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{{Short description|Production scheduling model}}
{{Corporate finance}}
'''Economic order quantity''' ('''EOQ''') is the order quantity that minimizes total inventory holding costs and ordering costs. It is one of the oldest classical production scheduling models. The framework used to determine this order quantity is also known as '''Wilson EOQ Model''' or '''Wilson Formula'''. The model was developed by Ford W. Harris in 1913,<ref>{{cite jstor|170962}}</ref> but R. H. Wilson, a consultant who applied it extensively, is given credit for his in-depth analysis.<ref name=Hax1984>{{Citation
'''Economic order quantity''' ('''EOQ'''), also known as '''financial purchase quantity''' or '''economic buying quantity''',{{cn|date=November 2023}} is the order quantity that minimizes the total [[holding cost]]s and [[ordering cost]]s in [[Field inventory management|inventory management]]. It is one of the oldest classical [[Scheduling (production processes)|production scheduling]] models. The model was developed by [[Ford W. Harris]] in 1913, but the consultant '''R. H. Wilson''' applied it extensively, and he and K. Andler are given credit for their in-depth analysis.<ref name="Hax1984">{{Citation
| title = Production and Operations Management
| title = Production and Operations Management
| url = http://catalogue.nla.gov.au/Record/772207
| url = http://catalogue.nla.gov.au/Record/772207
| year = 1984
| year = 1984
| author = Hax, AC and Candea, D.
|author1=Hax, AC |author2=Candea, D.
| publisher = Prentice-Hall, Englewood Cliffs, NJ
| publisher = Prentice-Hall |location=Englewood Cliffs, NJ
| pages = 135
| page = 135
| isbn = 9780137248803
}}</ref>
}}</ref>


==Overview==
==Overview==
The EOQ indicates the optimal number of units to order to minimize the total cost associated with the purchase, delivery, and storage of a product.
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.


EOQ applies only when [[demand]] for a product is constant over a period of time (such as a 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 quantity of items ordered; an order is assumed to contain only one type of inventory item. 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. Although the EOQ formulation is straightforward, factors such as transportation rates and quantity discounts factor into its real-world application.
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.
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 for a single item 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===
===Variables===
*<math>P</math> = purchase price, unit production cost
*<math>T</math> = total annual inventory cost
*<math>P</math> = purchase unit price, unit production cost
*<math>Q</math> = order quantity
*<math>Q</math> = order quantity
*<math>Q^*</math> = optimal order quantity
*<math>Q^*</math> = optimal order quantity
*<math>D</math> = annual demand quantity
*<math>D</math> = annual demand quantity
*<math>K</math> = fixed cost per order, setup cost (''not'' per unit, typically cost of ordering and shipping and handling. This is not the cost of goods)
*<math>K</math> = fixed cost per order, setup cost (''not'' per unit, typically cost of ordering and shipping and handling. This is not the cost of goods)
*<math>h</math> = 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)
*<math>h</math> = annual holding cost per unit, also known as carrying cost or storage cost (capital cost, warehouse space, refrigeration, insurance, [[opportunity cost]] (price x interest), etc. usually not related to the unit production cost)


===The Total Cost function===
===Total cost function and derivation of EOQ formula===

[[Image:Eoq inventory 0001.png|thumb|right|320px|Classic EOQ model: trade-off between ordering cost (blue) and [[holding cost]] (red). Total cost (green) admits a [[global optimum]]. Purchase cost is not a [[relevant cost]] for determining the optimal order quantity. ]]


The single-item EOQ formula finds the minimum point of the following cost function:
The single-item EOQ formula finds the minimum point of the following cost function:
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Total Cost = purchase cost or production cost + ordering cost + holding cost
Total Cost = purchase cost or production cost + ordering cost + holding cost


