideration past-due backlog?" I'm stumped, it seems like it should be covered because it's just part of our regular demand but if it all gets shipped at once we may run out of parts. Does anyone have any advice?

  4. gaurav dua April 14, 2016, 09:59 AM
    are CFR and CSL interchangeable in formula ? 95% CSL means 1.65 times stock but baseline CFR is 99% for most companies and using this as CSL means 4 times SD resulting in too much stocks
    For SS ,it is better to target CSL (95% for A and B category) and because that means carrying enough SS ,it can cater to 98% or more of CFR
    So what should one use in formula - CSL or CFR where the two measures are very different and baseline expectations are also different 95% vs 99%

  5. Peter L. King, CSCP April 14, 2016, 01:35 PM
    CSL (Cycle Service Level) and CFR (Customer Fill Rate) are two entirely different measures and are not interchangeable. The calculations are entirely different.  For each of the equations described in the article, the Z score depends on CSL. The calculation of CFR is more complex, and is described in my book, Lean for the Process Industries (Productivity Press, 2009), as well as Chopra and Meindl, Supply Chain Management (Pearson Prentice Hall, 2007).

    In many cases, the CFR will be much higher than CSL, and the situation you describe, a 99% CFR with a 95% CSL, is not uncommon. However, with high demand variability, the opposite may be true; CFR can actually be lower than CSL.
  6. Filippo August 02, 2016, 10:50 AM
    Really nice article.
    Although this, I still have a doubt:
    Why does the SS formula take into account ((PC/T1)* σD^2) and not ((PC/T1)* σD)^2, when both variabilities (demand and time variability) are considered?
    I have this doubt, because, theoretically, the variance of a product of two variables(σ(X*Y) is equal to the following formula:
    σ(X*Y)= σx^2*σy^2 +σx^2*(E(Y))^2+ σy^2*(E(X))^2
    And applying this formula to the demand and time variability, the SS should be:
    SS=Z* σ(D*LT)
    And, therefore, the final result should be:
    SS= Z*sqrt((PC/T1)*σD)^2+(σLT*D)^2)
  7. Peter L. King, CSCP August 03, 2016, 08:18 AM
    That’s a very good question, and it’s good you asked.  A lot of people assume the equation is incorrect based on the same reasoning you used, and just square the PC/T1 parameter, which is not correct. 

    The PC/T1 parameter is there to correct for situations where the time increments from which sD was calculated differ from the total lead time.  If for example, D is captured in weekly increments and sD is calculated from that data set, but the lead time is, say 17 days, you would have to multiply sD representing 7 day variability by SQRT (17/7) to approximate the sD for 17 day periods.  So SQRT (17/7) would go into the combined variance equation, and whenever everything gets squared, it goes to the first power.

    I hope that clarifies it for you.  Many people are confused by the equation, expecting it to be symmetrical, and when it is not, just square the (PC/T1) term to make it so.

    For a more thorough treatment of this general topic, it is covered in chapter 15 of my book, Lean for the Process Industries – Dealing with Complexity, Productivity Press, 2009.  In fact, the Crack the Code article was a heavily edited version of chapter 15.
  8. Jess August 21, 2016, 04:31 PM
    How would I find the safety stock if I'm given a MAPE instead?

    I'm having a hard time wrapping my head around this. How would I calculate SS with the following info below?
    Demand per month: Jan = 100, Feb = 350, Mar = 200, Apr = 700, May = 200, June = 800
    Lead-time: 4 weeks + variability of 10%
    Service Lvl: 98%

    Thanks for any help.
  9. Peter L. King, CSCP August 29, 2016, 08:20 AM
    I have never known of anyone using MAPE, or any average deviation or error, to set safety stock levels.  It is much better to use a statistical measure of variation or of error so you can calculate the likelihood of a stockout under various conditions and compare that to customer delivery targets.  So if you’re using forecasts to calculate cycle stock, you should use standard deviation of forecast error to set safety stock levels.   (In a pure statistical sense, this may not be a true standard deviation, but it is calculated in a similar fashion.)

