1) Formula question: Why is the Average Lead Time term not squared in the equation? Can you explain this in “layman’s” language, maybe with an example?

{Z * SQRT (Avg. Lead Time * Standard Deviation of Demand ^2 + Avg. Demand ^2 * Standard Deviation of Lead Time ^2}

2) Plant A uses Raw Material X (10 units day +/- 3 units). Raw material X is stored in Warehouse 1. Raw Material X is shipped by truck from Warehouse 1 every Friday to Plant A (1 day +/- 0.5 days). Raw Material X is sourced in Warehouse 1 at 50% by local supplier (truck arrives every 7 days +/- day) and 50% by import supplier(ship arrives every 30 days +/- 7 days). What would you recommend to calcualte the “total” system safety stock for Plant A to insure 95% service level and why? Do you make two calculations and add them together(i.e. one for local and one for imports)? Any help would be great.

I’d apprecaite your feedback and advice.
Thanks!

Short answer: Let’s address the first formula that you cited in your post, which we will call Formula (1) in what follows:
z * SQRT(avg lead time * std dev demand^2 + avg demand^2 * std dev lead time^2).

This formula is stated correctly, given the event-based service-level criterion that it intends to optimize (and with the understanding that the expressions involved are the means and standard deviations of the relevant theoretical distributions).

Expanded answer: Formula (1) is based on the distribution of total demand over lead time, where lead time is allowed to vary according to some probability distribution. It can be shown that the standard deviation of the total demand over lead time is precisely the square root given by the formula, namely, SQRT(avg lead time * std dev demand^2 + avg demand^2 * std dev lead time^2). We are currently preparing a white paper that discusses the derivation and shortcomings of this formula; this paper should appear on our Website shortly. If you (or anyone else) would like a copy, please contact me at http://topdownleansystems.com/contact.php.

Before discussing the other two formulas that you mentioned, we need to say that Formula (1) can give extremely misleading results. This formula’s computed safety-stock values can be excessively overstated (poor inventory velocity) or inadequate (poor actual service levels). These values are highly sensitive to the distribution of lead time. This is illustrated in the white paper that we are currently preparing. Again, if you are interested, please contact me for a copy.

Concerning the second formula that you listed:
z * SQRT(avg lead time^2 * std dev demand^2 + avg demand^2 * std dev lead time^2 + std dev demand^2 * std dev lead time^2).

We are not familiar with this formula. The square-root portion of this formula cannot represent the std dev of total demand over lead time, since, as mentioned above, the square root in Formula (1) is the correct expression for this. Might this be a std dev for a different situation? Could you provide an explanation of how this formula is derived, or where you might have seen it published? We are very interested in its source.

Concerning the MAD-based formula that you listed: In this formula, the std dev of demand in your second formula has been replaced by MAD. As you know, the use of MAD implies that demand is being forecast. So, whereas Formula (1) is built around actual demand values, using MAD in a safety-stock formula presumes that safety stock will be computed to guard against forecast error. Although it is true that MAD can be used to estimate the theoretical std dev of forecast error, it is a flawed approach that is inferior to simply using the sample std dev of forecast errors as an estimate of the theoretical std dev. However, whenever MAD is used to estimate the std dev of forecast error, it must be multiplied by a factor of approximately 1.25; in other words, the theoretical std dev of forecast errors is estimated by 1.25 * MAD.

Using forecast-error MAD to estimate the theoretical std dev of forecast error is suboptimal. The sample std dev of forecast errors gives a better estimate of the theoretical std dev of forecast errors than does MAD. Using MAD made sense when computing power was minimal, but today the use of MAD to estimate a std dev is not defensible.

Additionally, using MAD to estimate the theoretical std dev of forecast errors can be problematic for a number of reasons:

1. The typical use of MAD to estimate the theoretical std dev of forecast error requires that forecast errors be normally distributed with mean zero. However, actual demand values are often right-skewed, and sometimes highly so, especially in cases of sporadic demand, leading to non-normal forecast errors.

2. Forecasts are biased for various reasons, especially at the inventory-item level. For example, an inventory item’s forecast demand may be chronically under or over actual demand. In such cases, forecast errors will not have mean zero, making the computed safety stock-value suspect; this could lead to safety-stock levels that are much too high, or dangerously low.