Where:
- Purchase cost: This is the variable cost of goods: purchase unit price &times; annual demand quantity. This is P &times; 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 &times; D/Q
* Purchase cost: This is the variable cost of goods: purchase unit price &times; annual demand quantity. This is <math>P \times D</math>.
* Ordering cost: This is the cost of placing orders: each order has a fixed cost <math>K</math>, and we need to order <math>D/Q</math> times per year. This is <math>KD/Q</math>
* Holding cost: the average quantity in stock (between fully replenished and empty) is <math>Q/2</math>, so this cost is <math>hQ/2</math>


: <math>T = PD + K {\frac{D}{Q}} + h {\frac{Q}{2}}</math>.
- Holding cost: the average quantity in stock (between fully replenished and empty) is Q/2, so this cost is h &times; Q/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:
<math>TC = PD + {\frac{DK}{Q}} + {\frac{hQ}{2}}</math>.


: <math>{0} = -{\frac{DK}{Q^2}}+{\frac{h}{2}}</math>
To determine the minimum point of the total cost curve, partially differentiate the total cost with respect to Q (assume all other variables are constant) and set to 0:

<math>{0} = -{\frac{DK}{Q^2}}+{\frac{h}{2}}</math>


Solving for Q gives Q* (the optimal order quantity):
Solving for Q gives Q* (the optimal order quantity):


<math>Q^{*2}={\frac{2DK}{h}}</math>
: <math>Q^{*2}={\frac{2DK}{h}}</math>


Therefore:
Therefore:
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Q* is independent of P; it is a function of only K, D, h.
Q* is independent of P; it is a function of only K, D, h.


The optimal value Q* may also be found by recognizing that
The optimal value Q* may also be found by recognising that<ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/0925-5273(95)00109-3 | title = Modelling production opportunities — an historical overview | journal = International Journal of Production Economics| volume = 41| pages = 1–14 | year = 1995}}</ref>


<math>TC = {\frac{DK}{Q}} + {\frac{hQ}{2}} + PD ={\frac{h}{2Q}}(Q - \sqrt{2DK/h})^2 + \sqrt{2hDK} +PD, </math>
: <math>T = {\frac{DK}{Q}} + {\frac{hQ}{2}} + PD ={\frac{h}{2Q}}(Q - \sqrt{2DK/h})^2 + \sqrt{2hDK} +PD, </math>
where the non-negative quadratic term disappears for <math>Q = \sqrt{2DK/h}, </math> which provides the cost minimum <math>TC_{min} = \sqrt{2hDK} + PD. </math>
where the non-negative quadratic term disappears for <math display="inline">Q = \sqrt{2DK/h}, </math> which provides the cost minimum <math>T_{min} = \sqrt{2hDK} + PD. </math>


==Quantity Discounts==
=== Example ===
*Annual requirement quantity (D) = 10000 units
An important extension to the EOQ model of Wilson is to accommodate quantity discounts. There are two main types of quantity discounts: (1) all-units and (2) incremental.<ref>{{Cite book | title = Production and operations analysis | last1 = Nahmias | first1 = Steven | year = 2005 | publisher = McGraw Hill Higher Education }}</ref> Here is a numerical example.
*Cost per order (K) = 40
*Cost per unit (P)= 50
*Yearly carrying cost per unit = 4
*Market interest = 2%

Economic order quantity = <math> \sqrt{\frac{2D\cdot K}{h}} </math> <math> = \sqrt{\frac{2\cdot 10000\cdot 40}{4 + 50 \cdot 2\%}} = \sqrt{\frac{2\cdot 10000 \cdot 40}{5}}</math> = 400 units

Number of orders per year (based on EOQ) <math> = {\frac{10000}{400}} = 25 </math>

Total cost <math> = P\cdot D + K (D/EOQ) + h (EOQ/2) </math>

Total cost <math> = 50\cdot 10000 + 40\cdot (10000/400) + 5\cdot (400/2) = 502000 </math>

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 <math> = 50\cdot 10000 + 40\cdot (10000/500) + 5\cdot (500/2) = 502050 </math>

