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									Economic Order Quantity

Economic order quantity
Economic Order Quantity (also known as the Wilson EOQ Model or simply the EOQ Model) is a model that defines the optimal quantity to order that minimizes total variable costs required to order and hold inventory. The model was originally developed by F. W. Harris in 1915, though R. H. Wilson is credited for his early in-depth analysis of the model.

Underlying assumptions
1. the monthly demand for the item is known, deterministic and constant 2. the lead time is zero, i.e., delivery is immediate 3. the receipt of the order occurs in a single instant and immediately after ordering it 4. quantity discounts are not calculated as part of the model 5. the setup cost is constant Note that deterministic does not imply the constancy of the demand. For instance, the sine function is deterministic, but not constant.

Some consequences
1. Shortages or stockouts do not occur, as delivery of the order is immediate.

What is EOQ
Inventory is held to avoid the nuisance, the time and the cost etc. of constant replenishment. However, to replenish inventory only infrequently would necessitate the holding of very large inventories. It is therefore apparent that some balance or trade-off or compromise is needed in deciding how much inventory to hold, and therefore how much inventory to order. There are costs of holding inventory and there are costs of re-ordering inventory and these two costs need to be balanced. The purpose of the EOQ model is to minimise the total costs of inventory. The important costs are the ordering cost, the cost of placing an order, and the cost of carrying or holding a unit of inventory in stock. All other costs such as, for example, the purchase cost of the inventory itself, are constant and therefore not relevant to the model.

Cost Components
Annual Usage/Demand: Expressed in units this is generally the easiest part of the equation. You simply input your forecasted annual usage.

Order Cost: Also known as purchase cost or set up cost, this is the sum of the fixed costs that are incurred each time an item is ordered. These costs are not associated with the quantity ordered but primarily with physical activities required to process the order. For purchased items these would include the cost to enter the Purchase Order and/or Requisition, any approval steps, the cost to process the receipt, incoming inspection, invoice processing and vendor payment, and in some cases a portion of the inbound freight may also be included in order cost. It is important to understand that these are costs associated with the frequency of the orders and not the quantities ordered. For example in your receiving department the time spent checking in the receipt, entering the receipt and doing any other related paperwork would be included while the time spent repacking materials, unloading trucks, and delivery to other departments would likely not be included. If you have inbound quality inspection where you inspect a percentage of the quantity received you would include the time to get the specs and process the paperwork and not include time spent actually inspecting, however if you inspect a fixed quantity per receipt you would then include the entire time including inspecting, repacking, etc. In the purchasing department you would include all time associated with creating the purchase order, approval steps, contacting the vendor, expediting, and reviewing order reports, you would not include time spent reviewing

forecasts, sourcing, getting quotes (unless you get quotes each time you order), and setting up new items. All time spent dealing with vendor invoices would be included in order cost. Associating actual costs to the activities associated with order cost is where many an EOQ formula runs afoul. Do not make a list of all of the activities and then ask the people performing the activities "how long does it take you to do this?" The results of this type of measurement are rarely even close to accurate. I have found it to be more accurate to determine what percentage of time within the department is consumed performing the specific activities and multiplying this by the total labor costs for a certain time period (usually a month) and then dividing by the line items processed during that same period. It is extremely difficult to associate inbound freight costs with order costs in an automated EOQ program and I suggest it only if the inbound freight cost has a significant effect on unit cost and its effect on unit cost varies significantly based upon the order quantity. In manufacturing the Order cost would include the time to initiate the work order, time associated with picking and issuing components excluding time associated with counting and handling specific quantities, all production scheduling time, machine set up time, and inspection time. Production scrap directly associated with the machine setup should also be included in order cost as would be any tooling that is discarded after each production run. There may be times when you want to artificially inflate or deflate set up costs. If you lack the capacity to meet the production schedule using the EOQ you may want to artificially increase set up costs to increase lot sizes and reduce overall set up time. If you have excess capacity you may want to artificially decrease set up costs, this will increase overall set up time and reduce inventory investment. The idea being that if you are paying for the labor and machine overhead anyway it would make sense to take advantage of the savings in reduced inventories. For the most part Order cost is primarily the labor associated with processing the order however you can include the other costs such as the costs of phone calls, faxes, postage, envelopes, etc.

