The document discusses inventory management. It defines inventory as items kept by an organization to meet demand. The purpose of inventory management is to determine how much inventory to keep and when to replenish it. Types of inventory include raw materials, work in progress, finished goods, and replacement parts. Inventory serves functions like meeting demand, production smoothing, and protecting against stock-outs. The objective is to balance customer service and inventory costs. Effective management requires tracking inventory levels, forecasting demand, and estimating costs of ordering, carrying, and shortages.

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Inventory is a stock ofitems keptby an organization to meetinternal or external customer demand. The purpose of
inventory managementis to determine the amount of inventory to keep in stock –how much to order and when to
replenish or order.
Types ofInventories:
 Raw materials & purchased parts
 Partially completed goods called work in progress
 Finished-goods inventories (manufacturing firms) or merchandise (retail stores)
 Replacementparts, tools, & supplies
 Goods-in-transit to warehouses or customers (pipeline inventory)
Functions ofInventory:
 To meetanticipated demand
 To smooth production requirements
 To decouple operations
 To protectagainst stock-outs
 To take advantage of order cycles
 To help hedge againstprice increases
 To permitoperations
 To take advantage of quantity discounts
Objective ofInventory Control:
To achieve satisfactory levels ofcustomer service while keeping inventory costs within reasonable bounds
 Level ofcustomer service (to have the right goods, in sufficientquantities, in the right place, atthe
right time)
 Costs of ordering and carrying inventory
Effective Inventory Management:
 A system to keep track ofinventory
 A reliable forecastofdemand
 Knowledge oflead times
 Reasonable estimates ofinventory
 Holding costs
 Ordering costs
 Shortage costs
 A classification system
Inventory Counting Systems:
 Periodic System
Physical countof items made at periodic intervals
 Perpetual Inventory System
System that keeps track
of removals from inventory
continuously, thus
monitoring
current levels of
each item
 Two-Bin System - Two containers ofinventory; reorder when the first is empty

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 Universal Bar Code - Bar code
printed on a label that has
information aboutthe item
to which it is attached
ABC Classification System:
Classifying inventory according to some measure ofimportance and allocating control efforts accordingly.
A - very important
B - moderate important
C - least important
Nature of Inventory Problem:
The nature of the inventory problem consists ofrepeatedly placing and receiving orders ofgiven sizes atset
intervals. From this stand pointan inventory policy answers the following two questions.
1. How much to order?
2. When to order?
How much to order determines the Economic Order Quantity (EOQ) model by minimizing the cost.
When to order highly connected and represents ordering and placing time.
When to order includes: 1. Periodic order / review (discrete)
2. Continuous order / review.
If the system requires periodic review (e.g every week or month) the time for receiving a new order coincides with the
start of each period.
Alternately if the system is based on continuous review, new orders are placed when the inventory level drops to a
pre specified level, called the reorder point.
The deterministic model ofinventory are oftwo types: 1. Static, which have constant demand over time.
2. Dynamic, in which the demand varies.
Inventory cost:
Total Inventory Cost = Purchasing cost+ Setup cost+ Holding cost+ Shortage cost
All costmust be expressed in terms ofthe desired order quantity and the time between orders.
Purchasing cost: The price per unit ofthe item. It may be constant, or it may be offered ata discountthat
depends on the size ofthe order.
Setup cost: Represents the fixed charge incurred when and order is placed. This is independentofthe size of
the order.
Holding Cost: Represents the costofmaintaining the inventory in stock. It includes storage, intereston capital,
maintenance and handling.

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Shortage Cost: Shortage costis penalty incurred when we run out ofstock.
EOQ model with static demand:
The simplestofthe inventory model involves constantrate demand with instantaneous order replacementand no
shortage.
y = Order Quantity (number ofunits)
D= Demand rate (units per unit time)
t0 = Ordering cycle length (time units)
Inventory
level Points in time at which orders are received.
y
Reorder
Point
Average Inventory
= y/ 2
to = y / D L L
Time
An order ofsize y units is placed and received instantaneously when the inventory level is zero. And the
demand rate is constant.
The ordering cycle for this pattern is to = y/D time units.
The average inventory level is y/2 units
There are two costparameters
K = Setup costassociated with the placementofan order.
h = Holding cost(dollars per inventory unitper unit time)