Similarly, if we choose 300 for the order quantity, then

Total cost <math> = 50\cdot 10000 + 40\cdot (10000/300) + 5\cdot (300/2) = 502083.33 </math>

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.<ref>{{Cite book | title = Production and operations analysis | last1 = Nahmias | first1 = Steven | year = 2005 | publisher = McGraw Hill Higher Education }}{{page needed|date=January 2017}}</ref><ref>Zipkin, Paul H, Foundations of Inventory Management, McGraw Hill 2000{{page needed|date=January 2017}}</ref> 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.
* 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.
* All units discount: An order of 1-1000 units costs $30 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.


In order to find the optimal order quantity under different quantity discount schemes, one should use algorithms; these algorithms are developed under the assumption that the EOQ policy is still optimal with quantity discounts. Perera et al. (2017)<ref>{{cite journal |doi=10.1016/j.ijpe.2016.09.017 |title=Optimality of (s,S) policies in EOQ models with general cost structures |journal=International Journal of Production Economics |volume=187 |pages=216–228 |year=2017 |last1=Perera |first1=Sandun |last2=Janakiraman |first2=Ganesh |last3=Niu |first3=Shun-Chen }}</ref> establish this optimality and fully characterize the (s,S) optimality within the EOQ setting under general cost structures.
===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.<ref>{{cite doi | 10.1287/mnsc.1070.0829}}</ref>


===Design of optimal quantity discount schedules===
==Other Extensions==
In presence of a strategic customer, who responds optimally to discount schedules, the design of an 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.<ref>{{cite journal |doi=10.1287/mnsc.1070.0829 |jstor=20122426 |title=Quantity Discounts Under Demand Uncertainty |journal=Management Science |volume=54 |issue=4 |pages=777–92 |year=2008 |last1=Altintas |first1=Nihat |last2=Erhun |first2=Feryal |last3=Tayur |first3=Sridhar }}</ref>
Several extensions can be made to the EOQ model developed by Mr. Pankaj Mane, 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.


===Backordering costs and multiple items===
A version of the model, the [[Baumol-Tobin]] model, has also been used to determine the [[Money demand#Inventory models|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.<ref name=jep>Andrew Caplin and John Leahy, "Economic Theory and the World of Practice: A Celebration of the (S,s) Model", ''[[Journal of Economic Perspectives]]'', Winter 2010, V 24, N 1</ref>
Several extensions can be made to the EOQ model, including backordering costs<ref>{{cite journal |doi=10.1016/j.ijpe.2016.09.017 |title=Optimality of (s,S) policies in EOQ models with general cost structures |journal=International Journal of Production Economics |volume=187 |pages=216–228 |year=2017 |last1=Perera |first1=Sandun |last2=Janakiraman |first2=Ganesh |last3=Niu |first3=Shun-Chen }}</ref> and multiple items. In the case backorders are permitted, the inventory carrying costs per cycle are:<ref>[[Thomson M. Whitin|T. Whitin]], G. Hadley, Analysis of Inventory Systems, Prentice Hall 1963</ref>


: <math>IC \int\limits_{0}^{T_1}(Q-s-\lambda t)\,dt = \frac{IC}{2 \lambda} (Q-s)^2,</math>
== Example ==


where s is the number of backorders when order quantity Q is delivered and <math>\lambda</math> is the rate of demand. The backorder cost per cycle is:
*Suppose annual requirement quantity (D) = 10000 units
*Cost per order (K) = $2
*Cost per unit (P)= $8
*Carrying cost percentage (h/P)(percentage of P) = 0.02
*Annual carrying cost per unit (h) = $0.16


