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]]>The instantaneous overall service level mentioned above is, roughly speaking, equal to the percentage of *fast moving* items which are in stock. To be precise, it is the sum of the demand transaction rates for all items which are in stock divided by the same type of sum for all items. As mentioned above, the instantaneous service level can change quickly. For that reason, I suggest that the service level be measured by dividing the sum of the square roots of the demand rates for items which are in stock by the same type of sum for all items. This will give a more stable estimate of the overall service level.

In many organisations, especially retailers, it is not possible to capture customer demands. Fortunately, the method of measuring the service level described above is not normally greatly affected by using sales data instead of demand data.

There is one important limitation. That is that the computer records need to give a reasonably reliable indication as to which items are in stock. Methods of achieving that reliability will be given in later articles.

There is open source software available for measuring overall service levels in the manner described in this article. For information concerning that software, click here.

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]]>Click on the appropriate link below depending on what spreadsheet software you are using:

Open document format (for use with modern spreadsheet software)

Excel 97 format (for use with pre-2013 versions of Microsoft Excel)

You will then see a spreadsheet showing the weightings given to the demand in each period. You can download it and open it with your own spreadsheet software. Alternatively, you can view it online by clicking on “Open”. If the periods are months then the spreadsheet will show the weightings given to the demand last month, the second to last month, the third to last month, etc. The weighting given to the demand in the most recent period is called the “smoothing constant” or “smoothing factor” or “smoothing coefficient” (α). The periods can be any length of time, e.g, months or weeks or days. Try changing the smoothing constant in the yellow cell (Cell B3). Fig.1 below shows what you will see for a smoothing constant of 0.1:

Fig.1 – Exponential smoothing weightings with a smoothing constant of 0.1

Suppose that it is to be used to average the monthly demands for an item and that average is to be updated each month. That average can be treated as a forecast if the demand rate is reasonably stable.

The technique is as follows: At the end of each month, the average demand per month is adjusted taking into account the most recent month’s demand. Suppose, for example, the smoothing constant is 0.2 (i.e. 20%). Then 20% of the weighting is given to the demand in the most recent month and 80% of the weighting is given to the old average, i.e.

New average = α(latest month’s demand) + (1 – α)(old average)

This formula can be applied using this calculator.

If the old average is 30, the latest month’s demand is 40 and the smoothing coefficient is 0.2 then

New average = 0.2 x 40 + (1 – 0.2) * 30

= 0.2 x 40 + 0.8 x 30

= 8 + 24

= 32

Here is another way to look at it: For α=0.2 (20%), the average is adjusted by 20% of the error in using the old average as a forecast of the latest month’s demand (i.e. by 20% of the difference between the new demand and the old average).

New average = old average + α(error in the forecast of the latest month’s demand)

i.e. New average = old average + α(latest month’s demand – old average)

which is equivalent to the first formula.

Using the above data,

New average = 30 + 0.2 x (40 – 30)

= 30 + 0.2 x 10

= 30 + 2

= 32

In this case, if the old average had been used as a forecast of the latest month’s demand then that forecast would have had an error of 40-30, i.e. 10. The average is then increased by 20% of that forecasting error to give a new average of 32.

The exponential smoothing formula can be applied using this calculator.

If the smoothing constant is small then the estimated demand rate will take a long time to catch up with changes in the demand rate. If it is high then the demand rate will tend to be sensitive to random fluctuations in demand. Consequently, choice of a smoothing constant involves a compromise. Ideally, the smoothing constant should be set by means of simulation of the effects on your overall service level and overall investment in inventory.

The following formula will usually give a reasonably appropriate smoothing constant (α):

α = 1/(4*L* + 1)

where *L* is the expected lead time in the event of the inventory position falling below the reorder point just after a reorder review. The above formula can be applied using this calculator. The formula is compromise between keeping up with a changing demand rate and being overly sensitive to random fluctuations in demand. The lead time and reorder review period (time between reorder recommendations reports) should both be in months if the exponential smoothing is applied to monthly demands, weeks if weekly demands are used, etc.

Suppose that the lead time, ignoring the reorder review period, is two months and that reordering recommendations are produced monthly. Suppose that the total of the other components of the lead time is one month. Then an appropriate smoothing constant is given by

α = 1/(4(2 + 1 + 1) + 1)

= 1/17

= 0.058

The (2 + 1 + 1) in the above example is the supplier lead time (2 months) plus the reorder review period (1 month) plus the other components of the lead time (1 month).

Try using the calculator with the above worked example. If a month is treated as being 30 days then the answer will be 0.0667.

