Inventory Management OPIM 310-Lecture #5 Instructor: Jose Cruz

Inventory ManagementInventory Management

OPIM 310-Lecture #5

Instructor: Jose Cruz

InventoryInventoryStock of items held to meet Stock of items held to meet

future demandfuture demand Inventory management answers Inventory management answers

two questionstwo questions How much to orderHow much to order When to orderWhen to order

Types of InventoryTypes of Inventory Raw materialsRaw materials Purchased parts and suppliesPurchased parts and supplies LaborLabor In-process (partially completed) productsIn-process (partially completed) products Component partsComponent parts Working capitalWorking capital Tools, machinery, and equipmentTools, machinery, and equipment

Reasons to Hold Reasons to Hold InventoryInventory

Meet unexpected demandMeet unexpected demandSmooth seasonal or cyclical demandSmooth seasonal or cyclical demandMeet variations in customer demandMeet variations in customer demandTake advantage of Take advantage of

price discountsprice discountsHedge against price Hedge against price

increasesincreasesQuantity discountsQuantity discounts

Two Forms of DemandTwo Forms of Demand

DependentDependent Items used to produce final productsItems used to produce final products

IndependentIndependent Items demanded by external customersItems demanded by external customers

Inventory CostsInventory Costs

Carrying CostCarrying Cost Cost of holding an item in inventoryCost of holding an item in inventory

Ordering CostOrdering Cost Cost of replenishing inventoryCost of replenishing inventory

Shortage CostShortage Cost Temporary or permanent loss of Temporary or permanent loss of

sales when demand cannot be metsales when demand cannot be met

Inventory Control Inventory Control SystemsSystems

Continuous system (fixed-order-Continuous system (fixed-order-quantity)quantity)

Constant amount ordered when Constant amount ordered when inventory declines to predetermined inventory declines to predetermined levellevel

Periodic system (fixed-time-period)Periodic system (fixed-time-period) Order placed for variable amount Order placed for variable amount

after fixed passage of timeafter fixed passage of time

Assumptions of Basic Assumptions of Basic EOQ ModelEOQ Model

Demand is known with certainty Demand is known with certainty and is constant over timeand is constant over time

No shortages are allowedNo shortages are allowedLead time for the receipt of orders Lead time for the receipt of orders

is constantis constantThe order quantity is received all The order quantity is received all

at onceat once

The Inventory Order CycleThe Inventory Order Cycle

Demand Demand raterate

TimeTimeLead Lead timetime

Lead Lead timetime

Order Order placedplaced

Order Order placedplaced

Order Order receiptreceipt

Order Order receiptreceipt

Inve

nto

ry L

evel

Inve

nto

ry L

evel

Reorder point, Reorder point, RR

Order quantity, Order quantity, QQ

00

Figure 10.1Figure 10.1

EOQ Cost ModelEOQ Cost ModelCCoo - cost of placing order - cost of placing order DD - annual demand - annual demand

CCcc - annual per-unit carrying cost - annual per-unit carrying cost QQ - order quantity - order quantity

Annual ordering cost =Annual ordering cost =CCooDD

QQ

Annual carrying cost =Annual carrying cost =CCccQQ

22

Total cost = +Total cost = +CCooDD

QQ

CCccQQ

22

EOQ Cost ModelEOQ Cost ModelCCoo - cost of placing order - cost of placing order DD - annual demand - annual demand

CCcc - annual per-unit carrying cost - annual per-unit carrying cost QQ - order quantity - order quantity

Annual ordering cost =Annual ordering cost =CCooDD

QQ

Annual carrying cost =Annual carrying cost =CCccQQ

22

Total cost = +Total cost = +CCooDD

QQ

CCccQQ

22

TC = +CoD

Q

CcQ

2

= +CoD

Q2

Cc

2

TC

Q

0 = +C0D

Q2

Cc

2

Qopt =2CoD

Cc

Deriving QoptProving equality of costs at optimal point

=CoD

Q

CcQ

2

Q2 =2CoD

Cc

Qopt =2CoD

Cc

EOQ Cost ModelEOQ Cost Model


Order Quantity, Order Quantity, QQ

Annual Annual cost ($)cost ($)

Ordering Cost =Ordering Cost =CCooDD

QQ




Carrying Cost =Carrying Cost =CCccQQ

22


QQ



Slope = 0Slope = 0

Total CostTotal Cost



Minimum Minimum total costtotal cost

Optimal orderOptimal order QQoptopt

Carrying Cost =Carrying Cost =CCccQQ

22


QQ


EOQ ExampleEOQ ExampleCCcc = $0.75 per yard = $0.75 per yard CCoo = $150 = $150 DD = 10,000 yards = 10,000 yards

