7. Managing Flow Variability: Safety Inventory
Chapter 7
1
Managing Flow Variability: Safety Inventory
7.1
Demand Forecasts and Forecast Errors
7.2
Safety Inventory and Service Level
7.3
Optimal Service Level – The Newsvendor
Problem
7.4
Lead Time Demand Variability
7.5
Pooling Efficiency through Aggregation
7.6
Shortening the Forecast Horizon
7.7
Levers for Reducing Safety Inventory
7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecast and Forecast Errors
2
In review, we have 3 stages of a process:
1.
Input (e.g. raw materials)
2.
Process
3.
Output (finished goods)
This is important to forecasting because it will allow us to
more closely match outputs to inputs and vice versa.
7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecasts and Forecast Errors
3
We have previously assumed demand is known
and is constant.
 Demand varies in predictable and unpredictable
ways.
 Unpredictable, random factors affecting demand is
referred to as “noise”.
“As a process of predicting future demand,
forecasting is, among other things, an effort to
deal with NOISE.”
7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecast and Forecast Errors
4
Why do we forecast?
We forecast so that we can make decisions about the
future.
We need to make rational decisions about process
inventory.
* How to spend money and how not to spend money.
* When to buys more widgets.
* When to hire more workers.
* How to avoid stockouts (upset customers =
business losses)
* How to avoid holding excess inventory (= $ lost)
7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecasts and Forecast Errors
5
Forecasting methods
 Subjective – Based on judgement and experience
• Surveys and expert judgements
 Objective – Based on data analysis
• Causal models - Forecast methods that assume that in
addition to data, there are other factors that influence
demand (eg. Consumer prices.)
• Time series analyses - Methods that rely solely on past
data.
7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecasts and Forecast Errors
6
4 Characteristics of Forecasts
 Forecasts are usually wrong.
Because of random noise – forecasts are inaccurate.
 Forecasts should be accompanied by a measure of forecast
error.
A measure of forecast error quantifies the manager’s degree of
confidence in the forecast.
 Aggregate forecasts are more accurate than individual forecasts.
Aggregate forecasts reduce the amount of variability – relative to the
aggregate mean demand.
 Long-range forecasts are less accurate than short-range
forecasts.
Precise forecasting of events far out in the future are much more
difficult to predict than something that will occur in a matter of moments
from now.
7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecasts and Forecast Errors
7
Forecasts should incorporate hard quantitaive data as
well as qualitative factors such as managerial judgement,
intuition, and market savvy.
Forecasting is as much art as science.
7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecast and Forecast Errors
8
Safety Inventory cushions the process against
supply disruptions or surges in demand.
Having adequate Safety Inventory reduces the
uncertainty in supply and demand.
Ensuring reliable suppliers and stable demand
eliminates the need for Safety Inventory.
7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
9
Objective:
Review of common terms and a discussion of Service
Level
Where: SL = f (Safety Inventory, I safety)
And some math using Excel…
7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
10
SL:
Service Level
I safety:
Safety Inventory (or Safety Stock)
I cycle:
Cycle Inventory
LTD:
Lead Time Demand
ROP:
Re-order Point
L:
Replenishment Lead Time
Q:
Order Size
NORMDIST:
Standard Normal Tables
NORMSINV:
Standard Normal Tables
NORMINV:
Standard Normal Tables
7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
11
Inventory, I (t)
ORDER
ORDER
ROP
LTD, # of Units used
during lead time
Safety Inventory
(I safety)
0
Time, t
L
L
7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
12
I safety = ROP – LTD
Inventory, I (t)
ROP
LTD, # of Units used
during lead time
Safety Inventory
0
(I safety)
7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
13
An Inventory with an Order Size = Q
Average Inventory = Q/2
I cycle = Q/2
I = I cycle + I safety = Q/2 + I safety
Average Flow Rate = R
Average Flow Time as expressed by Little’s Law
T = I /R = (Q/2 + I safety )
R
7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
14
The Service Level for a given ROP is given by:
SL = Prob (LTD < ROP)
To calculate SL, recall first that if LTD is normally distributed
with mean LTD and standard deviation sLTD then
I safety = z x sLTD , where z is a multiple of sLTD
Or the number of standard deviations
7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
15
Example:
At GE Lighting’s Paris warehouse,
LTD (average Lead Time Demand) = 20,000 lamps
Actual Demand varies daily and sLTD = 5,000
The warehouse re-orders whenever ROP = 24,000
Therefore, I safety = ROP – LTD = 24,000 – 20,000 = 4,000
And:
And:
z = I safety / sLTD = 4,000 / 5,000 = 0.8
SL= Prob (Z< 0.8) from Appendix II
SL= 0.7881
7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
16
A
1
2
3
B
C
z = 0.80
SL (z<0.8)
SL = 0.78814
4
5
6
7
8
Service Level
D
E
F
EXCEL
7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
17
A
1
2
B
SL = 0.78814
z = 0.80
3
4
5
6
7
8
Safety
Inventory
C
D
E
F
EXCEL
7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
18
A
1
2
3
4
B
SL = 0.78814
LTD =
20,000
sLTD =
5,000
ROP 24,000
5
6
7
8
Reorder
Point
C
D
E
F
EXCEL
7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
19
So Far…
Safety inventory has been defined for a desired level
of customer service.
