Koç University
OPSM 301 Operations Management
Class 15:
Inventory Management
EOQ Model
Zeynep Aksin
zaksin@ku.edu.tr
Inventory
 “The stock of any item or resource used in
an organization”
 “All the money that the system has
invested in purchasing things it intends to
sell”
The Material Flow Cycle
Types of Inventories
 Inputs - Raw Materials
 Processes - Work-in-Progress
 Outputs - Finished Goods
Why do we need Inventory?
 Variability (uncertainty)
– Demand
– Capacity availability
– Materials and lead times
– Processing times
 Time
– Delivery lead time, production lead time
 Economies of Scale
– Purchasing, production
Functions Provided by Inventories
Purpose
/Reason
Type
Cost
Transportation
Pipeline
Transportation Costs
Economies in Setups
Cycle Stocks
Setup/Order Costs
Seasonality in Demand
Seasonal Stock
Smoothing Costs
Uncertainty in Demand
Safety Stock
Shortage/Stock-out
Costs
Economies in Purchase
Cycle Stocks
Price Discounts
Inflation and/or
Price Fluctuations
Speculative Stock Costs due to Price
Inventory Costs
 Purchase Cost
 Ordering Cost
– Receiving and inspection
– Transportation
 Holding (Carrying) Cost
–
–
–
–
Cost of money
Insurance
Taxes
Shrinkage, spoilage, obsolescence
 Stock-out (Shortage) Cost
– Lost sales, customers etc.
– Emergency shipment costs
Economies of Scale:
Inventory Management for a Retailer
The South Face retail shop in the John Hancock Tower has
observed a stable monthly demand for its line of Gore-Tex jackets
on the order of 100 jackets per month. The retail shop incurs a fixed
cost of $2,000 every time it places an order to the Berkeley
warehouse for stock replenishment. The marginal cost of a jacket is
$200, and South Face’s cost of capital is approximately 25%.
What order size would you recommend for The South Face?
warehouse
retailer
Parameters EOQ Model
D
C
S
H
Q
demand rate (units per year)
unit production cost, not counting
setup or inventory costs
(dollars per unit)
fixed or setup cost to place an order
(dollars)
holding cost (dollars per year); if the
holding cost is consists entirely of
interest on money tied up in
inventory, then H = iC where i is an
annual interest rate.
the unknown size of the order or lot
size
Inventory Usage Over Time
Inventory Level
Order quantity = Q
(maximum inventory
level)
Minimum
inventory 0
Usage Rate
Average
Inventory
(Q*/2)
Time
Cost Minimization Goal
C
O
S
T
Total Cost (TC)
Holding Cost
Annual Cost of
Items
Ordering Cost
QOPT
D
Q
TC = DC +
S+
H
Q
2
Order Quantity (Q)
Total Annual Cost
Annual
Annual
Total Annual Cost = Purchasing + Ordering +
Cost
Cost
Annual
Holding
Cost
D
Q
TC = DC +
S+
H
Q
2
 Using calculus, we can take the derivative of
the total cost function and set the derivative
(slope) equal to zero
 We can also use economic intuition
Find most economical order quantity:
Spreadsheet for The South Face
Number of units
per order/batch
Q
50
100
150
200
250
260
270
280
290
300
310
320
330
340
350
400
500
600
700
Number of
Batches per
Year: R/Q
24
12
8
6
5
5
4
4
4
4
4
4
4
4
3
3
2
2
2
Annual
Setup Cost
48000
24000
16000
12000
9600
9231
8889
8571
8276
8000
7742
7500
7273
7059
6857
6000
4800
4000
3429
Annual
Holding Cost
1250
2500
3750
5000
6250
6500
6750
7000
7250
7500
7750
8000
8250
8500
8750
10000
12500
15000
17500
Annual
Total Cost
49250
26500
19750
17000
15850
15731
15639
15571
15526
15500
15492
15500
15523
15559
15607
16000
17300
19000
20929
Deriving the EOQ
QOPT
QOPT
2DS
=
H
2(Annual Demand)(Or der or Setup Cost)
=
Annual Holding Cost
EOQ Model: if there is a lead time L
# Units on hand
Qopt
ROP
L
ROP = Reorder point
L = Lead time (constant)
Q = Economic order quantity
L
Time
EOQ Example
•
•
•
•
•
•
Annual Demand = 1,000 units
Days per year considered in average daily demand = 250
Cost to place an order = $10
Holding cost per unit per year = $0.50
Lead time = 7 days
Cost per unit = $15
Determine the economic order quantity and the
reorder point
An EOQ Example
Determine optimal number of needles to order
D = 1,000 units
S = $10 per order
H = $.50 per unit per year
Q* =
2DS
H
Q* =
2(1,000)(10)
=
0.50
40,000 = 200 units
An EOQ Example
Determine optimal number of needles to order
D = 1,000 units
Q* = 200 units
S = $10 per order
H = $.50 per unit per year
Expected
Demand
D
number of = N =
=
Order
quantity
Q*
orders
1,000
N=
= 5 orders per year
200
An EOQ Example
Determine optimal number of needles to order
D = 1,000 units
Q* = 200 units
S = $10 per order
N = 5 orders per year
H = $.50 per unit per year
Number of working
Expected
days per year
time between = T =
N
orders
T=
250
= 50 days between orders
5
An EOQ Example
Determine optimal number of needles to order
D = 1,000 units
Q* = 200 units
S = $10 per order
N = 5 orders per year
H = $.50 per unit per year
T = 50 days
Total annual cost = Setup cost + Holding cost
D
Q
S +
H
Q
2
1,000
200
TC =
($10) +
($.50)
200
2
TC =
TC = (5)($10) + (100)($.50) = $50 + $50 = $100
Inventory level (units)
Reorder Point Curve
Figure 12.5
Q*
Slope = units/day = d
ROP
(units)
Time (days)
Lead time = L
Reorder Point Example
Demand = 8,000 iPods per year
250 working day year
Lead time for orders is 3 working days
D
d=
Number of working days in a year
= 8,000/250 = 32 units
ROP = d x L
= 32 units per day x 3 days = 96 units
Economic Order Quantity (EOQ) Model
 Economic Order Quantity (EOQ)
Model
– Robust, widely used
– Insensitive to errors in estimating
parameters (40-20-2 Rule):
• 40% error in one of the parameters
• 20% error in Q
• < 2% of total cost penalty
An EOQ Example
Management underestimated demand by 50%
D = 1,000 units 1,500 units Q* = 200 units
S = $10 per order
N = 5 orders per year
H = $.50 per unit per year
T = 50 days
D
Q
TC =
S +
H
Q
2
1,500
200
TC =
($10) +
($.50) = $75 + $50 = $125
200
2
Total annual cost increases by only 25%
An EOQ Example
Actual EOQ for new demand is 244.9 units
D = 1,000 units 1,500 units Q* = 244.9 units
S = $10 per order
H = $.50 per unit per year
D
Q
TC =
S +
H
Q
2
1,500
244.9
TC =
($10) +
($.50)
244.9
2
TC = $61.24 + $61.24 = $122.48