Safety Stocks: Beware of FormulasA 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."

Michel BaudinFascinated with the art of making things,

Michel is on a mission to improve it. Trained

in engineering and applied math, he got his

feet wet in production in the early 1980s,

and later apprenticed under master

Japanese consultant Kei Abe for eight years,

starting his own group in 1996.

He has been consulting since 1987,

teaching courses and writing technical

books. He intends to keep working with like-

minded partners in the Takt Times Group

and contributing improvements in the

management and technology of

manufacturing as a consultant, trainer, and

writer.

1. The safety stock formula for the reorder point methodSafety 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.

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:

Figure 1. The reorder point inventory model

Where:

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 level    C

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

2. Case study: Misapplication of the safety stock formulaThis 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.

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:

C*SQRT((AJ4*AL4^2)+(AI4^2*AM4^2))

Figure 2. Actual versus normal cumulative

distribution

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 σDxμL2 instead of μLxσD

2.

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.