Transcript Production and Operations Management: Manufacturing and

1
Inventory Management
and
Control
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AMAZON.com
• Jeff Bezos, in 1995, started AMAZON.com as a
“virtual” retailer – no inventory, no warehouses, no
overhead; just a bunch of computers.
• Growth forced AMAZON.com to excel in inventory
management!
• AMAZON is now a worldwide leader in warehouse
management and automation.
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Order Fulfillment at AMAZON (1 of 2)
1. You order items; computer assigns your order to
distribution center [closest facility that has the
product(s)]
2. Lights indicate products ordered to workers who
retrieve product and reset light.
3. Items placed in crate with items from other orders,
and crate is placed on conveyor. Bar code on item is
scanned 15 times – virtually eliminating error.
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Order Fulfillment at AMAZON (2 of 2)
4. Crates arrive at a central point where items are boxed
and labeled with new bar code.
5. Gift wrapping done by hand (30 packages per hour)
6. Box is packed, taped, weighed and labeled before
leaving warehouse in a truck.
7. Order appears on your doorstep within a week
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Inventory (Definition of)
• Inventory is the stock of any item or resource held to
meet future demand and can include: raw materials,
finished products, component parts, supplies, and workin-process
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Inventory Classifications
Inventory
Process
stage
Raw Material
WIP
Finished Goods
Number
& Value
Demand
Type
A Items
B Items
C Items
Independent
Dependent
Other
Maintenance
Operating
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Inventories by Demand Type
Independent Demand : Demand for the final end-product that are ready to be
sold or used.
Demand not related to other items; demand created by
external customers); eg. Demand for computers
Finished product:
eg: Computer
A
Independent demand is uncertain
Dependent demand is certain
B(4)
D(1)
C(2)
E(1
E(2)
)
Components: eg. parts that
make up the computer
B(1)
E(3)
Dependent Demand : Derived demand for
components of finished
products (parts, raw
materials, subassemblies)
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Types of Inventories (1 of 2)
• Raw materials & purchased parts
• Partially completed goods called
work in process
• Finished-goods inventories
(manufacturing firms)
or merchandise
(retail stores)
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Types of Inventories (2 of 2)
• Maintenance and repairs (MRO) inventory,
replacement parts, tools, & supplies
• Goods-in-transit to warehouses or customers (pipeline
inventory)
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The Material Flow Cycle (1 of 2)
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The Material Flow Cycle (2 of 2)
Input
Wait
Time
Move
Time
Queue
Time
Setup
Time
Run
Time
Output
Cycle Time
Run time: Job is at machine and being worked on
Setup time: Job is at the work station, and the work station is
being "setup."
Queue time: Job is where it should be, but is not being
processed because other work precedes it.
Move time: The time a job spends in transit
Wait time: When one process is finished, but the job is waiting
to be moved to the next work area.
Other: "Just-in-case" inventory.
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Performance Measures
• Inventory turnover (the ratio of annual cost of
goods sold to average inventory investment)
• Days of inventory on hand (expected number of
days of sales that can be supplied from existing
inventory)
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Functions of Inventories (1 of 2)
1. To meet variation in product demand and to protect
against stock-outs
2. To “decouple” operations or separate various parts of
the production process, ie. to maintain independence
of operations
3. To meet unexpected demand & to provide high levels
of customer service
3. To smooth production requirements by meeting
seasonal or cyclical variations in demand
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Functions of Inventories (2 of 2)
4. To provide a safeguard for variation in raw material
delivery time
5. To provide a stock of goods that will provide a
“selection” for customers
6. To take advantage of economic purchase-order size
7. To take advantage of quantity discounts
8. To take advantage of order cycles
9. To hedge against price increases
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Disadvantages of Inventories
• Higher costs
– Item cost (if purchased)
– Ordering (or setup) cost
– Holding (or carrying) cost
• Difficult to control
• Hides production problems
• May decrease flexibility
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Inventory Costs
 Holding (or carrying) costs
Costs for storage, handling, insurance, etc
 Setup (or production change) costs
Costs to prepare a machine or process for
manufacturing an order, eg. arranging specific
equipment setups, etc
 Ordering costs (costs of replenishing inventory)
Costs of placing an order and receiving goods
 Shortage costs
Costs incurred when demand exceeds supply
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Holding (Carrying) Costs
•
•
•
•
•
•
•
•
Obsolescence
Insurance
Extra staffing
Interest
Pilferage
Damage
Warehousing
Etc.
