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Order Quantities - Examples Order Quantities Order Quantities -- Examples Examples

Outline:Outline:I. Basic Inventory Planning QuestionsI. Basic Inventory Planning QuestionsII. Inventory ModelsII. Inventory ModelsIII. EOQ Model III. EOQ Model IV. EOQ Model ExampleIV. EOQ Model ExampleV. Production Order Quantity ModelV. Production Order Quantity ModelVI. POQ Model ExampleVI. POQ Model ExampleVII. Fixed Period Model.VII. Fixed Period Model.

KuliahKuliah keke--7: Rabu/5 Nov 20087: Rabu/5 Nov 2008

I. Basic Inventory Planning Questions

I. Basic Inventory Planning I. Basic Inventory Planning QuestionsQuestions

How much to order?How much to order?Purchase Order

Description Qty.Microwave 1000

How much to order?How much to order?When to order?When to order? Purchase Order

Description Qty.Microwave 1000

II. Inventory ModelII. Inventory ModelII. Inventory Model

We need Inventory Models:We need Inventory Models:1.1. Economic order quantityEconomic order quantity2.2. Production order quantityProduction order quantity3.3. Quantity discountQuantity discount

Help answer the inventory planning questions!

Help answer the inventory planning questions!

Why Carrying Cost Increases?Why Carrying Cost Increases?Why Carrying Cost Increases?

More units must be stored if more More units must be stored if more orderedordered

Purchase OrderDescription Qty.Microwave 1000


Order Order quantityquantity


Why Order Cost Decreases?Why Order Cost Decreases?Why Order Cost Decreases?

Cost is spread over more unitsCost is spread over more units

Example: You need 1000 microwave ovens Example: You need 1000 microwave ovens






1 Order (Postage $ 0.32)1 Order (Postage $ 0.32) 1000 Orders (Postage $320)1000 Orders (Postage $320)


III. EOQ ModelIII. EOQ ModelIII. EOQ Model

Order QuantityOrder Quantity

Annual CostAnnual Cost 1. How much to order?

EOQ ModelEOQ ModelEOQ Model


Annual CostAnnual Cost

Carrying CostCarrying Cost

1. How much to order?





Order (Setup) CostOrder (Setup) Cost






Total Cost CurveTotal Cost Curve







Total Cost CurveTotal Cost Curve


Optimal Optimal Order Quantity (Q*)Order Quantity (Q*)


EOQ Model: When to Order?EOQ Model: When to Order?EOQ Model: When to Order?

Inventory LevelInventory Level

TimeTime

2. When to order?

EOQ Model EOQ Model EOQ Model


Optimal Optimal Order Order Quantity Quantity (Q*)(Q*)

TimeTime

2. When to order?




Decrease due to Decrease due to constant demandconstant demand

TimeTime

2. When to order?




Instantaneous Instantaneous receipt of optimal receipt of optimal order quantityorder quantity

TimeTime

2. When to order?




TimeTime

2. When to order?




Lead TimeLead Time TimeTime

2. When to order?




Lead TimeLead Time

Reorder Reorder Point Point

(ROP)(ROP)TimeTime

2. When to order?

Reorder Reorder Point Point

(ROP)(ROP)TimeTime



Optimal Optimal Order Order QuantityQuantity(Q*)(Q*)

AverageAverageInventory (Q*/2)Inventory (Q*/2)

Lead TimeLead Time

2. When to order?

IV. EOQ Model Example IV. EOQ Model Example IV. EOQ Model Example

Problem 1Problem 1::When the inventory of microwaves gets When the inventory of microwaves gets down to 15 units (reorder point), order down to 15 units (reorder point), order 35 units (EOQ).35 units (EOQ).

15 left


Problem 2:Problem 2:YouYou’’re a buyer for Walre a buyer for Wal--Mart. Mart. WalWal--Mart needs Mart needs 10001000 coffee coffee makers per year. The cost of makers per year. The cost of each coffee maker is each coffee maker is $78$78. . Ordering cost is Ordering cost is $100$100 per order. per order. Carrying cost is Carrying cost is 40%40% of per unit of per unit cost. Lead time is cost. Lead time is 55 days. Waldays. Wal--Mart is open Mart is open 365365 days/yr. What days/yr. What is the is the optimal order quantityoptimal order quantity & & ROPROP??

