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optimisation_quantitecommande_eoq Version traduite (française). En ligne 29/02/2004


Calcul de la quantité optimal de commande, Je fais un lien en Doc. Word Texte Classement par taille de sort  pour ceux qui veulent poursuivre, c'est bien expliqué, malgré en attente d'être complètement traduit. Le 18/07/2002

Révision de traduction au coup par coup. Merci. 29/02/2004

Optimizing Economic Order Quantity (EOQ)

By Dave Piasecki

As published in the January, 2001 issue of solutions.gif (1683 bytes)

Inventory models for calculating optimal order quantities and reorder points have been in existence long before the arrival of the computer.  When the first Model T Fords were rolling off the assembly line, manufacturers were already reaping the financial benefits of inventory management by determining the most cost effective answers to the questions of  When? and How much?.  Yes long before JIT, TQM, TOC, and MRP, companies were using these same (then unnamed) concepts in managing their production and inventory.  I recently read Purchasing and Storing, a textbook that was part of a “Modern Business Course” at the Alexander Hamilton Institute in New York.  The textbook published in 1931 (that’s right 1931) was essentially a how to book on inventory management in a manufacturing environment.  If you’re wondering why I would want to read a 70-year-old business text, my answer would be that the fundamental concepts of managing a business change very little with time, and reading about these concepts in a vintage text is a great way to reinforce the value of the fundamentals.  The occasional reference to “The War” (referring to WWI) also keeps it interesting and the complete absence of acronyms is refreshing.

As you may have guessed, this 70-year-old book contained a section on Minimum Cost Quantity, which is what we now refer to as Economic Order Quantity (EOQ).  I can imagine that in the 1930’s an accountant (or more likely a room full of accountants) would have calculated EOQ or other inventory related formulas one item at a time in a dimly lit office using the inventory books, a mechanical adding machine and a slide rule.  Time consuming as this was, some manufacturers of the time recognized the financial benefits of taking a scientific approach to making these inventory decisions.

So why is it that, in these days of advanced information technology, many companies are still not taking advantage of these fundamental inventory models?  Part of the answer lies in poor results received due to inaccurate data inputs.  Accurate product costs, activity costs, forecasts, history, and lead times are crucial in making inventory models work. Ironically, software advancements may also in part to blame. Many ERP packages come with built in calculations for EOQ which calculate automatically. Often the users do not understand how it is calculated and therefore do not understand the data inputs and system setup which controls the output. When the output appears to be "out of whack" it is simply ignored.  This sometimes creates a situation in which the executives who had purchased the software incorrectly assume the material planners and purchasing clerks are ordering based upon the systems recommendations.  I should also note that many operations will find these built-in EOQ calculations inadequate and in need of modifications to deal with the diversity of their product groups and processes.

Corporate goals and strategies may sometimes conflict with EOQ.  Measuring performance solely by inventory turns is one of the most prolific mistakes made in the name of inventory management.  Many companies have achieved aggressive goals in increasing inventory turns only to find their bottom line has shrunk due to increased operational costs.  

EOQ is essentially an accounting formula that determines the point at which the combination of order costs and inventory carrying costs are the least. The result is the most cost effective quantity to order.  In purchasing this is known as the order quantity, in manufacturing it is known as the production lot size.

While EOQ may not apply to every inventory situation, most organizations will find it beneficial in at least some aspect of their operation.  Anytime you have repetitive purchasing or planning of an item, EOQ should be considered. Obvious applications for EOQ are purchase-to-stock distributors and make-to-stock manufacturers, however, make-to-order manufacturers should also consider EOQ when they have multiple orders or release dates for the same items and when planning components and sub-assemblies.  Repetitive buy maintenance, repair, and operating (MRO) inventory is also a good application for EOQ.  Though EOQ is generally recommended in operations where demand is relatively steady, items with demand variability such as seasonality can still use the model by going to shorter time periods for the EOQ calculation.  Just make sure your usage and carrying costs are based on the same time period.

Doesn’t EOQ conflict with Just-In-Time?  While I don’t want to get into a long discussion on the misconceptions of what Just-In-Time (JIT) is, I will address the most common misunderstanding in which JIT is assumed to mean all components should arrive in the exact run quantities “just in time” for the production run.  JIT is actually a quality initiative with the goal of eliminating wasted steps, wasted labor, and wasted cost. EOQ should be one of the tools used to achieve this. EOQ is used to determine which components fit into this JIT model and what level of JIT is economically advantageous for your operation.  As an example, let us assume you are a lawn equipment manufacturer and you produce 100 units per day of a specific model of lawn mower.  While it may be cost effective to have 100 engines arrive on your dock each day, it would certainly not be cost effective to have 500 screws (1 days supply) used to mount a plastic housing on the lawn mower shipped to you daily.  To determine the most cost effective quantities of screws or other components you will need to use the EOQ formula.