Don’t be tempted to use a higher smoothing constant to keep up with trends. There are better methods of dealing with that and one of them will be discussed in a later article.

The simulator described in the article entitled “An Educational Inventory Management Simulator” can be used to investigate the effects when used with your own data. Try entering your own demand history into that simulator for at least one of your fast moving items and at least one of your infrequently moving items. In each case, try more than one smoothing constant including the one suggested by the above-mentioned calculator and one which is considerably higher than that. Note the large orders which tend to result from relatively high demands and note how that problem is exacerbated by use of a high smoothing constant.

Fig. 2 below illustrates the effect of using a smoothing constant of 0.5 if the lead time is three months.

Notice how erratic the estimated mean demand is in spite of the fact that the simulation was carried out with a constant demand rate (constant mean demand per month).

The “inventory level” is the stock on hand minus customer back orders so yellow points above the axis indicate stock on hand and those below the axis represent customer back orders.

Notice the excessively high order quantities in months 7, 26 and 30 resulting from the relatively high demands in months 5, 24 and 28. Not all peaks in demand cause orders to be placed on the supplier. This is because ordering only takes place when the inventory position falls below the reorder point. For example the relatively high demands in periods 21 and 37 do not result in ordering because at those times the inventory levels are already high as a result of earlier over-ordering. With regard to the shortages early on, see the section entitled “Initialisation” below. These shortages could be prevented by the means of safety stock but that would make the problems of over-ordering and of excessive stocks even greater.

Fig.3 below shows how much more stable the demand rate estimate is with a smoothing constant of 0.1. Notice however that there is still the problem of over-ordering after relatively high demands. The reasons are given in the article entitled “Evaluating Forecasting Algorithms“.

Fig.2. – Effects of a high smoothing constant (0.5)

Fig.3 – Effects of reducing the smoothing constant to 0.1

In Figures 2 and 3,note the prolonged shortages when the item is new. This problem can be dealt with by manual entry of the initial average demand per month.

- It uses all of the demand history, even that which is no longer stored in the system.
- It will not result in zero demand rate estimates if there has ever been any demand.
- The greatest weightings are given to the most recent demands.
- It facilitates manual intervention when the demand rate is expected to change.

- As with most demand rate estimation techniques and forecasting techniques, relatively high demands tend to result in over-ordering.
- Selection of a smoothing constant involves a compromise between responsiveness and stability.
- The optimal smoothing constant is not the same for all items.
- The algorithm requires modification if there are trends or seasonal effects.

The initial old average demand per period needs to be set. If the exponential smoothing is implemented when there is already some demand history available then all of that history should be used in order to ensure that the initial average is not zero if there has ever been any demand.

Use whatever relevant information you can obtain from the supplier and also any other relevant data. In a multiple store operation, if an item is introduced into one store when there is already some history of the item in other stores, then appropriate use should be made of that history.

If no useful information is available then the smoothing “constant” (better called the “smoothing factor”) should initially be fairly high but *only for a short time*.

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]]>Any row which is concerned with only partial fulfilment of a purchase order is weighted accordingly.

For each row, for the purpose of testing the null hypothesis that the dates are correct, I assumed a lognormal distribution of supplier lead times, an example of which is shown in the following graph:

The estimation of the natural logarithms of the mean and standard deviation of the lead times was done separately for each row using all rows except the one concerned. That was necessary in order to prevent outliers or multi-modal lead time distributions from severely adversely affecting the hypothesis testing. Another thing which I did for the same reasons was to base the estimate of the standard deviation of the logarithms of the lead times on mean absolute deviation (MAD) because of its robustness.

Cells L4 and Q5 were calculated using the fact that the variance of the sum of independently and identically distributed random variables is equal to the sum of the variances of the individual variables. For manually set mean supplier lead times, it is assumed that the amount of historical data used in setting them is the same as in the spreadsheet.

The testing which I have done appears to show that the techniques which I have used are effective at ensuring that suspect lead times are flagged, either immediately or after the initially flagged rows are checked and corrected.

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]]>Transaction history held in your computer could be used for the lead time analysis if it is sufficiently reliable. Unfortunately, in most companies, it is not. Many ERP systems do not keep records of order dates and do not match up goods received with the purchase orders. It is essential for the records of order dates to be reliable if they are to be used for lead time calculations. Unless the dates of order placement and of receipt into store are reliable, I suggest that sample data be used. The sample should be small enough to enable it to be checked. Use of a small amount of reliable data is better than use of a large amount of unreliable data.