Carrying cost (Inventory Holding Costs): Also called Holding cost, carrying cost is the cost associated with having inventory on hand. It is primarily made up of the costs associated with the inventory investment and storage cost. For the purpose of the EOQ calculation, if the cost does not change based upon the quantity of inventory on hand it should not be included in carrying cost. In the EOQ formula, carrying cost is represented as the annual cost per average on hand inventory unit. Below are the primary components of carrying cost.

Interest - If you had to borrow money to pay for your inventory, the interest rate would be part of the carrying cost. If you did not borrow on the inventory however have loans on other capital items, you can use the interest rate on those loans since a

reduction in inventory would free up money that could be used to pay these loans. If by some miracle you are debt free you would need to determine how much you could make if the money was invested.

Insurance - Since insurance costs are directly related to the total value of the inventory, you would include this as part of carrying cost.

Taxes - If you are required to pay any taxes on the value of your inventory they would also be included.

Storage Costs - Mistakes in calculating storage costs are common in EOQ implementations. Generally companies take all costs associated with the warehouse and divide it by the average inventory to determine a storage cost percentage for the EOQ calculation. This tends to include costs that are not directly affected by the inventory levels and does not compensate for storage characteristics. Carrying costs for the purpose of the EOQ calculation should only include costs that are variable based upon inventory levels. If you are running a pick/pack operation where you have fixed picking locations assigned to each item where the locations are sized for picking efficiency and are not designed to hold the entire inventory, this portion of the warehouse should not be included in carrying cost since changes to inventory levels do not effect costs here. Your overflow storage areas would be included in carrying cost. Operations that use purely random storage for their product would include the entire storage area in the calculation. Areas such as shipping/receiving and staging areas are usually not included in the storage calculations, however if you have to add an additional warehouse just for overflow inventory then you would include all areas of the second warehouse as well as freight and labor costs associated with moving the material between the warehouses. Since storage costs are generally applied as a percentage of the inventory value you may need to classify your inventory based upon a ratio of storage space requirements to value in order to assess storage costs accurately. For example let's say you have just opened a new E-business called "BobsWeSellEverything.com". You calculated that overall your annual storage costs were 5% of your average inventory value, and applied this to your entire inventory in the EOQ calculation. Your average inventory on a particular piece of software and on 80 lb. bags of concrete mix both came to $10,000. The EOQ formula applied a $500 storage cost to the average quantity of each of these items even though the software actually took up only 1 pallet position while the concrete mix consumed 75 pallet positions. Categorizing these items would place the software in a category with minimal storage costs (1% or less) and the concrete in a category with extreme storage costs (50%) that would then allow the EOQ formula to work correctly. There are situations where you may not want to include any storage costs in your EOQ calculation. If your operation has excess storage space of which it has no other

uses you may decide not to include storage costs since reducing your inventory does not provide any actual savings in storage costs. As your operation grows near a point at which you would need to expand your physical operations you may then start including storage in the calculation. A portion of the time spent on cycle counting should also be included in carrying cost, remember to apply costs which change based upon changes to the average inventory level. So in cycle counting you would include the time spent physically counting and not the time spent filling out paperwork, data entry, and travel time between locations. Other costs that can be included in carrying cost are risk factors associated with obsolescence, damage, and theft. Do not factor in these costs unless they are a direct result of the inventory levels and are significant enough to change the results of the EOQ equation.

Assumptions of the Model
1. Demand is known and is deterministic, ie. constant. 2. The lead time, ie. the time between the placement of the order and the receipt of the order is known and constant. 3. The receipt of inventory is instantaneous. In other words the inventory from an order arrives in one batch at one point in time. 4. Quantity discounts are not possible, in other words it does not make any difference how much we order, the price of the product will still be the same. (for the Basic EOQ-Model) 5. That the only costs pertinent to the inventory model are the cost of placing an order and the cost of holding or storing inventory over time Important Note: When calculating the Economic Order Quantity, be aware of the assumptions mentioned above!

Variations
There are many variations on the basic EOQ model. I have listed the most useful ones below.