    If your forecast demand varies as much as you show, then the cycle stock target should be adjusted month by month to agree with the forecast.
    The lead time variability should be stated as a standard deviation around the 4 weeks, rather than as 10%, again so you can estimate the likelihood of lead time variation causing a stockout for various levels of safety stock.
    I hope that helps.  If not, I’ll be happy to take another crack at explaining it.
  10. Gaurav December 20, 2016, 09:57 AM
    Thanks for this useful article. I have two questions
    1. As per the equation 
    (X*Y)= σx^2*σy^2 +σx^2*(E(Y))^2+ σy^2*(E(X))^2
    SS should have been equal to 
    SS = Z*sqrt((PC/T1)^2*σD^2+(PC/T1)*σD^2+(σLT*D)^2) however the first term in the RHS under sqrt is not provided for. Can you please explain why?
    2. Shouldn't (PC/T1)*σD^2  still be multiplied by E(LT)^2 as per the equation given above? Am I missing something very basic?

    Thanks for your time

  11. Peter L. King, CSCP December 20, 2016, 10:55 AM
    The equation is correct as it stands.  A lot of people question the fact that it is asymmetrical, and think that if three terms are squared, the fourth should also be.  But the PC/T1 term begins as SQRT (PC/T1) and is a statistical adjustment made whenever the Performance Cycle (PC) differs from the time periods (T1) from which the σD was calculated.  So when that term gets squared, it goes to the first power.
    There is no reason why (PC/T1)*σD^2 should be multiplied by E(LT)^2.  That term is there to protect against variation in demand, and the premise is that it is independent of lead time.  If that premise is not true, then the two variations cannot be combined, as was pointed out in the article.
  12. Peter L. King, CSCP December 20, 2016, 11:00 AM
    The equation is correct as it stands.  A lot of people question the fact that it is asymmetrical, and think that if three terms are squared, the fourth should also be.  But the PC/T1 term begins as SQRT (PC/T1) and is a statistical adjustment made whenever the Performance Cycle (PC) differs from the time periods (T1) from which the σD was calculated.  So when that term gets squared, it goes to the first power.
    There is no reason why (PC/T1)*σD^2 should be multiplied by E(LT)^2.  That term is there to protect against variation in demand, and the premise is that it is independent of lead time.  If that premise is not true, then the two variations cannot be combined, as was pointed out in the article.
  13. Omar January 03, 2017, 03:13 PM
    Peter, this is an amazing article, we have been using a lot!
    However, I have seen two different equations for Safety Stock with demand and lead time variable.
    The only difference between both formulas is the inclusion of the Time increment (T1)

    Formula 1:
    SS= Z*sqrt (((PC/T1)*σD^2)+(σLT*D)^2)

    Formula 2:
    SS= Z*sqrt ((PC*σD^2)+(σLT*D)^2)

    What is the difference between both formulas?
    How should we interpret the variables, in both formulas?
    What is the meaning and/or calculation of PC and σD in both formulas?

    Thank you so much!
  14. Peter L. King, CSCP January 09, 2017, 09:05 AM
    Formula 1 is the one I use, and is the most comprehensive.  Formula 2 would be correct, and consistent with Formula 1, in cases where T1 = 1.  If, for example, the Performance Cycle (PC) is three weeks and T1 is one week, then T1 would not be necessary in the formula.  Similarly, if T1 = 1 month, and PC is some number of months, T1 would not be needed in the formula.

    PC is the performance Cycle, is the total lead time, the total .time at risk, i.e., the time between making a determination on how much to purchase or produce, and the time to make the next determination and have it realized.

    When procuring raw materials, the performance cycle includes the time to:
    -- Decide what to order (order interval or review period)
    -- Communicate the order to the supplier
    -- Manufacture or process the material
    -- Deliver the material
    -- Perform a store-in

    Inside our own manufacturing facility, the performance cycle includes the time to:
    -- Decide what to produce
    -- Manufacture the material
    -- Release the material to the downstream inventory
    -- Return to the next cycle

    If we are carrying inventory in a finished product warehouse, and customers allow a delivery lead time greater than the time needed to deliver to the customer, then the remaining customer lead time can be subtracted from the Performance Cycle

    Sigma D is the standard deviation of demand.  T1 represents the time periods included in the Std Dev calculation.  For example, if demand is captured in weekly totals, and the Std Dev is based on the variation from week to week to week, then T1 is 1 week, or 7 days.  It is important that T1 and PC are expressed in the same time units, i.e., days or weeks or months.
  15. enrique March 16, 2017, 10:18 AM
    apics rocks
  16. Juliano Serpa May 16, 2017, 09:05 AM
    Calculo de Safety stock
  17. Karol June 30, 2017, 11:56 AM
    This is a great article thanks.

