Economic order quantity - wikipedia Economic order quantity In inventory management, economic order quantity ( EOQ ) is the order quantity that minimizes the total holding costs and ordering costs. It is one of the oldest classical production scheduling models. The model was developed by Ford W. Harris in 1913, but R.H. Wilson, a consultant who applied it extensively, and K. Andler are given credit for their in - depth analysis. Contents 1 Overview 1.1 Variables 1.2 The total cost function and derivation of EOQ formula 1.3 Example 2 Extensions of the EOQ model 2.1 Quantity discounts 2.2 Design of optimal quantity discount schedules 2.3 Backordering costs and multiple items 2.4 Imperfect quality 3 For improving fuel economy of internal combustion engines 4 See also 5 References 6 Further reading 7 External links Overview EOQ applies only when demand for a product is constant over the year and each new order is delivered in full when inventory reaches zero. There is a fixed cost for each order placed, regardless of the number of units ordered. There is also a cost for each unit held in storage, commonly known as holding cost, sometimes expressed as a percentage of the purchase cost of the item. We want to determine the optimal number of units to order so that we minimize the total cost associated with the purchase, delivery and storage of the product. The required parameters to the solution are the total demand for the year, the purchase cost for each item, the fixed cost to place the order and the storage cost for each item per year. Note that the number of times an order is placed will also affect the total cost, though this number can be determined from the other parameters. Variables P ( displaystyle P ) = purchase unit price, unit production cost Q ( displaystyle Q ) = order quantity. Q ∗ ( displaystyle Q ^ ( * ) ) = optimal order quantity. D ( displaystyle D ) = annual demand quantity. K ( displaystyle K ) = fixed cost per order, setup cost ( not per unit, typically cost of ordering and shipping and handling. This is not the cost of goods ) h ( displaystyle h ) = annual holding cost per unit, also known as carrying cost or storage cost ( capital cost, warehouse space, refrigeration, insurance, etc. usually not related to the unit production cost ) The total cost function and derivation of EOQ formula The single - item EOQ formula finds the minimum point of the following cost function : Total Cost = purchase cost or production cost + ordering cost + holding cost Where : Purchase cost : This is the variable cost of goods : purchase unit price × annual demand quantity. This is P × D Ordering cost : This is the cost of placing orders : each order has a fixed cost K, and we need to order D / Q times per year. This is K × D / Q Holding cost : the average quantity in stock ( between fully replenished and empty ) is Q / 2, so this cost is h × Q / 2 T C = P D + D K Q + h Q 2 ( displaystyle TC = PD+ ( \frac ( DK ) ( Q ) ) + ( \frac ( hQ ) ( 2 ) ) ). To determine the minimum point of the total cost curve, calculate the derivative of the total cost with respect to Q ( assume all other variables are constant ) and set it equal to 0 : 0 = − D K Q 2 + h 2 ( displaystyle ( 0 ) = - ( \frac ( DK ) ( Q ^ ( 2 ) ) ) + ( \frac ( h ) ( 2 ) ) ) Solving for Q gives Q * ( the optimal order quantity ) : Q ∗ 2 = 2 D K h ( displaystyle Q ^ ( * 2 ) = ( \frac ( 2DK ) ( h ) ) ) Therefore : Economic Order Quantity Q ∗ = 2 D K h ( displaystyle Q ^ ( * ) = ( sqrt ( \frac ( 2DK ) ( h ) ) ) ) Q * is independent of P ; it is a function of only K, D, h. The optimal value Q * may also be found by recognising that T C = D K Q + h Q 2 + P D = h 2 Q ( Q − 2 D K / h ) 2 + 2 h D K + P D, ( displaystyle TC = ( \frac ( DK ) ( Q ) ) + ( \frac ( hQ ) ( 2 ) ) + PD = ( \frac ( h ) ( 2Q ) ) ( Q - ( sqrt ( 2DK / h ) ) ) ^ ( 2 ) + ( sqrt ( 2hDK ) ) + PD, ) where the non-negative quadratic term disappears for Q = 2 D K / h, ( displaystyle Q = ( sqrt ( 2DK / h ) ), ) which provides the cost minimum T C m i n = 2 h D K + P D. ( displaystyle TC_ ( min ) = ( sqrt ( 2hDK ) ) + PD. ) Example annual requirement quantity ( D ) = 10000 units Cost per order ( K ) = 40 Cost per unit ( P ) = 50 Yearly carrying cost per unit ( h ) = 5 Economic order quantity = 2 D ∗ K h ( displaystyle ( sqrt ( \frac ( 2D * K ) ( h ) ) ) ) = 2 ∗ 10000 ∗ 40 5 ( displaystyle = ( sqrt ( \frac ( 2 * 