Managing Uncertainty with
Inventory I
John H. Vande Vate
Spring, 2007
1
1
Topics
• Integrate Obermeyer (wholesaler) with the
Retail Game (retail pricing)
• Continuous Review Inventory Management
• Periodic Review Inventory Management
• Safety Lead Time
2
2
The Retail Game Revisited
• How much inventory to bring to the
Item
Season Sales
market? 2000?That’s not demand!
1
1034
2
1942
1097
• What will demand It’s
be?supply 3
4
1068
5
1578
• How to estimate it?
6
2000
• How to estimate
demand for this item?
7
8
9
10
11
12
13
14
15
16
Average
1429
1145
1571
1248
2000
1708
1770
1537
1611
2000
3
1546
3
Estimating Demand
• How fast was it
selling?
Week Inventory
1
2000
2
1906
3
1821
4
1651
5
1496
6
1370
7
1306
8
1201
9
972
10
719
11
540
12
377
13
154
14
0
15
0
16
0
Price
Weeks Sales
60
94
60
85
60
170
60
155
60
126
60
64
60
105
48
229
48
253
48
179
48
163
48
223
48
154
48
0
48
0
• So an estimate of
season demand for
Average
this item is
209/week
• 2473 = 2000 – 154 + 3*209
4
4
New Estimate
• Should we order 1664? Item
• What are the issues?
• If salvage value exceeds
our cost?
• If salvage value is less
than our cost?
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Average
Season Sales
1034
1942
1097
1068
1578
2473
1429
1145
1571
1248
2490
1708
1770
1537
1611
2927
1664
5
5
Risk & Return
• Will Demand be 1664?
• How to measure our uncertainty about
demand?
– Method 1: Standard deviation of diverse
forecasts
– Method 2: Historical A/F ratios + Point
forecast
• Trade off Risks (out of stock and
overstock) vs Return (sales)
6
6
Swimsuit Case p 49
•
•
•
•
•
•
•
•
Fixed Production Cost $100K
Variable Production Cost $80
Selling Price $125
Salvage Value $20
Profit is $125 - $80 = $45
Risk is $80 - $20 = $60
Profit + Risk is $125 - $20 = $105
Order to an expected stock out probability
57% = 1-$45/$105 = 1-43%
• Several Sales Forecasts
7
7
Forecasts
30%
25%
20%
15%
10%
5%
0%
8,000
10,000
12,000
14,000
16,000
18,000
8
8
Inferred Cum. Probability
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
8,000
10,000
12,000
14,000
16,000
18,00
9
9
Net Profit as a function of
Gross Profits
Quantity
$600
from sales
$500
Net Profits
$400
Thousands
$300
$200
Costs of
liquidations
$100
$8,000
$(100)
$(200)
$(300)
$(400)
9,000
10,000
11,000
12,000
13,000
14,000
15,000
16,000
17,000
18,000
What to order?
• So, we want P to be
(Selling Price – Cost)
(Selling Price– Salvage)
• Assume Cost = $30,
• But what’s the selling price?
• In a wholesale environment this is easier.
In a retail environment, it is messier
Some protection from vendor some times
11
11
The Value of P as a function of
Average Selling Price
• If Cost is $30
88%
86%
84%
82%
80%
78%
76%
74%
$48
$50
$52
$54
$56
$58
$60
12
12
The Quantity as a function of
Average Selling Price
• If Cost is $30
2,300
2,250
2,200
2,150
2,100
2,050
2,000
$48
$50
$52
$54
$56
$58
$60
13
13
Not Overly Sensitive
$50,000
$45,000
2100
2150
$40,000
2200
2250
Differences
are small
$35,000
$30,000
$25,000
$20,000
$15,000
$10,000
$5,000
$$48
$50
$52
$54
$56
$58
14
$60
14
Extend Idea
• Ship too little, you have to EXPEDITE the
rest
• Ship Q
• If demand < Q
– We sell demand and salvage (Q – demand)
• If demand > Q
– We sell demand and expedite (demand – Q)
• What’s the strategy?
