## Introduction

Claims that particular software packages provide optimal inventory management are commonplace. Such claims, by themselves, are meaningless. They only become meaningful if there is a full description of the theoretical assumptions involved and of the objective function (whatever is to be minimised or maximised).

## Total Cost as the Objective Function

Total cost is the objective function which is used most often. It can be used in relation to a single item or the overall inventory. It should be the total of *all* relevant costs which includes, at least

- holding costs
- ordering costs
- shortage costs
- deterioration cost
- obsolescence cost

How close reordering is to optimal is probably limited by the least accurate of the types of costs involved. The main exceptions are that if neither deterioration nor obsolescence are major problems then inaccuracies in the costs related to them might not matter much.

Let’s look at those costs individually:

### Holding Costs

This is often expressed as a percentage, per annum, of the cost price. One of components is the opportunity cost of capital. This is often more that the interest rate minus the inflation rate, especially if there are difficulties raising more capital or if money is needed for other purposes or if there is a desire to reduce debt.

Each item needs storage space and, hence, contributes to the warehouse rental cost or the opportunity cost of the warehouse space involved which could, perhaps, be used for other purposes. Some items are more bulky than others and, consequently, treating the holding cost as a percentage, per annum, of the cost price is not always appropriate.

There is the cost of insurance of the inventory and of the warehouse. It might be appropriate to treat the insurance of the inventory as a percentage, per annum, of the cost of the inventory. The same does not apply to the insurance of the warehouse because bulky items have more effect on the required warehouse size than is the case for less bulky items. If there are dangerous goods then the insurance cost is affected simply by the fact that there are dangerous goods in the warehouse. The resulting effect on insurance premiums should be attributed to the dangerous items alone.

The fact that every item stocked needs to be counted from time to time is a cost which can probably, reasonably, be looked upon as a cost per item per annum.

The holding costs mentioned above are, perhaps, the main ones but not the only ones.

Holding costs are commonly treated as a percentage of cost price per annum in spite of the fact that doing so is not completely appropriate.

### Ordering Cost

This is often treated as being the same for each order placed regardless of whether or not it is the same for all items at all times. If an order is being prepared to send to a supplier for a number of items then the ordering cost per item ordered might be lower than if an order is to be placed for only one item.

If there is a quantity discount then first find the optimal order quantity without the discount being taken into account. If the resulting order quantity is less than that required for the discount, re-optimise with the order quantity set equal to that required for the discount and with the ordering cost reduced by the discount for that quantity. Compare the value of the objective function with the value obtained previously to ascertain whether or not the quantity discount should be taken advantage of.

### Shortage Cost

You might be thinking that you can’t ascertain accurately what the shortage costs are and that is likely to be correct to some extent. If the shortage cost is the least accurate then that is the cost which should have the most attention. Whenever you aim for a particular service level for a particular item, that indicates that you have some idea of the cost of shortages.

Most of the theoretical work in relation to inventory management which I have seen assumes that the shortage cost is a cost per unit short per day and that quantities which cannot be supplied to customers immediately are backordered. Under these circumstances, if the holding cost per unit per day is *h* then the shortage cost per unit per day (*s*) and service level (*l*) are connected using the following two formulae which are equivalent to each other:

\[l=\frac{s}{s+h}\]

\[s=\frac{hl}{1-l}\]

For example, suppose that the cost of an item is $300, and the holding cost is 18% per annum.

\[h=\frac{0.18\times300}{365}=$0.148\text{ per unit per day}\]

The shortage cost per unit per day which corresponds to a service level of 90% is given by

\[s=\frac{0.148\times0.9}{1-0.9}=$1.33\text{ per unit per day}\]

The service level which corresponds to a shortage cost of $1.33 per unit per day is given by

\[l=\frac{1.33}{1.33+0.148}=0.9=90\text{%}\]

Treating the shortage cost as a cost per unit short per day that the customer has to wait might be reasonably appropriate for a distributor. It is certainly not appropriate for a retailer because shortages usually result in lost sales. In the case of a retailer, the profit margin is a better indication of the shortage cost. That will vary from item to item. Treating it as a percentage of the cost price might be appropriate in some cases. However, inexpensive items are likely to have a higher percentage mark up than expensive items. Shortages affect the reputation of a company. Unfortunately, it is difficult to quantify the financial value of reputation.

### Deterioration Cost

This is the cost of an item multiplied by the probability that it will not be sold before its expiry date.

### Obsolescence Cost

This is the expected cost resulting from an item not being sold before it becomes obsolete. This is usually difficult to model although history of items becoming obsolete provides an indication of the magnitude of this cost.

## Other Objective Functions

Profit can be used as an objective function. All of the costs mentioned above should be taken into account except that, instead of forfeiture of gross margin being treated as a component of shortage costs, the gross margin on sales should be treated as a profit.

Overall service level and total inventory value are often used as indicators of inventory management performance. However, the overall service level which can be achieved with a given investment in inventory or the investment in inventory required to achieve a given overall service level are not suitable for use as objective functions. Optimising on this basis would result in a small total inventory value in relation to the overall service level provided but the profitability could be adversely affected. To illustrate, on this basis, a car dealer would stock spark plugs but not cars. The reason for this not being appropriate is that the profit on the sale of a car is far greater than the profit on the sale of a spark plug.

## Effects of Theoretical Assumptions

The usefulness of the results of any optimisation depends of the validity of the theoretical assumptions involved.

### Case Study – Economic Order Quantities (EOQ)

The classical EOQ formula is

\[\text{EOQ}=\sqrt{\frac{2DK}{h}}\]

where

*D* = demand per annum,

*K* = cost of placing an order and

*h* = annual holding cost.

It is obtained by minimising the total of the ordering and holding costs in a continuous review system. Those are only two of the five costs mentioned above. It does not take into account

- the fact that the order quantities affect the service level,
- the fact that, for a perishable item, part of the quantity purchased might expire before it is sold,
- the fact that the item might become obsolete before all of the quantity purchased is sold and
- quantity discounts.

The demand rate is assumed to be constant forever.

As mentioned above, it is assumed that a continuous review system is used. If a periodic review system is used then the order quantity should depend on the quantities on hand, on supplier order and on customer backorder.

## Conclusion

Do not be fooled by claims of optimality unless the assumptions involved and the objective function are stated. If they are stated then ascertain the applicability of the assumptions to your company and check to see that all of the relevant costs have been taken into account..

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