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Inventory models
Nur Aini Masruroh
Outline
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Introduction
Deterministic model
Probabilistic model
Introduction
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What is inventory?
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Over stocking
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It is usable but idle resource
Keeping of physical goods or commodities for the purpose of
satisfying demand over a specified time horizon.
Higher invested capital per unit time
Less frequent occurrence of placement of orders
Less frequent occurrence of shortages
Under stocking
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Lower invested capital per unit time
Increase the frequency of ordering
Higher risk of running out of stock
The Functions of Inventory
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To ”decouple” or separate various parts of the
production process
To provide a stock of goods that will provide a
“selection” for customers
To take advantage of quantity discounts
To hedge against inflation and upward price changes
Disadvantages of Inventory
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Higher costs
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Item cost (if purchased)
Ordering (or setup) cost
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Holding (or carrying) cost
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Costs of forms, clerks’ wages etc.
Building lease, insurance, taxes etc.
Difficult to control
Hides production problems
Inventory Classifications
Inventory
Process
stage
Raw Material
WIP
Finished Goods
© 1984-1994
T/Maker Co.
Number
& Value
Demand
Type
Other
A Items
B Items
C Items
Independent
Dependent
Maintenance
Operating
ABC Analysis
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Divides on-hand inventory into 3 classes
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Basis is usually annual $ volume
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A class, B class, C class
$ volume = Annual demand x Unit cost
Policies based on ABC analysis
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Develop class A suppliers more
Give tighter physical control of A items
Forecast A items more carefully
Classifying Items as ABC
Class
A
B
C
% Annual $ Usage
100
80
60
% $ Vol
80
15
5
A
40
B
20
C
0
0
50
100
% of Inventory Items
% Items
15
30
55
Independent versus Dependent Demand
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Independent demand - demand for item is independent
of demand for any other item
Dependent demand - demand for item is dependent
upon the demand for some other item
Inventory decision
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How should it be ordered for stocking?
When should it be ordered?
Basic characteristics of inventory systems
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Economic parameters
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Holding cost
Ordering cost
Setup cost
Shortage cost
Purchase price
Selling price
Demand
Ordering cycle
Delivery lags or lead times
Economic parameters
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Holding costs - associated with holding or “carrying”
inventory over time
Ordering (setup) costs - Involve the fixed charge
associated with the placement of an order or with the
initial preparation of a production system.
Shortage costs - The penalty costs incurred as a result of
running out of stock
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Loss in customers goodwill
Loss in sales, etc.
Purchase price - This parameter is of interest when
quantity discounts or price breaks can be secured for
orders above a certain quantity
Selling price- This parameter is of interest when the
demand on the commodity may be affected by the quantity
stocked
Holding (Carrying) Costs
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Obsolescence
Insurance
Extra staffing
Interest
Pilferage
Damage
Warehousing
Etc.
Ordering Costs
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Supplies
Forms
Order processing
Clerical support
Etc.
Symbols of inventory model
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Q = Order quantity per order
K = Setup cost per order
d = Demand rate
h = Holding cost per unit per unit time
c = Purchase price or cost per unit
s = Shortage cost per unit per unit time
t = Inventory cycle
Basic EOQ model
EOQ assumptions:
 Known and constant demand
 Known and constant lead time
 Instantaneous receipt of material
 No quantity discounts
 Only order (setup) cost and holding cost
 No stockouts
Inventory Usage Over Time
Inventory Level
Order quantity =
Q (maximum
inventory level)
Minimum
inventory 0
Usage Rate
Average
Inventory
(Q*/2)
Time
EOQ Model
How Much to Order?
Annual Cost
Minimum
total cost
Order (Setup) Cost Curve
Optimal
Order Quantity (Q*)
Order quantity
Why Holding Costs Increase
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More units must be stored if more are ordered
Purchase Order
Description Qty.
Microwave
1
Order
quantity
Purchase Order
Description Qty.
Microwave 1000
Order
quantity
Why Order Costs Decrease
Cost is spread over more units
Example:You need 1000 microwave ovens
1 Order (Postage $ 0.33)
1000 Orders (Postage $330)
Purchase Order
Description
Qty.
Microwave
1000
PurchaseOrder
Order
Purchase
Description
Qty.
Purchase
Order
Description
Qty.
Description
Qty.1
Microwave
Description
Qty.
Microwave
1
Microwave
1
Microwave
1
Order
quantity
EOQ Model When To Order
Inventory Level
Average
Inventory
(Q*/2)
Optimal
Order
Quantity
(Q*)
Reorder
Point
(ROP)
Time
Lead Time
Inventory holding cost per cycle:
hQ2/(2d)
Deriving EOQ Model
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Total cost per cycle
TC(Q) = Acquisition costs + Holding cost = K + cQ + hQ2/(2d)
Total cost per unit time
TCU(Q) = TC(Q)/t = Kd/Q + cd + hQ/2
where t = Q/d.
To minimize the total cost per unit time, we differentiate
TCU(Q) with respect to Q and set it equal to zero
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This gives
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The order cycle length,
The Reorder Point (ROP) Curve
Inventory level (units)
Q*
Slope = units/day = d
ROP
(Units)
Time (days)
Lead time = L
EOQ Model with Shortages Allowed
Assumptions:
 A continuous, constant and known rate of demand d
 Constant lead time L
 Constant unit price or cost c
 No interaction between items
 Infinite planning horizon
 Unfilled demands are backlogged at the cost of s per unit
per unit time
 Split delivery not allowed
 No limit on capital availability
EOQ Model with Shortages Allowed
EOQ Model with Shortages Allowed
Single item economic production quantity
(EPQ)
Assumptions:
 Continuous, constant and known rate of demand d
 Continuous, constant and known rate of production p
 P>d
 Constant lead time L
 Constant unit price or cost c
 No interaction between items
 Infinite planning horizon
 Shortages are not permitted
 Production and consumption can occur simultaneously
 No limit on production capacity
Single item economic production quantity
(EPQ)
Single item economic production quantity
(EPQ)
Quantity Discount Model
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Answers how much to order &
when to order
Allows quantity discounts
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Reduced price when item is purchased in larger
quantities
Other EOQ assumptions apply
Trade-off is between lower price & increased holding
cost
Quantity Discount Schedule
Discount Discount Quantity Discount
Number
(%)
1
0 to 999
No discount
Discount
Price (P)
$5.00
2
1,000 to 1,999
4
$4.80
3
2,000 and over
5
$4.75
Quantity Discount – How Much to
Order
Probabilistic Models
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Answer how much & when to order
Allow demand to vary
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Follows normal distribution
Other EOQ assumptions apply
Consider service level & safety stock
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Service level = 1 - Probability of stockout
Higher service level means more safety stock
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More safety stock means higher ROP
Probabilistic Models When to Order?
Inventory Level
Frequency
Service
Level
P(Stockout)
Optimal
Order
Quantity
SS
X
ROP
Reorder
Point
(ROP)
Safety Stock (SS)
Place
order
Lead Time
Receive
order
Time
Fixed Period Model
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Answers how much to order
Orders placed at fixed intervals
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No continuous inventory count
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Inventory brought up to target amount
Amount ordered varies
Possibility of stockout between intervals
Useful when vendors visit routinely
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Example: P&G representative calls every 2
weeks
Inventory Level in a Fixed Period
System
Various amounts (Qi) are ordered at regular time intervals (p) based on the
quantity necessary to bring inventory up to target maximum
Target maximum
Q1
Q4
Q2
On-Hand
Inventory
Q3
p
p
p
Time
Fixed Period Model When to Order?
Inventory Level
Period
Target maximum
Period
Period
Time