15
15
Same idea
• Ignore profit from sales – that’s independent
of Q
• Focus on salvage and expedite costs
• Look at last item
– Chance we salvage it is P
– Chance we expedite it is (1-P)
• Balance these costs
– Unit salvage cost * P = Unit expedite cost (1-P)
– P = expedite/(expedite + salvage)
16
16
Safety Stock
• Protection against variability
–
–
–
–
Variability in demand and
Variability in lead time
Typically described as days of supply
Should be described as standard deviations in lead
time demand
– Example: BMW safety stock
• For axles only protects against lead time variability
• For option parts protects against usage variability too
17
17
Inventory
• Inventory On-hand
• Inventory Position: On-hand and on-order
18
18
Continuous Review Basics
Order Up to Level
Inventory
On
Hand
Position
Lead
Time
If lead time
is long, …
EOQ
Reorder Point
Actual Lead
Time Demand
Order
placed
Avg LT
Demand
Safety Stock
Time
19
19
Assumptions
• Fixed Order Cost
• Constant average demand
• Typically assume Normally distributed
lead time demand
20
20
Safety Stock Basics
• Lead time demand N(m, s)
• Typically Normal with
– Average lead time demand m
– Standard Deviation in lead time demand s
• Setting Safety Stock
– Choose z from N(0,1) to get correct
probability that lead time demand exceeds z,
– Safety stock is zs
21
21
Only Variability in Demand
Sq. Root because
• If Lead Times are reliable
we are adding up L
– Average Lead Time Demand
independent (daily)
L*D
demands.
– Standard Deviation in lead time demand
sL = LsD
– Sqrt of Lead time * Standard Deviation in
demand
– Units (Example)
• L is the Lead Time in days,
• sD is the standard deviation in daily demand
22
22
Implementation
• Inventory On-hand
• Inventory Position: On-hand and on-order
• When Inventory Position reaches a re-order point
(ROP), order the EOQ
• This takes the Inventory Position to the OrderUp-To Level: EOQ + ROP
• That’s because review is continuous – we always
re-order at the ROP
• Often called a (Q,r) policy (when inventory
reaches r, order Q)
23
23
Example 3-7 page 61
Order cost
Cost of TV
Holding cost
Lead time
Month
Sales
$4,500 (e.g., transport cost)
$250
18%
2 weeks
September
200
October
152
November
100
December
221
Avg Monthly Demand
Std Deviation in Monthly Demand
191.17 Units
66.53 Units
Avg Weekly Demand
Std Deviation in weekly Demand
44.12 Units
32.09 Units
Service Level
z value
Safety Stock
Economic Order Quantity
Reorder Point
Order-Up-To Level
Average Inventory Position
Average Inventory On Hand
Average Pipeline Inventory
January February
287
176
March
151
April
198
May
246
June
309
July
98
August
156
97% This is the fraction of time we expect to run out of stock before the next order arrives
1.88
85.34
677
174
851
512.2
424.0
88.23
Standard Deviations
Units
Units
Units
Units
The inventory position rises and falls between the Reorder Point and the Order-Up-To Level
The inventory on hand rises and falls between the Safety Stock and the Economic Order Quantity plus the Safety Stock
The difference is the average pipeline inventory
24
24
Lead Time Variability
If Lead Times are variable
• D = Average (daily) demand
• sD = Std. Dev. in (daily) demand
• L = Average lead time (days)
• sL = Std. Dev. in lead time (days)
• Average lead time demand
– DL
• Std. Dev. in lead time demand
– sL = Ls2D + D2 s2L
• Remember: Std. Dev. in lead time demand drives
safety stock
25
25
Levers to Pull
• Std. Dev. in lead time demand
– sL = Ls2D + D2 s2L
Reduce
Lead
Time
Reduce
Variability in
Demand
Reduce
Variability in
Lead Time
26
26
Periodic Review
• Orders can only be triggered at certain
times
• Examples
– Batched transmissions (e.g., every night,
week, …)
– Imposed by transportation (e.g., weekly
vessel)
• Examples of Continuous Review?
27
27
No Ordering Cost
• Example?
• Cost typically viewed as
– Inventory cost
• Service Level seen as a constraint
– Probability of stock out in an order cycle
• Key Assumption: NO COST TO CHANGE
ORDER SIZE
• Is this typically the case?
28
28
Order-Up-To Policy
• Order-up-to Policy: At each period place
an order to bring inventory position up to a
level S
• What problem might we encounter?
29
29
(S,s) Policy
• To avoid small orders
• In each period, if the inventory position is
below s, place an order to bring it up to S.
30
30
Order Up To Policy
Target Inventory Position
Stock on hand
Reorder Point
Reorder Point
Actual Lead
Time Demand
Actual Lead
Time Demand
Actual Lead
Time Demand
Actual Lead
Time Demand
Order Quantity
Lead
Time
Order
placed
Time
31
How much stock is available to cover demand in this period?
31
Order Up To Policy: Inventory
Reorder Point
Stock on hand
On Average this is
the Expected
demand between
orders
Order Quantity
Reorder Point
On Average this is
the safety stock
Time
So average on-hand inventory is DT/2+ss
32
32
Order Up To Policy: Inventory
Reorder Point
Reorder Point
Stock on hand
After an order is
placed, it is the
Order up to level
Order Quantity
Before an order is
placed it is smaller
by the demand in
the period
Time
So average Pipeline inventory is S – DT/2
33
33
Safety Stock in Periodic Review
• Probability of stock out is the probability demand
in T+L exceeds the order up to level, S
• Set a time unit, e.g., days
• T = Time between orders (fixed)
• L = Lead time, mean E[L], std dev sL
• Demand per time unit has mean D, std dev sD
• Assume demands in different periods are
independent
• Let sD denote the standard deviation in demand
per unit time
• Let sL denote the standard deviation in the lead
34
34
time.
