Transcript 確率的在庫モデル - LOG OPT

Bullwhip Effect and Risk Pooling
Tokyo University of
Marine Science and Technology
Mikio Kubo
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.
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.
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.
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.
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.
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
Quantifying the Bullwhip
Effect
One stage model
For each period t=1,2…, let
Retailer
Ordering
quantity q[t]
Inventory I[t]
Customer
Demand D[t]
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]
Arrive the items
items ordered 4) Order q[t]
in period t+L+1 （L=3)
in period t-L-1
Demand process
• d: a constant term of the demand process
• ρ: a parameter that represents the correlation
between two consecutive periods 　(1    1)
• t (t  1,2,) : An error parameter in period t; it
has an independent distribution with mean 0 and
standard deviation σ
• Dt: the demand in period t
Dt  d  Dt 1   t
An example of demand process
d=80,ρ=0.5,ε[t]=[-10,10]
=80+0.5*B2+(RAND()*(-20)+10)
200
150
100
50
41
39
37
35
33
31
29
27
25
23
21
19
17
15
13
11
9
7
5
0
3
1
2
3
4
5
6
7
8
9
10
250
1
期 t
需要量 D(t)=d+
ρ*D(t-1)+ε
80
146.4349107
166.2490253
181.946823
200.6561255
210.0359644
202.0940006
200.3971697
193.985555
194.6002961
Ordering quantity q[t]
• Forecasting （p period moving average）
p
dˆt 
D
j 1
t j
p
We denotedˆt 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,…
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,…
Excel Simulation (bull.xls)
=(B5+B4+B3+B2)/4
=C6*2
=E7-E6+B6
=D6+1
=G5-B6+F3
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
Demand, ordering quantity, and
demand processes
350
300
250
需要量 D(t)=d+e*D(t1)+epsilon
発注量 q(t)=y(t)-y(t1)+D(t-1)
在庫量 I(t)=I(t-1)D(t)+q(t-3)
200
150
100
50
-100
41
37
33
29
25
21
17
9
13
-50
5
1
0
Asymptotic analysis:
expectation,variance, and Covariance)
d
E ( D[t ]) 
1 
2

Var( D[t ]) 
2
1 
By solving E[D]=d+ρE[D]
By solving
Var[D]=ρ2 Var[D]+σ2
 
Cov( D[t ], D[t  p]) 
2
1 
p
2
Expansion of ordering quantity
q[t ] 
D[t ]  LF [t  1]  LF [t ]
p

D[t ] 
L D[t  1  j ]
j 1
p
L
L
 (1  ) D[t ]  D[t  p ]
p
p
p

L D[t  j ]
j 1
p
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 

2

 2 (1   ) Var( D[t ])
1  
p 
  p

 2 L 2 L2 
Var( q[t ])
2


 1 
 2 (1   )
Var( D[t ])
p 
 p
Observations
 2L 2L
Var (q[t ])
 1  
 2
Var ( D[t ])
p
 p
2
• When

(1   ) 2

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.
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）
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
Coping with the Bullwhip Effect
３．Batch ordering
• Reduction of fixed ordering cost using EDI
and CAO
• 3PL（Third Party Logistics）
• VMI
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.
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, HewlettPackard）
• Share the inventory and production
information of makers with retailers and
wholesalers. （Hewlett-Packard，Motorola）