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Inventory Holding Costs
(Approximate Ranges)
Category
Cost as a
% of Inventory Value
Housing costs (building rent, depreciation,
operating cost, taxes, insurance)
6%
(3 - 10%)
Material handling costs (equipment, lease or
depreciation, power, operating cost)
3%
(1 - 3.5%)
Labor cost from extra handling
Investment costs (borrowing costs, taxes,
and insurance on inventory)
Pilferage, scrap, and obsolescence
Overall carrying cost
3%
(3 - 5%)
11%
(6 - 24%)
3%
(2 - 5%)
26%
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Ordering Costs
•
•
•
•
Supplies
Forms
Order processing
Clerical support, etc.
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Setup Costs
• Clean-up costs
• Re-tooling costs
• Adjustment costs, etc.
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Shortage Costs
• Backordering cost
• Cost of lost sales
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Inventory Control System
Defined
 An inventory system is the set of policies and controls
that monitor levels of inventory and determine what
levels should be maintained, when stock should be
replenished and how large orders should be
 Management has two basic functions concerning
inventory:
Establish a system for tracking items in inventory
Make decisions about:
 When to order?
 How much to order?
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Objective of Inventory Control
To achieve satisfactory levels of customer service while
keeping inventory costs within reasonable bounds
Level of customer service
Costs of ordering and carrying inventory
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Requirements of an Effective Inventory
Management
A system to keep track of inventory
A reliable forecast of demand
Knowledge of lead time and lead time variability
Reasonable estimates of
 Holding costs
 Ordering costs
 Shortage costs
A classification system for inventory items
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Inventory Counting (Control) Systems
• Periodic System
Physical count of items made at periodic intervals;
order is placed for a variable amount after fixed
passage of time
• Perpetual (Continuous) Inventory System
System that keeps track
of removals from inventory
continuously, thus
monitoring current levels of
each item (constant amount is
ordered when inventory
declines to a predetermined level)
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Inventory Accuracy and
Cycle Counting
• Inventory accuracy refers to how well the inventory records
agree with physical count.
• Cycle Counting
Physically counting a sample of total inventory on a regular
basis
• Used often with ABC classification
– A items counted most often (e.g., daily)
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Cycle Counting Management
• Cycle counting management
– How much accuracy is needed?
• A items: ± 0.2 percent
• B items: ± 1 percent
• C items: ± 5 percent
– When should cycle counting be performed?
– Who should do it?
12-27
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Advantages of Cycle Counting
• Eliminates shutdown and interruption of production
necessary for annual physical inventories
• Eliminates annual inventory adjustments
• Provides trained personnel to audit the accuracy of
inventory
• Allows the cause of errors to be identified and
remedial action to be taken
• Maintains accurate inventory records
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Inventory Counting Technologies
• Universal product code (UPC)
– Bar code printed on a label that has information
about the item to which it is attached
• Radio frequency identification (RFID) tags
– A technology that uses radio waves to identify
objects, such as goods in supply chains
12-29
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Demand Forecasts and Lead Time
• Forecasts
– Inventories are necessary to satisfy customer demands,
so it is important to have a reliable estimates of the
amount and timing of demand
• Lead time
– Time interval between ordering and receiving the order
• Point-of-sale (POS) systems
– A system that electronically records actual sales
– Such demand information is very useful for enhancing
forecasting and inventory management
12-30
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ABC Classification System
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ABC Classification System
• Demand volume and value of items vary
• Items kept in inventory are not of equal importance in
terms of:
–
dollars invested
–
profit potential
–
sales or usage volume
–
stock-out penalties
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ABC Classification System
Classifying inventory according to some measure of
importance and allocating control efforts accordingly.