For the above-mentioned reasons, a spreadsheet analysis is appropriate. I have developed a spreadsheet for this purpose. There are four versions. The appropriate version should be downloaded by clicking on the appropriate link below. When the spreadsheet appears, click on the three dots near the top right hand corner and then click on “Download”. You will probably find that an “Enable editing” button appears, in which case, click on it.

Open document format with UK dates

Open document format with US dates

Excel 97 format with UK dates

Excel 97 format with US dates

These were updated on 19 February 2017.

Almost all modern spreadsheet software should be able to handle the open document format versions. It is recommended that you use that version unless you are using a pre-2013 version of Microsoft Excel, in which case, it is recommended that you use an Excel 97 version of the spreadsheet. All of the spreadsheets were developed using Libre Office.

The yellow cells are where you can enter data. The spreadsheet calculations cannot work with fewer than three rows of data. Some cells will show errors until three rows have been entered. The supplier (Cell B3) is just for your information. The item code (product code) can be entered in Columns A and B respectively for your information. Enter the order date and date received in Columns C and D respectively. It is important for these dates to be correct. In Column F, enter the quantity received in the shipment concerned. There is no need to enter the “order quantity” (Column E) if it is the same as the quantity received.

The “Comment” column (Column I) indicates action which needs to be taken.

The “rejection confidence” (Column H) is an estimate of the confidence with which it can be said that the row concerned contains an incorrect date. If it is greater than 90% then the dates in that row should be checked thoroughly. This situation is indicated by the word “Check” in the “Comment” column (Column I). The most important things to check are that

- the order date as shown in Column C is correct and
- the date received as shown in Column D is correct.

Use Column J (“Checked”) to indicate that the data in the row concerned has been checked thoroughly. Any corrections which you make will result in recalculation of all of the rejection confidences. As a result, some more rows might need to be checked. For that reason, it is important for you to use the “Checked” column (Column J) so that you know which rows have already been checked.

The results of the analysis are shown at the top of Columns K to Q. The spreadsheet is concerned with the supplier lead times (i.e. from order placement until receipt) which are only one component of the lead time. The lead time starts when the inventory position of the item concerned falls below the reorder point and ends when the replenishment stock is available for picking. The coefficient of variation is the standard deviation of the supplier lead times divided by the mean supplier lead time. However, when setting safety stocks, it is the standard deviation of the errors in forecasting lead times which should be used. This is higher than the standard deviation of the lead times. Consequently, the number in Cell L5 or Q5 should be used instead of the coefficient of variation in practice. The number in Cell Q5 is for use in relation to items for which the mean lead time has been set manually rather than using the mean supplier lead time in Cell L1. If less historical data is used in manual setting of a lead time than the amount of data in the spreadsheet then the number in Cell Q5 will be lower than it should be. When using the online Monte Carlo simulator which is accessed by means of the “Simulator” tab in the menu, the number in Cell L5 or Q5 as appropriate should be entered in place of the coefficient of variation.

Part 2 of this article will contain information concerning the mathematics involved in the spreadsheet and is intended for readers who have some knowledge of statistical mathematics.

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]]>In order to reduce investment in inventory and improve the service level, there are a number of things which need to be looked at. Initially, it is better to tackle all of the important ones to some extent than to just concentrate on a few of them. More work can then be done in those areas which will lead to further cost-effective improvement. Things which will result in improvement include the following:

- Ensure that the data used for inventory management purposes is accurate and up to date.
- Ensure that the computer provides adequate and appropriate exception reporting.
- Ensure that any problems are identified and dealt with when they occur.
- Reduce the average effective lead time.
- Reduce the amount of variation in lead times.
- Improve the forecasting of lead times.
- Use the supplier as the stockist when appropriate.
- Improve the forecasting of demands.
- Do not aim at providing the same service level for all items. Instead, for each item, take into account
- the effects of shortages,
- the effects and likelihood of losing sales,
- the cost of providing a high service level for the item concerned,
- whether or not customers normally need the item urgently,
- whether or not the item can be obtained quickly in an emergency,
- whether or not there is an alternative item.

- When setting safety stocks, take into account
- the feasibility and costs of obtaining additional supplies quickly when necessary for the purpose of preventing shortages,
- the effect of the order quantity on the service level and
- the risks of obsolescence and deterioration.

- When setting order quantities, take into account the risks of obsolescence and deterioration.
- When considering cost saving measures such as quantity discounts and consolidation of orders, take into account the effects on the investment in inventory.

I will now elaborate on several of the above.

** 1. Ensure that the data used for inventory management purposes is accurate and up to date.**

The reasons for this are obvious. However, achieving this objective in a cost-effective manner is far from being simple. See the article entitled “Data Accuracy“.