But…
How do we choose what level of service a firm
should offer?
Examples:
• Newspapers / Magazines
• Perishables (fish, produce, bread, milk, etc.)
• Seasonal Items (Summer & Winter Apparel)
7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
20
Cost of Holding
Extra Inventory
Improved
Service
Optimal Service Level??
The Newsvendor Problem
Decision making under uncertainty whereby the decision
maker balances the expected costs of ordering too much
with the expected costs of ordering too little to determine
the optimal order quantity.
7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
21
Predicted Demand for HDTV’s
Cost: $1,800
Demand
Probability
Cumulative
Probability
Complementary Cumulative
Probability
r
Prob(R = r)
Prob(R ≤ r)
Prob(R > r)
100
0.02
0.02
0.98
110
0.05
0.07
0.93
120
0.08
0.15
0.85
130
0.09
0.24
0.76
140
0.11
0.35
0.65
150
0.16
0.51
0.49
160
0.2
0.71
0.29
170
0.15
0.86
0.14
180
0.08
0.94
0.06
190
0.05
0.99
0.01
200
0.01
1
0
Price: $2,500
Salvage: $1,700
Profit:
Loss:
$700
$100
Mean: 151.6
Std. Dev: 22.44
7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
22
Quantity Ordered
Demand
100
110
120
130
140
150
160
170
180
190
200
100
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
110
$69,000
$77,000
$77,000
$77,000
$77,000
$77,000
$77,000
$77,000
$77,000
$77,000
$77,000
120
$68,000
$76,000
$84,000
$84,000
$84,000
$84,000
$84,000
$84,000
$84,000
$84,000
$84,000
130
$67,000
$75,000
$83,000
$91,000
$91,000
$91,000
$91,000
$91,000
$91,000
$91,000
$91,000
140
$66,000
$74,000
$82,000
$90,000
$98,000
$98,000
$98,000
$98,000
$98,000
$98,000
$98,000
150
$65,000
$73,000
$81,000
$89,000
$97,000
$105,000
$105,000
$105,000
$105,000
$105,000
$105,000
160
$64,000
$72,000
$80,000
$88,000
$96,000
$104,000
$112,000
$112,000
$112,000
$112,000
$112,000
170
$63,000
$71,000
$79,000
$87,000
$95,000
$103,000
$111,000
$119,000
$119,000
$119,000
$119,000
180
$62,000
$70,000
$78,000
$86,000
$94,000
$102,000
$110,000
$118,000
$126,000
$126,000
$126,000
190
$61,000
$69,000
$77,000
$85,000
$93,000
$101,000
$109,000
$117,000
$125,000
$133,000
$133,000
200
$60,000
$68,000
$76,000
$84,000
$92,000
$100,000
$108,000
$116,000
$124,000
$132,000
$140,000
=IF($A3>B$1,B$1*700-($A3-B$1)*100,$A3*700)
100 x $700 – (110-100) x $100 = $69,000
7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
23
Quantity Ordered
Demand
100
110
120
130
140
150
160
170
180
190
200
100
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
$70,000
110
$69,000
$77,000
$77,000
$77,000
$77,000
$77,000
$77,000
$77,000
$77,000
$77,000
$77,000
120
$68,000
$76,000
$84,000
$84,000
$84,000
$84,000
$84,000
$84,000
$84,000
$84,000
$84,000
130
$67,000
$75,000
$83,000
$91,000
$91,000
$91,000
$91,000
$91,000
$91,000
$91,000
$91,000
140
$66,000
$74,000
$82,000
$90,000
$98,000
$98,000
$98,000
$98,000
$98,000
$98,000
$98,000
150
$65,000
$73,000
$81,000
$89,000
$97,000
$105,000
$105,000
$105,000
$105,000
$105,000
$105,000
160
$64,000
$72,000
$80,000
$88,000
$96,000
$104,000
$112,000
$112,000
$112,000
$112,000
$112,000
170
$63,000
$71,000
$79,000
$87,000
$95,000
$103,000
$111,000
$119,000
$119,000
$119,000
$119,000
180
$62,000
$70,000
$78,000
$86,000
$94,000
$102,000
$110,000
$118,000
$126,000
$126,000
$126,000
190
$61,000
$69,000
$77,000
$85,000
$93,000
$101,000
$109,000
$117,000
$125,000
$133,000
$133,000
200
$60,000
$68,000
$76,000
$84,000
$92,000
$100,000
$108,000
$116,000
$124,000
$132,000
$140,000
$69,000(0.02) + $77,000(0.05) + …+ $77,000(0.01) = $76,840
7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
24
Order Quantity (Q)
Expected Profit
100
$70,000
110
$76,840
120
$83,280
130
$89,080
140
$94,160
150
$98,360
160
$101,280
170
$102,600
180
$102,720
190
$102,200
200
$101,280
7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
25