A - very important (10 to 20 percent of the number of items
in inventory and about 60 to 70 percent of the annual dollar
value
B - mod. important
High
C - least important
(50 to 60 percent of
A
Annual
$ value
the number of items
of items
B
in inventory but only
C
Low
about 10 to 15 percent of the
Few
Number of Items Many
annual dolar value
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ABC Analysis
 Classify inventory into 3 categories typically on the
basis of the dollar value to the firm
$ volume =
Annual demand x Unit cost
 A class, B class, C class Policies based on ABC
analysis
– Develop class A suppliers more carefully
– Give tighter physical control of A items
– Forecast A items more carefully
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Classifying Items as ABC
Class
A
B
C
% Annual $ Usage
100
80
60
40
20
% $ Vol
70-80
15
5-10
A
B
C
0
50
100
% of Inventory Items
% Items
5-15
30
50-60
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ABC Classification
PART
UNIT COST
ANNUAL USAGE
1
2
3
4
5
6
7
8
9
10
$ 60
350
30
80
30
20
10
320
510
20
90
40
130
60
100
180
170
50
60
120
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ABC Classification
PART
9
8
2
1
4
3
6
5
10
7
TOTAL
PART
VALUE
$30,600
1
16,000
2
14,000
3
5,400
4
4,800
5
3,900
3,600
6
3,000
7
2,400
8
1,700
9
$85,400
10
% OF TOTAL % OF TOTAL
UNIT
ANNUAL
USAGE
VALUECOSTQUANTITY
% CUMMULATIVE
35.9
$ 60
18.7
350
16.4
30
6.3
5.680
4.630
4.220
3.510
2.8
320
2.0
510
20
6.0
5.0
4.0
9.0
6.0
10.0
18.0
13.0
12.0
17.0
90
40
130
60
100
180
170
50
60
120
6.0
11.0
15.0
24.0
30.0
40.0
58.0
71.0
83.0
100.0
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ABC Classification
PART
9
8
2
1
4
3
6
5
10
7
TOTAL
PART
VALUE
$30,600
1
16,000
2
14,000
3
5,400
4
4,800
5
3,900
3,600
6
3,000
7
2,400
8
1,700
9
$85,400
10
% OF TOTAL % OF TOTAL
UNIT
ANNUAL
VALUECOSTQUANTITY
% USAGE
CUMULATIVE
35.9
$ 60
18.7
350
16.4
30
6.3
5.680
4.630
4.220
3.510
2.8
320
2.0
510
20
6.0
5.0
4.0
9.0
6.0
10.0
18.0
13.0
12.0
17.0
90
A
40
130
60
B
100
180
170
C
50
60
120
6.0
11.0
15.0
24.0
30.0
40.0
58.0
71.0
83.0
100.0
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ABC Classification
PART
TOTAL
PART
VALUE
9 $30,600
1
8
16,000
2
2
14,000
3
1 CLASS
5,400
4
4
4,800
A3,900
5
3
B3,600
6
6
C3,000
5
7
10
2,400
8
7
1,700
9
$85,400
10
% OF TOTAL % OF TOTAL
UNIT
ANNUAL
USAGE
VALUECOSTQUANTITY
% CUMMULATIVE
35.9
6.0
$ 60
18.7
5.0
350
16.4 % OF TOTAL
4.0
30
6.3
ITEMS
VALUE9.0
5.680
6.0
9, 8, 2 4.630
71.010.0
1, 4, 3 4.220
16.518.0
6, 5, 10,3.5
7
12.513.0
10
2.8
12.0
320
2.0
17.0
510
20
6.0
90
11.0
A
40
15.0
% OF TOTAL
130
24.0
QUANTITY
60
B 15.030.0
100
40.0
180 25.058.0
60.071.0
170
C
83.0
50
100.0
60
120
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ABC Classification
C
100 –
B
% of Value
80 –
60 –
A
40 –
20 –
0 |–
0
|
20
|
40
|
60
% of Quantity
|
80
|
100
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Inventory Models
 Multi-Period Inventory Models
 Fixed-Order Quantity Models
• Event triggered (Example: running out of stock)
 Fixed-Time Period Models
• Time triggered (Example: Monthly sales call by
sales representative)
 Single-Period Inventory Model
 One time purchasing decision (Example: vendor selling
t-shirts at a football game)
 Seeks to balance the costs of inventory overstock and
under stock
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Multi-Period Inventory Models
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Multi-Period Inventory Models
Fixed-Order Quantity Models (Types of)
The Basic Economic Order Quantity Model
Economic Production Order Quantity (Economic
Lot Size) Model
Economic Order Quantity Model with Quantity
Discounts
Fixed Time Period (Fixed Order Interval) Models
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Fixed Order Quantity Models:
Economic Order Quantity Model
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Economic Order Quantity Model
The basic EOQ Model is used to find a fixed order quantity that will
minimize total annual inventory costs
Assumptions:
• Only one product is involved
• Demand for the product is known with certainty, is constant and
uniform (even) throughout the period
• Lead time (time from ordering to receipt) is known and constant
• Price per unit of product is constant (no quantity discounts)
• Inventory holding cost is based on average inventory
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Economic Order Quantity Model
• Ordering or setup costs are constant
• All demands for the product will be satisfied (no back
orders are allowed)
• No stockouts (shortages) are allowed
• The order quantity is received all at once. (Instantaneous
receipt of material in a single lot)
The goal is to calculate the order quantitiy that
minimizes total cost
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Basic Fixed-Order Quantity Model and
Reorder Point Behavior
1. You receive an order quantity Q.
Number
of units
on hand
(Inv.
Level)
Q
4. The cycle then repeats.
Q
Q
R
2. You start using them up
over time.(usage rate)
L
Place
order
Time
3. When you reach down to a level
of inventory of R, you place your
next Q sized order.