  

Quantity discount logic can be programmed to work in conjunction with the EOQ formula to determine optimum order quantities. Most systems will require this additional programming. Additional logic can be programmed to determine max quantities for items subject to spoilage or to prevent obsolescence on items reaching the end of their product life cycle. When used in manufacturing to determine lot sizes where production runs are very long (weeks or months) and finished product is being released to stock and consumed/sold throughout the production run you may need to take into account the ratio of production to consumption to more accurately represent the average inventory level. Your safety stock calculation may take into account the order cycle time that is driven by the EOQ. If so, you may need to tie the cost of the change in safety stock levels into the formula.



Graphical Solution
If we minimize the sum of the ordering and carrying costs, we are also minimizing the total costs. To help visualize this we can graph the ordering cost and the holding cost as shown in the chart below:

This chart shows costs on the vertical axis or Y axis and the order quantity on the horizontal or X axis. The straight line which commences at the origin is the carrying cost curve, the total cost of carrying units of inventory. As expected, as we order more on the X axis, the carrying cost line increases in a proportionate manner. The downward sloping curve which commences high on the Y axis and decreases as it approaches the X axis and moves to the right is the ordering cost curve. This curve represents the total ordering cost depending on the size of the order quantity. Obviously the ordering cost will decrease as the order quantity is increased thereby causing there to be fewer orders which need to be made in any particular period of time. The point at which these two curves intersect is the same point which is the minimum of the curve which represents the total cost for the inventory system. Thus the sum of the carrying cost curve and the ordering cost curve is represented by the total cost curve and the minimum point of the total cost curve corresponds to the same point where the carrying cost curve and the ordering cost curve intersect.

How to calculate
Basic EOQ: The objective is to determine the quantity to order which minimizes the total annual inventory management cost. Thus: Minimize! Total cost per period = inventory holding costs per period + order costs per period where Order Cost = The Number of Orders Placed in the period x Order Costs and Carrying Cost = Average Inventory Level x the Carrying Costs of 1 unit of Stock for one period with:
   

Q = order quantity A = demand per time period (e.g. Annual Demand) S = Carrying / Holding Cost of 1 unit of Stock for one period P = Order Cost

and the derivation set to zero we get the following formula:

So we can see that the two cost elements at the economic order quantity are equal, one to the other; (compare with the graphical solution!) If we now isolate the Q, we get the following Basic EOQ-Formula:

Production EOQ: Instead of instantaneous replenishment, we include the finite Production Rate R which leads to the following formula: (You can see, that production rate must be greater than demand rate, in order to fulfill the demand!) EOQ = sqrt ( 2 * A * P / (S*(1-A/R))

Backlogging EOQ: By including the Backlogging Cost B, which is the cost of back-logging one unit per period, we get the following formula: EOQ = sqrt (2 * A * P * (S+B) / S * B)

Extensions
Economic production quantity
Economic Production Quantity model (also known as the EPQ model) is an extension of the Economic Order Quantity model. The difference being that the EPQ model assumes orders are received incrementally during the production process. The function of this model is to balance the inventory holding cost and the average fixed ordering cost.

Variables
     

K = ordering cost D = demand rate F = holding cost T = cycle length P = production rate

Formula

Case
Stop n Slurp Convenience Store Annual Demand: 5200 Cases of Coca Cola Fixed Ordering Cost: 500 per order Cost Per Case: 100 Holding Cost: 20% of value of inventory per year Suppose EOQ assumptions hold (Constant demand, no lag/lead time, no shortages) How Much Coca Cola should Stop n Slurp Order?

Solution
EOQ Parameters: Annual Usage = 5200 Order Cost = 500 Annual Carrying Cost = 20% of 100 = 20 Optimum Ordering Policy: Q = sqrt(2 x 5200 x 500)/(20) = 509.9 ~ 510 Cases

Bibliography
Materials Management an integrated approach – P. Gopalakrishnan Cost Accounting http://www.pafis.shh.fi/~stecon02/afis/ws2/ http://www.inventoryops.com/economic_order_quantity.htm http://www.caam.rice.edu/~timredl/caam376/ http://en.wikipedia.org/wiki/Economic_order_quantity


								
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