Economic Order Quantity (EOQ) Model

Economic Order Quantity (EOQ) Model

Introduction
The cost of inventory, including inventories and excess stocks, has been estimated worldwide at more than $1.1 trillion.
Roughly 43% of small businesses either do not keep track of inventory or use a manual system, e.g. US retailers only have an inventory accuracy of 63%.
Human error, such as lack of training, internal theft, and registration errors, cost companies around the world $259.1 billion annually.
The emergence of the EOQ method
To prevent these losses from occurring, organizations have sought to establish a system of inventory, purchase orders or requisition since reducing expenses such as purchasing inventory and holding costs can increase the profit margin of the organization.
To accurately calculate the optimal order quantity, the proven EOQ method was used. This method aims to reduce ordering and warehousing costs by analyzing inventory preparation cost, production cost and order rate.
Ford W. Harris developed the original model for the first time in 1915, which kept the level of demand and prices constant. This method has been modified several times to account for different production levels, order breaks, and variable costs for large organisations.

Assumptions of the Economic Order Quantity (EOQ) Model
The volume of demand is known and the rate of consumption is constant.
The supply period (purchasing) and the cost of executing/issuing the order are known and fixed.
Storage cost is known and fixed.
There is no discount on the quantities we buy.
Average inventory = 0.5 order volume, i.e. average inventory = Q ÷ 2
The cost of storage is directly proportional to the amount of inventory.
Inventory types
Raw Material and Production Supplies
Semi-manufactured or in-process materials
Finished Goods
Spare Parts. Spare Parts
Inventory related costs
1- The cost of issuing an ordering cost
When we place a supply order, we bear some of the costs of the supply order other than the price of the goods we purchase. These costs include labor costs that work for several hours or days to determine specifications, tender and study them, correspond to suppliers, bear transportation costs, insurance costs, inspection costs...
Increasing the number of sales orders leads to an increase in the cost of supply, while a decrease in the number of orders leads to an increase in inventory due to the supply of large quantities in each supply order. Therefore, the stock must be handled carefully so that it does not exceed or less than the appropriate limit
2- Carrying Cost:
Capital Cost: The money with which we buy inventory is frozen money, that is, uninvested. This stock is the reason for reducing the available cash equal to the value of the stock. This cost increases with the increase in inventory value
Storage Cost In order to store spare parts, raw materials or finished products, we incur a storage cost. This cost includes the rent of the stores, the cost of the labor supervising the storage, the cost of lighting and transportation within the stores, and the expenses of maintaining the inventory in a good condition of heating or cooling or the like. This cost increases with increasing stock
Obselence and Deterioration Cost Inventory Damage may suffer from long storage or poor storage and the stock may become old, making it worthless. This cost is sometimes called the risk cost. This cost is high in some types of inventory, such as foodstuffs, products associated with rapid technological development and products associated with rapid change in tastes.
3- Stockout cost
When a customer orders a product and we cannot fulfill his order because the finished product or some raw materials are out of stock, we are actually losing the profits that we could have made. In addition to the unavailability of the product, the customer may turn to other competitors or cause the customer to be dissatisfied with our services, which affects our reputation and future sales. In some cases, we may have to pay a late fine to the customer and we may have to supply some raw materials urgently, causing us to bear a higher cost. These costs increase as the number of customers whose requests we cannot meet or whose requests we delay fulfilling increases

4- The cost of the inefficiency of the production process:
Availability of inventory causes errors in the production process to be overlooked and problems not to be analyzed seriously to eliminate their roots. When we have a lot of stock of raw materials, semi-finished products and finished products, production problems do not surface because it is just wastage but it does not cause delays in fulfilling customer orders. But when errors cause delays in fulfilling customer requests, these errors are taken care of and an attempt to eliminate them is one of the pillars of the Just In Time waste reduction policy.

Types of inventory suitable for economic demand
This model is suitable for the materials in which the assumptions are fulfilled. If the volume of demand is variable or unknown, then this model is not suitable. If there is a decrease in price versus an increase in quantity we can use another mathematical model and if there is some error it is better than relying on pure guesswork.
We can use this model for materials that are in constant and unchanging demand. In this case, we can determine the average demand size and the average lead time. In this way, we obtain the optimum order quantity within the limits of the information available to us. You must always remember to use a mathematical method.
There are materials for which the demand is very variable. Sometimes the demand is high and at other times it is much less. This model is not suitable for them.
Inventory costs:
The figure shows that the total cost is the sum of the increasing and decreasing costs in terms of order volume, and this cost is at its lowest when the cost curve intersects 