Safety Stock in Periodic Review
• Probability of stock out is the probability demand
in T+L exceeds the order up to level, S
• Expected Demand in T + L
 D(T+E[L])
• Variance in Demand in T+L
 (T+E[L]) sD2 +D2 sL2
• Order Up to Level: S= D(T+E[L]) + safety stock
• Question: What happens to service level if we
hold safety stock constant, but increase frequency?
35
35
Impact of Frequency
• What if we double frequency, but hold safety stock
constant?
• Expected Demand in T/2 + L
 D(T/2+E[L])
• Variance in Demand in T/2+L
 (T/2+E[L]) sD2 +D2 sL2
• Order Up to Level:
 S = D(T/2+E[L]) + safety stock
This is reduced by
TsD2/2
 But now we face the risk of failure twice as often
36
36
Example
• Time period is a day
• Frequency is once per week
 T=7
• Daily demand
 Average 105
 Std Dev 67
• Lead time
 Average 2 days
 Std Dev 2 days
• Expected Demand in T+L
 D (T + E[L]) = 105 (7 + 2) = 945
• Variance in Demand in T+L
 (T+E[L]) sD2 +D2 sL2 = (7+2)*672 + (1052)*22

= 40,401 + 44,100 = 84,501
 Std Deviation = 291
37
37
Example Cont’d
Expected Demand in T+L
 D (T + E[L]) = 105 (7 + 2) = 945
 If we ship twice a week this drops to 578
 If we ship thrice a week this drops to 456
• Variance in Demand in T+L
 (T+E[L]) sD2 +D2 sL2 = (7+28)*672 + (1052)*22

= 40,401 + 44,100 = 84,501
 Std Deviation = 291
 If we ship twice a week this drops to 262
 If we ship thrice a week this drops to 252
38
38
Example Cont’d
• With weekly shipments: To have a 98% chance of no
stockouts in a year, we need .9996 chance of no
stockouts in a week
 .999652 ~ .98
• With twice a week shipments, we need .9998 chance
of no stockouts between two shipments
 .9998104 ~ .98
• With thrice a week shipments, we need .9999 chance
of no stockouts between two shipments
 .9999156 ~ .98
39
39
Example Cont’d
• Assume Demand in L+T is Normal
• Hold risk constant 98% chance of no
shortages all year
D(T+E[L])
Std dev in Demand
Order up to Level
Safety Stock
On Hand Inventory
% Reduction
Average Inventory
% Reduction
Once a week
945
291
1,920
975
1,342
0%
1,552
-
Twice a week
578
262
1,506
929
1,112
17.1%
1,322
14.8%
Thrice a week
455
252
1,392
938
1,060
21.0%
1,270
18.2%
40
40
Lead time = 28
• When lead time is long relative to T
• Safety stock is less clear (Intervals of
Assume
L+T overlap)
• Very Conservative Estimate independence
Once a week Twice a week Thrice a week
D(T+E[L])
3,675
3,308
3,185
Std dev in Demand
449
431
425
Order up to Level
5,179
4,832
4,764
Safety Stock
1,504
1,525
1,579
On Hand Inventory
1,871
1,708
1,701
% Reduction
0%
8.7%
9.1%
Average Inventory
4,811
4,648
4,641
% Reduction
0.0%
3.4%
3.5%
41
41
Lead time = 28
• When lead time is long relative to T
• Safety stock is less clear (Intervals of
L+T overlap)
• Aggressive Estimate: Hold safety stock
constant
D(T+E[L])
Std dev in Demand
Order up to Level
Safety Stock
On Hand Inventory
% Reduction
Average Inventory
% Reduction
Once a week
3,675
449
5,179
1,504
1,871
0.0%
4,811
0.0%
Twice a week
Thrice a week
3,308
3,185
431
425
4,811
4,689
1,504
1,504
1,688
1,626
9.8%
13.1%
4,628
4,566
3.8%
5.1%
42
42
Periodic Review against
a Forecast
• A forecast of day-to-day or week-to-week
requirements
• Two sources of error
– Forecast error (from demand variability)
– Lead time variability
• Safety Lead Time replaces/augments Safety Stock
• Example 6 days Safety Lead Time
• Safety Lead Time translates into a quantity
through the forecast, e.g., the next 6 days of
forecasted requirements (remember the forecast
changes)
43
43
2/
1/
20
03
2/
3/
20
03
2/
5/
20
03
2/
7/
20
03
2/
9/
20
03
2/
11
/2
00
3
2/
13
/2
00
3
2/
15
/2
00
3
2/
17
/2
00
3
2/
19
/2
00
3
2/
21
/2
00
3
2/
23
/2
00
3
2/
25
/2
00
3
2/
27
/2
00
3
3/
1/
20
03
3/
3/
20
03
3/
5/
20
03
3/
7/
20
03
3/
9/
20
03
3/
11
/2
00
3
3/
13
/2
00
3
3/
15
/2
00
3
3/
17
/2
00
3
3/
19
/2
00
3
3/
21
/2
00
3
3/
23
/2
00
3
3/
25
/2
00
3
3/
27
/2
00
3
Safety Lead Time as a quantity
700
600
500
400
300
200
100
0
Safety Lead Time:
The next X days of
44
forecasted demand
44
The Ship-to-Forecast Policy
• Periodic shipments every T days
• Safety lead time of S days
• Each shipment is planned so that after
it arrives we should have S + T days of
coverage.