A = 1,000 unitA = 1,000 unitS = $ 100 per orderS = $ 100 per orderc = $ 78 per unitc = $ 78 per uniti = 0.40i = 0.40L = 5 daysL = 5 daysWorking days/year = 365 daysWorking days/year = 365 days

Question:Question:a.a. EOQ = ?EOQ = ?b.b. ROP = ?ROP = ?

EOQ Solution*EOQ Solution*EOQ Solution*

Qi X c

d A

ROP d L

*.

=2

.

. .

= =

= = =

= ⋅ = =

X1000X10040 (78)

80

1000365

2 74

2 74 X 5 13 7

units

units / day

units

Working Days/ Year

2 X A X S

V. Production Order Quantity ModelV. Production Order Quantity ModelV. Production Order Quantity Model

1.1. Answers how much to order & when to orderAnswers how much to order & when to order2.2. Allows partial receipt of material (other EOQ Allows partial receipt of material (other EOQ

assumptions apply)assumptions apply)3.3. Suited for production environmentSuited for production environment

•• Material produced, used immediatelyMaterial produced, used immediately•• Provides production lot sizeProvides production lot size

4.4. Lower carrying cost than EOQ model.Lower carrying cost than EOQ model.

POQ Model: Inventory LevelsPOQ Model: Inventory LevelsPOQ Model: Inventory Levels

TimeTime



TimeTime


Supply Supply BeginsBegins


TimeTime



Supply Supply EndsEnds


TimeTime




Inventory level with NO Inventory level with NO demand during supply of demand during supply of optimum order quantityoptimum order quantity


TimeTime




Inventory level with NO Inventory level with NO demand during supply of demand during supply of optimum order quantityoptimum order quantity

Q*Q*

Q* is optimum Q* is optimum order qtyorder qty


TimeTime




Q*Q* Quantity used before Quantity used before becoming inventorybecoming inventory


TimeTime




Decrease due to Decrease due to no supply & no supply & constant demandconstant demand


TimeTime

Inventory LevelInventory LevelNext CycleNext Cycle


TimeTime



Next CycleNext Cycle


TimeTime





TimeTime


Max. Inventory Max. Inventory Q*Q*··(1(1-- d/pd/p))

POQ Model EquationsPOQ Model EquationsPOQ Model Equations

Max. Inventory Level

Order Cost

Carrying Cost =

= −

=

Q dp

x 1

AQ

SxAA = Demand per year= Demand per yearSS = Order cost= Order costHH = Carrying cost (i X c)= Carrying cost (i X c)dd = Demand per day= Demand per dayp p = Production per day= Production per day

Optimal Order Quantity = Qp* = 2 x A x S

H x (1- d/p)

Q 1 - (d/p) H

VI. POQ Model ExampleVI. POQ Model ExampleVI. POQ Model Example

YouYou’’re a production planner re a production planner for Stanley Tools. Stanley for Stanley Tools. Stanley Tools makes Tools makes 30,00030,000 screw screw drivers per year. Demand is drivers per year. Demand is 100100 screw drivers per day & screw drivers per day & production is production is 300 300 per day. per day. Production setup cost is Production setup cost is $150 $150 per order. Carrying cost is per order. Carrying cost is $1.50 $1.50 per screw driver. What per screw driver. What is the is the optimal lot sizeoptimal lot size? ? Max. Max. Inv. Level?Inv. Level?

Production Order Quantity Model Solution*Production Order Quantity Model Solution*Production Order Quantity Model Solution*

Q A S

H dp

p * =⋅ ⋅

⋅=

⋅ ⋅

⋅ −=

= ⋅ − =

2 2 30000 150

1.5 1 100300

3000

3000 100300

2000Max. Inventory Level

AA = Demand Per Year = 30000 unit= Demand Per Year = 30000 unitSS = Order Cost = $ 150 per order= Order Cost = $ 150 per orderHH = Carrying Cost = $ 1,5 per unit= Carrying Cost = $ 1,5 per unitdd = Demand Per Day = 100 unit= Demand Per Day = 100 unitp p = Production Per Day = 300 unit= Production Per Day = 300 unit

1 -

1

Answers how much to orderAnswers how much to orderOrders placed at fixed intervalsOrders placed at fixed intervals

Inventory brought up to target amountInventory brought up to target amountAmount ordered variesAmount ordered varies

No continuous inventory countNo continuous inventory countPossibility of Possibility of stockoutstockout between intervalsbetween intervals

