Economic Order Quantities Economic Order Quantities (EOQs)(EOQs)

The concept of an economic-order The concept of an economic-order quantity addresses the question of quantity addresses the question of how much to order at one time.how much to order at one time.

DefinitionDefinition:: The Economic order quantity The Economic order quantity (EOQ) is the optimum ordering quantity for (EOQ) is the optimum ordering quantity for an item of stock that minimizes cost.an item of stock that minimizes cost.

To calculate the EOQ a mathematical model of To calculate the EOQ a mathematical model of reality must be constructed. reality must be constructed.

Some assumptions are made to simplify reality .Some assumptions are made to simplify reality . When an assumption is modified or deleted a new When an assumption is modified or deleted a new

model must be constructedmodel must be constructed..

Assumptions for EOQ Basic Assumptions for EOQ Basic ModelModel

Demand is uniform i.e. constant and Demand is uniform i.e. constant and continuous over time.continuous over time.

There is no limit on order size due either to There is no limit on order size due either to stores capacity or other constraints.stores capacity or other constraints.

The cost of placing is independent of the size The cost of placing is independent of the size of order and delivery charge is also of order and delivery charge is also independent of the size of order.independent of the size of order.

The cost of holding a unit of stock does The The cost of holding a unit of stock does The lead time is constant and certain.lead time is constant and certain.

All prices are constant and certain. There are All prices are constant and certain. There are no bulk purchase discounts.no bulk purchase discounts.

Exactly the same quantity is ordered each Exactly the same quantity is ordered each time that a purchase is made.time that a purchase is made.

Algebraic Formula for EOQAlgebraic Formula for EOQby Ford Harris in1913by Ford Harris in1913

The basic EOQ formula is developed from a total cost The basic EOQ formula is developed from a total cost equation involving procurement cost and inventory equation involving procurement cost and inventory carrying cost. It is expressed ascarrying cost. It is expressed as

______ ______ √ √ 2ca /h2ca /h Q= order quantity in unitsQ= order quantity in units A= Annual usage in unitsA= Annual usage in units C= Acquisition cost per unitC= Acquisition cost per unit P= price per unitP= price per unit H= holding cost per unitH= holding cost per unit If the holding cost is given as percentage of for exampleIf the holding cost is given as percentage of for example

ExampleExample

A manufacturer has an annual requirement of 6000 A manufacturer has an annual requirement of 6000 units of a component.units of a component.

Each component costs 2 L.E.Each component costs 2 L.E. Ordering costs 15L.E.per orderOrdering costs 15L.E.per order Holding cost of one unit is 5% of its price (cost)Holding cost of one unit is 5% of its price (cost) There are no bulk discountsThere are no bulk discounts Calculate the optimum order quantity by examining Calculate the optimum order quantity by examining

10 possibilities from ordering once a year to ten 10 possibilities from ordering once a year to ten times a year.times a year.

Use a graphical method to show your answer.Use a graphical method to show your answer. Calculate the EOQ using the Algebraic FormulaCalculate the EOQ using the Algebraic Formula

Number of orders per yearQuantity ordered (units)Average Stock (units)cost of ordering (L.E.)Holding cost (L.E.)Total cost (L.E.)1 6000 3000 15 300 3152 3000 1500 30 150 1803 2000 1000 45 100 1454 1500 750 60 75 1355 1200 600 75 60 1356 1000 500 90 50 1407 847 428 105 42.8 147.58 750 375 120 27.5 157.59 667 333 135 33.3 168.3

10 600 300 150 30 180

Economic Order Quantity

0

50

100

150

200

250

300

350

600030002000150012001000847750667600

Order Quantity

Cost

in L

.E.

cost of ordering (L.E.) Holding cost (L.E.) Total cost (L.E.)