: <math>\pi s + \hat{\pi} \int\limits_{0}^{T_2}\lambda t dt =\pi s +\frac{1}{2} \hat{\pi} \lambda T^{2}_{2} = \pi s + \frac{ \hat{\pi} s^{2}}{2\lambda},</math>
Economic order quantity = <math> \sqrt{\frac{2D*K}{h}} </math> <math> = \sqrt{\frac{2*10000*2}{8*0.02}} </math> = 500 units


where <math>\pi</math> and <math>\hat{\pi}</math> are backorder costs, <math>T_{2}=T-T_{1}</math>, T being the cycle length and <math>T_{1}=(Q-s) / \lambda</math>. The average annual variable cost is the sum of order costs, holding inventory costs and backorder costs:
Number of orders per year (based on EOQ) <math> = {\frac{10000}{500}} = 20 </math>


: <math>\mathcal{K} = \frac{\lambda}{Q} A+\frac{1}{2Q} IC (Q-s)^2+\frac{1} {Q} [ \pi \lambda s+ \frac{1}{2} \hat{\pi} s^{2}]</math>
Total cost <math> = P*D + K (D/EOQ) + h (EOQ/2) </math>


To minimize <math>\mathcal{K}</math> impose the [[partial derivatives]] equal to zero:
Total cost <math> = 8*10000 + 2 (10000/500) + 0.16 (500/2) = $80080 </math>


: <math>\frac{\partial \mathcal{K}}{\partial Q} =- \frac{1}{Q^2} \left[ {\lambda} A+\frac{1}{2} IC (Q-s)^2+\pi \lambda s+ \frac{1}{2} \hat{\pi} s^{2} \right]+\frac{IC}{Q}(Q-s)=0</math>
If we check the total cost for any order quantity other than 500(=EOQ), we will see that the cost is higher. For instance, supposing 600 units per order, then
: <math>\frac{\partial \mathcal{K}}{\partial s} =-\frac{IC}{Q}(Q-s) + \frac{1}{Q} \pi \lambda + \frac{1}{Q} \hat{\pi} s =0</math>


Substituting the second equation into the first gives the following [[quadratic equation]]:
Total cost <math> = 8*10000 + 2 (10000/600) + 0.16 (600/2) = $80081 </math>


: <math>[\hat{\pi} ^{2} + \hat{\pi} IC] s^2 +2\pi \hat{\pi} \lambda s+(\pi \lambda) ^2 -2 \lambda A IC=0</math>
Similarly, if we choose 300 for the order quantity then


If <math>\hat{\pi}=0</math> either s=0 or <math>s=\infty</math> is optimal. In the first case the optimal lot is given by the classic EOQ formula, in the second case an order is never placed and minimum yearly cost is given by <math>\pi \lambda</math>. If <math>\pi > \sqrt{\frac{2AIC}{\lambda}} =\delta</math> or <math>\pi \lambda > K_{w}</math> <math>s^*=0</math> is optimal, if <math>\pi<\delta</math> then there shouldn't be any inventory system. If <math>\hat{\pi}\ne0</math> solving the preceding quadratic equation yields:
Total cost <math> = 8*10000 + 2 (10000/300) + 0.16 (300/2) = $80091</math>


: <math>s^* = [\hat {\pi} + IC] ^{-1} \left ( -\pi \lambda + \left [ (2\lambda AIC) \left ( 1 + \frac{IC} {\hat{\pi}} \right)- \frac{IC}{\hat{\pi}}(\pi \lambda )^{2} \right]^{1/2} \right ) </math>
This illustrates that the Economic Order Quantity is always in the best interests of the entity.
: <math>Q^* = \left [ \frac{\hat{\pi}+IC}{ \hat{\pi}} \right]^{1/2} \left [ \frac{2 \lambda A}{IC} -\frac{(\pi \lambda)^2}{IC(\hat{\pi}+IC)} \right]^{1/2}</math>


If there are [[backorder]]s, the reorder point is: <math>r^*_{h} = \mu - mQ^* - s^*</math>; with m being the largest integer <math>m \leq \frac{\tau}{T}</math> and μ the lead time demand.
==Multi-Criteria EOQ==