    However I am not feeling comfortable about the PC/T1 factor.

    Namely...imagine a situation where lead time PC is much lower than T1 (period used to calculate standard deviation). I will give you an example of an ATM machine where usually Cash Delivery Lead Time = 1 and the Standard Deviation is calculated over 90 days.
    While running several analysis I noticed that the less days I use for the Standard Deviation, the higher is the Lead Time factor i.e. sqrt(PC/T1)...meaning the shorter period for Standard Deviation the higher Safety Stock. And it could be thousdands of $. There is no Lead Time variability I ignore this parameter.

    In your view, should the T1 be used in the calculation at all or just SQRT(PC). If using T1 makes sense, then in your opinion what is the best period of T1 to use? 30 days, 60 days, 90 days or more? Thanks

  18. Paulo July 04, 2017, 11:51 PM
    Great article. Very helpful.

    I'm actually having a little more trouble calculating what should be the easiest part.
    I'm trying to get the average cycle stock, considering that I have a fixed review period and fixed batch size (the order could be a multiple of this batch). These two limitations are making impossible to bring me an average cycle stock of Q/2.

    Is there a formula I can use to calculate Average cycle stock using as my only variables the Review Period (R), Leadtime (L), Batch Size (B) and Average Demand (D)? (EOQ is not an issue here. My optimal model focuses only on having the lowest inventory possible across an specific time frame).

    Can I dissociate the extra inventory I'm carrying across the review periods from Safety Stock?

    Thank you!
  19. Peter L. King, CSCP July 07, 2017, 01:16 PM
    Hi Karol,
    The PC/T1 factor is always an approximation, to estimate the standard deviation for time periods different from that over which the data is being collected, and it is even more of an approximation when PC is much lower than T1. 
    If I understand your situation, you have an ATM cash machine which is replenished daily, and you want to know how much safety stock cash will be required to handle the variation in daily withdrawals.  If you have a history of what was withdrawn from the ATM machine each day over some period of time, then that daily data should be used to calculate the standard deviation, and T1 and the PC will both be one day, and the PC/T1 factor becomes 1.   That will give you the best indication of how much safety stock cash you need   If you only have data to calculate standard deviations on a 90 day basis, then it’s nearly impossible to predict the daily variation from the 90 day variation.
    I hope that answered it for you; if not, you can reach me at
    Regards, Pete King

  20. Peter L. King, CSCP July 07, 2017, 01:17 PM
    Hi Paulo,
    The first thing I need to know is whether you are using a fixed interval replenishment, where the inventory is replenished on some standard time frame like weekly or monthly, or on a Re-Order Point (ROP) basis, where the inventory is replenished whenever the current amount drops to or below the re-order point.
    If you are using a fixed interval model, then the amount replenished will vary from cycle to cycle based on the demand during each cycle, and the average cycle stock will be the average demand over a number of order intervals, or the forecast average over a number of periods.  The factor which must generally be considered here is not the review period, but the replenishment lead time.  That won’t affect cycle stock, but it will the order quantity.
    Order Quantity = Demand During Lead Time + Cycle Stock + Safety Stock – Current Inventory.
    If a ROP model is being used, then the cycle stock is whatever quantity you have decided to order whenever the ROP is hit.   Here, the thing that must be understood is how to compute the ROP:
    Re-Order Point = Demand During Review Period + Demand During Lead Time + Safety Stock.
    This is all covered in more detail, with charts and diagrams, in my book, Lean for the Process Industries, Dealing With Complexity (Productivity Press, 2009)
    I hope that answered it for you; if not, you can reach me at
    Regards, Pete King
  21. Flavia July 14, 2017, 04:32 AM
    Great article! 
    The notion that we should always consider the lead time in the same base as the demand seems obvious but is not always considered in the formulas that you find in the theories. I also have a question about the lead time deviation, in the example that you gave there was a deviation of 0,5 days that you translated to 0.07 weeks. Does it means that we also need adjust the PC deviation based on the demand period we are using? In this case the demand you used was based in weeks, so you split 0.5 days / 7 days, right? If I have a demand for 30 days and my PC is 60 days, should I divide both PC and PC deviation by 30? 
    Thank you very much!
  22. Peter L. King July 21, 2017, 05:43 PM
    One very important consideration is that generally all factors should be in the same time units (days or weeks or hours, etc.)  But if the numerator and denominator are in the same time units, they cancel so they don’t have to be the same units as factors in other parts of the equation. 

