Lesson 1, Topic 1
In Progress

4.6. A system for managing stock in a business unit is applied with reference to ensuring sufficient stock, control of costs and the value of the stock.

ryanrori January 14, 2021

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Stock Control Mathematics

Quantity discounts

The order cost is spread over more items (larger batches) with discounts! But there are higher holding costs

  • orders arrive in one batch with a known lead time arriving just as stock is exhausted.
  • some order placement/receiving costs are independent of order quantity, others are dependent.
  • economies or diseconomies of scale do not apply to holding stock (assumption = holding costs are a fixed % of stock value). Order quantities should not exceed shelf-life expectations, room for storage or what is affordable (eg. at the end of a year – a school or hospital department may buy small (retail) quantities, as the budget is depleted).
  • someone can buy 10000 in error when usage is in 10’s.
  • EOQ assumes good information on variable costs e.g. we know that by placing few big orders, R950.00 per order, say, can be saved or that with smaller quantities the opportunity cost of tied up capital and associated warehouse space costs are saved. But such data on actual costs may be at best a good estimate. This is less of a problem as EOQ is not very sensitive to error so long as the magnitude of the cost/demand data is reasonably accurate.

Safety stock and service levels

We may run out of stock because of a re-supply delay or higher than anticipated usage. If we can predict demand then we merely place EOQ orders on time. The figure shows the predictable, timely re-ordered stock movements.

But we risk a stock-out with unpredictable demand, usage and resupply so introducing a safety or buffer stock reduces the risks of variable demand/lead time.

Lead time

– the time between a replenishment need arising and new deliveries being ready for use. It includes e.g. time to

  • detect then authorise replenishment
  • establish supplier contact and complete admin/paperwork
  • obtain, produce and have the goods delivered
  • goods inwards/receiving and quality checking time

Service Levels

Rather than guarantee 100% stock availability for any foreseeable need, inventory managers would normally agree an inventory service level (% probability of stock availability to meet demand). This mediates the estimated costs of stock-outs with the cost of carrying a safety reserve. If demand and lead times are normally distributed, the safety stock formula is:

Safety stock levels

Calculating safety stocks requires understanding of demand and lead time. Assuming that these are normally distributed then

Safety stock = ( L Dv+D 2Lv )

  • L = mean lead time, D = mean demand (in the lead time) Lvar = variance of lead time, Dvar = variance of demand


  • For a safety stock level in a service agreement, an Sdev (standard deviation) value of 1.6 gives 95% stock availability and 2.3. gives 99%.

The controllable aspects of lead time should be investigated. It is seldom that normally distributed and improved control over the length and variability of lead time will reduce the need to maintain safety stocks.

Serving items from stock

In order to serve materials from stock, replenishment must occur to replace materials that have been issued or are known to have future requirements.

Economical order quantity EOQ

For every item an economical reorder quantity EOQ can be calculated, that will lead to the lowest total cost of ordering and stockholding. The calculation of the EOQ is based on general economical conditions such as consumption, cost of ordering and cost of holding the item in stock.

However, the EOQ calculation does not include considerations such as maximum stockable quantity, limited shelf-life, dimensions of the material other than the consumption unit, or difficulties in obtaining the material. When the item is re-ordered, other constraints such as the supplier’s minimum, and multiple, are included in the re-order quantity.

Reorder point ROP

Re-ordering of items takes place whenever the available stock quantity drops below a certain level, the re-order point (ROP). This reorder point ROP represents the buffer quantity of items required to ensure that material requests will continue to be served, during the time between the ordering of new material and the arrival of the ordered new material. The time between the moment a proposition for re-ordering of new material and the availability in stock of the ordered new material is the so-called re-ordering delay or lead-time.

The re-order point ROP is calculated as the sum of 2 components

* A stock reserve or buffer stock quantity SR which is based on the normal average expected consumption during the lead-time.

 * An extra quantity, the so-called safety or security stock SS, to ensure that the item is available up to the pre-defined required service level, even when the re-ordered material arrives later then expected, or the fluctuations in demand during the lead-time cause the demand to be larger then expected.

The safety stock is the most important component of the re-order point. The average stock value is determined for a large part by the safety stock. Good safety stock ensures a high service level.

Reorder cycle and lead-time

The lead-time of the re-order cycle is composed of the following components:

1.   Review time needed to confirm a replenishment proposal.

2.   Time needed to process and send an order.

3.   The supply time or delivery time, defined as the time needed for the supplier to deliver the requested materials after the order has been received. This time includes assembling or manufacturing, when needed.