10000 * 40 ) ( 5 ) ) ) ) = 400 units Number of orders per year ( based on EOQ ) = 10000 400 = 25 ( displaystyle = ( \frac ( 10000 ) ( 400 ) ) = 25 ) Total cost = P ∗ D + K ( D / E O Q ) + h ( E O Q / 2 ) ( displaystyle = P * D + K ( D / EOQ ) + h ( EOQ / 2 ) ) Total cost = 50 ∗ 10000 + 40 ∗ ( 10000 / 400 ) + 5 ∗ ( 400 / 2 ) = 502000 ( displaystyle = 50 * 10000 + 40 * ( 10000 / 400 ) + 5 * ( 400 / 2 ) = 502000 ) If we check the total cost for any order quantity other than 400 ( = EOQ ), we will see that the cost is higher. For instance, supposing 500 units per order, then Total cost = 50 ∗ 10000 + 40 ∗ ( 10000 / 500 ) + 5 ∗ ( 500 / 2 ) = 502050 ( displaystyle = 50 * 10000 + 40 * ( 10000 / 500 ) + 5 * ( 500 / 2 ) = 502050 ) Similarly, if we choose 300 for the order quantity then Total cost = 50 ∗ 10000 + 40 ∗ ( 10000 / 300 ) + 5 ∗ ( 300 / 2 ) = 502083.33 ( displaystyle = 50 * 10000 + 40 * ( 10000 / 300 ) + 5 * ( 300 / 2 ) = 502083.33 ) This illustrates that the economic order quantity is always in the best interests of the firm. Extensions of the EOQ model Quantity discounts An important extension to the EOQ model is to accommodate quantity discounts. There are two main types of quantity discounts : ( 1 ) all - units and ( 2 ) incremental. Here is a numerical example : Incremental unit discount : Units 1 -- 100 cost $30 each ; Units 101 -- 199 cost $28 each ; Units 200 and up cost $26 each. So when 150 units are ordered, the total cost is $30 * 100 + $28 * 50. All units discount : an order of 1 -- 1000 units costs $50 each ; an order of 1001 -- 5000 units costs $45 each ; an order of more than 5000 units costs $40 each. So when 1500 units are ordered, the total cost is $45 * 1500. Design of optimal quantity discount schedules In presence of a strategic customer, who responds optimally to discount schedule, the design of optimal quantity discount scheme by the supplier is complex and has to be done carefully. This is particularly so when the demand at the customer is itself uncertain. An interesting effect called the `` reverse bullwhip '' takes place where an increase in consumer demand uncertainty actually reduces order quantity uncertainty at the supplier. Backordering costs and multiple items Several extensions can be made to the EOQ model, including backordering costs and multiple items. Additionally, the economic order interval can be determined from the EOQ and the economic production quantity model ( which determines the optimal production quantity ) can be determined in a similar fashion. A version of the model, the Baumol - Tobin model, has also been used to determine the money demand function, where a person 's holdings of money balances can be seen in a way parallel to a firm 's holdings of inventory. Malakooti ( 2013 ) has introduced the multi-criteria EOQ models where the criteria could be minimizing the total cost, Order quantity ( inventory ), and Shortages. A version taking the time - value of money into account was developed by Trippi and Lewin. Imperfect quality Another important extension of EOQ model is to consider items with imperfect quality. Salameh and Jaber ( 2000 ) are the first to study the imperfect items in an EOQ model very thoroughly. They consider an inventory problem in which the demand is deterministic and there is a fraction of imperfect items in the lot and are screened by the buyer and sold by them at the end of the circle at discount price. Imperfect quality items have also been considered in a decentralized supply chain and the problem has also been studied with game theoretical models. For improving fuel economy of internal combustion engines Recently an interesting similarity between EOQ of Melon picking and fuel injection in Gasoline Direction Injection has been proposed. See also Constant fill rate for the part being produced : Economic production quantity Demand is random : classical Newsvendor model Demand varies over time : Dynamic lot size model Several products produced on the same machine : Economic lot scheduling problem Reorder point Revised Wilson Formula by Daniel CRETOIS ( 1 ) References Jump up ^ Hax, AC ; Candea, D. ( 1984 ), Production and Operations Management, Englewood Cliffs, NJ : Prentice - Hall, p. 135 Jump up ^ Grubbström, Robert W. ( 1995 ). `` Modelling production opportunities -- an historical overview ''. International Journal of Production Economics. 