3. Many common MAD-based safety-stock formulas also assume that the forecast errors are independent. Often, forecasts are based on demand values from previous periods. Although this does not necessarily mean that the errors will fail to be independent, in many cases, the errors could fail the independence requirement, again making the computed safety-stock value suspect.

4. The time frame of typical forecast periods decreases the number of forecast-to-actual deviation data points. For example, if the forecast period is one month, one year’s worth of historical forecast-to-actual comparisons provides a relatively unreliable 12 deviation values, and even a one-week forecast period provides only 52 values. Historical daily demand variation based on customer-requested fulfillment, not forecast-to-actual-demand forecast error, provides the best indicator of potential service-level failures.

Even if all of the assumptions on which Formula (1) and similar formulas are based were satisfied, we contend that these formulas are not “correct,” in part because they optimize the wrong service-level criterion. These formulas aim to limit the risk of a stockout occurring over a lead-time period. This is an event-oriented criterion. This is not how service level is measured at most companies. Most companies measure service level as a fill rate (see http://en.wikipedia.org/wiki/Service_level), where the “beta” and “gamma” criteria are described).

We have developed a safety-stock approach that is fill-rate based. Beyond this, it integrates the many other parameters that affect inventory levels and thus safety stock. Our approach is correct and comprehensive, in contrast to the many simple formulas that are available. For example:

None of the safety-stock formulas you cited provide a confidence interval, or range of possible results based on the variation of all the variables. Instead, the implied confidence of these formulas is only 50%. This means that even if the assumptions behind the formulas were satisfied, the safety-stock level itself would, on average, achieve the desired service level only 50% of the time. This low confidence would result in a financially-unfavorable combination of expediting and unhappy customers for one month, quarter or year out of every two, on average. Our safety-stock approach allows you to specify your desired confidence level (we suggest 90-95%).

All of the safety-stock formulas you cited assume that actual demand is normally distributed. Actual demand is often right-skewed and may also be quite sporadic. Actual demand has a natural lower bound of zero, but no upper bound. The upper range of a demand distribution poses the potential for service-level failures. Our safety-stock approach accommodates actual demand-distribution skewness and sporadicity.

Our approach adjusts safety-stock levels for positive or negative forecast bias, if your acquisition is driven by forecast (such as MRP).

Our approach also includes these factors that definitely affect safety stock and service levels: Historical daily demand patterns; quantity-based fill-rate targets; realistic lead times; MOQs / EOQs / package sizes / reorder-review intervals; probability of unfulfilled-demand cancellation; high confidence of consistently achieving fill-rate targets; expected near-future trend and/or seasonality.

I’ll be happy to perform our correct, comprehensive safety-stock analysis on up to 30 of your items as a free trial, just as I’ve offered at http://www.resourcesystemsconsulting.com/blog/blog (about the third posting down from the top).

See our white papers on safety stock at http://topdownleansystems.com/white.htm. Take the Safety Stock Quiz, at http://topdownleansystems.com/quiz.htm. Contact me at http://topdownleansystems.com/contact.php.

David McPhetrige, TopDown Lean Systems

A question about the following formula:

{Z * SQRT (Avg. Lead Time * Standard Deviation of Demand ^2 + Avg. Demand ^2 * Standard Deviation of Lead Time ^2}

Shouldn’t the correct one be {Z * SQRT (Avg. Lead Time ^2 * Standard Deviation of Demand ^2 + Avg. Demand ^2 * Standard Deviation of Lead Time ^2 + Standard Deviation of Demand ^2 * Standard Deviation of Lead Time ^2)}?

Also, has anyone studied the following variation:

{Z * SQRT (Avg. Lead Time ^2 * MAD ^2 + Avg. Demand ^2 * Standard Deviation of Lead Time ^2 + MAD ^2 * Standard Deviation of Lead Time ^2)}

Where MAD = (Sum(|Real Demand – Forecast|)/n) where n = periods

I’d really apreciate your comments.

Thanks

First, your L(z) question about the Quick MBA service-level formula, which seeks to find a safety-stock quantity that provides a quantity-based, as opposed to a stockout-event-based, service level. This formula is based on the idea of computing the expected number of units late over a time period, given a distribution that represents the demand. In the Quick MBA formula, that time period is the lead time (l), adjusted by the reorder period (p).

