Inventory
Management
Inventory management
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A subsystem of logistics
Inventory: a stock of materials or other goods
to facilitate production or to satisfy customer
demand
Main decisions:
Which items should be carried in stock?
 How much should be ordered?
 When should an order be placed?
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The need to hold stocks 1
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Buffer between Supply and Demand
To keep down production costs: to achieve low unit costs,
production have to run as long as possible (setting up machines
is tend to be costly)
To take account of variable supply times: safety stock to cover
delivery delays from suppliers
To minimize buying costs associated with raising an order
To accommodate variations (on the short run) in demand (to
avoid stock-outs)
To account for seasonal fluctuations:
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There are products popular only in peak times
There are goods produced only at a certain time of the year
Adaptation o the fluctuation of
demand with building up stocks
DEMAND
Inventory
accumulation
CAPACITY
Inventory
reduction
The need to hold stocks 2
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To take advantage of quantity discounts (buying in bulk)
To allow for price fluctuations/speculation: to buy large
quantities when a good is cheaper
To help production and distribution operations run smoothly:
to increase the independence of these activities
To provide immediate service for customers
To minimize production delays caused by lack of spare parts
(for maintenance, breakdowns)
Work-in-progress: facilitating production process by providing
semi-finished stocks between different processes
Types of Stock-holding/Inventory
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raw material, component and packaging stock
in-process stocks (work-in-progress; WIP)
finished products (finished goods inventory; FGI)
pipeline stocks: held in the distribution chain
general stores: contains a mixture of products to
support
spare parts:
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Consumables (nuts, bolts, etc.)
Rotables and repairables
Independent vs. dependent demand
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Independent demand:
Influenced only by market conditions
 Independent from operations
 Example: finished goods
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Dependent demand:
Related to the demand for another item (with
independent demand)
 Example: product components, raw material
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Another typology of stocks
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working stock: reflects the actual demand
cycle stock: follows the production (or demand)
cycles
safety stock: to cover unexpected fluctuations in
demand
speculative stock: built up on expectations
seasonal stock: goods stockpiled before peaks
Inventory cost
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Item cost: the cost of buying or producing inventory
items
Ordering cost: does not depend on the number of
items ordered. Form typing the order to transportation
and receiving costs.
Holding (carrying) cost:
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Capital cost: the opportunity cost of tying up capital
Storage cost: space, insurance, tax
Cost of obsolescence, deterioration and loss
Stockout cost: economic consequences of running out
of stock (lost profit and/or goodwill)
Economic Order Quantity (EOQ)
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Assumptions of the model:
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Demand rate is constant, recurring and known
The lead time (from order placement and order delivery) is
constant and known
No stockouts are allowed
Goods are ordered and produced in lots, and the lot is placed
into inventory all at one time
Unit item cost is constant, carrying cost is linear function of
average inventory level
Ordering cost is independent of the number of items in a lot
The item is a single product (no interaction with other
products)
The „SAW-TOOTH”
Inventory level
Order inteval
Order
quantity
(Q)
Average inventory leve
= Q/2
Time
Total cost of inventory
(tradeoff between ordering frequency and inventory level)
Total cost
Holding cost
(H ∙ Q/2)
Minimum
cost
Ordering cost
(S ∙ D/Q)
EOQ
Calculating the total cost of
inventory
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Let…
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S be the ordering cost (setup cost) per oder
D be demanded items per planning period
H be the stock holding cost per unit
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H=i∙C, where C is the unit cost of an item, and i is the carrying rate
Q be the ordered quantity per order (= lot)
TC = S ∙ (D/Q) + H ∙ (Q/2)
(D/Q) is the number of orders per period
(Q/2) is the average inventory level in this model
The minimum cost (EOQ)
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TC = S ∙ (D/Q) + H ∙ (Q/2)
бTC/бQ = 0
0 = - S ∙ (D/Q 2) + H/2
H/2 = S ∙ (D/Q 2)
Q 2 = (2 ∙ S ∙ D)/H
EOQ = √ (2 ∙ S ∙ D)/H
Example
D = 1000 units per year
S = 100 euro per order
H = 20 euro per unit
Find the economic order quantity!
(we assume a saw-tooth model)
EOQ = √ (2 ∙ 1,000units ∙ 100euro)/20euro/unit
EOQ = √ 10,000units = 100units
2
Reordering (or replenishment) point
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When to start the ordering process?
It depends on the…
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Stock position: stock on-hand (+ stock on-order)
in a simple saw-tooth model it is Q ,
 in some cases, there can be an initial stock (Q0), that is different
from Q. In this case the first order depends on Q0
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lead time (L): the time interval from setting up order to the
start of using up the ordered stock
 Average demand per day (d)
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R = Q/d – L
Examples
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Q0 = 600 tons
Q = 200 tons
d = 10 tons per day
L = 8 days
R1 = 600ts /10ts/ds – 8ds = 52. day
Q0 = Q = 400 tons
d = 16 tons per day
L =20 days
R1 = 400ts /16ts/ds – 20ds = 5. day
R2 = 200ts /10ts/ds – 8ds = 12 days
after the arrival of the first order
= 52+8+12=72. day
Example on both EOQ and R
D = 2,000 tons
S = 100 euros per order
H = 25 euros per order
Q0 = 1,000 tons
L = 12 days
N = 250 days
Calculate the following:
EOQ
d
R1
R2
EOQ = √ (2 ∙ 2,000ts ∙ 100euro)/25euro/ts = 126,49ts
d = 2,000ts/250ds = 8 ts/ds
R1 = 1,000ts/8ts/ds – 12ds = 113. day
R2 = 126,49ts /8ts/ds – 12ds = 3,81 = 3 days after the arrival of the first
order = 113+12+3 = 128.