L
Receive
order
R = Reorder point
Q = Economic order quantity
L = Lead time
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EOQ Model
Inventory Level
Order
Quantity
(Q)
Average
Inventory
(Q/2)
Demand
rate
Reorder
Point
(ROP)
Order placed
Lead Time
Order received
Time
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Total Annual Cost
Total
Annual =
Cost
Annual
Annual
Annual
Purchase + Ordering + Holding
Cost
Cost
Cost
D
Q
TC = DC + S + H
Q
2
TC=Total annual cost
D =Annual demand
C =Cost per unit
Q =Order quantity in
units
S =Cost of placing an
order or setup cost
R =Reorder point
L =Lead time
H=Annual holding
(carrying) and storage
cost per unit of
inventory
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EOQ Cost Model: How Much to Order?
By adding the holding and ordering costs together, we determine the total cost curve,
which in turn is used to find the optimal order quantity that minimizes total costs
The total cost curve is U-Shaped
Annual
cost ($)
Q
D
TC  H  S
2
Q
Total Cost
Slope = 0
HQ
Carrying Cost =
Minimum
total cost
2
SD
Ordering Cost =
Q
Optimal order
Qopt
Order Quantity, Q
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Why Holding Costs Increase?
• 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
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Why Ordering 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
PurchaseOrder
Order
Description
Qty.
Purchase
Description
Qty.
Description
Qty.1
Microwave
Description
Qty.
Microwave 11
Microwave
Microwave
1
Order
quantity
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Basic Fixed-Order Quantity (EOQ) Model
Formula
Total
Annual =
Cost
Annual
Annual
Annual
Purchase + Ordering + Holding
Cost
Cost
Cost
D
Q
TC = DC + S + H
Q
2
TC=Total annual cost
D =Annual demand
C =Cost per unit
Q =Order quantity
S =Cost of placing an
order or setup cost
R =Reorder point
L =Lead time
H=Annual holding and
storage cost per unit
of inventory
54
EOQ Cost Model
Using calculus, we take the first derivative of the total cost function with respect to Q,
and set the derivative (slope) equal to zero, solving for the optimized (cost minimized)
value of Qopt . The total cost curve reaches its minimum where the carrying and
ordering costs are equal
SD
Annual ordering cost =
Q
SD
HQ
TC =
+
HQ
Q
2
Annual carrying cost =
2
SD
H
TC
=
2 +
Q
2
SD
HQ
Q
Total cost =
+
Q
2
SD
H
0=
+
Q2
2
Deriving Qopt
Qopt =
2SD
H
Proving equality
of costs at
optimal point
SD HQ
=
Q
2
Q2
2S D
=
H
Qopt =
2SD
H
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Deriving the EOQ
How much to order?:
Q OPT =
2DS
=
H
2(Annual Demand)(Order or Setup Cost)
Annual Holding Cost
When to order?
_
We also need a
reorder point to tell
us when to place an
order
R eo rd er p o in t, R = d L
_
d = average daily demand (constant)
L = Lead time (constant)
56
EOQ Model Equations
Expected Number of Orders = N = D
*
Q
Expected Time Between Orders
d =
D
Working Days / Year
ROP = d × L
=T =
Working Days / Year
N
57
EOQ Example 1 (1 of 3)
Given the information below, what are the EOQ and
reorder point?
Annual Demand = 1,000 units
Days per year considered in average daily demand = 365
Cost to place an order = $10
Holding cost per unit per year = $2.50
Lead time = 7 days
Cost per unit = $15
58
EOQ Example 1(2 of 3)
Q OPT =
2DS
=
H
2(1,000 )(10)
= 89.443 units or 90 units
2.50
1,000 units / year
d =
= 2.74 units / day
365 days / year
_
Reorder point, R = d L = 2.74units / day (7days) = 19.18 or 20 units
In summary, you place an optimal order of 90 units. In the
course of using the units to meet demand, when you only have
20 units left, place the next order of 90 units.
59
EOQ Example 1(3 of 3)
Orders per year = D/Qopt
= 1000/90
= 11 orders/year
TCmin =
TCmin =
SD
Q
+
(10)(1,000)
90
TCmin = $ 111 + $111 = 222 $
Order cycle time = 365/(D/Qopt)
= 365/11
= 33.1days
HQ
2
+
(2,5)(90)
2
60
EOQ Example 2(1 of 2)
Determine the economic order quantity
and the reorder point given the following…
Annual Demand = 10,000 units
Days per year considered in average daily demand = 365
Cost to place an order = $10
Holding cost per unit per year = 10% of cost per unit
Lead time = 10 days
Cost per unit = $15
61
EOQ Example 2(2 of 2)
Q OPT =
2D S
=
H
2(10,000 )(10)
= 365.148 un its, or 366 u n its
1.50
10,000 units / year
d=
= 27.397 units / day
365 days / year
_
R = d L = 27.397 units / day (10 days) = 273.97 or 274 units
Place an order for 366 units. When in the course of
using the inventory you are left with only 274 units,
place the next order of 366 units.
