Transcript Inventory Management

Inventory Management
Agenda
• Independent Demand Inventory
– Dependent vs. independent demand
– Basic Economic Order Quantity (EOQ) model.
Also known as Economic Lot Size Model
– Models with Demand and Supply Uncertainty
• Fixed ordering costs: the base-stock model (s,S)
• No fixed ordering costs: the base-stock model (S)
– Risk pooling
Why do companies hold inventory?
Why might they avoid doing so?
• WHY?
– To meet anticipated customer demand
– To account for differences in production timing (smoothing)
– To protect against uncertainty (demand surge, price increase,
lead time slippage)
– To maintain independence of operations (buffering)
– To take advantage of economic purchase order size
• WHY NOT?
– Requires additional space
– Opportunity cost of capital
– Spoilage / obsolescence
Independent vs. Dependent Demand
Independent Demand (Demand not related to other
items or the final end-product)
Ford
Taurus
Body
Assy.
Wheel
Assy. (4)
E(1
)
Wheel (1)
Tire (1)
Dependent
Demand
(Derived demand
items for
component parts,
subassemblies,
raw materials,
etc.)
Two Decisions in Inventory
Management
• When is it time to reorder?
• If it is time to reorder, how much?
Economic Order Quantity Model:
Where it all started….
Time Between Orders
On-hand Inventory
(Cycle Time) T = Q/D
Q
Q/2
Demand
Rate, D
Average Cycle
Inventory, Q/2
Time
Basic EOQ Assumptions
•
•
•
•
Constant Demand Rate
Instantaneous replenishment
Orders received in full after lead-time.
Constant Unit Price (no discounts)
Economic Order Quantity Cost Model:
Constant Demand, No Shortages
TC
D
Q
K
h
=
=
=
=
=
total annual inventory cost
annual demand (units / year)
order quantity (units)
cost of placing an order or setup cost ($)
annual inventory carrying cost ($ / unit /year)
Total Annual
Inventory =
Cost
TC
=
Annual
Ordering
Cost
Annual
+ Holding
Cost
(D / Q) K + (Q / 2) h
Many orders,
low inventory
level
On-hand Inventory
Trade-off in EOQ Model:
Inventory Level vs. Number of Orders
Q
Time
Few orders,
high inventory
level
On-hand Inventory
Q
Time
Cost Relationships for Basic EOQ
(Constant Demand, No Shortages)
Total
Cost
Carrying
Cost
Ordering
Cost
Q*
EOQ balances carrying
costs and ordering
costs in this model.
Order Quantity (how much)
EOQ Results (How Much to Order)
(Constant Demand, No Shortages)
Economic Order Quantity = Q* =
2DK
h
Number of Orders per year = D / Q*
Length of order cycle T = Q* / D
Total cost = TC = (D / Q*) K + (Q* / 2) h
EOQ Example (How Much)
D = 1,000 units per year
BE CAREFUL!
K = $20 per order
h = $8.33 per unit per month = $100 per unit per year
2 1000  20 
 20
EOQ: Q * 
100
Number of orders per year = 1000/20 = 50 orders
Length of order cycle = T = 365 days/50 orders = 7.3 days
Total cost = 20(1000/20)+100*20/2 = $2,000
EOQ Example (cont.)
(D = 1,000, K = $20, h = $100)
Question: What if the company can only order in multiples
of 12? (That is, order either 0 or 12 or 24 or 36, etc…)?
Answer: Q* = 20. Closest matches (above and below)
are 12 and 24. Need to compute TC for both, and decide:
TC(Q) = (D / Q) K + (Q / 2) h
Q = 12  TC(12) = 20(1000/12)+100*12/2 = $2,266.67
Q = 24  TC(24) = 20(1000/24)+100*24/2 = $2,033.33
So, the company should order in lots of Q = 24
Robustness of EOQ model
Very Flat Curve - Good!!
Total
Cost
DTC
Q*-DQ
Q*
Q*+DQ
Order Quantity
Would have to mis-specify Q* by quite a bit
before total annual inventory costs would
change significantly.
Example
Example: EOQ Robustness
• Suppose that in the last problem, you have mis-specified
the order costs by 100% and the holding costs by 50%.
That is,
– K used in the computation = $40/order (actual cost = $20 / order)
– h used in computation = $150 / unit / year (actual = $ 100)
– Then, using these wrong costs, you would have gotten
2(1,000)40
Q' 
 23.1
150
Your actual TC (computed substituting Q’ into TC, using correct costs of K = $20, and h = $100:
TC 
1,000
23
20  100  $2,019
23
2
Only 1% above minimum TC!
Variations of EOQ
Some assumptions so far
• Instantaneous replenishment (zero lead time)
• Certain and constant demand rate
• Constant price
Some variations of EOQ
• Positive lead times and uncertain lead times
• Uncertain demand
• EOQ with quantity discounts
EOQ with Positive Lead Time
Time Between Orders
On-hand Inventory
(Cycle Time) T = Q/D
Q
Demand
Rate, D
Average Cycle
Inventory, Q/2
Q/2
Reorder
Point, s
Place
Order
Receive
order
Lead
Time,
L
Time
Determining When to Reorder
• Quantity to order (how much…) determined by EOQ
• Reorder point (when…)determined by finding the
inventory level that is adequate to protect the
company from running out during delivery lead time
• With constant demand and constant lead time,
(EOQ assumptions), the reorder point is exactly the
amount that will be sold during the lead time.
Example:
Daily demand (d) = 1,000/365 = 2.74 /day
Delivery lead time (L) of 2 days
s = d*L = (2.74) (2) = 5.5  6 units
Effects of Demand / Lead Time Variability on
Reorder Point (When)
Variable demand
Expected demand at
average demand rate d
QUESTION: How
much inventory is
needed during lead
time L?
s
Safety Stock level
Place
order
Receive
order
L
KEY POINT: s is larger
when there is uncertainty
about demand or L
Calculation of Appropriate Safety
Stock Level
• Safety stock: stock carried to provide a level of protection
against stockouts due to uncertainty of demand during
lead time
• Stockout Criterion: Find s such that the probability of
stockout (during the lead time) is 
• Demand during lead time is a random variable
– Estimate distribution from historical data (build
histogram of demand + frequencies)
– Normal is frequently used if distribution is unknown
Computing s …
Assumption: Demand during lead-time is normally distributed
Probability {Demand during lead-time < s} = 1- 
Service Level
Probability distribution of
demand during lead time
1- 

