1. The bullwhip effect occurs in supply chains where demand appears more variable upstream than downstream due to factors like demand forecasting, lead times, batch ordering, price variability, and supply issues. 2. A study by Procter & Gamble found that retailer sales were fairly stable but orders to factories fluctuated more, showing the bullwhip effect. 3. Ways to mitigate the bullwhip effect include reducing forecast error and lead times, continuous replenishment, centralized demand information, and allocating supply based on sales rather than orders.

1. Bullwhip Effect and Risk Pooling
Tokyo University of
Marine Science and Technology
Mikio Kubo

2. Bullwhip effect
• Key concept for understanding the SCM
• Procter & Gamble noticed an interesting
phenomenon that retail sales of the
product were fairly uniform, but
distributors’ orders placed to the factory
fluctuated much more than retail sales.

3. Why the bullwhip effect occurs?
１． Demand Forecasting
• One day, the manager of a retailer observed a
larger demand (sales) than expected.
• He increased the inventory level because he
expected more demand in the future (forecasting).
• The manager of his wholesaler observed more
demand (some of which are not actual demand)
than usual and increased his inventory.
• This caused more (non-real) demand to his maker;
the manager of the maker increased his inventory,
and so on. This is the basic reason of the bull
whip effect.

4. Why the bullwhip effect occurs?
２． Lead time
• With longer lead times, a small change
in the estimate of demand variability
implies a significant change in safety
stock, reorder level, and thus in order
quantities.
• Thus a longer lead time leads to an
increase in variability and the bull whip
effect.

5. Why the bullwhip effect occurs?
３． Batch Ordering
• When using a min-max inventory policy, then
the wholesaler will observe a large order,
followed by several periods of no orders,
followed by another large order, and so on.
• The wholesaler sees a distorted and highly
variable pattern of orders.
• Thus, batch ordering increases the bull whip
effect.

6. Why the bullwhip effect occurs?
４． Variability of Price
• Retailers (or wholesalers or makers)
offer promotions and discounts at
certain times or for certain quantities.
• Retailers (or customers) often attempt
to stock up when prices are lower.
• It increases the variability of demands and
the bull whip effect.

7. Why the bullwhip effect occurs?
５． Lack of supply and supply
allocation
• When retailers suspect that a product
will be in short supply, and therefore
anticipate receiving supply proportional
to the amount ordered (supply
allocation).
• When the period of shortage is over,
the retailer goes back to its standard
orders, leading to all kinds of distortions

8. Quantifying the Bullwhip
Effect
One stage model
For each period t=1,2…, let
Retailer Customer
Ordering
quantity q[t] Inventory I[t] Demand D[t]

9. Discrete time model
(Periodic ordering system)
Lead time L
Items ordered at the end of period t will arrive at the
beginning of period t+L+1.
2)
Demand
D[t]
occurs
t t+1 t+2 t+3 t+4
１） Arrive the 3) Forecast demand F[t+1]
items ordered 4) Order q[t] Arrive the items
in period t-L-1 in period t+L+1 （ L=3)

10. Demand process
• d: a constant term of the demand process
• ρ: a parameter that represents the correlation
between two consecutive periods ρ 1 < ρ < 1)
(−
• ε t = 1,2, ) : An error parameter in period t; it
(t
has an independent distribution with mean 0 and
standard deviation σ
• Dt: the demand in period t
Dt = d + ρDt −1 + ε t

11. An example of demand process
d=80,ρ=0.5,ε[t]=[-10,10]
=80+0.5*B2+(RAND()*(-20)+10)
250
需要量 D(t )=d +
期 t ρ * D(t - 1 )+ε
200
1 80
2 1 46.43491 07
3 1 66.2490253 150
4 1 81 .946823
5 200.6561 255 100
6 21 0.0359644
7 202.0940006
50
8 200.3971 697
9 1 93.985555
10 1 94.6002961 0
3
5
7
9
1
15
19
25
27
33
37
39
13
17
23
29
35
11
21
41
31

12. Ordering quantity q[t]
• Forecasting （ p period moving average ）
p
∑D
j =1
t− j
ˆ
dt =
p
ˆ
We denote d t and Dt by F [t ] and D[t ], respectively.
• Ordering quantity q[t] of period t is:
q[t]=D[t]+L (F[t+1]-F[t]) ,t=1,2,…

13. Inventory I[t]
• Inventory flow conservation equation:
Final inventory (period t)=
Final inventory (period t-1)-Demand ＋ Arrival
Volume
I[0]=A Safety Stock Level
I[t] =I[t-1] –D[t] +q[t-L-1],t=1,2,…