62
EOQ Example 3
H = $0.75 per yard
Qopt =
2SD
H
Qopt =
2(150)(10,000)
(0.75)
Qopt = 2,000 yards
S = $150
D = 10,000 yards
SD
HQ
TCmin =
+
Q
2
TCmin
(150)(10,000) (0.75)(2,000)
=
+
2,000
2
TCmin = $750 + $750 = $1,500
Orders per year = D/Qopt
Order cycle time =311 days/(D/Qopt)
= 10,000/2,000
= 311/5
= 5 orders/year
= 62.2 store days
63
When to Reorder with EOQ Ordering ?
Reorder Point
– When the quantity on hand of an item drops to this
amount, the item is reordered
ROP = d . L
where:
d= demand rate (units per period, per day, per week)
L= lead time (in the same unts as d)
– Determinants of the reorder point
1. the rate of demand
2. the lead time
3. the extent of demand and/or lead time variability
4. the degree of stockout risk acceptable to management
64
Reorder Point Example
Demand = 10,000 yards/year
Store open 311 days/year
Daily demand = 10,000 / 311 = 32.154 yards/day
Lead time = L = 10 days
R = dL = (32.154)(10) = 321.54 yards
65
Reorder Point: Under Uncertainty
• Demand or lead time uncertainty creates the
possibility that demand will be greater than available
supply
• To reduce the likelihood of a stockout, it becomes
necessary to carry safety stock
Safety Stock
• Stock that is held in excess of expected demand due to
variable demand and/or lead time
Expecteddemand
ROP 
 SafetyStock
during lead time
12-65
66
Quantity
Safety Stock
Maximum probable demand
during lead time
Expected demand
during lead time
ROP
Safety stock
Safety stock reduces risk of
stockout during lead time
LT
Time
67
Variable Demand with
a Reorder Point
Inventory level
Q
Reorder
point, R
0
LT
LT
Time
68
Inventory level
Reorder Point with a Safety Stock
Q
Reorder
point, R
Safety Stock
0
LT
LT
Time
69
Safety Stock?
• As the amount of safety stock carried increases, the risk of
stockout decreases.
– This improves customer service level
• Service level
– The probability that demand will not exceed supply during
lead time (probability that inventory available during lead
time will meet demand)
– Service level = 100% - Stockout risk (probability of stockout)
- Higher service level means more safety stock
- More safety stock means higher ROP
12-69
70
How Much Safety Stock?
• The amount of safety stock that is appropriate for a
given situation depends upon:
1. The average demand rate and average lead time
2. Demand and lead time variability
3. The desired service level
Expect eddemand
R
 z dLT
during lead t ime
where
z  Number of st andarddeviat ions
 dLT  T hest andarddeviat ionof lead t imedemand
12-70
71
Reorder Point for Service Level
Probability of
a stockout
Safety stock
zd L
dL
Expected Demand
R
The reorder point based on a normal distribution of LT demand
72
Reorder Point
Probability of meeting demand during
lead time (Probability of no stockout)
= service level
Probability of a
stockout
Service level
Expected
ROP
demand
Safety stock
Quantity
zd √ L
0
z
z-scale
The ROP based on a normal distribution of lead time demand
12-72
73
Reorder Point With Variable Demand
R = dL + zd L
where
d = average daily demand
L = lead time (same time units as average demand)
d = the standard deviation of daily demand(same time units as average demand)
z = number of standard deviations corresponding to the service level
probability
zd L = safety stock
Note: dLT   d L
74
Reorder Point for Variable Demand
(Example)
The carpet store wants a reorder point with a
95% service level and a 5% stockout probability
d = 30 yards per day
L = 10 days
d = 5 yards per day
For a 95% service level, z = 1.65
R = dL + z d L
Safety stock = z d L
= 30(10) + (1.65)(5)( 10)
= (1.65)(5)( 10)
= 326.1 yards
= 26.1 yards
75
Reorder Point: Lead Time Uncertainty
ROP  d  LT  zd LT
where
z  Number of standarddeviations
d  Demandper period(per day, per week)
 LT  T hestddev.of lead time(same timeunits as d )