Malakooti (2013) <ref>{{cite book|last1=Malakooti|first1=B|title=Operations and Production Systems with Multiple Objectives|date=2013|publisher=John Wiley & Sons|isbn=978-1-118-58537-5}}</ref> has introduced the multi-criteria EOQ models where the criteria could be minimizing the total cost, Order quantity (inventory), and Shortages.
Additionally, the economic order interval<ref>{{Cite journal |doi = 10.1016/0167-188X(87)90025-5|title = A simple heuristic method for determining economic order interval for linear demand|journal = Engineering Costs and Production Economics|volume = 11|pages = 53–57|year = 1987|last1 = Goyal|first1 = S.K.}}</ref> 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#Inventory models|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.<ref name=jep>{{cite journal |doi=10.1257/jep.24.1.183 |first1=Andrew |last1=Caplin |first2=John |last2=Leahy |year=2010 |title=Economic Theory and the World of Practice: A Celebration of the (s, S) Model |journal=The Journal of Economic Perspectives |volume=24 |issue=1 |pages=183–201 |jstor=25703488 |citeseerx=10.1.1.730.8784 }}</ref>

[[Behnam Malakooti|Malakooti]] (2013)<ref>{{cite book|last1=Malakooti|first1=B|title=Operations and Production Systems with Multiple Objectives|date=2013|publisher=John Wiley & Sons|isbn=978-1-118-58537-5}}{{page needed|date=January 2017}}</ref> 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.<ref name="tri">{{cite journal |doi=10.1111/j.1540-5915.1974.tb00592.x |title=A Present Value Formulation of the Classical Eoq Problem |journal=Decision Sciences |volume=5 |issue=1 |pages=30–35 |year=1974 |last1=Trippi |first1=Robert R. |last2=Lewin |first2=Donald E. }}</ref>

=== Imperfect quality ===
Another important extension of the EOQ model is to consider items with imperfect quality. Salameh and Jaber (2000) were 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.<ref>{{Cite journal|last1=Salameh|first1=M.K.|last2=Jaber|first2=M.Y.|date=March 2000|title=Economic production quantity model for items with imperfect quality|journal=International Journal of Production Economics|volume=64|issue=1–3|pages=59–64|doi=10.1016/s0925-5273(99)00044-4|issn=0925-5273}}</ref>

==Criticisms==
The EOQ model and its sister, the [[economic production quantity]] model (EPQ), have been criticised for "their restrictive set[s] of assumptions.<ref>Tao, Z., A. L. Guiffrida, and M. D. Troutt, "A green cost based economic production/order quantity model", in ''Proceedings of the 1st Annual Kent State International Symposium on Green Supply Chains'', Canton, Ohio, US, 29–30 July 2010</ref> Guga and Musa make use of the model for an Albanian business case study and conclude that the model is "perfect theoretically, but not very suitable from the practical perspective of this firm".<ref>Guga, E. and Musa, O. (2015) in [https://ijecm.co.uk/wp-content/uploads/2015/12/31211.pdf Inventory Management through EOQ Model], ''International Journal of Economics, Commerce & Management'', Vol. III, Issue 12, December 2015, accessed 9 February 2024</ref> However, James Cargal notes that the formula was developed when business calculations were undertaken "by hand", or using [[logarithmic table]]s or a [[slide rule]]. Use of [[spreadsheet]]s and specialist software allows for more versatility in the use of the formula and adoption of "assumptions which are more realistic" than in the original model.<ref>Cargal, J. M. (2003), [http://www.cargalmathbooks.com/The%20EOQ%20Formula.pdf The EOQ Formula], ''Troy University'', accessed 9 February 2024</ref>{{Self-published inline|date=February 2024|certain=y|reason=Commended by Guga, E. and Musa, O. (2015) in previous citation}}


