A brief explanation on Bullwhip effect. Presentation.

1. ISITECH B-SCHOOL
Hyderabad
Kishore

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 and variations

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
 : An error parameter in period t; it has an independent distribution with
mean 0 and standard deviation σ
 Dt: the demand in period t
ρ 1 < ρ < 1)
(−
ε t = 1,2,)
(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
denote ˆ
We 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 ）