Inventory Models

Inventory (stocks) is an integral part of every business operation. Inventories occur in all forms and for the most diverse purposes. In applications involving inventory, the manager must answer two important questions: 1. How much should be ordered? and 2. When should it be ordered? The inventories are normally considered as goods for sales, raw materials for production, work in progress held for later production stages, and fi nished goods for supporting activities and customer service. They need to be controlled for being in limits before they become liabilities. For many organizations inventories are a major investment. Inventory management is an important function in many organizations even in the Internet age. The fundamental questions in inventory control are when to order and how much to order. A fi rm has to keep some stocks of raw materials to enable uninterrupted production operations. Depending on the nature of production operations, it may keep small or large volumes of semi-processed stocks for quick conversion to fi nished goods. The fi rm has to keep stocks of fi nished goods to meet the needs of customers on demand.

 The need for inventory depends on the following factors:

  • Variations in demand for the fi nished goods
  • Variations in production lead time and production rates
  • Variations in raw material supply lead time
  • Demand and capacity conditions
  • Loss of customers and goodwill due to shortages or delays

There are three basic motivations for keeping inventory:

  1. Transaction motive: This is the desire to ensure that the business of meeting demands on time (sales) is carried out effi ciently. In other words, to keep suffi cient stocks so that no sale is lost and not too much stock remains unsold.
  2. Precautionary motive: It means the need for protection against uncertainties in demands, in lead times of production, purchase and distribution. Market demands are uncertain. Should the demand over a period be more than what was forecast, then to avoid losing the extra demand some extra (safety or buffer) stocks would be desirable. Similarly, if the lead time for receiving raw material or producing goods or distribution should be more than normal, then extra stocks would be desirable to cover the extra delays in time.
  3. Speculative motive: At a time of rapid changes in prices or supplies, it may be desirable to increase or decrease the stock holdings to gain some advantage. If prices are likely to fall in the near future, then stocks may be run down to the barest minimum and replenished to normal levels later availing the lower price. In contrast, if prices are likely to increase, stocks may be built up at the lower current prices; similarly, with respect to supplies, stocks may be increased or decreased.

As regards costs involved in inventory systems, there may be four different types of costs to be considered in a general inventory:

  1. Holding Cost: These are the costs associated with having possession of inventory, and the components are:
    • Cost of money tied up in inventory (opportunity loss of capital tied up or interest paid on capital borrowed). Usually this component will be the major one, of the order of 20 per cent per year
    • Cost of storing the inventory (warehouse rental/depreciation and maintenance)
    • Rates, taxes and insurance
    • Security
    • Loss due to pilferage/shortages
    • Loss due to obsolescence and deterioration
  2. Ordering Cost: These are costs incurred in acquiring inventory—purchase from external sources or in-house production. The components are:
    • Cost of information processing on inventory status
    • Cost of negotiations with suppliers
    • Cost of transmission of order
    • Cost associated with the receipt and inspection of stocks
  3. Shortage Costs: These costs are relatively more difficult to assess. Yet they may be very perti­nent to the decisions on inventory.
    • Cost of losing sales due to non-availability of item, that is, profit that was not earned if there could have been a sale
    • Cost of back order—in case the sale is not lost, that is, the customer is prepared to wait for some time to receive the stock
    • Cost of goodwill—if the firm frequently goes out of stock in the market or frequently has to back-order, in the long run the customers may not be impressed with the service and may forever be lost for all future sales potential
  1. Cost of Usage or Consumption: This is the total purchase cost of the items or the total cost of production.