LT  Averagelead time(same timeunits as d )
12-75
76
Fixed Order Quantity Models:
-Noninstantaneous ReceiptProduction Order Quantity
(Economic Lot Size)
Model
Economic Production Quantity (EPQ)
or Economic Order Quantity
or Economic Lot Size
77
• Assumptions
– Only one product is involved
– Annual demand requirements are known
– Usage rate is constant
– Usage occurs continually, but production occurs
periodically
– The production rate is constant
– Lead time does not vary
– There are no quantity discounts
12-77
78
Production Order Quantity Model
Production done in batches or lots
Capacity to produce a part exceeds that part’s
usage or demand rate
Allows partial receipt of material
 Other EOQ assumptions apply
Suited for production environment
 Material produced, used immediately
 Provides production lot size
Lower holding cost than EOQ model
Answers how much to order and when to order
79
EOQ EPQ: Inventory Profile
Q
Q*
Production
and usage
Usage
only
Production
and usage
Usage
only
Production
and usage
Cumulative
production
Imax
Amount
on hand
Time
12-79
80
EOQ POQ Model
When To Order
Inventory Level
Average
Inventory
Optimal
Order
Quantity
(Q*)
Reorder
Point
(ROP)
Lead Time
Time
81
POQ Model Inventory Levels
Inventory Level
Inventory level with no demand
Production
Portion of
Cycle
Q*
Supply Supply
Begins Ends
Max. Inventory Level
Q·(1- u/p)
Demand portion of
cycle with no supply
Average
inventory
(Q/2)(1- u/p)
Time
82
EPQ – Total Cost
T C  CarryingCost  Set up Cost
D
 I max 

H  S
Q
 2 
where
I max  Maximuminvent ory
Q
  p  u
p
p  P roduct ionor deliveryrat e
u  Usage rat e
12-82
83
POQ Model Equations
Production Order Quantity = Q * =
p
Q0 
2 DS
H
2*D*S
u
H* 1 p
( )
p
p u
84
Production Order Quantity Example
(1 of 2)
H = $0.75 per yard
S = $150
u = 10,000/311 = 32.2 yards per day
2SD
POQopt=
H 1- u
p
SD HQ
u
TC = Q + 2 1 - p
D = 10,000 yards
p = 150 yards per day
2(150)(10,000)
=
32.2
0.75 1 150
= 2,256.8 yards
= $1,329
2,256.8
Q
Production run =
=
= 15.05 days per order
150
p
85
Production Quantity Example
(2 of 2)
H = $0.75 per yard
S = $150
u= 10,000/311 = 32.2 yards per day
2CoD
Qopt =
Cc 1 - d
p
D = 10,000 yards
p = 150 yards per day
2(150)(10,000)
=
CoD CcQ
d
TC = Q + 2 1 - p
32.2
0.75 1 150
= 2,256.8 yards
= $1,329
2,256.8
Q
Production run =
=
= 15.05 days per order
150
p
86
Production Quantity Example
(2 of 2)
Number of production runs =
10,000
D
=
= 4.43 runs/year
2,256.8
Q
Maximum inventory level = Q (1 - u ) = 2,256.8 ( 1 - 32.2
p
150
= 1,772 yards
87
Fixed-Order Quantity Models:
Economic Order Quantity Model
with Quantity Discounts
88
Quantity Discount Model
• Answers how much to order & when to order
• Allows quantity discounts
– Price reduction of fered to customers for
placing large orders, ie. Price per unit decreases
as order quantity increases
– Other EOQ assumptions apply
• Trade-off is between lower price & increased
holding cost
Total cost with purchasing cost
SD
Q
TC =
+ H
Q
2
+ PD
Where P: Unit Price
89
Price-Break Model Formula
Based on the same assumptions as the EOQ model,
the price-break model has a similar Qopt formula:
QOPT
2DS
2(AnnualDemand)(Order or Setup Cost)
=
=
iC
AnnualHoldingCost
i = percentage of unit cost attributed to carrying inventory
C = cost per unit
Since “C” changes for each price-break, the formula above will
have to be used with each price-break cost value
90
Cost
Total Costs with PD
Adding Purchasing cost
doesn’t change EOQ
TC with PD
TC without PD
PD
0
EOQ
Quantity
91
Total Cost with Constant Carrying Costs
Total Cost
TCa
TCb
Decreasing
Price
TCc
CC a,b,c
OC
EOQ
Quantity
92
Quantity Discounts
12-92
93
Quantity Discount Schedule
Discount
Number
Discount
Quantity
Discount
(%)
Discount
Price (P)
1
0 to 999
No discount
$5.00
2
1,000 to 1,999
4
$4.80
3
2,000 and over
5
$4.75
94
Quantity Discount – How Much to Order?