==See also==
==See also==


* Costant 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]]
* [[Reorder point]]
* [[Safety stock]]
*[[Economic production quantity]]
*[[Newsvendor model]]
*[[Dynamic lot size model]]
*[[Economic lot scheduling problem]]


==References==
==References==
{{Reflist}}
<references/>


==Further reading==
==Further reading==
* Harris, Ford W. ''Operations Cost'' (Factory Management Series), Chicago: Shaw (1915)
* Harris, Ford W. ''Operations Cost'' (Factory Management Series), Chicago: Shaw (1915)
*{{cite journal |last1=Harris |first1=Ford W. |title=How many parts to make at once |journal=Factory, the Magazine of Management |volume=10 |pages=135–136, 152 |year=1913 }}
* Wilson, R. H. "A Scientific Routine for Stock Control", Harvard Business Review, 13, 116-128 (1934)
* Camp, W. E. "Determining the production order quantity", Management Engineering, 1922
* Camp, W. E. "Determining the production order quantity", Management Engineering, 1922
*{{cite journal |last1=Wilson |first1=R. H. |title=A Scientific Routine for Stock Control |journal=Harvard Business Review |volume=13 |pages=116–28 |year=1934 }}
* Plossel, George. Orlicky's Material Requirement's Planning. Second Edition. McGraw Hill. 1984. (first edition 1975)
* Plossel, George. Orlicky's Material Requirement's Planning. Second Edition. McGraw Hill. 1984. (first edition 1975)
*{{cite journal |doi=10.1016/j.ijpe.2013.12.008 |title=Ford Whitman Harris's economical lot size model |journal=International Journal of Production Economics |volume=155 |pages=12–15 |year=2014 |last1=Erlenkotter |first1=Donald |s2cid=153794306 |url=https://escholarship.org/uc/item/4s8482p2 }}
*{{cite journal |doi=10.1016/j.ijpe.2016.09.017 |title=Optimality of (s,S) policies in EOQ models with general cost structures |journal=International Journal of Production Economics |volume=187 |pages=216–228 |year=2017 |last1=Perera |first1=Sandun |last2=Janakiraman |first2=Ganesh |last3=Niu |first3=Shun-Chen }}
*{{cite journal |doi=10.1111/poms.12795 |title=Optimality of (s, S) Inventory Policies under Renewal Demand and General Cost Structures |journal=Production and Operations Management |volume=27 |issue=2 |pages=368–383 |year=2018 |last1=Perera |first1=Sandun |last2=Janakiraman |first2=Ganesh |last3=Niu |first3=Shun-Chen |hdl=2027.42/142450 |hdl-access=free }}
* 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}}.
* {{cite journal |doi=10.1016/j.applthermaleng.2016.02.024 |title=Optimization of fuel injection in GDI engine using economic order quantity and Lambert W function |journal=Applied Thermal Engineering |volume=101 |pages=112–20 |year=2016 |last1=Ventura |first1=Robert |last2=Samuel |first2=Stephen |url=https://radar.brookes.ac.uk/radar/items/cc17b1a1-d1c6-4b76-a34a-c8689ea0270f/1/ }}
* Renewal Demand and (s, S) Optimality by Perera, Janakiraman, and Niu [https://onlinelibrary.wiley.com/doi/full/10.1111/poms.12795]


==External links==
==External links==
*[http://www.logisitik.com/learning-center/inventory-management/item/466-economic-order-quantity-eoq-model.html The EOQ Model]
*[http://www.logisitik.com/learning-center/inventory-management/item/466-economic-order-quantity-eoq-model.html The EOQ Model]
* http://www.inventoryops.com/economic_order_quantity.htm
*Piasecki, D., [http://www.inventoryops.com/economic_order_quantity.htm Vol. III, Issue 12, December 2015]

*http://www.scmfocus.com/supplyplanning/2014/04/10/economic-order-quantity-calculator/
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[[de:Klassische Losformel]]

Latest revision as of 16:51, 15 December 2024

Economic order quantity (EOQ), also known as financial purchase quantity or economic buying quantity,[citation needed] is the order quantity that minimizes the total holding costs and ordering costs in inventory management. It is one of the oldest classical production scheduling models. The model was developed by Ford W. Harris in 1913, but the consultant R. H. Wilson applied it extensively, and he and K. Andler are given credit for their in-depth analysis.[1]

Overview

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The EOQ indicates the optimal number of units to order to minimize the total cost associated with the purchase, delivery, and storage of a product.