In the study of inventory, decisions have to be taken as to:

  • How much stock to buy (or produce)?—Q factor (quantity)
  • When to buy or (produce)?—P factor (time)

The conditions and assumptions under which the above two decisions are taken may vary from situation to situation. Each situation leads to a model of the inventory system. Many models—from the very simple to the most complex—have been developed and studied, depending on the specific needs of the operations of the firm. These models can take the following forms:

  • Q fixed, P fixed (fixed order quantity policy)
  • Q fixed, P variable (regular replenishment of a fixed quantity)
  • Q variable, P fixed (optional replenishment policy)
  • Q variable, P variable (general replenishment policy)

1. Basic Stock Model—Economic Order Quantity Model (EOQ)

This is the earliest model developed. While it has a very limited use in practice, yet the model is informative to understand the interplay of factors that influence inventory decision. It can be used in certain restricted areas of purchase of general-purpose consumables where there are many sup­pliers of standard products. The assumptions in EOQ model are:

  • The demand for the item of inventory is known with certainty. The rate of usage of the item is nearly constant over time
  • The lead time for supply is known and constant
  • Stockouts are not allowed
  • Delivery of supplies is instantaneous in one lot, that is, the time to receive the stock is very short
  • The price of the item is independent of the quantity ordered. In an EOQ system the stock is very short
  • The price of the item is independent of the quantity ordered. In an EOQ system the stock levels will repeatedly fluctuate as shown in the graph

In the literature, this formula is sometimes referred to as Wilson’s formula.


D: Annual usage of the item (in units) Co: Cost of ordering (Rs. per order)

Cc: Cost of carrying (holding) (Rs. per item per year)

P: Unit price of goods (Rs. per unit)

I: Inventory-carrying cost (expressed as per cent per annum), that is, marginal rate of interest on working capital finance

Q: Order quantity

N: Number of orders per year

Q*: Optimal order quantity, that is, EOQ

In this model the total cost of consumption or usage over the year will remain constant, irre­spective of the decision on the quantity Q ordered, since the Cc does not change with Q. Hence, this cost can be ignored. Since the demand is known and fixed and the lead time for receipt of stocks is also fixed, there will never be a stockout. Hence, stockout cost will not feature in the model. The two costs that will be relevant to an optimal decision are: the cost of ordering and the cost of holding. These costs can be graphically depicted as under:

The graph clearly brings out the features of the interplay of ordering/holding costs related to order quantity Q. If Q is large, fewer orders will be required per year and the ordering cost will fall, but the quantity per order being large, the holding cost will increase.

Conversely, if the quantity Q is small, while the holding cost will fall the number of orders per year and the ordering cost will increase. At the level Q* we note that the two curves of ordering and holding costs cross (both attain equal value) to yield the minimum total cost as seen from the graph.

In the above formula for EOQ, we have used Cc as the cost of holding inventory. This is an absolute value in rupees per unit per year. In a company where there are thousands of items of inventory, it would be tedious to compile the absolute holding cost for each item.

Even if one were to do so, and if at a later time one of the components of holding cost (e.g., cost of capital or storage) were to change, then all the costs will have to be recomputed.

To avoid such a situation, it is the usual practice to use what is called an inventory-carrying rate (I), uniformly applicable to all items. It is expressed as a percentage of the cost of the item. Thus, the holding cost Cc = I • P (where P is the per unit cost of the item).

If P is large then the holding cost will also be large, and if P is small then the holding cost will be small too. We deem that each item, so to say, will bear a holding cost in proportion to its cost or value. This is partly justifiable since out of the 30 per cent cost per annum, the cost of money tied up usually accounts for 20 per cent. Thus, the EOQ formula usually used in practice is:

It can be seen from the graph of the total cost that the curve becomes nearly flat around the region of the EOQ. That is, the change in total cost will not be very significant if the order quantity Q were to be slightly different from the EOQ. Further, it can be seen that the total cost is not very sensitive to changes in D, Co or Cc. This is a very useful feature advantageous to the operation of an EOQ system.

Sensitivity analysis of EOQ is carried out at the planning stage, much before the start of the financial year. As such, the values of Co, Cc, price P and interest I are all determined on the basis of forecasting. It is quite likely that when the actual operations for the year start, the actual values of all parameters may be different.

Inventory models under uncertainty risks depend on probabilities of the occurrence of the parameters. For example, the level of safety stocks varies with probabilistic demand during lead time. In addition, service level is probability of stock availability and stockout risk (SOR) is prob­ability of shortage. The risks are very high, when both daily demand and lead time are probabilistic.

Source: Sople V.V (2013), Logistics Management, Pearson Education India; Third edition.

2 thoughts on “Inventory Models

  1. zoritoler imol says:

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