    I’m not sure I understand part of your question: “If I have demand for 30 days…” do you mean that demand data is collected in 30 day buckets?  If so, then PC could be in days so that the PC/T1 factor would be 60/30 =2.  Or in months it would be 2/1, again = 2.

    Hope this is helpful and answers your question.
  23. Mayur August 09, 2017, 10:42 AM
    very useful article!

    I would like to know how to calculate Standard Deviation of demand. So to calculate safety stock we Need to calculate Standard Deviation of demand (demand variability). For example, if I want to calculate safety stock for next month i.e September 2017, so should I consider following steps to calculate demand variability:
    1. Note down Forecast for September 2017 from APO (eg. 745 Units)
    2. Note down Actual sales for sembtember 2016 (eg. 840 Units)
    3. Take difference of Forecast vs Actual but take absolute value. (95 Units) 
    and then use this value to calculate safety stock.

    Is this the right way to calculate demand variability?
  24. Peter L. King, CSCP August 11, 2017, 09:51 AM
    If you're producing to a forecast, the thing you need for safety stock calculations is not demand variability but forecast error.  You use demand variability for safety stock when you are basing production on average demand from the demand history over the past, say, 12 months.

    When producing to a forecast, you need forecast error for the past, say, 12 months.  You need to make sure bias has been removed, then you need to calculate something that looks like a standard deviation of the forecast errors.  Here is an article from Wikipedia about calculating standard deviation: 
  25. Lucas August 16, 2017, 03:31 PM

    This article has been very useful for us! Thank you very much! We are building the major part of our inventory policy based in this article.

    I just have one question about Lead Time.

    The time for manucturing  1 box or 1.000.000 box has differents Lead Time.

    Whatis the quantity of products should I measure the Lead time? (ex: 1 box, 1 week demand, 1 day demand)
  26. Emily Gao August 27, 2017, 10:04 PM
    Peter, this is an educational article about inventory management!

    One question regarding the formula. Does it assume normal distribution? If demand or lead time variability is in other forms of distribution, can we still set safety stock with this formula? If not, do you have other suggestions?

  27. Emily Gao August 27, 2017, 10:20 PM

    Thanks for the amazing article. I have another question as I read thru it more times. In the article, you mentioned in the case where demand variability and lead time variability are independent, statistical calculations can combine to give a lower total safety stock. Could you explain how you get the final formula and are sure that it gives a lower and most importantly reasonable safety stock? 

    I have this question because in the business world,  both the internal and external customers care about "Why"? Need to explain to the audience if I were to cite this formula to solve real world problems.

  28. Lucas de Oliveira Alves August 31, 2017, 08:46 AM

    That's a excellent article! But I'm not sure about one point.

    What's exactly the Performance Cycle (PC)?

    Is the Lead Time (the time's length of a production start until the product is ready to be sold), or the Replenishment Time (the interval between one prodction and another of the same product)? 

    Thanks very much

  29. Peter L. King, CSCP September 05, 2017, 01:30 PM
    Lucas: In response to your first question, the Lead Time should be expressed in terms of a production lot, a group of product that moves together.  Thus if you make one box at a time, and then release it to the next step or into inventory, then lead time is measured on one box.  If you make 100 boxes as a group, then lead time is based on the time for 100 boxes.

    In response to your second question, in most cases, the Performance Cycle and Lead Time are the same. When procuring raw materials, the performance cycle (PC) includes the time to:
    -- Decide what to order  (Order interval or Review period)
    -- Communicate the order to the supplier
    -- Manufacture or process the material
    -- Have the material delivered
    -- Perform a store-in. 

    Inside our own manufacturing facility, the performance cycle includes the time to:
    -- Decide what to produce
    -- Manufacture the material
    -- Release the material to Finished Product Inventory
    -- Return to the next cycle. 

    Emily: In response to your first question, the formula assumes a normal distribution.  If the distribution is non-normal, but close, the formula should do a pretty good job.  As the data gets more non-normal, the results get to be less accurate.  There are statistical methods of converting various distributions to approximate a normal distribution, but I don’t have much experience in that area.