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So, the total cost per unit time
TCU (y) = Setup costper unit time + Holding costper unittime

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
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

2/
2
2
0
0
00
00
y
h
Dy
K
y
h
t
K
t
y
t
h
t
K
t
CostHolding
t
CostSetup
To find the optimum value ofy (minimizing y) setfirst derivative equal to 0.
h
kD
y
h
kD
y
h
y
KD
impliesWhich
h
y
KDhy
Y
KD
dy
d
dy
yTCUd
22
2
,
0
22
))((
2
2
2










The condition is also sufficient because TCU(y) is convex. The solution ofthe equation yields the EOQ y* as
y* =
h
kD2
i.e. Order y* =
h
kD2
units every t0* = y* / D time units.
Example:1 Neon lights on the U ofA campus are replaced atthe rate of100 units per day. The physical plant
orders the neon lights periodically.Itcosts $100 to initiate a purchase order. A neon lightkeptin storage is estimated
to costabout$0.02 per day. The lead time between placing and receiving an order is 12 days. Determine the optimal
inventory policy for ordering the neon light.
Solution: We have D = 100 units / day
K = $ 100 per order
h =$0.02 per unit / day
L = 12 days
Optimum order quantity y* =
h
kD2
= 10001000000
02.0
1001002


neon light

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The associated cycle length is t0* = y* / D = 1000 / 100 = 10 days
Since L = 12 > t0* = 10 days , we have to fine the effective lead time Le .
Where Le = L – nt0* Consider n = ( Largest integer) ≤
0t
L
= 12 – (1)10 = Largestinteger ≤
10
12
= 2 days = 1
The reorder pointthus occurs when the inventory level drops to LeD = 2 x 100 = 200 units.
So, the inventory policy for ordering the neon lightis order 1000 units whenever the inventory level drops to
200 units.
Also, the daily inventory costassociated with the proposed inventory policy is
TCU (y) = KD / y + h (y / 2)
= (100 x 100) / 1000 + .02 ( 1000 / 2)
= 10 + 10 = $ 20 per day.
Q: 1 A company stocks an item that is consumed atthe rate of50 units per day. It costs the company $ 20 each
time an order is placed.An inventory unit held in stock for a week will cost$ 0.35.
a. Determine the optimum inventory policy, assuming a lead time of1 week.
b. Determine the optimum number of orders per year (365 days)
[ Ans: a. Order 200 units when ever inventory drops to 150 units. TCU (y) = $ 10 / day. ]
Q: 2 In each ofthe following cases, no shortage is allowed and the lead time between placing and receiving an
order is 30 days. Determine the optimal inventory policy and the associated costper day.
a. K = $ 100, h = $ 0.05, D = 30 units per day, Also L = 30 [Ans: Order 346.5 units when ever inventory
drops to 207 units. TCU = 17.32 per day]
b. K = $ 50, h = $ 0.05, D = 30 units per day, L = 30 [ Ans: Order 245 units when ever inventory
drops to165 units. TCU = $12.246 per day]
c. K = $ 100, h = $ 0.01, D = 40 units per day, L = 30 [ Ans: Order 894 units when ever inventory
drops to 305 units; TCU = $ 8.94 per day]
d. K = $100, h = $0.04, D = 20 units, L = 30 [ Ans: Order 316 units when ever inventory drops
to 284 units. TCU = $12.65 per day]

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# The EOQ Model with Non instantaneous Receipt
The non instantaneous receiptmodel is an inventory system in which an order is received gradually, as inventory is
depleted.
In the basic EOQ Model, average inventory was half the maximum inventory level or y/2, butin this model variation
the maximum inventory level is notsimply y; itis an amount somewhatlower than y, adjusted for the fact the order
quantity is depleted during the order receiptperiod.
In order to determine the average inventory level, we define the following parameters unique to this model.
P = Daily rate at which the order is received over time (production rate)
d= Daily rate at which inventory is demanded.
As we know no shortages are possible. The demand cannotexceed production rate. i.e p≥ d
The time required to receive and order is the order quantity divided by the rate at which the order is received or y/p.
For example ifthe order size is 100 units and the production received in 5 days. The amount of inventory that will be
depleted or used up during this time period is determined by multiplying by the demand rate: y/p(d)
For example, ifit takes 5 days to receive the order and during this time inventory is depleted atthe rate of2 units per
day, then 10 units are used. As a result, the maximum amount ofinventory on hand is the order size minus the
amount depleted during the receiptperiod, computed as
Maximum inventory level = )1()(
p
d
yd
p
y
y 
Hence average inventory level = 