** 2. Ensure that the computer provides adequate and appropriate exception reporting.**

Whenever, for any reason, action needs to be taken, that fact should be brought to someone’s attention promptly, thereby facilitating prompt appropriate action. However, too many false alarms can greatly reduce the benefits of exception reports.

** 3. Ensure that any problems are identified and dealt with when they occur.**

This can be greatly facilitated by good exception reporting (See “2” above). The exception reporting needs to be prompt and acted upon promptly. Adequate monitoring of indicators of overall inventory system performance should also be carried out. Such indicators include total inventory value, total value of orders on suppliers and customer service level. The reasons for any unexpected changes in any of these should be investigated thoroughly.

**4 Reduce the average effective lead time.**

The lead time is the time taken to obtain replenishment supplies. It effectively starts when the item should be ordered. It effectively ends when the replenishment supplies are available for issue. The greater the lead time, the greater are the errors in forecasts of lead time demands and, consequently, the required amount of safety stock. The risks of being left with obsolete stock and of deterioration are also affected. See the articles entitled “Reducing Lead Times“, “Reducing Reorder Review Periods” and “Reducing Ordering Delays“.

**5. Reduce the amount of variation in lead times.**

Uncertainty in future lead times contributes to the uncertainty in the lead time demand and, consequently, to the required safety stock.

**6. Improve the forecasting of lead times.**

Part of each reorder point is the forecast lead time demand. In order to forecast that, one of the requirements is a forecast of the lead time. Averaging recent lead times is helpful in this regard. However, if for any reason, future lead times are likely to differ from those in the past, this needs to be taken into account. If there have been few purchases of the item from the supplier then the past lead times are likely to be of little use. For that reason, the historical lead times of all items obtained from the same supplier might provide useful information. As in any forecasting, it is better to forecast the distribution of future lead times rather than producing a forecast as a single number. For example, if, on average, the lead time for an item is 15 days but it is longer than 40 days on 10% of occasions, that information is useful. Unfortunately, what can be done in this regard is probably limited by data availability.

**7. Use the supplier as the stockist when appropriate.**

This benefit of this in terms of your investment in inventory is obvious.

**8. Improve the forecasting of demands.**

If the demands are, to some extent, predictable then you should be able to achieve a very high turnover ratio (i.e. a small amount of inventory in relation to sales). If the demands are unpredictable, as is usually the case, then they should be forecast as accurately as possible given the information available. Demand history, market knowledge and market indicators should all be used as appropriate. Good results can, as a general rule, only be achieved by choosing a forecasting technique on the basis of the peculiarities of your own company. Use of whatever forecasting technique is supplied with your inventory software will not usually produce good results, even if it is an adaptive technique (such as focus forecasting). As with lead time forecasting (See “6” above), it is helpful to produce each forecast as a distribution rather than a single number. In relation to forecasting of demands, see the articles entitled “Evaluating Forecasting Algorithms” and “Demand Rate Estimation“.

**9. Do not aim at providing the same service level for all items.**

This is usually a major source of potential improvement. The cost of providing good service is not the same for all items. Also, the effects of shortages are more serious for some items than for others.

**10. Setting of safety stocks**

Safety stock is the stock which is held in case of higher than expected demand and/or a longer than expected lead time. Appropriate setting of safety stocks is the main, but not the only, means by which the service levels are controlled. Improving the manner in which they are set is usually a major source of potential improvement.

**11. When setting order quantities, take into account the risks of obsolescence and deterioration.**

For some types of items, this is particularly important. Large order quantities can result in stock being on the shelves for a long time. This can be a problem for perishable items. Also, if an item becomes obsolete, there could be a substantial amount of “dead” stock as a result. One of the shortcomings of the classical “economic order quantity” formula is that it does not take these problems into account.

**12. When considering cost saving measures such as quantity discounts and consolidation of orders, take into account the effects on the investment in inventory.**

Such cost saving measures are often very tempting. They should only be used after a realistic analysis of both the short and long term effects on your inventory levels.

No mention has been made so far in this article about distribution through multiple stores. Achieving near optimal inventory management under these circumstances is far more complex than in a single store operation. However, the principles mentioned above can be extended to multiple store operations. In a multiple store operation, several of them are often even more relevant. For example, if a store normally obtains an item directly from the supplier, then the most appropriate method of obtaining the item in an emergency might be to obtain it from another store.

Inventory management improvement should be tackled on many fronts. That will usually produce much better results than concentrating on just a few improvements. An exception is if those improvements have been identified as the ones which will have the greatest effects.

It should now be apparent that good inventory management is far from being simple and that there are many inter-related things to be considered.