    In response to your second question,  the formula is correct and should give you reasonable approximation of the safety stock required.  The basic concept behind the math of combining the two components is that it is very unlikely that you will have the longest lead times simultaneously with the highest demands.  So the calculation says that the combined probability is less than the sum of the two individual probabilities.  It is possible that you can be out on the end of the distribution tail on both distributions, but less likely.

    One very important point that was covered briefly in the article is that these formulas give safety stock requirements based on Cycle Service Level.  What many businesses are interested is not that, but Fill Rate.  Please see the link at the end of this article, which will connect you to a PDF document that describes this point in more detail. 
  30. Mukul September 20, 2017, 08:25 AM
    Hi Peter,

    Wonderful article, and even more wonderful insights given by you in Q&A.

    Whereas I have no doubt in understanding what you've mentioned here, where I'm facing a challenge is in understanding the impact of receipt period in safety stock factor 'Z'. I've seen this factor vary a lot due to receipt period, which in my particular case varies from daily to once a month across SKUs. My question is, are there any scenarios where not to consider impact of receipt period on 'Z', and instead assume Receipt size as one day's demand?

    Thanks & Regards
  31. Dhananjay December 01, 2017, 06:42 AM
    While considering Safety stock calculation demand variability calculation as per Customer Relations level will give same value of difference if deviation is on positive side or negative side .
    Safety stock=Z*SQRT(PC/T1) * Sigma D
    Means 1st time demand 80 Forecast 100 As well demand 100 & forecast 80 . Deviation is 20 both times.Service level factor being same will not it give wrong calculation for addition of demand variability factor.
  32. Rick Catalano, CSCP December 07, 2017, 01:23 PM
    Hello, I've been tasked with reviewing demand and calculating statistical safety stocks. I've used the formula "Safety stock = Z x Sqrt(PC/T * standard deviation of D

    I've used forecast, shipments, and variation shipments minus forecast for the 'demand' piece of the equation. My manager scratches his head when he reviews the numbers due to his gut saying that the numbers are too low. We also assume no variability in leadtime thus why the second part of the equation is left off.

    Is there a way you could send an excel example of calculating statistical safety stock so I can validate my formula?

    Any assistance is greatly appreciated. thank you,
    Rick Catalano, CSCP
    Master Production Scheduler
    Hologic, Inc.

  33. Cheki December 17, 2017, 07:16 AM
    Important article, we adopt this theory for our effective calculation...
    My question is almost linked to specific case where I have two locations, one location is supposed to be satellite warehouse , second one is effectively the main Distribution center..
    The replenishment LT from central DC to satellite warehouse is 2 weeks , the replenishment LT to Central DC is 12 weeks.. the variances of demand
     Are kept in the satellite what , and artificially injected the avrg inside the central DC.. this mean when we calculated the SS , we cover higher buffer in the satellite and effectively lower buffer in the Central.. I come to the point that with this may we survive in the satellite against demand and at any point we can crack the stock holding at the central since our buffer becomes minimum and hence we are not covering 10 weeks variances properly.
    Is this good interpretation , is this could increase the chances to be in short ?

  34. BobWare January 01, 2018, 01:34 PM
    Just wanted to read the article
  35. Jonathan Miranda January 09, 2018, 04:42 PM
    Good infomation for basics supply learning
  36. Iker January 17, 2018, 10:22 AM
    Hi Peter, Excelent article, the most clarifying one I read related to SS calculation!

    My doubt is concerning to the calculation of the standard deviation of demand, what is not usually discussed as I have seen it solved in different ways.

    I want to calculate a SS for Procuring Raw Material. However, the raw material is not consumed regularly, as the production lines not always fabricate the same finished products. Lets say that "A Material" is consumed 50 days spread in one year. Do I have to calculate the std. dev. only for the demand during those 50 days or over the 365 days of the year ?

    It applies the same for the average demand if I want to calculate the safety stock for variabily of supplier lead time. Do I have to consider average only during those 50 days of usage (higher) or over the whole year (lower)?

    Thanks a lot in advance!
  37. Ravi April 10, 2018, 02:16 PM
    I have past usage data for 12 months
    Lead time for part LT is 90 days

    If I taking Standard deviation of demand over 12 months period, T in above calculation should be 365? as time horizon against which Standard deviation was calculated was 12 months i.e. 365 days?
  38. Cristian April 20, 2018, 03:51 AM
    Dear Peter,

    Great value in your article.
    I face a different situation - it seems I need an adaptation of your formula.
    We import and distribute + sell in our own store network. The retail and the wholesale have different statistics - the fluctuations are much higher for wholesale, due to the grocery channels which are almost unpredictable in purchase. That's why I separated the average sales and standard deviation of demand per retail/wholesale. So I add the safety stocks - of course a fluctuation of demand in wholesale influences the service level both for wholesale and retail.

