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Dr. RAVI SHANKARProfessor
Department of Management Studies
Indian Institute of Technology DelhiHauz Khas, New Delhi 110 016, India
Phone: +91-11-26596421 (O); 2659-1991(H); (0)-+91-9811033937 (m)Fax: (+91)-(11) 26862620
Email: [email protected]://web.iitd.ac.in/~ravi1
SESSION#3: TUTORIAL ON RISK POOLING (CFVG: 2012)
A TUTORIAL ON RISK POOLING
RISK POOLINGRisk pooling is an important concept in supply chain management. The idea of risk pooling is executed by a centralized distribution system which caters to the requirements of all the markets in a given region instead of separate warehouse allocated for different markets.
Market Two
Risk Pooling
• Consider these two systems:
Supplier
Warehouse One
Warehouse Two
Market One
Market Two
Supplier Warehouse
Market One
Supplier
Warehouse
Retailers
Centralized Systems
Decentralized System
Supplier
Warehouses
Retailers
Demand Forecasts
• The three principles of all forecasting techniques:
– Forecasting is always wrong
– The longer the forecast horizon the worst is the
forecast
– Aggregate forecasts are more accurate
The Effect of
Demand Uncertainty• Most companies treat the world as if it were predictable:
– Production and inventory planning are based on forecasts of demand made far in advance of the selling season
– Companies are aware of demand uncertainty when they create a forecast, but they design their planning process as if the forecast truly represents reality
• Recent technological advances have increased the level
of demand uncertainty:
– Short product life cycles
– Increasing product variety
Market one
Market two
Factory
Central
warehouse
Warehouse 1
Warehouse 2
Factory
Decentralized Warehouses
Market one
Market two
Factory
Centralised
warehouse at
Ayutthaya
Market Two
ABC Chiang Pai
Market One
Market Two
ABC Chiang Pai
Market One
Prachin Buri Warehouse
Pathumthani Warehouse
Central
warehouse:
Ayutthaya
Market Pathumthani
Market Prachin Buri
Factory: ABC
Central
warehouse
Market Two
ABC company
Market One
Market Two
ABC company
Market One
Prachin Buri Warehouse
Pathumthani Warehouse
Central
warehouse
(Ayutthaya)
Market one
Market two
Market one
Market two
WEEK 1 2 3 4 5 6 7 8
Pathumthani 68(-17) 37(+14) 45(+6) 58(-7) 16(+35) 32(+19) 72(-21) 80(-29)
Prachin Buri 87(-27) 62(-3) 55(+4) 67(-8) 12(+47) 42(+17) 69(-10) 81(-22)
TOTAL 155(-45) 99(+11) 100(+10) 125(-15) 28(+82) 74(+36) 141(-31) 161(-51)
PRODUCT A
0
10
20
30
40
50
60
70
80
90
100
1 2 3 4 5 6 7 8
WEEK
AV
ER
AG
E W
EE
KL
Y D
EM
AN
D
DEMAND Pathumthani
DEMAND Prachin Buri
HISTORICAL DEMAND DATA
51
59
110
Average
Theoretical Approach
• Consider two markets
– Risk Polling by Aggregating Demand by
Centralized procurement, centralized
warehousing, centralized distribution like
super stores etc
– Risk Polling by Aggregating time horizon by
combining orders as discussed in previous
slide
A Detail Analysis of
RISK POOLING Case
The Basic EOQ Model
We assumed that, we will only keep half the inventory over a year then
The total carry cost/yr = Cc x (Q/2). Total order cost = Co x (D/Q)
Then , Total cost = 2Q
CQDCTC co +=
Finding optimal Q*
Cost Relationships for Basic EOQ(Constant Demand, No Shortages)
TC
–A
nn
ual
Co
st
Total Cost
CarryingCost
OrderingCost
EOQ balances carryingcosts and ordering costs in this model.
Q* Order Quantity (how much)
The Basic EOQ Model
• EOQ occurs where total cost curve is at minimum value and carrying cost equals
ordering cost:
•Where is Q* located in our model?
c
o
co
CDCQ
QCQDCTC
2
2
*
min
=
+=
(How to obtain this?)Then, *c
o
co
CDCQ
QCQDCTC
2
2
*
min
=
+=
A Revision of model discussed in Sesion-3:
Model with “re-order points”• The reorder point is the inventory level at which a new order is placed.
• Order must be made while there is enough stock in place to cover demand during lead time.
• Formulation: R = dL, where d = demand rate per time period, L = lead time
Then R = dL = (10,000/311)(10) = 321.54
Working days/yr
Reorder Point• Inventory level might be depleted at slower or faster rate during lead time.
• When demand is uncertain, safety stock is added as a hedge against stockout.
Two possible scenarios
Safety stock!
No Safety
stocks!
We should then ensure
Safety stock is secured!