Net Marginal Benefit:
MB = p – c
MB = $2,500 - $1,800 = $700
Net Marginal Cost:
MC = c - v
MC = $1,800 - $1,700 = $100
We receive Marginal Benefit when R > Q, therefore at any order quantity Q,
Expected MB = MB x Prob(R > Q)
We receive Marginal Cost when R ≤ Q, therefore at any order quantity Q,
Expected MC = MC x Prob(R ≤ Q)
MC x Prob(R ≤ Q*) ≥ MB x Prob(R > Q*)
7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
26
Time for Algebra…
MC x Prob(R ≤ Q*) ≥ MB x Prob(R > Q*)
Since,
Prob(R > Q) = 1 – Prob(R ≤ Q)
We can write,
MC x Prob(R ≤ Q*) ≥ MB x [1 – Prob(R ≤ Q*)]
MB
After rearranging, Prob(R ≤ Q*) ≥
MB  MC
MB
Newsvendor formula: SL* = Prob(R ≤ Q*) = MB  MC
7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
27
Going back to the example…
MB
$700
SL* 

 0.875
MB  MC $700  $100
So what quantity corresponds to this service level ?
If we assume demand is normally distributed then,
Q*  R  z  s R
7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
28
Probability Less than Upper Bound is 0.87493
0.4
0.35
0.3
Density
0.25
0.2
z = 1.15
0.15
0.1
0.05
0
-4
-3
-2
-1
0
Critical Value (z)
1
2
3
4
Q*  R  z  s R  151.6  1.15  22.44  177.41
7. Managing Flow Variability: Safety Inventory
7.4 Lead Time Demand Variability
29
Average Lead Time Demand:
LTD  L  R
Variability in Periodic Demand:
2
s LTD
 L  s R2
Variability in Lead Time:
2
s LTD
 R 2  s L2
Variability in Demand and Lead Time:
s LTD  Ls  R s
2
R
2
2
L
7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
30
Third characteristic of forecasts
Aggregation: pooling demand for several similar
products
Aggregate sales
Safety Inventory: Uncertain demand
Assume Decentralized: Warehouses operates
independently
Imbalance of inventory - Customer demand not satisfied
7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
31
Physical Centralization: that the firm can consolidate all its stock in
one location from which is can serve all its customers.
ELIMINATES stock imbalance
BETTER customer service
SAME total inventory
LESS inventory
Location 1
Lead times demands:
Mean of LTD
Location 2
LTD1
LTD2
Standard Deviation
s LTD
7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
32
LTD1 and LTD2: statistically identically distributed
To provide desired level of service, SL each location must carry
Safety Inventory:
I safety    s LTD
Z determined by the desired service level
Each facility: Identical demand and service levels
Total safety inventory decentralized system:
d
I safety
 2    s LTD
7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
33
Independent Demands
Centralizing the two locations in one location when
lead time demands at the two locations are
independent.
LTD = LTD1 + LTD2 Centralized Pool
The mean of total lead time demand is:
LTD + LTD = 2 LTD
Variance is:
s 2 LTD  s 2 LTD  2s 2 LTD
Standard Deviation is:
2s LTD
7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
34
d
Comparing safety inventories of decentralized ( I safety
)
c
and centralized ( I safety
) systems.
Safety Inventory in Centralized system is
 in a 2
location decentralized system by a factor of
1
2
7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
35
Centralization of N locations:
Safety Inventory needed is
c
I safety
   N s LTD
Centralization will reduce inventory by factor of
N
7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
36
Example
GE lighting operating 7 warehouses
Consolidated in to one centralized warehouse
Replenishment lead time remain at 10 days
What will be the impact of accepting the task
force recommendations?
7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
37
A warehouse with average lead time demand of 20,000 units
with a standard deviation of 5,000 units needs to carry a safety
inventory I safety  8,246
to provide a 95% service level.