95
Price-Break Example 1 (1 of 3)
ORDER SIZE
0 - 99
100 - 199
200+
PRICE
$10
8 (d1)
6 (d2)
For this problem holding cost is given as a constant value,
not as a percentage of price, so the optimal order quantity is
the same for each of the price ranges. (see the figure)
96
Price Break Example 1 (2 of 3)
TC = ($10 )
TC (d1 = $8 )
Inventory cost ($)
TC (d2 = $6 )
Carrying cost
Ordering cost
Q(d1 ) = 100 Qopt
Q(d2 ) = 200
97
Price Break Example 1 (3 of 3)
TC = ($10 )
TC (d1 = $8 )
Inventory cost ($)
TC (d2 = $6 )
Carrying cost
Ordering cost
Q(d1 ) = 100 Qopt
Q(d2 ) = 200
The lowest total cost is at the second price break
98
Price Break Example 2
QUANTITY
1 - 49
50 - 89
90+
Qopt =
PRICE
$1,400
1,100
900
2SD
=
H
S = $2,500
H = $190 per computer
D = 200
2(2500)(200)
= 72.5 PCs
190
For Q = 72.5
H Qopt
SD
TC =
+
2 + PD = $233,784
Qopt
For Q = 90
HQ
SD
TC =
+ 2 + PD = $194,105
Q
99
Price-Break Example 3
(1 of 4)
A company has a chance to reduce their inventory
ordering costs by placing larger quantity orders using the
price-break order quantity schedule below. What should
their optimal order quantity be if this company purchases
this single inventory item with an e-mail ordering cost of
$4, a carrying cost rate of 2% of the inventory cost of the
item, and an annual demand of 10,000 units?
Order Quantity(units) Price/unit($)
0 to 2,499
$1.20
2,500 to 3,999 1.00
4,000 or more .98
100
Price-Break Example (2 of 4)
First, plug data into formula for each price-break value of “C”
Annual Demand (D)= 10,000 units
Cost to place an order (S)= $4
Carrying cost % of total cost (i)= 2%
Cost per unit (C) = $1.20, $1.00, $0.98
Next, determine if the computed Qopt values are feasible or not
Interval from 4000 & more, the
Qopt value is not feasible
Interval from 2500-3999, the
Qopt value is not feasible
Interval from 0 to 2499, the
Qopt value is feasible
QOPT =
QOPT =
QOPT =
2DS
=
iC
2DS
=
iC
2DS
=
iC
2(10,000)(
4)
= 2,020units
0.02(0.98)
2(10,000)(
4)
= 2,000units
0.02(1.00)
2(10,000)(
4)
= 1,826units
0.02(1.20)
101
Price-Break Example 2 (3 of 4)
Since the feasible solution occurred in the first pricebreak, it means that all the other true Qopt values occur
at the beginnings of each price-break interval. Why?
Because the total annual cost function is
a “u” shaped function
Total
annual
costs
So the candidates
for the pricebreaks are 1826,
2500, and 4000
units
0
1826
2500
4000
Order Quantity
102
Price-Break Example 2 (4 of 4)
Next, we plug the true Qopt values into the total cost annual cost
function to determine the total cost under each price-break
D
Q
T C = DC +
S+
iC
Q
2
TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20)
= $12,043.82
TC(2500-3999)= $10,041
TC(4000&more)= $9,949.20
Finally, we select the least costly Qopt, which in this
problem occurs in the 4000 & more interval. In summary,
our optimal order quantity is 4000 units
103
Multi-period Inventory Models:
Fixed Time Period
(Fixed-Order- Interval)
Models
104
Fixed-Order-Interval Model
 Orders are placed at fixed time intervals
 Order quantity for next interval? (inventory is brought up to
target amount, amount ordered varies)
 Risk of stockout between intervals
 Reasons for using the FOI model
– Supplier’s policy may encourage its use
– Grouping orders from the same supplier can produce savings
in ordering, packing and shipping costs.
– Some circumstances do not lend themselves to continuously
monitoring inventory position
– Requires only periodic checks of inventory levels (no
continous monitoring is required)
105
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
d Inventory
Q3
p
p
p
Time
106
Fixed-Interval Disadvantages
 Requires a larger safety stock
 Increases carrying cost
 Costs of periodic reviews
107
Fixed-Quantity vs. Fixed-Interval
Ordering
12-107
108
FOI Model
Amount  T argetInventory InventoryPosition
to Order
Level
at T imeof Order
Q  T  IP
where
Q  Amount toorder
T  T argetinventorylevel
IP  Inventorypositionat timeof order
12-108
109
Fixed-Time Period Model with Safety Stock
Formula
q = Average demand + Safety stock – Inventory currently on hand
q = d(T + L) + Z  T + L - I
Where:
q = quant it iyt o be ordered
T = lengt hof t imebet ween orders (orderint erval)
L = lead t imein days
d = forecast averagedaily demand
z = t henumber of st andarddeviat ionsfor a specifiedserviceprobability
 T + L = st andarddeviat ionof demandover t hereviewand lead t ime
I = currentinvent orylevel(includesit emson order)
110
Fixed-Time Period Model:
Determining the Value of T+L
 T+ L =
 
T+ L
i 1
di

2
Since each day is independent and  d is constant,
 T+ L =
(T + L) d 2
The standard deviation of a sequence of random
events equals the square root of the sum of the
variances
111
Order Quantity for a
Periodic Inventory System
where
Q = d(T + L) + zd
d
T
L
d
T+L -I
= average demand rate
= the fixed time between orders
= lead time
= standard deviation of demand
zd T + L = safety stock
I = inventory level
z = the number of standard deviations
for a specified service level
112
Fixed-Period Model with Variable
Demand (Example 1)
d
d
T
L
I
z
= 6 bottles per day
= 1.2 bottles
= 60 days
= 5 days
= 8 bottles
= 1.65 (for a 95% service level)
Q = d(T + L) + zd
T+L -I
= (6)(60 + 5) + (1.65)(1.2)
= 397.96 bottles
60 + 5 - 8
113
Fixed-Time Period Model with
Variable Demand (Example 2)(1 of 3)
Given the information below, how many units should be
ordered?