Determining Safety Stocks Using Service Levels
• We apply the Z test to secure its safety level,
)( LZLdR dσ+=
Reorder point
Safety stock
Average sample demand
How these values are represented in the diagram of normal distribution?
Reorder Point with Variable Demand
stocksafety
yprobabilit level service toingcorrespond deviations standard ofnumber
demanddaily ofdeviation standard the
timelead
demanddaily average
pointreorder
where
=
=
=
=
=
=
+=
LZ
Z
L
d
R
LZLdR
d
d
d
σ
σ
σ
Reorder Point with Variable Demand
Example
Example: determine reorder point and safety stock for service level of 95%.
26.1. : formulapoint reorder in termsecond isstock Safety
yd 1.3261.26300)10)(5)(65.1()10(30
1.65 Zlevel, service 95%For
dayper yd 5 days, 10 L day,per yd 30 d
=+=+=+=
=
===
LZLdR
d
dσ
σ
A detail treatment of
this case study
TERMINOLOGY
• AVG: Average daily demand faced by the distributor.
• STD: standard deviation of the daily demand faced by the distributor.
• L: Replenishment lead time from the supplier to the distributor in days
• K: Fixed cost (set up cost) incurred every time the warehouse places an order, it includes transportation cost.
• h: Cost of holding one unit of the product in the inventory for one day at the warehouse.
• α: Service level -the probability of not stocking out during lead time.
• Average demand during lead time=L×AVG. This ensures that if a distributor places an order the system has enough inventory to cover expected demand during lead time.
• Safety stock= z×STD× this is the amount of inventory distributor needs to keep to meet deviations from average demand during lead time.
• z: Safety factor which is chosen from statistical table to ensure that probability of stock out is exactly 1-α
• Reorder level (s) = average demand during lead time + safety stock
=L×AVG + z×STD×Whenever the inventory level drops below reorder
level the distributor should place new order to raise itsinventory.
L
L
• . Order quantity (Q): It is the number of items ordered each time places an order that minimizes the average total cost per unit of time distributor.
Q=
• Order-up-to level (S): Since there is variability in demand the distributor places an order for Q items whenever inventory is below reorder level (s).
S= Q + s
2K AVG
h
×
• Average inventory = Q/2 + z STD
• Coefficient of variation =
×× L
STD
AVG L×
A View of (s, S) Policy
Time
Inven
tory
Lev
el
S
s
0
Lead
Time
Lead
Time
Inventory Position
EXAMPLE OF RISK
POOLINGLet us illustrate this with an example of a Chiang Paibased company ABC that produces certain type of products and distributes them in the South Thailand region .The current distribution system partitions S-Thailand region into two markets each of which has a warehouse.
1. One warehouse is located in Prachin Buri
2. Another one located in Pathumthani.
alternative strategy of centralized distribution system replaces two warehouses by a single warehouse located between the two cities in Ayutthaya that will serve all customer orders in both markets
Market Two
Consider these two systems:
ABC company
Pathumthani Warehouse
Prachin Buri. Warehouse
Market One
Market Two
ABC companyCentral
warehouse
Market OneMarket one
Market two
Market two
Market one
Chiang Rai
Chiang Rai
ASSUMPTIONS
• Manufacturing facility has sufficient capacity to
satisfy any warehouse demand
• Lead time for delivery to each warehouse is
about one week and is assumed to be constant.
• Delivery time does not change significantly if we
adopt a centralized distribution system.
• Service level of 95% that is the probability of
stocking out is 5% is maintained.
DATA ANALYSIS
Now with analysis of weekly demand for two different products, product A and product B produced by ABC company for last 8 weeks in both market zones we will be able to decide which distribution strategy will be more efficient and cost effective.
WEEK 1 2 3 4 5 6 7 8
Pathum 68 37 45 58 16 32 72 80
Prachine 87 62 55 67 12 42 69 81
TOTAL 155 99 100 125 28 74 141 161
PRODUCT A
0
10
20
30
40
50
60
70
80
90
100
1 2 3 4 5 6 7 8
WEEK
AV
ER
AG
E W
EE
KL
Y D
EM
AN
D
DEMAND Pathum
DEMAND Prachine
HISTORICAL DEMAND DATA FOR PRODUCT A
WEEK 1 2 3 4 5 6 7 8
Pathum 0 0 1 3 2 4 0 1
Prachine 1 0 2 0 0 3 1 1
TOTAL 1 0 3 3 2 7 1 2
PRODUCT B
00.5
1
1.52
2.5
33.5
4
4.5
1 2 3 4 5 6 7 8
WEEK
AV
ER
AG
E D
EM
AN
D
DEMAND Pathum DEMAND Prachine
HISTORICAL DEMAND DATA FOR PRODUCT B
ANALYSIS OF HISTORICAL DATA
PRODUCT AVERAGEDEMAND
STANDARDDEVIATION
COEFFICIENT OF
VARIATION
Pathum A 51 20.70 0.41
Prachin B 1.38 1.41 1.02
Pathum A 59.38 22.23 0.32
Prachin B 1 1 1
CENTRAL A 110.38 39.14 0.35
CENTRAL B 2.38 1.99 0.84
SAMPLE CALCULATIONS
FOR PRODUCT A IN Pathumthani WAREHOUSE
1. Average demand = (68+37+45+58+16+32+72+80)/8=51
2. Standard deviation of demand =
= 20.7
3. Coefficient of variation = 20.7/51 = 0.41
2 2 2(68 51) (51 37) .............. (80 51)
8
− + − + −
GENERALIZATIONS
• average demand for product A is much higher than product B which is a slow moving product.
