SMA 6305 1 4th February 2005 A/P Sivakumar

Economic Order Quantity modeland

News Vendor model

SMA6305Class on 4th February 2005

Assoc. Prof. Sivakumar

SMA 6305 2 4th February 2005 A/P Sivakumar

Reasons for Holding Inventories

Economies of ScaleUncertainty in delivery leadtimesSpeculation. Changing Costs Over TimeSmoothing.Demand UncertaintyCosts of Maintaining Control System

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Economic Order Quantity modelconstant and known demand over time

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Relevant Costs

Holding Costs - Costs proportional to the quantity of inventory held. Includes Typical proportions [Nahmias]

a) Physical Cost of Space (3%)b) Taxes and Insurance (2 %)c) Breakage Spoilage and Deterioration (1%)

*d) Opportunity Cost of alternative investment. (18%)(Total: 24%

Note: Since inventory may be changing on a continuous basis, holding cost is proportional to the area under the inventory curve.

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Relevant Costs (continued)

Penalty or Shortage Costs. All costs that accrue when insufficient stock is available to meet demand. These include:

– Loss of revenue for lost demand– Costs of bookeeping for backordered demands– Loss of goodwill for being unable to satisfy demands when they occur.

– Generally assume cost is proportional to number of units of excess demand.

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EOQ Economic Order QuantityBasic and simple model is EOQAn item having constant and known demand over time There is a cost involved each time an order is placedThere is a cost to hold the item in the storesThere are multiple periods

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EOQ AssumptionsDemand rate is constant = λInventory holding cost ($) = h per each unit held per unit timeCost per piece in $ = cOrder (setup cost) per each order placed=KNo shortages are permittedNo order lead timeOn hand inventory at t=0 is zero

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Inventory levels for EOQ model

Order and delivery is assumed to take place at discrete interval with no lead time.

InventoryQuantity

Q

Order & Deliver Inventory slope = - λ = - Q / T

Average inventory= (Q/2)

Timet=0

T

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Average Cost of OperationOrdering cost per time period T =Ordering cost + (Quantity X $ per piece) = (K+ c x Q)Ordering cost per unit time = (K+ c x Q)/ TInventory holding cost per unit time =Cost of avg. inv. × Inv holding cost $ per each unit held per

unit time. =(Q / 2) × hTotal cost per unit time = Ordering cost + Inventory holding cost

Since

Average cost per unit time

}2/{}/){()( hQTcQkQG ++=}2/{}/){()( hQQcQkQG ++= λλ/QT =

2//)( hQcQkQG ++= λλ

SMA 6305 10 4th February 2005 A/P Sivakumar

Minimized costHow to find Q to minimize G(Q); :

Differentiating

and for Q>0

And therefore G(Q) is a convex function of Q

3/)( QkQG λ=′′

2//)( 2 hQkQG +−=′ λ2/)( hG =∞′ ∞=′ )0(G

2//)( hQcQkQG ++= λλ

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Minimized cost

Optimal occurs when

Or when

that is when

Optimal Q

2//0)( 2 hQkQG +−==′ λ

2// 2 hQk =λ

hkQ /22 λ=

hkQ λ2* =

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The Average Annual Cost Function G(Q) [Nahmias]

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EOQK= $75 λ= 3120 h= 75

Order Ordering Holding TotalQty (Q) cost Kλ/Q cost hQ/2 cost G(Q)-cλ

20 $11,700 750 12,450 40 $5,850 1500 7,350 60 $3,900 2250 6,150 80 $2,925 3000 5,925

100 $2,340 3750 6,090 120 $1,950 4500 6,450 140 $1,671 5250 6,921 160 $1,463 6000 7,463

Cost-Orderquantity

$0

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

$14,000

0 20 40 60 80 100 120 140Order qty

Cos

t $

Ordering costKλ/QHolding costhQ/2Total cost G(Q)-cλ

797531207522* === XX

hkQ λ

daysQT 24.93120/79*365/** === λ

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EOQ - factorsThere is an order quantity in which the total cost is a minimum Q*=EOQ.This leads to an optimal cycle length T*This minimum (EOQ) occurs at the point where cost of ordering is equal to cost of carrying. (true for equations of the form y = ax+b/xwhere x = q).The total cost varies little over a wide range of order sizes around the EOQ.EOQ is a good approximation to the nearest package / pallet / case /quantity.