• Coverage: Inventory on hand should
meet S+T days of forecasted demand
45
45
3/
7/
03
3/
5/
03
3/
3/
03
3/
1/
03
Planned
Inventory
100
0
Safety Lead Time:
The next X days of
46
forecasted demand
3/
9/
03
3/
11
/0
3
3/
13
/0
3
3/
15
/0
3
3/
17
/0
3
3/
19
/0
3
3/
21
/0
3
3/
23
/0
3
3/
25
/0
3
3/
27
/0
3
700
2/
9/
03
2/
11
/0
3
2/
13
/0
3
2/
15
/0
3
2/
17
/0
3
2/
19
/0
3
2/
21
/0
3
2/
23
/0
3
2/
25
/0
3
2/
27
/0
3
2/
7/
03
2/
5/
03
2/
3/
03
2/
1/
03
If all goes as planned
Ship to this
level
600
500
400
300
200
46
Safety Stock Basics
• n customers
• Each with lead time demand N(m, s)
• Individual safety stock levels
– Choose z from N(0,1) to get correct
probability that lead time demand exceeds z,
– Safety stock for each customer is zs
– Total safety stock is nzs
47
47
Safety Stock Basics
• Collective Lead time demand N(nm, ns)
• This is true if their demands and lead times are
independent!
• Collective safety stock is nzs
• Typically demands are negatively or positively
correlated
• What happens to the collective safety stock if
demands are
– positively correlated?
– Negatively correlated?
48
48
Risk Pooling Case 3.3 p 64
Historical Data for Product A
Massachusetts
New Jersey
Pooled
Massachusetts
New Jersey
Pooled
Week
1
33
46
79
Average
39.25
38.63
77.88
2
45
35
80
Std Dev
13.18
12.05
20.71
3
37
41
78
4
38
40
78
Avg. Lead
time
Coeff of
Var
Demand
0.34
39.25
0.31
38.63
0.27
77.88
Historical Data for Product B
Massachusetts
New Jersey
Pooled
Massachusetts
New Jersey
Pooled
5
55
26
81
Safety
Stock
24.78
22.66
38.95
Week
1
0
2
2
Average
1.13
1.25
2.38
2
2
4
6
3
3
0
3
Std Dev
1.36
1.58
1.92
Coeff of
Var
1.21
1.26
0.81
4
0
0
0
Avg. Lead
time
Demand
1.13
1.25
2.38
5
0
3
3
Safety
Stock
2.55
2.97
3.62
Inventory Comparison
Product A
Product B
Total
Massachusetts New Jersey
91
88
14
15
105
103
Total
179
28
207
Pooled
132
20
152
Reduction
26%
30%
27%
The6 impact7 is less 8than
30
18
58
48the sqrt
18 of 2 law
55
78
36
113
It predicts that if 2 DCs
need 47 units
then aAvg.
Reorder
Order Up
Point
To Level
single EOQ
DC will
needInventory
33
64
61
117
132
131
186
196
192
303
91
88
132
The impact is greater
than
the 7sqrt of 28 law
6
It 11predicts30 that if 002 DCs
2
0
need
5.53 units then
a
single DC will need 4
Reorder
Point
EOQ
4
4
6
22
24
32
Order Up
Avg.
To Level Inventory
26
14
28
15
38
20
Pooling Inventory can
reduce safety49stock
49
Inventory (Risk) Pooling
• Centralizing inventory can reduce
safety stock
• Best results with high variability and
uncorrelated or negatively correlated
demands
• Postponement ~ risk pooling across
products
50
50
Next Time
• Read Mass Customization Article
• Read To Pull or Not To Pull by Spearman
51
51