Important notes
The cost of holding inventory usually ranges between 15% and 25%, because the main component of this cost is the cost of freezing the capital, and we usually do not invest in a project unless its return is higher than the return of the banks.
The percentage of stock deterioration varies according to the quality of the stock and according to its storage period, as it is as little as 1% in most cases, except in some materials that spoil quickly or products that age quickly.
The cost of stores and labor is usually a small percentage such as 3% or 5% of the cost of stored materials. So the cost of holding inventory is usually between 15% to 25% mostly.
The cost of the order or one supply order Order cost includes transportation costs, insurance, and administrative expenses for preparing specifications, supply order, discussion with suppliers, the cost of opening a credit, examination cost, and so on. These have to be estimated to arrive at their true value.
The algebraic formula for the economic volume of a purchase order
The annual cost of purchase orders (orders) = (the number of orders per year) x (the cost of issuing one order).
Annual cost of purchase orders: (𝐃) item of annual order volume 𝐐 order volume (D) variety of annual order volume)/( (Q) order volume) x cost of issuing a single order (R) = (DR)/Q
The annual cost of holding inventory: x inventory2 inventory/2 unit holding cost (H) = QH/2
When the cost of storage is equal to the cost of the order (QH/2 = DR/Q) and with some simple mathematical operations, we arrive at the optimal value of the order size (EOQ), which is:
√(2DR/H) = EOQ

The sum of the annual order cost and the annual inventory cost we get the total cost of inventory:
QH/2 + DR/Q = TC

Economical order volume
Example: The annual needs of a factory of a certain type are estimated (18,000) units, the cost of issuing one purchase order (20) riyals, and the cost of storing one unit annually is (2) riyals.
What is required: calculate the economic quantity of the demand by algebraic method, and calculate the total cost of inventory.
Example: The annual needs of an industrial workshop of a certain type are estimated at (1200) units, and the cost of issuing one purchase order is (10) riyals, and the cost of maintaining inventory is 12% annually of the unit purchase price of (5) riyals.

Example 3: Calculating the economic quantity of an order by algebraic method

Al Jafal Pharmaceutical Industries uses annually (1000) units of a specific class of medicines, the purchase price of one unit of the brand from the supplier is (2) monetary units, the cost of preparing one order is (10) monetary units, and the cost of keeping inventory per unit is (0.5) monetary units annually.
Required :

Calculate the economic volume of the purchase order.
Calculate the total cost of inventory
Tabular or arithmetic method
The tabular method is based on calculating the total costs for quantities and orders of different sizes in order to arrive at the lowest possible cost.
The corresponding order volume represents the economic quantity of the order.
Example: The estimated annual needs D for one of the industrial workshops of a specific category (1200) units, and the cost of issuing an order
One purchase is R (10) riyals, and the cost of maintaining inventory H is 12% annually of the unit purchase price of (5) riyals.
The solution of Tabular or arithmetic method:

For quantity of 50
number orders: 1200 ÷ 50 = 24

orders cost: 24 x 10 = 240

Storage cost: (50 ÷ 2) (0.12 x 5)

= 25 x 0.6 = 15

Total costs: 240 + 15 = 255

Example: An organization needs 1,000 units of item X per month, and estimates the cost of issuing a purchase order (20 riyals per unit), and estimates the cost of maintaining inventory (0.5 riyals per unit), and the average inventory per month is half the size of the order. It is required to calculate the economic volume of the purchase order using the tabular method, and comment on the results.

Tabular or arithmetic method
Total cost (order cost + holding cost)

Storage cost ((order size ÷ 2) x storage cost per unit)

Cost of purchase orders (number of orders to buy x cost of issuing one order)

Tabular or Arithmitic method

Finding the EOQ using the Graphic Approach
It is noted that when the size of the order is small, the total costs of issuing orders are high, due to the recurrence of purchase orders and the increase in their number, while storage costs decrease, due to the lack of stock available at one time.
If the volume of the order is large, it will lead to a decrease in the total costs of issuing orders and an increase in storage costs, due to the high average stock available at a time.
In view of this, the curve of total costs of inventory is the lowest possible at the point of intersection of the cost curve of the order to purchase and the curve of storage cost
Of course, any point other than the economic quantity point will have a higher total inventory cost

Methods for calculating the economic volume of a purchase order: the graphing method
Example: The annual needs of an industrial workshop of a certain type are estimated at (1200) units, the cost of issuing a single purchase order is (10) riyals, and the cost of maintaining inventory is 12% annually of the unit purchase price of (5) riyals. The following table includes the calculation of the total costs, the costs of issuing the order, and the cost of warehousing for different volumes of orders.
It is required to draw the curves for issuing orders, the cost of storage, and the total costs.
Indicating the economic quantity of the order, and the corresponding total costs graphically
Methods for calculating the economic volume of a purchase order?

EOQ Model with Discount Model
Organizations give their customers a quantity discount in order to increase sales, and the quantity discount is defined as a reduction in the price of one unit when purchasing in large quantities, and companies often prepare tables showing the purchase quantities and the price associated with each quantity.
Example: The annual needs of a specific type amounted to 180,000 units imported by Al Rayan Company, and the supplying company submitted an offer to it that includes a quantity discount of 10% of the unit purchase price of (2) riyals in the case of purchasing a quantity of (9,000) units or more. The cost of issuing the supply order is (20) riyals, while the cost of maintaining inventory is estimated at 10% of the unit purchase price. What is the decision advised by the management of the organization?
Determining the economic size of the purchase order in the event of a quantity discount
Example: The Rumaila Industrial Corporation estimated its annual needs of one of the items at the size of (18000) units, and the basic purchase price for one unit is (10) riyals. The cost of issuing a single supply order was estimated at (20) riyals, while the cost of storing one unit per year was estimated at 2 riyals. The supplier presented her with an attractive selling offer that includes a quantitative discount on the unit price and according to the quantity purchased each time, as shown in the following table:

What is the alternative recommended by the organization?

