1 7. managing flow variability: safety inventory chapter 7 managing flow variability: safety...
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7. Managing Flow Variability: Safety Inventory
Chapter 7
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
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7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecast and Forecast Errors
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.
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7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecasts and Forecast Errors
We have previously assumed demand is knownand 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.”
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7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecast and Forecast Errors
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)
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7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecasts and Forecast Errors
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.
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7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecasts and Forecast Errors
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.
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7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecasts and Forecast Errors
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.
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7. Managing Flow Variability: Safety Inventory
7.1 Demand Forecast and Forecast Errors
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.
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7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
Objective:
Review of common terms and a discussion of Service
Level
Where: SL = f (Safety Inventory, I safety)
And some math using Excel…
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7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
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
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7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
Inventory, I (t)
ROP
0
Safety Inventory
(I safety)
LTD, # of Units used during lead time
ORDER ORDER
L LTime, t
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7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
I safety = ROP – LTD
0
ROP
LTD, # of Units used during lead time
Safety Inventory
(I safety)
Inventory, I (t)
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7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
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
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7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
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 LTD then
I safety = z x LTD , where z is a multiple of LTD
Or the number of standard deviations
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7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
Example:
At GE Lighting’s Paris warehouse,
LTD (average Lead Time Demand) = 20,000 lamps
Actual Demand varies daily and LTD = 5,000
The warehouse re-orders whenever ROP = 24,000
Therefore, I safety = ROP – LTD = 24,000 – 20,000 = 4,000
And: z = I safety / LTD = 4,000 / 5,000 = 0.8
And: SL= Prob (Z< 0.8) from Appendix II
SL= 0.7881
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7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
A B C D E F
1 z = 0.802 SL (z<0.8)3 SL = 0.788144
5
678
EXCEL
Service Level
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7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
EXCEL
Safety Inventory
A B C D E F
1 SL = 0.788142 z = 0.803
4
5
678
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7. Managing Flow Variability: Safety Inventory
7.2 Safety Inventory and Service Level
EXCEL
Reorder Point
A B C D E F
1 SL = 0.788142 LTD = 20,000
3
LTD = 5,0004 ROP 24,0005
678
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7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
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)
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7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
Cost of Holding Extra Inventory
Improved Service
Optimal Service Level??
The Newsvendor ProblemDecision 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.
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7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
Cost:
Price:
Salvage:
$1,800
$2,500
$1,700
Demand ProbabilityCumulative 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
Predicted Demand for HDTV’s
Profit:
Loss:
$700
$100
Mean:
Std. Dev:
151.6
22.44
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7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
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)
Demand
Qua
ntity
Ord
ered
100 x $700 – (110-100) x $100 = $69,000
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7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
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
Demand
Qua
ntity
Ord
ered
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7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
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
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7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
Net Marginal Benefit:
Net Marginal Cost:
MB = p – c
MC = c - v
MB = $2,500 - $1,800 = $700
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*)
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7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
MB
MB MC
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*)]
After rearranging, Prob(R ≤ Q*) ≥
Newsvendor formula: SL* = Prob(R ≤ Q*) = MB
MB MC
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7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
$700* 0.875
$700 $100
MBSL
MB MC
Going back to the example…
So what quantity corresponds to this service level ?
If we assume demand is normally distributed then,
* RQ R z
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7. Managing Flow Variability: Safety Inventory
7.3 Optimal Service Level: The Newsvendor Problem
-4 -3 -2 -1 0 1 2 3 40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Probability Less than Upper Bound is 0.87493
Den
sity
Critical Value (z)
z = 1.15
* 151.6 1.15 22.44 177.41RQ R z
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7. Managing Flow Variability: Safety Inventory
7.4 Lead Time Demand Variability
LTD L R 2 2LTD RL
2 2 2LTD LR
Average Lead Time Demand:
Variability in Periodic Demand:
Variability in Lead Time:
Variability in Demand and Lead Time:
2 2 2LTD R LL R
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7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
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
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7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
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 Location 2
Lead times demands: LTD1 LTD2
Mean of LTD Standard Deviation LTD
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7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
LTD1 and LTD2: statistically identically distributed
To provide desired level of service, SL each location must carry
Safety Inventory:
Z determined by the desired service level
Each facility: Identical demand and service levels
Total safety inventory decentralized system:
LTDsafetyI
LTDdsafetyI 2
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7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
Independent DemandsCentralizing the two locations in one location when
lead time demands at the two locations are independent.
LTD = LTD1 + LTD2 Centralized PoolThe mean of total lead time demand is:
LTD + LTD = 2 LTDVariance is:
LTDLTDLTD222 2
LTD2Standard Deviation is:
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7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
Comparing safety inventories of decentralized
and centralized systems.
Safety Inventory in Centralized system is in a 2
location decentralized system by a factor of
)( dsafetyI
)( csafetyI
2
1
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7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
Centralization of N locations:
Safety Inventory needed is
Centralization will reduce inventory by factor of
N
LTDcsafety NI
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7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
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?
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7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
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 to provide a 95% service level.
Total safety inventory across 7 warehouses:
Task force accepted, single central warehouse will face total lead
time demand with mean and standard deviation of:
722,57246,87 dsafetyI
246,8safetyI
80.228,13000,57
000,140000,207
LTD
LTD
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7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
95% service level, the central warehouse must carry a safety
inventory:
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
828,2180.228,1365.165.1 LTDcsafetyI
65.27
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7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
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?
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7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
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:
Total safety in centralized system is:
LTDLTDLTDLTD2222 )1(22
LTDcsafetyI )1(2
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7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
The total safety inventory in the decentralized system:
The safety inventory in the two-location decentralized
system is larger than in the centralized system by a factor
of
If demand is positively correlated
centralization offers no benefits in the reduction of safety
inventory
)1(
2
LTDdsafetyI 2
)1.,.( ei
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7. Managing Flow Variability: Safety Inventory
7.5: Pooling Efficiency through Aggregation
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
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7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
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
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7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
Virtual Centralization
Specialization
Component Commonality
Product Substitute
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7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
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
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7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
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.
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7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
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
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7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
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
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7. Managing Flow Variability: Safety Inventory
7.5.2: Principle of Aggregation and Pooling Inventory
Disadvantage Make-to-Order Strategy
Customer must wait for firm to produce product
Advantage Make-to-Stock Strategy
Available for immediate consumption
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7. Managing Flow Variability: Safety Inventory
7.6: Shortening the Forecast Horizon through Postponement
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?
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7. Managing Flow Variability: Safety Inventory
7.6: Shortening the Forecast Horizon through Postponement
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
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7. Managing Flow Variability: Safety Inventory
7.6: Shortening the Forecast Horizon through Postponement
Process B has the advantage
Aggregation Reduces Variability1. Aggregates demands by color in the first phase2. Requires shorter-range forecasts of individual T-shirts
needed by color in the second phase.
Less Demand Variability Less Total Safety Inventory
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7. Managing Flow Variability: Safety Inventory
7.7: Levers for Reducing Safety Inventory
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
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7. Managing Flow Variability: Safety Inventory
7.7: Levers for Reducing Safety Inventory
5. Exploit product substitution
6. Use common components
7. Postpone product-differentiation processing until
closer to the point of actual demand