Useful when vendors visit routinelyUseful when vendors visit routinelyExample: P&G representative calls every 2 Example: P&G representative calls every 2 weeksweeks

VII. Fixed Period ModelVII. Fixed Period ModelVII. Fixed Period Model

Inventory Level in a Fixed Period SystemInventory Level in a Fixed Period SystemInventory Level in a Fixed Period System

Various amounts (Qi) are ordered at regular time intervals (p) based on the quantity necessary to bring inventory up to

target maximum

pp pp pp

QQ11 QQ22

QQ33

QQ44

Target maximum

TimeTime

OnOn-- H

and I

nven

tory

Hand

Inve

ntory

Waktu

Inventory Level Target maximum

Periode PeriodePeriode

Sistem P (Periode Tetap)Kapan memesan?

SistemSistem P (P (PeriodePeriode TetapTetap))KapanKapan memesanmemesan??

Q1 Q2 Q3

VII. Quantity Discount ModelVII. Quantity Discount ModelVII. Quantity Discount Model

Answers how much to order & Answers how much to order & when to orderwhen to orderAllows quantity discountsAllows quantity discounts

Reduced price when item is purchased Reduced price when item is purchased in larger quantitiesin larger quantitiesOther EOQ assumptions applyOther EOQ assumptions apply

TradeTrade--off is between lower price & off is between lower price & increased holding costincreased holding cost

Quantity Discount ModelsQuantity Discount Models

•• Reduced prices are often available when Reduced prices are often available when larger quantities are purchasedlarger quantities are purchased

•• TradeTrade--off is between reduced product cost off is between reduced product cost and increased holding costand increased holding cost

Total cost = Setup cost + Holding cost + Product costTotal cost = Setup cost + Holding cost + Product cost

TC = S + + TC = S + + cAcAAAQQ

QHQH22

Quantity Discount Model: How Much to Order?

Quantity Discount Model: Quantity Discount Model: How Much to Order?How Much to Order?

Order Order QuantityQuantity

Total CostTotal Cost





Discount Discount Quantity 1Quantity 1

Price 1Price 1







Price 1Price 1 Price 2Price 2







Price 1Price 1 Price 2Price 2 Price 3Price 3




Total CostTotal Cost TC for TC for Discount 1Discount 1











Q*Q*






TC for TC for Discount 2Discount 2








Disc Disc Qty 1Qty 1




Q*Q*










Q*Q*

Outside Outside discount discount rangerange

Outside Outside discount rangediscount range









Q*Q*Discount Discount Quantity 1Quantity 1











Outside Outside discount rangediscount range Q*Q*











Lowest cost not Lowest cost not in discount rangein discount range

Quantity Quantity OrderedOrdered


Quantity Discount Model StepsQuantity Discount Model StepsQuantity Discount Model Steps

Compute EOQ for each quantity Compute EOQ for each quantity discount pricediscount priceIs computed EOQ in discount range?Is computed EOQ in discount range?

If not, use the lowest cost quantity in If not, use the lowest cost quantity in discount rangediscount range

Compute total cost for EOQ or lowest Compute total cost for EOQ or lowest cost quantity in discount rangecost quantity in discount rangeSelect quantity with lowest total costSelect quantity with lowest total cost

Contoh: Quantity Discount ScheduleContoh: Quantity Discount Schedule

Discount Number

Discount Quantity

Discount (%)

Discount Price (P)

1 0 to 999 No discount

$5.00

2 1,000 to 1,999 4 $4.803 2,000 and over 5 $4.75

Quantity Discount ModelsQuantity Discount Models

1.1. For each discount, calculate Q*For each discount, calculate Q*2.2. If Q* for a discount doesnIf Q* for a discount doesn’’t qualify, t qualify,

choose the smallest possible order size choose the smallest possible order size to get the discountto get the discount

3.3. Compute the total cost for each Q* or Compute the total cost for each Q* or adjusted value from Step 2adjusted value from Step 2

4.4. Select the Q* that gives the lowest total Select the Q* that gives the lowest total costcost

Steps in analyzing a quantity discountSteps in analyzing a quantity discount

Quantity Discount ExampleQuantity Discount Example

Calculate Q* for every discountCalculate Q* for every discount Q* = 2Sic

QQ11* * = = 700= = 700 cars ordercars order2(5,000)(49)2(5,000)(49)(.2)(5.00)(.2)(5.00)