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]]>Click here to obtain a copy in Open Document Format. That is a standard format which can be read using almost all modern spreadsheet software. If you are using a pre-2013 version of Microsoft Office then click here instead to obtain an Excel 97 version of the spreadsheet. That version has not been tested as thoroughly as the Open Document Format version.

You cannot use the spreadsheet online so you will need to download a copy. To do so, click on the icon with three dots near top right hand corner of the browser window and then click “Download”.

If you encounter any problems then please contact me and let me know what spreadsheet software you are using. You are least likely to encounter problems if you use Libre Office which is a free download.

The yellow cells are the cells in which you can enter data.

Cell E4 should contain the number of the forecasting algorithm which is to be used. At present it should contain 1 which is exponential smoothing. The Croston, Croston Median and Holt-Winters algorithms are to be added later.

In Cell E5, enter the desired smoothing constant for exponential smoothing. The smoothing constant is the weighting to be given to the most recent period’s demand and should be between 0 and 1.

In Cell E6, enter the lead time in periods starting from when the order is sent to the supplier. It must be 1 or 2 or 3 or 4 or 5 because of the limitations of spreadsheets. The length of a period can be anything you like (e.g. month or week or day). At the beginning of each period, if the inventory position is strictly less than the reorder point then an order is sent to the supplier to increase the inventory position to the “maximum” level. The inventory position is the quantity on hand minus the quantity on customer back order plus the quantity on supplier order. The lead time is treated as being deterministic (fixed). If you want to use stochastic (varying) lead times then you will need to use the online simulator.

In Cell E7, enter the reorder point expressed as a number of periods supply,

In Cell E8, enter the “maximum” level expressed as a number of periods supply. It should not be less than the entry in cell E7.

In Cell E9, enter the mean demand per period to be used when the spreadsheet is used to carry out Monte Carlo simulations.

If no entries are made in Column C, Monte Carlo simulations will be carried out. Changing the contents of a yellow cell or re-entering what is already there will result in a new Monte Carlo simulation. If you want simulations to be carried out using your own demand history, enter that history in the yellow cells in Column C, starting from the oldest period. Deleting Cell C12 will result in a return to Monte Carlo simulation. The Monte Carlo simulation of demands is carried out in the same manner as in the online simulator.

In Cell D12, enter the assumed initial estimate of the mean demand per period.

In Cell E12, enter the assumed starting inventory level (stock on hand minus the quantity on customer back order).

Columns E to J show the situation at the beginning of each period.

The estimated mean demand per period is updated, using exponential smoothing, at the end of each period

The results of the simulation are shown in the two graphs and in cells L5 to P6.

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]]>The simulations are carried out on a daily basis and continue until there have been at least 1000 customer demands.

In the event of a shortage, customer backordering takes place. I intend to allow for lost sales at a later date.

The daily customer demands are assumed to be independent and identically distributed negative binomial random variates. That type of distribution is a realistic one for modeling daily demands. It is a type of stuttering Poisson distribution. In other words, the daily number of customer requests is treated as having a Poisson distribution and the quantity requested by each customer can vary. The Poisson distribution is a special case. For items which move fairly fast, the negative binomial distribution can be approximated by a gamma distribution. For very fast moving items, it can be approximated by a normal distribution.

The figure below shows the probability function of a negative distribution with a mean of 1 and a standard deviation (a measure of spread) of 1.8. It is typical of the probability function of the single period demand of an item for which the mean demand per period is 1.

The figure below shows the probability function of a negative distribution with a mean of 1 and a standard deviation of 5.3. It is typical of the probability function of the single period demand of an item for which the mean demand per period is 6.

From the above two graphs, it can be seen that the normal distribution is not suitable as a model of demands for slow moving items. Customer demands distributions are skewed and negative demands would not normally occur.

The standard deviation of daily demands is modeled using the equation

\[\sigma=a\mu^b\]

where is the mean daily demand and *a* and *b* are constants. This model performs remarkably well in practice. That is, perhaps, not too surprising considering that the standard deviation of a Poisson distribution is the square root of the mean and that daily customer demands tend to have higher standard deviations than for a Poisson distribution. The reasons for being higher are that some customers might request quantities of greater than one and sometimes customers might arrive in groups.

Supplier lead times in days are also assumed to be independent and identically distributed negative binomial random variates.

Reordering takes place at any reorder review at which the inventory position is less than the reorder point. The quantity ordered is then that which is required to increase the inventory position to the “maximum”.

Measurement of the service level and the mean stock on hand in days supply takes place after the simulation has had time to settle down.

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