The basic Economic Order Quantity (EOQ) formula is as follows:  


The Inputs

While the calculation itself is fairly simple the task of determining the correct data inputs to accurately represent your inventory and operation is a bit of a project.  Exaggerated order costs and carrying costs are common mistakes made in EOQ calculations.  Using all costs associated with your purchasing and receiving departments to calculate order cost or using all costs associated with storage and material handling to calculate carrying cost will give you highly inflated costs resulting in inaccurate results from your EOQ calculation.  I also caution against using benchmarks or published industry standards in calculations.  I have frequently seen references to average purchase order costs of $100 to $150 in magazine articles and product brochures.  Often these references trace back to studies performed by advocacy agencies working for business that directly benefit from these exaggerated (my opinion) costs used in ROI calculations for their products or services.  I am not denying that some operations may have purchase costs in this range, especially if you are frequently re-sourcing, re-quoting, and/or buying from overseas vendors. However if your operation is primarily involved with repetitive buying from domestic vendors which is more common you’ll likely see your purchase order costs in the substantially lower $10 to $30 range. 

As you prepare to undertake this project keep in mind that even though accuracy is crucial, small variances in the data inputs generally have very little effect on the outputs.  The following breaks down the data inputs in more detail and gives insight into the aspects of each.


Annual Usage

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 effective to determine the percentage of time within the department 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

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, but 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 with 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.



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.


Implementing EOQ

There are primarily two ways to implement EOQ. Both methods obviously require that you have already determined the associated costs.  The simplest method is to set up your calculation in a spreadsheet program, manually calculate EOQ one item at a time, and then manually enter the order quantity into your inventory system.  If your inventory has fairly steady demand and costs and you have less than one or two thousand SKUs you can probably get by using this method once per year.  If you have more than a couple thousand SKUs and/or higher variability in demand and costs you will need to program the EOQ formula into your existing inventory system.  This allows you to quickly re-calculate EOQ automatically as often as needed.  You can also use a hybrid of the two systems by downloading your data to a spreadsheet or database program, perform the calculations and then update your inventory system either manually or through a batch program.  Whichever method you use you should make sure to follow the following steps:

·        Test the formula.  Prior to final implementation you must test the programming and setup.  Run the EOQ program and then manually check the results using sample items that are representative of the variations of your inventory base.

·        Project results.  You'll need to run a simulation or use a representative sampling of items to determine the overall short-term and long-term effects the EOQ calculation will have on warehouse space, cash flow, and operations.  Dramatic increases in inventory levels may not be immediately feasible, if this is the case you may temporarily adjust the formula until arrangements can be made to handle the additional storage requirements and compensate for the effects on cash flow.  If the projection shows inventory levels dropping and order frequency increasing, you may need to evaluate staffing, equipment, and process changes to handle the increased activity.

·        Maintain EOQ.  The values for Order cost and Carrying cost should be evaluated at least once per year taking into account any changes in interest rates, storage costs, and operational costs.

 A related calculation is the Total Annual Cost calculation.  This calculation can be used to prove the EOQ calculation.  Total Annual Cost = [(annual usage in units)/(order quantity)(order cost)]+{[.5(order quantity)+(safety stock)]*(annual carrying cost per unit)}.  This formula is also very useful when comparing quotes where vendors offer different minimum order quantities, price breaks, lead times, transportation costs.

Use it!  The EOQ calculation is "Hard Science", if you have accurate inputs the output is the most cost-effective quantity to order based upon your current operational costs.  To further increase inventory turns you will need to reduce the order costs.  E-procurement, vendor-managed inventories, bar coding, and vendor certification programs can reduce the costs associated with processing an order.  Equipment enhancements and process changes can reduce costs associated with manufacturing set up.  Increasing forecast accuracy and reducing lead times which result in the ability to operate with reduced safety stock can also reduce inventory levels.


  optimisation_quantitecommande_eoq Version traduite (française). En ligne 29/02/2004


Calcul de la quantité optimal de commande, Je fais un lien en Doc. Word Texte Classement par taille de sort  pour ceux qui veulent poursuivre, c'est bien expliqué, malgré en attente d'être complètement traduit. Le 18/07/2002

Révision de traduction au coup par coup. Merci. 28/02/2004


Page Up

07.10.2010 19:57


Also read my article on Calculating Safety Stock



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