1. Dr. RAVI SHANKAR
Professor
Department of Management Studies
Indian Institute of Technology Delhi
Hauz Khas, New Delhi 110 016, India
Phone: +91-11-26596421 (O); 2659-1991(H); (0)-+91-9811033937 (m)
Fax: (+91)-(11) 26862620
Email: r.s.research@gmail.com
http://web.iitd.ac.in/~ravi1
SESSION#3: TUTORIAL ON RISK POOLING (CFVG: 2012)
A TUTORIAL ON RISK POOLING
2. RISK POOLING
Risk 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.
3. Market Two
Risk Pooling
• Consider these two systems:
Supplier
Warehouse One
Warehouse Two
Market One
Market Two
Supplier Warehouse
Market One
6. 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
7. 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
11. 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
12. 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
14. 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
16. 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 = 2
QC
Q
DCTC co += Finding optimal Q*
17. Cost Relationships for Basic EOQ
(Constant Demand, No Shortages)
TC–AnnualCost
Total
Cost
Carrying
Cost
Ordering
Cost
EOQ balances carrying
costs and ordering
costs in this model.
Q* Order Quantity (how much)
18. 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
c
o
C
DCQ
QC
Q
DCTC
2
2
*
min
=
+=
(How to obtain this?)Then, *
c
o
c
o
C
DCQ
QC
Q
DCTC
2
2
*
min
=
+=
19. 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
20. 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!
21. 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?
22. Reorder Point with Variable Demand
stocksafety
yprobabilitlevelservicetoingcorresponddeviationsstandardofnumber
demanddailyofdeviationstandardthe
timelead
demanddailyaverage
pointreorder
where
=
=
=
=
=
=
+=
LZ
Z
L
d
R
LZLdR
d
d
d
σ
σ
σ
23.
24. Reorder Point with Variable Demand
Example
Example: determine reorder point and safety stock for service level of 95%.
26.1.:formulapointreorderintermsecondisstockSafety
yd1.3261.26300)10)(5)(65.1()10(30
1.65Zlevel,service95%For
dayperyd5days,10Lday,peryd30 d
=+=+=+=
=
===
LZLdR
d
dσ
σ
26. 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.
27. • 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 its
inventory.
L
L
28. • . 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
×
29. • Average inventory = Q/2 + z STD
• Coefficient of variation =
×× L
STD
AVG L×
30. A View of (s, S) Policy
Time
InventoryLevel
S
s
0
Lead
Time
Lead
Time
Inventory Position
31. EXAMPLE OF RISK
POOLING
Let us illustrate this with an example of a Chiang Pai
based 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
32. Market Two
Consider these two systems:
ABC company
Pathumthani Warehouse
Prachin Buri. Warehouse
Market One
Market Two
ABC company
Central
warehouse
Market OneMarket one
Market two
Market two
Market one
Chiang Rai
Chiang Rai
33. 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.
34. 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.
37. ANALYSIS OF HISTORICAL DATA
PRODUCT AVERAGE
DEMAND
STANDARD
DEVIATION
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
38. 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
− + − + −
39. 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.
40. 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.
41. INVENTORY LEVELS
PRODUCT AVERAGE
DEMAND
DURING
LEAD TIME
SAFETY
STOCK
(SS)
REORDER
POINT
(s)
ORDER
QUANTITY
(Q)
ORDER
UPTO
LEVEL
(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
42. 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 CALCULATIONS
FOR PRODUCT A IN Pathumthani WAREHOUSE
43. % 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)
+ −
×
+
44. NORMAL DISTRIBUTION
Average mean = 0
Standard deviation = 1
X axis- safety factor
Shaded area under curve= service level
Z=1.65
P(z)=.95
Z=0
45. Demand Variability: Example 1
Product Demand
150
75
225
100
150
50
125
61
48 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)
47. 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
48. PERCENTAGE REDUCTION IN
AVERAGE INVENTORY VS
SERVICE LEVEL
0
5
10
15
20
25
30
90 93 96 99
SERVICE LEVEL
%REDUCTIONINAVG
INVENTORY
PRODUCT
A
PRODUCT
B
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
49. 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.
50. 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