Using the FormulaUsing the Formula

EOQ =EOQ =square root of 2ca/hsquare root of 2ca/h square root of square root of

2x15x6000/5%x2=13422x15x6000/5%x2=1342

Demand Ordering Demand Ordering SystemSystem

Demand ordering systems answers Demand ordering systems answers the question of when to place a the question of when to place a replacement order at one time. replacement order at one time.

An order may be placed when An order may be placed when needed or at a specific time needed or at a specific time ( ( e.g.e.g.every monthevery month) or when stock ) or when stock falls to a predetermined level.falls to a predetermined level.

Basic reorder systemsBasic reorder systems

Three basic systems are used to Three basic systems are used to determine when to order determine when to order Order point systemOrder point system periodic review systemperiodic review system material requirements planning.material requirements planning.

The first two are for independent The first two are for independent demand items, the last is for demand items, the last is for dependent demand items.dependent demand items.

Order Point SystemOrder Point System

When a quantity of an item onhand falls When a quantity of an item onhand falls to a predetermined level, called an to a predetermined level, called an order point, an order is placed. The order point, an order is placed. The quantity ordered is usually quantity ordered is usually precalculated and based on EOQ precalculated and based on EOQ concepts.concepts.

The item is ordered when the quantity The item is ordered when the quantity on hand falls to a level equal to the on hand falls to a level equal to the demand during the lead time plus the demand during the lead time plus the safety stock.safety stock.

OP=DDLTOP=DDLT++SSSS

where where OP= order pointOP= order point

DDLT= demand during the lead timeDDLT= demand during the lead time

SS= safety stock.SS= safety stock.

ExampleExample

Demand is 200 units a week, the Demand is 200 units a week, the lead time is three weeks, and safety lead time is three weeks, and safety stock is 300 units. Calculate the stock is 300 units. Calculate the order point.order point.

AnswerAnswerOP= DDLT + SSOP= DDLT + SS

=200x3=200x3++300300

=900 units=900 units

Determining Safety Determining Safety StockStock

Safety stock increases as the Safety stock increases as the uncertainty increasesuncertainty increases

Uncertainty is reflected in the Uncertainty is reflected in the deviation of actual demand from the deviation of actual demand from the forecast demand forecast demand

The standard deviation Sigma isThe standard deviation Sigma is used to used to measure how closely the measure how closely the individual values cluster about the individual values cluster about the averageaverage

How to calculate How to calculate Standard deviation Standard deviation

(Sigma(Sigma)) Calculate the deviation for each period by Calculate the deviation for each period by subtracting the actual demand from the subtracting the actual demand from the forecast demandforecast demand

Square each deviatiionSquare each deviatiion Add the squares of the deviationsAdd the squares of the deviations Divide the value is step3 by the number of Divide the value is step3 by the number of

periods to determine the average of the periods to determine the average of the squared deviations.squared deviations.

Calculate the square root of the value Calculate the square root of the value calculated in step 4. This is the standard calculated in step 4. This is the standard deviation.deviation.

ExampleExample

Given the data in the following table Given the data in the following table calculate the standard deviation (sigma) calculate the standard deviation (sigma) and usand usee the answer to the answer to calculate the calculate the safety stock for an 84%(1 Sigma) service safety stock for an 84%(1 Sigma) service level.level.

If a safety stock equal to two standard If a safety stock equal to two standard deviations is carried, calculate the deviations is carried, calculate the safety stock and the order point if the safety stock and the order point if the demand during the lead time is 1000 demand during the lead time is 1000 units.units.