Average daily demand for a product is 20 units. The
review period is 30 days, and lead time is 10 days.
Management has set a policy of satisfying 96 percent of
demand from items in stock. At the beginning of the
review period there are 200 units in inventory. The
standard deviation of daily demand is 4 units.
114
Fixed-Time Period Model with Variable
Demand (Example 2)(2 of 3)
 T+ L =
(T + L) d =
2
 30 + 10  4  2 = 25.298
So, by looking at the value from the Table, we have a
probability of 0.9599, which is given by a z = 1.75
115
Fixed-Time Period Model with Variable
Demand (Example 2) (3 of 3)
q = d(T + L) + Z  T + L - I
q = 20(30+ 10) + (1.75)(25.
298)- 200
q = 800  44.272- 200 = 644.272,or 645u n i ts
So, to satisfy 96 percent of the demand, you should
place an order of 645 units at this review period
116
Miscellaneous Systems:
Optional Replenishment System
Maximum Inventory Level, M
q=M-I
Actual Inventory Level, I
M
I
Q = minimum acceptable order quantity
If q > Q, order q, otherwise do not order any.
117
Single-Period Inventory Model
118
Single Period Model
• Single period model: model for ordering of
perishables and other items with limited useful lives
• Shortage cost: generally the unrealized profits per
unit
• Cshortage = Cs = Revenue per unit – Cost per unit
• Excess cost: difference between purchase cost and
salvage value of items left over at the end of a period
• Cexcess = Ce = Cost per unit – Salvage value per unit
119
Single-Period Model
• The goal of the single-period model is to identify
the order quantity that will minimize the long-run
excess and shortage costs
• Two categories of problem:
– Demand can be characterized by a continuous
distribution
– Demand can be characterized by a discrete distribution
12-119
120
Single Period Model
• Continuous stocking levels
– Identifies optimal stocking levels
– Optimal stocking level balances unit shortage and
excess cost
• Discrete stocking levels
– Service levels are discrete rather than continuous
– Desired service level is equaled or exceeded
121
Single-Period Model
Cs
P
Cs  Ce
This model states that we should
continue to increase the size of the
inventory so long as the probability of
selling the last unit added is equal to
or greater than the ratio of: Cs/Cs+Ce
Where:
Ce  Cost per unit of demandoverestimated
Cs  Cost per unit of demandunder estimated
P  Probability that theunit will be sold
122
Optimal Stocking Level
Service level =
Cs
Cs + Ce
Cs = Shortage cost per unit
Ce = Excess cost per unit
Ce
Cs
Service Level
Quantity
So
Balance point
(Optimum Stocking Quantity)
123
Single Period Example 1
•
•
•
•
Ce = $0.20 per unit
Cs = $0.60 per unit
Service level = Cs/(Cs+Ce) = .6/(.6+.2)
Service level = .75
Ce
Cs
Service Level = 75%
Quantity
Stockout risk = 1.00 – 0.75 = 0.25
124
Single Period Model Example 2
Our college basketball team is playing in a tournament
game this weekend. Based on our past experience we
sell on average 2,400 shirts with a standard deviation
of 350. We make $10 on every shirt we sell at the
game, but lose $5 on every shirt not sold. How many
shirts should we make for the game?
Cs = $10 and Ce = $5; P ≤ $10 / ($10 + $5) = .667
Z.667 = .432
therefore we need 2,400 + .432(350) = 2,551 shirts
125
Last Words
Inventories have certain functions.
But too much inventory
- Tends to hide problems
- Costly to maintain
So it is desired
• Reduce lot sizes
• Reduce safety stocks