Variability
The Bull Whip Effect
A Vicious Cycle
Build-to-Order, Lean, JIT, …
Managing Variability: A different view of
inventory
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Example
Procter & Gamble: Pampers
Smooth consumer demand
Fluctuating sales at retail stores
Highly variable demand on distributors
Wild swings in demand on
manufacturing
Greatest swings in demand on suppliers
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Illustration
Consumer Sales at Retailer
Consumer demand
1000
900
800
700
600
500
400
300
200
41
39
37
35
33
31
29
27
25
23
21
19
17
13
15
11
9
5
7
3
0
1
100
Retailer's Orders to Distributor
1000
800
700
600
500
400
300
200
41
39
37
35
33
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29
27
25
23
21
19
17
15
13
9
11
7
5
0
3
100
1
Retailer Order
900
3
Illustration
Retailer's Orders to Distributor
1000
Retailer Order
900
800
700
600
500
400
300
200
41
39
37
35
33
31
29
27
25
23
21
19
17
15
13
9
11
7
5
3
0
1
100
Distributor's Orders to P&G
900
800
700
600
500
400
300
200
41
39
37
35
33
31
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27
25
23
21
19
17
15
13
11
9
7
5
0
3
100
1
Distributor Order
1000
4
Illustration
Distributor’s Orders to P&G
Distributor Order
1000
900
800
700
600
500
400
300
200
41
39
37
35
33
31
29
27
25
23
21
19
17
15
13
9
11
7
5
3
0
1
100
P&G's Orders with 3M
1000
900
800
P&G Order
700
600
500
400
300
200
40
37
34
31
22
19
16
13
10
7
4
1
28
0
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100
5
Illustration
Consumer Sales at Retailer
Consumer demand
1000
900
800
700
600
500
400
300
200
41
39
37
35
33
31
29
27
25
23
21
19
17
13
15
11
9
5
7
3
0
1
100
P&G's Orders with 3M
1000
900
800
P&G Order
700
600
500
400
300
200
40
37
34
31
28
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22
19
16
10
7
4
1
0
13
100
6
What Are the Effects?
What problems, costs, challenges does
this create for the players in the
supply chain?
What problems does this create for the
product in the market place?
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Forecasting
More variability
Poorer forecasts
Less reliable supply
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Causes of Bullwhip Today
Product Proliferation/Mass
Customization
More varieties of products
Build-to-Order
Prohibits pooling orders to smooth
requirements
Lean
Prevents pooling releases to smooth
demand on the supply chain
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Why Lean (Just-In-Time)?
Reduces inventory
Capital requirements
Etc
Reduces handling
Direct-to-Line
Improves Quality
See problems quickly
Increases launch speed
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Why Not Lean?
Capacity
Changes in requirements create upstream
inventory
Changes in requirements raise transport
costs
Reliability
Distant suppliers subject to disruption
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Release Variability
Daily Receipt
1400
1200
1000
800
600
400
200
0
01-Apr-02
06-Apr-02
11-Apr-02
16-Apr-02
21-Apr-02
26-Apr-02
Managing Variability
Capacity Buffer
Inventory Buffer
Time Buffer
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Required rate of conveyances
Freight Cost
Capacity
Utilized Capacity
Time
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Required rate of conveyances
Expediting
Expedited Service
Lower Capacity
Utilized Capacity
Time
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Ideal Supply Chain
Same requirements every day
No excess capacity
No inventory No service failures
Minimum Cost
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Buffer Inventory with Constant Supply
Volume
Variation in Demand
Variation in Buffer
Time
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A Financial Model
From Revenues
Cash Expenses
Cash Acct
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A Financial Model
From Revenues
Cash Expenses
Invest
Sell Assets
Cash Acct
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Controls
Invest
enough
to bring it
to here
When Cash
balance reaches
here
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Controls
Sell assets
to bring it
to here
When Cash
balance
falls to here
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Controls
T
t
b
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Trade-offs
Opportunity cost of Cash Balance
Transaction costs of investing and
selling assets
Set the controls, T, t and b to
balance these costs
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Inventory Analogy
Cash Expenses
From Revenue
Sell Assets
Invest Excess
Daily Production reqs.
Constant supplies
Expedited order
Curtailed order
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Trade-offs
Cost of holding
Inventory
Supply chain
costs of
expediting
and curtailing
orders
Set the controls, T, t and b
to balance these costs
Opportunity
cost of Cash Balance
Transaction
costs of
investing and
selling assets
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The Traditional Model
1
1-(p1+ q1) p1
p2
2
…
1
0
q1
q2
pk
k
…
qk
…
M
1
A Markov Model
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Challenges
Data intensive: pi’s and qi’s
Computationally intensive
Alternative
Brownian motion
Inventory behaves like
a random walk
Model of a particle in space
Two parameters: mean and variance
Advanced calculus methods make it “easy” to work with
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Release Variability
Daily Receipt
1400
68% of time usage is
between 0 and 800
1200
1000
800
σ = 412
600
400
µ = 367
200
0
01-Apr-02
06-Apr-02
11-Apr-02
16-Apr-02
21-Apr-02
26-Apr-02
The EOQ as a Special Case
Average usage rate µ
Variance in usage σ2
Nominal release rate λ < µ
Since we order less than we consume,
inventory drifts downward
How much should we “expedite” when it
reaches 0?
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Trade offs
Expediting disrupts the supply chain
Fixed cost F for each time we expedite
Variable cost f for each item in the order
holding cost h for inventory
Larger orders mean less frequent but larger disruptions and more inventory
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EOQ as Special Case
Order Q
Time between orders is Q/(µ – λ)
Order frequency is (µ – λ)/Q
Average Inventory is Q/2 + σ2/2(µ – λ)
Average Cost is
Expediting Cost:
Inventory Cost:
(F+fQ)(µ – λ)/Q
h(Q/2 + σ2/2(µ – λ))
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The Total Cost Formula
hQ/2 + F(µ – λ)/Q + hσ2/2(µ – λ) + f(µ – λ)
EOQ: Transaction
EOQ: Inventory
Constant that doesn’t depend on Q
The best Q balances inventory and ordering
costs:
hQ/2 = F(µ – λ)/Q
Q2 = 2F(µ – λ)/h
Q = √2F(µ – λ)/h
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Fixed Expediting Quantity
Find the best nominal release rate λ
to get the right frequency of expediting
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The Total Cost Formula
hQ/2 + hσ2/2r + fr + Fr/Q
EOQ: Transaction
EOQ: Inventory
Constant that doesn’t depend on r
The best drift rate r = µ − λ balances
inventory and ordering costs:
hσ2/2r = fr + Fr/Q
r2 = hσ2Q/2(F+fQ)
r = √hσ2Q/2(F+fQ)
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Two-sided Version
If inventory grows too large, curtail
shipments
What’s too large?
How much should we curtail?
If expediting is expensive
create a positive drift
order more than you need
curtail shipments when inventory is too high
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Differences
Constant Stream of Releases punctuated
by Expediting and Curtailing
If supplier can see inventory,
can anticipate expedited and curtailed orders
Have to set a lower bound > 0 to protect
against disruptions – safety stock
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8
1
Example: Shipments
300
250
200
150
100
50
0
37
Example: Inventory
1000
900
800
700
600
500
400
300
200
100
0
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