QQoptopt = =22CCooDD

CCcc

QQoptopt = =2(150)(10,000)2(150)(10,000)

(0.75)(0.75)

QQoptopt = 2,000 yards = 2,000 yards

TCTCminmin = + = +CCooDD

QQ

CCccQQ

22

TCTCminmin = + = +(150)(10,000)(150)(10,000)

2,0002,000(0.75)(2,000)(0.75)(2,000)

22

TCTCminmin = $750 + $750 = $1,500 = $750 + $750 = $1,500

Orders per year =Orders per year = DD//QQoptopt

== 10,000/2,00010,000/2,000

== 5 orders/year5 orders/year

Order cycle time =Order cycle time = 311 days/(311 days/(DD//QQoptopt))

== 311/5311/5

== 62.2 store days62.2 store daysExample 10.2Example 10.2

EOQ with EOQ with Noninstantaneous ReceiptNoninstantaneous Receipt

QQ(1-(1-d/pd/p))

InventoryInventorylevellevel

(1-(1-d/pd/p))QQ22

TimeTime00

OrderOrderreceipt periodreceipt period

BeginBeginorderorder

receiptreceipt

EndEndorderorder

receiptreceipt

MaximumMaximuminventory inventory levellevel

AverageAverageinventory inventory levellevel


EOQ with EOQ with Noninstantaneous ReceiptNoninstantaneous Receipt

pp = production rate = production rate dd = demand rate = demand rate

Maximum inventory level =Maximum inventory level = QQ - - dd

== QQ 1 - 1 -

QQpp

ddpp

Average inventory level = Average inventory level = 1 - 1 -QQ22

ddpp

TCTC = + 1 - = + 1 -ddpp

CCooDD

QQ

CCccQQ

22

QQoptopt = =22CCooDD

CCcc 1 - 1 - ddpp

Production QuantityProduction QuantityCCcc = $0.75 per yard = $0.75 per yard CCoo = $150 = $150 DD = 10,000 yards = 10,000 yards

dd = 10,000/311 = 32.2 yards per day = 10,000/311 = 32.2 yards per day pp = 150 yards per day = 150 yards per day

QQoptopt = = = 2,256.8 yards = = = 2,256.8 yards

22CCooDD

CCcc 1 - 1 - ddpp

2(150)(10,000)2(150)(10,000)

0.75 1 - 0.75 1 - 32.232.2150150

TCTC = + 1 - = $1,329 = + 1 - = $1,329ddpp

CCooDD

QQ

CCccQQ

22

Production run = = = 15.05 days per orderProduction run = = = 15.05 days per orderQQpp

2,256.82,256.8150150

Example 10.3Example 10.3

Production QuantityProduction QuantityCCcc = $0.75 per yard = $0.75 per yard CCoo = $150 = $150 DD = 10,000 yards = 10,000 yards

dd = 10,000/311 = 32.2 yards per day = 10,000/311 = 32.2 yards per day pp = 150 yards per day = 150 yards per day

QQoptopt = = = 2,256.8 yards = = = 2,256.8 yards

22CCooDD

CCcc 1 - 1 - ddpp

2(150)(10,000)2(150)(10,000)

0.75 1 - 0.75 1 - 32.232.2150150

TCTC = + 1 - = $1,329 = + 1 - = $1,329ddpp

CCooDD

QQ

CCccQQ

22

Production run = = = 15.05 days per orderProduction run = = = 15.05 days per orderQQpp

2,256.82,256.8150150

Number of production runs = = = 4.43 runs/yearDQ

10,0002,256.8

Maximum inventory level = Q 1 - = 2,256.8 1 -

= 1,772 yards

dp

32.2150


Quantity DiscountsQuantity Discounts Price per unit decreases as order Price per unit decreases as order

quantity increasesquantity increases

TCTC = + + = + + PDPDCCooDD

QQ

CCccQQ

22

wherewhere

PP = per unit price of the item = per unit price of the itemDD = annual demand = annual demand

Quantity DiscountsQuantity Discounts Price per unit decreases as order Price per unit decreases as order

quantity increasesquantity increases

TCTC = + + = + + PDPDCCooDD

QQ

CCccQQ

22

wherewhere

PP = per unit price of the item = per unit price of the itemDD = annual demand = annual demand

