 # Economic Order Quantity

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In the system based on the information level, it was assumed that the size of the ordered lot was constant. It was also pointed out that an undoubted advantage of this method is the possibility of placing orders in the amount that minimizes the total costs of replenishment and storage of stocks. In the literature on the subject many models allowing to calculate the amount of orders were described, taking into account different criteria and boundary conditions of optimization [Engming 1980; Yu-Sheng 1992; Dmytrów 2003a; Ragsdale 2004; Balakrishnan, Pangburn and Stavrulaki 2004; Dmytrów 2013]. The starting point for the derivation of a formula for the size of an order is the division of all costs related to replenishment and maintenance of stock on the basis of two criteria. The first one is the dependence of costs on the size of supplies and the level of inventories. On the basis of this criterion, replenishment and maintenance costs can be divided into two categories:

• costs depending on the size of the supply, the size of the stock, the size of the shortage,
• costs independent of the size of the delivery, the size of the stock, the size of the shortage.

The second criterion for the allocation of costs of replenishment and holding is their allocation to the following categories:

• ordering costs,
• carrying costs,
• stockout costs.

As a consequence of the overlap between the two classifications above, the stock cost is divided into six categories listed in Table 1.

As the deciding variable in the Wilson model is the size of the ordered lot, all components that do not depend on the decision variable are moved from the above formula.

Supposing, moreover, that the model does not assume the possibility of occurrence of shortages, the formula for the total cost of replenishing and holding inventories takes the form:

The classic and most commonly used method of determining delivery is the Harris-Wilson [Wilson 1934] model, in which the total cost of replenishment and stockholding is used as a function of the objective.

It is a deterministic model in which the following assumptions were made [Krzyżaniak 2008]:

• the demand for a given product at time O is known and amounts to N,
• the consumption of the product is constant over time,
• the quantity of a single delivery is constant over the period considered and amounts to ,
• deliveries are made at fixed intervals T,
• the cost of storage of a unit of goods is known, unchangeable in the period of time and amounts to n ,
• The cost of order completion is known, unchangeable in time and amounts to T,
• orders shall be placed in advance so that the next consignment is delivered at the time of the total consumption of the previous consignment,
• a supply shortage is unacceptable.

The model assumes that the variable cost of order completion is the product of the cost of a unit order completion and the number of orders completed in the period in question. Taking into account that the number of contracts completed in the period considered is equal to the ratio of the total demand for this good to the size of the unit contract, the cost of performing the contract may be expressed as a formula:

The model also assumes that the variable cost of holding inventories is the product of the unit cost of storage and the average level of inventories in the period considered. Assuming that the inventory is used evenly over time, its average level in the warehouse is equal to half of the order volume. The variable cost of stockholding can therefore be expressed as a formula:

Once the two patterns are combined, the total cost of replenishment and stockholding takes on a form:

Fig. 9 shows the functions of the two component costs and the function of the overall total cost.

Economic Order Quantity, which derived from the Wilson model, is a minimum of the total cost of replenishment and stockholding and is calculated from a formula:

The optimal time between orders is calculated from the formula:

When using the formula on the EOQ in business conditions, it is essential to remember about the accepted limitations in which the model operates. In the literature on the subject one can find references and developments of the Wilson model [Herron 1967; Hill Jr. 1976].

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Author of the article: Radosław Śliwka