p
d
y 1
2
1
Total holding cost = 






p
dy
h 1
2
.
TCU = 






p
dhy
y
KD
1
2
Solving this function for the optimal value of y
)1(
2
p
d
h
kD
y


Example: 2 The 1-75 Carpet Discount store in North Georgia stocks carpetin its warehouse and sells itthrough
an adjoining showroom. The store keeps several brands and styles ofcarpetin stock; however, its biggest seller is
super shag carpet. The store wants to determine the optimal order size and total inventory costfor this brand of
carpetgiven an estimated annual demand of10,000 yards ofcarpet, and annual carrying costof0.75 per yard and
an ordering costof $150. The store like to know the number of orders that will made annually and time between

7. Page7 of 8
orders, given thatthe store is open every day exceptSunday. Thanksgiving Day and Christmas day (which is not a
Sunday)
Solution: Demand D = 10,000 yards / year
Holding Cost h = 0.75/year
Ordering Cost K = 150/ order
The optimum order qty yards
h
kD
y 2000
75.0
1000015022



Time between orders years
D
y
t 2.0
10000
2000
0 
Given that the store is open 311 days annually (365- 52 Sundays – Thanks giving and Christmas)
Hence to = 0.2 x 311 = 62.2 storedays
No. of orders yearorders
y
D
to
/5
2000
100001

TCU 1500750750
2
200075.0
2000
10000150
2





hy
y
KD
Example: 3 (Quantity Discount with constant carrying cost)
Comptek Computers wants to reduce a large stock ofPCs it is discontinuing. It has offered the university bookstore
at Tech a quantity discountpricing schedule as follows:
Quantity Price
1-49 $1400
50-89 $1100
90+ $900
The annual carrying cost for the bookstore for a PC is $190, the ordering cost is $2500 and annual
demand for this particular model is estimated to be 200 units. The bookstore wants to determine if it
should take advantage of this discount or order the basic EOQ order size.
Solution: K =2500
D =200 units / year
h =109/ year
pcs
h
kD
y 55.72
190
200250022



The order size is eligible for the first discount$1100. Therefore, this price is used to compute total cost
TCU 2337842200002.689279.6891)2001100(
2
55.72190
55.72
2002500
2




 PD
hy
y
KD
Since there is a discountfor a larger order size than 50 units. This total costof233784 must be compared with total
costwith an order of90 and discounted price of$900

8. Page8 of 8
TCU 194105$180000855055.5555)200900(
2
90190
90
2002500
2




 PD
hy
y
KD
Since the total costis lower, the maximum discountprice should be taken and 90 units should be ordered.
Example: 4 The 1-75 Outlet has its own manufacturing facility in which it produces super shag carpet. The
ordering costk is $costofsetting up the production process to make super shag carpet. Recall that h = $0.75 per
yard and D= 10,000 yards per year. The manufacturing facility operates the same days the store is open( i.e 311
days) and produces 150 yards ofcarpetper day. Determining the optimal order size, total inventory cost, the length
of time to receive an order, the number oforders per year and the maximum inventory level.
Solution: K =150/order
D =10000/year
h =0.75/ yard/ year
d = 10000/ 311 = 32.2 yards / day
p = 150 yards per day
The optimum order qty yards
p
d
h
kD
y 8.2256
)
150
2.32
1(75.0
100001502
)1(
2






63.66465.664)7854.0(3.84665.664
150
2.32
1
2
8.225675.0
8.2256
10000150
1
2

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












p
dhy
y
KD
TCU
= 1329.27
The length oftime to receive an order for this type ofmanufacturing operation is commonly called the length ofthe
production run.
Production run orderperdays
p
y
05.15
150
8.2256

Given that the store is open 311 days annually (365- 52 Sundays – Thanks giving and Christmas)
No. of orders per year yearruns
y
D
/43.4
8.2256
10000

Maximum Inventory level yards
p
d
y 1772)
150
2.32
1(8.2256)1( 
Exercise: Taylor, p-483, No. 10-1, 10-3, 10-4, 10-6, 10-7, 10-9, 10-20, 10-25