EOQ applies only when demand for a product is constant over a period of time (such as a 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 quantity of items ordered; an order is assumed to contain only one type of inventory item. 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. Although the EOQ formulation is straightforward, factors such as transportation rates and quantity discounts factor into its real-world application.

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 for a single item 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

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  • = total annual inventory cost
  • = purchase unit price, unit production cost
  • = order quantity
  • = optimal order quantity
  • = annual demand quantity
  • = fixed cost per order, setup cost (not per unit, typically cost of ordering and shipping and handling. This is not the cost of goods)
  • = annual holding cost per unit, also known as carrying cost or storage cost (capital cost, warehouse space, refrigeration, insurance, opportunity cost (price x interest), etc. usually not related to the unit production cost)

Total cost function and derivation of EOQ formula

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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 .
  • Ordering cost: This is the cost of placing orders: each order has a fixed cost , and we need to order times per year. This is
  • Holding cost: the average quantity in stock (between fully replenished and empty) is , so this cost is
.

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:

Solving for Q gives Q* (the optimal order quantity):

Therefore:

Economic Order Quantity

Q* is independent of P; it is a function of only K, D, h.

The optimal value Q* may also be found by recognizing that

where the non-negative quadratic term disappears for which provides the cost minimum

Example

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  • Annual requirement quantity (D) = 10000 units
  • Cost per order (K) = 40
  • Cost per unit (P)= 50
  • Yearly carrying cost per unit = 4
  • Market interest = 2%

Economic order quantity = = 400 units

Number of orders per year (based on EOQ)

Total cost

Total cost

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

Similarly, if we choose 300 for the order quantity, then

Total cost

This illustrates that the economic order quantity is always in the best interests of the firm.

Extensions of the EOQ model

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Quantity discounts

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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.[2][3] 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.

In order to find the optimal order quantity under different quantity discount schemes, one should use algorithms; these algorithms are developed under the assumption that the EOQ policy is still optimal with quantity discounts. Perera et al. (2017)[4] establish this optimality and fully characterize the (s,S) optimality within the EOQ setting under general cost structures.

Design of optimal quantity discount schedules

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In presence of a strategic customer, who responds optimally to discount schedules, the design of an 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.[5]

Backordering costs and multiple items

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Several extensions can be made to the EOQ model, including backordering costs[6] and multiple items. In the case backorders are permitted, the inventory carrying costs per cycle are:[7]

where s is the number of backorders when order quantity Q is delivered and is the rate of demand. The backorder cost per cycle is:

where and are backorder costs, , T being the cycle length and . The average annual variable cost is the sum of order costs, holding inventory costs and backorder costs:

To minimize impose the partial derivatives equal to zero:

Substituting the second equation into the first gives the following quadratic equation:

If either s=0 or is optimal. In the first case the optimal lot is given by the classic EOQ formula, in the second case an order is never placed and minimum yearly cost is given by . If or is optimal, if then there shouldn't be any inventory system. If solving the preceding quadratic equation yields:

If there are backorders, the reorder point is: ; with m being the largest integer and μ the lead time demand.