    What I need to understand: We order periodically for each supplier, we don't use ROP but rather a standard time frame. In some cases the periodicity is smaller than the purchasing cycle (e.g order from China each month, delivery time ~3-5 months), in other cases the periodicity is bigger (suppliers from Europe, articles with low sales).
    I calculate the order qty in 2 steps:
    Step 1 - expected qty in warehouse before receiving the order = actual stock-reservations+goods in transit-(average weekly sales in wholesale+retail)*PC(days)/7. If result is negative then obvious the expected stock will be 0 - we'll probably have a gap.

    Step 2 - Order = (average weekly sales in wholesale+retail)*OrderRhitmicity(days)/7+Zretail*Sqrt((PC+OrderRhythmicity)/7*SIGMAretail^2+(SIGMA_PC/7*AverageRetailSales)^2)+ a similar calculation for wholesale).
    As you can see I replaced PC/7 with PC+Order rhythmicity/7, as what we order today should last until (today+PC+Order rhythmicity) when next order will arrive in our warehouse.
    Is there any more accurate solution to my problem?

    Many thanks for your great work,
  39. Julia May 19, 2018, 06:33 AM

    This is a great article. I have seen no one has explained the safety stock calculation logic better than you do. I like particularly how you explained the SS is the result of applying the Z-values on two variabilities that are assumed to be normal distributed - LT and demand. That explains everything. 

    I have a question though on how to apply this to a more complicated situation where we want to bring in the third variability - as you said, the forecast inaccuracy into the formula. How should I approach it? 

    Thank you.


  40. Asis July 10, 2018, 02:37 AM
    Very interesting article Pete! Excellent read.

    In my case, orders follow a monthly cycle and are forecasted n+5 periods, (lets assume no variance in demand or supply). Would it be safe to say that we would have to keep a safety stock of 2.05x the forecasted cycle stock for each period if we wanted to attain a 98% service level? Would this mean that the target stock for the beggining of each period be 3.05x cycle stock (1 cycle stock + 2.05 safety stock)?

    Thank you!
  41. Heiko July 18, 2018, 11:21 AM
    Dear Peter,

    thanks for this very helpful article.
    I would come to a similar question like Jess - how to understand the equation PC/T1
    For example: We are calculation the Demand accuracy in monthly buckets while the lead time is measured in working days.
    Would I need to devide the lead time by 20 (working day per month)? Or how can I get this two time periodes on the same level.
    Thanks in advance!
  42. Karol August 09, 2018, 02:36 AM
    Hi Peter,

    Thank you for your previous help.

    If you hopefully recall I asked you in the past about SS for cash management industry and in particular how to calculate it for ATM Withdrawals.

    Now I am puzzled on how to calculate it for a Branch. Namely ATMs only Withdraws cash (at least standard ATMs) and we just need to supply enough cash to go through the cycle. But branch, gets Withdrawals and Deposits, how would you approach SS calculation for an instance that gets cash out and cash in?

    Thank you,
  43. Kriti August 10, 2018, 07:36 AM
    Very informative article. Q & A has explored it better. Thanks Peter, you are star. Please comment on my inventory stock.
    I need to calculate the safety stock  where lead time is 1 day, my distribution  is not normal and have lots of demand variability. I have historical data for last 7 months and have forecast data for next three months as well. I understand from your reasons that - 
    1. I need to find the forecast error (historical data - forecast for that period), and draw the distribution of forecast errors. 
    2.  If its not normal distribution , I have an option to convert it to normal distribution using Central limit theorem Only in case I am looking for more accurate stock) and find the SD.
    3. SS = z * SD (forecast error)

    Did I put it correctly?

    Do I need to top up safety stock more?

  44. Clinton October 18, 2018, 02:21 PM
    I have a general question.  

    In the event that your data points are not normally distributed, how do you approach safety stock?  I know there are z-scores, t-scores, etc., but will replacing the z score (with a t score, as an example) be beneficial in some situations?  Does anyone have any experience with this?

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