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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
Annual carrying cost =Annual carrying cost =CCccQQ
22
Total cost = +Total cost = +CCooDD
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
Annual carrying cost =Annual carrying cost =CCccQQ
22
Total cost = +Total cost = +CCooDD
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
Figure 10.2Figure 10.2
Order Quantity, Order Quantity, QQ
Annual Annual cost ($)cost ($)
Ordering Cost =Ordering Cost =CCooDD
EOQ Cost ModelEOQ Cost Model
Order Quantity, Order Quantity, QQ
Annual Annual cost ($)cost ($)
Carrying Cost =Carrying Cost =CCccQQ
22
Ordering Cost =Ordering Cost =CCooDD
Figure 10.2Figure 10.2
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
Figure 10.2Figure 10.2
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
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
Figure 10.3Figure 10.3
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
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
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
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
Example 10.3Example 10.3
Quantity DiscountsQuantity Discounts Price per unit decreases as order Price per unit decreases as order
quantity increasesquantity increases
TCTC = + + = + + PDPDCCooDD
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
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
CCccQQ
22
For For QQ = 90 = 90
Example 10.4Example 10.4
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
Example 10.5Example 10.5
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
Figure 10.5Figure 10.5
ReorderReorderpoint, point, RR
LTLT
TimeTimeLTLT
Inve
nto
ry le
vel
Inve
nto
ry le
vel
00
Reorder Point with Reorder Point with a Safety Stocka Safety Stock
Figure 10.6Figure 10.6
ReorderReorderpoint, point, RR
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
Figure 10.7Figure 10.7
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
Example 10.6Example 10.6
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
Example 10.1Example 10.1
ABC ClassificationABC Classification
Example 10.1Example 10.1
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
ABC ClassificationABC Classification
Example 10.1Example 10.1
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
AA
BB
CC
ABC ClassificationABC Classification
Example 10.1Example 10.1
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
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
ABC ClassificationABC Classification
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
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