ORDER SIZE PRICE

0 - 99 $10

100 - 199 8 (d1)

200+ 6 (d2)

Quantity Discount ModelQuantity Discount Model

Figure 10.4Figure 10.4QQoptopt

Carrying cost Carrying cost

Ordering cost Ordering cost

Inve

nto

ry c

ost

($)

Inve

nto

ry c

ost

($)

QQ((dd1 1 ) = 100) = 100 QQ((dd2 2 ) = 200) = 200

TC TC ((dd2 2 = $6 ) = $6 )

TCTC ( (dd1 1 = $8 )= $8 )

TC TC = ($10 )= ($10 )

Quantity Discount ModelQuantity Discount Model

Figure 10.4Figure 10.4QQoptopt

Carrying cost Carrying cost

Ordering cost Ordering cost

Inve

nto

ry c

ost

($)

Inve

nto

ry c

ost

($)

QQ((dd1 1 ) = 100) = 100 QQ((dd2 2 ) = 200) = 200

TC TC ((dd2 2 = $6 ) = $6 )

TCTC ( (dd1 1 = $8 )= $8 )

TC TC = ($10 )= ($10 )

Quantity DiscountQuantity DiscountQUANTITYQUANTITY PRICEPRICE

1 - 491 - 49 $1,400$1,400

50 - 8950 - 89 1,1001,100

90+90+ 900900

CCoo = = $2,500 $2,500

CCcc = = $190 per computer $190 per computer

DD = = 200200

QQoptopt = = = 72.5 PCs = = = 72.5 PCs22CCooDD

CCcc

2(2500)(200)2(2500)(200)190190

TCTC = + + = + + PD PD = $233,784 = $233,784 CCooDD

QQoptopt

CCccQQoptopt

22

For For QQ = 72.5 = 72.5

TCTC = + + = + + PD PD = $194,105= $194,105CCooDD

QQ

CCccQQ

22

For For QQ = 90 = 90


When to OrderWhen to OrderReorder Point is the level of inventory Reorder Point is the level of inventory at which a new order is placed at which a new order is placed

RR = = dLdL

wherewhere

dd = demand rate per period = demand rate per periodLL = lead time = lead time

Reorder Point ExampleReorder Point Example

Demand = 10,000 yards/yearDemand = 10,000 yards/year

Store open 311 days/yearStore open 311 days/year

Daily demand = 10,000 / 311 = 32.154 yards/dayDaily demand = 10,000 / 311 = 32.154 yards/day

Lead time = L = 10 daysLead time = L = 10 days

R = dL = (32.154)(10) = 321.54 yardsR = dL = (32.154)(10) = 321.54 yards


Safety Stocks Safety Stocks

Safety stockSafety stock buffer added to on hand inventory during buffer added to on hand inventory during

lead timelead time

Stockout Stockout an inventory shortagean inventory shortage

Service level Service level probability that the inventory available probability that the inventory available

during lead time will meet demandduring lead time will meet demand

Variable Demand with Variable Demand with a Reorder Pointa Reorder Point


ReorderReorderpoint, point, RR

QQ

LTLT

TimeTimeLTLT

Inve

nto

ry le

vel

Inve

nto

ry le

vel

00

Reorder Point with Reorder Point with a Safety Stocka Safety Stock


ReorderReorderpoint, point, RR

QQ

LTLT

TimeTimeLTLT

Inve

nto

ry le

vel

Inve

nto

ry le

vel

00

Safety Stock

Reorder Point With Reorder Point With Variable DemandVariable Demand

RR = = dLdL + + zzdd L L

wherewhere

dd == average daily demandaverage daily demandLL == lead timelead time

dd == the standard deviation of daily demand the standard deviation of daily demand

zz == number of standard deviationsnumber of standard deviationscorresponding to the service levelcorresponding to the service levelprobabilityprobability

zzdd L L == safety stocksafety stock

Reorder Point for Reorder Point for a Service Levela Service Level

Probability of Probability of meeting demand during meeting demand during lead time = service levellead time = service level

Probability of Probability of a stockouta stockout

RR

Safety stock

ddLLDemandDemand

zd L


Reorder Point for Reorder Point for Variable DemandVariable Demand

The carpet store wants a reorder point with a The carpet store wants a reorder point with a 95% service level and a 5% stockout probability95% service level and a 5% stockout probability

dd = 30 yards per day= 30 yards per dayLL = 10 days= 10 daysdd = 5 yards per day= 5 yards per day