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Lead time and Safety stockOrder and delivery can be considered with a lead time.

InventoryQuantity

Q

Order & Deliver

t=0

Inventory slope = - λ = - Q / T

Lead Time

Average inventory= (Q/2)

Time

T

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Safety stock & Order point

Lead Time

Quantity

Order Quantity

Order Point

Time

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Safety stock & Order point

Lead Time

Quantity

Order Quantity

Order Point

Safety Stock

Time

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Determining safety stockSpecification of required service level (i. e. frequency of stock shortage) help to determine the safety or buffer stock level.

Demand

Order Quantity

Mean Demand during lead time

Safetystock

Frequency

Probability of shortage(Assume normal distribution)

1σ 16 %2σ 2.5 %3σ 0.15 %

Demand up to lead time

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Two-Bin System & Perpetual Inventory System

Derivations of the Inventory Systems.– Two-Bin System:. – Periodic review Inventory System– Perpetual or continuous review Inventory System

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News Vendor modelRandom demand over the time

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The News vendor ModelAt the start of each day, a news vendor must decide the number of papers to purchase. Daily sales cannot be predicted exactly, and are represented by the random variable, D. Although demand history is discrete and empirical probability functions can be used, it is more convenient and effective to approximate the demand Di to continuous distribution

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The News vendor Model

The most popular distribution for inventory application is normal distribution. (must watch out for the negative observation)Normal distribution is defined by mean & variance µ σ 2

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Mean and Variances

Suppose variables D1 , D2 , …Dn are the past n observations of demand for the news papers.

Sample Mean= is a point estimate for

Sample variance =

is an unbiased estimator of

)1(

])([ 2

)(2 1

−

−=∑=

n

D nDnS

n

i

i

µ

σ 2

n

DD

n

ii

n

∑== 1

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The News vendor Model Assumptions

Single productCosts: co = unit cost of overage

cu = unit cost of underageDemand D is a continuous non- negative random variable with density function f(x) and cumulative distribution function F(x).Decision variable is Q, number of units to purchase at the beginning of the period.The goal is to determine Q to minimize the expected cost at the end of the period.

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Cost functionDefine G(Q,D) = total underage and overage cost.

Define expected cost function

Using basic statistics rules (See Nahmias Appendix 5_A)

),0max()(,0max(),( QDCDQcDQG uo −+−=

)),(()( DQGEQG =

∫∫∞

−+−=Q

Q

dxxfQxdxxfxQcQG )()()()()(0

0

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Optimal policyValue of Q that minimizes the expected cost G(Q) is obtained by differentiating G(Q)

As G(Q) can proved to be convex the optimal solution Q* occurs when the above =0

Rearranging:

∫∫ −+==Q

u

Q

dxxfCdxxfcQGdQQdG

000 )()1()(1)(')(

))(1()(0 QFcQFc u −−=

*)()/(*)(0*)()(*)('

0

0

QdemandPcccQFcQFccQG

uu

uu

≤=+==−+=

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Optimal policyThe ratio cu /(cu+co) is referred as critical fractile ratio µσ += ∗zQ*

xµ

f(x) Area =cu /(cu+co)

Q*

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Determine Optimal Q*When demand is assumed normal we can find Q* in the following ways:1.Using normal tables to obtain standardized values

of z and substitute in2.Using excel

3. Using excel obtain& substitute in

µσ += zQ*

µσ += zQ*

),),((* σµCFRNORMINVQ =

)(* CFRNORMSINVz =

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Example News Vendor ModelNahmias Example 5.1

Demand is normal with µ =11.73 and standard deviation σ = 4.74If papers are bought at 25 Cents and sold at 75 cents and the salvage value of unsold copies is 10 centsThen co =25-10=15 cents cu=75-25=50 centscritical fractile, cu /(cu+co) =50/(50+15)=0.77.

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Wrap-upA powerful but Basic and simple model with constant demand, EOQ is described as a basis for analyzing complex systems in the next few classesNews Vendor model, another important but a simple model with Random demand assumption is described as a basis for analyzing complex systems in the next few classes