14. Excel Simulation (bull.xls)
=E7-E6+B6
=(B5+B4+B3+B2)/4 =D6+1 =G5-B6+F3
=C6*2
リードタイム中の 発注量 在庫量
需要量 D(t )=d+ 移動平均法による 需要量予測 目標在庫レベル q(t )=y(t )- y(t - I(t )=I(t - 1 )-
期 t ρ * D(t - 1 )+ε 予測 F(t ):p=4 F(t ) * :L, L=2 y(t )= F[t ]* L+ z *σ 1 )+D(t - 1 ) D(t )+q(t - 3)
1 80 80 0
2 127.81847 80 0
3 144.8770316 80 0
4 152.9420471 80 300
5 157.4258033 126.4093872 252.8187744 254.8187744 196.138705 222.5741967
6 151.3785902 145.765838 291.5316761 293.5316761 163.1586503 151.1956064
7 161.1899679 151.6558681 303.3117361 305.3117361 169.3464361 70.00563851
8 158.4760476 155.7341022 311.4682043 313.4682043 161.2430479 107.6682959
9 164.937867 157.1176023 314.2352046 316.2352046 168.6938988 105.8890792
10 156.4019926 158.9956182 317.9912364 319.9912364 158.9136938 118.8335227

15. Demand, ordering quantity, and
demand processes
350
300
250
200 需要量 D(t )=d+e * D(t -
1 )+e ps ilo n
1 50 発注量 q(t )=y(t )- y(t -
1 00 1 )+D(t - 1 )
在庫量 I )=I - 1 )-
(t (t
50 D(t )+q(t - 3)
0
5
9
13
17
29
25
33
37
1
21
41
- 50
- 1 00

16. Asymptotic analysis:
expectation,variance, and Covariance)
d
E ( D[t ]) = By solving E[D]=d+ρE[D]
1− ρ
σ 2
Var ( D[t ]) = By solving
1− ρ 2 Var[D]=ρ2 Var[D]+σ2
ρ σ
p 2
Cov ( D[t ], D[t − p ]) =
1− ρ 2

17. Expansion of ordering quantity
q[t ] = D[t ] + LF [t + 1] − LF [t ]
p p
L ∑ D[t + 1 − j ] L ∑ D[t − j ]
j =1 j =1
= D[t ] + −
p p
L L
= (1 + ) D[t ] − D[t − p ]
p p

18. Variance of ordering quantity
L 2 L 2
Var ( q[t ]) = (1 + ) Var ( D[t ]) + ( ) Var ( D[t − p ])
p p
L L
− 2(1 + )( )Cov ( D[t ], D[t − p ])
p p
  2 L 2 L2  
=  p + p 2 (1 − ρ ) Var ( D[t ])
1 + 
2

   
Var ( q[t ])  2 L 2 L2 
=1+ 
 p + 2 (1 − ρ ) 2

Var ( D[t ])  p 

19. Observations
Var (q[t ])  2 L 2 L2 

= 1+  + 2  (1 − ρ ) 2
Var ( D[t ])  p p 
• When p is large, and L is small, the bullwhip
effect due to forecasting error is negligible.
• The bullwhip effect is magnified as we increase
the lead time and decrease p.
• A positive correlation DECRESES the bull
whip effect.

20. Coping with the Bullwhip Effect
１． Demand uncertainty
• Adjust the forecasting parameters, e.g.,
larger p for the moving average method.
• Centralizing demand information; by
providing each stage of the supply
chain with complete information on
actual customer demand (POS: Point-Of-
Sales data ）
• Continuous replenishment
• VMI （ Vender Managed Inventory:
VMI ）

21. Coping with the Bullwhip Effect
２． Lead time
• Lead time reduction
• Information lead time can be reduced ujsing
EDI （ Electric Data Interchange ） or
CAO （ Computer Assisted Ordering ） .
• QR （ Quick Response ） in apparel
industry

22. Coping with the Bullwhip Effect
３． Batch ordering
• Reduction of fixed ordering cost using EDI
and CAO
• 3PL （ Third Party Logistics ）
• VMI

23. Coping with the Bullwhip Effect
４． Variability of Price
• EDLP: Every Day Low Price （ P&G ）
• Remark that the same strategy does not
work well in Japan.

24. Coping with the Bullwhip Effect
５． Lack of supply and supply
allocation
• Allocate the lacking demand due to sales
volume and/or market share instead of order
volume. （ General Motors ， Saturn,
Hewlett-Packard ）
• Share the inventory and production
information of makers with retailers and
wholesalers. （ Hewlett-
Packard ， Motorola ）