Period Forecast demandActual demandDeviation deviation squared1 1000 1200 200 400002 1000 1000 0 03 1000 800 -200 400004 1000 900 -100 100005 1000 1400 400 1400006 1000 1100 100 100007 1000 1100 100 100008 1000 700 -300 900009 1000 1000 0 0

10 1000 800 -200 40000

AnswerAnswer

Average of the square deviation Average of the square deviation =380000/10=38000=380000/10=38000

Sigma=square root of 38000=194.9=195 Sigma=square root of 38000=194.9=195 unitsunits

Safety stock if service level is 84%= Safety stock if service level is 84%= 1Sigma=1x195=195 units1Sigma=1x195=195 units

Order point=DDLT+SSOrder point=DDLT+SS =1000+195=1195=1000+195=1195 with this order point and level of safety stock on the with this order point and level of safety stock on the

average taverage there are no stock outs 84% of the time here are no stock outs 84% of the time when a stockout is possiblewhen a stockout is possible

Safety StockSafety Stock and Service and Service levellevel

Safety Stock is a calculated extra amount Safety Stock is a calculated extra amount of stock carried and is generally used to of stock carried and is generally used to protect against quantity uncertainty protect against quantity uncertainty during the lduring the leead time.ad time.

The service level is a statement of the The service level is a statement of the percentage of time there is no stock outpercentage of time there is no stock out when a stock out is possiblewhen a stock out is possible..

The service level is directly related to the The service level is directly related to the number of standard deviations provided number of standard deviations provided as safety stock called ( safety factoras safety stock called ( safety factor))

Table of safety factorTable of safety factor

Service Level % Safety factor

507580859094959697989999.8699.99

0.000.670.841.041.281.561.671.751.882.052.333.004.00

ExampleExample

If the standard deviation is 200 If the standard deviation is 200 units, what safety stock should be units, what safety stock should be carried to provide a service level of carried to provide a service level of 90%? If the expected demand during 90%? If the expected demand during the lead time is 1500 units, what is the lead time is 1500 units, what is the order pointthe order point

AnswerAnswer

From the table of of safety factor, the From the table of of safety factor, the safety factor for a service level of 90% safety factor for a service level of 90% is 1.28. Therefore,is 1.28. Therefore,

Safety stock= sigmaSafety stock= sigma x safety factorx safety factor

= 200x = 200x 1.281.28

=256=256

Order Point = DDLT+SSOrder Point = DDLT+SS

=1500+256=1500+256

=1756=1756

Periodic ReviewPeriodic Review Using the periodic review system the quantity on hand Using the periodic review system the quantity on hand

of a particular item is determined at specified, fixed-of a particular item is determined at specified, fixed-time intervals,and an order is placed time intervals,and an order is placed

The review period is fixed and the order quantity is The review period is fixed and the order quantity is allowed to vary.allowed to vary.

A maximum level of inventory is set that covers the A maximum level of inventory is set that covers the demand during the review period and the safety stock demand during the review period and the safety stock this is the maximum inventory level.this is the maximum inventory level.

The quantity ordered will be this Maximum - the The quantity ordered will be this Maximum - the quantity on handquantity on hand

Q= M-qQ= M-q Q quantity ordered=M maximum inventory level- Q quantity ordered=M maximum inventory level-

quantity on handquantity on hand

Order Quantity for a Order Quantity for a Periodic Inventory Periodic Inventory

SystemSystemExampleExample

QQ = = dd((ttbb + + LL) + ) + zzdd ttbb + + LL - - II

wherewhere

dd = average demand rate= average demand ratettbb = the fixed time between orders= the fixed time between orders

LL = lead time= lead timedd = standard deviation of demand= standard deviation of demand

zzdd ttbb + + LL = safety stock= safety stock

II = inventory level= inventory level

Periodic Review SystemPeriodic Review System

15/05/1430 Operations Management 345 32

Fixed-Period Model Example

d

t

Q d t L z t L I

d

b

b d b

= bottles 6per day

bottle 1.2s

days 60

L = days 5

I = bottles 8

z = for) 1.65servic 95%(e level

bot 397.96tles

) )( ( ) . )( . (6 60 5 165 12 60 5 8

AdvantagesAdvantages of the of the periodic review systemperiodic review system

More convenient to review More convenient to review inventories on a definite scheduleinventories on a definite schedule

A large number of items can be A large number of items can be ordered from the same supplier thus ordered from the same supplier thus reducing transport costs reducing transport costs and and ordering costordering cost