This formula first requires your desired fill-rate service level, say f = 0.98. Next, let’s suppose your demand distribution is approximately normal, with mean = 50 and standard deviation = 10. Also, let’s say that the reorder period is 5 days, and that lead time is 2 days. Then L(z) = ((1 – 0.98)*50*5)/(10*(5+2)^.5) = 0.189.

Next, look up L(z) in a table of normal-loss-function values. The corresponding z-value is approximately 0.53. Take this value and multiply it by the standard deviation of demand times the adjusted lead time (the square root of the total period of 7 days): S = 0.53*10*7^.5 = 14.0, which implies that safety stock of about 14 units is needed to obtain a 98% fill rate.

This safety-stock calculation, based on the unit normal loss function, reflects a quantity-based service measure. In our example, we expect our demand quantity during lead time to be 100, and our quantity-based service level is 98%, so we would like to have sufficient safety stock to ensure that 98 of these are fulfilled on time. The Quick MBA formula uses L(z) to obtain a value that corresponds to the desired 98% fill rate.

In contrast to this calculation that finds a safety stock value intended to achieve a desired fill rate, formulas such as the one from Inventory Management Review seek to ensure a level of stockout (or no-stockout) events. So, you might calculate safety stock to ensure a 98% no-stockout rate. Note that the stockout could involve 4 items or 4000 items – this formula does not distinguish, it looks only at the probability that a stockout occurs.

One practical concern with the stockout-event approach is that it rarely represents the way you measure your actual service-level performance. For example, you may track actual demand quantities delivered on time as a percentage of total demand quantities, a typical “on-time delivery” measure. Likely, you also have a strategic target for this quantity-based on-time-delivery measure. Let’s say that your on-time-delivery target is 98%. If you use the Management Review formula for your calculation, you are actually determining the safety-stock level required to achieve 98% probability of no stockouts. This is very different from saying that 98% of the demand quantities will be on time.

Generally, event-based safety-stock formulas provide higher safety stock values than do fill-rate based formulas and methods. If your actual service-level performance measure is quantity-based, and your safety-stock level is based on the probability of no stockout events, your safety stock levels may be too high.

Both formulas we have discussed assume that total demand over the period of interest is approximately normal. In reality, demand data is rarely normally-distributed. It follows that the central limit theorem may not apply, especially when the lead time is short, the demand distribution is significantly skewed, and/or demand is sporadic.

Next, your question on demand-data intervals: From a practical perspective, daily demand data is best, for several reasons. 1) Usually, one wants to have safety stock to cover demand each day. This requires that the estimate of demand variability (standard deviation, for example) be based on daily demand data. If, say, the daily data are binned by week and the standard deviation of the weekly data is computed, this estimate will conceal variability that occurs on a daily basis, thus giving incorrect results when used in calculations. 2) Trends, seasonality, other patterns and special causes can be concealed by weekly or monthly data. With short product life cycles and market volatility, your representative historical demand-data time frame may be only 12-24 months. If this data is in weekly or monthly bins, you will have only 52-104 weekly, or 12-24 monthly, data points. By contrast, daily data will provide you with several hundred data points.

Finally, your question on “beta”: As Lawrence Loucka indicates in his posting dated September 11, 2009, beta is intended to adjust lead time for forecast interval, and forecast interval for order cycle interval. I suggest you contact Kent Linford, the author of the formula, to get his explanation of beta.

Importantly, the Linford formula acknowledges that MOQ, EOQ or reorder-review interval can have a significant impact on the safety-stock level actually required to achieve a target service level. Intuitively, this makes sense: A large MOQ provides “de facto” safety stock, so that little or no additional calculated safety stock may be needed to achieve the target service level.

I’ll be happy to discuss these various safety-stock factors with you. The best way to contact me is on http://topdownleansystems.com/contact.php. Also, I’m the only “David McPhetrige” on Skype.

David McPhetrige, TopDown Lean Systems

Short answer: Since your demand data is in monthly intervals, and your lead-time is a whole-number-month multiple, be sure that all of the factors in your safety-stock calculation are expressed and/or calculated as monthly values.

