Chapter 12Chapter 12

Inventory Inventory ManagementManagement

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

Inventory CostsInventory Costs

Carrying CostCarrying CostCost of holding an item in inventory,Cost of holding an item in inventory,

May include cost of obsolescence May include cost of obsolescence

Ordering CostOrdering CostCost of replenishing inventoryCost of replenishing inventory

Shortage CostShortage CostTemporary or permanent loss of sales Temporary or permanent loss of sales

when demand cannot be metwhen 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 after Order placed for variable amount after

fixed passage of timefixed passage of time

Assumptions of Basic Assumptions of Basic EEconomic conomic OOrder rder QQuantity uantity

ModelModelDemand is known with certainty Demand is known with certainty

and is constant over timeand is constant over timeNo 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

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

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

Slope = 0Slope = 0

Total CostTotal Cost

Order Quantity, Order Quantity, QQ

Annual Annual cost ($)cost ($)

Minimum Minimum total costtotal cost

Optimal orderOptimal order QQoptopt

Carrying Cost =Carrying Cost =CCccQQ

22

Ordering Cost =Ordering Cost =CCooDD

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 days

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

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

1 - 99 $10

100 - 199 8

200+ 6

Quantity Discount ModelQuantity Discount Model

QQoptopt

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

QQoptopt

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 )