The economical size of the purchase order with a quantity discount
Exercise: Al-Gharafa Trading Company wishes to purchase a kind of children’s toys supplied to it by the producing company in China, and it has presented an offer that includes a quantitative discount on the price according to specific quantities as in the following table:

2000 - and above 4.75The cost of issuing one order amounted to (49) riyals, and this company annually imports (5000) toys, and the cost of maintaining the inventory was estimated at 20% of the purchase price of one unit, bearing in mind that the purchase price of one unit is (5) riyals. What is the alternative that the company recommends taking?

ROP: Determining when the order is triggered
The organization must have a stockpile of the material that covers consumption during the supply/waiting period, i.e. from the launch of the order until it is received, which requires determining the reorder point for each item. We have assumed in the previous model of economic demand quantity that the order arrives as soon as it is launched. But in practice, the organization waits for a few days, weeks, or even months until the order arrives, and this period is called the lead time (L), and it is also called the supply period.
Reorder Point/Level: It is the stock balance level at which the Procurement Department is informed of reordering in order for the service or production process not to stop.
To determine the level of re-ordering, you must specify:
Daily or monthly consumption rates of the item.
The period of time that the buy-back process takes (the supply period of the material).
STOCK CONTROL SYSTEMS:
MINIMUM STOCK LEVEL: It means the least amount of merchandise that must be kept to ensure meeting production and sales requirements. To this level of stock can be added the so-called SAFETY STOCK as a precaution for any change that may occur in the prevailing production and sale rates to face the possibility of any delay in the suppliers’ delivery of the institution’s requests.
MAXIMUM STOCK LEVEL: It is the maximum stock level that can be kept within the organization at any time, and any increase in stock above this level will lead to unjustified costs to bear.
RE-ORDER LEVEL: It is the level at which the institution must take the initiative to send new orders for goods.
ROP: Determining when the order is triggered
Determining stock levels represents, as it is clear from the above, one of the tools for controlling stock. In light of determining these levels, materials are disbursed, as well as the demand from the purchasing function to repurchase materials.
Therefore, the following inventory levels must be determined:
Reorder level = Maximum daily usage x Maximum delivery time in days.
Minimum Stock Level = Reorder Level – (Daily Usage Rate x Average Delivery Time in Days)
Maximum Stock Level = Reorder Level + Order Quantity – (Minimum Daily Use x Minimum Delivery).
The reorder point is calculated as follows:
Re-order point = demand or daily requirement x supply period (waiting)
ROP = d × L
Since:
ROP = Redial Point

d = demand / daily requirement and we get it by dividing the annual demand D by the number of working days in the year.
L = supply/waiting period
Safety stock (reserve): It is the limit or quantity that the balance of any item (risk limit, emergency) cannot be less than, in order to ensure the continuity of work. It is equal to the minimum amount of storage to be kept.

Example ROP: Determining when the order is triggered

The annual requirement of the Al Khor Foundation of a specific type is estimated at 8000 units, bearing in mind that the daily requirement of this type is 40 units, and the average supply period is 3 working days.

•Required:

Define the redial point.

Recalculate the reorder point if you know the organization has decided to maintain a safety stock of 50 units.

•The solution:

Re-order point = daily need/consumption rate x supply period = 40 x 3 = 120 units.

Reorder point = (m

Adjusted daily consumption x supply period) + safety stock = (40 x 3) + 50 = 170 units.

•Example :

An organization needs 1,000 units per week, bearing in mind that the year is 52 weeks with a 4-week maintenance period, and the waiting period is estimated at 4 weeks, while the economic demand is estimated at half the annual need, and the stockpile is sufficient for 3 months’ consumption.

•Required:

•Calculation of minimum inventory.

Calculation of the reorder level.

Calculation of the annual requirement.

Calculation of the upper limit of stock.

The solution:

1- Safety stock (reserve) = 1000 units per week x 3 months (12 weeks) = 1000 x 12 = 12000 units.

Minimum = consumption rate x supply period = 1000 * 4 = 4000

2- Re-order level = (consumption rate x supply period) + safety stock = (1000 x 4) + 12000 = 16000 units.

3- Annual need = 1000 x (52 - 4) = 48000 units.

4- The upper limit = re-order level + order size = 12000 + (48000/2) = 36000 units.

General note:

To determine the quantity and level of the safety stock, the usage rates for each item must be determined during a certain period of time, with determining whether the usage rates are fixed or variable.