For a 95% service level, For a 95% service level, zz = 1.65 = 1.65

RR = = dLdL + + zz dd L L

= 30(10) + (1.65)(5)( 10)= 30(10) + (1.65)(5)( 10)

= 326.1 yards= 326.1 yards

Safety stockSafety stock = = zz dd L L

= (1.65)(5)( 10)= (1.65)(5)( 10)

= 26.1 yards= 26.1 yards


Order Quantity for a Order Quantity for a Periodic Inventory SystemPeriodic Inventory System

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

Fixed-Period Model with Fixed-Period Model with Variable DemandVariable Demand

dd = 6 bottles per day= 6 bottles per daydd = 1.2 bottles= 1.2 bottles

ttbb = 60 days= 60 days

LL = 5 days= 5 daysII = 8 bottles= 8 bottleszz = 1.65 (for a 95% service level)= 1.65 (for a 95% service level)

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

= (6)(60 + 5) + (1.65)(1.2) 60 + 5 - 8= (6)(60 + 5) + (1.65)(1.2) 60 + 5 - 8

= 397.96 bottles= 397.96 bottles

ABC Classification ABC Classification SystemSystem

Demand volume and value of items varyDemand volume and value of items vary Classify inventory into 3 categories, Classify inventory into 3 categories,

typically on the basis of the dollar value typically on the basis of the dollar value to the firmto the firm

PERCENTAGEPERCENTAGE PERCENTAGEPERCENTAGECLASSCLASS OF UNITSOF UNITS OF DOLLARSOF DOLLARS

AA 5 - 155 - 15 70 - 8070 - 80BB 3030 1515CC 50 - 6050 - 60 5 - 105 - 10

ABC ClassificationABC Classification

11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060

1010 2020 120120

PARTPART UNIT COSTUNIT COST ANNUAL USAGEANNUAL USAGE




11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060

1010 2020 120120

PARTPART UNIT COSTUNIT COST ANNUAL USAGEANNUAL USAGETOTAL % OF TOTAL % OF TOTALPART VALUE VALUE QUANTITY % CUMMULATIVE

9 $30,600 35.9 6.0 6.08 16,000 18.7 5.0 11.02 14,000 16.4 4.0 15.01 5,400 6.3 9.0 24.04 4,800 5.6 6.0 30.03 3,900 4.6 10.0 40.06 3,600 4.2 18.0 58.05 3,000 3.5 13.0 71.0

10 2,400 2.8 12.0 83.07 1,700 2.0 17.0 100.0

$85,400



11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060

1010 2020 120120


9 $30,600 35.9 6.0 6.08 16,000 18.7 5.0 11.02 14,000 16.4 4.0 15.01 5,400 6.3 9.0 24.04 4,800 5.6 6.0 30.03 3,900 4.6 10.0 40.06 3,600 4.2 18.0 58.05 3,000 3.5 13.0 71.0

10 2,400 2.8 12.0 83.07 1,700 2.0 17.0 100.0

$85,400

AA

BB

CC



11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060

1010 2020 120120


9 $30,600 35.9 6.0 6.08 16,000 18.7 5.0 11.02 14,000 16.4 4.0 15.01 5,400 6.3 9.0 24.04 4,800 5.6 6.0 30.03 3,900 4.6 10.0 40.06 3,600 4.2 18.0 58.05 3,000 3.5 13.0 71.0

10 2,400 2.8 12.0 83.07 1,700 2.0 17.0 100.0

$85,400

AA

BB

CC

% OF TOTAL % OF TOTALCLASS ITEMS VALUE QUANTITY

A 9, 8, 2 71.0 15.0B 1, 4, 3 16.5 25.0C 6, 5, 10, 7 12.5 60.0


100 100 –

80 80 –

60 60 –

40 40 –

20 20 –

0 0 –| | | | | |00 2020 4040 6060 8080 100100

% of Quantity% of Quantity

% o

f V

alu

e%

of

Val

ue

AA

BBCC

Assumptions of Basic Assumptions of Basic EOQ ModelEOQ Model

Demand is known with certainty Demand is known with certainty and is constant over timeand is constant over time

No shortages are allowedNo shortages are allowedLead time for the receipt of orders Lead time for the receipt of orders

is constantis constantThe order quantity is received all The order quantity is received all

at onceat once

Documents

Inventory Management OPIM 310-Lecture #5 Instructor: Jose Cruz