Longer answer:
1. Monthly data may not offer very many data points, and statistical reliability will suffer. For instance, a year’s worth of demand data is only 12 points, and this is not a very good sample size for several reasons. For one thing, your standard deviation estimate will only reflect month-to-month variability, and this will most likely be smaller than the amount of variability that you would see in weekly or daily data. Keep in mind that you can run out of stock any time during a month, and using monthly buckets treats demand as if all that matters is whether it is late in that month. It does not distinguish whether a stock-out occurs early in the month or late in the month, since within-the-month variability is not acknowledged. Also, monthly values conceal interesting patterns that may occur during a month, for example, trending, cyclical effects, or simply high-volume periods. If there’s any way you can get daily or weekly demand data, you’ll have many more data points to work with, and more reliable results.

2. The formula you posted uses a normality-based z value. This assumes that your demand over lead time is approximately normally distributed. In reality, demand is typically right-skewed, and may also be sporadic, meaning that normality-based calculations may result in incorrect safety-stock levels.

3. Make sure your actual service-level-measurement criterion matches the criterion you use in your safety stock approach. (See the Wikipedia entry for “Service Level”.) Likely, you measure your actual service-level performance as a quantity-based fill rate, such as on-time delivery (demand quantity fulfilled on time / total demand quantity). By contrast, a typical z-table shows the cumulative probability of no stock-out events. This z value is a different service-level criterion than your actual, quantity-based measure.

4. The formula you posted suggests that service level, lead time, demand variation and lead time variation are the only factors that significantly affect safety stock. However, MOQ (or EOQ, or reorder-review interval) also affects safety stock. Additionally, past-due demand backlog can make a big difference in safety stock requirements. In contrast, your posted formula assumes that demand not fulfilled on time is canceled.

5. A comprehensive safety-stock approach should provide you with a range of safety-stock levels that reflect various likelihoods of achieving the service level you desire. The value that you select from that range should represent the confidence you need to have that the safety-stock level will consistently achieve your service-level target.

I have another post on this forum, “Safety Stock Optimization”, at http://resourcesystemsconsulting.com/blog/blog. As it says, we will provide a complete, comprehensive free trial safety-stock analysis on up to 30 of your inventory items.

David McPhetrige, TopDown Lean Systems

Whatever time bucket you use, just be consistent. If you have lead time in days and demand in months, the math will be wrong.

For your question on Beta, why not write to Kent Linford(kent.linford@moen.com)?

I’m trying to calculate the safety stock (see formulas on top of this page) but this really doesn’t work for me.

For calculating L(z), what do they mean with desired fill rate? Is this the same as the customer service level?? And what do they mean with the demand (daily, weekly, yearly)??? When I calculate this, I calculate that L(z) = 497. But is this possible because the table only goes until 3,09.

And then I calculated the formula from Apics but this formula doesn’t work either. What do they mean with this “beta” (between 0,5 and 0,7). What do Which beta da I have to use?

Can someone please help me with this problem?

Thanks everyone

Please forgive me for answering your question with more questions. Might I ask for some clarification, please:

On this 25% of orders that are known, do you have enough visibility of demand (the customer allows you enough lead time) so that you can acquire (make or buy) what you need to fulfill this demand “to order”, instead of from stock? If this is true, do you actually acquire this 25% of demand “to order”, or allocate/reserve inventory for this demand as soon as the demand is visible, so that the inventory is unavailable for other orders?

What is the time interval for your historical demand data that you’re using in your estimation of standard deviation? Daily, weekly, monthly…?

I don’t understand the logic for adding R (replenishment interval), converted to weeks, to your lead-time days, and then taking the square root of that total. Could you explain this, please? As a general rule, longer replenishment intervals require less safety stock than do shorter intervals. This is because the larger reorder quantity, which is driven by the longer replenishment interval, provides some degree of “de facto” safety stock. By contrast, the formula in your post indicates that a longer replenishment interval effectively increases lead time, and therefore also makes the resulting safety-stock level higher.

Is your z value of 2.326 intended to represent a 99% service level? If so, how do you measure your actual service-level performance? If your actual measure is a quantity-based fill rate, such as units delivered on time as a percentage of total units (on-time delivery), then your z value causes a mismatch between the calculation’s event-based probability of no stockouts and your actual quantity-based measure.

Is your historical demand data actually normally distributed? In my experience, demand data is often significantly right-skewed, and is also often sporadic. If this is true of your data, and especially with a five-day lead time, then you cannot depend on the central limit theorem to justify using a normality-based z value.

Thanks!

David McPhetrige, TopDown Lean Systems