4.   Inspection and entering the item to stock.

Both the stock reserve SR and the safety stock SS quantity components of the reorder point ROP are very dependent on the delivery time and the reliability of the supplier.

If the delivery time changes, this will highly influence the re-order point and thus the moment of re-ordering.

Serving material requests by direct delivery from the supplier

Economical considerations such as short delivery time of the supplier, or low demand frequency of expensive items may lead to the decision not to keep the items available in stock, but to handle requests for these items only by direct delivery of the item from the supplier. Direct ordering for non-stocked items can be handled electronically by electronic data interchange EDI or normal ordering by purchase orders.

Exceptional large non-recurrent demands

Occasionally, material requests for exceptionally large quantities will disturb the servicing of material requests for normal quantities.

Exceptional requests handling

Correct handling of these non-recurrent or infrequent exceptional requests is very important. The penalties in costs and service level can be very high if these exceptional large demands are not correctly identified and handled.

Serving a demand for an exceptionally large quantity from stock will generally lead to stock-out, and backorders for the normal requested quantities.

In order to avoid this the material requests for an exceptional large quantity can be handled by a special order.

The infrequent or non-recurrent requests for exceptionally large quantities often originate from capital projects, where these material requests are planned well in advance. The non servicing of these planned materials on the requested delivery date, may disturb the project planning.

Exception threshold level

Requests for exceptional large quantities are identified by comparison of the requested quantities with a so-called exception threshold level.

If the demanded quantity is larger than this threshold level the request will be considered as an exceptional demand.

Occasionally a request for an exceptional quantity of an item can be served normally from stock, without endangering serving of the normal requests for the item. This is the case when after serving the exceptional quantity at least one re-order point is left.

Handling of requests for exceptional large quantities involves:

*    Check if serving the exceptional quantity from stock will disturb the normal stock keeping.

*    If the exceptional quantity can not be served from stock, a special order is created for the material request with a delivery date, such that the delivery date of the material request can be met. Notify requester if this date can not be met.

Exceptional requests and safety stock

If the historical exceptionally large requests are not correctly identified for the statistical calculations, they will dominate the normal statistical requests.

The statistical domination of the normal requests by the exceptionally large non-recurrent demands may result in very high stock values caused by unnecessary high security stocks.

NOTE: Careful definition of the exceptional threshold value is very important: it has a direct influence on:

* The actual stock operations

* The service level

* The stock value

Logical constraints

The stockholding, reordering and exceptional handling of items are controlled by a number of logistics parameters: the reorder point, economical order quantity and exception threshold level. These logistics parameters are calculated based on a mathematical model, representing an approximation of the real world, and statistical information from a number of sources such as historical consumption, material needs of future capital projects, and the estimates of the various costs of the steps and processes involved. However, the values calculated for the logistic parameters can never be more accurate than the model involved and the information used. The statistical standard model based on historical consumption will normally give very satisfactory results in 95% percent of the cases. However, in exceptional non-standard situations the normal statistical information (historical consumption) may be insufficient and lead to unsatisfactory results, e.g. in cases like:

*    You are informed by the supplier that the delivery time (supply time) for a given item or range of items will change.

      e.g. your supplier is not stocking the item anymore in the future, or there is a temporary production failure.

*    The product is technically sensitive: a new version is due after a certain date, or the old version is not valid anymore after a particular date (e.g. computers, calendars, agendas, etc.).

*    There is a certain duration of life, limiting the shelf-life of the product (e.g. ink, markers, medicines, etc.).

*    There is a non-statistical, expected growth, which is known to happen in the future.

*    The dimensions of the requested piece of raw material may be more important than the quantity requested as criterion if this request can be satisfied from a given unit of raw material or not. For raw materials the statistical reorder point quantity may not be very meaningful as a means to express if a given request can still be served from this quantity or not (see picture).

The dimensions of the requested piece of material may be such that the request can only be served from a complete stock unit of the material.

In order to cope with special situations, such as raw materials, limited shelf-life, change of delivery time, etc. additional information can be entered in the form of logical constraints

* item shelf-life

* New version date

* Maximum stockable quantity

* Supplier delivery time + valid date

* Lot quantity

* Minimum level

* Exceptional threshold level for direct delivery

* Expected growth factor + valid date

These logical constraints will influence the moment and quantity of the reorder proposition, or the handling of the material request.

The quantity that is proposed to reorder

* Shelf-life (indirect by expected consumption in shelf-life)

* Lot quantity

* Maximum stock

* Expected growth

The moment in time on which a proposition for re-ordering is created

* Supplier delivery time (indirect by new reorder point ROP)

* Minimum level

* New version date

* Expected growth

The handling of the material requests