s
Computing s: Taking Advantage of the
Normal Distribution
Probability distribution
of demand during lead
time:
Mean = ; Std Dev = 
1- 


1-
.90
.95
.98
.99
.999
z
1.28
1.64
2.05
2.33
3.09
s
z
1- 

From normal table
or, in Excel, use:
=normsinv (0.90)
0
z
s

 s    z
Issue…
• The parameters  and  refer to mean and standard
deviation of demand during lead time
• Normally, companies have statistics on demand and
lead time per unit of time (say, days, weeks, months)
–
–
–
–
–
AVG = average demand per unit of time
STD = standard deviation of demand per unit of time
AVGL = average lead time
STDL = standard deviation of lead time
Just be consistent: if demand is given on a certain time unit, say, days,
then use lead time in the same time unit (in this case, days)
• How to we compute  and  from AVG, STD, AVGL,
and STDL?
More specifically….
Mean demand
during lead
time ()
Safety
factor (std
normal
table)
Standard
deviation of
demand during
lead time ()
Safety stock SS
s  AVG  AVGL  z  STDL2  AVG2  STD2  AVGL
•If lead time is constant, STDL  0
•If demand is constant, STD  0
s  AVG  AVGL  z  STD AVGL
s  AVG  AVGL  z  STDL  AVG
Note: This is a very good approximation even when demand is not normally distributed.
Inventory
The (s,S) Policy: When There Are
Fixed Ordering Costs
S
sR
Average demand
during lead time
Safety Stock
L
Order
placed
Order
arrives
Time
s should be set to cover the lead time demand and together with a
safety stock that insures the stock out probability is  (When)
S depends on the fixed order cost – EOQ (How much)
The (s,S) Policy: Fixed Ordering Costs
• Need to define inventory position (IP)
IP = On-Hand + On-Order– Backorder
• Order when: inventory position (IP) drops below s
s  AVG  AVGL  z  STDL2  AVG2  STD2  AVGL
• Order how much: bring IP to S (“big S”)
• Compute Q using the EOQ formula, using mean
demand D = AVG (be careful about units…):
• Set S = s + Q
2  K  AVG
Q
h
Example: (s,S) Model
• Consider inventory management for a certain SKU at
Home Depot. Supply lead time is variable (since it
depends on order consolidation with other stores) and
has a mean of 5 days and standard deviation of 2 days.
• Daily demand for the item is variable with a mean of 30
units and a coefficient of variation of 0.20.
• Assume a 95% service level.
• There are fixed ordering costs that are estimated at $50.
Assume that holding costs are 15% of the product cost
($80) per year. Also, assume that the store is open 360
days a year. Propose an inventory policy for this SKU.
Solution
• Variable definitions and preliminary
calculations:
STDL  2
AVG  30;
STD  0.2(30)  6 AVGL  5;
h  (.15)80 / 360  0.0333;
K  50
Assume 95% service level  z = 1.64
• Compute s
s  AVG  AVGL  z  STDL2  AVG 2  STD 2  AVGL
 30  5  1.64 22  302  62  5  150  100.8  251
• Compute Q
2  AVG  K
2(30)50
Q