Additionally, the economic order interval[8] 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.[9]

Malakooti (2013)[10] 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.[11]

Imperfect quality

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Another important extension of the EOQ model is to consider items with imperfect quality. Salameh and Jaber (2000) were 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.[12]

Criticisms

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The EOQ model and its sister, the economic production quantity model (EPQ), have been criticised for "their restrictive set[s] of assumptions.[13] Guga and Musa make use of the model for an Albanian business case study and conclude that the model is "perfect theoretically, but not very suitable from the practical perspective of this firm".[14] However, James Cargal notes that the formula was developed when business calculations were undertaken "by hand", or using logarithmic tables or a slide rule. Use of spreadsheets and specialist software allows for more versatility in the use of the formula and adoption of "assumptions which are more realistic" than in the original model.[15][self-published source]

See also

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References

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  1. ^ Hax, AC; Candea, D. (1984), Production and Operations Management, Englewood Cliffs, NJ: Prentice-Hall, p. 135, ISBN 9780137248803
  2. ^ Nahmias, Steven (2005). Production and operations analysis. McGraw Hill Higher Education.[page needed]
  3. ^ Zipkin, Paul H, Foundations of Inventory Management, McGraw Hill 2000[page needed]
  4. ^ Perera, Sandun; Janakiraman, Ganesh; Niu, Shun-Chen (2017). "Optimality of (s,S) policies in EOQ models with general cost structures". International Journal of Production Economics. 187: 216–228. doi:10.1016/j.ijpe.2016.09.017.
  5. ^ 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.
  6. ^ Perera, Sandun; Janakiraman, Ganesh; Niu, Shun-Chen (2017). "Optimality of (s,S) policies in EOQ models with general cost structures". International Journal of Production Economics. 187: 216–228. doi:10.1016/j.ijpe.2016.09.017.
  7. ^ T. Whitin, G. Hadley, Analysis of Inventory Systems, Prentice Hall 1963
  8. ^ Goyal, S.K. (1987). "A simple heuristic method for determining economic order interval for linear demand". Engineering Costs and Production Economics. 11: 53–57. doi:10.1016/0167-188X(87)90025-5.
  9. ^ 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. CiteSeerX 10.1.1.730.8784. doi:10.1257/jep.24.1.183. JSTOR 25703488.
  10. ^ Malakooti, B (2013). Operations and Production Systems with Multiple Objectives. John Wiley & Sons. ISBN 978-1-118-58537-5.[page needed]
  11. ^ 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.
  12. ^ Salameh, M.K.; Jaber, M.Y. (March 2000). "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.
  13. ^ Tao, Z., A. L. Guiffrida, and M. D. Troutt, "A green cost based economic production/order quantity model", in Proceedings of the 1st Annual Kent State International Symposium on Green Supply Chains, Canton, Ohio, US, 29–30 July 2010
  14. ^ Guga, E. and Musa, O. (2015) in Inventory Management through EOQ Model, International Journal of Economics, Commerce & Management, Vol. III, Issue 12, December 2015, accessed 9 February 2024
  15. ^ Cargal, J. M. (2003), The EOQ Formula, Troy University, accessed 9 February 2024

Further reading

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  • Harris, Ford W. Operations Cost (Factory Management Series), Chicago: Shaw (1915)
  • Harris, Ford W. (1913). "How many parts to make at once". Factory, the Magazine of Management. 10: 135–136, 152.
  • 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)
  • 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. S2CID 153794306.
  • Perera, Sandun; Janakiraman, Ganesh; Niu, Shun-Chen (2017). "Optimality of (s,S) policies in EOQ models with general cost structures". International Journal of Production Economics. 187: 216–228. doi:10.1016/j.ijpe.2016.09.017.
  • Perera, Sandun; Janakiraman, Ganesh; Niu, Shun-Chen (2018). "Optimality of (s, S) Inventory Policies under Renewal Demand and General Cost Structures". Production and Operations Management. 27 (2): 368–383. doi:10.1111/poms.12795. hdl:2027.42/142450.
  • 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.
  • Renewal Demand and (s, S) Optimality by Perera, Janakiraman, and Niu [1]
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