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 fo 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
 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:
 Capital cost: the opportunity cost of tying up capital
 Storage cost: space, insurance, tax
 Cost of obsolescence, deterioration and loss
 Sometimes designated by management rather than
computed
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
Marginal holding cost is constant
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
The EOQ zone
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 (LT): 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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ROP = d (LT ) + safety stock
Examples
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Q0 = 600 tons
ROP = 80 tons
Q = 200 tons
d = 10 tons per day
(600 – 80)/10 = 52. day
LT = 8 days
What is the ROP and when 52 + 8 + (200 – 80)/10 = 72. day
we reach that level?
What is the time of the next
reorder?
Q0 = Q = 400 tons
d = 16 tons per day
LT = 20 days
ROP = ?
First rearder time?
ROP = 16 ∙ 20 = 320 tons
First reorder: (400 – 320)/16 = 5. day
Examples
D = 2,000 tons
S = 100 euros per order
H = 25 euros per order
Initial stock = 1,000 tons
LT = 12 days
N = 250 days
Calculate the following:
EOQ
d
ROP
first and second reorder
time
EOQ = √ (2 ∙ 2,000ts ∙ 100euro)/25euro/ts = 126.49ts
d = 2,000ts/250ds = 8 ts/ds; ROP = 12 ∙ 8 = 96 tons
Reroder1 = (1,000 – 96)/8 = 113
Reorder2 = 113 + 12 + (126.49 – 96)/8 = 128.81 = 128
The SAW-TOOTH
with safety stock
Inventory level
Continuous demand
Order
quantity
b
Safety stock or buffer stock
Time
Buffer stock depends
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Demand rate and lead time
Variability of demand and lead time
Desired service level
Service level
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The probability that demand will not exceed
supply during lead time.
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Service level = 100 percent - stockout risk
Buffer (safety) stock
b=z∙σ
where
z = safety factor from the (normal) distribution
σ = sandard deviation of demand over lead time
Let z be 1,65 (95%), and the standard deviation of
demand is 200 units/lead time.
b = 1,65 ∙ 200units = 330units
Example
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Lead time = 10 days
Average demand over lead time: 300 tons
Standard deviation over lead time: 20 tons
Accepted risk level: 5%
Safety stock = ? Reorder quantity = ?
b = z * σ = 1,65 * 20 = 33 tons
ROP = 300 + 33 = 333 tons
Examples
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Q0 = 600 tons
Q = 200 tons
d = 10 tons per day
LT = 8 days
b = 33 tons
ROP = 8 * 10 + 33 = 113
Q0 = Q = 400 tons
d = 16 tons per day
LT =20 days
b = 66 tons
ROP = 386
Economic production quantity
(EPQ)
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Production of spare parts /materials done in batches (lots)
Only on item
Annual demand is known
Usage rate (u) is constant
Usage occurs continually but production occurs
periodically
Production rate is constant (p)
Lead time (LT) does not vary
No quantity (Q) discounts
Setup cost instead of ordering cost
EPQ with incremental inventory
replenishment
Total cost in EPQ
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TCEPQ = carrying cost + setup cost =
= (Imax/2)H + (D/Q)S
The economic run quantity:
QEPQ = (2DS/H)0,5 ∙ [p/(p – u)]0,5
Cycle time = QEPQ / u
Run time = QEPQ / p
Imax = (QEPQ / p) (p – u)
Iaverage = Imax / 2
EOQ example
Quantity discounts
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Price reductions for large orders
Example:
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The buyers total cost curve to minimze:
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Advantages
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For the buyer
Fewer order set-ups (D*S/Q)
 Cheaper price (P*D)
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For the seller:
Decreased holding costs (I*H/2)
 Decreased administrative costs (FC)
 Lower opportunity cost
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A TC görbe, ha szerepeltetjük a
beszerzési költséget is
Allowances and TC curves
Constant carrying costs
Carrying costs are stated as a
percentage of unit price
Cost
Quantity
Finding EOQ with
constant holding cost
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D = 816 pieces/year
S = 12 dollars/order
H = 4 dollars/piece/year
Prices:
20 dollars 1-49 pieces,
 18 dollars 50-79 pieces,
 17 dollars 80-99 pieces,
 16 dollars over 100 pieces
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Calculating EOQ
Solution
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QEOQ = (2*816*12/4)0,5 = 69.97 = 70 pieces
TC70 = (70/2)*4 + (816/70)*12 + 18*816 =
= 14.968 dollars
TC80 = 14.154
TC100 = 13.354
With non-constant holding costs
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D = 4.000 pieces, S = 30 dollars, H = 0.4*P
Prices:
1-499 pieces 0.9 dollar;
 500-999 0.85 dollar;
 over 1.000 pieces 0.8 dollar.
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Solution
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3.
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QEOQ(0,8)=866 pieces → not feasible
QEOQ(0,85)=840 pieces → feasible
TC840=30*(4000/840)+0,4*0,85*(840/2)+4000*0,85=
=3685,66 dollars→ is it profitable to order a greater lot?
TC1000=30*(4000/1000)+0,4*0,8*(1000/2)+4000*0,8=
=3480 dollars→ yes, it is profitable.
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
Example
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Initial number of warehouses: 6
Initial sum of safety stocks: 6,000
New number of warehouses: 2
What is the new sum of safety stocks?
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6000*(2/6)0.5=3464.10
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
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