• Both standard deviation (absolute) and coefficient of variation (relative to average demand) are measure of variability of demand but we find that STD for product A is higher but coefficient of variation of product B is higher.
• For centralized distribution average demand is simply the sum of the demand faced by each of existing warehouse
• However the variability of demand as measured by STD or COV faced by central warehouse is lower than that faced by the two existing ones.
NUMERICAL VALUES
• Safety factor (Z) =1.65
• Fixed cost for both the products (Co) = Rs 3500
• Inventory holding cost (Cc) = Rs 18.5 per unit per week.
• Cost of transportation from warehouse to a customer – Current distribution system = Rs 50 per product
– Centralized distribution system = Rs 60 per product.
INVENTORY LEVELS
PRODUCT AVERAGE DEMAND DURINGLEAD TIME
SAFETY STOCK
(SS)
REORDERPOINT
(s)
ORDERQUANTITY
(Q)
ORDERUPTOLEVEL(S)
AVERAGE
INVENTORY
Pathum A 51 34.16 85 139 224 104
Prachine B 1.38 2.33 4 23 27 14
Pathum A 59.38 36.68 96 150 246 112
Prachine B 1 1.65 3 19 22 11
CENTRAL A 110.38 64.58 175 204 379 167
CENTRAL B 2.38 3.28 6 30 36 18
4. Safety stock =1.65 20.7 = 34.16
5. Reorder point = 51 + 34.16 = 85.16
6. Order quantity = = 139
7. Order up to level = 139 +85 = 224
8. Average inventory = 139/2 +34.16 = 103.66
× × 1
2 3500 51
18.5
× ×
SAMPLE CALCULATIONSFOR PRODUCT A IN Pathumthani WAREHOUSE
% REDUCTION IN
INVENTORY
REDUCTION IN AVERAGE INVENTORY
PRODUCT A = = 22.7%
PRODUCT B = = 28%
(104 112 167)100
(104 112)
+ −×
+
(14 11 18)100
(14 11)
+ −×
+
NORMAL DISTRIBUTIONAverage mean = 0
Standard deviation = 1
X axis- safety factor
Shaded area under curve= service level
Z=1.65P(z)=.95
Z=0
Demand Variability: Example 1
Product Demand
150
75
225
100
150
50
125
6148 53
104
45
0
50
100
150
200
250
Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar
Month
Demand
(000's)
Reminder:
The Normal Distribution
0 10 20 30 40 50 60Average = 30
Standard Deviation = 5
Standard Deviation = 10
ANALYSIS AT DIFFERENT
SERVICE LEVELS
When average inventory for different level of service is calculated corresponding to varying value of z it was found that there exists a trade-off between service level and reduction in inventory through risk pooling.
SERVICE LEVEL (%)
90 91 92 93 94 95 96 97 98 99 99.9
Z 1.29 1.34 1.41 1.48 1.56 1.65 1.75 1.88 2.05 2.33 3.08
PERCENTAGE REDUCTION IN AVERAGE INVENTORY VS
SERVICE LEVEL
0
5
10
15
20
25
30
90 93 96 99
SERVICE LEVEL
% R
ED
UC
TIO
N I
N A
VG
INV
EN
TO
RY
PRODUCTAPRODUCTB
SERVICE
LEVEL (%)90 91 92 93 94 95 96 97 98 99 99.9
PRODUCT A
24 23.7 23.4 23.1 23 22.7 22.3 21.8 21.7 21.2 19.5
PRODUCT B
27.12 27.07 27.0 26.94 26.89 26.82 26.72 26.59 26.44 26.2 25.65
% REDUCTION IN AVERAGE INVENTORY
Following generalizations are made
• If a company goes for higher level of service it has to compromise with the % of reduction in the inventory level and vice versa.
• To provide high service level company has to maintain high inventory too.
• % reduction in inventory decreases with increase in service level.
IDEAL SITUATION
This works best for
– High coefficient of variation, which reduces required
safety stock.
– Negatively correlated demand as in such a case the
high demand from one customer will be offset by low demand from another