 300
h
0.0333
S  s  Q  251 300  551
Example: (s,S) Model
• Consider inventory management for a certain SKU at
WalMart. Supply lead time is variable and has a mean
of 1 week and standard deviation of 2 weeks.
• Weekly demand for the item is variable with a mean of
125 units and a standard deviation of 50.
• Assume a 90% service level.
• There are fixed ordering costs that are estimated at $30.
Assume that holding costs are 20% of the product cost
($40) per year. Also, assume that the store is open 52
weeks a year. Propose an inventory policy for this SKU.
Solution
• Variable definitions and preliminary
calculations:
AVG  125;
STD  50
AVGL  1;
STDL  2
h  (.20)40 / 52  0.1538;
K  30
Assume 90% service level  z = 1.28
• Compute s
s  AVG  AVGL  z  STDL2  AVG 2  STD 2  AVGL
 125  1  1.28 22  1252  502  1  125  326.3  452
• Compute Q
2  AVG  K
2(125)30
Q

 221
h
0.1538
S  s  Q  452  221  673
The Base-Stock Policy s:
No Fixed Ordering Costs
• Inventory policy: keep IP constant at s units (s is called
the base stock level).
– When: IP drops below s
• So, s is also the reorder point for this model
– How much: order to bring IP back to s
• Example: suppose inventory level on-hand is 10, s = 20,
and there are 2 units already in order. Then,
IP = 10 + 2 = 12 units.
The firm should order 20 – 12 = 8 units.
Example: Base-Stock Model s
Consider inventory management for a certain SKU at King Soopers
Supply lead time is variable and has a mean of 2 days and standard
deviation of 4 days. Daily demand for the item is variable with a
mean of 24 units and a coefficient of variation of 0.30. Propose an
inventory policy for this SKU. Assume a 98% service level.
AVG  24;
STD  0.3(24)  8 AVGL  2;
STDL  4
Assume 98% service level  z = 2.05
s  AVG  AVGL  z  STDL2  AVG 2  STD 2  AVGL
 24  2  2.05 42  242  82  2  48  198.2  247
Summary of Inventory Models
Use EOQ
•How much: Q (EOQ formula)
•When: d*L (reorder point)
yes
Is demand rate and
lead time constant?
Are there fixed
ordering costs?
no
no
yes
Use (s, S) policy
•How much: necessary to bring IP
back to S, where S = s + Q (Q is
from EOQ formula)
•When: IP drops below s (basestock policy formula)
Use base stock (s) policy
•When: IP drops below s
•How much: necessary to
bring IP back to s
Risk Pooling
• Means and variances are additive
• Stock is based on std. Deviations
– Square root law: stock for combined demands is less
than the combined stocks

2
X Y
  
2
X
2
Y
 X Y   X2   Y2
HP Example:
Benefits of a Universal Product
Because of a different power supplies, HP had two laser printers, one for
Europe and one for N. America. A universal product (with a universal
power supply) has been proposed, but costs $30 extra. Is it worthwhile?
Below is monthly demand for HP for the two markets (in thousands).
Assume a one-month constant lead-time (STDL = 0) for both markets.
N. America
Europe
N(200,60)
N(150,50)
Consider z = 2 (~ 98% of service level)
sNA  AVGNA  AVGLNA  z  STDNA AVGLNA  200(1)  2(60) 1  320
sE  AVGE  AVGLE  z  STDE AVGLE  150(1)  2(50) 1  250
sNA  sE  320  250  570
HP Example (cont.):
Benefits of a Universal Product
Demand seen by HP (NA and Europe)
N (200  150, 502  602 )  N (350,78.1)
s  AVG  AVGL  z  STD AVGL  350(1)  2(78.1) 1  506.2