1,0001,000 —— adjustedadjusted

2,0002,000 —— adjustedadjusted

Quantity Discount ExampleQuantity Discount Example

Discount Number

Unit Price

Order Quantity

Annual Product

Cost

Annual Ordering

Cost

Annual Holding

Cost Total

1 $5.00 700 $25,000 $350 $350 $25,700

2 $4.80 1,000 $24,000 $245 $480 $24,725

3 $4.75 2,000 $23.750 $122.50 $950 $24,822.50

Table 12.3Table 12.3

Choose the price and quantity that gives Choose the price and quantity that gives the lowest total costthe lowest total cost

Buy Buy 1,0001,000 units at units at $4.80$4.80 per unitper unit

VIII. Probabilistic Models and Safety Stock

VIII. Probabilistic Models and Safety Stock

Answer how much and when to order Answer how much and when to order Used when demand is not constant or Used when demand is not constant or certaincertainUse safety stock to achieve a desired Use safety stock to achieve a desired service level and avoid stockoutsservice level and avoid stockouts

ROP ROP == d x Ld x L + + ssss

Annual stockout costs = the sum of the units short Annual stockout costs = the sum of the units short x the probability x the stockout cost/unit x the probability x the stockout cost/unit

x the number of orders per yearx the number of orders per year

Safety Stock ExampleSafety Stock Example

Number of Units Probability30 .240 .2

ROP 50 .360 .270 .1

1.0

ROP ROP = 50= 50 unitsunits Stockout cost Stockout cost = $40= $40 per frameper frameOrders per year Orders per year = 6= 6 Carrying cost Carrying cost = $5= $5 per frame per yearper frame per year

Safety Stock ExampleSafety Stock Example

ROP ROP = 50= 50 unitsunits Stockout cost Stockout cost = $40= $40 per frameper frameOrders per year Orders per year = 6= 6 Carrying cost Carrying cost = $5= $5 per frame per yearper frame per year

Safety Stock

Additional Holding Cost Stockout Cost

Total Cost

20 (20)($5) = $100 $0 $100

10 (10)($5) = $50 (10)(.1)($40)(6) = $240 $290

0 $0 (10)(.2)($40)(6) + (20)(.1)($40)(6) = $960 $960

A safety stock of A safety stock of 2020 frames gives the lowest total costframes gives the lowest total costROP ROP = 50 + 20 = 70= 50 + 20 = 70 framesframes

Safety stock 16.5 units

ROP ROP

Place Place orderorder

Probabilistic DemandProbabilistic DemandIn

vent

ory

leve

lIn

vent

ory

leve

l

TimeTime00

Minimum demand during lead timeMinimum demand during lead time

Maximum demand during lead timeMaximum demand during lead time

Mean demand during lead timeMean demand during lead time

Normal distribution probability of Normal distribution probability of demand during lead timedemand during lead time

Expected demand during lead time Expected demand during lead time (350(350 kitskits))

ROP ROP = 350 += 350 + safety stock of safety stock of 16.5 = 366.516.5 = 366.5

Receive Receive orderorder

Lead Lead timetime

Figure 12.8Figure 12.8

Probabilistic DemandProbabilistic Demand

Use prescribed service levels to set safety Use prescribed service levels to set safety stock when the cost of stockouts cannot be stock when the cost of stockouts cannot be determineddetermined

ROP = demand during lead time + ZROP = demand during lead time + Zσσdltdlt

wherewhere Z Z == number of standard deviationsnumber of standard deviationsσσdltdlt == standard deviation of demand standard deviation of demand

during lead timeduring lead time

Probabilistic ExampleProbabilistic Example

Average demand Average demand = = μμ = 350= 350 kitskitsStandard deviation of demand during lead time Standard deviation of demand during lead time = = σσdltdlt = 10= 10 kitskits5%5% stockout policy stockout policy ((service level service level = 95%)= 95%)

Using Appendix I, for an area under the curve Using Appendix I, for an area under the curve of of 95%,95%, the Z the Z = 1.65= 1.65

Safety stock Safety stock == ZZσσdltdlt = 1.65(10) = 16.5= 1.65(10) = 16.5 kitskits

Reorder pointReorder point == expected demand during lead time expected demand during lead time + safety stock+ safety stock

== 350350 kits kits + 16.5+ 16.5 kits of safety stockkits of safety stock== 366.5366.5 or or 367367 kitskits

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