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# Economic order quantity

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### Economic order quantity

1. 1. Economic Order Quantity
2. 2. Economic order quantityEconomic Order Quantity (also known as the Wilson EOQ Model or simply theEOQ Model) is a model that defines the optimal quantity to order that minimizestotal variable costs required to order and hold inventory.The model was originally developed by F. W. Harris in 1915, though R. H. Wilson iscredited for his early in-depth analysis of the model.Underlying assumptions1. the monthly demand for the item is known, deterministic and constant2. the lead time is zero, i.e., delivery is immediate3. the receipt of the order occurs in a single instant and immediately afterordering it4. quantity discounts are not calculated as part of the model5. the setup cost is constantNote that deterministic does not imply the constancy of the demand. For instance, thesine function is deterministic, but not constant.Some consequences1. Shortages or stockouts do not occur, as delivery of the order is immediate.
3. 3. What is EOQInventory is held to avoid the nuisance, the time and the cost etc. of constantreplenishment. However, to replenish inventory only infrequently would necessitatethe holding of very large inventories. It is therefore apparent that some balance ortrade-off or compromise is needed in deciding how much inventory to hold, andtherefore how much inventory to order. There are costs of holding inventory and thereare costs of re-ordering inventory and these two costs need to be balanced. Thepurpose 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 ofcarrying 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 themodel.Cost ComponentsAnnual Usage/Demand:Expressed in units this is generally the easiest part of the equation. You simply inputyour forecasted annual usage.Order Cost:Also known as purchase cost or set up cost, this is the sum of the fixed costs that areincurred each time an item is ordered. These costs are not associated with the quantityordered but primarily with physical activities required to process the order.For purchased items these would include the cost to enter the Purchase Order and/orRequisition, any approval steps, the cost to process the receipt, incoming inspection,invoice processing and vendor payment, and in some cases a portion of the inboundfreight may also be included in order cost. It is important to understand that these arecosts associated with the frequency of the orders and not the quantities ordered. Forexample in your receiving department the time spent checking in the receipt, enteringthe receipt and doing any other related paperwork would be included while the timespent repacking materials, unloading trucks, and delivery to other departments wouldlikely not be included. If you have inbound quality inspection where you inspect apercentage of the quantity received you would include the time to get the specs andprocess the paperwork and not include time spent actually inspecting, however if youinspect a fixed quantity per receipt you would then include the entire time includinginspecting, repacking, etc. In the purchasing department you would include all timeassociated with creating the purchase order, approval steps, contacting the vendor,expediting, and reviewing order reports, you would not include time spent reviewing
4. 4. forecasts, sourcing, getting quotes (unless you get quotes each time you order), andsetting up new items. All time spent dealing with vendor invoices would be includedin order cost.Associating actual costs to the activities associated with order cost is where many anEOQ formula runs afoul. Do not make a list of all of the activities and then ask thepeople performing the activities "how long does it take you to do this?" The results ofthis type of measurement are rarely even close to accurate. I have found it to be moreaccurate to determine what percentage of time within the department is consumedperforming the specific activities and multiplying this by the total labor costs for acertain time period (usually a month) and then dividing by the line items processedduring that same period.It is extremely difficult to associate inbound freight costs with order costs in anautomated EOQ program and I suggest it only if the inbound freight cost has asignificant effect on unit cost and its effect on unit cost varies significantly basedupon 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 withcounting and handling specific quantities, all production scheduling time, machine setup time, and inspection time. Production scrap directly associated with the machinesetup should also be included in order cost as would be any tooling that is discardedafter each production run. There may be times when you want to artificially inflate ordeflate set up costs. If you lack the capacity to meet the production schedule using theEOQ you may want to artificially increase set up costs to increase lot sizes and reduceoverall set up time. If you have excess capacity you may want to artificially decreaseset 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 itwould make sense to take advantage of the savings in reduced inventories.For the most part Order cost is primarily the labor associated with processing theorder 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 inventoryon hand. It is primarily made up of the costs associated with the inventory investmentand storage cost. For the purpose of the EOQ calculation, if the cost does not changebased upon the quantity of inventory on hand it should not be included in carryingcost. In the EOQ formula, carrying cost is represented as the annual cost per averageon 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 ratewould be part of the carrying cost. If you did not borrow on the inventory howeverhave loans on other capital items, you can use the interest rate on those loans since a
5. 5. reduction in inventory would free up money that could be used to pay these loans. Ifby some miracle you are debt free you would need to determine how much you couldmake if the money was invested.Insurance - Since insurance costs are directly related to the total value of theinventory, you would include this as part of carrying cost.Taxes - If you are required to pay any taxes on the value of your inventory theywould also be included.Storage Costs - Mistakes in calculating storage costs are common in EOQimplementations. Generally companies take all costs associated with the warehouseand divide it by the average inventory to determine a storage cost percentage for theEOQ calculation. This tends to include costs that are not directly affected by theinventory levels and does not compensate for storage characteristics. Carrying costsfor the purpose of the EOQ calculation should only include costs that are variablebased upon inventory levels.If you are running a pick/pack operation where you have fixed picking locationsassigned to each item where the locations are sized for picking efficiency and are notdesigned to hold the entire inventory, this portion of the warehouse