Total safety inventory across 7 warehouses:
d
I safety
 7  8,246  57,722
Task force accepted, single central warehouse will face total lead
time demand with mean and standard deviation of:
LTD  7  20,000  140,000
s LTD  7  5,000  13,228.80
7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
38
95% service level, the central warehouse must carry a safety
inventory:
c
I safety
 1.65 s LTD  1.65 13,228.80  21,828
Safety inventory with the single central warehouses is 35,894
less than that required under the current decentralized network
of 7 warehouses.
Decrease in safety inventory by a factor of
7  2.65
7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
39
Square Root Law
States that the total safety inventory required to provide
a specific level of service increases by the square root of
the number of locations in which it is held.
Previous example
Correlated Demands
Does centralization offer similar benefits when demands
in multiple locations are correlated?
7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
40
LTD1 and LTD2 are statistically identically distributed
but correlated.
Correlation between two locations with coefficient

Mean of total lead time: LTD + LTD = 2 LTD
Variance is: s 2 LTD  s 2 LTD  2 s 2 LTD  2(1   )s 2 LTD
Total safety in centralized system is:
c
I safety
   2(1   )s LTD
7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
41
The total safety inventory in the decentralized system:
I
d
safety
 2    s LTD
The safety inventory in the two-location decentralized
system is larger than in the centralized system by a factor
of
2
(1   )
If demand is positively correlated (i.e.,   1)
centralization offers no benefits in the reduction of safety
inventory
7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
42
Advantages
1. Centralized systems

as the demand on the two locations
become negatively correlated.
2. Centralized systems diminishes as the demand in the two locations
become positively correlated
Disadvantages of Centralization
1. Response time to Customers
2. Shipping Cost
7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
43
Statistical Principle
Principle Aggregation: the standard deviation of the sum
of random variables is less than the sum of the
individual standard deviations.
Pooling inventory: available inventory is shared among
various sources of demand
Pooling inventory applied in other ways other than physical
centralization
7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
44
Virtual Centralization
Specialization
Component Commonality
Product Substitute
7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
45
Virtual Centralization
Distribution System
Location A
Location B
Exceeds Available stock
Available
1. Information about product demand and availability must
be available at both locations
2. Shipping the product from one location to a customer at
another location must be fast and cost effective
7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
46
Correlation is less than one – Pooling is Effective
Inventory Decentralized instead of physically
consolidated
Virtual Centralization: is a system in which inventory
pooling in a network of locations is facilitated using
information regarding availability of goods and
subsequent transshipment of goods between locations to
satisfy demand.
7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
47
Specialization
Each product only one specialized warehouse
EXAMPLE
Location A
Location B
P1
P2
Safety Inventory is reduced because each inventory is now
centralized at one location
7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
48
Component Commonality
Aggregating demand across various products.
Computer companies with models that vary.
Make-to-stock: produce in anticipation of product
demand
Make-to-Order: Produce in response to customer
orders
Reduce inventory investment maintaining the same level
of service and product variety
7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
49
Disadvantage Make-to-Order Strategy
Customer must wait for firm to produce product
Advantage Make-to-Stock Strategy
Available for immediate consumption
7. Managing Flow Variability: Safety Inventory
7.6: Shortening the Forecast Horizon through Postponement
50
Postponement (or Delayed Differentiation): More Effective
Short-Range forecast more accurate
Two Alternative processes (both two weeks)
Process A: Coloring the fabric, assembling
Process B: Assembling T-shirts, coloring
Does one have the advantage over the other?
7. Managing Flow Variability: Safety Inventory
7.6: Shortening the Forecast Horizon through Postponement
51
By Reversing: assembling and dyeing process
Process B postponed the color difference until one week closer
to the time of sale
Postponement: the practice of delaying part of a process in order
to reduce the need for safety inventory
7. Managing Flow Variability: Safety Inventory
7.6: Shortening the Forecast Horizon through Postponement
52
Process B has the advantage
Aggregation Reduces Variability
1. Aggregates demands by color in the first phase
2. Requires shorter-range forecasts of individual T-shirts
needed by color in the second phase.
Less Demand Variability
Less Total Safety Inventory
7. Managing Flow Variability: Safety Inventory
7.7: Levers for Reducing Safety Inventory
53
Levers for Reducing Flow Variability and the Required
Safety Inventory
1. Reduce demand variability through improved
forecasting
2. Reduce replenishment lead time
3. Reduce variability in replenishment lead time
4. Pool safety inventory for multiple locations or products
7. Managing Flow Variability: Safety Inventory
7.7: Levers for Reducing Safety Inventory
54
5. Exploit product substitution
6. Use common components
7. Postpone product-differentiation processing until
closer to the point of actual demand