The SAW-TOOTH
with safety stock
Inventory level
Continuous demand
Order
quantity
b
Safety stock or buffer stock
Time
Buffer (safety) stock
b=z∙σ
where
z = safety factor from the (normal) distribution
σ = sandard deviation of daily demand over lead time
Let z be 1,65 (95%), and the standard deviation of daily
demand is 200 units/days.
b = 1,65 ∙ 200units/ds = 330units
When to order, when there is a buffer
stock?
R = (Q – b)/d – L or R1 = (Q0 – b)/d – L
Where (Q – b) is the available stock.
If Q0 = 100tons, Q = 60tons, L = 2 days, b = 10tons,
D = 300tons, N = 100 days, then R1, R2=?
R1 = (100 – 10)/(300/100) – 2 = 28. day
R2 = 60/3 – 2 = 18 days after the arrival of the first
order = 28 + 2 + 18 = 48. day
Alternative models 1
Periodic review system:
Stock level is examined at regular intervals
 Size of the order depends on the quantity on stock.
it should bring the inventory to a predetermined
level
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Stock
on hand
Q
Q
Q
time
T
L
L
L
T
T
Alternative models 2
Fixed-order-quantity system:
A predetermined stock level (reorder point) is given,
at which the replenishement order will be placed
 The order quantity is constant
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Stock
on hand
Q
Q
R
L
L
L
Elements of demand patterns
(forecasting)
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Actual demand:
Trend line
 Seasonal fluctuacion
 Weekly fluctuation
 (Daily fluctuation)
 Random fluctuation
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Inventory decisions and Multiple
Distribution Centres / warehouses
The ‘square root law’:
 A rule of thumb
 The total safety-stock holding in a distribution system is proportional to
the square root of the number of depot locations
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Example: If we reduce the number of DCs from 10 to 5, the savings in safety
stock is:
1 – (√5 / √10) = 29%
Pareto’s law or the ’80/20 rule’:
 A rule of thumb
 Approximately 20% of storage items account for 80% of the inventory
value measured in money.
 ABC analysis (or Pareto analysis):
 ‘A’ lines: fast movers (20%) – 80% of money usage
 ‘B’ lines: medium movers (30%) – 15% of money usage
 ‘C’ lines: slow movers (C+D 50%) – 5% of money usage
 ‘D’ lines: obsolete / dead stock
ABC analysis
ABC analysis
ABC analysis exercise
Item
number
Annual demand Price
Annual dollar value
1
2.500
360
?
2
1.000
70
?
3
2.400
500
?
4
1.500
100
?
5
700
70
?
6
1.000
1000
?
7
200
210
?
8
1.000
4000
?
9
8.000
10
?
500
200
?
10
?
ABC analysis exercise
Item
number
Annual demand Price
Annual dollar value
1
2.500
360
900.000
2
1.000
70
70.000
3
2.400
500
1.200.000
4
1.500
100
150.000
5
700
70
49.000
6
1.000
1000
1.000.000
7
200
210
42.000
8
1.000
4000
4.000.000
9
8.000
10
80.000
500
200
100.000
10
∑ 7.591.000
Solution
Item
number
Classification
Annual dollar
8
4.000.000
3
1.200.000
6
1.000.000
1
900.000
4
150.000
10
100.000
9
80.000
2
70.000
5
49.000
7
42.000
Percentage of
items
Percentage of
annual dollar
value
Solution
Item
number
Classification
Annual dollar
8
4.000.000
A
3
1.200.000
B
6
1.000.000
B
1
900.000
B
4
150.000
C
10
100.000
C
9
80.000
C
2
70.000
C
5
49.000
C
7
42.000
C
?
Percentage of
items
Percentage of
annual dollar
value
10
52,7
30
40,8
60
6,5
?
Data Capture
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Techniques and error rates
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Written entry – 25,000 in 3,000,000
Keyboard entry – 10,000 in 3,000,000
Optical character recognition (OCR) – 100 in 3,000,000
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labels that are both machine- and human-readable
for example: license plates
Bar code (code 39) – 1 in 3,000,000
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(Rushton et al. 2006)
fast, accurate and fairly robust
reliable and cheap technique
Transponders (radio frequency tags) – 1 in 30,000,000
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a tag (microchip + antenna) affixed to the goods or container
receiver antenna
reader
host station that relays the data to the server
can be passive or active
Some more examples
Calculate R1 and EOQ, if…
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L = 2 days, b = 12tons, D = 300tons, N = 100 days, S
= 50 euro, H = 20 euro/tons, Q0 = 60tons
L = 12 days, b = 20tons, D = 1300tons, N = 80 days, S
= 10 euro, H = 25 euro/tons, Q0 = 300tons
D = 1000 units, N = 500 days, S = 110 euro, H = 100
euro/units, L = 20 days, b = 50tons, Q0 =100tons