should not beincluded 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 usepurely random storage for their product would include the entire storage area in thecalculation. Areas such as shipping/receiving and staging areas are usually notincluded in the storage calculations, however if you have to add an additionalwarehouse just for overflow inventory then you would include all areas of the secondwarehouse as well as freight and labor costs associated with moving the materialbetween the warehouses.Since storage costs are generally applied as a percentage of the inventory value youmay need to classify your inventory based upon a ratio of storage space requirementsto value in order to assess storage costs accurately. For example lets say you have justopened a new E-business called "BobsWeSellEverything.com". You calculated thatoverall your annual storage costs were 5% of your average inventory value, andapplied this to your entire inventory in the EOQ calculation. Your average inventoryon 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 ofeach of these items even though the software actually took up only 1 pallet positionwhile the concrete mix consumed 75 pallet positions. Categorizing these items wouldplace the software in a category with minimal storage costs (1% or less) and theconcrete in a category with extreme storage costs (50%) that would then allow theEOQ formula to work correctly.There are situations where you may not want to include any storage costs in yourEOQ calculation. If your operation has excess storage space of which it has no other
6. 6. uses you may decide not to include storage costs since reducing your inventory doesnot provide any actual savings in storage costs. As your operation grows near a pointat which you would need to expand your physical operations you may then startincluding 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 inventorylevel. So in cycle counting you would include the time spent physically counting andnot 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 withobsolescence, damage, and theft. Do not factor in these costs unless they are a directresult of the inventory levels and are significant enough to change the results of theEOQ equation.Assumptions of the Model1. Demand is known and is deterministic, ie. constant.2. The lead time, ie. the time between the placement of the order and the receiptof the order is known and constant.3. The receipt of inventory is instantaneous. In other words the inventory from anorder arrives in one batch at one point in time.4. Quantity discounts are not possible, in other words it does not make anydifference 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 anorder and the cost of holding or storing inventory over timeImportant Note: When calculating the Economic Order Quantity, be aware of theassumptions mentioned above!VariationsThere 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 EOQformula to determine optimum order quantities. Most systems will require thisadditional programming.• Additional logic can be programmed to determine max quantities for items subject tospoilage or to prevent obsolescence on items reaching the end of their product lifecycle.• When used in manufacturing to determine lot sizes where production runs are verylong (weeks or months) and finished product is being released to stock andconsumed/sold throughout the production run you may need to take into account theratio of production to consumption to more accurately represent the averageinventory level.
7. 7. • Your safety stock calculation may take into account the order cycle time that is drivenby the EOQ. If so, you may need to tie the cost of the change in safety stock levelsinto the formula.Graphical SolutionIf we minimize the sum of the ordering and carrying costs, we are also minimizing thetotal costs. To help visualize this we can graph the ordering cost and the holding costas shown in the chart below:This chart shows costs on the vertical axis or Y axis and the order quantity on thehorizontal or X axis. The straight line which commences at the origin is the carryingcost curve, the total cost of carrying units of inventory. As expected, as we order moreon the X axis, the carrying cost line increases in a proportionate manner. Thedownward sloping curve which commences high on the Y axis and decreases as itapproaches the X axis and moves to the right is the ordering cost curve. This curverepresents the total ordering cost depending on the size of the order quantity.Obviously the ordering cost will decrease as the order quantity is increased therebycausing there to be fewer orders which need to be made in any particular period oftime.The point at which these two curves intersect is the same point which is the minimumof the curve which represents the total cost for the inventory system. Thus the sum ofthe carrying cost curve and the ordering cost curve is represented by the total costcurve and the minimum point of the total cost curve corresponds to the same pointwhere the carrying cost curve and the ordering cost curve intersect.
8. 8. How to calculateBasic EOQ:The objective is to determine the quantity to order which minimizes the totalannual inventory management cost.Thus: Minimize! Total cost per period = inventory holding costs per period +order costs per periodwhere Order Cost = The Number of Orders Placed in the period x Order Costsand Carrying Cost = Average Inventory Level x the Carrying Costs of 1 unit ofStock for one periodwith:• 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 Costand 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:
9. 9. Instead of instantaneous replenishment, we include the finite Production Rate Rwhich leads to the following formula: (You can see, that production rate must begreater 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 unitper period, we get the following formula:EOQ = sqrt (2 * A * P * (S+B) / S * B)ExtensionsEconomic production quantityEconomic Production Quantity model (also known as the EPQ model) is anextension of the Economic Order Quantity model. The difference being that theEPQ model assumes orders are received incrementally during the productionprocess. The function of this model is to balance the inventory holding cost and theaverage fixed ordering cost.Variables• K = ordering cost• D = demand rate• F = holding cost• T = cycle length• P = production rate•Formula
10. 10. CaseStop n Slurp Convenience StoreAnnual Demand: 5200 Cases of Coca ColaFixed Ordering Cost: 500 per orderCost Per Case: 100Holding Cost: 20% of value of inventory per yearSuppose EOQ assumptions hold (Constant demand, no lag/lead time, no shortages)How Much Coca Cola should Stop n Slurp Order?SolutionEOQ Parameters:Annual Usage = 5200Order Cost = 500Annual Carrying Cost = 20% of 100 = 20Optimum Ordering Policy:Q = sqrt(2 x 5200 x 500)/(20) = 509.9 ~ 510 CasesBibliographyMaterials Management an integrated approach – P. GopalakrishnanCost Accountinghttp://www.pafis.shh.fi/~stecon02/afis/ws2/http://www.inventoryops.com/economic_order_quantity.htmhttp://www.caam.rice.edu/~timredl/caam376/http://en.wikipedia.org/wiki/Economic_order_quantity