41 : 1 -- 14. doi : 10.1016 / 0925 - 5273 ( 95 ) 00109 - 3. Jump up ^ Nahmias, Steven ( 2005 ). Production and operations analysis. McGraw Hill Higher Education. Jump up ^ Zipkin, Paul H, Foundations of Inventory Management, McGraw Hill 2000 Jump up ^ Altintas, Nihat ; Erhun, Feryal ; Tayur, Sridhar ( 2008 ). `` Quantity Discounts Under Demand Uncertainty ''. Management Science. 54 ( 4 ) : 777 -- 92. doi : 10.1287 / mnsc. 1070.0829. JSTOR 20122426. Jump up ^ Engineering Costs and Production Economics Volume 11, Issue 1, S.K. Goyal ( April 1987 ), Pages 53 - 57. Jump up ^ Caplin, Andrew ; Leahy, John ( 2010 ). `` Economic Theory and the World of Practice : A Celebration of the ( S, s ) Model ''. The Journal of Economic Perspectives. 24 ( 1 ) : 183 -- 201. doi : 10.1257 / jep. 24.1. 183. JSTOR 25703488. Jump up ^ Malakooti, B ( 2013 ). Operations and Production Systems with Multiple Objectives. John Wiley & Sons. ISBN 978 - 1 - 118 - 58537 - 5. Jump up ^ Trippi, Robert R. ; Lewin, Donald E. ( 1974 ). `` A Present Value Formulation of the Classical Eoq Problem ''. Decision Sciences. 5 ( 1 ) : 30 -- 35. doi : 10.1111 / j. 1540 - 5915.1974. tb00592. x. Jump up ^ Salameh, M.K. ; Jaber, M.Y. ( 2000 - 03 ). `` Economic production quantity model for items with imperfect quality ''. International Journal of Production Economics. 64 ( 1 - 3 ) : 59 -- 64. doi : 10.1016 / s0925 - 5273 ( 99 ) 00044 - 4. ISSN 0925 - 5273. Check date values in : date = ( help ) Jump up ^ Elyasi, Milad ; Khoshalhan, Farid ; Khanmirzaee, Mohammad ( 2014 ). `` Modified economic order quantity ( EOQ ) model for items with imperfect quality : Game - theoretical approaches ''. international journal of industrial engineering computations. 5 ( 2 ) : 211 -- 222. doi : 10.5267 / j. ijiec. 2014.1. 003. ISSN 1923 - 2926. Jump up ^ Ventura, Robert ; Samuel, Stephen ( 2016 ). `` Optimization of fuel injection in GDI engine using economic order quantity and Lambert W function ''. Applied Thermal Engineering. 101 : 112 -- 20. doi : 10.1016 / j. applthermaleng. 2016.02. 024. Further reading Harris, Ford W. Operations Cost ( Factory Management Series ), Chicago : Shaw ( 1915 ) Camp, W.E. `` Determining the production order quantity '', Management Engineering, 1922 Wilson, R.H. ( 1934 ). `` A Scientific Routine for Stock Control ''. Harvard Business Review. 13 : 116 -- 28. Plossel, George. Orlicky 's Material Requirement 's Planning. Second Edition. McGraw Hill. 1984. ( first edition 1975 ) Andriolo, Alessandro ; Battini, Daria ; Grubbström, Robert W. ; Persona, Alessandro ; Sgarbossa, Fabio ( 2014 ). `` A century of evolution from Harris ׳ s basic lot size model : Survey and research agenda ''. International Journal of Production Economics. 155 : 16 -- 38. doi : 10.1016 / j. ijpe. 2014.01. 013. Erlenkotter, Donald ( 2014 ). `` Ford Whitman Harris 's economical lot size model ''. International Journal of Production Economics. 155 : 12 -- 15. doi : 10.1016 / j. ijpe. 2013.12. 008. Tsan - Ming Choi ( Ed. ) Handbook of EOQ Inventory Problems : Stochastic and Deterministic Models and Applications, Springer 's International Series in Operations Research and Management Science, 2014. doi : 10.1007 / 978 - 1 - 4614 - 7639 - 9. Ventura, Robert ; Samuel, Stephen ( 2016 ). `` Optimization of fuel injection in GDI engine using economic order quantity and Lambert W function ''. Applied Thermal Engineering. 101 : 112 -- 20. doi : 10.1016 / j. applthermaleng. 2016.02. 024. External links The EOQ Model http://www.inventoryops.com/economic_order_quantity.htm http://www.scmfocus.com/supplyplanning/2014/04/10/economic-order-quantity-calculator/ Retrieved from `` https://en.wikipedia.org/w/index.php?title=Economic_order_quantity&oldid=854229875 '' Categories : Inventory optimization Hidden categories : Wikipedia articles needing page number citations from January 2017 CS1 errors : dates Talk Contents About Wikipedia Azərbaycanca Deutsch Español Euskara Français Italiano עברית Nederlands नेपाली 日本 語 Norsk Polski Português Русский Suomi Svenska Українська 中文 10 more Edit links This page was last edited on 9 August 2018, at 20 : 48 ( UTC ). Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. 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Question: what is basic eoq model of inventory management
Answer: the order quantity that minimizes the total holding costs and ordering costs